Daily Papers

Interpreting individual neurons or directions in activations space is an important component of mechanistic interpretability. As such, many algorithms have been proposed to automatically produce neuron explanations, but it is often not clear how reliable these explanations are, or which methods produce the best explanations. This can be measured via crowd-sourced evaluations, but they can often be noisy and expensive, leading to unreliable results. In this paper, we carefully analyze the evaluation pipeline and develop a cost-effective and highly accurate crowdsourced evaluation strategy. In contrast to previous human studies that only rate whether the explanation matches the most highly activating inputs, we estimate whether the explanation describes neuron activations across all inputs. To estimate this effectively, we introduce a novel application of importance sampling to determine which inputs are the most valuable to show to raters, leading to around 30x cost reduction compared to uniform sampling. We also analyze the label noise present in crowd-sourced evaluations and propose a Bayesian method to aggregate multiple ratings leading to a further ~5x reduction in number of ratings required for the same accuracy. Finally, we use these methods to conduct a large-scale study comparing the quality of neuron explanations produced by the most popular methods for two different vision models.

Natural language is an appealing medium for explaining how large language models process and store information, but evaluating the faithfulness of such explanations is challenging. To help address this, we develop two modes of evaluation for natural language explanations that claim individual neurons represent a concept in a text input. In the observational mode, we evaluate claims that a neuron a activates on all and only input strings that refer to a concept picked out by the proposed explanation E. In the intervention mode, we construe E as a claim that the neuron a is a causal mediator of the concept denoted by E. We apply our framework to the GPT-4-generated explanations of GPT-2 XL neurons of Bills et al. (2023) and show that even the most confident explanations have high error rates and little to no causal efficacy. We close the paper by critically assessing whether natural language is a good choice for explanations and whether neurons are the best level of analysis.

While deep learning has led to huge progress in complex image classification tasks like ImageNet, unexpected failure modes, e.g. via spurious features, call into question how reliably these classifiers work in the wild. Furthermore, for safety-critical tasks the black-box nature of their decisions is problematic, and explanations or at least methods which make decisions plausible are needed urgently. In this paper, we address these problems by generating images that optimize a classifier-derived objective using a framework for guided image generation. We analyze the decisions of image classifiers by visual counterfactual explanations (VCEs), detection of systematic mistakes by analyzing images where classifiers maximally disagree, and visualization of neurons and spurious features. In this way, we validate existing observations, e.g. the shape bias of adversarially robust models, as well as novel failure modes, e.g. systematic errors of zero-shot CLIP classifiers. Moreover, our VCEs outperform previous work while being more versatile.

We propose a technique for producing "visual explanations" for decisions from a large class of CNN-based models, making them more transparent. Our approach - Gradient-weighted Class Activation Mapping (Grad-CAM), uses the gradients of any target concept, flowing into the final convolutional layer to produce a coarse localization map highlighting important regions in the image for predicting the concept. Grad-CAM is applicable to a wide variety of CNN model-families: (1) CNNs with fully-connected layers, (2) CNNs used for structured outputs, (3) CNNs used in tasks with multimodal inputs or reinforcement learning, without any architectural changes or re-training. We combine Grad-CAM with fine-grained visualizations to create a high-resolution class-discriminative visualization and apply it to off-the-shelf image classification, captioning, and visual question answering (VQA) models, including ResNet-based architectures. In the context of image classification models, our visualizations (a) lend insights into their failure modes, (b) are robust to adversarial images, (c) outperform previous methods on localization, (d) are more faithful to the underlying model and (e) help achieve generalization by identifying dataset bias. For captioning and VQA, we show that even non-attention based models can localize inputs. We devise a way to identify important neurons through Grad-CAM and combine it with neuron names to provide textual explanations for model decisions. Finally, we design and conduct human studies to measure if Grad-CAM helps users establish appropriate trust in predictions from models and show that Grad-CAM helps untrained users successfully discern a 'stronger' nodel from a 'weaker' one even when both make identical predictions. Our code is available at https://github.com/ramprs/grad-cam/, along with a demo at http://gradcam.cloudcv.org, and a video at youtu.be/COjUB9Izk6E.

Safety fine-tuning algorithms are commonly used to fine-tune language models to reduce harmful outputs, but the exact internal mechanisms of how those models achieve this remain unclear. In studying direct preference optimisation (DPO) for toxicity reduction, current explanations claim that DPO works by dampening the most toxic MLP neurons to learn an offset to avert toxic regions in the residual stream. However, by ablating the most toxic neurons and applying activation patching, we find this explanation incomplete. By projecting neuron activation changes onto a toxicity probe, we find that only 31.8\% of toxicity reduction comes from dampened toxic neurons. Instead, DPO reduces toxicity by accumulating effects across multiple neuron groups, both reducing writing in the toxic direction and promoting anti-toxicity in the residual stream. Moreover, DPO gives noisy adjustments to neuron activations, with many neurons actually increasing toxicity. This indicates that DPO is a balancing process between opposing neuron effects to achieve toxicity reduction.

This article is about the neural conundrum behind the slowness of human behavior. The information throughput of a human being is about 10 bits/s. In comparison, our sensory systems gather data at ~10^9 bits/s. The stark contrast between these numbers remains unexplained and touches on fundamental aspects of brain function: What neural substrate sets this speed limit on the pace of our existence? Why does the brain need billions of neurons to process 10 bits/s? Why can we only think about one thing at a time? The brain seems to operate in two distinct modes: the "outer" brain handles fast high-dimensional sensory and motor signals, whereas the "inner" brain processes the reduced few bits needed to control behavior. Plausible explanations exist for the large neuron numbers in the outer brain, but not for the inner brain, and we propose new research directions to remedy this.

Sensory systems across all modalities and species exhibit adaptation to continuously changing input statistics. Individual neurons have been shown to modulate their response gains so as to maximize information transmission in different stimulus contexts. Experimental measurements have revealed additional, nuanced sensory adaptation effects including changes in response maxima and minima, tuning curve repulsion from the adapter stimulus, and stimulus-driven response decorrelation. Existing explanations of these phenomena rely on changes in inter-neuronal synaptic efficacy, which, while more flexible, are unlikely to operate as rapidly or reversibly as single neuron gain modulations. Using published V1 population adaptation data, we show that propagation of single neuron gain changes in a recurrent network is sufficient to capture the entire set of observed adaptation effects. We propose a novel adaptive efficient coding objective with which single neuron gains are modulated, maximizing the fidelity of the stimulus representation while minimizing overall activity in the network. From this objective, we analytically derive a set of gains that optimize the trade-off between preserving information about the stimulus and conserving metabolic resources. Our model generalizes well-established concepts of single neuron adaptive gain control to recurrent populations, and parsimoniously explains experimental adaptation data.

While the activations of neurons in deep neural networks usually do not have a simple human-understandable interpretation, sparse autoencoders (SAEs) can be used to transform these activations into a higher-dimensional latent space which may be more easily interpretable. However, these SAEs can have millions of distinct latent features, making it infeasible for humans to manually interpret each one. In this work, we build an open-source automated pipeline to generate and evaluate natural language explanations for SAE features using LLMs. We test our framework on SAEs of varying sizes, activation functions, and losses, trained on two different open-weight LLMs. We introduce five new techniques to score the quality of explanations that are cheaper to run than the previous state of the art. One of these techniques, intervention scoring, evaluates the interpretability of the effects of intervening on a feature, which we find explains features that are not recalled by existing methods. We propose guidelines for generating better explanations that remain valid for a broader set of activating contexts, and discuss pitfalls with existing scoring techniques. We use our explanations to measure the semantic similarity of independently trained SAEs, and find that SAEs trained on nearby layers of the residual stream are highly similar. Our large-scale analysis confirms that SAE latents are indeed much more interpretable than neurons, even when neurons are sparsified using top-k postprocessing. Our code is available at https://github.com/EleutherAI/sae-auto-interp, and our explanations are available at https://huggingface.co/datasets/EleutherAI/auto_interp_explanations.

A basic question within the emerging field of mechanistic interpretability is the degree to which neural networks learn the same underlying mechanisms. In other words, are neural mechanisms universal across different models? In this work, we study the universality of individual neurons across GPT2 models trained from different initial random seeds, motivated by the hypothesis that universal neurons are likely to be interpretable. In particular, we compute pairwise correlations of neuron activations over 100 million tokens for every neuron pair across five different seeds and find that 1-5\% of neurons are universal, that is, pairs of neurons which consistently activate on the same inputs. We then study these universal neurons in detail, finding that they usually have clear interpretations and taxonomize them into a small number of neuron families. We conclude by studying patterns in neuron weights to establish several universal functional roles of neurons in simple circuits: deactivating attention heads, changing the entropy of the next token distribution, and predicting the next token to (not) be within a particular set.

A central goal for mechanistic interpretability has been to identify the right units of analysis in large language models (LLMs) that causally explain their outputs. While early work focused on individual neurons, evidence that neurons often encode multiple concepts has motivated a shift toward analyzing directions in activation space. A key question is how to find directions that capture interpretable features in an unsupervised manner. Current methods rely on dictionary learning with sparse autoencoders (SAEs), commonly trained over residual stream activations to learn directions from scratch. However, SAEs often struggle in causal evaluations and lack intrinsic interpretability, as their learning is not explicitly tied to the computations of the model. Here, we tackle these limitations by directly decomposing MLP activations with semi-nonnegative matrix factorization (SNMF), such that the learned features are (a) sparse linear combinations of co-activated neurons, and (b) mapped to their activating inputs, making them directly interpretable. Experiments on Llama 3.1, Gemma 2 and GPT-2 show that SNMF derived features outperform SAEs and a strong supervised baseline (difference-in-means) on causal steering, while aligning with human-interpretable concepts. Further analysis reveals that specific neuron combinations are reused across semantically-related features, exposing a hierarchical structure in the MLP's activation space. Together, these results position SNMF as a simple and effective tool for identifying interpretable features and dissecting concept representations in LLMs.

Deep neural networks (DNNs) have demonstrated state-of-the-art results on many pattern recognition tasks, especially vision classification problems. Understanding the inner workings of such computational brains is both fascinating basic science that is interesting in its own right - similar to why we study the human brain - and will enable researchers to further improve DNNs. One path to understanding how a neural network functions internally is to study what each of its neurons has learned to detect. One such method is called activation maximization (AM), which synthesizes an input (e.g. an image) that highly activates a neuron. Here we dramatically improve the qualitative state of the art of activation maximization by harnessing a powerful, learned prior: a deep generator network (DGN). The algorithm (1) generates qualitatively state-of-the-art synthetic images that look almost real, (2) reveals the features learned by each neuron in an interpretable way, (3) generalizes well to new datasets and somewhat well to different network architectures without requiring the prior to be relearned, and (4) can be considered as a high-quality generative method (in this case, by generating novel, creative, interesting, recognizable images).

The truthfulness of existing explanation methods in authentically elucidating the underlying model's decision-making process has been questioned. Existing methods have deviated from faithfully representing the model, thus susceptible to adversarial attacks. To address this, we propose a novel eXplainable AI (XAI) method called SRD (Sharing Ratio Decomposition), which sincerely reflects the model's inference process, resulting in significantly enhanced robustness in our explanations. Different from the conventional emphasis on the neuronal level, we adopt a vector perspective to consider the intricate nonlinear interactions between filters. We also introduce an interesting observation termed Activation-Pattern-Only Prediction (APOP), letting us emphasize the importance of inactive neurons and redefine relevance encapsulating all relevant information including both active and inactive neurons. Our method, SRD, allows for the recursive decomposition of a Pointwise Feature Vector (PFV), providing a high-resolution Effective Receptive Field (ERF) at any layer.

There is an analogy that is often made between deep neural networks and actual brains, suggested by the nomenclature itself: the "neurons" in deep neural networks should correspond to neurons (or nerve cells, to avoid confusion) in the brain. We claim, however, that this analogy doesn't even type check: it is structurally flawed. In agreement with the slightly glib summary of Hebbian learning as "cells that fire together wire together", this article makes the case that the analogy should be different. Since the "neurons" in deep neural networks are managing the changing weights, they are more akin to the synapses in the brain; instead, it is the wires in deep neural networks that are more like nerve cells, in that they are what cause the information to flow. An intuition that nerve cells seem like more than mere wires is exactly right, and is justified by a precise category-theoretic analogy which we will explore in this article. Throughout, we will continue to highlight the error in equating artificial neurons with nerve cells by leaving "neuron" in quotes or by calling them artificial neurons. We will first explain how to view deep neural networks as nested dynamical systems with a very restricted sort of interaction pattern, and then explain a more general sort of interaction for dynamical systems that is useful throughout engineering, but which fails to adapt to changing circumstances. As mentioned, an analogy is then forced upon us by the mathematical formalism in which they are both embedded. We call the resulting encompassing generalization deeply interacting learning systems: they have complex interaction as in control theory, but adaptation to circumstances as in deep neural networks.

A neuroscience method to understanding the brain is to find and study the preferred stimuli that highly activate an individual cell or groups of cells. Recent advances in machine learning enable a family of methods to synthesize preferred stimuli that cause a neuron in an artificial or biological brain to fire strongly. Those methods are known as Activation Maximization (AM) or Feature Visualization via Optimization. In this chapter, we (1) review existing AM techniques in the literature; (2) discuss a probabilistic interpretation for AM; and (3) review the applications of AM in debugging and explaining networks.

Do large language models (LLMs) solve reasoning tasks by learning robust generalizable algorithms, or do they memorize training data? To investigate this question, we use arithmetic reasoning as a representative task. Using causal analysis, we identify a subset of the model (a circuit) that explains most of the model's behavior for basic arithmetic logic and examine its functionality. By zooming in on the level of individual circuit neurons, we discover a sparse set of important neurons that implement simple heuristics. Each heuristic identifies a numerical input pattern and outputs corresponding answers. We hypothesize that the combination of these heuristic neurons is the mechanism used to produce correct arithmetic answers. To test this, we categorize each neuron into several heuristic types-such as neurons that activate when an operand falls within a certain range-and find that the unordered combination of these heuristic types is the mechanism that explains most of the model's accuracy on arithmetic prompts. Finally, we demonstrate that this mechanism appears as the main source of arithmetic accuracy early in training. Overall, our experimental results across several LLMs show that LLMs perform arithmetic using neither robust algorithms nor memorization; rather, they rely on a "bag of heuristics".

Interpretability researchers have attempted to understand MLP neurons of language models based on both the contexts in which they activate and their output weight vectors. They have paid little attention to a complementary aspect: the interactions between input and output. For example, when neurons detect a direction in the input, they might add much the same direction to the residual stream ("enrichment neurons") or reduce its presence ("depletion neurons"). We address this aspect by examining the cosine similarity between input and output weights of a neuron. We apply our method to 12 models and find that enrichment neurons dominate in early-middle layers whereas later layers tend more towards depletion. To explain this finding, we argue that enrichment neurons are largely responsible for enriching concept representations, one of the first steps of factual recall. Our input-output perspective is a complement to activation-dependent analyses and to approaches that treat input and output separately.

The static synaptic connectivity of neuronal circuits stands in direct contrast to the dynamics of their function. As in changing community interactions, different neurons can participate actively in various combinations to effect behaviors at different times. We introduce an unsupervised approach to learn the dynamic affinities between neurons in live, behaving animals, and to reveal which communities form among neurons at different times. The inference occurs in two major steps. First, pairwise non-linear affinities between neuronal traces from brain-wide calcium activity are organized by non-negative tensor factorization (NTF). Each factor specifies which groups of neurons are most likely interacting for an inferred interval in time, and for which animals. Finally, a generative model that allows for weighted community detection is applied to the functional motifs produced by NTF to reveal a dynamic functional connectome. Since time codes the different experimental variables (e.g., application of chemical stimuli), this provides an atlas of neural motifs active during separate stages of an experiment (e.g., stimulus application or spontaneous behaviors). Results from our analysis are experimentally validated, confirming that our method is able to robustly predict causal interactions between neurons to generate behavior. Code is available at https://github.com/dyballa/dynamic-connectomes.

The approximation capability of ANNs and their RNN instantiations, is strongly correlated with the number of parameters packed into these networks. However, the complexity barrier for human understanding, is arguably related to the number of neurons and synapses in the networks, and to the associated nonlinear transformations. In this paper we show that the use of biophysical synapses, as found in LTCs, have two main benefits. First, they allow to pack more parameters for a given number of neurons and synapses. Second, they allow to formulate the nonlinear-network transformation, as a linear system with state-dependent coefficients. Both increase interpretability, as for a given task, they allow to learn a system linear in its input features, that is smaller in size compared to the state of the art. We substantiate the above claims on various time-series prediction tasks, but we believe that our results are applicable to any feedforward or recurrent ANN.

Some neurons in deep networks specialize in recognizing highly specific perceptual, structural, or semantic features of inputs. In computer vision, techniques exist for identifying neurons that respond to individual concept categories like colors, textures, and object classes. But these techniques are limited in scope, labeling only a small subset of neurons and behaviors in any network. Is a richer characterization of neuron-level computation possible? We introduce a procedure (called MILAN, for mutual-information-guided linguistic annotation of neurons) that automatically labels neurons with open-ended, compositional, natural language descriptions. Given a neuron, MILAN generates a description by searching for a natural language string that maximizes pointwise mutual information with the image regions in which the neuron is active. MILAN produces fine-grained descriptions that capture categorical, relational, and logical structure in learned features. These descriptions obtain high agreement with human-generated feature descriptions across a diverse set of model architectures and tasks, and can aid in understanding and controlling learned models. We highlight three applications of natural language neuron descriptions. First, we use MILAN for analysis, characterizing the distribution and importance of neurons selective for attribute, category, and relational information in vision models. Second, we use MILAN for auditing, surfacing neurons sensitive to human faces in datasets designed to obscure them. Finally, we use MILAN for editing, improving robustness in an image classifier by deleting neurons sensitive to text features spuriously correlated with class labels.

DeConvNet, Guided BackProp, LRP, were invented to better understand deep neural networks. We show that these methods do not produce the theoretically correct explanation for a linear model. Yet they are used on multi-layer networks with millions of parameters. This is a cause for concern since linear models are simple neural networks. We argue that explanation methods for neural nets should work reliably in the limit of simplicity, the linear models. Based on our analysis of linear models we propose a generalization that yields two explanation techniques (PatternNet and PatternAttribution) that are theoretically sound for linear models and produce improved explanations for deep networks.

Artificial neural networks for motor control usually adopt generic architectures like fully connected MLPs. While general, these tabula rasa architectures rely on large amounts of experience to learn, are not easily transferable to new bodies, and have internal dynamics that are difficult to interpret. In nature, animals are born with highly structured connectivity in their nervous systems shaped by evolution; this innate circuitry acts synergistically with learning mechanisms to provide inductive biases that enable most animals to function well soon after birth and learn efficiently. Convolutional networks inspired by visual circuitry have encoded useful biases for vision. However, it is unknown the extent to which ANN architectures inspired by neural circuitry can yield useful biases for other AI domains. In this work, we ask what advantages biologically inspired ANN architecture can provide in the domain of motor control. Specifically, we translate C. elegans locomotion circuits into an ANN model controlling a simulated Swimmer agent. On a locomotion task, our architecture achieves good initial performance and asymptotic performance comparable with MLPs, while dramatically improving data efficiency and requiring orders of magnitude fewer parameters. Our architecture is interpretable and transfers to new body designs. An ablation analysis shows that constrained excitation/inhibition is crucial for learning, while weight initialization contributes to good initial performance. Our work demonstrates several advantages of biologically inspired ANN architecture and encourages future work in more complex embodied control.

Generating explanations for neural networks has become crucial for their applications in real-world with respect to reliability and trustworthiness. In natural language processing, existing methods usually provide important features which are words or phrases selected from an input text as an explanation, but ignore the interactions between them. It poses challenges for humans to interpret an explanation and connect it to model prediction. In this work, we build hierarchical explanations by detecting feature interactions. Such explanations visualize how words and phrases are combined at different levels of the hierarchy, which can help users understand the decision-making of black-box models. The proposed method is evaluated with three neural text classifiers (LSTM, CNN, and BERT) on two benchmark datasets, via both automatic and human evaluations. Experiments show the effectiveness of the proposed method in providing explanations that are both faithful to models and interpretable to humans.

The Spiking Neural Network (SNN) has attracted more and more attention recently. It adopts binary spike signals to transmit information. Benefitting from the information passing paradigm of SNNs, the multiplications of activations and weights can be replaced by additions, which are more energy-efficient. However, its ``Hard Reset" mechanism for the firing activity would ignore the difference among membrane potentials when the membrane potential is above the firing threshold, causing information loss. Meanwhile, quantifying the membrane potential to 0/1 spikes at the firing instants will inevitably introduce the quantization error thus bringing about information loss too. To address these problems, we propose to use the ``Soft Reset" mechanism for the supervised training-based SNNs, which will drive the membrane potential to a dynamic reset potential according to its magnitude, and Membrane Potential Rectifier (MPR) to reduce the quantization error via redistributing the membrane potential to a range close to the spikes. Results show that the SNNs with the ``Soft Reset" mechanism and MPR outperform their vanilla counterparts on both static and dynamic datasets.

These handouts are designed for people who is just starting involved with the topic artificial neural networks. We show how it works a single artificial neuron (McCulloch & Pitt model), mathematically and graphically. We do explain the delta rule, a learning algorithm to find the neuron weights. We also present some examples in MATLAB/Octave. There are examples for classification task for lineal and non-lineal problems. At the end, we present an artificial neural network, a feed-forward neural network along its learning algorithm backpropagation. ----- Estos apuntes est\'an dise\~nados para personas que por primera vez se introducen en el tema de las redes neuronales artificiales. Se muestra el funcionamiento b\'asico de una neurona, matem\'aticamente y gr\'aficamente. Se explica la Regla Delta, algoritmo deaprendizaje para encontrar los pesos de una neurona. Tambi\'en se muestran ejemplos en MATLAB/Octave. Hay ejemplos para problemas de clasificaci\'on, para problemas lineales y no-lineales. En la parte final se muestra la arquitectura de red neuronal artificial conocida como backpropagation.

Historically, artificial intelligence has drawn much inspiration from neuroscience to fuel advances in the field. However, current progress in reinforcement learning is largely focused on benchmark problems that fail to capture many of the aspects that are of interest in neuroscience today. We illustrate this point by extending a T-maze task from neuroscience for use with reinforcement learning algorithms, and show that state-of-the-art algorithms are not capable of solving this problem. Finally, we point out where insights from neuroscience could help explain some of the issues encountered.

We can better understand deep neural networks by identifying which features each of their neurons have learned to detect. To do so, researchers have created Deep Visualization techniques including activation maximization, which synthetically generates inputs (e.g. images) that maximally activate each neuron. A limitation of current techniques is that they assume each neuron detects only one type of feature, but we know that neurons can be multifaceted, in that they fire in response to many different types of features: for example, a grocery store class neuron must activate either for rows of produce or for a storefront. Previous activation maximization techniques constructed images without regard for the multiple different facets of a neuron, creating inappropriate mixes of colors, parts of objects, scales, orientations, etc. Here, we introduce an algorithm that explicitly uncovers the multiple facets of each neuron by producing a synthetic visualization of each of the types of images that activate a neuron. We also introduce regularization methods that produce state-of-the-art results in terms of the interpretability of images obtained by activation maximization. By separately synthesizing each type of image a neuron fires in response to, the visualizations have more appropriate colors and coherent global structure. Multifaceted feature visualization thus provides a clearer and more comprehensive description of the role of each neuron.

Why do neurons communicate through spikes? By definition, spikes are all-or-none neural events which occur at continuous times. In other words, spikes are on one side binary, existing or not without further details, and on the other can occur at any asynchronous time, without the need for a centralized clock. This stands in stark contrast to the analog representation of values and the discretized timing classically used in digital processing and at the base of modern-day neural networks. As neural systems almost systematically use this so-called event-based representation in the living world, a better understanding of this phenomenon remains a fundamental challenge in neurobiology in order to better interpret the profusion of recorded data. With the growing need for intelligent embedded systems, it also emerges as a new computing paradigm to enable the efficient operation of a new class of sensors and event-based computers, called neuromorphic, which could enable significant gains in computation time and energy consumption -- a major societal issue in the era of the digital economy and global warming. In this review paper, we provide evidence from biology, theory and engineering that the precise timing of spikes plays a crucial role in our understanding of the efficiency of neural networks.

Neural networks often pack many unrelated concepts into a single neuron - a puzzling phenomenon known as 'polysemanticity' which makes interpretability much more challenging. This paper provides a toy model where polysemanticity can be fully understood, arising as a result of models storing additional sparse features in "superposition." We demonstrate the existence of a phase change, a surprising connection to the geometry of uniform polytopes, and evidence of a link to adversarial examples. We also discuss potential implications for mechanistic interpretability.

Why do brains have inhibitory connections? Why do deep networks have negative weights? We propose an answer from the perspective of representation capacity. We believe representing functions is the primary role of both (i) the brain in natural intelligence, and (ii) deep networks in artificial intelligence. Our answer to why there are inhibitory/negative weights is: to learn more functions. We prove that, in the absence of negative weights, neural networks with non-decreasing activation functions are not universal approximators. While this may be an intuitive result to some, to the best of our knowledge, there is no formal theory, in either machine learning or neuroscience, that demonstrates why negative weights are crucial in the context of representation capacity. Further, we provide insights on the geometric properties of the representation space that non-negative deep networks cannot represent. We expect these insights will yield a deeper understanding of more sophisticated inductive priors imposed on the distribution of weights that lead to more efficient biological and machine learning.

Mechanistic Interpretability (MI) promises a path toward fully understanding how neural networks make their predictions. Prior work demonstrates that even when trained to perform simple arithmetic, models can implement a variety of algorithms (sometimes concurrently) depending on initialization and hyperparameters. Does this mean neuron-level interpretability techniques have limited applicability? We argue that high-dimensional neural networks can learn low-dimensional representations of their training data that are useful beyond simply making good predictions. Such representations can be understood through the mechanistic interpretability lens and provide insights that are surprisingly faithful to human-derived domain knowledge. This indicates that such approaches to interpretability can be useful for deriving a new understanding of a problem from models trained to solve it. As a case study, we extract nuclear physics concepts by studying models trained to reproduce nuclear data.

Deep neural networks are widely used for classification. These deep models often suffer from a lack of interpretability -- they are particularly difficult to understand because of their non-linear nature. As a result, neural networks are often treated as "black box" models, and in the past, have been trained purely to optimize the accuracy of predictions. In this work, we create a novel network architecture for deep learning that naturally explains its own reasoning for each prediction. This architecture contains an autoencoder and a special prototype layer, where each unit of that layer stores a weight vector that resembles an encoded training input. The encoder of the autoencoder allows us to do comparisons within the latent space, while the decoder allows us to visualize the learned prototypes. The training objective has four terms: an accuracy term, a term that encourages every prototype to be similar to at least one encoded input, a term that encourages every encoded input to be close to at least one prototype, and a term that encourages faithful reconstruction by the autoencoder. The distances computed in the prototype layer are used as part of the classification process. Since the prototypes are learned during training, the learned network naturally comes with explanations for each prediction, and the explanations are loyal to what the network actually computes.

It has long been known in both neuroscience and AI that ``binding'' between neurons leads to a form of competitive learning where representations are compressed in order to represent more abstract concepts in deeper layers of the network. More recently, it was also hypothesized that dynamic (spatiotemporal) representations play an important role in both neuroscience and AI. Building on these ideas, we introduce Artificial Kuramoto Oscillatory Neurons (AKOrN) as a dynamical alternative to threshold units, which can be combined with arbitrary connectivity designs such as fully connected, convolutional, or attentive mechanisms. Our generalized Kuramoto updates bind neurons together through their synchronization dynamics. We show that this idea provides performance improvements across a wide spectrum of tasks such as unsupervised object discovery, adversarial robustness, calibrated uncertainty quantification, and reasoning. We believe that these empirical results show the importance of rethinking our assumptions at the most basic neuronal level of neural representation, and in particular show the importance of dynamical representations.

Large Language Models (LLMs) typically have billions of parameters and are thus often difficult to interpret in their operation. Such black-box models can pose a significant risk to safety when trusted to make important decisions. The lack of interpretability of LLMs is more related to their sheer size, rather than the complexity of their individual components. The TARS method for knowledge removal (Davies et al 2024) provides strong evidence for the hypothesis that that linear layer weights which act directly on the residual stream may have high correlation with different concepts encoded in the residual stream. Building upon this, we attempt to decode neuron weights directly into token probabilities through the final projection layer of the model (the LM-head). Firstly, we show that with Llama 3.1 8B we can utilise the LM-head to decode specialised feature neurons that respond strongly to certain concepts, with examples such as "dog" and "California". This is then confirmed by demonstrating that these neurons can be clamped to affect the probability of the concept in the output. This extends to the fine-tuned assistant Llama 3.1 8B instruct model, where we find that over 75% of neurons in the up-projection layers have the same top associated token compared to the pretrained model. Finally, we demonstrate that clamping the "dog" neuron leads the instruct model to always discuss dogs when asked about its favourite animal. Through our method, it is possible to map the entirety of Llama 3.1 8B's up-projection neurons in less than 15 minutes with no parallelization.

In the artificial neuron, I replace the dot product with the weighted Lehmer mean, which may emulate different cases of a generalized mean. The single neuron instance is replaced by a multiplet of neurons which have the same averaging weights. A group of outputs feed forward, in lieu of the single scalar. The generalization parameter is typically set to a different value for each neuron in the multiplet. I further extend the concept to a multiplet taken from the Gini mean. Derivatives with respect to the weight parameters and with respect to the two generalization parameters are given. Some properties of the network are investigated, showing the capacity to emulate the classical exclusive-or problem organically in two layers and perform some multiplication and division. The network can instantiate truncated power series and variants, which can be used to approximate different functions, provided that parameters are constrained. Moreover, a mean case slope score is derived that can facilitate a learning-rate novelty based on homogeneity of the selected elements. The multiplet neuron equation provides a way to segment regularization timeframes and approaches.

One of the roadblocks to a better understanding of neural networks' internals is polysemanticity, where neurons appear to activate in multiple, semantically distinct contexts. Polysemanticity prevents us from identifying concise, human-understandable explanations for what neural networks are doing internally. One hypothesised cause of polysemanticity is superposition, where neural networks represent more features than they have neurons by assigning features to an overcomplete set of directions in activation space, rather than to individual neurons. Here, we attempt to identify those directions, using sparse autoencoders to reconstruct the internal activations of a language model. These autoencoders learn sets of sparsely activating features that are more interpretable and monosemantic than directions identified by alternative approaches, where interpretability is measured by automated methods. Ablating these features enables precise model editing, for example, by removing capabilities such as pronoun prediction, while disrupting model behaviour less than prior techniques. This work indicates that it is possible to resolve superposition in language models using a scalable, unsupervised method. Our method may serve as a foundation for future mechanistic interpretability work, which we hope will enable greater model transparency and steerability.

Recurrent neural networks (RNNs) are widely used throughout neuroscience as models of local neural activity. Many properties of single RNNs are well characterized theoretically, but experimental neuroscience has moved in the direction of studying multiple interacting areas, and RNN theory needs to be likewise extended. We take a constructive approach towards this problem, leveraging tools from nonlinear control theory and machine learning to characterize when combinations of stable RNNs will themselves be stable. Importantly, we derive conditions which allow for massive feedback connections between interacting RNNs. We parameterize these conditions for easy optimization using gradient-based techniques, and show that stability-constrained "networks of networks" can perform well on challenging sequential-processing benchmark tasks. Altogether, our results provide a principled approach towards understanding distributed, modular function in the brain.

In a recent paper, Erasmus et al. (2021) defend the idea that the ambiguity of the term "explanation" in explainable AI (XAI) can be solved by adopting any of four different extant accounts of explanation in the philosophy of science: the Deductive Nomological, Inductive Statistical, Causal Mechanical, and New Mechanist models. In this chapter, I show that the authors' claim that these accounts can be applied to deep neural networks as they would to any natural phenomenon is mistaken. I also provide a more general argument as to why the notion of explainability as it is currently used in the XAI literature bears little resemblance to the traditional concept of scientific explanation. It would be more fruitful to use the label "understandable AI" to avoid the confusion that surrounds the goal and purposes of XAI. In the second half of the chapter, I argue for a pragmatic conception of understanding that is better suited to play the central role attributed to explanation in XAI. Following Kuorikoski & Ylikoski (2015), the conditions of satisfaction for understanding an ML system are fleshed out in terms of an agent's success in using the system, in drawing correct inferences from it.

Many learning algorithms used as normative models in neuroscience or as candidate approaches for learning on neuromorphic chips learn by contrasting one set of network states with another. These Contrastive Learning (CL) algorithms are traditionally implemented with rigid, temporally non-local, and periodic learning dynamics that could limit the range of physical systems capable of harnessing CL. In this study, we build on recent work exploring how CL might be implemented by biological or neurmorphic systems and show that this form of learning can be made temporally local, and can still function even if many of the dynamical requirements of standard training procedures are relaxed. Thanks to a set of general theorems corroborated by numerical experiments across several CL models, our results provide theoretical foundations for the study and development of CL methods for biological and neuromorphic neural networks.

Human brains respond to semantic features of presented stimuli with different neurons. It is then curious whether modern deep neural networks admit a similar behavior pattern. Specifically, this paper finds a small cluster of neurons in a diffusion model corresponding to a particular subject. We call those neurons the concept neurons. They can be identified by statistics of network gradients to a stimulation connected with the given subject. The concept neurons demonstrate magnetic properties in interpreting and manipulating generation results. Shutting them can directly yield the related subject contextualized in different scenes. Concatenating multiple clusters of concept neurons can vividly generate all related concepts in a single image. A few steps of further fine-tuning can enhance the multi-concept capability, which may be the first to manage to generate up to four different subjects in a single image. For large-scale applications, the concept neurons are environmentally friendly as we only need to store a sparse cluster of int index instead of dense float32 values of the parameters, which reduces storage consumption by 90\% compared with previous subject-driven generation methods. Extensive qualitative and quantitative studies on diverse scenarios show the superiority of our method in interpreting and manipulating diffusion models.

Deep neural networks have proved to be a very effective way to perform classification tasks. They excel when the input data is high dimensional, the relationship between the input and the output is complicated, and the number of labeled training examples is large. But it is hard to explain why a learned network makes a particular classification decision on a particular test case. This is due to their reliance on distributed hierarchical representations. If we could take the knowledge acquired by the neural net and express the same knowledge in a model that relies on hierarchical decisions instead, explaining a particular decision would be much easier. We describe a way of using a trained neural net to create a type of soft decision tree that generalizes better than one learned directly from the training data.

Diagrams matter. Unfortunately, the deep learning community has no standard method for diagramming architectures. The current combination of linear algebra notation and ad-hoc diagrams fails to offer the necessary precision to understand architectures in all their detail. However, this detail is critical for faithful implementation, mathematical analysis, further innovation, and ethical assurances. I present neural circuit diagrams, a graphical language tailored to the needs of communicating deep learning architectures. Neural circuit diagrams naturally keep track of the changing arrangement of data, precisely show how operations are broadcast over axes, and display the critical parallel behavior of linear operations. A lingering issue with existing diagramming methods is the inability to simultaneously express the detail of axes and the free arrangement of data, which neural circuit diagrams solve. Their compositional structure is analogous to code, creating a close correspondence between diagrams and implementation. In this work, I introduce neural circuit diagrams for an audience of machine learning researchers. After introducing neural circuit diagrams, I cover a host of architectures to show their utility and breed familiarity. This includes the transformer architecture, convolution (and its difficult-to-explain extensions), residual networks, the U-Net, and the vision transformer. I include a Jupyter notebook that provides evidence for the close correspondence between diagrams and code. Finally, I examine backpropagation using neural circuit diagrams. I show their utility in providing mathematical insight and analyzing algorithms' time and space complexities.

We propose a formal mathematical model for sparse representations and active dendrites in neocortex. Our model is inspired by recent experimental findings on active dendritic processing and NMDA spikes in pyramidal neurons. These experimental and modeling studies suggest that the basic unit of pattern memory in the neocortex is instantiated by small clusters of synapses operated on by localized non-linear dendritic processes. We derive a number of scaling laws that characterize the accuracy of such dendrites in detecting activation patterns in a neuronal population under adverse conditions. We introduce the union property which shows that synapses for multiple patterns can be randomly mixed together within a segment and still lead to highly accurate recognition. We describe simulation results that provide further insight into sparse representations as well as two primary results. First we show that pattern recognition by a neuron with active dendrites can be extremely accurate and robust with high dimensional sparse inputs even when using a tiny number of synapses to recognize large patterns. Second, equations representing recognition accuracy of a dendrite predict optimal NMDA spiking thresholds under a generous set of assumptions. The prediction tightly matches NMDA spiking thresholds measured in the literature. Our model matches many of the known properties of pyramidal neurons. As such the theory provides a mathematical framework for understanding the benefits and limits of sparse representations in cortical networks.

Biological nervous systems are created in a fundamentally different way than current artificial neural networks. Despite its impressive results in a variety of different domains, deep learning often requires considerable engineering effort to design high-performing neural architectures. By contrast, biological nervous systems are grown through a dynamic self-organizing process. In this paper, we take initial steps toward neural networks that grow through a developmental process that mirrors key properties of embryonic development in biological organisms. The growth process is guided by another neural network, which we call a Neural Developmental Program (NDP) and which operates through local communication alone. We investigate the role of neural growth on different machine learning benchmarks and different optimization methods (evolutionary training, online RL, offline RL, and supervised learning). Additionally, we highlight future research directions and opportunities enabled by having self-organization driving the growth of neural networks.

Mechanistic interpretability aims to understand the behavior of neural networks by reverse-engineering their internal computations. However, current methods struggle to find clear interpretations of neural network activations because a decomposition of activations into computational features is missing. Individual neurons or model components do not cleanly correspond to distinct features or functions. We present a novel interpretability method that aims to overcome this limitation by transforming the activations of the network into a new basis - the Local Interaction Basis (LIB). LIB aims to identify computational features by removing irrelevant activations and interactions. Our method drops irrelevant activation directions and aligns the basis with the singular vectors of the Jacobian matrix between adjacent layers. It also scales features based on their importance for downstream computation, producing an interaction graph that shows all computationally-relevant features and interactions in a model. We evaluate the effectiveness of LIB on modular addition and CIFAR-10 models, finding that it identifies more computationally-relevant features that interact more sparsely, compared to principal component analysis. However, LIB does not yield substantial improvements in interpretability or interaction sparsity when applied to language models. We conclude that LIB is a promising theory-driven approach for analyzing neural networks, but in its current form is not applicable to large language models.

Neural Module Networks, originally proposed for the task of visual question answering, are a class of neural network architectures that involve human-specified neural modules, each designed for a specific form of reasoning. In current formulations of such networks only the parameters of the neural modules and/or the order of their execution is learned. In this work, we further expand this approach and also learn the underlying internal structure of modules in terms of the ordering and combination of simple and elementary arithmetic operators. Our results show that one is indeed able to simultaneously learn both internal module structure and module sequencing without extra supervisory signals for module execution sequencing. With this approach, we report performance comparable to models using hand-designed modules.

This paper discusses a simple and explicit toy-model example of the categorical Hopfield equations introduced in previous work of Manin and the author. These describe dynamical assignments of resources to networks, where resources are objects in unital symmetric monoidal categories and assignments are realized by summing functors. The special case discussed here is based on computational resources (computational models of neurons) as objects in a category of DNNs, with a simple choice of the endofunctors defining the Hopfield equations that reproduce the usual updating of the weights in DNNs by gradient descent.

The greatest ambition of mechanistic interpretability is to completely rewrite deep neural networks in a format that is more amenable to human understanding, while preserving their behavior and performance. In this paper, we attempt to partially rewrite a large language model using simple natural language explanations. We first approximate one of the feedforward networks in the LLM with a wider MLP with sparsely activating neurons - a transcoder - and use an automated interpretability pipeline to generate explanations for these neurons. We then replace the first layer of this sparse MLP with an LLM-based simulator, which predicts the activation of each neuron given its explanation and the surrounding context. Finally, we measure the degree to which these modifications distort the model's final output. With our pipeline, the model's increase in loss is statistically similar to entirely replacing the sparse MLP output with the zero vector. We employ the same protocol, this time using a sparse autoencoder, on the residual stream of the same layer and obtain similar results. These results suggest that more detailed explanations are needed to improve performance substantially above the zero ablation baseline.

The Backpropagation algorithm, widely used to train neural networks, has often been criticised for its lack of biological realism. In an attempt to find a more biologically plausible alternative, and avoid to back-propagate gradients in favour of using local learning rules, the recently introduced Forward-Forward algorithm replaces the traditional forward and backward passes of Backpropagation with two forward passes. In this work, we show that internal representations obtained with the Forward-Forward algorithm organize into robust, category-specific ensembles, composed by an extremely low number of active units (high sparsity). This is remarkably similar to what is observed in cortical representations during sensory processing. While not found in models trained with standard Backpropagation, sparsity emerges also in networks optimized by Backpropagation, on the same training objective of Forward-Forward. These results suggest that the learning procedure proposed by Forward-Forward may be superior to Backpropagation in modelling learning in the cortex, even when a backward pass is used.

Attending to what is relevant is fundamental to both the mammalian brain and modern machine learning models such as Transformers. Yet, determining relevance remains a core challenge, traditionally offloaded to learning algorithms like backpropagation. Inspired by recent cellular neurobiological evidence linking neocortical pyramidal cells to distinct mental states, this work shows how models (e.g., Transformers) can emulate high-level perceptual processing and awake thought (imagination) states to pre-select relevant information before applying attention. Triadic neuronal-level modulation loops among questions (Q), clues (keys, K), and hypotheses (values, V) enable diverse, deep, parallel reasoning chains at the representation level and allow a rapid shift from initial biases to refined understanding. This leads to orders-of-magnitude faster learning with significantly reduced computational demand (e.g., fewer heads, layers, and tokens), at an approximate cost of O(N), where N is the number of input tokens. Results span reinforcement learning (e.g., CarRacing in a high-dimensional visual setup), computer vision, and natural language question answering.

We analyze a family of large language models in such a lightweight manner that can be done on a single GPU. Specifically, we focus on the OPT family of models ranging from 125m to 66b parameters and rely only on whether an FFN neuron is activated or not. First, we find that the early part of the network is sparse and represents many discrete features. Here, many neurons (more than 70% in some layers of the 66b model) are "dead", i.e. they never activate on a large collection of diverse data. At the same time, many of the alive neurons are reserved for discrete features and act as token and n-gram detectors. Interestingly, their corresponding FFN updates not only promote next token candidates as could be expected, but also explicitly focus on removing the information about triggering them tokens, i.e., current input. To the best of our knowledge, this is the first example of mechanisms specialized at removing (rather than adding) information from the residual stream. With scale, models become more sparse in a sense that they have more dead neurons and token detectors. Finally, some neurons are positional: them being activated or not depends largely (or solely) on position and less so (or not at all) on textual data. We find that smaller models have sets of neurons acting as position range indicators while larger models operate in a less explicit manner.

How neuronal circuits achieve credit assignment remains a central unsolved question in systems neuroscience. Various studies have suggested plausible solutions for back-propagating error signals through multi-layer networks. These purely functionally motivated models assume distinct neuronal compartments to represent local error signals that determine the sign of synaptic plasticity. However, this explicit error modulation is inconsistent with phenomenological plasticity models in which the sign depends primarily on postsynaptic activity. Here we show how a plausible microcircuit model and Hebbian learning rule derived within an adaptive control theory framework can resolve this discrepancy. Assuming errors are encoded in top-down dis-inhibitory synaptic afferents, we show that error-modulated learning emerges naturally at the circuit level when recurrent inhibition explicitly influences Hebbian plasticity. The same learning rule accounts for experimentally observed plasticity in the absence of inhibition and performs comparably to back-propagation of error (BP) on several non-linearly separable benchmarks. Our findings bridge the gap between functional and experimentally observed plasticity rules and make concrete predictions on inhibitory modulation of excitatory plasticity.

We interpret the function of individual neurons in CLIP by automatically describing them using text. Analyzing the direct effects (i.e. the flow from a neuron through the residual stream to the output) or the indirect effects (overall contribution) fails to capture the neurons' function in CLIP. Therefore, we present the "second-order lens", analyzing the effect flowing from a neuron through the later attention heads, directly to the output. We find that these effects are highly selective: for each neuron, the effect is significant for <2% of the images. Moreover, each effect can be approximated by a single direction in the text-image space of CLIP. We describe neurons by decomposing these directions into sparse sets of text representations. The sets reveal polysemantic behavior - each neuron corresponds to multiple, often unrelated, concepts (e.g. ships and cars). Exploiting this neuron polysemy, we mass-produce "semantic" adversarial examples by generating images with concepts spuriously correlated to the incorrect class. Additionally, we use the second-order effects for zero-shot segmentation and attribute discovery in images. Our results indicate that a scalable understanding of neurons can be used for model deception and for introducing new model capabilities.

We create a reusable Transformer, BrainBERT, for intracranial recordings bringing modern representation learning approaches to neuroscience. Much like in NLP and speech recognition, this Transformer enables classifying complex concepts, i.e., decoding neural data, with higher accuracy and with much less data by being pretrained in an unsupervised manner on a large corpus of unannotated neural recordings. Our approach generalizes to new subjects with electrodes in new positions and to unrelated tasks showing that the representations robustly disentangle the neural signal. Just like in NLP where one can study language by investigating what a language model learns, this approach opens the door to investigating the brain by what a model of the brain learns. As a first step along this path, we demonstrate a new analysis of the intrinsic dimensionality of the computations in different areas of the brain. To construct these representations, we combine a technique for producing super-resolution spectrograms of neural data with an approach designed for generating contextual representations of audio by masking. In the future, far more concepts will be decodable from neural recordings by using representation learning, potentially unlocking the brain like language models unlocked language.

The nature of abstract reasoning is a matter of debate. Modern artificial neural network (ANN) models, like large language models, demonstrate impressive success when tested on abstract reasoning problems. However, it has been argued that their success reflects some form of memorization of similar problems (data contamination) rather than a general-purpose abstract reasoning capability. This concern is supported by evidence of brittleness, and the requirement of extensive training. In our study, we explored whether abstract reasoning can be achieved using the toolbox of ANNs, without prior training. Specifically, we studied an ANN model in which the weights of a naive network are optimized during the solution of the problem, using the problem data itself, rather than any prior knowledge. We tested this modeling approach on visual reasoning problems and found that it performs relatively well. Crucially, this success does not rely on memorization of similar problems. We further suggest an explanation of how it works. Finally, as problem solving is performed by changing the ANN weights, we explored the connection between problem solving and the accumulation of knowledge in the ANNs.

There has recently been a surge of work in explanatory artificial intelligence (XAI). This research area tackles the important problem that complex machines and algorithms often cannot provide insights into their behavior and thought processes. XAI allows users and parts of the internal system to be more transparent, providing explanations of their decisions in some level of detail. These explanations are important to ensure algorithmic fairness, identify potential bias/problems in the training data, and to ensure that the algorithms perform as expected. However, explanations produced by these systems is neither standardized nor systematically assessed. In an effort to create best practices and identify open challenges, we provide our definition of explainability and show how it can be used to classify existing literature. We discuss why current approaches to explanatory methods especially for deep neural networks are insufficient. Finally, based on our survey, we conclude with suggested future research directions for explanatory artificial intelligence.

In this work we identify the dormant neuron phenomenon in deep reinforcement learning, where an agent's network suffers from an increasing number of inactive neurons, thereby affecting network expressivity. We demonstrate the presence of this phenomenon across a variety of algorithms and environments, and highlight its effect on learning. To address this issue, we propose a simple and effective method (ReDo) that Recycles Dormant neurons throughout training. Our experiments demonstrate that ReDo maintains the expressive power of networks by reducing the number of dormant neurons and results in improved performance.

Training recurrent neural networks (RNNs) is a high-dimensional process that requires updating numerous parameters. Therefore, it is often difficult to pinpoint the underlying learning mechanisms. To address this challenge, we propose to gain mechanistic insights into the phenomenon of abrupt learning by studying RNNs trained to perform diverse short-term memory tasks. In these tasks, RNN training begins with an initial search phase. Following a long period of plateau in accuracy, the values of the loss function suddenly drop, indicating abrupt learning. Analyzing the neural computation performed by these RNNs reveals geometric restructuring (GR) in their phase spaces prior to the drop. To promote these GR events, we introduce a temporal consistency regularization that accelerates (bioplausible) training, facilitates attractor formation, and enables efficient learning in strongly connected networks. Our findings offer testable predictions for neuroscientists and emphasize the need for goal-agnostic secondary mechanisms to facilitate learning in biological and artificial networks.

The deep neural nets of modern artificial intelligence (AI) have not achieved defining features of biological intelligence, including abstraction, causal learning, and energy-efficiency. While scaling to larger models has delivered performance improvements for current applications, more brain-like capacities may demand new theories, models, and methods for designing artificial learning systems. Here, we argue that this opportunity to reassess insights from the brain should stimulate cooperation between AI research and theory-driven computational neuroscience (CN). To motivate a brain basis of neural computation, we present a dynamical view of intelligence from which we elaborate concepts of sparsity in network structure, temporal dynamics, and interactive learning. In particular, we suggest that temporal dynamics, as expressed through neural synchrony, nested oscillations, and flexible sequences, provide a rich computational layer for reading and updating hierarchical models distributed in long-term memory networks. Moreover, embracing agent-centered paradigms in AI and CN will accelerate our understanding of the complex dynamics and behaviors that build useful world models. A convergence of AI/CN theories and objectives will reveal dynamical principles of intelligence for brains and engineered learning systems. This article was inspired by our symposium on dynamical neuroscience and machine learning at the 6th Annual US/NIH BRAIN Initiative Investigators Meeting.

The parameter space for any fixed architecture of feedforward ReLU neural networks serves as a proxy during training for the associated class of functions - but how faithful is this representation? It is known that many different parameter settings can determine the same function. Moreover, the degree of this redundancy is inhomogeneous: for some networks, the only symmetries are permutation of neurons in a layer and positive scaling of parameters at a neuron, while other networks admit additional hidden symmetries. In this work, we prove that, for any network architecture where no layer is narrower than the input, there exist parameter settings with no hidden symmetries. We also describe a number of mechanisms through which hidden symmetries can arise, and empirically approximate the functional dimension of different network architectures at initialization. These experiments indicate that the probability that a network has no hidden symmetries decreases towards 0 as depth increases, while increasing towards 1 as width and input dimension increase.

Neural networks for NLP are becoming increasingly complex and widespread, and there is a growing concern if these models are responsible to use. Explaining models helps to address the safety and ethical concerns and is essential for accountability. Interpretability serves to provide these explanations in terms that are understandable to humans. Additionally, post-hoc methods provide explanations after a model is learned and are generally model-agnostic. This survey provides a categorization of how recent post-hoc interpretability methods communicate explanations to humans, it discusses each method in-depth, and how they are validated, as the latter is often a common concern.

Sparse Autoencoders (SAEs) have emerged as a useful tool for interpreting the internal representations of neural networks. However, naively optimising SAEs for reconstruction loss and sparsity results in a preference for SAEs that are extremely wide and sparse. We present an information-theoretic framework for interpreting SAEs as lossy compression algorithms for communicating explanations of neural activations. We appeal to the Minimal Description Length (MDL) principle to motivate explanations of activations which are both accurate and concise. We further argue that interpretable SAEs require an additional property, "independent additivity": features should be able to be understood separately. We demonstrate an example of applying our MDL-inspired framework by training SAEs on MNIST handwritten digits and find that SAE features representing significant line segments are optimal, as opposed to SAEs with features for memorised digits from the dataset or small digit fragments. We argue that using MDL rather than sparsity may avoid potential pitfalls with naively maximising sparsity such as undesirable feature splitting and that this framework naturally suggests new hierarchical SAE architectures which provide more concise explanations.

The ability of neural networks to represent more features than neurons makes interpreting them challenging. This phenomenon, known as superposition, has spurred efforts to find architectures that are more interpretable than standard multilayer perceptrons (MLPs) with elementwise activation functions. In this note, I examine bilinear layers, which are a type of MLP layer that are mathematically much easier to analyze while simultaneously performing better than standard MLPs. Although they are nonlinear functions of their input, I demonstrate that bilinear layers can be expressed using only linear operations and third order tensors. We can integrate this expression for bilinear layers into a mathematical framework for transformer circuits, which was previously limited to attention-only transformers. These results suggest that bilinear layers are easier to analyze mathematically than current architectures and thus may lend themselves to deeper safety insights by allowing us to talk more formally about circuits in neural networks. Additionally, bilinear layers may offer an alternative path for mechanistic interpretability through understanding the mechanisms of feature construction instead of enumerating a (potentially exponentially) large number of features in large models.

Within the complex neuroarchitecture of the brain, astrocytes play crucial roles in development, structure, and metabolism. These cells regulate neural activity through tripartite synapses, directly impacting cognitive processes such as learning and memory. Despite the growing recognition of astrocytes' significance, traditional Spiking Neural Network (SNN) models remain predominantly neuron-centric, overlooking the profound influence of astrocytes on neural dynamics. Inspired by these biological insights, we have developed an Astrocyte-Modulated Spiking Unit (AM-SU), an innovative framework that integrates neuron-astrocyte interactions into the computational paradigm, demonstrating wide applicability across various hardware platforms. Our Astrocyte-Modulated Spiking Neural Network (AstroSNN) exhibits exceptional performance in tasks involving memory retention and natural language generation, particularly in handling long-term dependencies and complex linguistic structures. The design of AstroSNN not only enhances its biological authenticity but also introduces novel computational dynamics, enabling more effective processing of complex temporal dependencies. Furthermore, AstroSNN shows low latency, high throughput, and reduced memory usage in practical applications, making it highly suitable for resource-constrained environments. By successfully integrating astrocytic dynamics into intelligent neural networks, our work narrows the gap between biological plausibility and neural modeling, laying the groundwork for future biologically-inspired neural computing research that includes both neurons and astrocytes.

Despite widespread adoption, machine learning models remain mostly black boxes. Understanding the reasons behind predictions is, however, quite important in assessing trust, which is fundamental if one plans to take action based on a prediction, or when choosing whether to deploy a new model. Such understanding also provides insights into the model, which can be used to transform an untrustworthy model or prediction into a trustworthy one. In this work, we propose LIME, a novel explanation technique that explains the predictions of any classifier in an interpretable and faithful manner, by learning an interpretable model locally around the prediction. We also propose a method to explain models by presenting representative individual predictions and their explanations in a non-redundant way, framing the task as a submodular optimization problem. We demonstrate the flexibility of these methods by explaining different models for text (e.g. random forests) and image classification (e.g. neural networks). We show the utility of explanations via novel experiments, both simulated and with human subjects, on various scenarios that require trust: deciding if one should trust a prediction, choosing between models, improving an untrustworthy classifier, and identifying why a classifier should not be trusted.

We study the ability of foundation models to learn representations for classification that are transferable to new, unseen classes. Recent results in the literature show that representations learned by a single classifier over many classes are competitive on few-shot learning problems with representations learned by special-purpose algorithms designed for such problems. In this paper we provide an explanation for this behavior based on the recently observed phenomenon that the features learned by overparameterized classification networks show an interesting clustering property, called neural collapse. We demonstrate both theoretically and empirically that neural collapse generalizes to new samples from the training classes, and -- more importantly -- to new classes as well, allowing foundation models to provide feature maps that work well in transfer learning and, specifically, in the few-shot setting.

We introduce Explanatory Learning (EL), a framework to let machines use existing knowledge buried in symbolic sequences -- e.g. explanations written in hieroglyphic -- by autonomously learning to interpret them. In EL, the burden of interpreting symbols is not left to humans or rigid human-coded compilers, as done in Program Synthesis. Rather, EL calls for a learned interpreter, built upon a limited collection of symbolic sequences paired with observations of several phenomena. This interpreter can be used to make predictions on a novel phenomenon given its explanation, and even to find that explanation using only a handful of observations, like human scientists do. We formulate the EL problem as a simple binary classification task, so that common end-to-end approaches aligned with the dominant empiricist view of machine learning could, in principle, solve it. To these models, we oppose Critical Rationalist Networks (CRNs), which instead embrace a rationalist view on the acquisition of knowledge. CRNs express several desired properties by construction, they are truly explainable, can adjust their processing at test-time for harder inferences, and can offer strong confidence guarantees on their predictions. As a final contribution, we introduce Odeen, a basic EL environment that simulates a small flatland-style universe full of phenomena to explain. Using Odeen as a testbed, we show how CRNs outperform empiricist end-to-end approaches of similar size and architecture (Transformers) in discovering explanations for novel phenomena.

Explaining deep learning models is essential for clinical integration of medical image analysis systems. A good explanation highlights if a model depends on spurious features that undermines generalization and harms a subset of patients or, conversely, may present novel biological insights. Although techniques like GradCAM can identify influential features, they are measurement tools that do not themselves form an explanation. We propose a human-machine-VLM interaction system tailored to explaining classifiers in computational pathology, including multi-instance learning for whole-slide images. Our proof of concept comprises (1) an AI-integrated slide viewer to run sliding-window experiments to test claims of an explanation, and (2) quantification of an explanation's predictiveness using general-purpose vision-language models. The results demonstrate that this allows us to qualitatively test claims of explanations and can quantifiably distinguish competing explanations. This offers a practical path from explainable AI to explained AI in digital pathology and beyond. Code and prompts are available at https://github.com/nki-ai/x2x.

Convolutional neural networks (CNN) are built upon the classical McCulloch-Pitts neuron model, which is essentially a linear model, where the nonlinearity is provided by a separate activation function. Several researchers have proposed enhanced neuron models, including quadratic neurons, generalized operational neurons, generative neurons, and super neurons, with stronger nonlinearity than that provided by the pointwise activation function. There has also been a proposal to use Pade approximation as a generalized activation function. In this paper, we introduce a brand new neuron model called Pade neurons (Paons), inspired by the Pade approximants, which is the best mathematical approximation of a transcendental function as a ratio of polynomials with different orders. We show that Paons are a super set of all other proposed neuron models. Hence, the basic neuron in any known CNN model can be replaced by Paons. In this paper, we extend the well-known ResNet to PadeNet (built by Paons) to demonstrate the concept. Our experiments on the single-image super-resolution task show that PadeNets can obtain better results than competing architectures.

Traveling waves of neural activity have been observed throughout the brain at a diversity of regions and scales; however, their precise computational role is still debated. One physically inspired hypothesis suggests that the cortical sheet may act like a wave-propagating system capable of invertibly storing a short-term memory of sequential stimuli through induced waves traveling across the cortical surface, and indeed many experimental results from neuroscience correlate wave activity with memory tasks. To date, however, the computational implications of this idea have remained hypothetical due to the lack of a simple recurrent neural network architecture capable of exhibiting such waves. In this work, we introduce a model to fill this gap, which we denote the Wave-RNN (wRNN), and demonstrate how such an architecture indeed efficiently encodes the recent past through a suite of synthetic memory tasks where wRNNs learn faster and reach significantly lower error than wave-free counterparts. We further explore the implications of this memory storage system on more complex sequence modeling tasks such as sequential image classification and find that wave-based models not only again outperform comparable wave-free RNNs while using significantly fewer parameters, but additionally perform comparably to more complex gated architectures such as LSTMs and GRUs.

Latest insights from biology show that intelligence not only emerges from the connections between neurons but that individual neurons shoulder more computational responsibility than previously anticipated. This perspective should be critical in the context of constantly changing distinct reinforcement learning environments, yet current approaches still primarily employ static activation functions. In this work, we motivate why rationals are suitable for adaptable activation functions and why their inclusion into neural networks is crucial. Inspired by recurrence in residual networks, we derive a condition under which rational units are closed under residual connections and formulate a naturally regularised version: the recurrent-rational. We demonstrate that equipping popular algorithms with (recurrent-)rational activations leads to consistent improvements on Atari games, especially turning simple DQN into a solid approach, competitive to DDQN and Rainbow.

As artificial intelligence models have exploded in scale and capability, understanding of their internal mechanisms remains a critical challenge. Inspired by the success of dynamical systems approaches in neuroscience, here we propose a novel framework for studying computations in deep learning systems. We focus on the residual stream (RS) in transformer models, conceptualizing it as a dynamical system evolving across layers. We find that activations of individual RS units exhibit strong continuity across layers, despite the RS being a non-privileged basis. Activations in the RS accelerate and grow denser over layers, while individual units trace unstable periodic orbits. In reduced-dimensional spaces, the RS follows a curved trajectory with attractor-like dynamics in the lower layers. These insights bridge dynamical systems theory and mechanistic interpretability, establishing a foundation for a "neuroscience of AI" that combines theoretical rigor with large-scale data analysis to advance our understanding of modern neural networks.

State-of-the-art systems neuroscience experiments yield large-scale multimodal data, and these data sets require new tools for analysis. Inspired by the success of large pretrained models in vision and language domains, we reframe the analysis of large-scale, cellular-resolution neuronal spiking data into an autoregressive spatiotemporal generation problem. Neuroformer is a multimodal, multitask generative pretrained transformer (GPT) model that is specifically designed to handle the intricacies of data in systems neuroscience. It scales linearly with feature size, can process an arbitrary number of modalities, and is adaptable to downstream tasks, such as predicting behavior. We first trained Neuroformer on simulated datasets, and found that it both accurately predicted simulated neuronal circuit activity, and also intrinsically inferred the underlying neural circuit connectivity, including direction. When pretrained to decode neural responses, the model predicted the behavior of a mouse with only few-shot fine-tuning, suggesting that the model begins learning how to do so directly from the neural representations themselves, without any explicit supervision. We used an ablation study to show that joint training on neuronal responses and behavior boosted performance, highlighting the model's ability to associate behavioral and neural representations in an unsupervised manner. These findings show that Neuroformer can analyze neural datasets and their emergent properties, informing the development of models and hypotheses associated with the brain.

We find arithmetic ability resides within a limited number of attention heads, with each head specializing in distinct operations. To delve into the reason, we introduce the Comparative Neuron Analysis (CNA) method, which identifies an internal logic chain consisting of four distinct stages from input to prediction: feature enhancing with shallow FFN neurons, feature transferring by shallow attention layers, feature predicting by arithmetic heads, and prediction enhancing among deep FFN neurons. Moreover, we identify the human-interpretable FFN neurons within both feature-enhancing and feature-predicting stages. These findings lead us to investigate the mechanism of LoRA, revealing that it enhances prediction probabilities by amplifying the coefficient scores of FFN neurons related to predictions. Finally, we apply our method in model pruning for arithmetic tasks and model editing for reducing gender bias. Code is on https://github.com/zepingyu0512/arithmetic-mechanism.

Deep learning models have achieved remarkable success in different areas of machine learning over the past decade; however, the size and complexity of these models make them difficult to understand. In an effort to make them more interpretable, several recent works focus on explaining parts of a deep neural network through human-interpretable, semantic attributes. However, it may be impossible to completely explain complex models using only semantic attributes. In this work, we propose to augment these attributes with a small set of uninterpretable features. Specifically, we develop a novel explanation framework ELUDE (Explanation via Labelled and Unlabelled DEcomposition) that decomposes a model's prediction into two parts: one that is explainable through a linear combination of the semantic attributes, and another that is dependent on the set of uninterpretable features. By identifying the latter, we are able to analyze the "unexplained" portion of the model, obtaining insights into the information used by the model. We show that the set of unlabelled features can generalize to multiple models trained with the same feature space and compare our work to two popular attribute-oriented methods, Interpretable Basis Decomposition and Concept Bottleneck, and discuss the additional insights ELUDE provides.

In this manuscript, we show that any neural network with any activation function can be represented as a decision tree. The representation is equivalence and not an approximation, thus keeping the accuracy of the neural network exactly as is. We believe that this work provides better understanding of neural networks and paves the way to tackle their black-box nature. We share equivalent trees of some neural networks and show that besides providing interpretability, tree representation can also achieve some computational advantages for small networks. The analysis holds both for fully connected and convolutional networks, which may or may not also include skip connections and/or normalizations.

Identifying cell types and understanding their functional properties is crucial for unraveling the mechanisms underlying perception and cognition. In the retina, functional types can be identified by carefully selected stimuli, but this requires expert domain knowledge and biases the procedure towards previously known cell types. In the visual cortex, it is still unknown what functional types exist and how to identify them. Thus, for unbiased identification of the functional cell types in retina and visual cortex, new approaches are needed. Here we propose an optimization-based clustering approach using deep predictive models to obtain functional clusters of neurons using Most Discriminative Stimuli (MDS). Our approach alternates between stimulus optimization with cluster reassignment akin to an expectation-maximization algorithm. The algorithm recovers functional clusters in mouse retina, marmoset retina and macaque visual area V4. This demonstrates that our approach can successfully find discriminative stimuli across species, stages of the visual system and recording techniques. The resulting most discriminative stimuli can be used to assign functional cell types fast and on the fly, without the need to train complex predictive models or show a large natural scene dataset, paving the way for experiments that were previously limited by experimental time. Crucially, MDS are interpretable: they visualize the distinctive stimulus patterns that most unambiguously identify a specific type of neuron.

Interpreting the inner function of neural networks is crucial for the trustworthy development and deployment of these black-box models. Prior interpretability methods focus on correlation-based measures to attribute model decisions to individual examples. However, these measures are susceptible to noise and spurious correlations encoded in the model during the training phase (e.g., biased inputs, model overfitting, or misspecification). Moreover, this process has proven to result in noisy and unstable attributions that prevent any transparent understanding of the model's behavior. In this paper, we develop a robust interventional-based method grounded by causal analysis to capture cause-effect mechanisms in pre-trained neural networks and their relation to the prediction. Our novel approach relies on path interventions to infer the causal mechanisms within hidden layers and isolate relevant and necessary information (to model prediction), avoiding noisy ones. The result is task-specific causal explanatory graphs that can audit model behavior and express the actual causes underlying its performance. We apply our method to vision models trained on classification tasks. On image classification tasks, we provide extensive quantitative experiments to show that our approach can capture more stable and faithful explanations than standard attribution-based methods. Furthermore, the underlying causal graphs reveal the neural interactions in the model, making it a valuable tool in other applications (e.g., model repair).

The utility of a learned neural representation depends on how well its geometry supports performance in downstream tasks. This geometry depends on the structure of the inputs, the structure of the target outputs, and the architecture of the network. By studying the learning dynamics of networks with one hidden layer, we discovered that the network's activation function has an unexpectedly strong impact on the representational geometry: Tanh networks tend to learn representations that reflect the structure of the target outputs, while ReLU networks retain more information about the structure of the raw inputs. This difference is consistently observed across a broad class of parameterized tasks in which we modulated the degree of alignment between the geometry of the task inputs and that of the task labels. We analyzed the learning dynamics in weight space and show how the differences between the networks with Tanh and ReLU nonlinearities arise from the asymmetric asymptotic behavior of ReLU, which leads feature neurons to specialize for different regions of input space. By contrast, feature neurons in Tanh networks tend to inherit the task label structure. Consequently, when the target outputs are low dimensional, Tanh networks generate neural representations that are more disentangled than those obtained with a ReLU nonlinearity. Our findings shed light on the interplay between input-output geometry, nonlinearity, and learned representations in neural networks.

We propose a novel nonlinear bidirectionally coupled heterogeneous chain network whose dynamics evolve in discrete time. The backbone of the model is a pair of popular map-based neuron models, the Chialvo and the Rulkov maps. This model is assumed to proximate the intricate dynamical properties of neurons in the widely complex nervous system. The model is first realized via various nonlinear analysis techniques: fixed point analysis, phase portraits, Jacobian matrix, and bifurcation diagrams. We observe the coexistence of chaotic and period-4 attractors. Various codimension-1 and -2 patterns for example saddle-node, period-doubling, Neimark-Sacker, double Neimark-Sacker, flip- and fold-Neimark Sacker, and 1:1 and 1:2 resonance are also explored. Furthermore, the study employs two synchronization measures to quantify how the oscillators in the network behave in tandem with each other over a long number of iterations. Finally, a time series analysis of the model is performed to investigate its complexity in terms of sample entropy.

A central goal of cognitive modeling is to develop models that not only predict human behavior but also provide insight into the underlying cognitive mechanisms. While neural network models trained on large-scale behavioral data often achieve strong predictive performance, they typically fall short in offering interpretable explanations of the cognitive processes they capture. In this work, we explore the potential of pretrained large language models (LLMs) to serve as dual-purpose cognitive models--capable of both accurate prediction and interpretable explanation in natural language. Specifically, we employ reinforcement learning with outcome-based rewards to guide LLMs toward generating explicit reasoning traces for explaining human risky choices. Our findings demonstrate that this approach produces high-quality explanations alongside strong quantitative predictions of human decisions.

A mechanistic understanding of how MLPs do computation in deep neural networks remains elusive. Current interpretability work can extract features from hidden activations over an input dataset but generally cannot explain how MLP weights construct features. One challenge is that element-wise nonlinearities introduce higher-order interactions and make it difficult to trace computations through the MLP layer. In this paper, we analyze bilinear MLPs, a type of Gated Linear Unit (GLU) without any element-wise nonlinearity that nevertheless achieves competitive performance. Bilinear MLPs can be fully expressed in terms of linear operations using a third-order tensor, allowing flexible analysis of the weights. Analyzing the spectra of bilinear MLP weights using eigendecomposition reveals interpretable low-rank structure across toy tasks, image classification, and language modeling. We use this understanding to craft adversarial examples, uncover overfitting, and identify small language model circuits directly from the weights alone. Our results demonstrate that bilinear layers serve as an interpretable drop-in replacement for current activation functions and that weight-based interpretability is viable for understanding deep-learning models.

Interpretation and explanation of deep models is critical towards wide adoption of systems that rely on them. In this paper, we propose a novel scheme for both interpretation as well as explanation in which, given a pretrained model, we automatically identify internal features relevant for the set of classes considered by the model, without relying on additional annotations. We interpret the model through average visualizations of this reduced set of features. Then, at test time, we explain the network prediction by accompanying the predicted class label with supporting visualizations derived from the identified features. In addition, we propose a method to address the artifacts introduced by stridded operations in deconvNet-based visualizations. Moreover, we introduce an8Flower, a dataset specifically designed for objective quantitative evaluation of methods for visual explanation.Experiments on the MNIST,ILSVRC12,Fashion144k and an8Flower datasets show that our method produces detailed explanations with good coverage of relevant features of the classes of interest

Inspired by the diversity of biological neurons, quadratic artificial neurons can play an important role in deep learning models. The type of quadratic neurons of our interest replaces the inner-product operation in the conventional neuron with a quadratic function. Despite promising results so far achieved by networks of quadratic neurons, there are important issues not well addressed. Theoretically, the superior expressivity of a quadratic network over either a conventional network or a conventional network via quadratic activation is not fully elucidated, which makes the use of quadratic networks not well grounded. Practically, although a quadratic network can be trained via generic backpropagation, it can be subject to a higher risk of collapse than the conventional counterpart. To address these issues, we first apply the spline theory and a measure from algebraic geometry to give two theorems that demonstrate better model expressivity of a quadratic network than the conventional counterpart with or without quadratic activation. Then, we propose an effective training strategy referred to as ReLinear to stabilize the training process of a quadratic network, thereby unleashing the full potential in its associated machine learning tasks. Comprehensive experiments on popular datasets are performed to support our findings and confirm the performance of quadratic deep learning. We have shared our code in https://github.com/FengleiFan/ReLinear.

The out-of-distribution (OOD) problem generally arises when neural networks encounter data that significantly deviates from the training data distribution, i.e., in-distribution (InD). In this paper, we study the OOD problem from a neuron activation view. We first formulate neuron activation states by considering both the neuron output and its influence on model decisions. Then, to characterize the relationship between neurons and OOD issues, we introduce the neuron activation coverage (NAC) -- a simple measure for neuron behaviors under InD data. Leveraging our NAC, we show that 1) InD and OOD inputs can be largely separated based on the neuron behavior, which significantly eases the OOD detection problem and beats the 21 previous methods over three benchmarks (CIFAR-10, CIFAR-100, and ImageNet-1K). 2) a positive correlation between NAC and model generalization ability consistently holds across architectures and datasets, which enables a NAC-based criterion for evaluating model robustness. Compared to prevalent InD validation criteria, we show that NAC not only can select more robust models, but also has a stronger correlation with OOD test performance.

Exposing meaningful and interpretable neural interactions is critical to understanding neural circuits. Inferred neural interactions from neural signals primarily reflect functional interactions. In a long experiment, subject animals may experience different stages defined by the experiment, stimuli, or behavioral states, and hence functional interactions can change over time. To model dynamically changing functional interactions, prior work employs state-switching generalized linear models with hidden Markov models (i.e., HMM-GLMs). However, we argue they lack biological plausibility, as functional interactions are shaped and confined by the underlying anatomical connectome. Here, we propose a novel prior-informed state-switching GLM. We introduce both a Gaussian prior and a one-hot prior over the GLM in each state. The priors are learnable. We will show that the learned prior should capture the state-constant interaction, shedding light on the underlying anatomical connectome and revealing more likely physical neuron interactions. The state-dependent interaction modeled by each GLM offers traceability to capture functional variations across multiple brain states. Our methods effectively recover true interaction structures in simulated data, achieve the highest predictive likelihood with real neural datasets, and render interaction structures and hidden states more interpretable when applied to real neural data.

Neural networks have achieved remarkable success across various fields. However, the lack of interpretability limits their practical use, particularly in critical decision-making scenarios. Post-hoc interpretability, which provides explanations for pre-trained models, is often at risk of robustness and fidelity. This has inspired a rising interest in self-interpretable neural networks, which inherently reveal the prediction rationale through the model structures. Although there exist surveys on post-hoc interpretability, a comprehensive and systematic survey of self-interpretable neural networks is still missing. To address this gap, we first collect and review existing works on self-interpretable neural networks and provide a structured summary of their methodologies from five key perspectives: attribution-based, function-based, concept-based, prototype-based, and rule-based self-interpretation. We also present concrete, visualized examples of model explanations and discuss their applicability across diverse scenarios, including image, text, graph data, and deep reinforcement learning. Additionally, we summarize existing evaluation metrics for self-interpretability and identify open challenges in this field, offering insights for future research. To support ongoing developments, we present a publicly accessible resource to track advancements in this domain: https://github.com/yangji721/Awesome-Self-Interpretable-Neural-Network.

Methods for understanding the decisions of and mechanisms underlying deep neural networks (DNNs) typically rely on building intuition by emphasizing sensory or semantic features of individual examples. For instance, methods aim to visualize the components of an input which are "important" to a network's decision, or to measure the semantic properties of single neurons. Here, we argue that interpretability research suffers from an over-reliance on intuition-based approaches that risk-and in some cases have caused-illusory progress and misleading conclusions. We identify a set of limitations that we argue impede meaningful progress in interpretability research, and examine two popular classes of interpretability methods-saliency and single-neuron-based approaches-that serve as case studies for how overreliance on intuition and lack of falsifiability can undermine interpretability research. To address these concerns, we propose a strategy to address these impediments in the form of a framework for strongly falsifiable interpretability research. We encourage researchers to use their intuitions as a starting point to develop and test clear, falsifiable hypotheses, and hope that our framework yields robust, evidence-based interpretability methods that generate meaningful advances in our understanding of DNNs.

Vision Transformer models exhibit immense power yet remain opaque to human understanding, posing challenges and risks for practical applications. While prior research has attempted to demystify these models through input attribution and neuron role analysis, there's been a notable gap in considering layer-level information and the holistic path of information flow across layers. In this paper, we investigate the significance of influential neuron paths within vision Transformers, which is a path of neurons from the model input to output that impacts the model inference most significantly. We first propose a joint influence measure to assess the contribution of a set of neurons to the model outcome. And we further provide a layer-progressive neuron locating approach that efficiently selects the most influential neuron at each layer trying to discover the crucial neuron path from input to output within the target model. Our experiments demonstrate the superiority of our method finding the most influential neuron path along which the information flows, over the existing baseline solutions. Additionally, the neuron paths have illustrated that vision Transformers exhibit some specific inner working mechanism for processing the visual information within the same image category. We further analyze the key effects of these neurons on the image classification task, showcasing that the found neuron paths have already preserved the model capability on downstream tasks, which may also shed some lights on real-world applications like model pruning. The project website including implementation code is available at https://foundation-model-research.github.io/NeuronPath/.

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the i-th neuron in a nonlinear operator layer is defined by mathcal O_i(u) = sigmaleft( sum_j mathcal W_{ij} u + mathcal B_{ij}right). Here, mathcal W_{ij} denotes the bounded linear operator connecting j-th input neuron to i-th output neuron, and the bias mathcal B_{ij} takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

Mathematical reasoning is an increasingly important indicator of large language model (LLM) capabilities, yet we lack understanding of how LLMs process even simple mathematical tasks. To address this, we reverse engineer how three mid-sized LLMs compute addition. We first discover that numbers are represented in these LLMs as a generalized helix, which is strongly causally implicated for the tasks of addition and subtraction, and is also causally relevant for integer division, multiplication, and modular arithmetic. We then propose that LLMs compute addition by manipulating this generalized helix using the "Clock" algorithm: to solve a+b, the helices for a and b are manipulated to produce the a+b answer helix which is then read out to model logits. We model influential MLP outputs, attention head outputs, and even individual neuron preactivations with these helices and verify our understanding with causal interventions. By demonstrating that LLMs represent numbers on a helix and manipulate this helix to perform addition, we present the first representation-level explanation of an LLM's mathematical capability.

Neuromorphic computing with spiking neural networks is promising for energy-efficient artificial intelligence (AI) applications. However, different from humans who continually learn different tasks in a lifetime, neural network models suffer from catastrophic forgetting. How could neuronal operations solve this problem is an important question for AI and neuroscience. Many previous studies draw inspiration from observed neuroscience phenomena and propose episodic replay or synaptic metaplasticity, but they are not guaranteed to explicitly preserve knowledge for neuron populations. Other works focus on machine learning methods with more mathematical grounding, e.g., orthogonal projection on high dimensional spaces, but there is no neural correspondence for neuromorphic computing. In this work, we develop a new method with neuronal operations based on lateral connections and Hebbian learning, which can protect knowledge by projecting activity traces of neurons into an orthogonal subspace so that synaptic weight update will not interfere with old tasks. We show that Hebbian and anti-Hebbian learning on recurrent lateral connections can effectively extract the principal subspace of neural activities and enable orthogonal projection. This provides new insights into how neural circuits and Hebbian learning can help continual learning, and also how the concept of orthogonal projection can be realized in neuronal systems. Our method is also flexible to utilize arbitrary training methods based on presynaptic activities/traces. Experiments show that our method consistently solves forgetting for spiking neural networks with nearly zero forgetting under various supervised training methods with different error propagation approaches, and outperforms previous approaches under various settings. Our method can pave a solid path for building continual neuromorphic computing systems.

It is typically understood that the training of modern neural networks is a process of fitting the probability distribution of desired output. However, recent paradoxical observations in a number of language generation tasks let one wonder if this canonical probability-based explanation can really account for the empirical success of deep learning. To resolve this issue, we propose an alternative utility-based explanation to the standard supervised learning procedure in deep learning. The basic idea is to interpret the learned neural network not as a probability model but as an ordinal utility function that encodes the preference revealed in training data. In this perspective, training of the neural network corresponds to a utility learning process. Specifically, we show that for all neural networks with softmax outputs, the SGD learning dynamic of maximum likelihood estimation (MLE) can be seen as an iteration process that optimizes the neural network toward an optimal utility function. This utility-based interpretation can explain several otherwise-paradoxical observations about the neural networks thus trained. Moreover, our utility-based theory also entails an equation that can transform the learned utility values back to a new kind of probability estimation with which probability-compatible decision rules enjoy dramatic (double-digits) performance improvements. These evidences collectively reveal a phenomenon of utility-probability duality in terms of what modern neural networks are (truly) modeling: We thought they are one thing (probabilities), until the unexplainable showed up; changing mindset and treating them as another thing (utility values) largely reconcile the theory, despite remaining subtleties regarding its original (probabilistic) identity.

Representation learning, and interpreting learned representations, are key areas of focus in machine learning and neuroscience. Both fields generally use representations as a means to understand or improve a system's computations. In this work, however, we explore surprising dissociations between representation and computation that may pose challenges for such efforts. We create datasets in which we attempt to match the computational role that different features play, while manipulating other properties of the features or the data. We train various deep learning architectures to compute these multiple abstract features about their inputs. We find that their learned feature representations are systematically biased towards representing some features more strongly than others, depending upon extraneous properties such as feature complexity, the order in which features are learned, and the distribution of features over the inputs. For example, features that are simpler to compute or learned first tend to be represented more strongly and densely than features that are more complex or learned later, even if all features are learned equally well. We also explore how these biases are affected by architectures, optimizers, and training regimes (e.g., in transformers, features decoded earlier in the output sequence also tend to be represented more strongly). Our results help to characterize the inductive biases of gradient-based representation learning. These results also highlight a key challenge for interpretability - or for comparing the representations of models and brains - disentangling extraneous biases from the computationally important aspects of a system's internal representations.

A Perceptron is a fundamental building block of a neural network. The flexibility and scalability of perceptron make it ubiquitous in building intelligent systems. Studies have shown the efficacy of a single neuron in making intelligent decisions. Here, we examined and compared two perceptrons with distinct mechanisms, and developed a quantum version of one of those perceptrons. As a part of this modeling, we implemented the quantum circuit for an artificial perception, generated a dataset, and simulated the training. Through these experiments, we show that there is an exponential growth advantage and test different qubit versions. Our findings show that this quantum model of an individual perceptron can be used as a pattern classifier. For the second type of model, we provide an understanding to design and simulate a spike-dependent quantum perceptron. Our code is available at https://github.com/ashutosh1919/quantum-perceptron

Gradient descent is the method of choice for training large artificial intelligence systems. As these systems become larger, a better understanding of the mechanisms behind gradient training would allow us to alleviate compute costs and help steer these systems away from harmful behaviors. To that end, we suggest utilizing the circuit perspective brought forward by mechanistic interpretability. After laying out our intuition, we illustrate how it enables us to design a curriculum for efficient learning in a controlled setting. The code is available at https://github.com/facebookresearch/pal.

Recently, interest has grown in exploring the hypothesis that neural activity conveys information through precise spiking motifs. To investigate this phenomenon, various algorithms have been proposed to detect such motifs in Single Unit Activity (SUA) recorded from populations of neurons. In this study, we present a novel detection model based on the inversion of a generative model of raster plot synthesis. Using this generative model, we derive an optimal detection procedure that takes the form of logistic regression combined with temporal convolution. A key advantage of this model is its differentiability, which allows us to formulate a supervised learning approach using a gradient descent on the binary cross-entropy loss. To assess the model's ability to detect spiking motifs in synthetic data, we first perform numerical evaluations. This analysis highlights the advantages of using spiking motifs over traditional firing rate based population codes. We then successfully demonstrate that our learning method can recover synthetically generated spiking motifs, indicating its potential for further applications. In the future, we aim to extend this method to real neurobiological data, where the ground truth is unknown, to explore and detect spiking motifs in a more natural and biologically relevant context.

The lack of transparency of Deep Neural Networks continues to be a limitation that severely undermines their reliability and usage in high-stakes applications. Promising approaches to overcome such limitations are Prototype-Based Self-Explainable Neural Networks (PSENNs), whose predictions rely on the similarity between the input at hand and a set of prototypical representations of the output classes, offering therefore a deep, yet transparent-by-design, architecture. So far, such models have been designed by considering pointwise estimates for the prototypes, which remain fixed after the learning phase of the model. In this paper, we introduce a probabilistic reformulation of PSENNs, called Prob-PSENN, which replaces point estimates for the prototypes with probability distributions over their values. This provides not only a more flexible framework for an end-to-end learning of prototypes, but can also capture the explanatory uncertainty of the model, which is a missing feature in previous approaches. In addition, since the prototypes determine both the explanation and the prediction, Prob-PSENNs allow us to detect when the model is making uninformed or uncertain predictions, and to obtain valid explanations for them. Our experiments demonstrate that Prob-PSENNs provide more meaningful and robust explanations than their non-probabilistic counterparts, thus enhancing the explainability and reliability of the models.

While neural networks can be approximated by linear models as their width increases, certain properties of wide neural networks cannot be captured by linear models. In this work we show that recently proposed Neural Quadratic Models can exhibit the "catapult phase" [Lewkowycz et al. 2020] that arises when training such models with large learning rates. We then empirically show that the behaviour of neural quadratic models parallels that of neural networks in generalization, especially in the catapult phase regime. Our analysis further demonstrates that quadratic models can be an effective tool for analysis of neural networks.

Interpretability and explainability of deep neural networks are challenging due to their scale, complexity, and the agreeable notions on which the explaining process rests. Previous work, in particular, has focused on representing internal components of neural networks through human-friendly visuals and concepts. On the other hand, in real life, when making a decision, human tends to rely on similar situations and/or associations in the past. Hence arguably, a promising approach to make the model transparent is to design it in a way such that the model explicitly connects the current sample with the seen ones, and bases its decision on these samples. Grounded on that principle, we propose in this paper an explainable, evidence-based memory network architecture, which learns to summarize the dataset and extract supporting evidences to make its decision. Our model achieves state-of-the-art performance on two popular question answering datasets (i.e. TrecQA and WikiQA). Via further analysis, we show that this model can reliably trace the errors it has made in the validation step to the training instances that might have caused these errors. We believe that this error-tracing capability provides significant benefit in improving dataset quality in many applications.

As one of the energy-efficient alternatives of conventional neural networks (CNNs), spiking neural networks (SNNs) have gained more and more interest recently. To train the deep models, some effective batch normalization (BN) techniques are proposed in SNNs. All these BNs are suggested to be used after the convolution layer as usually doing in CNNs. However, the spiking neuron is much more complex with the spatio-temporal dynamics. The regulated data flow after the BN layer will be disturbed again by the membrane potential updating operation before the firing function, i.e., the nonlinear activation. Therefore, we advocate adding another BN layer before the firing function to normalize the membrane potential again, called MPBN. To eliminate the induced time cost of MPBN, we also propose a training-inference-decoupled re-parameterization technique to fold the trained MPBN into the firing threshold. With the re-parameterization technique, the MPBN will not introduce any extra time burden in the inference. Furthermore, the MPBN can also adopt the element-wised form, while these BNs after the convolution layer can only use the channel-wised form. Experimental results show that the proposed MPBN performs well on both popular non-spiking static and neuromorphic datasets. Our code is open-sourced at https://github.com/yfguo91/MPBN{MPBN}.

Going beyond 'dendritic democracy', we introduce a 'democracy of local processors', termed Cooperator. Here we compare their capabilities when used in permutation-invariant neural networks for reinforcement learning (RL), with machine learning algorithms based on Transformers, such as ChatGPT. Transformers are based on the long-standing conception of integrate-and-fire 'point' neurons, whereas Cooperator is inspired by recent neurobiological breakthroughs suggesting that the cellular foundations of mental life depend on context-sensitive pyramidal neurons in the neocortex which have two functionally distinct points. We show that when used for RL, an algorithm based on Cooperator learns far quicker than that based on Transformer, even while having the same number of parameters.

Neural networks are a group of neurons stacked together in multiple layers to mimic the biological neurons in a human brain. Neural networks have been trained using the backpropagation algorithm based on gradient descent strategy for several decades. Several variants have been developed to improve the backpropagation algorithm. The loss function for the neural network is optimized through backpropagation, but several local minima exist in the manifold of the constructed neural network. We obtain several solutions matching the minima. The gradient descent strategy cannot avoid the problem of local minima and gets stuck in the minima due to the initialization. Particle swarm optimization (PSO) was proposed to select the best local minima among the search space of the loss function. The search space is limited to the instantiated particles in the PSO algorithm, and sometimes it cannot select the best solution. In the proposed approach, we overcome the problem of gradient descent and the limitation of the PSO algorithm by training individual neurons separately, capable of collectively solving the problem as a group of neurons forming a network. Our code and data are available at https://github.com/dipkmr/train-nn-wobp/

In living organisms, homeostasis is the natural regulation of internal states aimed at maintaining conditions compatible with life. Typical artificial systems are not equipped with comparable regulatory features. Here, we introduce an artificial neural network that incorporates homeostatic features. Its own computing substrate is placed in a needful and vulnerable relation to the very objects over which it computes. For example, artificial neurons performing classification of MNIST digits or Fashion-MNIST articles of clothing may receive excitatory or inhibitory effects, which alter their own learning rate as a direct result of perceiving and classifying the digits. In this scenario, accurate recognition is desirable to the agent itself because it guides decisions to regulate its vulnerable internal states and functionality. Counterintuitively, the addition of vulnerability to a learner does not necessarily impair its performance. On the contrary, self-regulation in response to vulnerability confers benefits under certain conditions. We show that homeostatic design confers increased adaptability under concept shift, in which the relationships between labels and data change over time, and that the greatest advantages are obtained under the highest rates of shift. This necessitates the rapid un-learning of past associations and the re-learning of new ones. We also demonstrate the superior abilities of homeostatic learners in environments with dynamically changing rates of concept shift. Our homeostatic design exposes the artificial neural network's thinking machinery to the consequences of its own "thoughts", illustrating the advantage of putting one's own "skin in the game" to improve fluid intelligence.

Much of the excitement in modern AI is driven by the observation that scaling up existing systems leads to better performance. But does better performance necessarily imply better internal representations? While the representational optimist assumes it must, this position paper challenges that view. We compare neural networks evolved through an open-ended search process to networks trained via conventional stochastic gradient descent (SGD) on the simple task of generating a single image. This minimal setup offers a unique advantage: each hidden neuron's full functional behavior can be easily visualized as an image, thus revealing how the network's output behavior is internally constructed neuron by neuron. The result is striking: while both networks produce the same output behavior, their internal representations differ dramatically. The SGD-trained networks exhibit a form of disorganization that we term fractured entangled representation (FER). Interestingly, the evolved networks largely lack FER, even approaching a unified factored representation (UFR). In large models, FER may be degrading core model capacities like generalization, creativity, and (continual) learning. Therefore, understanding and mitigating FER could be critical to the future of representation learning.

A hallmark of intelligence is the ability to autonomously learn new flexible, cognitive behaviors - that is, behaviors where the appropriate action depends not just on immediate stimuli (as in simple reflexive stimulus-response associations), but on contextual information that must be adequately acquired, stored and processed. While many meta-learning algorithms can design agents that autonomously learn new tasks, cognitive tasks adds another level of learning and memory to typical ``learning-to-learn'' problems. Here we evolve neural networks, endowed with plastic connections and neuromodulation, over a sizable set of simple cognitive tasks adapted from a computational neuroscience framework. The resulting evolved networks can automatically modify their own connectivity to acquire a novel simple cognitive task, never seen during evolution, from stimuli and rewards alone, through the spontaneous operation of their evolved neural organization and plasticity system. Our results emphasize the importance of carefully considering the multiple learning loops involved in the emergence of intelligent behavior.

This paper describes MAIA, a Multimodal Automated Interpretability Agent. MAIA is a system that uses neural models to automate neural model understanding tasks like feature interpretation and failure mode discovery. It equips a pre-trained vision-language model with a set of tools that support iterative experimentation on subcomponents of other models to explain their behavior. These include tools commonly used by human interpretability researchers: for synthesizing and editing inputs, computing maximally activating exemplars from real-world datasets, and summarizing and describing experimental results. Interpretability experiments proposed by MAIA compose these tools to describe and explain system behavior. We evaluate applications of MAIA to computer vision models. We first characterize MAIA's ability to describe (neuron-level) features in learned representations of images. Across several trained models and a novel dataset of synthetic vision neurons with paired ground-truth descriptions, MAIA produces descriptions comparable to those generated by expert human experimenters. We then show that MAIA can aid in two additional interpretability tasks: reducing sensitivity to spurious features, and automatically identifying inputs likely to be mis-classified.

This paper presents an unsupervised method to learn a neural network, namely an explainer, to interpret a pre-trained convolutional neural network (CNN), i.e., explaining knowledge representations hidden in middle conv-layers of the CNN. Given feature maps of a certain conv-layer of the CNN, the explainer performs like an auto-encoder, which first disentangles the feature maps into object-part features and then inverts object-part features back to features of higher conv-layers of the CNN. More specifically, the explainer contains interpretable conv-layers, where each filter disentangles the representation of a specific object part from chaotic input feature maps. As a paraphrase of CNN features, the disentangled representations of object parts help people understand the logic inside the CNN. We also learn the explainer to use object-part features to reconstruct features of higher CNN layers, in order to minimize loss of information during the feature disentanglement. More crucially, we learn the explainer via network distillation without using any annotations of sample labels, object parts, or textures for supervision. We have applied our method to different types of CNNs for evaluation, and explainers have significantly boosted the interpretability of CNN features.

Neural networks often exhibit emergent behavior, where qualitatively new capabilities arise from scaling up the amount of parameters, training data, or training steps. One approach to understanding emergence is to find continuous progress measures that underlie the seemingly discontinuous qualitative changes. We argue that progress measures can be found via mechanistic interpretability: reverse-engineering learned behaviors into their individual components. As a case study, we investigate the recently-discovered phenomenon of ``grokking'' exhibited by small transformers trained on modular addition tasks. We fully reverse engineer the algorithm learned by these networks, which uses discrete Fourier transforms and trigonometric identities to convert addition to rotation about a circle. We confirm the algorithm by analyzing the activations and weights and by performing ablations in Fourier space. Based on this understanding, we define progress measures that allow us to study the dynamics of training and split training into three continuous phases: memorization, circuit formation, and cleanup. Our results show that grokking, rather than being a sudden shift, arises from the gradual amplification of structured mechanisms encoded in the weights, followed by the later removal of memorizing components.

A large set of the explainable Artificial Intelligence (XAI) literature is emerging on feature relevance techniques to explain a deep neural network (DNN) output or explaining models that ingest image source data. However, assessing how XAI techniques can help understand models beyond classification tasks, e.g. for reinforcement learning (RL), has not been extensively studied. We review recent works in the direction to attain Explainable Reinforcement Learning (XRL), a relatively new subfield of Explainable Artificial Intelligence, intended to be used in general public applications, with diverse audiences, requiring ethical, responsible and trustable algorithms. In critical situations where it is essential to justify and explain the agent's behaviour, better explainability and interpretability of RL models could help gain scientific insight on the inner workings of what is still considered a black box. We evaluate mainly studies directly linking explainability to RL, and split these into two categories according to the way the explanations are generated: transparent algorithms and post-hoc explainaility. We also review the most prominent XAI works from the lenses of how they could potentially enlighten the further deployment of the latest advances in RL, in the demanding present and future of everyday problems.

Recent studies in interpretability have explored the inner workings of transformer models trained on tasks across various domains, often discovering that these networks naturally develop highly structured representations. When such representations comprehensively reflect the task domain's structure, they are commonly referred to as "World Models" (WMs). In this work, we identify WMs in transformers trained on maze-solving tasks. By using Sparse Autoencoders (SAEs) and analyzing attention patterns, we examine the construction of WMs and demonstrate consistency between SAE feature-based and circuit-based analyses. By subsequently intervening on isolated features to confirm their causal role, we find that it is easier to activate features than to suppress them. Furthermore, we find that models can reason about mazes involving more simultaneously active features than they encountered during training; however, when these same mazes (with greater numbers of connections) are provided to models via input tokens instead, the models fail. Finally, we demonstrate that positional encoding schemes appear to influence how World Models are structured within the model's residual stream.

Large language models (LLMs) have demonstrated remarkable prediction performance for a growing array of tasks. However, their rapid proliferation and increasing opaqueness have created a growing need for interpretability. Here, we ask whether we can automatically obtain natural language explanations for black box text modules. A "text module" is any function that maps text to a scalar continuous value, such as a submodule within an LLM or a fitted model of a brain region. "Black box" indicates that we only have access to the module's inputs/outputs. We introduce Summarize and Score (SASC), a method that takes in a text module and returns a natural language explanation of the module's selectivity along with a score for how reliable the explanation is. We study SASC in 3 contexts. First, we evaluate SASC on synthetic modules and find that it often recovers ground truth explanations. Second, we use SASC to explain modules found within a pre-trained BERT model, enabling inspection of the model's internals. Finally, we show that SASC can generate explanations for the response of individual fMRI voxels to language stimuli, with potential applications to fine-grained brain mapping. All code for using SASC and reproducing results is made available on Github.

Developing Intelligent Systems involves artificial intelligence approaches including artificial neural networks. Here, we present a tutorial of Deep Neural Networks (DNNs), and some insights about the origin of the term "deep"; references to deep learning are also given. Restricted Boltzmann Machines, which are the core of DNNs, are discussed in detail. An example of a simple two-layer network, performing unsupervised learning for unlabeled data, is shown. Deep Belief Networks (DBNs), which are used to build networks with more than two layers, are also described. Moreover, examples for supervised learning with DNNs performing simple prediction and classification tasks, are presented and explained. This tutorial includes two intelligent pattern recognition applications: hand- written digits (benchmark known as MNIST) and speech recognition.

Transformers demonstrate impressive performance on a range of reasoning benchmarks. To evaluate the degree to which these abilities are a result of actual reasoning, existing work has focused on developing sophisticated benchmarks for behavioral studies. However, these studies do not provide insights into the internal mechanisms driving the observed capabilities. To improve our understanding of the internal mechanisms of transformers, we present a comprehensive mechanistic analysis of a transformer trained on a synthetic reasoning task. We identify a set of interpretable mechanisms the model uses to solve the task, and validate our findings using correlational and causal evidence. Our results suggest that it implements a depth-bounded recurrent mechanisms that operates in parallel and stores intermediate results in selected token positions. We anticipate that the motifs we identified in our synthetic setting can provide valuable insights into the broader operating principles of transformers and thus provide a basis for understanding more complex models.

Causal abstraction is a promising theoretical framework for explainable artificial intelligence that defines when an interpretable high-level causal model is a faithful simplification of a low-level deep learning system. However, existing causal abstraction methods have two major limitations: they require a brute-force search over alignments between the high-level model and the low-level one, and they presuppose that variables in the high-level model will align with disjoint sets of neurons in the low-level one. In this paper, we present distributed alignment search (DAS), which overcomes these limitations. In DAS, we find the alignment between high-level and low-level models using gradient descent rather than conducting a brute-force search, and we allow individual neurons to play multiple distinct roles by analyzing representations in non-standard bases-distributed representations. Our experiments show that DAS can discover internal structure that prior approaches miss. Overall, DAS removes previous obstacles to conducting causal abstraction analyses and allows us to find conceptual structure in trained neural nets.

The role of hidden units in recurrent neural networks is typically seen as modeling memory, with research focusing on enhancing information retention through gating mechanisms. A less explored perspective views hidden units as active participants in the computation performed by the network, rather than passive memory stores. In this work, we revisit bi-linear operations, which involve multiplicative interactions between hidden units and input embeddings. We demonstrate theoretically and empirically that they constitute a natural inductive bias for representing the evolution of hidden states in state tracking tasks. These are the simplest type of task that require hidden units to actively contribute to the behavior of the network. We also show that bi-linear state updates form a natural hierarchy corresponding to state tracking tasks of increasing complexity, with popular linear recurrent networks such as Mamba residing at the lowest-complexity center of that hierarchy.

Neuro-symbolic learning was proposed to address challenges with training neural networks for complex reasoning tasks with the added benefits of interpretability, reliability, and efficiency. Neuro-symbolic learning methods traditionally train neural models in conjunction with symbolic programs, but they face significant challenges that limit them to simplistic problems. On the other hand, purely-neural foundation models now reach state-of-the-art performance through prompting rather than training, but they are often unreliable and lack interpretability. Supplementing foundation models with symbolic programs, which we call neuro-symbolic prompting, provides a way to use these models for complex reasoning tasks. Doing so raises the question: What role does specialized model training as part of neuro-symbolic learning have in the age of foundation models? To explore this question, we highlight three pitfalls of traditional neuro-symbolic learning with respect to the compute, data, and programs leading to generalization problems. This position paper argues that foundation models enable generalizable neuro-symbolic solutions, offering a path towards achieving the original goals of neuro-symbolic learning without the downsides of training from scratch.

Universality is a key hypothesis in mechanistic interpretability -- that different models learn similar features and circuits when trained on similar tasks. In this work, we study the universality hypothesis by examining how small neural networks learn to implement group composition. We present a novel algorithm by which neural networks may implement composition for any finite group via mathematical representation theory. We then show that networks consistently learn this algorithm by reverse engineering model logits and weights, and confirm our understanding using ablations. By studying networks of differing architectures trained on various groups, we find mixed evidence for universality: using our algorithm, we can completely characterize the family of circuits and features that networks learn on this task, but for a given network the precise circuits learned -- as well as the order they develop -- are arbitrary.

Many interpretable AI approaches have been proposed to provide plausible explanations for a model's decision-making. However, configuring an explainable model that effectively communicates among computational modules has received less attention. A recently proposed shared global workspace theory showed that networks of distributed modules can benefit from sharing information with a bottlenecked memory because the communication constraints encourage specialization, compositionality, and synchronization among the modules. Inspired by this, we propose Concept-Centric Transformers, a simple yet effective configuration of the shared global workspace for interpretability, consisting of: i) an object-centric-based memory module for extracting semantic concepts from input features, ii) a cross-attention mechanism between the learned concept and input embeddings, and iii) standard classification and explanation losses to allow human analysts to directly assess an explanation for the model's classification reasoning. We test our approach against other existing concept-based methods on classification tasks for various datasets, including CIFAR100, CUB-200-2011, and ImageNet, and we show that our model achieves better classification accuracy than all baselines across all problems but also generates more consistent concept-based explanations of classification output.

Despite their remarkable success and deployment across diverse workflows, language models sometimes produce untruthful responses. Our limited understanding of how truthfulness is mechanistically encoded within these models jeopardizes their reliability and safety. In this paper, we propose a method for identifying representations of truthfulness at the neuron level. We show that language models contain truth neurons, which encode truthfulness in a subject-agnostic manner. Experiments conducted across models of varying scales validate the existence of truth neurons, confirming that the encoding of truthfulness at the neuron level is a property shared by many language models. The distribution patterns of truth neurons over layers align with prior findings on the geometry of truthfulness. Selectively suppressing the activations of truth neurons found through the TruthfulQA dataset degrades performance both on TruthfulQA and on other benchmarks, showing that the truthfulness mechanisms are not tied to a specific dataset. Our results offer novel insights into the mechanisms underlying truthfulness in language models and highlight potential directions toward improving their trustworthiness and reliability.

Biological cortical neurons are remarkably sophisticated computational devices, temporally integrating their vast synaptic input over an intricate dendritic tree, subject to complex, nonlinearly interacting internal biological processes. A recent study proposed to characterize this complexity by fitting accurate surrogate models to replicate the input-output relationship of a detailed biophysical cortical pyramidal neuron model and discovered it needed temporal convolutional networks (TCN) with millions of parameters. Requiring these many parameters, however, could stem from a misalignment between the inductive biases of the TCN and cortical neuron's computations. In light of this, and to explore the computational implications of leaky memory units and nonlinear dendritic processing, we introduce the Expressive Leaky Memory (ELM) neuron model, a biologically inspired phenomenological model of a cortical neuron. Remarkably, by exploiting such slowly decaying memory-like hidden states and two-layered nonlinear integration of synaptic input, our ELM neuron can accurately match the aforementioned input-output relationship with under ten thousand trainable parameters. To further assess the computational ramifications of our neuron design, we evaluate it on various tasks with demanding temporal structures, including the Long Range Arena (LRA) datasets, as well as a novel neuromorphic dataset based on the Spiking Heidelberg Digits dataset (SHD-Adding). Leveraging a larger number of memory units with sufficiently long timescales, and correspondingly sophisticated synaptic integration, the ELM neuron displays substantial long-range processing capabilities, reliably outperforming the classic Transformer or Chrono-LSTM architectures on LRA, and even solving the Pathfinder-X task with over 70% accuracy (16k context length).

We explore the effects of various plasticity functions on assemblies of neurons. To bridge the gap between experimental and computational theories we make use of a conceptual framework, the Assembly Calculus, which is a formal system for the description of brain function based on assemblies of neurons. The Assembly Calculus includes operations for projecting, associating, and merging assemblies of neurons. Our research is focused on simulating different plasticity functions with Assembly Calculus. Our main contribution is the modification and evaluation of the projection operation. We experiment with Oja's and Spike Time-Dependent Plasticity (STDP) rules and test the effect of various hyper-parameters.

Mechanistic interpretability aims to understand the computational mechanisms underlying neural networks' capabilities in order to accomplish concrete scientific and engineering goals. Progress in this field thus promises to provide greater assurance over AI system behavior and shed light on exciting scientific questions about the nature of intelligence. Despite recent progress toward these goals, there are many open problems in the field that require solutions before many scientific and practical benefits can be realized: Our methods require both conceptual and practical improvements to reveal deeper insights; we must figure out how best to apply our methods in pursuit of specific goals; and the field must grapple with socio-technical challenges that influence and are influenced by our work. This forward-facing review discusses the current frontier of mechanistic interpretability and the open problems that the field may benefit from prioritizing.

Research in mechanistic interpretability seeks to explain behaviors of machine learning models in terms of their internal components. However, most previous work either focuses on simple behaviors in small models, or describes complicated behaviors in larger models with broad strokes. In this work, we bridge this gap by presenting an explanation for how GPT-2 small performs a natural language task called indirect object identification (IOI). Our explanation encompasses 26 attention heads grouped into 7 main classes, which we discovered using a combination of interpretability approaches relying on causal interventions. To our knowledge, this investigation is the largest end-to-end attempt at reverse-engineering a natural behavior "in the wild" in a language model. We evaluate the reliability of our explanation using three quantitative criteria--faithfulness, completeness and minimality. Though these criteria support our explanation, they also point to remaining gaps in our understanding. Our work provides evidence that a mechanistic understanding of large ML models is feasible, opening opportunities to scale our understanding to both larger models and more complex tasks.

Spiking Neural Networks (SNNs) as one of the biology-inspired models have received much attention recently. It can significantly reduce energy consumption since they quantize the real-valued membrane potentials to 0/1 spikes to transmit information thus the multiplications of activations and weights can be replaced by additions when implemented on hardware. However, this quantization mechanism will inevitably introduce quantization error, thus causing catastrophic information loss. To address the quantization error problem, we propose a regularizing membrane potential loss (RMP-Loss) to adjust the distribution which is directly related to quantization error to a range close to the spikes. Our method is extremely simple to implement and straightforward to train an SNN. Furthermore, it is shown to consistently outperform previous state-of-the-art methods over different network architectures and datasets.

Prior studies on the emergence in large models have primarily focused on how the functional capabilities of large language models (LLMs) scale with model size. Our research, however, transcends this traditional paradigm, aiming to deepen our understanding of the emergence within LLMs by placing a special emphasis not just on the model size but more significantly on the complex behavior of neuron interactions during the training process. By introducing the concepts of "self-organization" and "multifractal analysis," we explore how neuron interactions dynamically evolve during training, leading to "emergence," mirroring the phenomenon in natural systems where simple micro-level interactions give rise to complex macro-level behaviors. To quantitatively analyze the continuously evolving interactions among neurons in large models during training, we propose the Neuron-based Multifractal Analysis (NeuroMFA). Utilizing NeuroMFA, we conduct a comprehensive examination of the emergent behavior in LLMs through the lens of both model size and training process, paving new avenues for research into the emergence in large models.

Artificial neural networks (ANN) were inspired by the architecture and functions of the human brain and have revolutionised the field of artificial intelligence (AI). Inspired by studies on the latent geometry of the brain we posit that an increase in the research and application of hyperbolic geometry in machine learning will lead to increased accuracy, improved feature space representations and more efficient models across a range of tasks. We look at the structure and functions of the human brain, highlighting the alignment between the brain's hierarchical nature and hyperbolic geometry. By examining the brain's complex network of neuron connections and its cognitive processes, we illustrate how hyperbolic geometry plays a pivotal role in human intelligence. Empirical evidence indicates that hyperbolic neural networks outperform Euclidean models for tasks including natural language processing, computer vision and complex network analysis, requiring fewer parameters and exhibiting better generalisation. Despite its nascent adoption, hyperbolic geometry holds promise for improving machine learning models and advancing the field toward AGI.

We extend the capabilities of neural networks by coupling them to external memory resources, which they can interact with by attentional processes. The combined system is analogous to a Turing Machine or Von Neumann architecture but is differentiable end-to-end, allowing it to be efficiently trained with gradient descent. Preliminary results demonstrate that Neural Turing Machines can infer simple algorithms such as copying, sorting, and associative recall from input and output examples.

The generalization mystery in deep learning is the following: Why do over-parameterized neural networks trained with gradient descent (GD) generalize well on real datasets even though they are capable of fitting random datasets of comparable size? Furthermore, from among all solutions that fit the training data, how does GD find one that generalizes well (when such a well-generalizing solution exists)? We argue that the answer to both questions lies in the interaction of the gradients of different examples during training. Intuitively, if the per-example gradients are well-aligned, that is, if they are coherent, then one may expect GD to be (algorithmically) stable, and hence generalize well. We formalize this argument with an easy to compute and interpretable metric for coherence, and show that the metric takes on very different values on real and random datasets for several common vision networks. The theory also explains a number of other phenomena in deep learning, such as why some examples are reliably learned earlier than others, why early stopping works, and why it is possible to learn from noisy labels. Moreover, since the theory provides a causal explanation of how GD finds a well-generalizing solution when one exists, it motivates a class of simple modifications to GD that attenuate memorization and improve generalization. Generalization in deep learning is an extremely broad phenomenon, and therefore, it requires an equally general explanation. We conclude with a survey of alternative lines of attack on this problem, and argue that the proposed approach is the most viable one on this basis.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

Biological brains demonstrate complex neural activity, where the timing and interplay between neurons is critical to how brains process information. Most deep learning architectures simplify neural activity by abstracting away temporal dynamics. In this paper we challenge that paradigm. By incorporating neuron-level processing and synchronization, we can effectively reintroduce neural timing as a foundational element. We present the Continuous Thought Machine (CTM), a model designed to leverage neural dynamics as its core representation. The CTM has two core innovations: (1) neuron-level temporal processing, where each neuron uses unique weight parameters to process a history of incoming signals; and (2) neural synchronization employed as a latent representation. The CTM aims to strike a balance between oversimplified neuron abstractions that improve computational efficiency, and biological realism. It operates at a level of abstraction that effectively captures essential temporal dynamics while remaining computationally tractable for deep learning. We demonstrate the CTM's strong performance and versatility across a range of challenging tasks, including ImageNet-1K classification, solving 2D mazes, sorting, parity computation, question-answering, and RL tasks. Beyond displaying rich internal representations and offering a natural avenue for interpretation owing to its internal process, the CTM is able to perform tasks that require complex sequential reasoning. The CTM can also leverage adaptive compute, where it can stop earlier for simpler tasks, or keep computing when faced with more challenging instances. The goal of this work is to share the CTM and its associated innovations, rather than pushing for new state-of-the-art results. To that end, we believe the CTM represents a significant step toward developing more biologically plausible and powerful artificial intelligence systems.

The differential diagnosis of neurodegenerative dementias is a challenging clinical task, mainly because of the overlap in symptom presentation and the similarity of patterns observed in structural neuroimaging. To improve diagnostic efficiency and accuracy, deep learning-based methods such as Convolutional Neural Networks and Vision Transformers have been proposed for the automatic classification of brain MRIs. However, despite their strong predictive performance, these models find limited clinical utility due to their opaque decision making. In this work, we propose a framework that integrates two core components to enhance diagnostic transparency. First, we introduce a modular pipeline for converting 3D T1-weighted brain MRIs into textual radiology reports. Second, we explore the potential of modern Large Language Models (LLMs) to assist clinicians in the differential diagnosis between Frontotemporal dementia subtypes, Alzheimer's disease, and normal aging based on the generated reports. To bridge the gap between predictive accuracy and explainability, we employ reinforcement learning to incentivize diagnostic reasoning in LLMs. Without requiring supervised reasoning traces or distillation from larger models, our approach enables the emergence of structured diagnostic rationales grounded in neuroimaging findings. Unlike post-hoc explainability methods that retrospectively justify model decisions, our framework generates diagnostic rationales as part of the inference process-producing causally grounded explanations that inform and guide the model's decision-making process. In doing so, our framework matches the diagnostic performance of existing deep learning methods while offering rationales that support its diagnostic conclusions.

Deep Neural Networks (DNNs) have revolutionized various fields by enabling task automation and reducing human error. However, their internal workings and decision-making processes remain obscure due to their black box nature. Consequently, the lack of interpretability limits the application of these models in high-risk scenarios. To address this issue, the emerging field of eXplainable Artificial Intelligence (XAI) aims to explain and interpret the inner workings of DNNs. Despite advancements, XAI faces challenges such as the semantic gap between machine and human understanding, the trade-off between interpretability and performance, and the need for context-specific explanations. To overcome these limitations, we propose a novel multimodal framework named VALE Visual and Language Explanation. VALE integrates explainable AI techniques with advanced language models to provide comprehensive explanations. This framework utilizes visual explanations from XAI tools, an advanced zero-shot image segmentation model, and a visual language model to generate corresponding textual explanations. By combining visual and textual explanations, VALE bridges the semantic gap between machine outputs and human interpretation, delivering results that are more comprehensible to users. In this paper, we conduct a pilot study of the VALE framework for image classification tasks. Specifically, Shapley Additive Explanations (SHAP) are used to identify the most influential regions in classified images. The object of interest is then extracted using the Segment Anything Model (SAM), and explanations are generated using state-of-the-art pre-trained Vision-Language Models (VLMs). Extensive experimental studies are performed on two datasets: the ImageNet dataset and a custom underwater SONAR image dataset, demonstrating VALEs real-world applicability in underwater image classification.

This paper studies the problem of training a two-layer ReLU network for binary classification using gradient flow with small initialization. We consider a training dataset with well-separated input vectors: Any pair of input data with the same label are positively correlated, and any pair with different labels are negatively correlated. Our analysis shows that, during the early phase of training, neurons in the first layer try to align with either the positive data or the negative data, depending on its corresponding weight on the second layer. A careful analysis of the neurons' directional dynamics allows us to provide an O(log n{mu}) upper bound on the time it takes for all neurons to achieve good alignment with the input data, where n is the number of data points and mu measures how well the data are separated. After the early alignment phase, the loss converges to zero at a O(1{t}) rate, and the weight matrix on the first layer is approximately low-rank. Numerical experiments on the MNIST dataset illustrate our theoretical findings.

Adaptive behavior often requires predicting future events. The theory of reinforcement learning prescribes what kinds of predictive representations are useful and how to compute them. This paper integrates these theoretical ideas with work on cognition and neuroscience. We pay special attention to the successor representation (SR) and its generalizations, which have been widely applied both as engineering tools and models of brain function. This convergence suggests that particular kinds of predictive representations may function as versatile building blocks of intelligence.

A key goal in mechanistic interpretability is circuit analysis: finding sparse subgraphs of models corresponding to specific behaviors or capabilities. However, MLP sublayers make fine-grained circuit analysis on transformer-based language models difficult. In particular, interpretable features -- such as those found by sparse autoencoders (SAEs) -- are typically linear combinations of extremely many neurons, each with its own nonlinearity to account for. Circuit analysis in this setting thus either yields intractably large circuits or fails to disentangle local and global behavior. To address this we explore transcoders, which seek to faithfully approximate a densely activating MLP layer with a wider, sparsely-activating MLP layer. We successfully train transcoders on language models with 120M, 410M, and 1.4B parameters, and find them to perform at least on par with SAEs in terms of sparsity, faithfulness, and human-interpretability. We then introduce a novel method for using transcoders to perform weights-based circuit analysis through MLP sublayers. The resulting circuits neatly factorize into input-dependent and input-invariant terms. Finally, we apply transcoders to reverse-engineer unknown circuits in the model, and we obtain novel insights regarding the greater-than circuit in GPT2-small. Our results suggest that transcoders can prove effective in decomposing model computations involving MLPs into interpretable circuits. Code is available at https://github.com/jacobdunefsky/transcoder_circuits.

There has been significant debate in the NLP community about whether or not attention weights can be used as an explanation - a mechanism for interpreting how important each input token is for a particular prediction. The validity of "attention as explanation" has so far been evaluated by computing the rank correlation between attention-based explanations and existing feature attribution explanations using LSTM-based models. In our work, we (i) compare the rank correlation between five more recent feature attribution methods and two attention-based methods, on two types of NLP tasks, and (ii) extend this analysis to also include transformer-based models. We find that attention-based explanations do not correlate strongly with any recent feature attribution methods, regardless of the model or task. Furthermore, we find that none of the tested explanations correlate strongly with one another for the transformer-based model, leading us to question the underlying assumption that we should measure the validity of attention-based explanations based on how well they correlate with existing feature attribution explanation methods. After conducting experiments on five datasets using two different models, we argue that the community should stop using rank correlation as an evaluation metric for attention-based explanations. We suggest that researchers and practitioners should instead test various explanation methods and employ a human-in-the-loop process to determine if the explanations align with human intuition for the particular use case at hand.

In this work, we develop a technique to produce counterfactual visual explanations. Given a 'query' image I for which a vision system predicts class c, a counterfactual visual explanation identifies how I could change such that the system would output a different specified class c'. To do this, we select a 'distractor' image I' that the system predicts as class c' and identify spatial regions in I and I' such that replacing the identified region in I with the identified region in I' would push the system towards classifying I as c'. We apply our approach to multiple image classification datasets generating qualitative results showcasing the interpretability and discriminativeness of our counterfactual explanations. To explore the effectiveness of our explanations in teaching humans, we present machine teaching experiments for the task of fine-grained bird classification. We find that users trained to distinguish bird species fare better when given access to counterfactual explanations in addition to training examples.

Recent research developing neural network architectures with external memory have often used the benchmark bAbI question and answering dataset which provides a challenging number of tasks requiring reasoning. Here we employed a classic associative inference task from the memory-based reasoning neuroscience literature in order to more carefully probe the reasoning capacity of existing memory-augmented architectures. This task is thought to capture the essence of reasoning -- the appreciation of distant relationships among elements distributed across multiple facts or memories. Surprisingly, we found that current architectures struggle to reason over long distance associations. Similar results were obtained on a more complex task involving finding the shortest path between nodes in a path. We therefore developed MEMO, an architecture endowed with the capacity to reason over longer distances. This was accomplished with the addition of two novel components. First, it introduces a separation between memories (facts) stored in external memory and the items that comprise these facts in external memory. Second, it makes use of an adaptive retrieval mechanism, allowing a variable number of "memory hops" before the answer is produced. MEMO is capable of solving our novel reasoning tasks, as well as match state of the art results in bAbI.

There is renewed interest in modeling and understanding the nervous system of the nematode Caenorhabditis elegans (C. elegans), as this small model system provides a path to bridge the gap between nervous system structure (connectivity) and function (physiology). However, existing physiology datasets, whether involving passive recording or stimulation, are in distinct formats, and connectome datasets require preprocessing before analysis can commence. Here we compile and homogenize datasets of neural activity and connectivity. Our neural activity dataset is derived from 11 C. elegans neuroimaging experiments, while our connectivity dataset is compiled from 9 connectome annotations based on 3 primary electron microscopy studies and 1 signal propagation study. Physiology datasets, collected under varying protocols, measure calcium fluorescence in labeled subsets of the worm's 300 neurons. Our preprocessing pipeline standardizes these datasets by consistently ordering labeled neurons and resampling traces to a common sampling rate, yielding recordings from approximately 900 worms and 250 uniquely labeled neurons. The connectome datasets, collected from electron microscopy reconstructions, represent the entire nervous system as a graph of connections. Our collection is accessible on HuggingFace, facilitating analysis of the structure-function relationship in biology using modern neural network architectures and enabling cross-lab and cross-animal comparisons.

The Linear Representation Hypothesis (LRH) states that neural networks learn to encode concepts as directions in activation space, and a strong version of the LRH states that models learn only such encodings. In this paper, we present a counterexample to this strong LRH: when trained to repeat an input token sequence, gated recurrent neural networks (RNNs) learn to represent the token at each position with a particular order of magnitude, rather than a direction. These representations have layered features that are impossible to locate in distinct linear subspaces. To show this, we train interventions to predict and manipulate tokens by learning the scaling factor corresponding to each sequence position. These interventions indicate that the smallest RNNs find only this magnitude-based solution, while larger RNNs have linear representations. These findings strongly indicate that interpretability research should not be confined by the LRH.

In theoretical neuroscience, recent work leverages deep learning tools to explore how some network attributes critically influence its learning dynamics. Notably, initial weight distributions with small (resp. large) variance may yield a rich (resp. lazy) regime, where significant (resp. minor) changes to network states and representation are observed over the course of learning. However, in biology, neural circuit connectivity could exhibit a low-rank structure and therefore differs markedly from the random initializations generally used for these studies. As such, here we investigate how the structure of the initial weights -- in particular their effective rank -- influences the network learning regime. Through both empirical and theoretical analyses, we discover that high-rank initializations typically yield smaller network changes indicative of lazier learning, a finding we also confirm with experimentally-driven initial connectivity in recurrent neural networks. Conversely, low-rank initialization biases learning towards richer learning. Importantly, however, as an exception to this rule, we find lazier learning can still occur with a low-rank initialization that aligns with task and data statistics. Our research highlights the pivotal role of initial weight structures in shaping learning regimes, with implications for metabolic costs of plasticity and risks of catastrophic forgetting.

Our understanding of the generalization capabilities of neural networks (NNs) is still incomplete. Prevailing explanations are based on implicit biases of gradient descent (GD) but they cannot account for the capabilities of models from gradient-free methods nor the simplicity bias recently observed in untrained networks. This paper seeks other sources of generalization in NNs. Findings. To understand the inductive biases provided by architectures independently from GD, we examine untrained, random-weight networks. Even simple MLPs show strong inductive biases: uniform sampling in weight space yields a very biased distribution of functions in terms of complexity. But unlike common wisdom, NNs do not have an inherent "simplicity bias". This property depends on components such as ReLUs, residual connections, and layer normalizations. Alternative architectures can be built with a bias for any level of complexity. Transformers also inherit all these properties from their building blocks. Implications. We provide a fresh explanation for the success of deep learning independent from gradient-based training. It points at promising avenues for controlling the solutions implemented by trained models.

Despite a sea of interpretability methods that can produce plausible explanations, the field has also empirically seen many failure cases of such methods. In light of these results, it remains unclear for practitioners how to use these methods and choose between them in a principled way. In this paper, we show that for moderately rich model classes (easily satisfied by neural networks), any feature attribution method that is complete and linear -- for example, Integrated Gradients and SHAP -- can provably fail to improve on random guessing for inferring model behaviour. Our results apply to common end-tasks such as characterizing local model behaviour, identifying spurious features, and algorithmic recourse. One takeaway from our work is the importance of concretely defining end-tasks: once such an end-task is defined, a simple and direct approach of repeated model evaluations can outperform many other complex feature attribution methods.

Inspired by the Kolmogorov-Arnold representation theorem, we propose Kolmogorov-Arnold Networks (KANs) as promising alternatives to Multi-Layer Perceptrons (MLPs). While MLPs have fixed activation functions on nodes ("neurons"), KANs have learnable activation functions on edges ("weights"). KANs have no linear weights at all -- every weight parameter is replaced by a univariate function parametrized as a spline. We show that this seemingly simple change makes KANs outperform MLPs in terms of accuracy and interpretability. For accuracy, much smaller KANs can achieve comparable or better accuracy than much larger MLPs in data fitting and PDE solving. Theoretically and empirically, KANs possess faster neural scaling laws than MLPs. For interpretability, KANs can be intuitively visualized and can easily interact with human users. Through two examples in mathematics and physics, KANs are shown to be useful collaborators helping scientists (re)discover mathematical and physical laws. In summary, KANs are promising alternatives for MLPs, opening opportunities for further improving today's deep learning models which rely heavily on MLPs.

We propose CX-ToM, short for counterfactual explanations with theory-of mind, a new explainable AI (XAI) framework for explaining decisions made by a deep convolutional neural network (CNN). In contrast to the current methods in XAI that generate explanations as a single shot response, we pose explanation as an iterative communication process, i.e. dialog, between the machine and human user. More concretely, our CX-ToM framework generates sequence of explanations in a dialog by mediating the differences between the minds of machine and human user. To do this, we use Theory of Mind (ToM) which helps us in explicitly modeling human's intention, machine's mind as inferred by the human as well as human's mind as inferred by the machine. Moreover, most state-of-the-art XAI frameworks provide attention (or heat map) based explanations. In our work, we show that these attention based explanations are not sufficient for increasing human trust in the underlying CNN model. In CX-ToM, we instead use counterfactual explanations called fault-lines which we define as follows: given an input image I for which a CNN classification model M predicts class c_pred, a fault-line identifies the minimal semantic-level features (e.g., stripes on zebra, pointed ears of dog), referred to as explainable concepts, that need to be added to or deleted from I in order to alter the classification category of I by M to another specified class c_alt. We argue that, due to the iterative, conceptual and counterfactual nature of CX-ToM explanations, our framework is practical and more natural for both expert and non-expert users to understand the internal workings of complex deep learning models. Extensive quantitative and qualitative experiments verify our hypotheses, demonstrating that our CX-ToM significantly outperforms the state-of-the-art explainable AI models.

Redundancies and correlations in the responses of sensory neurons seem to waste neural resources but can carry cues about structured stimuli and may help the brain to correct for response errors. To assess how the retina negotiates this tradeoff, we measured simultaneous responses from populations of ganglion cells presented with natural and artificial stimuli that varied greatly in correlation structure. We found that pairwise correlations in the retinal output remained similar across stimuli with widely different spatio-temporal correlations including white noise and natural movies. Meanwhile, purely spatial correlations tended to increase correlations in the retinal response. Responding to more correlated stimuli, ganglion cells had faster temporal kernels and tended to have stronger surrounds. These properties of individual cells, along with gain changes that opposed changes in effective contrast at the ganglion cell input, largely explained the similarity of pairwise correlations across stimuli where receptive field measurements were possible.

Backpropagation (BP) is the most successful and widely used algorithm in deep learning. However, the computations required by BP are challenging to reconcile with known neurobiology. This difficulty has stimulated interest in more biologically plausible alternatives to BP. One such algorithm is the inference learning algorithm (IL). IL has close connections to neurobiological models of cortical function and has achieved equal performance to BP on supervised learning and auto-associative tasks. In contrast to BP, however, the mathematical foundations of IL are not well-understood. Here, we develop a novel theoretical framework for IL. Our main result is that IL closely approximates an optimization method known as implicit stochastic gradient descent (implicit SGD), which is distinct from the explicit SGD implemented by BP. Our results further show how the standard implementation of IL can be altered to better approximate implicit SGD. Our novel implementation considerably improves the stability of IL across learning rates, which is consistent with our theory, as a key property of implicit SGD is its stability. We provide extensive simulation results that further support our theoretical interpretations and also demonstrate IL achieves quicker convergence when trained with small mini-batches while matching the performance of BP for large mini-batches.

Circuit analysis is a promising technique for understanding the internal mechanisms of language models. However, existing analyses are done in small models far from the state of the art. To address this, we present a case study of circuit analysis in the 70B Chinchilla model, aiming to test the scalability of circuit analysis. In particular, we study multiple-choice question answering, and investigate Chinchilla's capability to identify the correct answer label given knowledge of the correct answer text. We find that the existing techniques of logit attribution, attention pattern visualization, and activation patching naturally scale to Chinchilla, allowing us to identify and categorize a small set of `output nodes' (attention heads and MLPs). We further study the `correct letter' category of attention heads aiming to understand the semantics of their features, with mixed results. For normal multiple-choice question answers, we significantly compress the query, key and value subspaces of the head without loss of performance when operating on the answer labels for multiple-choice questions, and we show that the query and key subspaces represent an `Nth item in an enumeration' feature to at least some extent. However, when we attempt to use this explanation to understand the heads' behaviour on a more general distribution including randomized answer labels, we find that it is only a partial explanation, suggesting there is more to learn about the operation of `correct letter' heads on multiple choice question answering.

A growing literature in computational neuroscience leverages gradient descent and learning algorithms that approximate it to study synaptic plasticity in the brain. However, the vast majority of this work ignores a critical underlying assumption: the choice of distance for synaptic changes - i.e. the geometry of synaptic plasticity. Gradient descent assumes that the distance is Euclidean, but many other distances are possible, and there is no reason that biology necessarily uses Euclidean geometry. Here, using the theoretical tools provided by mirror descent, we show that the distribution of synaptic weights will depend on the geometry of synaptic plasticity. We use these results to show that experimentally-observed log-normal weight distributions found in several brain areas are not consistent with standard gradient descent (i.e. a Euclidean geometry), but rather with non-Euclidean distances. Finally, we show that it should be possible to experimentally test for different synaptic geometries by comparing synaptic weight distributions before and after learning. Overall, our work shows that the current paradigm in theoretical work on synaptic plasticity that assumes Euclidean synaptic geometry may be misguided and that it should be possible to experimentally determine the true geometry of synaptic plasticity in the brain.

We put forward the dynamical study of a novel higher-order small network of Chialvo neurons arranged in a ring-star topology, with the neurons interacting via linear diffusive couplings. This model is perceived to imitate the nonlinear dynamical properties exhibited by a realistic nervous system where the neurons transfer information through higher-order multi-body interactions. We first analyze our model using the tools from nonlinear dynamics literature: fixed point analysis, Jacobian matrix, and bifurcation patterns. We observe the coexistence of chaotic attractors, and also an intriguing route to chaos starting from a fixed point, to period-doubling, to cyclic quasiperiodic closed invariant curves, to ultimately chaos. We numerically observe the existence of codimension-1 bifurcation patterns: saddle-node, period-doubling, and Neimark Sacker. We also qualitatively study the typical phase portraits of the system and numerically quantify chaos and complexity using the 0-1 test and sample entropy measure respectively. Finally, we study the collective behavior of the neurons in terms of two synchronization measures: the cross-correlation coefficient, and the Kuramoto order parameter.

In large language models (LLMs), certain neurons can store distinct pieces of knowledge learned during pretraining. While knowledge typically appears as a combination of relations and entities, it remains unclear whether some neurons focus on a relation itself -- independent of any entity. We hypothesize such neurons detect a relation in the input text and guide generation involving such a relation. To investigate this, we study the Llama-2 family on a chosen set of relations with a statistics-based method. Our experiments demonstrate the existence of relation-specific neurons. We measure the effect of selectively deactivating candidate neurons specific to relation r on the LLM's ability to handle (1) facts whose relation is r and (2) facts whose relation is a different relation r' neq r. With respect to their capacity for encoding relation information, we give evidence for the following three properties of relation-specific neurons. (i) Neuron cumulativity. The neurons for r present a cumulative effect so that deactivating a larger portion of them results in the degradation of more facts in r. (ii) Neuron versatility. Neurons can be shared across multiple closely related as well as less related relations. Some relation neurons transfer across languages. (iii) Neuron interference. Deactivating neurons specific to one relation can improve LLM generation performance for facts of other relations. We will make our code publicly available at https://github.com/cisnlp/relation-specific-neurons.

The field of explainable artificial intelligence (XAI) aims to uncover the inner workings of complex deep neural models. While being crucial for safety-critical domains, XAI inherently lacks ground-truth explanations, making its automatic evaluation an unsolved problem. We address this challenge by proposing a novel synthetic vision dataset, named FunnyBirds, and accompanying automatic evaluation protocols. Our dataset allows performing semantically meaningful image interventions, e.g., removing individual object parts, which has three important implications. First, it enables analyzing explanations on a part level, which is closer to human comprehension than existing methods that evaluate on a pixel level. Second, by comparing the model output for inputs with removed parts, we can estimate ground-truth part importances that should be reflected in the explanations. Third, by mapping individual explanations into a common space of part importances, we can analyze a variety of different explanation types in a single common framework. Using our tools, we report results for 24 different combinations of neural models and XAI methods, demonstrating the strengths and weaknesses of the assessed methods in a fully automatic and systematic manner.

Explainability has become an important topic in computer science and artificial intelligence, leading to a subfield called Explainable Artificial Intelligence (XAI). The goal of providing or seeking explanations is to achieve (better) 'understanding' on the part of the explainee. However, what it means to 'understand' is still not clearly defined, and the concept itself is rarely the subject of scientific investigation. This conceptual article aims to present a model of forms of understanding for XAI-explanations and beyond. From an interdisciplinary perspective bringing together computer science, linguistics, sociology, philosophy and psychology, a definition of understanding and its forms, assessment, and dynamics during the process of giving everyday explanations are explored. Two types of understanding are considered as possible outcomes of explanations, namely enabledness, 'knowing how' to do or decide something, and comprehension, 'knowing that' -- both in different degrees (from shallow to deep). Explanations regularly start with shallow understanding in a specific domain and can lead to deep comprehension and enabledness of the explanandum, which we see as a prerequisite for human users to gain agency. In this process, the increase of comprehension and enabledness are highly interdependent. Against the background of this systematization, special challenges of understanding in XAI are discussed.

Recent advances in Large Multimodal Models (LMMs) lead to significant breakthroughs in both academia and industry. One question that arises is how we, as humans, can understand their internal neural representations. This paper takes an initial step towards addressing this question by presenting a versatile framework to identify and interpret the semantics within LMMs. Specifically, 1) we first apply a Sparse Autoencoder(SAE) to disentangle the representations into human understandable features. 2) We then present an automatic interpretation framework to interpreted the open-semantic features learned in SAE by the LMMs themselves. We employ this framework to analyze the LLaVA-NeXT-8B model using the LLaVA-OV-72B model, demonstrating that these features can effectively steer the model's behavior. Our results contribute to a deeper understanding of why LMMs excel in specific tasks, including EQ tests, and illuminate the nature of their mistakes along with potential strategies for their rectification. These findings offer new insights into the internal mechanisms of LMMs and suggest parallels with the cognitive processes of the human brain.

AI led chess systems to a superhuman level, yet these systems heavily rely on black-box algorithms. This is unsustainable in ensuring transparency to the end-user, particularly when these systems are responsible for sensitive decision-making. Recent interpretability work has shown that the inner representations of Deep Neural Networks (DNNs) were fathomable and contained human-understandable concepts. Yet, these methods are seldom contextualised and are often based on a single hidden state, which makes them unable to interpret multi-step reasoning, e.g. planning. In this respect, we propose contrastive sparse autoencoders (CSAE), a novel framework for studying pairs of game trajectories. Using CSAE, we are able to extract and interpret concepts that are meaningful to the chess-agent plans. We primarily focused on a qualitative analysis of the CSAE features before proposing an automated feature taxonomy. Furthermore, to evaluate the quality of our trained CSAE, we devise sanity checks to wave spurious correlations in our results.

Several machine learning models, including neural networks, consistently misclassify adversarial examples---inputs formed by applying small but intentionally worst-case perturbations to examples from the dataset, such that the perturbed input results in the model outputting an incorrect answer with high confidence. Early attempts at explaining this phenomenon focused on nonlinearity and overfitting. We argue instead that the primary cause of neural networks' vulnerability to adversarial perturbation is their linear nature. This explanation is supported by new quantitative results while giving the first explanation of the most intriguing fact about them: their generalization across architectures and training sets. Moreover, this view yields a simple and fast method of generating adversarial examples. Using this approach to provide examples for adversarial training, we reduce the test set error of a maxout network on the MNIST dataset.

Graph Neural Networks (GNNs) have emerged as powerful tools to encode graph-structured data. Due to their broad applications, there is an increasing need to develop tools to explain how GNNs make decisions given graph-structured data. Existing learning-based GNN explanation approaches are task-specific in training and hence suffer from crucial drawbacks. Specifically, they are incapable of producing explanations for a multitask prediction model with a single explainer. They are also unable to provide explanations in cases where the GNN is trained in a self-supervised manner, and the resulting representations are used in future downstream tasks. To address these limitations, we propose a Task-Agnostic GNN Explainer (TAGE) that is independent of downstream models and trained under self-supervision with no knowledge of downstream tasks. TAGE enables the explanation of GNN embedding models with unseen downstream tasks and allows efficient explanation of multitask models. Our extensive experiments show that TAGE can significantly speed up the explanation efficiency by using the same model to explain predictions for multiple downstream tasks while achieving explanation quality as good as or even better than current state-of-the-art GNN explanation approaches. Our code is pubicly available as part of the DIG library at https://github.com/divelab/DIG/tree/main/dig/xgraph/TAGE/.

Despite its widespread use in neural networks, error backpropagation has faced criticism for its lack of biological plausibility, suffering from issues such as the backward locking problem and the weight transport problem. These limitations have motivated researchers to explore more biologically plausible learning algorithms that could potentially shed light on how biological neural systems adapt and learn. Inspired by the counter-current exchange mechanisms observed in biological systems, we propose counter-current learning (CCL), a biologically plausible framework for credit assignment in neural networks. This framework employs a feedforward network to process input data and a feedback network to process targets, with each network enhancing the other through anti-parallel signal propagation. By leveraging the more informative signals from the bottom layer of the feedback network to guide the updates of the top layer of the feedforward network and vice versa, CCL enables the simultaneous transformation of source inputs to target outputs and the dynamic mutual influence of these transformations. Experimental results on MNIST, FashionMNIST, CIFAR10, and CIFAR100 datasets using multi-layer perceptrons and convolutional neural networks demonstrate that CCL achieves comparable performance to other biologically plausible algorithms while offering a more biologically realistic learning mechanism. Furthermore, we showcase the applicability of our approach to an autoencoder task, underscoring its potential for unsupervised representation learning. Our work presents a direction for biologically inspired and plausible learning algorithms, offering an alternative mechanism of learning and adaptation in neural networks.

Recent success in training deep neural networks have prompted active investigation into the features learned on their intermediate layers. Such research is difficult because it requires making sense of non-linear computations performed by millions of parameters, but valuable because it increases our ability to understand current models and create improved versions of them. In this paper we investigate the extent to which neural networks exhibit what we call convergent learning, which is when the representations learned by multiple nets converge to a set of features which are either individually similar between networks or where subsets of features span similar low-dimensional spaces. We propose a specific method of probing representations: training multiple networks and then comparing and contrasting their individual, learned representations at the level of neurons or groups of neurons. We begin research into this question using three techniques to approximately align different neural networks on a feature level: a bipartite matching approach that makes one-to-one assignments between neurons, a sparse prediction approach that finds one-to-many mappings, and a spectral clustering approach that finds many-to-many mappings. This initial investigation reveals a few previously unknown properties of neural networks, and we argue that future research into the question of convergent learning will yield many more. The insights described here include (1) that some features are learned reliably in multiple networks, yet other features are not consistently learned; (2) that units learn to span low-dimensional subspaces and, while these subspaces are common to multiple networks, the specific basis vectors learned are not; (3) that the representation codes show evidence of being a mix between a local code and slightly, but not fully, distributed codes across multiple units.

Activity difference based learning algorithms-such as contrastive Hebbian learning and equilibrium propagation-have been proposed as biologically plausible alternatives to error back-propagation. However, on traditional digital chips these algorithms suffer from having to solve a costly inference problem twice, making these approaches more than two orders of magnitude slower than back-propagation. In the analog realm equilibrium propagation may be promising for fast and energy efficient learning, but states still need to be inferred and stored twice. Inspired by lifted neural networks and compartmental neuron models we propose a simple energy based compartmental neuron model, termed dual propagation, in which each neuron is a dyad with two intrinsic states. At inference time these intrinsic states encode the error/activity duality through their difference and their mean respectively. The advantage of this method is that only a single inference phase is needed and that inference can be solved in layerwise closed-form. Experimentally we show on common computer vision datasets, including Imagenet32x32, that dual propagation performs equivalently to back-propagation both in terms of accuracy and runtime.

Mapping behavioral actions to neural activity is a fundamental goal of neuroscience. As our ability to record large neural and behavioral data increases, there is growing interest in modeling neural dynamics during adaptive behaviors to probe neural representations. In particular, neural latent embeddings can reveal underlying correlates of behavior, yet, we lack non-linear techniques that can explicitly and flexibly leverage joint behavior and neural data. Here, we fill this gap with a novel method, CEBRA, that jointly uses behavioral and neural data in a hypothesis- or discovery-driven manner to produce consistent, high-performance latent spaces. We validate its accuracy and demonstrate our tool's utility for both calcium and electrophysiology datasets, across sensory and motor tasks, and in simple or complex behaviors across species. It allows for single and multi-session datasets to be leveraged for hypothesis testing or can be used label-free. Lastly, we show that CEBRA can be used for the mapping of space, uncovering complex kinematic features, and rapid, high-accuracy decoding of natural movies from visual cortex.

We aim to understand grokking, a phenomenon where models generalize long after overfitting their training set. We present both a microscopic analysis anchored by an effective theory and a macroscopic analysis of phase diagrams describing learning performance across hyperparameters. We find that generalization originates from structured representations whose training dynamics and dependence on training set size can be predicted by our effective theory in a toy setting. We observe empirically the presence of four learning phases: comprehension, grokking, memorization, and confusion. We find representation learning to occur only in a "Goldilocks zone" (including comprehension and grokking) between memorization and confusion. We find on transformers the grokking phase stays closer to the memorization phase (compared to the comprehension phase), leading to delayed generalization. The Goldilocks phase is reminiscent of "intelligence from starvation" in Darwinian evolution, where resource limitations drive discovery of more efficient solutions. This study not only provides intuitive explanations of the origin of grokking, but also highlights the usefulness of physics-inspired tools, e.g., effective theories and phase diagrams, for understanding deep learning.

Learning to remember over long timescales is fundamentally challenging for recurrent neural networks (RNNs). While much prior work has explored why RNNs struggle to learn long timescales and how to mitigate this, we still lack a clear understanding of the dynamics involved when RNNs learn long timescales via gradient descent. Here we build a mathematical theory of the learning dynamics of linear RNNs trained to integrate white noise. We show that when the initial recurrent weights are small, the dynamics of learning are described by a low-dimensional system that tracks a single outlier eigenvalue of the recurrent weights. This reveals the precise manner in which the long timescale associated with white noise integration is learned. We extend our analyses to RNNs learning a damped oscillatory filter, and find rich dynamical equations for the evolution of a conjugate pair of outlier eigenvalues. Taken together, our analyses build a rich mathematical framework for studying dynamical learning problems salient for both machine learning and neuroscience.

In this paper, we study two challenging problems in explainable AI (XAI) and data clustering. The first is how to directly design a neural network with inherent interpretability, rather than giving post-hoc explanations of a black-box model. The second is implementing discrete k-means with a differentiable neural network that embraces the advantages of parallel computing, online clustering, and clustering-favorable representation learning. To address these two challenges, we design a novel neural network, which is a differentiable reformulation of the vanilla k-means, called inTerpretable nEuraL cLustering (TELL). Our contributions are threefold. First, to the best of our knowledge, most existing XAI works focus on supervised learning paradigms. This work is one of the few XAI studies on unsupervised learning, in particular, data clustering. Second, TELL is an interpretable, or the so-called intrinsically explainable and transparent model. In contrast, most existing XAI studies resort to various means for understanding a black-box model with post-hoc explanations. Third, from the view of data clustering, TELL possesses many properties highly desired by k-means, including but not limited to online clustering, plug-and-play module, parallel computing, and provable convergence. Extensive experiments show that our method achieves superior performance comparing with 14 clustering approaches on three challenging data sets. The source code could be accessed at www.pengxi.me.

Recent work in mechanistic interpretability has shown that behaviors in language models can be successfully reverse-engineered through circuit analysis. A common criticism, however, is that each circuit is task-specific, and thus such analysis cannot contribute to understanding the models at a higher level. In this work, we present evidence that insights (both low-level findings about specific heads and higher-level findings about general algorithms) can indeed generalize across tasks. Specifically, we study the circuit discovered in Wang et al. (2022) for the Indirect Object Identification (IOI) task and 1.) show that it reproduces on a larger GPT2 model, and 2.) that it is mostly reused to solve a seemingly different task: Colored Objects (Ippolito & Callison-Burch, 2023). We provide evidence that the process underlying both tasks is functionally very similar, and contains about a 78% overlap in in-circuit attention heads. We further present a proof-of-concept intervention experiment, in which we adjust four attention heads in middle layers in order to 'repair' the Colored Objects circuit and make it behave like the IOI circuit. In doing so, we boost accuracy from 49.6% to 93.7% on the Colored Objects task and explain most sources of error. The intervention affects downstream attention heads in specific ways predicted by their interactions in the IOI circuit, indicating that this subcircuit behavior is invariant to the different task inputs. Overall, our results provide evidence that it may yet be possible to explain large language models' behavior in terms of a relatively small number of interpretable task-general algorithmic building blocks and computational components.

Recently, a race towards the simplification of deep networks has begun, showing that it is effectively possible to reduce the size of these models with minimal or no performance loss. However, there is a general lack in understanding why these pruning strategies are effective. In this work, we are going to compare and analyze pruned solutions with two different pruning approaches, one-shot and gradual, showing the higher effectiveness of the latter. In particular, we find that gradual pruning allows access to narrow, well-generalizing minima, which are typically ignored when using one-shot approaches. In this work we also propose PSP-entropy, a measure to understand how a given neuron correlates to some specific learned classes. Interestingly, we observe that the features extracted by iteratively-pruned models are less correlated to specific classes, potentially making these models a better fit in transfer learning approaches.

One of the most surprising puzzles in neural network generalisation is grokking: a network with perfect training accuracy but poor generalisation will, upon further training, transition to perfect generalisation. We propose that grokking occurs when the task admits a generalising solution and a memorising solution, where the generalising solution is slower to learn but more efficient, producing larger logits with the same parameter norm. We hypothesise that memorising circuits become more inefficient with larger training datasets while generalising circuits do not, suggesting there is a critical dataset size at which memorisation and generalisation are equally efficient. We make and confirm four novel predictions about grokking, providing significant evidence in favour of our explanation. Most strikingly, we demonstrate two novel and surprising behaviours: ungrokking, in which a network regresses from perfect to low test accuracy, and semi-grokking, in which a network shows delayed generalisation to partial rather than perfect test accuracy.

Neural networks surround us, in the form of large language models, speech transcription systems, molecular discovery algorithms, robotics, and much more. Stripped of anything else, neural networks are compositions of differentiable primitives, and studying them means learning how to program and how to interact with these models, a particular example of what is called differentiable programming. This primer is an introduction to this fascinating field imagined for someone, like Alice, who has just ventured into this strange differentiable wonderland. I overview the basics of optimizing a function via automatic differentiation, and a selection of the most common designs for handling sequences, graphs, texts, and audios. The focus is on a intuitive, self-contained introduction to the most important design techniques, including convolutional, attentional, and recurrent blocks, hoping to bridge the gap between theory and code (PyTorch and JAX) and leaving the reader capable of understanding some of the most advanced models out there, such as large language models (LLMs) and multimodal architectures.