Daily Papers

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.

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.

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.

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.

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.

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.

This study introduces a novel approach by replacing the traditional perceptron neuron model with a biologically inspired probabilistic meta neuron, where the internal neuron parameters are jointly learned, leading to improved classification accuracy of spiking neural networks (SNNs). To validate this innovation, we implement and compare two SNN architectures: one based on standard leaky integrate-and-fire (LIF) neurons and another utilizing the proposed probabilistic meta neuron model. As a second key contribution, we present a new biologically inspired classification framework that uniquely integrates SNNs with Lempel-Ziv complexity (LZC) a measure closely related to entropy rate. By combining the temporal precision and biological plausibility of SNNs with the capacity of LZC to capture structural regularity, the proposed approach enables efficient and interpretable classification of spatiotemporal neural data, an aspect not addressed in existing works. We consider learning algorithms such as backpropagation, spike-timing-dependent plasticity (STDP), and the Tempotron learning rule. To explore neural dynamics, we use Poisson processes to model neuronal spike trains, a well-established method for simulating the stochastic firing behavior of biological neurons. Our results reveal that depending on the training method, the classifier's efficiency can improve by up to 11.00%, highlighting the advantage of learning additional neuron parameters beyond the traditional focus on weighted inputs alone.

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).

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.

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 Wilson-Cowan model for metapopulation, a Neural Mass Network Model, treats different subcortical regions of the brain as connected nodes, with connections representing various types of structural, functional, or effective neuronal connectivity between these regions. Each region comprises interacting populations of excitatory and inhibitory cells, consistent with the standard Wilson-Cowan model. By incorporating stable attractors into such a metapopulation model's dynamics, we transform it into a learning algorithm capable of achieving high image and text classification accuracy. We test it on MNIST and Fashion MNIST, in combination with convolutional neural networks, on CIFAR-10 and TF-FLOWERS, and, in combination with a transformer architecture (BERT), on IMDB, always showing high classification accuracy. These numerical evaluations illustrate that minimal modifications to the Wilson-Cowan model for metapopulation can reveal unique and previously unobserved dynamics.

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 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.

The brain understands the external world through an internal model that generates predictions and refines them based on prediction errors. A complete prediction specifies what will happen, when it will happen, and with what probability, which we refer to as a "prediction object". Existing models typically capture only what and when, omit probabilities, and rely on biologically-implausible algorithms. Here we show that a single population of spiking neurons can jointly encode the prediction object through a biologically grounded learning mechanism. We implement a heterogeneous Izhikevich spiking reservoir with readouts trained by an error-modulated, attention-gated three-factor Hebbian rule and test it on a novel paradigm that controls both the timing and probability of upcoming stimuli. By integrating real-time learning of "when" with offline consolidation of "what", the model encodes the complete prediction object, firing at the correct times with magnitudes proportional to the probabilities. Critically, it rapidly adapts to changes in both stimulus timing and probability, an ability that global least-squares methods such as FORCE lack without explicit resets. During learning, the model self-organizes its readout weights into near-orthogonal subspaces for "what" and "when," showing that multiplexed encoding arises naturally from generic recurrent dynamics under local, error-gated modulation. These results challenge the view that "what" and "when" predictions require separate modules, suggesting instead that mixed selectivity within shared populations supports flexible predictive cognition. The model also predicts phase-specific neuromodulation and overlapping neural subspaces, offering a parsimonious alternative to hierarchical predictive-coding accounts.

We introduce NeuroSA, a neuromorphic architecture specifically designed to ensure asymptotic convergence to the ground state of an Ising problem using an annealing process that is governed by the physics of quantum mechanical tunneling using Fowler-Nordheim (FN). The core component of NeuroSA consists of a pair of asynchronous ON-OFF neurons, which effectively map classical simulated annealing (SA) dynamics onto a network of integrate-and-fire (IF) neurons. The threshold of each ON-OFF neuron pair is adaptively adjusted by an FN annealer which replicates the optimal escape mechanism and convergence of SA, particularly at low temperatures. To validate the effectiveness of our neuromorphic Ising machine, we systematically solved various benchmark MAX-CUT combinatorial optimization problems. Across multiple runs, NeuroSA consistently generates solutions that approach the state-of-the-art level with high accuracy (greater than 99%), and without any graph-specific hyperparameter tuning. For practical illustration, we present results from an implementation of NeuroSA on the SpiNNaker2 platform, highlighting the feasibility of mapping our proposed architecture onto a standard neuromorphic accelerator platform.

The liquid state machine (LSM) combines low training complexity and biological plausibility, which has made it an attractive machine learning framework for edge and neuromorphic computing paradigms. Originally proposed as a model of brain computation, the LSM tunes its internal weights without backpropagation of gradients, which results in lower performance compared to multi-layer neural networks. Recent findings in neuroscience suggest that astrocytes, a long-neglected non-neuronal brain cell, modulate synaptic plasticity and brain dynamics, tuning brain networks to the vicinity of the computationally optimal critical phase transition between order and chaos. Inspired by this disruptive understanding of how brain networks self-tune, we propose the neuron-astrocyte liquid state machine (NALSM) that addresses under-performance through self-organized near-critical dynamics. Similar to its biological counterpart, the astrocyte model integrates neuronal activity and provides global feedback to spike-timing-dependent plasticity (STDP), which self-organizes NALSM dynamics around a critical branching factor that is associated with the edge-of-chaos. We demonstrate that NALSM achieves state-of-the-art accuracy versus comparable LSM methods, without the need for data-specific hand-tuning. With a top accuracy of 97.61% on MNIST, 97.51% on N-MNIST, and 85.84% on Fashion-MNIST, NALSM achieved comparable performance to current fully-connected multi-layer spiking neural networks trained via backpropagation. Our findings suggest that the further development of brain-inspired machine learning methods has the potential to reach the performance of deep learning, with the added benefits of supporting robust and energy-efficient neuromorphic computing on the edge.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Understanding the inner workings of neural networks is essential for enhancing model performance and interpretability. Current research predominantly focuses on examining the connection between individual neurons and the model's final predictions. Which suffers from challenges in interpreting the internal workings of the model, particularly when neurons encode multiple unrelated features. In this paper, we propose a novel framework that transitions the focus from analyzing individual neurons to investigating groups of neurons, shifting the emphasis from neuron-output relationships to functional interaction between neurons. Our automated framework, NeurFlow, first identifies core neurons and clusters them into groups based on shared functional relationships, enabling a more coherent and interpretable view of the network's internal processes. This approach facilitates the construction of a hierarchical circuit representing neuron interactions across layers, thus improving interpretability while reducing computational costs. Our extensive empirical studies validate the fidelity of our proposed NeurFlow. Additionally, we showcase its utility in practical applications such as image debugging and automatic concept labeling, thereby highlighting its potential to advance the field of neural network explainability.

The precise timing of spikes emitted by neurons plays a crucial role in shaping the response of efferent biological neurons. This temporal dimension of neural activity holds significant importance in understanding information processing in neurobiology, especially for the performance of neuromorphic hardware, such as event-based cameras. Nonetheless, many artificial neural models disregard this critical temporal dimension of neural activity. In this study, we present a model designed to efficiently detect temporal spiking motifs using a layer of spiking neurons equipped with heterogeneous synaptic delays. Our model capitalizes on the diverse synaptic delays present on the dendritic tree, enabling specific arrangements of temporally precise synaptic inputs to synchronize upon reaching the basal dendritic tree. We formalize this process as a time-invariant logistic regression, which can be trained using labeled data. To demonstrate its practical efficacy, we apply the model to naturalistic videos transformed into event streams, simulating the output of the biological retina or event-based cameras. To evaluate the robustness of the model in detecting visual motion, we conduct experiments by selectively pruning weights and demonstrate that the model remains efficient even under significantly reduced workloads. In conclusion, by providing a comprehensive, event-driven computational building block, the incorporation of heterogeneous delays has the potential to greatly improve the performance of future spiking neural network algorithms, particularly in the context of neuromorphic chips.

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.

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).

The complete connectome of the Drosophila larva brain offers a unique opportunity to investigate whether biologically evolved circuits can support artificial intelligence. We convert this wiring diagram into a Biological Processing Unit (BPU), a fixed recurrent network derived directly from synaptic connectivity. Despite its modest size 3,000 neurons and 65,000 weights between them), the unmodified BPU achieves 98% accuracy on MNIST and 58% on CIFAR-10, surpassing size-matched MLPs. Scaling the BPU via structured connectome expansions further improves CIFAR-10 performance, while modality-specific ablations reveal the uneven contributions of different sensory subsystems. On the ChessBench dataset, a lightweight GNN-BPU model trained on only 10,000 games achieves 60% move accuracy, nearly 10x better than any size transformer. Moreover, CNN-BPU models with ~2M parameters outperform parameter-matched Transformers, and with a depth-6 minimax search at inference, reach 91.7% accuracy, exceeding even a 9M-parameter Transformer baseline. These results demonstrate the potential of biofidelic neural architectures to support complex cognitive tasks and motivate scaling to larger and more intelligent connectomes in future work.

The brain has computational capabilities that surpass those of modern systems, being able to solve complex problems efficiently in a simple way. Neuromorphic engineering aims to mimic biology in order to develop new systems capable of incorporating such capabilities. Bio-inspired learning systems continue to be a challenge that must be solved, and much work needs to be done in this regard. Among all brain regions, the hippocampus stands out as an autoassociative short-term memory with the capacity to learn and recall memories from any fragment of them. These characteristics make the hippocampus an ideal candidate for developing bio-inspired learning systems that, in addition, resemble content-addressable memories. Therefore, in this work we propose a bio-inspired spiking content-addressable memory model based on the CA3 region of the hippocampus with the ability to learn, forget and recall memories, both orthogonal and non-orthogonal, from any fragment of them. The model was implemented on the SpiNNaker hardware platform using Spiking Neural Networks. A set of experiments based on functional, stress and applicability tests were performed to demonstrate its correct functioning. This work presents the first hardware implementation of a fully-functional bio-inspired spiking hippocampal content-addressable memory model, paving the way for the development of future more complex neuromorphic systems.

Modern neural recording techniques allow neuroscientists to obtain spiking activity of multiple neurons from different brain regions over long time periods, which requires new statistical methods to be developed for understanding structure of the large-scale data. In this paper, we develop a bi-clustering method to cluster the neural spiking activity spatially and temporally, according to their low-dimensional latent structures. The spatial (neuron) clusters are defined by the latent trajectories within each neural population, while the temporal (state) clusters are defined by (populationally) synchronous local linear dynamics shared with different periods. To flexibly extract the bi-clustering structure, we build the model non-parametrically, and develop an efficient Markov chain Monte Carlo (MCMC) algorithm to sample the posterior distributions of model parameters. Validating our proposed MCMC algorithm through simulations, we find the method can recover unknown parameters and true bi-clustering structures successfully. We then apply the proposed bi-clustering method to multi-regional neural recordings under different experiment settings, where we find that simultaneously considering latent trajectories and spatial-temporal clustering structures can provide us with a more accurate and interpretable result. Overall, the proposed method provides scientific insights for large-scale (counting) time series with elongated recording periods, and it can potentially have application beyond neuroscience.

Recent advances in artificial neural networks for machine learning, and language modeling in particular, have established a family of recurrent neural network (RNN) architectures that, unlike conventional RNNs with vector-form hidden states, use two-dimensional (2D) matrix-form hidden states. Such 2D-state RNNs, known as Fast Weight Programmers (FWPs), can be interpreted as a neural network whose synaptic weights (called fast weights) dynamically change over time as a function of input observations, and serve as short-term memory storage; corresponding synaptic weight modifications are controlled or programmed by another network (the programmer) whose parameters are trained (e.g., by gradient descent). In this Primer, we review the technical foundations of FWPs, their computational characteristics, and their connections to transformers and state space models. We also discuss connections between FWPs and models of synaptic plasticity in the brain, suggesting a convergence of natural and artificial intelligence.

The brain is a noisy system subject to energy constraints. These facts are rarely taken into account when modelling artificial neural networks. In this paper, we are interested in demonstrating that those factors can actually lead to the appearance of robust associative memories. We first propose a simplified model of noise in the brain, taking into account synaptic noise and interference from neurons external to the network. When coarsely quantized, we show that this noise can be reduced to insertions and erasures. We take a neural network with recurrent modifiable connections, and subject it to noisy external inputs. We introduce an energy usage limitation principle in the network as well as consolidated Hebbian learning, resulting in an incremental processing of inputs. We show that the connections naturally formed correspond to state-of-the-art binary sparse associative memories.

We propose the APTx Neuron, a novel, unified neural computation unit that integrates non-linear activation and linear transformation into a single trainable expression. The APTx Neuron is derived from the APTx activation function, thereby eliminating the need for separate activation layers and making the architecture both computationally efficient and elegant. The proposed neuron follows the functional form y = sum_{i=1}^{n} ((alpha_i + tanh(beta_i x_i)) cdot gamma_i x_i) + delta, where all parameters alpha_i, beta_i, gamma_i, and delta are trainable. We validate our APTx Neuron-based architecture on the MNIST dataset, achieving up to 96.69% test accuracy within 11 epochs using approximately 332K trainable parameters. The results highlight the superior expressiveness and computational efficiency of the APTx Neuron compared to traditional neurons, pointing toward a new paradigm in unified neuron design and the architectures built upon 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.

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.

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}.

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.

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.

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.

Gradient descent has proven to be a powerful and effective technique for optimization in numerous machine learning applications. Recent advances in computational neuroscience have shown that learning in standard gradient descent optimization formulation is not consistent with learning in biological systems. This has opened up interesting avenues for building biologically inspired learning techniques. One such approach is inspired by Dale's law, which states that inhibitory and excitatory synapses do not swap roles during the course of learning. The resulting exponential gradient descent optimization scheme leads to log-normally distributed synaptic weights. Interestingly, the density that satisfies the Fokker-Planck equation corresponding to the stochastic differential equation (SDE) with geometric Brownian motion (GBM) is the log-normal density. Leveraging this connection, we start with the SDE governing geometric Brownian motion, and show that discretizing the corresponding reverse-time SDE yields a multiplicative update rule, which surprisingly, coincides with the sampling equivalent of the exponential gradient descent update founded on Dale's law. Furthermore, we propose a new formalism for multiplicative denoising score-matching, subsuming the loss function proposed by Hyvaerinen for non-negative data. Indeed, log-normally distributed data is positive and the proposed score-matching formalism turns out to be a natural fit. This allows for training of score-based models for image data and results in a novel multiplicative update scheme for sample generation starting from a log-normal density. Experimental results on MNIST, Fashion MNIST, and Kuzushiji datasets demonstrate generative capability of the new scheme. To the best of our knowledge, this is the first instance of a biologically inspired generative model employing multiplicative updates, founded on geometric Brownian motion.

Physics-informed neural networks (PINNs) leverage neural-networks to find the solutions of partial differential equation (PDE)-constrained optimization problems with initial conditions and boundary conditions as soft constraints. These soft constraints are often considered to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the challenge of training (i) persists even when the boundary conditions are strictly enforced, and (ii) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum regularization, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection-diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called "complexity" of the problems, compared to those in the traditional PINNs in the Eulerian framework.

This study introduces BrainTransformers, an innovative Large Language Model (LLM) implemented using Spiking Neural Networks (SNN). Our key contributions include: (1) designing SNN-compatible Transformer components such as SNNMatmul, SNNSoftmax, and SNNSiLU; (2) implementing an SNN approximation of the SiLU activation function; and (3) developing a Synapsis module to simulate synaptic plasticity. Our 3-billion parameter model, BrainTransformers-3B-Chat, demonstrates competitive performance across various benchmarks, including MMLU (63.2), BBH (54.1), ARC-C (54.3), and GSM8K (76.3), while potentially offering improved energy efficiency and biological plausibility. The model employs a three-stage training approach, including SNN-specific neuronal synaptic plasticity training. This research opens new avenues for brain-like AI systems in natural language processing and neuromorphic computing. Future work will focus on hardware optimization, developing specialized SNN fine-tuning tools, and exploring practical applications in energy-efficient computing environments.

Recurrent neural networks (RNNs) provide a powerful approach in neuroscience to infer latent dynamics in neural populations and to generate hypotheses about the neural computations underlying behavior. However, past work has focused on relatively simple, input-driven, and largely deterministic behaviors - little is known about the mechanisms that would allow RNNs to generate the richer, spontaneous, and potentially stochastic behaviors observed in natural settings. Modeling with Hidden Markov Models (HMMs) has revealed a segmentation of natural behaviors into discrete latent states with stochastic transitions between them, a type of dynamics that may appear at odds with the continuous state spaces implemented by RNNs. Here we first show that RNNs can replicate HMM emission statistics and then reverse-engineer the trained networks to uncover the mechanisms they implement. In the absence of inputs, the activity of trained RNNs collapses towards a single fixed point. When driven by stochastic input, trajectories instead exhibit noise-sustained dynamics along closed orbits. Rotation along these orbits modulates the emission probabilities and is governed by transitions between regions of slow, noise-driven dynamics connected by fast, deterministic transitions. The trained RNNs develop highly structured connectivity, with a small set of "kick neurons" initiating transitions between these regions. This mechanism emerges during training as the network shifts into a regime of stochastic resonance, enabling it to perform probabilistic computations. Analyses across multiple HMM architectures - fully connected, cyclic, and linear-chain - reveal that this solution generalizes through the modular reuse of the same dynamical motif, suggesting a compositional principle by which RNNs can emulate complex discrete latent dynamics.

We introduce a mathematically rigorous framework based on rough path theory to model stochastic spiking neural networks (SSNNs) as stochastic differential equations with event discontinuities (Event SDEs) and driven by c\`adl\`ag rough paths. Our formalism is general enough to allow for potential jumps to be present both in the solution trajectories as well as in the driving noise. We then identify a set of sufficient conditions ensuring the existence of pathwise gradients of solution trajectories and event times with respect to the network's parameters and show how these gradients satisfy a recursive relation. Furthermore, we introduce a general-purpose loss function defined by means of a new class of signature kernels indexed on c\`adl\`ag rough paths and use it to train SSNNs as generative models. We provide an end-to-end autodifferentiable solver for Event SDEs and make its implementation available as part of the diffrax library. Our framework is, to our knowledge, the first enabling gradient-based training of SSNNs with noise affecting both the spike timing and the network's dynamics.

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.

Advancements in LLMs have recently unveiled challenges tied to computational efficiency and continual scalability due to their requirements of huge parameters, making the applications and evolution of these models on devices with limited computation resources and scenarios requiring various abilities increasingly cumbersome. Inspired by modularity within the human brain, there is a growing tendency to decompose LLMs into numerous functional modules, allowing for inference with part of modules and dynamic assembly of modules to tackle complex tasks, such as mixture-of-experts. To highlight the inherent efficiency and composability of the modular approach, we coin the term brick to represent each functional module, designating the modularized structure as configurable foundation models. In this paper, we offer a comprehensive overview and investigation of the construction, utilization, and limitation of configurable foundation models. We first formalize modules into emergent bricks - functional neuron partitions that emerge during the pre-training phase, and customized bricks - bricks constructed via additional post-training to improve the capabilities and knowledge of LLMs. Based on diverse functional bricks, we further present four brick-oriented operations: retrieval and routing, merging, updating, and growing. These operations allow for dynamic configuration of LLMs based on instructions to handle complex tasks. To verify our perspective, we conduct an empirical analysis on widely-used LLMs. We find that the FFN layers follow modular patterns with functional specialization of neurons and functional neuron partitions. Finally, we highlight several open issues and directions for future research. Overall, this paper aims to offer a fresh modular perspective on existing LLM research and inspire the future creation of more efficient and scalable foundational models.

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.

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.

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 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.

We consider the solvable neural scaling model with three parameters: data complexity, target complexity, and model-parameter-count. We use this neural scaling model to derive new predictions about the compute-limited, infinite-data scaling law regime. To train the neural scaling model, we run one-pass stochastic gradient descent on a mean-squared loss. We derive a representation of the loss curves which holds over all iteration counts and improves in accuracy as the model parameter count grows. We then analyze the compute-optimal model-parameter-count, and identify 4 phases (+3 subphases) in the data-complexity/target-complexity phase-plane. The phase boundaries are determined by the relative importance of model capacity, optimizer noise, and embedding of the features. We furthermore derive, with mathematical proof and extensive numerical evidence, the scaling-law exponents in all of these phases, in particular computing the optimal model-parameter-count as a function of floating point operation budget.

The behavior of Large Language Models (LLMs) when facing contextual information that conflicts with their internal parametric knowledge is inconsistent, with no generally accepted explanation for the expected outcome distribution. Recent work has identified in autoregressive transformer models a class of neurons -- called entropy neurons -- that produce a significant effect on the model output entropy while having an overall moderate impact on the ranking of the predicted tokens. In this paper, we investigate the preliminary claim that these neurons are involved in inhibiting context copying behavior in transformers by looking at their role in resolving conflicts between contextual and parametric information. We show that entropy neurons are responsible for suppressing context copying across a range of LLMs, and that ablating them leads to a significant change in the generation process. These results enhance our understanding of the internal dynamics of LLMs when handling conflicting information.

AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN). We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP in terms of accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.

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.

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.

Clustering is a widely used unsupervised learning technique involving an intensive discrete optimization problem. Associative Memory models or AMs are differentiable neural networks defining a recursive dynamical system, which have been integrated with various deep learning architectures. We uncover a novel connection between the AM dynamics and the inherent discrete assignment necessary in clustering to propose a novel unconstrained continuous relaxation of the discrete clustering problem, enabling end-to-end differentiable clustering with AM, dubbed ClAM. Leveraging the pattern completion ability of AMs, we further develop a novel self-supervised clustering loss. Our evaluations on varied datasets demonstrate that ClAM benefits from the self-supervision, and significantly improves upon both the traditional Lloyd's k-means algorithm, and more recent continuous clustering relaxations (by upto 60% in terms of the Silhouette Coefficient).

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.

While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing PINNs formulations to respect the spatio-temporal causal structure that is inherent to the evolution of physical systems. We argue that this is a fundamental limitation and a key source of error that can ultimately steer PINN models to converge towards erroneous solutions. We address this pathology by proposing a simple re-formulation of PINNs loss functions that can explicitly account for physical causality during model training. We demonstrate that this simple modification alone is enough to introduce significant accuracy improvements, as well as a practical quantitative mechanism for assessing the convergence of a PINNs model. We provide state-of-the-art numerical results across a series of benchmarks for which existing PINNs formulations fail, including the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the Navier-Stokes equations in the turbulent regime. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such systems, introducing new opportunities for their applicability to problems of industrial complexity.

The relationship between computing systems and the brain has served as motivation for pioneering theoreticians since John von Neumann and Alan Turing. Uniform, scale-free biological networks, such as the brain, have powerful properties, including generalizing over time, which is the main barrier for Machine Learning on the path to Universal Reasoning Models. We introduce `Dragon Hatchling' (BDH), a new Large Language Model architecture based on a scale-free biologically inspired network of \n locally-interacting neuron particles. BDH couples strong theoretical foundations and inherent interpretability without sacrificing Transformer-like performance. BDH is a practical, performant state-of-the-art attention-based state space sequence learning architecture. In addition to being a graph model, BDH admits a GPU-friendly formulation. It exhibits Transformer-like scaling laws: empirically BDH rivals GPT2 performance on language and translation tasks, at the same number of parameters (10M to 1B), for the same training data. BDH can be represented as a brain model. The working memory of BDH during inference entirely relies on synaptic plasticity with Hebbian learning using spiking neurons. We confirm empirically that specific, individual synapses strengthen connection whenever BDH hears or reasons about a specific concept while processing language inputs. The neuron interaction network of BDH is a graph of high modularity with heavy-tailed degree distribution. The BDH model is biologically plausible, explaining one possible mechanism which human neurons could use to achieve speech. BDH is designed for interpretability. Activation vectors of BDH are sparse and positive. We demonstrate monosemanticity in BDH on language tasks. Interpretability of state, which goes beyond interpretability of neurons and model parameters, is an inherent feature of the BDH architecture.

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.

Statistical whitening transformations play a fundamental role in many computational systems, and may also play an important role in biological sensory systems. Existing neural circuit models of adaptive whitening operate by modifying synaptic interactions; however, such modifications would seem both too slow and insufficiently reversible. Motivated by the extensive neuroscience literature on gain modulation, we propose an alternative model that adaptively whitens its responses by modulating the gains of individual neurons. Starting from a novel whitening objective, we derive an online algorithm that whitens its outputs by adjusting the marginal variances of an overcomplete set of projections. We map the algorithm onto a recurrent neural network with fixed synaptic weights and gain-modulating interneurons. We demonstrate numerically that sign-constraining the gains improves robustness of the network to ill-conditioned inputs, and a generalization of the circuit achieves a form of local whitening in convolutional populations, such as those found throughout the visual or auditory systems.

We consider dense, associative neural-networks trained with no supervision and we investigate their computational capabilities analytically, via a statistical-mechanics approach, and numerically, via Monte Carlo simulations. In particular, we obtain a phase diagram summarizing their performance as a function of the control parameters such as the quality and quantity of the training dataset and the network storage, valid in the limit of large network size and structureless datasets. Moreover, we establish a bridge between macroscopic observables standardly used in statistical mechanics and loss functions typically used in the machine learning. As technical remarks, from the analytic side, we implement large deviations and stability analysis within Guerra's interpolation to tackle the not-Gaussian distributions involved in the post-synaptic potentials while, from the computational counterpart, we insert Plefka approximation in the Monte Carlo scheme, to speed up the evaluation of the synaptic tensors, overall obtaining a novel and broad approach to investigate neural networks in general.

Modelling the complex spatiotemporal patterns of large-scale brain dynamics is crucial for neuroscience, but traditional methods fail to capture the rich structure in modalities such as magnetoencephalography (MEG). Recent advances in deep learning have enabled significant progress in other domains, such as language and vision, by using foundation models at scale. Here, we introduce MEG-GPT, a transformer based foundation model that uses time-attention and next time-point prediction. To facilitate this, we also introduce a novel data-driven tokeniser for continuous MEG data, which preserves the high temporal resolution of continuous MEG signals without lossy transformations. We trained MEG-GPT on tokenised brain region time-courses extracted from a large-scale MEG dataset (N=612, eyes-closed rest, Cam-CAN data), and show that the learnt model can generate data with realistic spatio-spectral properties, including transient events and population variability. Critically, it performs well in downstream decoding tasks, improving downstream supervised prediction task, showing improved zero-shot generalisation across sessions (improving accuracy from 0.54 to 0.59) and subjects (improving accuracy from 0.41 to 0.49) compared to a baseline methods. Furthermore, we show the model can be efficiently fine-tuned on a smaller labelled dataset to boost performance in cross-subject decoding scenarios. This work establishes a powerful foundation model for electrophysiological data, paving the way for applications in computational neuroscience and neural decoding.

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.

We present a simple mathematical model for the mammalian low visual pathway, taking into account its key elements: retina, lateral geniculate nucleus (LGN), primary visual cortex (V1). The analogies between the cortical level of the visual system and the structure of popular CNNs, used in image classification tasks, suggests the introduction of an additional preliminary convolutional module inspired to precortical neuronal circuits to improve robustness with respect to global light intensity and contrast variations in the input images. We validate our hypothesis on the popular databases MNIST, FashionMNIST and SVHN, obtaining significantly more robust CNNs with respect to these variations, once such extra module is added.

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.

Recent advancements in the direct training of Spiking Neural Networks (SNNs) have demonstrated high-quality outputs even at early timesteps, paving the way for novel energy-efficient AI paradigms. However, the inherent non-linearity and temporal dependencies in SNNs introduce persistent challenges, such as temporal covariate shift (TCS) and unstable gradient flow with learnable neuron thresholds. In this paper, we present two key innovations: MP-Init (Membrane Potential Initialization) and TrSG (Threshold-robust Surrogate Gradient). MP-Init addresses TCS by aligning the initial membrane potential with its stationary distribution, while TrSG stabilizes gradient flow with respect to threshold voltage during training. Extensive experiments validate our approach, achieving state-of-the-art accuracy on both static and dynamic image datasets. The code is available at: https://github.com/kookhh0827/SNN-MP-Init-TRSG

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.

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.

The choice of activation function plays a critical role in neural networks, yet most architectures still rely on fixed, uniform activation functions across all neurons. We introduce SmartMixed, a two-phase training strategy that allows networks to learn optimal per-neuron activation functions while preserving computational efficiency at inference. In the first phase, neurons adaptively select from a pool of candidate activation functions (ReLU, Sigmoid, Tanh, Leaky ReLU, ELU, SELU) using a differentiable hard-mixture mechanism. In the second phase, each neuron's activation function is fixed according to the learned selection, resulting in a computationally efficient network that supports continued training with optimized vectorized operations. We evaluate SmartMixed on the MNIST dataset using feedforward neural networks of varying depths. The analysis shows that neurons in different layers exhibit distinct preferences for activation functions, providing insights into the functional diversity within neural architectures.

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.

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".

Understanding the properties of neural networks trained via stochastic gradient descent (SGD) is at the heart of the theory of deep learning. In this work, we take a mean-field view, and consider a two-layer ReLU network trained via SGD for a univariate regularized regression problem. Our main result is that SGD is biased towards a simple solution: at convergence, the ReLU network implements a piecewise linear map of the inputs, and the number of "knot" points - i.e., points where the tangent of the ReLU network estimator changes - between two consecutive training inputs is at most three. In particular, as the number of neurons of the network grows, the SGD dynamics is captured by the solution of a gradient flow and, at convergence, the distribution of the weights approaches the unique minimizer of a related free energy, which has a Gibbs form. Our key technical contribution consists in the analysis of the estimator resulting from this minimizer: we show that its second derivative vanishes everywhere, except at some specific locations which represent the "knot" points. We also provide empirical evidence that knots at locations distinct from the data points might occur, as predicted by our theory.

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 structure of biological neural circuits-modular, hierarchical, and sparsely interconnected-reflects an efficient trade-off between wiring cost, functional specialization, and robustness. These principles offer valuable insights for artificial neural network (ANN) design, especially as networks grow in depth and scale. Sparsity, in particular, has been widely explored for reducing memory and computation, improving speed, and enhancing generalization. Motivated by systems neuroscience findings, we explore how patterns of functional connectivity in the mouse visual cortex-specifically, ensemble-to-ensemble communication, can inform ANN design. We introduce G2GNet, a novel architecture that imposes sparse, modular connectivity across feedforward layers. Despite having significantly fewer parameters than fully connected models, G2GNet achieves superior accuracy on standard vision benchmarks. To our knowledge, this is the first architecture to incorporate biologically observed functional connectivity patterns as a structural bias in ANN design. We complement this static bias with a dynamic sparse training (DST) mechanism that prunes and regrows edges during training. We also propose a Hebbian-inspired rewiring rule based on activation correlations, drawing on principles of biological plasticity. G2GNet achieves up to 75% sparsity while improving accuracy by up to 4.3% on benchmarks, including Fashion-MNIST, CIFAR-10, and CIFAR-100, outperforming dense baselines with far fewer computations.

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.

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.

Equilibrium propagation (EP) is a compelling alternative to the backpropagation of error algorithm (BP) for computing gradients of neural networks on biological or analog neuromorphic substrates. Still, the algorithm requires weight symmetry and infinitesimal equilibrium perturbations, i.e., nudges, to estimate unbiased gradients efficiently. Both requirements are challenging to implement in physical systems. Yet, whether and how weight asymmetry affects its applicability is unknown because, in practice, it may be masked by biases introduced through the finite nudge. To address this question, we study generalized EP, which can be formulated without weight symmetry, and analytically isolate the two sources of bias. For complex-differentiable non-symmetric networks, we show that the finite nudge does not pose a problem, as exact derivatives can still be estimated via a Cauchy integral. In contrast, weight asymmetry introduces bias resulting in low task performance due to poor alignment of EP's neuronal error vectors compared to BP. To mitigate this issue, we present a new homeostatic objective that directly penalizes functional asymmetries of the Jacobian at the network's fixed point. This homeostatic objective dramatically improves the network's ability to solve complex tasks such as ImageNet 32x32. Our results lay the theoretical groundwork for studying and mitigating the adverse effects of imperfections of physical networks on learning algorithms that rely on the substrate's relaxation dynamics.

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.

Predictive coding networks are neuroscience-inspired models with roots in both Bayesian statistics and neuroscience. Training such models, however, is quite inefficient and unstable. In this work, we show how by simply changing the temporal scheduling of the update rule for the synaptic weights leads to an algorithm that is much more efficient and stable than the original one, and has theoretical guarantees in terms of convergence. The proposed algorithm, that we call incremental predictive coding (iPC) is also more biologically plausible than the original one, as it it fully automatic. In an extensive set of experiments, we show that iPC constantly performs better than the original formulation on a large number of benchmarks for image classification, as well as for the training of both conditional and masked language models, in terms of test accuracy, efficiency, and convergence with respect to a large set of hyperparameters.

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.

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.

Motivated by the paradigm of reservoir computing, we consider randomly initialized controlled ResNets defined as Euler-discretizations of neural controlled differential equations (Neural CDEs), a unified architecture which enconpasses both RNNs and ResNets. We show that in the infinite-width-depth limit and under proper scaling, these architectures converge weakly to Gaussian processes indexed on some spaces of continuous paths and with kernels satisfying certain partial differential equations (PDEs) varying according to the choice of activation function, extending the results of Hayou (2022); Hayou & Yang (2023) to the controlled and homogeneous case. In the special, homogeneous, case where the activation is the identity, we show that the equation reduces to a linear PDE and the limiting kernel agrees with the signature kernel of Salvi et al. (2021a). We name this new family of limiting kernels neural signature kernels. Finally, we show that in the infinite-depth regime, finite-width controlled ResNets converge in distribution to Neural CDEs with random vector fields which, depending on whether the weights are shared across layers, are either time-independent and Gaussian or behave like a matrix-valued Brownian motion.

In-context learning is a powerful emergent ability in transformer models. Prior work in mechanistic interpretability has identified a circuit element that may be critical for in-context learning -- the induction head (IH), which performs a match-and-copy operation. During training of large transformers on natural language data, IHs emerge around the same time as a notable phase change in the loss. Despite the robust evidence for IHs and this interesting coincidence with the phase change, relatively little is known about the diversity and emergence dynamics of IHs. Why is there more than one IH, and how are they dependent on each other? Why do IHs appear all of a sudden, and what are the subcircuits that enable them to emerge? We answer these questions by studying IH emergence dynamics in a controlled setting by training on synthetic data. In doing so, we develop and share a novel optogenetics-inspired causal framework for modifying activations throughout training. Using this framework, we delineate the diverse and additive nature of IHs. By clamping subsets of activations throughout training, we then identify three underlying subcircuits that interact to drive IH formation, yielding the phase change. Furthermore, these subcircuits shed light on data-dependent properties of formation, such as phase change timing, already showing the promise of this more in-depth understanding of subcircuits that need to "go right" for an induction head.

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.

A modern challenge of Artificial Intelligence is learning multiple patterns at once (i.e.parallel learning). While this can not be accomplished by standard Hebbian associative neural networks, in this paper we show how the Multitasking Hebbian Network (a variation on theme of the Hopfield model working on sparse data-sets) is naturally able to perform this complex task. We focus on systems processing in parallel a finite (up to logarithmic growth in the size of the network) amount of patterns, mirroring the low-storage level of standard associative neural networks at work with pattern recognition. For mild dilution in the patterns, the network handles them hierarchically, distributing the amplitudes of their signals as power-laws w.r.t. their information content (hierarchical regime), while, for strong dilution, all the signals pertaining to all the patterns are raised with the same strength (parallel regime). Further, confined to the low-storage setting (i.e., far from the spin glass limit), the presence of a teacher neither alters the multitasking performances nor changes the thresholds for learning: the latter are the same whatever the training protocol is supervised or unsupervised. Results obtained through statistical mechanics, signal-to-noise technique and Monte Carlo simulations are overall in perfect agreement and carry interesting insights on multiple learning at once: for instance, whenever the cost-function of the model is minimized in parallel on several patterns (in its description via Statistical Mechanics), the same happens to the standard sum-squared error Loss function (typically used in Machine Learning).

The neural Hawkes process (Mei & Eisner, 2017) is a generative model of irregularly spaced sequences of discrete events. To handle complex domains with many event types, Mei et al. (2020a) further consider a setting in which each event in the sequence updates a deductive database of facts (via domain-specific pattern-matching rules); future events are then conditioned on the database contents. They show how to convert such a symbolic system into a neuro-symbolic continuous-time generative model, in which each database fact and the possible event has a time-varying embedding that is derived from its symbolic provenance. In this paper, we modify both models, replacing their recurrent LSTM-based architectures with flatter attention-based architectures (Vaswani et al., 2017), which are simpler and more parallelizable. This does not appear to hurt our accuracy, which is comparable to or better than that of the original models as well as (where applicable) previous attention-based methods (Zuo et al., 2020; Zhang et al., 2020a).

The article explores an encoding and structural information processing approach using sparse bit vectors and fixed-length linear vectors. The following are presented: a discrete method of speculative stochastic dimensionality reduction of multidimensional code and linear spaces with linear asymptotic complexity; a geometric method for obtaining discrete embeddings of an organised code space that reflect the internal structure of a given modality. The structure and properties of a code space are investigated using three modalities as examples: morphology of Russian and English languages, and immunohistochemical markers. Parallels are drawn between the resulting map of the code space layout and so-called pinwheels appearing on the mammalian neocortex. A cautious assumption is made about similarities between neocortex organisation and processes happening in our models.

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.

Neurosymbolic (NeSy) frameworks combine neural representations and learning with symbolic representations and reasoning. Combining the reasoning capacities, explainability, and interpretability of symbolic processing with the flexibility and power of neural computing allows us to solve complex problems with more reliability while being data-efficient. However, this recently growing topic poses a challenge to developers with its learning curve, lack of user-friendly tools, libraries, and unifying frameworks. In this paper, we characterize the technical facets of existing NeSy frameworks, such as the symbolic representation language, integration with neural models, and the underlying algorithms. A majority of the NeSy research focuses on algorithms instead of providing generic frameworks for declarative problem specification to leverage problem solving. To highlight the key aspects of Neurosymbolic modeling, we showcase three generic NeSy frameworks - DeepProbLog, Scallop, and DomiKnowS. We identify the challenges within each facet that lay the foundation for identifying the expressivity of each framework in solving a variety of problems. Building on this foundation, we aim to spark transformative action and encourage the community to rethink this problem in novel ways.

The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers.

There has been a recent surge of interest in modeling neural networks (NNs) as Gaussian processes. In the limit of a NN of infinite width the NN becomes equivalent to a Gaussian process. Here we demonstrate that for an ensemble of large, finite, fully connected networks with a single hidden layer the distribution of outputs at initialization is well described by a Gaussian perturbed by the fourth Hermite polynomial for weights drawn from a symmetric distribution. We show that the scale of the perturbation is inversely proportional to the number of units in the NN and that higher order terms decay more rapidly, thereby recovering the Edgeworth expansion. We conclude by observing that understanding how this perturbation changes under training would reveal the regimes in which the Gaussian process framework is valid to model NN behavior.

We introduce a novel approach for analyzing the training dynamics of ReLU networks by examining the characteristic activation boundaries of individual ReLU neurons. Our proposed analysis reveals a critical instability in common neural network parameterizations and normalizations during stochastic optimization, which impedes fast convergence and hurts generalization performance. Addressing this, we propose Geometric Parameterization (GmP), a novel neural network parameterization technique that effectively separates the radial and angular components of weights in the hyperspherical coordinate system. We show theoretically that GmP resolves the aforementioned instability issue. We report empirical results on various models and benchmarks to verify GmP's theoretical advantages of optimization stability, convergence speed and generalization performance.

Mainstream Transformer-based large language models face major efficiency bottlenecks: training computation scales quadratically with sequence length, and inference memory grows linearly, limiting long-context processing. Building large models on non-NVIDIA platforms also poses challenges for stable and efficient training. To address this, we introduce SpikingBrain, a family of brain-inspired models designed for efficient long-context training and inference. SpikingBrain leverages the MetaX GPU cluster and focuses on three aspects: (1) Model Architecture: linear and hybrid-linear attention architectures with adaptive spiking neurons; (2) Algorithmic Optimizations: an efficient, conversion-based training pipeline and a dedicated spike coding framework; (3) System Engineering: customized training frameworks, operator libraries, and parallelism strategies tailored to MetaX hardware. Using these techniques, we develop two models: SpikingBrain-7B, a linear LLM, and SpikingBrain-76B, a hybrid-linear MoE LLM. These models demonstrate the feasibility of large-scale LLM development on non-NVIDIA platforms. SpikingBrain achieves performance comparable to open-source Transformer baselines while using only about 150B tokens for continual pre-training. Our models significantly improve long-sequence training efficiency and deliver inference with (partially) constant memory and event-driven spiking behavior. For example, SpikingBrain-7B attains over 100x speedup in Time to First Token for 4M-token sequences. Training remains stable for weeks on hundreds of MetaX C550 GPUs, with the 7B model reaching a Model FLOPs Utilization of 23.4 percent. The proposed spiking scheme achieves 69.15 percent sparsity, enabling low-power operation. Overall, this work demonstrates the potential of brain-inspired mechanisms to drive the next generation of efficient and scalable large model design.

An obstacle to artificial general intelligence is set by continual learning of multiple tasks of different nature. Recently, various heuristic tricks, both from machine learning and from neuroscience angles, were proposed, but they lack a unified theory ground. Here, we focus on continual learning in single-layered and multi-layered neural networks of binary weights. A variational Bayesian learning setting is thus proposed, where the neural networks are trained in a field-space, rather than gradient-ill-defined discrete-weight space, and furthermore, weight uncertainty is naturally incorporated, and modulates synaptic resources among tasks. From a physics perspective, we translate the variational continual learning into Franz-Parisi thermodynamic potential framework, where previous task knowledge acts as a prior and a reference as well. We thus interpret the continual learning of the binary perceptron in a teacher-student setting as a Franz-Parisi potential computation. The learning performance can then be analytically studied with mean-field order parameters, whose predictions coincide with numerical experiments using stochastic gradient descent methods. Based on the variational principle and Gaussian field approximation of internal preactivations in hidden layers, we also derive the learning algorithm considering weight uncertainty, which solves the continual learning with binary weights using multi-layered neural networks, and performs better than the currently available metaplasticity algorithm. Our proposed principled frameworks also connect to elastic weight consolidation, weight-uncertainty modulated learning, and neuroscience inspired metaplasticity, providing a theory-grounded method for the real-world multi-task learning with deep networks.

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.

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.

Spiking Neural Networks (SNNs) are well known as a promising energy-efficient alternative to conventional artificial neural networks. Subject to the preconceived impression that SNNs are sparse firing, the analysis and optimization of inherent redundancy in SNNs have been largely overlooked, thus the potential advantages of spike-based neuromorphic computing in accuracy and energy efficiency are interfered. In this work, we pose and focus on three key questions regarding the inherent redundancy in SNNs. We argue that the redundancy is induced by the spatio-temporal invariance of SNNs, which enhances the efficiency of parameter utilization but also invites lots of noise spikes. Further, we analyze the effect of spatio-temporal invariance on the spatio-temporal dynamics and spike firing of SNNs. Then, motivated by these analyses, we propose an Advance Spatial Attention (ASA) module to harness SNNs' redundancy, which can adaptively optimize their membrane potential distribution by a pair of individual spatial attention sub-modules. In this way, noise spike features are accurately regulated. Experimental results demonstrate that the proposed method can significantly drop the spike firing with better performance than state-of-the-art SNN baselines. Our code is available in https://github.com/BICLab/ASA-SNN.

Chaotic systems are intrinsically sensitive to small errors, challenging efforts to construct predictive data-driven models of real-world dynamical systems such as fluid flows or neuronal activity. Prior efforts comprise either specialized models trained separately on individual time series, or foundation models trained on vast time series databases with little underlying dynamical structure. Motivated by dynamical systems theory, we present Panda, Patched Attention for Nonlinear DynAmics. We train Panda on a novel synthetic, extensible dataset of 2 times 10^4 chaotic dynamical systems that we discover using an evolutionary algorithm. Trained purely on simulated data, Panda exhibits emergent properties: zero-shot forecasting of unseen real world chaotic systems, and nonlinear resonance patterns in cross-channel attention heads. Despite having been trained only on low-dimensional ordinary differential equations, Panda spontaneously develops the ability to predict partial differential equations without retraining. We demonstrate a neural scaling law for differential equations, underscoring the potential of pretrained models for probing abstract mathematical domains like nonlinear dynamics.

Neural populations exhibit latent dynamical structures that drive time-evolving spiking activities, motivating the search for models that capture both intrinsic network dynamics and external unobserved influences. In this work, we introduce LangevinFlow, a sequential Variational Auto-Encoder where the time evolution of latent variables is governed by the underdamped Langevin equation. Our approach incorporates physical priors -- such as inertia, damping, a learned potential function, and stochastic forces -- to represent both autonomous and non-autonomous processes in neural systems. Crucially, the potential function is parameterized as a network of locally coupled oscillators, biasing the model toward oscillatory and flow-like behaviors observed in biological neural populations. Our model features a recurrent encoder, a one-layer Transformer decoder, and Langevin dynamics in the latent space. Empirically, our method outperforms state-of-the-art baselines on synthetic neural populations generated by a Lorenz attractor, closely matching ground-truth firing rates. On the Neural Latents Benchmark (NLB), the model achieves superior held-out neuron likelihoods (bits per spike) and forward prediction accuracy across four challenging datasets. It also matches or surpasses alternative methods in decoding behavioral metrics such as hand velocity. Overall, this work introduces a flexible, physics-inspired, high-performing framework for modeling complex neural population dynamics and their unobserved influences.

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 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.

Recurrent neural networks (RNNs) in the brain and in silico excel at solving tasks with intricate temporal dependencies. Long timescales required for solving such tasks can arise from properties of individual neurons (single-neuron timescale, tau, e.g., membrane time constant in biological neurons) or recurrent interactions among them (network-mediated timescale). However, the contribution of each mechanism for optimally solving memory-dependent tasks remains poorly understood. Here, we train RNNs to solve N-parity and N-delayed match-to-sample tasks with increasing memory requirements controlled by N by simultaneously optimizing recurrent weights and taus. We find that for both tasks RNNs develop longer timescales with increasing N, but depending on the learning objective, they use different mechanisms. Two distinct curricula define learning objectives: sequential learning of a single-N (single-head) or simultaneous learning of multiple Ns (multi-head). Single-head networks increase their tau with N and are able to solve tasks for large N, but they suffer from catastrophic forgetting. However, multi-head networks, which are explicitly required to hold multiple concurrent memories, keep tau constant and develop longer timescales through recurrent connectivity. Moreover, we show that the multi-head curriculum increases training speed and network stability to ablations and perturbations, and allows RNNs to generalize better to tasks beyond their training regime. This curriculum also significantly improves training GRUs and LSTMs for large-N tasks. Our results suggest that adapting timescales to task requirements via recurrent interactions allows learning more complex objectives and improves the RNN's 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.

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.

Recent years have witnessed Spiking Neural Networks (SNNs) gaining attention for their ultra-low energy consumption and high biological plausibility compared with traditional Artificial Neural Networks (ANNs). Despite their distinguished properties, the application of SNNs in the computationally intensive field of image generation is still under exploration. In this paper, we propose the Spiking Diffusion Models (SDMs), an innovative family of SNN-based generative models that excel in producing high-quality samples with significantly reduced energy consumption. In particular, we propose a Temporal-wise Spiking Mechanism (TSM) that allows SNNs to capture more temporal features from a bio-plasticity perspective. In addition, we propose a threshold-guided strategy that can further improve the performances by up to 16.7% without any additional training. We also make the first attempt to use the ANN-SNN approach for SNN-based generation tasks. Extensive experimental results reveal that our approach not only exhibits comparable performance to its ANN counterpart with few spiking time steps, but also outperforms previous SNN-based generative models by a large margin. Moreover, we also demonstrate the high-quality generation ability of SDM on large-scale datasets, e.g., LSUN bedroom. This development marks a pivotal advancement in the capabilities of SNN-based generation, paving the way for future research avenues to realize low-energy and low-latency generative applications. Our code is available at https://github.com/AndyCao1125/SDM.

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.

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.

We present XMem, a video object segmentation architecture for long videos with unified feature memory stores inspired by the Atkinson-Shiffrin memory model. Prior work on video object segmentation typically only uses one type of feature memory. For videos longer than a minute, a single feature memory model tightly links memory consumption and accuracy. In contrast, following the Atkinson-Shiffrin model, we develop an architecture that incorporates multiple independent yet deeply-connected feature memory stores: a rapidly updated sensory memory, a high-resolution working memory, and a compact thus sustained long-term memory. Crucially, we develop a memory potentiation algorithm that routinely consolidates actively used working memory elements into the long-term memory, which avoids memory explosion and minimizes performance decay for long-term prediction. Combined with a new memory reading mechanism, XMem greatly exceeds state-of-the-art performance on long-video datasets while being on par with state-of-the-art methods (that do not work on long videos) on short-video datasets. Code is available at https://hkchengrex.github.io/XMem

This work is dedicated to the review and perspective of the new direction that we call "Neuropunk revolution" resembling the cultural phenomenon of cyberpunk. This new phenomenon has its foundations in advances in neuromorphic technologies including memristive and bio-plausible simulations, BCI, and neurointerfaces as well as unconventional approaches to AI and computing in general. We present the review of the current state-of-the-art and our vision of near future development of scientific approaches and future technologies. We call the "Neuropunk revolution" the set of trends that in our view provide the necessary background for the new generation of approaches technologies to integrate the cybernetic objects with biological tissues in close loop system as well as robotic systems inspired by the biological processes again integrated with biological objects. We see bio-plausible simulations implemented by digital computers or spiking networks memristive hardware as promising bridge or middleware between digital and (neuro)biological domains.

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.

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.

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.

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.

Previous literature offers limited clues on how to learn a periodic function using modern neural networks. We start with a study of the extrapolation properties of neural networks; we prove and demonstrate experimentally that the standard activations functions, such as ReLU, tanh, sigmoid, along with their variants, all fail to learn to extrapolate simple periodic functions. We hypothesize that this is due to their lack of a "periodic" inductive bias. As a fix of this problem, we propose a new activation, namely, x + sin^2(x), which achieves the desired periodic inductive bias to learn a periodic function while maintaining a favorable optimization property of the ReLU-based activations. Experimentally, we apply the proposed method to temperature and financial data prediction.

Electron microscopy (EM) enables the reconstruction of neural circuits at the level of individual synapses, which has been transformative for scientific discoveries. However, due to the complex morphology, an accurate reconstruction of cortical axons has become a major challenge. Worse still, there is no publicly available large-scale EM dataset from the cortex that provides dense ground truth segmentation for axons, making it difficult to develop and evaluate large-scale axon reconstruction methods. To address this, we introduce the AxonEM dataset, which consists of two 30x30x30 um^3 EM image volumes from the human and mouse cortex, respectively. We thoroughly proofread over 18,000 axon instances to provide dense 3D axon instance segmentation, enabling large-scale evaluation of axon reconstruction methods. In addition, we densely annotate nine ground truth subvolumes for training, per each data volume. With this, we reproduce two published state-of-the-art methods and provide their evaluation results as a baseline. We publicly release our code and data at https://connectomics-bazaar.github.io/proj/AxonEM/index.html to foster the development of advanced methods.

Stochastic neurons and hard non-linearities can be useful for a number of reasons in deep learning models, but in many cases they pose a challenging problem: how to estimate the gradient of a loss function with respect to the input of such stochastic or non-smooth neurons? I.e., can we "back-propagate" through these stochastic neurons? We examine this question, existing approaches, and compare four families of solutions, applicable in different settings. One of them is the minimum variance unbiased gradient estimator for stochatic binary neurons (a special case of the REINFORCE algorithm). A second approach, introduced here, decomposes the operation of a binary stochastic neuron into a stochastic binary part and a smooth differentiable part, which approximates the expected effect of the pure stochatic binary neuron to first order. A third approach involves the injection of additive or multiplicative noise in a computational graph that is otherwise differentiable. A fourth approach heuristically copies the gradient with respect to the stochastic output directly as an estimator of the gradient with respect to the sigmoid argument (we call this the straight-through estimator). To explore a context where these estimators are useful, we consider a small-scale version of {\em conditional computation}, where sparse stochastic units form a distributed representation of gaters that can turn off in combinatorially many ways large chunks of the computation performed in the rest of the neural network. In this case, it is important that the gating units produce an actual 0 most of the time. The resulting sparsity can be potentially be exploited to greatly reduce the computational cost of large deep networks for which conditional computation would be useful.

The human brain is a complex spiking neural network (SNN) that learns multimodal signals in a zero-shot manner by generalizing existing knowledge. Remarkably, the brain achieves this with minimal power consumption, using event-based signals that propagate within its structure. However, mimicking the human brain in neuromorphic hardware presents both hardware and software challenges. Hardware limitations, such as the slowdown of Moore's law and the von Neumann bottleneck, hinder the efficiency of digital computers. On the software side, SNNs are known for their difficult training, especially when learning multimodal signals. To overcome these challenges, we propose a hardware-software co-design that combines a fixed and random liquid state machine (LSM) SNN encoder with trainable artificial neural network (ANN) projections. The LSM is physically implemented using analogue resistive memory, leveraging the inherent stochasticity of resistive switching to generate random weights. This highly efficient and nanoscale in-memory computing approach effectively addresses the von Neumann bottleneck and the slowdown of Moore's law. The ANN projections are implemented digitally, allowing for easy optimization using contrastive loss, which helps to overcome the difficulties associated with SNN training. We experimentally implement this co-design on a 40nm 256Kb in-memory computing macro. We first demonstrate LSM-based event encoding through supervised classification and linear probing on the N-MNIST and N-TIDIGITS datasets.