arXiv subjects

Jonathan Kadmon

Publications and source records attributed to Jonathan Kadmon.

8 recordsLinked to original sources

arXiv2026

Predictable Mean-Field Chaos in Random Recurrent Neural Networks

Dynamical mean-field theory (DMFT) maps deterministic chaos in random recurrent neural networks to an effective Gaussian process, usually treated as an ensemble description rather than a predictor of individual trajectories. Contrary to this view, we show that the chaotic mean-field process of the Sompolinsky--Crisanti--Sommers model can be perfectly predictable at the level of a single neuron. When the nonlinearity has a Gaussian or faster Fourier decay, the exact continuous past of one realization determines its entire future: the conditional prediction error vanishes despite a positive Lyapunov exponent. We trace this result to complex-time singularities of the self-consistent covariance through the Paley--Wiener criterion. A Lanczos/Krylov representation turns the covariance into a state-space predictor and defines $\alpha$, the rate at which predictive information is transferred to higher temporal modes. For smooth odd nonlinearities in this predictable class, the Krylov rate and the largest Lyapunov exponent scale differently near the chaotic transition, demonstrating that predictive complexity and microscopic instability are distinct characteristics of the dynamics. Resolving the first $p$ Krylov modes yields a prediction horizon that grows as $\log p$ and enables finite-network forecasts from a single neuron's observed past, without knowing the connectivity and without observing the rest of the network. Thus, the mean-field power spectrum becomes a diagnostic of whether apparent variability reflects irreducible fluctuations or hidden deterministic structure encoded in the observed past.

cond-mat.dis-nn↗

arXiv2026

Discrete signaling mediates chaotic regularization in recurrent neural networks

Cortical circuits operate in a regime of intrinsic chaos, where even tiny changes in input can lead to divergent neural responses. Yet, remarkably, population codes in the brain vary smoothly with sensory stimuli, forming coherent representational manifolds. How can chaotic networks sustain such stable coding? Here, we develop a theoretical framework that links the microscopic chaos of recurrent networks to the macroscopic geometry of neural representations. Combining kernel methods with dynamical mean-field theory, we show that chaotic dynamics induce local roughness (introducing sharp distortions at small scales) while preserving global smoothness across larger stimulus variations. This structural property acts as an intrinsic regularizer, enhancing generalization while maintaining expressivity. Moreover, we show how chaotic networks naturally produce power-law spectral signatures, closely matching experimental observations in cortical recordings. These results explain how chaotic spiking networks can sustain smooth, differentiable population codes and establish a theoretical framework linking network dynamics, computational structure, and recorded neural activity.

q-bio.NC↗

arXiv2025

Curl Descent: Non-Gradient Learning Dynamics with Sign-Diverse Plasticity

Gradient-based algorithms are a cornerstone of artificial neural network training, yet it remains unclear whether biological neural networks use similar gradient-based strategies during learning. Experiments often discover a diversity of synaptic plasticity rules, but whether these amount to an approximation to gradient descent is unclear. Here we investigate a previously overlooked possibility: that learning dynamics may include fundamentally non-gradient "curl"-like components while still being able to effectively optimize a loss function. Curl terms naturally emerge in networks with inhibitory-excitatory connectivity or Hebbian/anti-Hebbian plasticity, resulting in learning dynamics that cannot be framed as gradient descent on any objective. To investigate the impact of these curl terms, we analyze feedforward networks within an analytically tractable student-teacher framework, systematically introducing non-gradient dynamics through neurons exhibiting rule-flipped plasticity. Small curl terms preserve the stability of the original solution manifold, resulting in learning dynamics similar to gradient descent. Beyond a critical value, strong curl terms destabilize the solution manifold. Depending on the network architecture, this loss of stability can lead to chaotic learning dynamics that destroy performance. In other cases, the curl terms can counterintuitively speed learning compared to gradient descent by allowing the weight dynamics to escape saddles by temporarily ascending the loss. Our results identify specific architectures capable of supporting robust learning via diverse learning rules, providing an important counterpoint to normative theories of gradient-based learning in neural networks.

cs.LG↗

arXiv2025

Training Large Neural Networks With Low-Dimensional Error Feedback

Training deep neural networks typically relies on backpropagating high dimensional error signals a computationally intensive process with little evidence supporting its implementation in the brain. However, since most tasks involve low-dimensional outputs, we propose that low-dimensional error signals may suffice for effective learning. To test this hypothesis, we introduce a novel local learning rule based on Feedback Alignment that leverages indirect, low-dimensional error feedback to train large networks. Our method decouples the backward pass from the forward pass, enabling precise control over error signal dimensionality while maintaining high-dimensional representations. We begin with a detailed theoretical derivation for linear networks, which forms the foundation of our learning framework, and extend our approach to nonlinear, convolutional, and transformer architectures. Remarkably, we demonstrate that even minimal error dimensionality on the order of the task dimensionality can achieve performance matching that of traditional backpropagation. Furthermore, our rule enables efficient training of convolutional networks, which have previously been resistant to Feedback Alignment methods, with minimal error. This breakthrough not only paves the way toward more biologically accurate models of learning but also challenges the conventional reliance on high-dimensional gradient signals in neural network training. Our findings suggest that low-dimensional error signals can be as effective as high-dimensional ones, prompting a reevaluation of gradient-based learning in high-dimensional systems. Ultimately, our work offers a fresh perspective on neural network optimization and contributes to understanding learning mechanisms in both artificial and biological systems.

cs.LG↗

arXiv2024

Efficient coding with chaotic neural networks: A journey from neuroscience to physics and back

This essay, derived from a lecture at "The Physics Modeling of Thought" workshop in Berlin in winter 2023, explores the mutually beneficial relationship between theoretical neuroscience and statistical physics through the lens of efficient coding and computation in cortical circuits. It highlights how the study of neural networks has enhanced our understanding of complex, nonequilibrium, and disordered systems, while also demonstrating how neuroscientific challenges have spurred novel developments in physics. The paper traces the evolution of ideas from seminal work on chaos in random neural networks to recent developments in efficient coding and the partial suppression of chaotic fluctuations. It emphasizes how concepts from statistical physics, such as phase transitions and critical phenomena, have been instrumental in elucidating the computational capabilities of neural networks. By examining the interplay between order and disorder in neural computation, the essay illustrates the deep connection between theoretical neuroscience and the statistical physics of nonequilibrium systems. This synthesis underscores the ongoing importance of interdisciplinary approaches in advancing both fields, offering fresh perspectives on the fundamental principles governing information processing in biological and artificial systems. This multidisciplinary approach not only advances our understanding of neural computation and complex systems but also points toward future challenges at the intersection of neuroscience and physics.

q-bio.NC↗

arXiv2020

Predictive coding in balanced neural networks with noise, chaos and delays

Biological neural networks face a formidable task: performing reliable computations in the face of intrinsic stochasticity in individual neurons, imprecisely specified synaptic connectivity, and nonnegligible delays in synaptic transmission. A common approach to combatting such biological heterogeneity involves averaging over large redundant networks of $N$ neurons resulting in coding errors that decrease classically as $1/\sqrt{N}$. Recent work demonstrated a novel mechanism whereby recurrent spiking networks could efficiently encode dynamic stimuli, achieving a superclassical scaling in which coding errors decrease as $1/N$. This specific mechanism involved two key ideas: predictive coding, and a tight balance, or cancellation between strong feedforward inputs and strong recurrent feedback. However, the theoretical principles governing the efficacy of balanced predictive coding and its robustness to noise, synaptic weight heterogeneity and communication delays remain poorly understood. To discover such principles, we introduce an analytically tractable model of balanced predictive coding, in which the degree of balance and the degree of weight disorder can be dissociated unlike in previous balanced network models, and we develop a mean field theory of coding accuracy. Overall, our work provides and solves a general theoretical framework for dissecting the differential contributions neural noise, synaptic disorder, chaos, synaptic delays, and balance to the fidelity of predictive neural codes, reveals the fundamental role that balance plays in achieving superclassical scaling, and unifies previously disparate models in theoretical neuroscience.

q-bio.NC↗

arXiv2018

Statistical mechanics of low-rank tensor decomposition

Often, large, high dimensional datasets collected across multiple modalities can be organized as a higher order tensor. Low-rank tensor decomposition then arises as a powerful and widely used tool to discover simple low dimensional structures underlying such data. However, we currently lack a theoretical understanding of the algorithmic behavior of low-rank tensor decompositions. We derive Bayesian approximate message passing (AMP) algorithms for recovering arbitrarily shaped low-rank tensors buried within noise, and we employ dynamic mean field theory to precisely characterize their performance. Our theory reveals the existence of phase transitions between easy, hard and impossible inference regimes, and displays an excellent match with simulations. Moreover, it reveals several qualitative surprises compared to the behavior of symmetric, cubic tensor decomposition. Finally, we compare our AMP algorithm to the most commonly used algorithm, alternating least squares (ALS), and demonstrate that AMP significantly outperforms ALS in the presence of noise.

stat.ML↗

arXiv2015

Transition to chaos in random neuronal networks

Firing patterns in the central nervous system often exhibit strong temporal irregularity and heterogeneity in their time averaged response properties. Previous studies suggested that these properties are outcome of an intrinsic chaotic dynamics. Indeed, simplified rate-based large neuronal networks with random synaptic connections are known to exhibit sharp transition from fixed point to chaotic dynamics when the synaptic gain is increased. However, the existence of a similar transition in neuronal circuit models with more realistic architectures and firing dynamics has not been established. In this work we investigate rate based dynamics of neuronal circuits composed of several subpopulations and random connectivity. Nonzero connections are either positive-for excitatory neurons, or negative for inhibitory ones, while single neuron output is strictly positive; in line with known constraints in many biological systems. Using Dynamic Mean Field Theory, we find the phase diagram depicting the regimes of stable fixed point, unstable dynamic and chaotic rate fluctuations. We characterize the properties of systems near the chaotic transition and show that dilute excitatory-inhibitory architectures exhibit the same onset to chaos as a network with Gaussian connectivity. Interestingly, the critical properties near transition depend on the shape of the single- neuron input-output transfer function near firing threshold. Finally, we investigate network models with spiking dynamics. When synaptic time constants are slow relative to the mean inverse firing rates, the network undergoes a sharp transition from fast spiking fluctuations and static firing rates to a state with slow chaotic rate fluctuations. When the synaptic time constants are finite, the transition becomes smooth and obeys scaling properties, similar to crossover phenomena in statistical mechanics

cond-mat.dis-nn↗