Drift-Diffusion Matching: Embedding dynamics in latent manifolds of asymmetric neural networks

TL;DR

This paper introduces drift-diffusion matching, a framework for training asymmetric recurrent neural networks to embed complex stochastic dynamical systems within low-dimensional manifolds, enabling modeling of rich biological neural dynamics.

Contribution

It presents a novel method for training continuous-time RNNs to represent arbitrary stochastic systems, extending neural network theory beyond symmetric and equilibrium assumptions.

Findings

RNNs can embed nonlinear and nonequilibrium stochastic dynamics.

Asymmetric connectivity enables modeling of chaotic attractors and transitions.

The framework unifies neural computation with concepts from statistical mechanics.

Abstract

Recurrent neural networks (RNNs) provide a theoretical framework for understanding computation in biological neural circuits, yet classical results, such as Hopfield's model of associative memory, rely on symmetric connectivity that restricts network dynamics to gradient-like flows. In contrast, biological networks support rich time-dependent behaviour facilitated by their asymmetry. Here we introduce a general framework, which we term drift-diffusion matching, for training continuous-time RNNs to represent arbitrary stochastic dynamical systems within a low-dimensional latent subspace. Allowing asymmetric connectivity, we show that RNNs can faithfully embed the drift and diffusion of a given stochastic differential equation, including nonlinear and nonequilibrium dynamics such as chaotic attractors. As an application, we construct RNN realisations of stochastic systems that transiently explore various attractors through both input-driven switching and autonomous transitions driven by nonequilibrium currents, which we interpret as models of associative and sequential (episodic) memory. To elucidate how these dynamics are encoded in the network, we introduce decompositions of the RNN based on its asymmetric connectivity and its time-irreversibility. Our results extend attractor neural network theory beyond equilibrium, showing that asymmetric neural populations can implement a broad class of dynamical computations within low-dimensional manifolds, unifying ideas from associative memory, nonequilibrium statistical mechanics, and neural computation.