Pattern Storage, Bifurcations and Higher-Order Correlation Structure of an Exactly Solvable Asymmetric Neural Network Model

TL;DR

This paper provides exact analytical solutions for an asymmetric neural network model, revealing its bifurcation structure, correlation patterns, and a new learning rule, with broad implications for understanding neural dynamics and memory storage.

Contribution

It introduces an exactly solvable asymmetric neural network model with analytical solutions for its dynamics, bifurcations, and correlation structures, advancing theoretical understanding.

Findings

Derived explicit joint probability distributions of membrane potentials and firing rates.

Identified bifurcation diagrams showing transitions between multistability and oscillations.

Proposed a new learning rule for storing attractors robustly in noisy environments.

Abstract

Exactly solvable neural network models with asymmetric weights are rare, and exact solutions are available only in some mean-field approaches. In this article we find exact analytical solutions of an asymmetric spin-glass-like model of arbitrary size and we perform a complete study of its dynamical and statistical properties. The network has discrete-time evolution equations, binary firing rates and can be driven by noise with any distribution. We find analytical expressions of the conditional and stationary joint probability distributions of the membrane potentials and the firing rates. The conditional probability distribution of the firing rates allows us to introduce a new learning rule to store safely, under the presence of noise, point and cyclic attractors, with important applications in the field of content-addressable memories. Furthermore, we study the neuronal dynamics in terms of the bifurcation structure of the network. We derive analytically examples of the codimension one and codimension two bifurcation diagrams of the network, which describe how the neuronal dynamics changes with the external stimuli. In particular, we find that the network may undergo transitions among multistable regimes, oscillatory behavior elicited by asymmetric synaptic connections, and various forms of spontaneous symmetry-breaking. On the other hand, the joint probability distributions allow us to calculate analytically the higher-order correlation structure of the network, which reveals neuronal regimes where, statistically, the membrane potentials and the firing rates are either synchronous or asynchronous. Our results are valid for networks composed of an arbitrary number of neurons, but for completeness we also derive the network equations in the mean-field limit and we study analytically their local bifurcations. All the analytical results are extensively validated by numerical simulations.