[Paper Review] Mean Field Analysis of Stochastic Neural Network Models with Synaptic Depression

Yasuhiko Igarashi, Masafumi Oizumi+1 more|arXiv (Cornell University)|Mar 5, 2010

Neural dynamics and brain function3 citations

TL;DR

This paper develops a mean field theory for stochastic binary neural networks with synaptic depression, showing that synaptic depression induces oscillatory instabilities through Hopf and Turing mechanisms. It identifies three distinct oscillatory states—uniform oscillation, rotating bump, and a previously unreported oscillatory bump—in ring networks with Mexican-hat connectivity, depending on connection strength.

ABSTRACT

We investigated the effects of synaptic depression on the macroscopic behavior of stochastic neural networks. Dynamical mean field equations were derived for such networks by taking the average of two stochastic variables: a firing state varialbe and a synaptic variable. In these equations, their average product is decoupled as the product of averaged them because the two stochastic variables are independent. We proved the independence of these two stochastic variables assuming that the synaptic weight is of the order of 1/N with respect to the number of neurons N. Using these equations, we derived macroscopic steady state equations for a network with uniform connections and a ring attractor network with Mexican hat type connectivity and investigated the stability of the steady state solutions. An oscillatory uniform state was observed in the network with uniform connections due to a Hopf instability. With the ring network, high-frequency perturbations were shown not to affect system stability. Two mechanisms destabilize the inhomogeneous steady state, leading two oscillatory states. A Turing instability leads to a rotating bump state, while a Hopf instability leads to an oscillatory bump state, which was previous unreported. Various oscillatory states take place in a network with synaptic depression depending on the strength of the interneuron connections.

Motivation & Objective

• To understand the macroscopic effects of synaptic depression on stochastic neural network dynamics.
• To derive a closed-form mean field theory for networks with stochastic spiking and dynamic synapses.
• To analyze the stability of steady state solutions in networks with uniform and Mexican-hat connectivity.
• To identify the mechanisms—Hopf and Turing instabilities—underlying oscillatory states induced by synaptic depression.
• To uncover previously unreported dynamic states, such as the oscillatory bump, in ring attractor networks.

Proposed method

• Derives microscopic dynamical mean field equations by averaging over stochastic spike realizations, assuming independence of firing and synaptic variables.
• Proves statistical independence of firing and synaptic variables under the assumption $ J_{ij} \sim O(1/N) $, enabling decoupling of their average product.
• Applies the mean field framework to two network types: a uniform connectivity network and a ring attractor network with Mexican-hat connectivity.
• Performs linear stability analysis on steady state solutions using Fourier-mode perturbations to identify Hopf and Turing instabilities.
• Reduces dimensionality of stability analysis by showing high-frequency perturbations do not affect stability when lateral inhibition is weak.
• Uses numerical and analytical methods to derive phase diagrams and characterize dynamic regimes in the ring network.

Experimental results

Research questions

• RQ1 How does synaptic depression affect the stability of steady states in stochastic neural networks?
• RQ2 What mechanisms—Hopf or Turing instability—lead to oscillatory states in networks with synaptic depression?
• RQ3 Can a novel oscillatory state, beyond uniform oscillation or rotating bumps, emerge in ring networks with synaptic depression?
• RQ4 Under what conditions do high-frequency perturbations influence system stability in such networks?
• RQ5 How does the strength of interneuron connections determine the emergence of different oscillatory states?

Key findings

• Synaptic depression induces a previously unreported oscillatory bump (OB) state in ring networks with Mexican-hat connectivity.
• In uniform networks, synaptic depression leads to a Hopf instability, resulting in a stable oscillatory uniform (OU) state.
• A rotating bump (RB) state emerges via a Turing instability when lateral-inhibitory connections are strong.
• High-frequency perturbations do not affect system stability when lateral inhibition is weak, enabling dimensionality reduction in stability analysis.
• The system exhibits multistable regimes, with distinct dynamic states (OU, RB, OB) emerging based on connection strength parameters.
• The mean field equations derived for stochastic binary networks with synaptic depression match those of analog networks, validating the approach.