Recurrent biological neural networks: The weak and noisy limit

TL;DR

This paper introduces a perturbative loop-expansion method to analyze the effects of recurrent synaptic interactions in noisy, weakly connected neural networks, providing analytic predictions validated by simulations.

Contribution

It develops a novel loop-expansion technique with a diagrammatic approach for analytic calculation of neural network interactions under noise and weak connectivity.

Findings

Accurately predicts the impact of recurrent connections on neuron activity.

Provides a convergent series expansion for noisy, weakly connected networks.

Validates analytic results with simulations across network parameters.

Abstract

A perturbative method is developed for calculating the effects of recurrent synaptic interactions between neurons embedded in a network. A series expansion is constructed that converges for networks with noisy membrane potential and weak synaptic connectivity. The terms of the series can be interpreted as loops of interactions between neurons, so the technique is called a loop-expansion. A diagrammatic method is introduced that allows for construction of analytic expressions for the parameter dependencies of the spike probability function and correlation functions. An analytic expression is obtained to predict the effect of the surrounding network on a neuron during an intracellular current injection. The analytic results are compared with simulations to test the range of their validity and significant effects of the the recurrent connections in network are accurately predicted by the loop-expansion.