Low dimensional dynamics of a sparse balanced synaptic network of quadratic integrate-and-fire neurons

TL;DR

This paper develops a low-dimensional model for a sparse inhibitory neural network with quadratic integrate-and-fire neurons, capturing complex collective dynamics beyond traditional diffusion approximations, and validated by numerical simulations.

Contribution

It introduces a rigorous two-cumulant reduction for the mean field equations, enabling accurate analysis of network regimes where diffusion approximation fails.

Findings

Low-dimensional model accurately predicts network dynamics.

Model captures regimes beyond diffusion approximation.

Reduced model explains low embedding dimensionality.

Abstract

Kinetics of a balanced network of neurons with a sparse grid of synaptic links is well representable by the stochastic dynamics of a generic neuron subject to an effective shot noise. The rate of delta-pulses of the noise is determined self-consistently from the probability density of the neuron states. Importantly, the most sophisticated (but robust) collective regimes of the network do not allow for the diffusion approximation, which is routinely adopted for a shot noise in mathematical neuroscience. These regimes can be expected to be biologically relevant. For the kinetics equations of the complete mean field theory of a homogeneous inhibitory network of quadratic integrate-and-fire neurons, we introduce circular cumulants of the genuine phase variable and derive a rigorous two cumulant reduction for both time-independent conditions and modulation of the excitatory current. The low dimensional model is examined with numerical simulations and found to be accurate for time-independent states and dynamic response to a periodic modulation deep into the parameter domain where the diffusion approximation is not applicable. The accuracy of a low dimensional model indicates and explains a low embedding dimensionality of the macroscopic collective dynamics of the network. The reduced model can be instrumental for theoretical studies of inhibitory-excitatory balanced neural networks.