Approximate Macroscopic Dynamics of Spiking Neural Networks Based on Solutions to the Transport Equation

TL;DR

This paper develops an analytical approximation for the evolution of firing rate fluctuations in neural networks, based on transport solutions to the Fokker-Planck system, considering slow, time-varying inputs.

Contribution

It introduces a novel transport-based mean field approach to model dynamic firing rate fluctuations in coupled integrate-and-fire neuron networks.

Findings

Predicts how firing rate fluctuations emerge from interactions of inputs and initial conditions.

Provides an analytical framework for understanding neural population dynamics under slow stimuli.

Extends previous mean field models by incorporating transport solutions to the Fokker-Planck system.

Abstract

Firing rate fluctuations in neural populations are observed experimentally over multiple time scales, in single neurons, across trials when elicited by stimuli, and across populations. In this work, we examine how firing rate fluctuations emerge in networks of coupled integrate-and-fire neurons as a function of the initial distribution of voltages in networks with time-varying inputs. We analytically derive an approximation for the evolution of the instantaneous population rate or flux as a function of the initial voltage distribution through a Fokker-Planck system. Unlike earlier mean field approaches based on asynchronous or constant flux steady state solutions to the Fokker-Planck system, the approach considered here is based on the transport solution to the advection equation and assumes that the time-varying inputs are slow, and the neurons are in the excitation-driven regime. The transport mean field system predicts how firing rate fluctuations emerge from a dynamic interaction between time-varying inputs, initial densities, and coupling in populations of neurons.