Efficient calculation of heterogeneous non-equilibrium statistics in coupled firing-rate models

Cheng Ly,Andrea K Barreiro,Woodrow L Shew

doi:10.1186/s13408-019-0070-7

Abstract

Understanding nervous system function requires careful study of transient (non-equilibrium) neural response to rapidly changing, noisy input from the outside world. Such neural response results from dynamic interactions among multiple, heterogeneous brain regions. Realistic modeling of these large networks requires enormous computational resources, especially when high-dimensional parameter spaces are considered. By assuming quasi-steady-state activity, one can neglect the complex temporal dynamics; however, in many cases the quasi-steady-state assumption fails. Here, we develop a new reduction method for a general heterogeneous firing-rate model receiving background correlated noisy inputs that accurately handles highly non-equilibrium statistics and interactions of heterogeneous cells. Our method involves solving an efficient set of nonlinear ODEs, rather than time-consuming Monte Carlo simulations or high-dimensional PDEs, and it captures the entire set of first and second order statistics while allowing significant heterogeneity in all model parameters.

Highlights

Advances in neural recording technologies have enabled experimentalists to simultaneously measure activity across different regions with cellular resolution [1,2,3,4]
We previously developed a fast approximation method [9] for the complete first and second order statistics of a firing-rate network model based on the Wilson–Cowan model [10], and applied it to the olfactory sensory pathway [11]
As detailed in the Introduction, the common assumption of equilibrium neural network responses is inaccurate in many neural systems

Summary

Introduction

Advances in neural recording technologies have enabled experimentalists to simultaneously measure activity across different regions with cellular resolution [1,2,3,4]. We previously developed a fast approximation method [9] for the complete first and second order statistics of a firing-rate network model based on the Wilson–Cowan model [10], and applied it to the olfactory sensory pathway [11]. Those methods assumed that the statistics of neural activity are stationary (i.e., in steady state). We present a method to approximate the non-equilibrium statistics of a general heterogeneous coupled firing-rate model of neural networks receiving background correlated noise, in which we: (i) assume weak coupling; equivalently, that neural activity is pairwise normal, and (ii) account for the entire probability distribution of inputs. Fk(σk(t)y1 + μk(t))y2 j,k(y1, y2) dy dy2 where the angular brackets · denotes averaging over realizations

Reduction of the Fokker–Planck equation

Moment closure methods

Results

Conclusion