Mean-field models for heterogeneous networks of two-dimensional integrate and fire neurons

Wilten Nicola,Sue Ann Campbell

doi:10.3389/fncom.2013.00184

Wilten Nicola, Sue Ann Campbell

Open Access

https://doi.org/10.3389/fncom.2013.00184

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Abstract

We analytically derive mean-field models for all-to-all coupled networks of heterogeneous, adapting, two-dimensional integrate and fire neurons. The class of models we consider includes the Izhikevich, adaptive exponential and quartic integrate and fire models. The heterogeneity in the parameters leads to different moment closure assumptions that can be made in the derivation of the mean-field model from the population density equation for the large network. Three different moment closure assumptions lead to three different mean-field systems. These systems can be used for distinct purposes such as bifurcation analysis of the large networks, prediction of steady state firing rate distributions, parameter estimation for actual neurons and faster exploration of the parameter space. We use the mean-field systems to analyze adaptation induced bursting under realistic sources of heterogeneity in multiple parameters. Our analysis demonstrates that the presence of heterogeneity causes the Hopf bifurcation associated with the emergence of bursting to change from sub-critical to super-critical. This is confirmed with numerical simulations of the full network for biologically reasonable parameter values. This change decreases the plausibility of adaptation being the cause of bursting in hippocampal area CA3, an area with a sizable population of heavily coupled, strongly adapting neurons.

Highlights

As computers become more powerful, there is a move to numerically simulate larger and larger model networks of neurons (Izhikevich and Edelman, 2008)
200 nS 50 nS 1200 pA 500 pA 100 pA 20 pA 50 ms 10 ms www.frontiersin.org networks of heterogeneous oscillators. This was accomplished through the derivation of three separate mean-field systems, mean-field one (MFI), MFII, and MFIII, with differing applications and regions of validity
We successfully applied numerical bifurcation analysis to MFI and MFII to aid in the understanding of the different behaviors that heterogeneous networks can display, and how they transition between these different types of behaviors

Summary

Introduction

As computers become more powerful, there is a move to numerically simulate larger and larger model networks of neurons (Izhikevich and Edelman, 2008). The tools of dynamical systems theory, such as bifurcation analysis can be useful in this regard when studying single neuron or small network models. They are not viable for large networks, especially if the neurons are not identical. A common approach is to try to extrapolate large network behavior from detailed analysis of the behavior of individual cells or small networks (Skinner et al, 2005). This can be problematic as networks can have behavior that is not present in individual cells. They found that bursting occurred in a larger range of parameters for larger networks (Dur-e-Ahmad et al, 2012, Figure 7)

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