# Bifurcations of large networks of two-dimensional integrate and fire neurons

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## Abstract

Recently, a class of two-dimensional integrate and fire models has been used to faithfully model spiking neurons. This class includes the Izhikevich model, the adaptive exponential integrate and fire model, and the quartic integrate and fire model. The bifurcation types for the individual neurons have been thoroughly analyzed by Touboul (SIAM J Appl Math 68(4):1045–1079, 2008). However, when the models are coupled together to form networks, the networks can display bifurcations that an uncoupled oscillator cannot. For example, the networks can transition from firing with a constant rate to burst firing. This paper introduces a technique to reduce a full network of this class of neurons to a mean field model, in the form of a system of switching ordinary differential equations. The reduction uses population density methods and a quasi-steady state approximation to arrive at the mean field system. Reduced models are derived for networks with different topologies and different model neurons with biologically derived parameters. The mean field equations are able to qualitatively and quantitatively describe the bifurcations that the full networks display. Extensions and higher order approximations are discussed.

## Keywords

Bifurcation theory Mean field Large networks Bursting Population density methods Integrate and fire## Notes

### Acknowledgments

This work benefitted from the support of the Natural Sciences and Engineering Research Council of Canada and the Ontario Graduate Scholarship program. The authors would like to thank F. Skinner for useful discussions. The authors would also like to thank the reviewers for their suggestions which improved the manuscript.

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