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arXiv · 1408.4767

Non-smooth Bifurcations of Mean Field Systems of Two-Dimensional Integrate and Fire Neurons

Wilten Nicola·Sue Ann Campbell
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Abstract

Mean-field systems have been recently derived that adequately predict the behaviors of large networks of coupled integrate-and-fire neurons [14]. The mean-field system for a network of neurons with spike frequency adaptation is typically a pair of differential equations for the mean adaptation and mean synaptic gating variable of the network. These differential equations are non-smooth, and in particular are piecewise smooth continuous (PWSC). Here, we analyze the smooth and non-smooth bifurcation structure of these equations and show that the system is organized around a pair of co-dimension two bifurcations that involve, respectively, the collision between a Hopf equilibrium point and a switching manifold, and a saddle-node equilibrium point and a switching manifold. These two co-dimension 2 bifurcations can coalesce into a co-dimension 3 non-smooth bifurcation. As the mean-field system we study is a non-generic piecewise smooth continuous system, we discuss possible regularizations of this system and how the bifurcations which occur are related to non-smooth bifurcations displayed by generic PWSC systems.

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BibTeXRIS

Wilten Nicola, Sue Ann Campbell. 2014-08-20. Non-smooth Bifurcations of Mean Field Systems of Two-Dimensional Integrate and Fire Neurons. https://arxiv.org/abs/1408.4767

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