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

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.

Appendix

1.1 Relationship Between the Flux and the Firing Rate

Here we show that in the limit that $N\rightarrow \infty$, the quantity

$$ j(t) = \frac{1}{N} \sum\limits_{j=1}^N\sum\limits_{t_{j,k}<t}\delta(t-t_{j,k})$$

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converges to the network averaged instantaneous firing rate. We then relate this firing rate to a flux described in Section 2.1. We will think of $j(t)$ and its limit as distributions.

Note that one needs to be careful in defining a mean or average firing rate, as there are at least three definitions in the literature. See Gerstner and Kistler (2002) for a good discussion of this. To define the appropriate rate for our purposes, we first define the function $n_{i}(t)$ to be the number of spikes fired by the ith neuron in the time interval $[0,t]$. One can then relate $j(t)$ to the average of $\langle n_{i}(t) \rangle$ across the network:

$$\begin{array}{rll}\langle n_{i}(t)\rangle &=& \lim\limits_{N\rightarrow\infty}\frac{1}{N} \sum\limits_{i=1}^N\int_{0}^{t} \sum_{t_{i,k}<t}\delta(x-t_{i,k}) \,dx\\ &=& \lim\limits_{N\rightarrow\infty}\int_{0}^{t} j(x)\,dx. \end{array}$$

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Then we define $\langle R_{i}(t) \rangle$ as

$$ \langle R_{i}(t)\rangle = \lim\limits_{\Delta t \rightarrow 0}\frac{1}{\Delta t}\lim\limits_{N\rightarrow\infty}\frac{1}{N} \sum\limits_{i=1}^N\frac{n_{i}(t+\Delta t) - n_{i}(t)}{N }.$$

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Thus, the network averaged firing rate in this sense is the limit of the population activity as $\Delta t \rightarrow 0$. However, rearranging the limits, this may also be written

$$\langle R_{i}(t) \rangle = \lim\limits_{\Delta t\rightarrow 0} \frac{\langle n_{i}(t+\Delta t)\rangle - \langle n_{i}(t)\rangle}{\Delta t} = \frac{d}{dt}\langle n_{i}(t)\rangle.$$

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Thus, in the limit $N\rightarrow \infty$, we can replace $j(t)$ in the synaptic coupling equation with $\langle R_{i}(t)\rangle$.

Now, all that remains is to relate the firing rate to the flux. For the integrate and fire models we are considering, there is a surface ${\mathcal F}=0$ which defines when a neuron has fired. As the flux is a kind of directional flow rate of the proportion of neurons at a particular point in phase space, the firing rate can be computed by integrating the flux vector over this “firing boundary”:

$$\langle R_{i}(t)\rangle = \int_{\mathcal F} \mathbf{J}\cdot \mathbf{n}\ dS,$$

where $\mathbf {n}$ is the outward normal to the firing boundary. For example, the flux for a system of one dimensional neurons, $J(v,t)$ is merely the proportion of neurons that flow across the point v in phase space per unit time at time t. Now for a network of linear integrate and fire neurons the firing boundary is the point $v=v_{t}$ in the v phase space, thus the firing rate is $J(v_{t},s,t)$, the flux through the threshold. For the class of neurons we are dealing with, however, the firing boundary is given by the line $v=v_{peak}$ in the $v,w$ phase space. Thus the surface integral reduces to integrating the v component of the flux, evaluated at $v=v_{peak}$, over the entire range of w in phase space:

$$\langle R_{i}(t) \rangle = \int_{W} J_V(v_{peak},w,s,t)\,dw.$$

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