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Biological models

# 2003 A simple model for spiking neurons

Christophe LETELLIER
16/04/2017
Eugene Izhikevich

Eugene M. Izhikevich presented a model that reproduces spiking and bursting behavior of known types of cortical neurons [1]. The model combines the biologically plausibility of the dynamics underlying the Hodgkin–Huxley model [2] and the computational efficiency of integrate-and-fire neurons. As initiated by Bard Ermentrout and Nancy Kopell [3], this model is made of an oscillator producing slow oscillations combined with a switching mechanism for reproducing the bursting phenomenon [4]. The model equations - as proposed in Ref. [1] - are

$\left\{&space;\begin{array}{l}&space;\dot{x}&space;=&space;a(by-x)&space;\\&space;\dot{y}&space;=&space;0.04y^2+5y+140-x+I_{\rm&space;syn}&space;\end{array}&space;\right.$

where the switching mechanism is introduced as follows

$\left|&space;\begin{array}{l}&space;x&space;=&space;x+d&space;\\[0.1cm]&space;y&space;=&space;c&space;\end{array}&space;\right.$

when y>30.

Variable x represents the membrane recovery which accounts for the activation of K+ ionic currents and inactivation of Na+ ionic currents, and it provides negative feedback to the membrane potential of the neuron y. Synaptic currents or injected dc-currents are delivered via the variable Isyn. The part 0.04y2+5y+140 was obtained by fitting the spike initiation dynamics of a cortical neuron so that the membrane potential is expressed in mV and the time in ms.

Using parameter values a=0.2, b=2, c=-56, d=-16 and Isyn=-99, and initial conditions as

$\left\{&space;\begin{array}{l}&space;x_0&space;=&space;-20&space;\\&space;y_0&space;=&space;-70&space;\end{array}&space;\right.$

a chaotic attractor can be obtained (Fig. 1). It is characterized by a first-return map to a Poincaré section made of four branches, more or less as can be found in the Rössler system [5].

Fig. 1. Chaotic attractor produced by Izhikevich’s model.

[1] E. M. Izhikevich, Simple model of spiking neurons, IEEE Transactions on Neural Networks, 14 (6), 1569-1572, 2003.

[2] A. L. Hodgkin & A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (4), 500-544, 1952.

[3] G. B. Ermentrout & N. Kopell, Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM Journal of Applied Mathematics, 46 (2), 233-253, 1984.

[4] E. M. Izhikevich, Neural, excitablity, spiking and bursting, International Journal of Bifurcation & Chaos, 10, 1171-1266, 2000.

[5] C. Letellier, P. Dutertre & B. Maheu, Unstable periodic orbits and templates of the Rössler system : toward a systematic topological characterization, Chaos, 5 (1), 272-281, 1995.

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