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

where the switching mechanism is introduced as follows *y* > 30 then
*x= x+d* and *y = c*.

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 *I*_{syn}. The
part 0.04*y*^{2}+5*y*+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 *I*_{syn}=-99, and initial conditions as

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.