Christophe LETELLIER
10/03/2016

To overcome the complexity of the Hodgkin-Huxley model [1], Richard FitzHugh introduced a second-order differential equations which appears to be too simplified [2]. Then James Hindmarsh and Malcom Rose added a third equation to limit the neuron firing and got the model [3]

where *x* is the membrane potential, *y* the recovery variable (quantifying
the transport of sodium and potassium through fast ion channels) and *z*
an adaptation current which gradually hyperpolarizes the cell (it corresponds
to the transport of other ions through slow channels). For appropriate
parameter values, the Hindmarsh-Rose system produces a chaotic attractor
(Fig. 1a) characterized by a first-return map to the Poincaré section

which is smooth and unimodal (Fig. 1b). This behavior is obtained after a period-doubling cascade when the applied current *I* is decreased. The parameter
values used for producing this chaotic attractor are *a*=1, *b*=3, *c*=1, *d*=5, *r*=0.001, *s*=4, and *I*=3.318.

**Fig. 1. Chaotic behavior produced by the Hindmarsh-Rose system.**

[1] **A. L. Hodgkin & A. F. Huxley**, A quantitative description of membrane current and its application to
conduction and excitation in nerve, *Journal of Physiology*, **177**, 500-544, 1952.

[2] **R. FitzHugh**, Impulses and physiological states in theoretical models of nerve membrane, *Biophysical Journal*, **1**, 445-466, 1961.

[3] **J. L. Hindmarsh & R. M. Rose**, A model of neuronal bursting using three coupled first order differential equations, *Proceedings of the Royal Society of London B*, **221**, 87-102, 1984.