1984 The Hindmarsh-Rose system
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]
![\left\{
\begin{array}{l}
\dot{x} = I + bx^2 - ax^3 + y - z \\[0.1cm]
\dot{y} = c - dx^2 - y \\[0.1cm]
\dot{z} = r \left[ \displaystyle s (x - X_c) -z \right]
\end{array}
\right.](local/cache-TeX/f318e3bf8c61ca2a1fd7b02f59564282.png "\left\{
\begin{array}{l}
\dot{x} = I + bx^2 - ax^3 + y - z \\[0.1cm]
\dot{y} = c - dx^2 - y \\[0.1cm]
\dot{z} = r \left[ \displaystyle s (x - X_c) -z \right]
\end{array}
\right.")
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.
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MàJ . 04/02/2024