1984 The Hindmarsh-Rose system

Christophe LETELLIER

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]

      \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]

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

  {\cal P}_{\rm HR} \equiv
  \left\{ (y_k, z_k) \in \mathbb{R}^2 ~|~ x_k = 0, \dot{x}_k < 0 \right\}

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, X_c = - \frac{1+\sqrt{5}}{2} and I=3.318.

JPG - 19.1 ko
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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