# Biological Neuron Model - Biological Abstraction - Hindmarsh-Rose

Hindmarsh-Rose

Building upon the FitzHugh-Nagumo model, Hindmarsh and Rose proposed in 1984 a model of neuronal activity described by three coupled first order differential equations:

$begin{array}{rcl} dfrac{d x}{d t} &=& y+3x^2-x^3-z+I \ \ dfrac{d y}{d t} &=& 1-5x^2-y \ \ dfrac{d z}{d t} &=& rcdot (4(x + tfrac{8}{5})-z) end{array}$

with r2 = x2 + y2 + z2, and r ≈ 10-2 so that the z variable only changes very slowly. This extra mathematical complexity allows a great variety of dynamic behaviors for the membrane potential, described by the x variable of the model, which include chaotic dynamics. This makes the Hindmarsh-Rose neuron model very useful, because being still simple, allows a good qualitative description of the many different patterns of the action potential observed in experiments.