Hindmarsh-Rose Model

The Hindmarsh-Rose model is a three-dimensional dynamical system for neuronal spiking and bursting (Hindmarsh and Rose 1984). In a parameterized form, its ordinary differential equations are

			(1)
			(2)
			(3)

where represents membrane voltage, is a recovery variable, is a slow adaptation variable, primes denote derivatives with respect to time , and is the applied current. The parameter sets the time scale of the adaptation. For small , the fast subsystem is obtained by holding fixed in the first two equations.

At an equilibrium point, and . Substitution leaves the cubic equation

(4)

For , , , , , , and , numerical solution from the initial state gives regular bursts after a transient. A burst is a group of rapid spikes separated from the next group by a much longer quiet interval. Linaro et al. (2012) used bifurcations of the fast subsystem and of the full model to analyze how changing parameters can add a spike to each burst.