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GeezerVison is showing mathworld.wolfram.com — Hindmarsh-Rose Model -- from Wolfram MathWorld

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Hindmarsh-Rose Model

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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

x^'=y-ax^3+bx^2-z+I
(1)
y^'=c-dx^2-y
(2)
z^'=r[s(x-u)-z],
(3)

where x(t) represents membrane voltage, y(t) is a recovery variable, z(t) is a slow adaptation variable, primes denote derivatives with respect to time t, and I is the applied current. The parameter r>0 sets the time scale of the adaptation. For small r, the fast subsystem is obtained by holding z fixed in the first two equations.

At an equilibrium point, y=c-dx^2 and z=s(x-u). Substitution leaves the cubic equation

ax^3+(d-b)x^2+sx=c+su+I.
(4)

For a=c=1, b=3, d=5, r=0.001, s=4, u=-1.6, and I=2, numerical solution from the initial state (x,y,z)=(-1.6,-12,2) 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.

HindmarshRoseBursts

The figure uses these parameters and initial conditions for 3000<=t<=4000. The left panel plots x(t) against dimensionless time t. The right panel shows the orthogonal projection of the trajectory with z horizontal and x vertical.

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.

See also

Bifurcation, Fitzhugh-Nagumo Equations, Periodic Orbit, Singular Perturbation

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References

Hindmarsh, J. L. and Rose, R. M. "A Model of Neuronal Bursting Using Three Coupled First Order Differential Equations." Proc. Roy. Soc. London B 221, 87-102, 1984. https://doi.org/10.1098/rspb.1984.0024.Linaro, D.; Champneys, A.; Desroches, M.; and Storace, M. "Codimension-Two Homoclinic Bifurcations Underlying Spike Adding in the Hindmarsh-Rose Burster." SIAM J. Appl. Dyn. Syst. 11, 939-962, 2012. https://doi.org/10.1137/110848931.

Cite this as:

Weisstein, Eric W. "Hindmarsh-Rose Model." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Hindmarsh-RoseModel.html

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