Abstract
The existence of exponential attractor for the diffusive Hindmarsh-Rose equations on a three-dimensional bounded domain in the study of neurodynamics is proved through uniform estimates and a new theorem on the squeezing property of the abstract reaction-diffusion equation established in this paper. This result on the exponential attractor infers that the global attractor whose existence has been proved in [<span class="xref"><a href="#b22" ref-type="bibr">22</a></span>] for the diffusive Hindmarsh-Rose semiflow has a finite fractal dimension.
Highlights
The Hindmarsh-Rose equations for neuronal spiking-bursting of the intracellular membrane potential observed in experiments was originally proposed in [14, 15]
We shall study in this paper the global dynamics in terms of the existence of an exponential attractor for the diffusive Hindmarsh-Rose equations:
A subset E ⊂ X is called an exponential attractor for this semiflow if the following three conditions are satisfied: 1) E is a compact in X with a finite fractal dimension
Summary
The Hindmarsh-Rose equations for neuronal spiking-bursting of the intracellular membrane potential observed in experiments was originally proposed in [14, 15] This mathematical model composed of three coupled ordinary differential equations has been studied through numerical simulations and mathematical analysis in recent years, cf [14, 15, 17, 21, 30, 39] and the references therein. FitzHugh-Nagumo equations [13] (1961-1962) provided a two-dimensional model for an excitable neuron with the membrane potential and the current variable This two-dimensional ODE model admits an exquisite phase plane analysis showing spikes excited by supra-threshold input pulses and sustained periodic spiking with refractory period, but due to the 2D nature FitzHugh-Nagumo equations exclude any chaotic solutions and chaotic dynamics so that no chaotic bursting can be generated.
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