Global and Stochastic Dynamics of Diffusive Hindmarsh-Rose Equations in Neurodynamics

This dissertation consisting of three parts is the study of the open problems of global dynamics of diffusive Hindmarsh-Rose equations, random dynamics of the stochastic Hindmarsh-Rose equations with multiplicative noise and additive noise respectively, and synchronization of boundary coupled Hindmarsh-Rose neuron networks. In Part I (Chapters 2, 3 and 4) of this dissertation, we study the global dynamics for the single neuron model of diffusive and partly diffusive Hindmarsh-Rose equations on a three-dimensional bounded domain. The existence of global attractors as well as its regularity and structure are established by showing the absorbing properties and the asymptotically compact characteristics, especially for the partly diffusive Hindmarsh-Rose equations by means of the Kolmogorov-Riesz theorem. A new general theorem on the squeezing property for reaction-diffusion equations is proved, through which we proved the existence of an exponential attractor and the finite fractal dimensionality of the global attractor. For nonautonomous diffusive Hindmarsh-Rose equations with translation bounded input, the existence of pullback exponential attractor is shown with the leverage of proving the $L^2$ to $H^1$ smoothing Lipschitz continuity in a long run of the nonautonomous solution process. In Part II (Chapters 5 and 6), we investigate the pullback long-term behavior of the random dynamical system or called cocycle of the stochastic Hindmarsh-Rose equations driven by multiplicative white noise on a 3D domain and by additive noise on a 2D domain, respectively. The existence of a random attractor for both problems is proved respectively through the exponential transformation and the additive transformation by means of the Ornstein-Uhlenbeck process. Through the sharp uniform estimates, we proved the pullback absorbing property and the pullback asymptotically compactness of these two dynamical system in the $L^2$ Hilbert space. Lastly in Part III (Chapters 7 and 8), the two new mathematical models of partly coupled neurons and of a boundary coupled neuron network are proposed in terms of the systems of partly diffusive Hindmarsh-Rose equations and with the coupling boundary conditions for the network. Through the absorbing and asymptotic analysis for the differencing Hindmarsh-Rose equations of the neuron network, the main result in Chapter 8 shows that the neuronal network is asymptotically synchronized at a uniform exponential rate provided that the combined boundary coupling strength and the stimulating signals exceed a quantified threshold explicitly in terms of the parameters