  Global dynamics of the diffusive Hindmarsh-Rose equations with memristor as a
new proposed model for neuron dynamics are investigated in this paper. We prove
the existence and regularity of a global attractor for the solution semiflow
through uniform analytic estimates showing the higher-order dissipative
property and the asymptotically compact characteristics of the solution
semiflow by the approach of Kolmogorov-Riesz theorem. The quantitative bounds
of the regions containing this global attractor respectively in the state space
and in the regular space are explicitly expressed by the model parameters.
