Global Existence and Uniqueness of a Three-dimensional Model of Cellular Electrophysiology

We study a three-dimensional model of cellular electrical activity, which is written as a pseudodifferential equation on a closed surface Γ in R coupled with a system of ordinary differential equations on Γ. Previously the existence of a global classical solution was not known, due mainly to the lack of a uniform L∞ bound. The main difficulty lies in the fact that, unlike the Laplace operator that appears in traditional models, the pseudodifferential operator in the present model does not satisfy the maximum principle. We overcome this difficulty by introducing the notion of “quasipositivity principle” and prove a uniform L∞ bound of solutions – hence the existence of global classical solutions – for a large class of nonlinearities including the FitzHughNagumo and the Hodgkin-Huxley kinetics. We then study the asymptotic behavior of solutions to show that the system possesses a finite dimensional global attractor consisting entirely of smooth functions despite the fact that the system is only partially dissipative. We also show that ordinary differential equation models without spatial extent, often used in modeling studies, can be obtained from the present model in the small-cell-size limit.