  In this paper we consider the classical differential equations of Hodgkin and
Huxley and a natural refinement of them to include a layer of stochastic
behavior, modeled by a large number of finite-state-space Markov processes
coupled to a simple modification of the original Hodgkin--Huxley PDE. We first
prove existence, uniqueness and some regularity for the stochastic process, and
then show that in a suitable limit as the number of stochastic components of
the stochastic model increases and their individual contributions decrease, the
process that they determine converges to the trajectory predicted by the
deterministic PDE, uniformly up to finite time horizons in probability. In a
sense, this verifies the consistency of the deterministic and stochastic
processes.
