The emergence of the deterministic Hodgkin--Huxley equations as a limit from the underlying stochastic ion-channel mechanism

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 finitestate-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. 1. Introduction: Ion channels of excitable membranes. Most neurons in most organisms have an axon: a long, narrow conduit connecting the central, roughly spherical part of the cell (the soma) to a network of smaller branches and ultimately to the synapses, which form connections with other neurons (principally at branched projections from the latter called dendrites). The axon connects the soma to synapses that may be a great distance away (often several cm) relative to the size of the soma or the diameter of the