Markov Chains and Nerve Impulses:An Interface Between Statistics and Neurophysiology

Abstract

Sodium channels are voltage-sensitive protein pores located in the cell membrane of nerves and other tissues. They play a central role in the generation of nerve impulses. The understanding of how an impulse is generated and the functions of ion channels was greatly enhanced by the invention of the patch-clamp technique by Neher and Sakman (1976). They were awarded the Nobel Prize in Physiology and Medicine in 1991. The technique allows recording, at the single channel level, of microscopic currents that flow through these channels. For such data, absorbing Markov chains have been widely used as models for studying the behavior of sodium channels; especially the "gating" behavior of channels. Among many statistical problems that arise in the channel studies, we discuss model identification and parameter estimation; and present some of our results. We use strong unimodality as a criterion for model identification. The problem of parameter estimation is complex due to the fact that the experimental data are only partially observable and mixed. The distribution of the data is a mixture of several absorbing Markov chains with hidden states. We use Le Cam's asymptotic procedure to estimate parameters and illustrate the results with experimental data.