Dissipative stochastic mechanics for capturing neuronal dynamics under the influence of ion channel noise: formalism using a special membrane

Abstract

Based on the idea conveyed in the author's prior study [Fluct. Noise Lett. 6, L147 (2006)], a physical approach for the description of neuronal dynamics under the influence of ion channel noise is developed in the realm of Nelson's stochastic mechanics when open to dissipative environments. The formalism therein is scrutinized using a special membrane with some tailored properties giving the Rose-Hindmarsh dynamics in the deterministic limit. Led by the presence of multiple number of gates in an ion channel, a dual viewpoint of channel noise is established. Then, stochastic mechanics is adopted to model those channel fluctuations emerging from the uncertainty in accessing the permissible topological states of open gates. A mutual interaction between the above fluctuations and the noise, emerging from the stochasticity in the movement of gating particles between the inner and the outer faces of the membrane, is portrayed within a system plus reservoir strategy. Induced by the interaction, renormalizations of the membrane capacitance and of a membrane voltage dependent potential are found to arise. Consequently, the equations of motion, for the expectation values of the variables and the pair correlation functions, are obtained in the collective membrane voltage space.