Current-induced coupling between conductance channels in membranes

The driven system of conducting channels in a nerve membrane is investigated. A current flow generates a coupling between the channels: the current through a channel is influenced by the presence of other conducting channels via the deformation of the equipotential surfaces within the media adjacent to the membrane. We derive an integral equation for the membrane voltageV(s) (s in the membrane plane) and solve it for different membrane conductance distributions(s) including models for stochastic distributions of conducting channels.V(s) is a nonlinear functional of(s). The system of coupled channels is compared with an Ising model. The system exhibits a multi-channel interaction which can be characterized by two different rangesd_int andD₁. For a mean channel distanced₀d_int interaction effects are negligible, and ford₀D₁ all channel-voltages are equal and thus represent a mean-field for the channels. Increasing conductivity of the medium decreasesd_int and increasesD₁. With experimental data on sodium channels in nerve membranes we find:d_intd₀, i.e. a 50% decrease of the channel-voltages by the interaction, andD₁10³10⁴d₀, which indicates mean-field behaviour of the channels. In a subsequent paper we shall treat the statistics of channels which open and close stochastically under the influence of the local membrane voltage.