Initiation of propagation in a one-dimensional excitable medium

This study examines the initiation of propagation in a one-dimensional fiber by local stimulation with a small electrode. The membrane dynamics is based on the generic FitzHugh-Nagumo model, reduced in a singular limit to a nonlinear heat equation. A steady-state solution of this nonlinear heat equation defines the critical nucleus, a time-independent distribution of potential that acts as a threshold for propagating wavefronts. The criterion for initiation of propagation from the initial conditions on potential is obtained by re-writing the nonlinear heat equation as a gradient flow of an energy and projecting this gradient flow onto an approximate solution space. Assuming that the evolving potential has a shape of a Gaussian pulse, the solution space consists of the amplitude of the pulse, a, and the inverse of its width, k. The evolution of the potential is visualized on the (a, k) phase plane in which the rest state is a stable node and the critical nucleus solution is a saddle point. The criterion for initiating propagation takes the form of a pair of separatrices that bisect all possible pulse widths, For a specific pulse width, the separatrices determine the minimum amplitude necessary to start propagation, Infinitely broad pulses (space-clamped fiber) require amplitude equal to the membrane excitation threshold. As the width of the pulse decreases, the requirement on the amplitude grows. In a limit of very narrow pulses, the pulse width and the amplitude are related by a linear relationship corresponding to a constant charge delivered by the pulse.