Initiation of propagation in a one-dimensional excitable medium

Abstract 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.