An approximate analytical solution for the excitation threshold in a one-dimensional Fitzhugh-Nagumo system

Understanding the nature of electrical excitation of a group of cells is important both in examining the onset of a cardiac arrhythmia and in designing the treatment for sudden cardiac arrest. In the past, several attempts have been made to understand the threshold for the excitation of a one-dimensional chain of cells from a mathematical viewpoint. However, obtaining an analytical solution to describe threshold phenomena has proven to be difficult as the equations in this problem are highly non-linear and resist solution by standard mathematical techniques. Here, we apply a method developed by Neu et al. where the time evolution of the width and amplitude of a pulse is approximately described by a gradient flow on a two-dimensional phase plane. Using this approach, we obtain a mathematical expression that successfully models the excitation threshold for an applied square current pulse in a simplified Fitzhugh-Nagumo system. We then analyze our solution to reveal how the excitation threshold depends on key physiological parameters.