An analytical approach to initiation of propagating fronts
Vadim N. Biktashev (University of Liverpool)
Thursday 19th March, 2009 14:00-15:00 Mathematics Building, 325
Excitable media have a spatially uniform equilibrium "resting state" that is stable against small perturbations, but can also sustain stable propagating waves (fronts and/or pulses) of fixed amplitude, given appropriate initial condition. Examples include biological (nerve and cardiac pulses) chemical (combustion waves) and physical (first-order phase transitions) systems. Whether or not an initial perturbation of the resting state will give rise to a propagating wave is of great pracrical importance. Mathematically, this question involves non-stationary, spatially non-uniform solutions to a strongly nonlinear equation or system of equations. Any analytical progress in this direction is therefore rather difficult, and this problem is mostly approached numerically. We suppose that in view of the importance of the question, even crude analytical answers could be very useful. We propose a theoretical approach to the problem of initiation based on the understanding of the threshold surface as a codimension-1 centre-stable manifold of a "critical solution". We analyze spatially one-dimensional simplified models of excitable media including Zeldovich-Frank-Kamenetsky (ZFK, aka Nagumo) equation, FitzHugh-Nagumo system, and a simplified two-component model of an INa-driven front in a typical cardiac excitation model we proposed earlier. The critical solutions relevant for these models are respectively "critical nucleus", "critical pulse" and "critical front". We demonstrate that the critical front idealization describes well the near-threshold behaviour in a detailed ionic model (Courtemanche et al 1998 human atrium model). An analytical approximation of the threshold manifold yields analytical criterion of initiation. We demonstrate this by deriving such criteria for ZFK and the two-component ionic front models, based on linear approximation of the threshold manifold near the relevant critical solution. Linearization around a critical solution is a workable method for determining the threshold for initiation of excitation wave. The nature of the initiation criteria in ionic models is different from the traditional excitable media models like FitzHugh-Nagumo.