Thursday 22 Nov 2012: Initiation of Excitation Waves
Prof Vadim Biktashev - University of Exeter
Harrison 203 14:00-15:00
Excitable medium is defined by existence of a spatially uniform stationary solution, so called resting state, which is stable with respect to small perturbation, but for which a localized perturbation above a certain threshold may elicit propagating so called excitation wave. Examples of excitation waves in nature are wildfire and electric pulses in nerve fibres and cardiac tissue. The question of what stimulation will or will not initiate an excitation wave is of fundamental importance in electrophysiology. 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. In view of the importance of the question, even crude analytical answers could be very useful.
I will talk about 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 caricature two-component model specific for early stage of cardiac action potential. The critical solutions relevant for these models are respectively "critical nucleus", "critical pulse" and "critical front" respectively. 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.