Calculation of the Threshold Surface for Nerve Equations

Abstract

In this paper calculations are presented for the threshold surface $\mathcal{M}$ in McKean’s caricature for nerve conduction $u_t = u_{xx} - u + H( {u - a} )$ and for the Fitzhugh–Nagumo equation $u_t = u_{xx} + u( {1 - u} )( {u - a} )$. The previous construction of this surface gave no information about data on $\mathcal{M}$, or, in particular, the value of the amplitude which makes data critical. This amplitude is computed by both a standard discretization of the partial differential equation and (for McKean’s equation) by means of an asymptotic calculation for the integral equation satisfied by the free boundary $m( t )$ defined by $u( {m( t ),t} ) = a$. On this numerical evidence, it is conjectured that for a large class of initial data $u_0 $ on $\mathcal{M}$ the value $\| {u_0 } \|L^1 $depends only ona.