Labouriau, Isabel Salgado Ruas, Maria Aparecida Soares
Appeared in:
Dynamical systems
Paging:
Volume 11 (1996) nr. 2 pages 91-108
Year:
1996
Contents:
We study the bifurcation of equilibrium points for a class of differential equations extending the Hodgkin and Huxley equations for the nerve impulse. These equations model the behaviour of excitable cells with an arbitrary number of ionic channels. We obtain a geometrical description of the subset in parameter space where the number of equilibria changes locally, called here multiple equilibria as they correspond to multiple zeros of a function of voltage, depending on the parameters in the equation. For generic functions governing ionic dynamics, the set of local bifurcation parameter values is shown to be a ruled submanifold, smooth at almost all points. For an equation with generic ion dynamics and N channels, i. e. with 2N + 3 parameters, we show that equilibria have multiplicity at most 2N + 2 and that there are always equilibria of multiplicity 2N + 1. For each equilibrium of multiplicity m, we show that there are nearby parameter values where the equations have m simple equilibria. The bifurcation set for the original Hodgkin and Huxley equations is studied numerically and we verify that it satisfies the genericity conditions. Until now, only regions in parameter space with one or three equilibria were described, we find here a region with five equilibria
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