Global Dynamics of Singularly Perturbed Hodgkin-Huxley Equations

Abstract

In their well-known work, Hodgkin and Huxley considered the following model for nerve impulse transmission across an axon:

$$ \frac{{\partial V}}{{\partial t}} - \frac{{{\partial ^2}V}}{{\partial {x^2}}} = {g_{Na}}{m^3}h\left( {{V_{Na}} - V} \right) + {g_K}{n^4}\left( {{V_K} - V} \right) + {g_L}\left( {{V_L} - V} \right) $$

(1)

$$ \frac{{\partial m}}{{\partial t}} = \left( {{m_\infty } - m} \right)/{T_m} $$

(2)

$$ \frac{{\partial h}}{{\partial t}}=\left({{h_\infty}-h}\right)/{T_h} $$

(3)

$$\ \frac{{\partial n}}{{\partial t}} = \left( {{n_\infty } - n} \right)/{T_n} $$

(4)

Supported by NSF Grant No.DMS 9207064