Positive and bounded nonstandard finite difference scheme for the Hodgkin–Huxley equations

In this work we consider the Hodgkin Huxley model in the form of a coupled system of one singularly perturbed partial differential equation and three ordinary differential equations. The existence of a small parameter, the nonlinearity and the coupling makes the numerical approximations using explicit finite difference schemes very difficult. In particular, spurious oscillations have been observed to exist. Here we propose the use of nonstandard finite difference to improve on the existing time step restrictions. In addition, we prove that the proposed scheme preserves positivity and is elementary stable. Numerical simulations will be given to support the performance of the proposed scheme.