Semi Analytic Solution of Hodgkin-Huxley Model by Homotopy Perturbation Method

Hodgkin-Huxley model is a system of four non-linear coupled differential equations which describes and explains the threshold and action potential by a stimulus arising in a single neuron. The solution and analysis of Hodgkin-Huxley equations is a formidable task because of the coupling between non-linear differential equations, lots of unknowns and their dependence on many physical parameters. Although this model has been solved by numerical methods, finding an analytic solution is interesting due to the challenges that the continuum model offers. In this paper, first order semi analytic solution of this model, in space-clamped situation, is derived by Homotopy Perturbation Method. We applied this technique in piece wise manner due to the strong and complex coupling between the variables in the model. Without this modification, finding an accurate analytic solution is impossible for this neural model. Results show that computed analytic solution has excellent agreement with higher order numerical solution. Robustness of the computed analytic solution in different physical scenarios is examined. Further, this analytic solution can describe many key properties such as the threshold potential, the action potential and the refractory period. MATLAB software is used to simulate the solution.