All Theses and Dissertations

Degree

Master of Science in Mathematics

Department

Department of Mathematical Sciences

Date of Award

Spring 2016

Advisor

Dr. Muhammad Shahid Qureshi

Committee Member 1

Dr. Muhammad Shahid Qureshi, Institute of Business Administration, Karachi

Project Type

Thesis

Access Type

Restricted Access

Pages

viii, 46

Abstract

In the last century, there has been extensive research on our brain and many mathematical models and theories have been developed which describe the dynamical behavior of neurons. One of them is the widely known, Hodgkin-Huxley model. The Hodgkin-Huxley model for space clamp situation (uniform voltage over a patch of nerve membrane) is a mathematical model consisting of 4 nonlinear ordinary differential equations that describe membrane action potentials in neurons. Before this work, these equations could only be solved by numerical techniques and analytical solutions were not found. In this work, efforts are put to find the analytic solution of the Hodgkin-Huxley model by using Homotopy Perturbation Method. Homotopy Perturbation Method was developed by Ji-Huan He (1998) by merging two techniques, the standard homotopy and the perturbation technique for solving linear, nonlinear, initial and boundary value problems. Further, the solution is compared with the experimental results found by Hodgkin and Huxley. In this work, the first-order approximate analytic solution of the space-clamped Hodgkin Huxley model has been computed and algorithm for a higher-order solution is given. For plotting the solution, MATHEMATICA is used. It is found that the first-order solution can describe many key properties of the Hodgkin-Huxley model. Further, besides some differences, the general agreement of the first-order solution of space-clamped Hodgkin-Huxley Equations by Homotopy Perturbation Method with experimental results is good. Homotopy Perturbation method is proved to be a convenient and efficient method to find an approximate or exact analytic solution of nonlinear differential equations.

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