Neurons are the fundamental units of the nervous system that receive stimuli as signals and pass on this information to other cells in different parts of the body. An action potential refers to the transmission of the electrical nerve impulse along the neuron. In their seminal work published in 1952, Alan L. Hodgkin and Andrew Huxley proposed a mathematical model of neuronal membrane action potentials based on a series of experiments they conducted using the giant squid neuron. This thesis is a study of the nature of the action potential used to transfer signals along the neuron based on the Hodgkin-Huxley (HH) model. The model consists of four coupled differential equations that contain non-linear terms and have no analytic solutions, and so numerical methods must be employed. In this work we developed MATLAB programs using the Runge-Kutta and Finite Difference Explicit Method to solve the space-clamped and full spatial and temporal HH equations respectively. Results illustrated that the solutions from these programs are consistent with current understanding of action potential behavior. The space-clamped calculations describe the behavior of an action potential as it evolves through time when a uniform potential is maintained in the neuron. The full spatial and temporal calculations describe how action potentials evolve in both space and time. The results can be interpreted as a type of non-linear diffusion of voltage, but with important differences compared to classical linear diffusion. Finally, some preliminary work on extensions of the HH model is provided.

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