Abstract
The current thesis is aimed at modeling the electrophysiological processes of neurons and myocytes using fractional differential equations (FDE). Using the framework of fractional differential equations, the refractory period, of the action potential generated in a biological cell can be modeled more accurately than that obtained from existing models based on ordinary differential equations (ODE).
Electrophysiological processes pertaining to bioelectric phenomenon are of great significance to biomedical engineers because these potentials are recorded in modern clinical practice for medical diagnostic applications on a regular basis. Some of the most common examples of such signals are ECG (Electrocardiogram), EMG (Electromyogram), EEG (Electroencephalogram), ENG (Electroneurogram), EOG (Electro-oculogram), ERG (Electroretinogram).
In order to reproduce experimental data of bioelectric phenomena quantitatively, several mathematical models have been proposed. Over the last few decades attempts have been made to make those models more and more accurate. One of the pioneering works in this field was done by Hodgkin and Huxley who developed the model of the axon from the Loligo squid. The Hodgkin-Huxley (H-H) model, although qualitatively correct, the H-H model cannot be applied to model biopotentials in other types of membranes. The cause for this limitation is that the H-H model takes into account only two ions (Na+ and K+), each with a single type of voltage-sensitive channel. However, it has been shown that other types of ions such as Ca2+ may be play a significant role in determining the biopotentials. In addition to the above it has also been empirically established that for every type of ion, there can be a variety of channels. The cardiac action potential, for example, shows how action potentials of various shapes can be generated in membranes with voltage-sensitive Ca2+ channels and different types of Na+/K+ channels. Inspired by the Hodgkin-Huxley model several models of the cardiac myocytes were developed. Though the models were fairly accurate, none of them were able to model the refractory period, duration, of the action potential accurately. The reason for this is that all the differential equation models assumed the membrane capacitance to be ideal, i.e., without any charge leakage, whereas in nature a capacitor can never be ideal. The current-voltage relationship of a capacitor cannot be described by an ordinary differential equation (ODE) accurately. The leaky nature of the capacitor can be modeled more accurately using a fractional differential equation (FDE).
The order of the FDE can be varied to obtain accurate values of refractory period, as measured empirically. In the current thesis, we aim to obtain a mathematical relation between refractory period and order of FDE. Using this relation the refractory period corresponding to a given fractional order can be predicted without the need to simulate the mathematical model. Using this relation the refractory period duration for fractional orders above 1 has also been predicted. This was not possible due to the physical limitation that the order of the FDE cannot exceed the number of initial conditions available, and we have only one set of initial conditions to solve the model equations. The Hodgkin-Huxley electrical circuit model of the neuron has also been modified by adding a thermopile, to account for the phenomenon of heat reabsorption in the neuron after an action potential spike has subsided.