In the 1960s and 1970s, mathematical biologists Sir Robert M. May, E.C. Pielou, and others utilized difference equations as models of ecological and epidemiological phenomena. Since then, with or without applications, the mathematics of difference equations has evolved into a field unto itself. Difference equations with the maximum (or the minimum or the "rank-type") function were rigorously investigated from the mid-1990s into the 2000s, without any applications in mind. These equations often involved arguments varying from reciprocal terms with parameters in the numerators to other special functions.
Recently, the authors of Analysis of a Model for Epilepsy: Application of a Max-Type Difference Equation to Mesial Temporal Lobe Epilepsy and their colleagues investigated the first known application of a "max-type" difference equation. Their equation is a phenomenological model of epileptic seizures. In this book, the authors expand on that research and present a more comprehensive development of mathematical, numerical, and biological results. Additionally, they describe the first documented instance of a novel dynamical behavior that they call rippled almost periodic behavior, which can be described as an unpredictable pseudo-periodic behavior.
Features:
- Suitable for researchers in mathematical neuroscience and potentially as supplementary reading for postgraduate students
- Thoroughly researched and replete with references
Table of Contents:
1. Introduction: Epilepsy. 1.1. Brief Overview. 1.2. Mesial Temporal Lobe Epilepsy and Other Examples. 2. The Model. 2.1. Model Description. 2.2. Connection to a Simpler Model. 2.3. Connection Between the Model and Epileptic Seizures. 2.4. Open Problem: Seizure Threshold as a Function of Time. 3. Eventual Periodicity of the Model. 3.1. Bounded and Persistent Solutions. 3.2. Eventually Periodic Solutions with Periods Multiples of Six. 3.3. Eventually Periodic Solutions with Period 4. 3.4. Partially and Completely Seizure-Free States. 4. Rippled Almost Periodic Solutions. 4.1. Rippled Behaviour. 4.2. Rippled Almost Periodic Solutions. 4.3. Lyapunov Exponent. 4.4. The State of Status Epilepticus. 4.5. On Termination of Repetition. 5. Numerical Results and Biological Conclusions. 5.1. Bifurcation Diagrams. 5.2. Variability in Seizure Characteristics. 5.3. A Case of Variability in Region 1. 5.4. The Hyperexcitable State. 5.5. Impact of Individual Historical Differences. 6. Epilogue
About the Author :
Since age seventeen, Candace M. Kent has had an intense interest in mathematical neuroscience, including the mathematical modeling of psychiatric and neurological disorders, adult neurogenesis, and ethology. In pursuit of this interest, she earned a B.A. in mathematics and molecular biology in 1979. She later enrolled in a graduate program in discrete dynamics at the University of Rhode Island, and earned an M.S. in mathematics in 1993 and a Ph.D. in mathematics in 1998. In between, while searching for the appropriate path to take for an eventual career in mathematical neuroscience, she attended the following universities:
- □Johns Hopkins University Graduate School, Biophysics, 1980-81.
- □ University of Iowa Medical College, M.D. Program, 1981-1984.
- □ Brown University, Applied Mathematics, 1986-88.
Consequently, she began her career late in life{in her early 40's as an Assistant Professor of Mathematics and Applied mathematics at Virginia Commonwealth University from 1998 to 2004. She was appointed Associate Professor from 2004 to 2018. She has been Associate Professor Emerita from 2018 to present, after taking early retirement because of ongoing family illness. During this relatively short time in her career, she has published forty-two peer-reviewed articles, all on a diversity of topics in difference equations and systems of difference equations (and some with applications to biology), including autonomous and nonautonomous reciprocal max-type difference equations. Post medical school, Candace Kent has been fortunate in having the skills to assimilate information in medical journals and books, especially in psychiatry and neurology.
David M. Chan earned a B.S. in Aerospace Engineering from Syracuse University in 1991. He then earned a M.A. in Mathematics at the University of Maine in 1994, and a Ph.D. in Applied Mathematics with a minor in Biomathematics from North Carolina State University in 1999. He is currently an associate professor at Virginia Commonwealth University. Using dynamical systems he models and studies problems that arise in medicine, epidemiology, ecology, and social/support networks.
