دانلود کتاب تجزیه و تحلیل یک مدل برای صرع

In the 1960s and 1970s, mathematical biologists Sir Robert M. May1, E.C. Pielou2, and others popularized the use of difference equations as models of real-life situations. At the time, their models were of population dynamics, involving ecology and epidemiology. 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, with- out any applications in mind. These equations often involved arguments vary- ing from reciprocal terms with parameters in the numerators to other spe- cial functions. Recently, Kent and colleagues3 in 2018 investigated the first known application of a “max-type” difference equation. Their equation is a phenomenological model of epileptic seizures. Here we expand on that research and present a more comprehensive development of mathematical, numerical, and biological results. We employ a nonautonomous (i.e., having periodic coefficient parameters), second-order, reciprocal, max-type difference equation incorporating a thresh- old function, specifically the Heaviside function, to represent the focal (i.e., local) seizures of a particularly common form of epilepsy called mesial, or medial, temporal lobe epilepsy. The Heaviside function, with its threshold ϵ, prevents solutions from becoming unbounded so that they can realistically represent the initiation and termination of focal seizures. We refer to ϵ as the “seizure threshold,” which is central to our mathematical, biological, and numerical analyses. Our model is phenomenological, and thus does not con- cern itself with seizures at the chemical or subcellular level. Instead, it seeks to describe the empirical connections between the seizure threshold and the elicitation or elimination of seizures. Its insights are not given in terms of mea- sured values of variables but instead are descriptions of observed relationships between phenomena that are general enough to pave the way for dialogue and further questions. Also, our model is recursive, and so nicely reflects what we maintain to be the recursive nature of oscillatory neuronal signaling processes. The numerical studies consist of bifurcation diagrams with respect to the parameter ϵ. We observe essentially three kinds of solutions over the spectrum of ϵ values disclosed by the bifurcation diagrams, and are mostly substantiated by mathematical analysis using in unique and intricate ways the tools at the disposal of difference equations such as semicycle analysis. Over the “middle” section of the observed spectrum, solutions are eventually periodic with rela- tively small periods representing the invariant recurrence of seizures. Over the right end of the observed spectrum, solutions are also eventually periodic, usu- ally with period 4, but represent a partially or completely seizure-free state. Over the left end of the spectrum, there exist anomalous solutions that may or may not be eventually periodic. Such anomalies are addressed as akin to but not exactly the same as almost periodic solutions of other types of differ- ence equations. These solutions at the left end of the observed ϵ spectrum also represent the presence of recurrent seizures whose characteristics are highly variable.