Mathematical models of muscle response to periodic stimuli

This thesis is based on two projects related to muscle physiology and pathology. One is a model of intracellular primary ions, calcium and hydrogen, in a muscle cell as part of a study of muscle fatigue. The other is a model of tension generation from rat muscles in the study of muscle performance as effected by hypertension. The first problem involves muscle fiber activities related to levels of intracellular calcium. During a muscle contraction, several metabolic cycles for different components and ions, such as Ca$\sp{2+}$ H$\sp+,$ and ATP, are triggered. The whole process has Ca$\sp{2+}$ playing a main role, especially as it is the activator for muscle contraction. Several questions involving calcium are examined here through mathematical models. The goal of this modeling work is to describe Ca$\sp{2+}$ release and uptake by SR and its modulation by H$\sp+.$ This enables one to examine the question of how much of the phenomena known as fatigue could be due to the (dynamic) levels of H$\sp+.$ A system of five nonlinear ordinary differential equations (ODE) was presented for this model. The existence and uniqueness theorem and the existence of periodic solutions was proved in this system. The numerical simulations are also able to show qualitative agreement with the experimental data. The second problem involves the effect of hypertension on muscles. The overall goal of this project is to develop and analyze mathematical models of muscle tension to investigate the performance of muscles under the influence of hypertensive conditions. Several models and procedures for parameter estimation are presented. The model will eventually be used to describe the different effects of hypertension on fast and slow twitch fibers. Instead of a theoretical analysis, Modeling techniques and the associated numerical methods in this problem are discussed. This model is built on a set of transcendental functions with up to 12 unknown parameters fit to 120,200 data points