Aproximações dos modelos de Hodgkin-Huxley e FitzHugh-Nagumo usando equações diferenciais com atraso

Abstract

To represent different phenomena and systems mathematical models are widely used. Many of them are based on systems of ordinary differential equations (ODEs), that is, they are based on sets of equalities involving dependent variables, their derivatives of first order and the independent variable. In this work, we study the modeling of action potential generation in excitable cells, such as neurons. There are two traditional and pioneering models that stand out in this area: Hodgkin-Huxley and FitzHugh-Nagumo. The objective of this dissertation is to evaluate the possibility of modeling the generation of the action potential via a single differential equation with delay. Differential equations with delay are important because of their capacity to reproduce a great diversity of phenomena. However, its use in modeling the action potential of excitable cells is still incipient. In this dissertation, the method used to achieve this goal was based on the development, initially, of an integral-differential equation that approximates the ODE system. Next, we develop an approximation for integrals that uses terms at both the current instant and the previous instant, i.e., time delayed. Thus, we show that it is possible to approximate each of the ODEs systems of the Hodgkin-Huxley and FitzHugh-Nagumo models by a single differential equation with delay. Finally, these new models are compared with the original ones, and directions are indicated for future works.