Abstract
We investigate a class of fractional time-partial differential equations describing the dynamics of the fast action potential process in contractile myocytes. The system is explored in both one and two dimensional cases. Homogeneous and nonhomogeneous solutions are derived. We also numerically simulate some of the proposed fractional solutions to provide a different modeling perspective on distinct phases of cardiac membrane potential. Results indicate that the fractional diffusion-wave equation may be employed to model membrane potential dynamics with the fractional order working as an extra asset to modulate electricity conduction, particularly for lower fractional order values.
Highlights
Bearing in mind the importance of better understanding the dynamics of the action potential in cardiac tissues, we investigate a class of fractional time-partial differential equations capable of modeling some stages of this phenomenon
The mathematical model of the 2-D action potential in cardiac tissue with membrane potential Vm proposed in this work is governed by the following fractional partial-time differential equation
In accordance with the exposed definitions, the solution of a fractional differential equation can be obtained from the solution of the corresponding homogeneous equation subject to initial conditions added to a particular solution of Equation (6)
Summary
Advancements in the understanding of heart functioning and related mechanisms have contributed to the progress and development of more appropriate clinical and surgical treatments of heart diseases [1]. The strong relation between fractional calculus and fractals has been long-debated and explored, with regards to whether this fractance refers to space (as in a complex geometry) or time (with heterogeneity and memory effects) [24]. Part of this relation is usually explored through physical and geometrical considerations with the intent of describing and predicting complex phenomena (such as physiology-related ones) [25]. Bearing in mind the importance of better understanding the dynamics of the action potential in cardiac tissues, we investigate a class of fractional time-partial differential equations capable of modeling some stages of this phenomenon.
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