Cardiac pacemaker: Fractional equations and frequency analysis

Abstract

Understanding the human body is fundamental due to its importance to living beings. The heart, as a vital organ, holds significant academic relevance. Among its main functions is the generation of electrical stimuli responsible for pumping blood. These stimuli originate from the Sinoatrial (SA) Node, the natural pacemaker, which transmits signals throughout the heart to enable circulation.A model representing pacemaker signals was developed using a relaxation oscillator, specifically the Van der Pol oscillator, which is described by a second-order differential equation. In this study, a more comprehensive investigation is proposed by introducing fractional-order differential equations. Varying the order of the system allows for a more refined analysis of the dynamic properties of the Van der Pol oscillator and, by extension, the cardiac pacemaker it models.The practical implementation of this model is carried out through numerical simulation in the Python programming language. Tools such as the Fast Fourier Transform (FFT) and Continuous Wavelet Transform (CWT) are employed to provide detailed insights into the frequency domain behaviors and time-frequency characteristics. These analyses highlight the nonlinear nature of the response and the effects of introducing a fractional parameter into the system.