Stability and Bifurcation in the Two-Dimensional Stochastic Zeeman Heartbeat Model

Area of Science:

• Mathematical Biology
• Dynamical Systems Theory
• Cardiac Electrophysiology

Background:

• Deterministic models of heartbeat dynamics are insufficient, necessitating stochastic approaches.
• The stochastic Zeeman heartbeat model requires further investigation, particularly regarding its bifurcations.
• Understanding bifurcations in cardiac models can aid in identifying abnormal heart rhythms.

Purpose of the Study:

• To analyze the stability and bifurcation properties of the two-dimensional stochastic Zeeman heartbeat model.
• To convert the classical Zeeman model into a stochastic Ito averaging diffusion system.
• To establish conditions for stochastic stability and different types of bifurcations.

Main Methods:

• Employing Taylor expansion and polar coordinate transformation.
• Utilizing a stochastic averaging procedure to derive an Ito averaging diffusion system.
• Applying theorems on drift and diffusion coefficients to determine stability and bifurcation criteria.

Main Results:

• Sufficient conditions for stochastic stability, D-bifurcation, and phenomenological bifurcation were established.
• The study successfully converted the classical system into an Ito averaging diffusion system.
• Numerical simulations demonstrated the impact of noise severity on the model's behavior.

Conclusions:

• The stochastic Zeeman model exhibits complex dynamics relevant to cardiac function.
• The theoretical findings on stability and bifurcations provide a foundation for further research.
• Numerical results validate the theoretical framework and highlight the role of noise in cardiac rhythm.