The research presented in this article was conducted by Dr. Beata Jackowska-Zduniak at the Department of Differential Equations and Applications of Mathematics, part of the Faculty of Applied Physics and Mathematics at Gdańsk University of Technology. The department, headed by Prof. Grzegorz Graff, focuses on mathematical methods and their applications to complex dynamical systems. This study explores how cellular automata can help explain electrical conduction in the heart under both normal and ischemic conditions.
The human heart beats continuously throughout our lives. Its rhythmic activity is controlled by electrical impulses generated in the sinoatrial node (SAN) and propagated through the cardiac conduction system.
But how does coordinated electrical activity emerge from thousands of interacting cells? And how can local tissue damage disrupt impulse propagation throughout the system?
In our work, we investigated these questions using a spatially heterogeneous cellular automata model, integrating cellular excitability, electrical coupling, and the structural organisation of cardiac tissue.
Cellular Automata Model
We developed an extended Greenberg–Hastings model consisting of five coupled cellular automata representing the SAN, three conduction pathways, and the atrioventricular node (AVN). The cellular automata are discrete dynamical systems in which the state of each cell evolves according to local transition rules determined by its current state and the states of its neighbours (typically including resting, excited, and refractory states, with additional states introduced depending on the modeled structure).
Unlike the classical formulation, our model incorporates tissue-specific excitation and recovery dynamics, heterogeneous distributions of cardiomyocytes, fibroblasts, and collagen, and different electrical resistances associated with connexins (Cx43, Cx45, and Cx40).
An important feature is cumulative, conductance-dependent excitation. Only cell in rest state accumulates electrical input from its active neighbours over successive time steps. The contribution of each connection depends on its electrical resistance, and excitation occurs only when the accumulated input reaches a prescribed threshold. The electrical conductance of each connection is defined as the inverse of its resistance, providing a physical basis for the weighted interactions between neighbouring cells.
This mechanism connects local cellular properties with the propagation of electrical waves. Importantly, directional conduction within the SAN emerges from tissue heterogeneity and the spatial organisation of electrical connections, without explicitly imposed conduction anisotropy.
Modelling Ischemia
To investigate post-infarction disturbances, we developed a spatiotemporal model of ischemic remodelling. A two-dimensional Gaussian function describes the spatial severity of ischemia, while its temporal progression affects cellular excitability, recovery times, intercellular resistance, connexin degradation, and tissue composition.
We considered two scenarios: gradually progressing ischemia and acute ischemia, characterised by different temporal patterns of tissue damage.
During progressive ischemia, excitation waves could bypass damaged regions through remaining conductive pathways, although their propagation was delayed. As electrical coupling deteriorated and tissue structure changed, local conduction blocks appeared.
Under acute ischemia, the dominant pacemaker region became electrically inactive, while the central part of the SAN spontaneously took over rhythm generation. The model also reproduced backup excitation from the AVN when impulses from the SAN were sufficiently delayed or absent.
A particularly interesting result was wavefront splitting and extinction. One part of the depolarisation wave propagated towards the AVN, while another entered the ischemic region and disappeared at the boundary of non-excitable tissue. This phenomenon resembles wave propagation failure observed in cerebral ischemia, suggesting interesting mathematical similarities between different excitable biological systems.
Our results demonstrate how the interaction of local excitation thresholds, heterogeneous electrical coupling, and evolving tissue structure can reorganise global cardiac conduction. By integrating these mechanisms within a single mathematical framework, the model reproduces physiological impulse propagation and reveals complex conduction disturbances arising under ischemic conditions.
In our future work, we plan to extend the model to investigate potential regenerative therapies and mechanisms of restoring electrical activity in damaged cardiac tissue.
Based on: B. Jackowska-Zduniak, Ischemic mechanisms after myocardial infarction in a cellular model of the cardiac conduction system, The European Physical Journal Special Topics 235 (2026), 8879–8895. doi:10.1140/epjs/s11734-025-02075-z