Using delay differential equations in models of cardiac electrophysiology

Abstract

In cardiac physiology, electrical alternans is a phenomenon characterized by long-short alternations in the action potential duration of cardiac myocytes that give rise to complex spatiotemporal dynamics in tissue. Experiments and clinical measurements indicate that alternans can be a precursor of life-threatening arrhythmias, such as cardiac _brillation. Despite the importance of alternans in the study of cardiac disease, many mathematical models developed to describe cardiac electrophysiology at the cellular level are not able to produce this phenomenon. As a potential remedy to this de_ciency, we introduce short time-delays in some formulations of existing cardiac cell models that are based on Ordinary Di_erential Equations (ODEs). Many processes within cardiac cells involve delays in sensing and responding to changes. In addition, delay di_erential equations (DDEs) are known to give rise to complex dynamical properties in mathematical models. In biological modeling, DDEs have been applied to epidemiology, population dynamics, immunology, and neural networks. Therefore, DDEs can potentially represent mechanisms that result in complex dynamics both at the cellular level and at the tissue level. In this thesis, we propose DDE-based formulations for ion channel models based on the Hodgkin-Huxley formalism that can induce alternans in single-cell simulations in many models found in the literature. We also show that these modi_cations can destabilize spiral waves and produce spiral breakups in two-dimensional simulations, which is a typical model of cardiac _brillation. However, the new DDE-based formulations introduce new computational challenges due to the need for storing and retrieving past values of variables. Therefore, we present novel numerical methods to overcome these challenges and enable e_cient DDE-based studies at the tissue level in standard computational environments. We _nd that the proposed methods decrease memory usage by up to 95% in cardiac tissue simulations compared to straightforward history management algorithms available in widely used DDE solvers.