Computing the stability of steady-state solutions of mathematical models of the electrical activity in the heart

Instabilities of the resting state in a mathematical model of calcium handling in cardiac myocytes

We analyze a recently published model of calcium handling in cardiac myocytes in order to find conditions for the presence of instabilities in the resting state of the model. Such instabilities can create calcium waves which in turn may be able to initiate cardiac arrhythmias. The model was developed by Swietach, Spitzer and Vaughan-Jones in order to study the effect, on calcium waves, of varying ryanodine receptor (RyR)-permeability, sarco/endoplasmic reticulum calcium ATPase (SERCA) and calcium diffusion. We study the model using the extracellular calcium concentration c(e) and the maximal velocity of the SERCA-pump v(SERCA) as control parameters. In the (c(e),v(SERCA))-domain we derive an explicit function v∗=v∗(c(e)), and we claim that any resting state based on parameters that lie above the curve (i.e. any pair (c(e),v(SERCA)) such that with v(SERCA) > v∗(c(e))) is unstable in the sense that small perturbations will grow and can eventually turn into a calcium wave. And conversely; any pair (c(e),v(SERCA)) below the curve is stable in the sense that small perturbations to the resting state will decay to rest. This claim is supported by analyzing the stability of the system in terms of computing the eigenmodes of the linearized model. Furthermore, the claim is supported by direct simulations based on the non-linear model. Since the curve separating stable from unstable states is given as an explicit function, we can show how stability depends on other parameters of the model.

Unstable eigenmodes are possible drivers for cardiac arrhythmias

The well-organized contraction of each heartbeat is enabled by an electrical wave traversing and exciting the myocardium in a regular manner. Perturbations to this wave, referred to as arrhythmias, can lead to lethal fibrillation if not treated within minutes. One manner in which arrhythmias originate is an ill-fated interaction of the regular electrical signal controlling the heartbeat, the sinus wave, with an ectopic stimulus. It is not fully understood how and when ectopic waves are generated. Based on mathematical models, we show that ectopic beats can be characterized in terms of unstable eigenmodes of the resting state.