Memory in a fractional-order cardiomyocyte model alters properties of alternans and spontaneous activity

Abstract

Cardiac memory is the dependence of electrical activity on the prior history of one or more system state variables, including transmembrane potential (V), ionic current gating, and ion concentrations. While prior work has represented memory either phenomenologically or with biophysical detail, in this study, we consider an intermediate approach of a minimal three-variable cardiomyocyte model, modified with fractional-order dynamics, i.e., a differential equation of order between 0 and 1, to account for history-dependence. Memory is represented via both capacitive memory, due to fractional-order V dynamics, that arises due to non-ideal behavior of membrane capacitance; and ionic current gating memory, due to fractional-order gating variable dynamics, that arises due to gating history-dependence. We perform simulations for varying V and gating variable fractional-orders and pacing cycle length and measure action potential duration (APD) and incidence of alternans, loss of capture, and spontaneous activity. In the absence of ionic current gating memory, we find that capacitive memory, i.e., decreased V fractional-order, typically shortens APD, suppresses alternans, and decreases the minimum cycle length (MCL) for loss of capture. However, in the presence of ionic current gating memory, capacitive memory can prolong APD, promote alternans, and increase MCL. Further, we find that reduced V fractional order (typically less than 0.75) can drive phase 4 depolarizations that promote spontaneous activity. Collectively, our results demonstrate that memory reproduced by a fractional-order model can play a role in alternans formation and pacemaking, and in general, can greatly increase the range of electrophysiological characteristics exhibited by a minimal model.

Key aspects for effective mathematical modelling of fractional-diffusion in cardiac electrophysiology: A quantitative study

“…As shown in Figure 2(a), the reduction of s affects the AP shape, e.g., by increasing the width of the upstroke foot (see related studies [18,38]). Following the approach used by Comlekoglu and Weinberg [21] to assess memory effects in their time-fractional model, through the introduction of a numerical non-local current characterising the difference between standard and fractional diffusion we can better understand the effect of non-locality on the AP morphology. While the effects during tissue activation can be shown to be fairly consistent across space (see E), likely due to the fast timescale on which depolarisation occurs, recovery is affected by the introduction of non-locality in different measure depending on spatial location, thus resulting in significant alterations of APD across the considered spatial domain.…”

“…We follow the approach Comlekoglu and Weinberg [21] used to assess memory effects in their time-fractional electrophysiological model and assume the non-local diffusion term in our models to be the sum of standard diffusion (corresponding to an ideally homogeneous tissue) and a hypothetical non-local current quantifying the effect of tissue heterogeneities on excitation dynamics. Specifically, we assume…”

“…Spacefractional differential operators allow to account for the multi-scale nature of the observed phenomenon, without the need of a detailed resolution of the spatial scale all the way down to the microscopic level [20]. In the context of cardiac electrophysiology, while fractional-order dynamics in time has been recently used to represent capacitive and ionic current gating memory in a single excitable cell model [21], to the best of our knowledge the first contribution considering a spatially nonlocal diffusion term (a fractional power of the classical Laplacian operator in one spatial dimension) was proposed by Bueno-Orovio et al [18]. Bueno-Orovio and colleagues motivate their approach by making a connection between the considered space-fractional operator and the electrical field generated by an applied stimulus in biological tissue characterised by the presence of various inhomogeneities on a variety of length scales.…”

Electrophysiological Mechanisms Underlying T-Wave Alternans and Their Role in Arrhythmogenesis

“…Kalb et al (2004) presented a new method to investigate the rate-and memory-dependent aspects of APD restitution, showing that none of these gradients approached unity for the persistent 2:2 response (alternans), and demonstrating that the gradient of the traditional restitution curve cannot be used to accurately predict the steady-state alternans. In their simulation, the gradient of the APD restitution curve alone is not an accurate parameter to predict alternans, which has been confirmed by mathematical models (Comlekoglu and Weinberg, 2017;Song and Qu, 2020). The effective refractory period (ERP) restitution (which is easier to measure than APD restitution in clinical settings) was also found to be a reliable indicator of alternans in the Courtemanche et al human atrial cell model (Xie et al, 2002) and ventricular myocytes experiments during hypokalemia (Osadchii et al, 2010).…”

“…When the maximum slope is greater than one, a small diastolic interval (DI) change will lead to the APD gradually drifting away from the equilibrium point, resulting in continuous and stable alternation. However, owing to the effect of cardiac memory, some other studies ( Comlekoglu and Weinberg, 2017 ; Song and Qu, 2020 ) have found that the APD restitution theory by itself is insufficient to produce stable alternans but instead involves more complex dynamic processes. Another potential mechanism is related to CaT alternans, which mainly involves the unstable state of calcium content in the sarcoplasmic reticulum (SR; Wang et al, 2014 ; Sun et al, 2018 ; Liu et al, 2019 ) or mitochondrial dysfunction ( Oropeza-Almazan and Blatter, 2020 ).…”