A mixture of mobility and meteorological data provides a high correlation with COVID-19 growth in an infection-naive population: a study for Spanish provinces.

Introduction: We use Spanish data from August 2020 to March 2021 as a natural experiment to analyze how a standardized measure of COVID-19 growth correlates with asymmetric meteorological and mobility situations in 48 Spanish provinces. The period of time is selected prior to vaccination so that the level of susceptibility was high, and during geographically asymmetric implementation of non-pharmacological interventions. Methods: We develop reliable aggregated mobility data from different public sources and also compute the average meteorological time series of temperature, dew point, and UV radiance in each Spanish province from satellite data. We perform a dimensionality reduction of the data using principal component analysis and investigate univariate and multivariate correlations of mobility and meteorological data with COVID-19 growth. Results: We find significant, but generally weak, univariate correlations for weekday aggregated mobility in some, but not all, provinces. On the other hand, principal component analysis shows that the different mobility time series can be properly reduced to three time series. A multivariate time-lagged canonical correlation analysis of the COVID-19 growth rate with these three time series reveals a highly significant correlation, with a median R-squared of 0.65. The univariate correlation between meteorological data and COVID-19 growth is generally not significant, but adding its two main principal components to the mobility multivariate analysis increases correlations significantly, reaching correlation coefficients between 0.6 and 0.98 in all provinces with a median R-squared of 0.85. This result is robust to different approaches in the reduction of dimensionality of the data series. Discussion: Our results suggest an important effect of mobility on COVID-19 cases growth rate. This effect is generally not observed for meteorological variables, although in some Spanish provinces it can become relevant. The correlation between mobility and growth rate is maximal at a time delay of 2-3 weeks, which agrees well with the expected 5?10 day delays between infection, development of symptoms, and the detection/report of the case.

Black and gray box learning of amplitude equations: Application to phase field systems.

We present a data-driven approach to learning surrogate models for amplitude equations and illustrate its application to interfacial dynamics of phase field systems. In particular, we demonstrate learning effective partial differential equations describing the evolution of phase field interfaces from full phase field data. We illustrate this on a model phase field system, where analytical approximate equations for the dynamics of the phase field interface (a higher-order eikonal equation and its approximation, the Kardar-Parisi-Zhang equation) are known. For this system, we discuss data-driven approaches for the identification of equations that accurately describe the front interface dynamics. When the analytical approximate models mentioned above become inaccurate, as we move beyond the region of validity of the underlying assumptions, the data-driven equations outperform them. In these regimes, going beyond black box identification, we explore different approaches to learning data-driven corrections to the analytically approximate models, leading to effective gray box partial differential equations.

Spatial Distribution of Calcium Sparks Determines Their Ability to Induce Afterdepolarizations in Human Atrial Myocytes.

Analysis of the spatio-temporal distribution of calcium sparks showed a preferential increase in sparks near the sarcolemma in atrial myocytes from patients with atrial fibrillation (AF), linked to higher ryanodine receptor (RyR2) phosphorylation at s2808 and lower calsequestrin-2 levels. Mathematical modeling, incorporating modulation of RyR2 gating, showed that only the observed combinations of RyR2 phosphorylation and calsequestrin-2 levels can account for the spatio-temporal distribution of sparks in patients with and without AF. Furthermore, we demonstrate that preferential calcium release near the sarcolemma is key to a higher incidence and amplitude of afterdepolarizations in atrial myocytes from patients with AF.

Multi-Scale Computational Modeling of Spatial Calcium Handling From Nanodomain to Whole-Heart: Overview and Perspectives.

Regulation of intracellular calcium is a critical component of cardiac electrophysiology and excitation-contraction coupling. The calcium spark, the fundamental element of the intracellular calcium transient, is initiated in specialized nanodomains which co-locate the ryanodine receptors and L-type calcium channels. However, calcium homeostasis is ultimately regulated at the cellular scale, by the interaction of spatially separated but diffusively coupled nanodomains with other sub-cellular and surface-membrane calcium transport channels with strong non-linear interactions; and cardiac electrophysiology and arrhythmia mechanisms are ultimately tissue-scale phenomena, regulated by the interaction of a heterogeneous population of coupled myocytes. Recent advances in imaging modalities and image-analysis are enabling the super-resolution reconstruction of the structures responsible for regulating calcium homeostasis, including the internal structure of nanodomains themselves. Extrapolating functional and imaging data from the nanodomain to the whole-heart is non-trivial, yet essential for translational insight into disease mechanisms. Computational modeling has important roles to play in relating structural and functional data at the sub-cellular scale and translating data across the scales. This review covers recent methodological advances that enable image-based modeling of the single nanodomain and whole cardiomyocyte, as well as the development of multi-scale simulation approaches to integrate data from nanometer to whole-heart. Firstly, methods to overcome the computational challenges of simulating spatial calcium dynamics in the nanodomain are discussed, including image-based modeling at this scale. Then, recent whole-cell models, capable of capturing a range of different structures (such as the T-system and mitochondria) and cellular heterogeneity/variability are discussed at two different levels of discretization. Novel methods to integrate the models and data across the scales and simulate stochastic dynamics in tissue-scale models are then discussed, enabling elucidation of the mechanisms by which nanodomain remodeling underlies arrhythmia and contractile dysfunction. Perspectives on model differences and future directions are provided throughout.

Ca2+-CaM Dependent Inactivation of RyR2 Underlies Ca2+ Alternans in Intact Heart.

RATIONALE: Ca2+ alternans plays an essential role in cardiac alternans that can lead to ventricular fibrillation, but the mechanism underlying Ca2+ alternans remains undefined. Increasing evidence suggests that Ca2+ alternans results from alternations in the inactivation of cardiac RyR2 (ryanodine receptor 2). However, what inactivates RyR2 and how RyR2 inactivation leads to Ca2+ alternans are unknown. OBJECTIVE: To determine the role of CaM (calmodulin) on Ca2+ alternans in intact working mouse hearts. METHODS AND RESULTS: We used an in vivo local gene delivery approach to alter CaM function by directly injecting adenoviruses expressing CaM-wild type, a loss-of-function CaM mutation, CaM (1-4), and a gain-of-function mutation, CaM-M37Q, into the anterior wall of the left ventricle of RyR2 wild type or mutant mouse hearts. We monitored Ca2+ transients in ventricular myocytes near the adenovirus-injection sites in Langendorff-perfused intact working hearts using confocal Ca2+ imaging. We found that CaM-wild type and CaM-M37Q promoted Ca2+ alternans and prolonged Ca2+ transient recovery in intact RyR2 wild type and mutant hearts, whereas CaM (1-4) exerted opposite effects. Altered CaM function also affected the recovery from inactivation of the L-type Ca2+ current but had no significant impact on sarcoplasmic reticulum Ca2+ content. Furthermore, we developed a novel numerical myocyte model of Ca2+ alternans that incorporates Ca2+-CaM-dependent regulation of RyR2 and the L-type Ca2+ channel. Remarkably, the new model recapitulates the impact on Ca2+ alternans of altered CaM and RyR2 functions under 9 different experimental conditions. Our simulations reveal that diastolic cytosolic Ca2+ elevation as a result of rapid pacing triggers Ca2+-CaM dependent inactivation of RyR2. The resultant RyR2 inactivation diminishes sarcoplasmic reticulum Ca2+ release, which, in turn, reduces diastolic cytosolic Ca2+, leading to alternations in diastolic cytosolic Ca2+, RyR2 inactivation, and sarcoplasmic reticulum Ca2+ release (ie, Ca2+ alternans). CONCLUSIONS: Our results demonstrate that inactivation of RyR2 by Ca2+-CaM is a major determinant of Ca2+ alternans, making Ca2+-CaM dependent regulation of RyR2 an important therapeutic target for cardiac alternans.

Cardiac computational modelling.

Cardiovascular diseases currently have a major social and economic impact, constituting one of the leading causes of mortality and morbidity. Personalized computational models of the heart are demonstrating their usefulness both to help understand the mechanisms underlying cardiac disease, and to optimize their treatment and predict the patient's response. Within this framework, the Spanish Research Network for Cardiac Computational Modelling (VHeart-SN) has been launched. The general objective of the VHeart-SN network is the development of an integrated, modular and multiscale multiphysical computational model of the heart. This general objective is addressed through the following specific objectives: a) to integrate the different numerical methods and models taking into account the specificity of patients; b) to assist in advancing knowledge of the mechanisms associated with cardiac and vascular diseases; and c) to support the application of different personalized therapies. This article presents the current state of cardiac computational modelling and different scientific works conducted by the members of the network to gain greater understanding of the characteristics and usefulness of these models.

Buffering and total calcium levels determine the presence of oscillatory regimes in cardiac cells.

Calcium oscillations and waves induce depolarization in cardiac cells which are believed to cause life-threathening arrhythimas. In this work, we study the conditions for the appearance of calcium oscillations in both a detailed subcellular model of calcium dynamics and a minimal model that takes into account just the minimal ingredients of the calcium toolkit. To avoid the effects of homeostatic changes and the interaction with the action potential we consider the somewhat artificial condition of a cell without pacing and with no calcium exchange with the extracellular medium. Both the full subcellular model and the minimal model present the same scenarios depending on the calcium load: two stationary states, one with closed ryanodine receptors (RyR) and most calcium in the cell stored in the sarcoplasmic reticulum (SR), and another, with open RyRs and a depleted SR. In between, calcium oscillations may appear. The robustness of these oscillations is determined by the amount of calsequestrin (CSQ). The lack of this buffer in the SR enhances the appearance of oscillations. The minimal model allows us to relate the stability of the oscillating state to the nullcline structure of the system, and find that its range of existence is bounded by a homoclinic and a Hopf bifurcation, resulting in a sudden transition to the oscillatory regime as the cell calcium load is increased. Adding a small amount of noise to the RyR behavior increases the parameter region where oscillations appear and provides a gradual transition from the resting state to the oscillatory regime, as observed in the subcellular model and experimentally.

Two-variable nullcline analysis of ionic general equilibrium predicts calcium homeostasis in ventricular myocytes.

Ventricular contraction is roughly proportional to the amount of calcium released from the Sarcoplasmic Reticulum (SR) during systole. While it is rather straightforward to measure calcium levels and contractibility under different physiological conditions, the complexity of calcium handling during systole and diastole has made the prediction of its release at steady state impossible. Here we approach the problem analyzing the evolution of intracellular and extracellular calcium fluxes during a single beat which is away from homeostatic balance. Using an in-silico subcellular model of rabbit ventricular myocyte, we show that the high dimensional nonlinear problem of finding the steady state can be reduced to a two-variable general equilibrium condition where pre-systolic calcium level in the cytosol and in the SR must fulfill simultaneously two different equalities. This renders calcium homeostasis as a problem that can be studied in terms of its equilibrium structure, leading to precise predictions of steady state from single-beat measurements. We show how changes in ion channels modify the general equilibrium, as shocks would do in general equilibrium macroeconomic models. This allows us to predict when an enhanced entrance of calcium in the cell reduces its contractibility and explain why SERCA gene therapy, a change in calcium handling to treat heart failure, might fail to improve contraction even when it successfully increases SERCA expression.

Influence of the tubular network on the characteristics of calcium transients in cardiac myocytes.

Transverse and axial tubules (TATS) are an essential ingredient of the excitation-contraction machinery that allow the effective coupling of L-type Calcium Channels (LCC) and ryanodine receptors (RyR2). They form a regular network in ventricular cells, while their presence in atrial myocytes is variable regionally and among animal species We have studied the effect of variations in the TAT network using a bidomain computational model of an atrial myocyte with variable density of tubules. At each z-line the t-tubule length is obtained from an exponential distribution, with a given mean penetration length. This gives rise to a distribution of t-tubules in the cell that is characterized by the fractional area (F.A.) occupied by the t-tubules. To obtain consistent results, we average over different realizations of the same mean penetration length. To this, in some simulations we add the effect of a network of axial tubules. Then we study global properties of calcium signaling, as well as regional heterogeneities and local properties of sparks and RyR2 openings. In agreement with recent experiments in detubulated ventricular and atrial cells, we find that detubulation reduces the calcium transient and synchronization in release. However, it does not affect sarcoplasmic reticulum (SR) load, so the decrease in SR calcium release is due to regional differences in Ca2+ release, that is restricted to the cell periphery in detubulated cells. Despite the decrease in release, the release gain is larger in detubulated cells, due to recruitment of orphaned RyR2s, i.e, those that are not confronting a cluster of LCCs. This probably provides a safeguard mechanism, allowing physiological values to be maintained upon small changes in the t-tubule density. Finally, we do not find any relevant change in spark properties between tubulated and detubulated cells, suggesting that the differences found in experiments could be due to differential properties of the RyR2s in the membrane and in the t-tubules, not incorporated in the present model. This work will help understand the effect of detubulation, that has been shown to occur in disease conditions such as heart failure (HF) in ventricular cells, or atrial fibrillation (AF) in atrial cells.

Computational Model of Calcium Signaling in Cardiac Atrial Cells at the Submicron Scale.

In cardiac cells, calcium is the mediator of excitation-contraction coupling. Dysfunctions in calcium handling have been identified as the origin of some cardiac arrhythmias. In the particular case of atrial myocytes, recent available experimental data has found links between these dysfunctions and structural changes in the calcium handling machinery (ryanodine cluster size and distribution, t-tubular network, etc). To address this issue, we have developed a computational model of an atrial myocyte that takes into account the detailed intracellular structure. The homogenized macroscopic behavior is described with a two-concentration field model, using effective diffusion coefficients of calcium in the sarcoplasmic reticulum (SR) and in the cytoplasm. The model reproduces the right calcium transients and dependence with pacing frequency. Under basal conditions, the calcium rise is mostly restricted to the periphery of the cell, with a large concentration ratio between the periphery and the interior. We have then studied the dependence of the speed of the calcium wave on cytosolic and SR diffusion coefficients, finding an almost linear relation with the former, in agreement with a diffusive and fire mechanism of propagation, and little dependence on the latter. Finally, we have studied the effect of a change in RyR cluster microstructure. We find that, under resting conditions, the spark frequency decreases slightly with RyR cluster spatial dispersion, but markedly increases when the RyRs are distributed in clusters of larger size, stressing the importance of RyR cluster organization to understand atrial arrhythmias, as recent experimental results suggest (Macquaide et al., 2015).

Minimal model for calcium alternans due to SR release refractoriness.

In the heart, rapid pacing rates may induce alternations in the strength of cardiac contraction, termed pulsus alternans. Often, this is due to an instability in the dynamics of the intracellular calcium concentration, whose transients become larger and smaller at consecutive beats. This alternation has been linked experimentally and theoretically to two different mechanisms: an instability due to (1) a strong dependence of calcium release on sarcoplasmic reticulum (SR) load, together with a slow calcium reuptake into the SR or (2) to SR release refractoriness, due to a slow recovery of the ryanodine receptors (RyR2) from inactivation. The relationship between calcium alternans and refractoriness of the RyR2 has been more elusive than the corresponding SR Ca load mechanism. To study the former, we reduce a general calcium model, which mimics the deterministic evolution of a calcium release unit, to its most basic elements. We show that calcium alternans can be understood using a simple nonlinear equation for calcium concentration at the dyadic space, coupled to a relaxation equation for the number of recovered RyR2s. Depending on the number of RyR2s that are recovered at the beginning of a stimulation, the increase in calcium concentration may pass, or not, over an excitability threshold that limits the occurrence of a large calcium transient. When the recovery of the RyR2 is slow, this produces naturally a period doubling bifurcation, resulting in calcium alternans. We then study the effects of inactivation, calcium diffusion, and release conductance for the onset of alternans. We find that the development of alternans requires a well-defined value of diffusion while it is less sensitive to the values of inactivation or release conductance.

Nonlinear physics of electrical wave propagation in the heart: a review.

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.

Calcium alternans is due to an order-disorder phase transition in cardiac cells.

Electromechanical alternans is a beat-to-beat alternation in the strength of contraction of a cardiac cell, which can be caused by an instability of calcium cycling. Using a distributed model of subcellular calcium we show that alternans occurs via an order-disorder phase transition which exhibits critical slowing down and a diverging correlation length. We apply finite size scaling along with a mapping to a stochastic coupled map model, to show that this transition in two dimensions is characterized by critical exponents consistent with the Ising universality class. These findings highlight the important role of cooperativity in biological cells, and suggest novel approaches to investigate the onset of the alternans instability in the heart.

Are SR Ca content fluctuations or SR refractoriness the key to atrial cardiac alternans?: insights from a human atrial model.

Despite the important role of electromechanical alternans in cardiac arrhythmogenesis, its molecular origin is not well understood. The appearance of calcium alternans has often been associated to fluctuations in the sarcoplasmic reticulum (SR) Ca loading. However, cytosolic calcium alternans observed without concurrent oscillations in the SR Ca content suggests an alternative mechanism related to a dysfunction in the dynamics of the ryanodine receptor (RyR2). We have investigated the effect of SR release refractoriness in the appearance of alternans, using a mathematical model of a single human atrial cell, based on the model by Nygren et al. (30), where we modified the dynamics of the RyR2 and of SR Ca release. The genesis of calcium alternans was studied stimulating the cell for different periods and values of the RyR2 recovery time from inactivation. At fast rates cytosolic calcium alternans were obtained without concurrent SR Ca content fluctuations. A transition from regular response to alternans was also observed, changing the recovery time from inactivation of the RyR2. This transition was found to be hysteretic, so for a given set of parameters different responses were observed. We then studied the relevance of RyR2 refractoriness for the generation of alternans, reproducing the same protocols as in recent experiments. In particular, restitution of Ca release during alternans was studied with a S1S2 protocol, obtaining a different response if the S2 stimulation was given after a long or a short release. We show that the experimental results can be explained by RyR2 refractoriness, arising from a slow RyR2 recovery from inactivation, stressing the role of the RyR2 in the genesis of alternans.

Dependency of calcium alternans on ryanodine receptor refractoriness.

BACKGROUND: Rapid pacing rates induce alternations in the cytosolic calcium concentration caused by fluctuations in calcium released from the sarcoplasmic reticulum (SR). However, the relationship between calcium alternans and refractoriness of the SR calcium release channel (RyR2) remains elusive. METHODOLOGY/PRINCIPAL FINDINGS: To investigate how ryanodine receptor (RyR2) refractoriness modulates calcium handling on a beat-to-beat basis using a numerical rabbit cardiomyocyte model. We used a mathematical rabbit cardiomyocyte model to study the beat-to-beat calcium response as a function of RyR2 activation and inactivation. Bi-dimensional maps were constructed depicting the beat-to-beat response. When alternans was observed, a novel numerical clamping protocol was used to determine whether alternans was caused by oscillations in SR calcium loading or by RyR2 refractoriness. Using this protocol, we identified regions of RyR2 gating parameters where SR calcium loading or RyR2 refractoriness underlie the induction of calcium alternans, and we found that at the onset of alternans both mechanisms contribute. At low inactivation rates of the RyR2, calcium alternans was caused by alternation in SR calcium loading, while at low activation rates it was caused by alternation in the level of available RyR2s. CONCLUSIONS/SIGNIFICANCE: We have mapped cardiomyocyte beat-to-beat responses as a function of RyR2 activation and inactivation, identifying domains where SR calcium load or RyR2 refractoriness underlie the induction of calcium alternans. A corollary of this work is that RyR2 refractoriness due to slow recovery from inactivation can be the cause of calcium alternans even when alternation in SR calcium load is present.

Cardiac dynamics: a simplified model for action potential propagation.

This paper analyzes a new semiphysiological ionic model, used recently to study reexitations and reentry in cardiac tissue [I.R. Cantalapiedra et al, PRE 82 011907 (2010)]. The aim of the model is to reproduce action potencial morphologies and restitution curves obtained, either from experimental data, or from more complex electrophysiological models. The model divides all ion currents into four groups according to their function, thus resulting into fast-slow and inward-outward currents. We show that this simplified model is flexible enough as to accurately capture the electrical properties of cardiac myocytes, having the advantage of being less computational demanding than detailed electrophysiological models. Under some conditions, it has been shown to be amenable to mathematical analysis. The model reproduces the action potential (AP) change with stimulation rate observed both experimentally and in realistic models of healthy human and guinea pig myocytes (TNNP and LRd models, respectively). When simulated in a cable it also gives the right dependence of the conduction velocity (CV) with stimulation rate. Besides reproducing correctly these restitution properties, it also gives a good fit for the morphology of the AP, including the notch typical of phase 1. Finally, we perform simulations in a realistic geometric model of the rabbit's ventricles, finding a good qualitative agreement in AP propagation and the ECG. Thus, this simplified model represents an alternative to more complex models when studying instabilities in wave propagation.

Supernormal conduction in cardiac tissue promotes concordant alternans and action potential bunching.

Supernormal conduction (SNC) in excitable cardiac tissue refers to an increase of pulse (or action potential) velocity with decreasing distance to the preceding pulse. Here we employ a simple ionic model to study the effect of SNC on the propagation of action potentials (APs) and the phenomenology of alternans in excitable cardiac tissue. We use bifurcation analysis and simulations to study attraction between propagating APs caused by SNC that leads to AP pairs and bunching. It is shown that SNC stabilizes concordant alternans in arbitrarily long paced one-dimensional cables. As a consequence, spiral waves in two-dimensional tissue simulations exhibit straight nodal lines for SNC in contrast to spiraling ones in the case of normal conduction.

Phase-2 reentry in cardiac tissue: role of the slow calcium pulse.

Phase-2 re-entry is thought to underlie many causes of idiopathic ventricular arrhythmias as, for instance, those occurring in Brugada syndrome. In this paper, we study under which circumstances a region of depolarized tissue can re-excite adjacent regions that exhibit shorter action potential duration (APD), eventually inducing reentry. For this purpose, we use a simplified ionic model that reproduces well the ventricular action potential. With the help of this model, we analyze the conditions that lead to very short action potentials (APs), as well as possible mechanisms for re-excitation in a cable. We then study the induction of re-entrant waves (spiral waves) in simulations of AP propagation in the heart ventricles. We show that re-excitation takes place via a slow pulse produced by calcium current that propagates into the region of short APs until it encounters excitable tissue. We calculate analytically the speed of the slow pulse, and also give an estimate of the minimal tissue size necessary for allowing reexcitation to take place.

Onset of sidebranching in directional solidification.

We use a computationally efficient phase-field formulation [B. Echebarria, Phys. Rev. E 70, 061604 (2004)] to investigate the origin and dynamics of sidebranching in directional solidification for realistic parameters of a dilute alloy previously studied experimentally [M. Gorgelin and A. Pocheau, Phys. Rev. E 57, 3189 (1998)]. Sidebranching is found to result either from noise amplification or from deterministic oscillations that exist both in two dimensions and in a three-dimensional thin-sample geometry. The oscillatory branch of growth solutions bifurcates subcritically from the main steady-state branch of solutions and exists over a finite range of large array spacings. In contrast, noise-induced sidebranching is associated with a smooth transition where the sidebranching amplitude increases exponentially with spacing up to nonlinear saturation due to the overlap of diffusion fields from neighboring cells, as observed experimentally. In the latter case where sidebranching is noise-induced, we find that increasing the externally imposed thermal gradient reduces the onset velocity and wavelength of sidebranching, as also observed experimentally. We show that this counterintuitive effect is due to tip blunting with increasing thermal gradient that promotes noise amplification in the tip region.

Reexcitation mechanisms in epicardial tissue: role of I(to) density heterogeneities and I(Na) inactivation kinetics.

Dispersion of action potential repolarization is known to be an important arrhythmogenic factor in cardiopathies such as Brugada syndrome. In this work, we analyze the effect of a variation in sodium current (I(Na)) inactivation and a heterogeneous rise of transient outward current (I(to)) in the probability of reentry in epicardial tissue. We use the Luo-Rudy model of epicardial ventricular action potential to study wave propagation in a one-dimensional fiber. Spatial dispersion in repolarization is introduced by splitting the fiber into zones with different strength of I(to). We then analyze the pro-arrhythmic effect of a variation in the relaxation time and steady-state of the sodium channel fast inactivating gate h. We quantify the probability of reentry measuring the percentage of reexcitations that occurs in 200 beats. We find that, for high stimulation rates, this percentage is negligible, but increases notably for pacing periods above 700ms. Surprisingly, with decreasing I(Na) inactivation time, the percentage of reexcitations does not grow monotonically, but presents vulnerable windows, separated by values of the I(Na) inactivation speed-up where reexcitation does not occur. By increasing the strength of L-type calcium current I(CaL) above a certain threshold, reexcitation disappears. Finally, we show the formation of reentry in stimulated two-dimensional epicardial tissue with modified I(Na) kinetics and I(to) heterogeneity. Thus, we confirm that while I(to) dispersion is necessary for phase-2 reentry, altered sodium inactivation kinetics influences the probability of reexcitation in a highly nonlinear fashion.