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In this paper, we study the propagation of the cardiac action potential in a one-dimensional fiber, where cells are electrically coupled through gap junctions (GJs). We consider gap junctional gate dynamics that depend on the intercellular potential. We find that different GJs in the tissue can end up in two different states: a low conducting state and a high conducting state. We first present evidence of the dynamical multistability that occurs by setting specific parameters of the GJ dynamics. Subsequently, we explain how the multistability is a direct consequence of the GJ stability problem by reducing the dynamical system's dimensions. The conductance dispersion usually occurs on a large time scale, i.e., thousands of heartbeats. The full cardiac model simulations are computationally demanding, and we derive a simplified model that allows for a reduction in the computational cost of four orders of magnitude. This simplified model reproduces nearly quantitatively the results provided by the original full model. We explain the discrepancies between the two models due to the simplified model's lack of spatial correlations. This simplified model provides a valuable tool to explore cardiac dynamics over very long time scales. That is highly relevant in studying diseases that develop on a large time scale compared to the basic heartbeat. As in the brain, plasticity and tissue remodeling are crucial parameters in determining the action potential wave propagation's stability. (C) 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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., Author summary Cardiomyocytes, upon voltage excitation, release calcium, which leads to cell contraction. However, under some pathological conditions, calcium handling is impaired. Recently, SERCA gene therapy, whose aim is to improve Ca2+ sequestration by the Sarcoplasmic Reticulum (SR), has failed to improve the prognosis of patients with Heart Failure. This, together with recent counterintuitive results in calcium handling, has highlighted the need for a framework to understand calcium homeostasis across species and pathologies. We show here that the proper framework is a general equilibrium approach of two independent variables. The development of this framework allows us to find a possible mechanism for the failure of SERCA gene therapy even when it manages to increase Ca SERCA expression.

Connexins are specialized ionic channels that control the action potential propagation between cardiac myocytes. In this paper, we study the connexin dynamics in a one-dimensional model of cardiac tissue. We show that the connexin dynamics may lead to a spatial organization of the gap junction conductance. In the numerical simulations presented in this paper we have found two different regimes for the spatial organization of the conductances: (a) a spatially uniform conductance; (b) a spatially complex pattern of local values of high and low conductances. In addition, we have observed that, locally, the two final states are limit cycles with a period equal to the period associated with the external excitation of the tissue strand. The conductance dispersion usually takes place on a very large time scale, i.e. thousands of heart beats, and on a very short spatial scale. Due to its simplicity, the one-dimensional setting allows a detailed study of the emerging structure and in particular very long simulations. We have studied the transition between the two aforementioned states as a function of the gap junction conductance characteristics. Furthermore, we have studied the effect of initially added noises on the outcome of the system. Finally, using spatial autocorrelation functions we have characterized the spatial dispersion in conductance values.

Atrial fibrillation (AF), a cardiac arrhythmia characterized by an abnormal heart rythm originated in the atria, is one of the most prevalent cardiac diseases. Although it may have diverse causes, genetic screening has shown that a percentage of pacients suffering of AF present a genetic variant related to disregulation of calcium-activated potassium (SK) channels. In this paper we review the main characteristics of these channels and use several mathematical models of human atrial cardiomyocytes to study their influence in the form of the atrial action potential. We show that an overexpression of SK channels results in decreased action potential duration and, under some circumstances, it may give rise to alternans, suggesting a pro-arrhythmic role of this current. This effect becomes more important at higher pacing rates. Nevertheless, we also find it to protect against spontaneous calcium release induced afterdepolarizations, acting in this case as an antiarrhythmic factor.

Green streets as a means to reduce urban noise are discussed in this chapter. Standard solutions used in highways or industrial environments, namely high noise barriers or wide space of dense vegetation, are frequently not feasible in urban spaces, and other solutions have to be considered. Hedges and medium-height greenery barriers can be interesting elements, due to their possibility to be placed near the noise source and the receiver. The presence of trees along the streets, and the integration of vegetation in walls, building facades and roofs increase the sound absorption, reducing the level of noise and the consequent annoyance. Furthermore, greenery elements present the advantage of landscape visual quality that physiologically affect acoustic perception, increasing the pleasantness and decreasing the annoyance impression. Findings from latest research referring to the efficacy of these kinds of barriers in main arteries of big cities are presented.

Among the different potassium channels present in cardiac myocytes, the small conductance Ca2+ activated potassium channels (SK channels) are particular because they are affected by changes in intracellular calcium. The functional role of these channels in cardiac electrophysiology is still under intense debate. While they do not seem to play an important role in healthy hearts – their associated current, IKCa, is smaller than other potassium currents -, there is increasing evidence that they may become relevant under pathological conditions. In fact, both pro- and anti-arrhythmic effects have been assigned to these channels, depending on the clinical situation. In this work, we have incorporated the current through SK channels, IKCa, into an electrophysiological ionic model of human atria myocyte. This allows us to evaluate changes in the action potential under different parameters affecting the kinetics of these channels. We observe a large dependence of the action potential duration with the conductance and gate dynamics of the channel. The dependence of SK channels with changes in intracellular calcium dynamics helps decreasing the proarrhythmic effect of spontaneous calcium release events

Electro-mechanical cardiac alternans consists in beat-to-beat changes in the strength of cardiac contraction. Despite its important role 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 calcium level (SR Ca load). 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). In this chapter we review recent results regarding the relative role of SR Ca content fluctuations and SR refractoriness for the appearance of alternans in both ventricular and atrial cells.

Green noise barriers have become an alternative means of reducing urban traffic noise. In this paper, the acoustic performance of a modular greenery noise barrier is evaluated. In situ measurements of noise reflection were performed using an experimental prototype to estimate the sound absorption coefficients. These coefficients were found to have values of approximately 0.7, higher than those previously found in laboratory measurements for a similar system with a lower vegetation density. The obtained values were input into software for predicting environmental noise to analyse the expected performance of such barriers, particularly in the case of a pair of parallel barriers. A comparison with the results for reflective barriers indicates a significant improvement in sound attenuation of up to 4 dBA. The values are similar and even superior to results reported by other authors regarding the effectiveness of absorptive treatments applied to parallel barriers, and furthermore, the proposed barriers offer an aesthetic element for environmental integration.

Constant conductances are often assumed when modeling cardiac tissue. However experimental evidences have shown that gap junctions (GJ) actually connect adjacent cardiac myocytes. These GJ are complex proteins of the connexin family (Cx40; Cx43; Cx45 are the most common in human). These GJ modify the conductances between cardiac cell through a dynamical process. The aim of this study is to develop a bidomain model of the cardiac tissue where the dynamics of the connexins is also included. In particular we will compare the differences associated with the use of a monodomain versus bidomain formulation in inducing intra-cellular conductance variations. We have found that the monodomain formulation gives conductance variations a factor four to five larger with respect to the bidomain formulation.

We perform simulations of both colloidal suspensions and planar arrays systems of superheated superconducting granules. Transitions induced by an increasing external field appear in both situations as an ordering mechanism. Concentrated planar arrays present an interval of external field for which no transitions are produced. These effects can be interesting in applications for superheated superconducting granule detectors.

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.

A diffusion-reaction model for the growth of bacterial colonies is presented. The often observed cooperative behavior developed by bacteria which increases their motility in adverse growth conditions is here introduced as a nonlinear diffusion term. The presence of this mechanism depends on a response which can present hysteresis. By changing only the concentrations of agar and initial nutrient, numerical integration of the proposed model reproduces the different patterns shown by {\it Bacillus subtilis} OG-01., 1 tex file and 6 ps figures. Accepted in Phys.Rev.E

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.

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.

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.

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.

We present results of simulations of transitions in planar arrays of superheated superconducting granules (PASS) immersed in an external magnetic field parallel to the system. Analysis of the behaviour of the system when the external field is slowly increased from zero shows the existence of an interval of external fields for which no transitions are produced. This gap is found to be a consequence of ordering induced by transitions. However, this ordering is different for diluted and for concentrated systems. Avalanches of different sizes with distributions depending on concentration are observed., 19 pages, 12 figures, to appear in Nuclear Instruments and Methods in Physics Research A

The rich variety of forms exhibited by bacterial colonies during their growth has drawn a great deal of attention [1–6]. Prom a theoretical point of view, the mathematical description of such diversity constitutes a stimulating non-equilibrium dynamical problem [2,7–13]. In general, the complexity of the growth pattern increases as the environmental conditions become less favorable. Bacteria respond to adverse growth conditions by developing sophisticated strategies and higher micro-level organization in order to cooperate more efficiently. Examples of these strategies are: the differentiation into longer-motile bacteria, the production of extracellular wetting fluid, the secretion of surfactants which change the surface tension or the chemotactic response to chemical agents produced by bacteria.

A numerical model of the rabbit ventricular cardiomyocyte is developed, that properly reproduces calcium alternans at high pacing frequency and experimental postrest-potentiation. We show that calcium alternans may appear due to the alternation in the number of RyR2 ready to open before each beat and not, despite the fact they are present, to oscillations in the SR Ca load. This mechanism is present when activation rates of RyR2 are low and is also favored by a slowing of the RyR2 recovery time from inactivation. Generally, changes in RyR2 refractoriness lead to alternans via two different mechanisms, (the already established) SR Ca2+ load and (a new one) recovery form inactivation. Drugs or mutations which target RyR2 refractoriness should be arrythmogenic via the activation of one of those mechanisms if they decrease the activation or inactivation rate.

Electromechanical alternans is a beat-to-beat alterna- tion in the strength of contraction of a cardiac cell which appears often due to an instability of calcium cycling. The global calcium signal in cardiomyocytes is the result of the combined effect of several thousand micron scale domains called Calcium Release Units (CaRU), coupled through diffusion, where the flow of calcium among different cell compartments is regulated by stochastic signaling involv- ing the ryanodine receptor (RyR). Recently, numerical sim- ulations have suggested that the transition from regular Ca cycling to alternans is an order-disorder phase transition consistent with the Ising universality class. Inside the cell, groups of CaRU form transient areas within the cell where alternans appear. However, global alternans appears only as a result of the synchronization of the oscillation phase among different subunits. We show here that this transi- tion is indeed robust and universal upon changes in the behavior of the RyR. Using three different set of parame- ters for the transition rates among open, closed and inacti- vated states in the RyR, we show that different RyR behav- ior leads to the same type of order-disorder transition.

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.