Anisotropic shortening in the wavelength of electrical waves promotes onset of electrical turbulence in cardiac tissue: An 
Nonlinear waves in the form of spirals occur in many excitable media, examples of which include Belousov-Zhabotinsky-type systems , calcium-ion waves in Xenopus oocytes , the aggregation of Dictyostelium discoideum by cyclic-AMP signaling , the oxidation of carbon monoxide on a platinum surface , and, most important of all, cardiac tissue . Understanding the development of such spiral waves and their spatiotemporal evolution is an important challenge in the study of extended dynamical systems, in general, and especially in cardiac tissue, where these waves are associated with abnormal rhythm disorders, which are also called arrhythmias. Cardiac tissue can support many patterns of nonlinear waves of electrical activation, like traveling waves, target waves, and spiral and scroll waves . The occurrence of spiral- and scroll-wave turbulence of electrical activation in cardiac tissue has been implicated in the precipitation of life-threatening cardiac arrhythmias like ventricular tachycardia (VT) and ventricular fibrillation (VF), which destroy the regular rhythm of a mammalian heart and render it incapable of pumping blood. These arrhythmias are the leading cause of death in the industrialized world . Biologically, VF can arise because of many complex mechanisms. Some of these are associated with the development of instability-induced spiral- or scroll-wave turbulence . One such instability-inducing factor is ionic heterogeneity , which arises from variations in the electrophysiological properties of cardiac cells (myocytes), like the morphology and duration of their action-potentials ( AP s) . Such variations may appear in cardiac tissue because of electrical remodeling , induced by alterations in ion-channel expression and activity, which arise, in turn, from diseases  like ischemia , some forms of cardiomyopathy , and the long-QT syndrome . To a certain extent, some heterogeneity is normal in healthy hearts; and it has an underlying physiological purpose ; but, if the degree of heterogeneity is more than is physiologically normal, it can be arrhythmogenic . It is important, therefore, to explore ionic-heterogeneity-induced spiral- or scroll-wave turbulence in mathematical models of cardiac tissue, which allow us to control this heterogeneity precisely, in order to be able to identify the nonlinear-wave instability that leads to such turbulence. We initiate such a study by examining the effects of this type of heterogeneity in three cardiac-tissue models, which are, in order of increasing complexity and biological realism, (a) the two-variable Aliev-Panfilov model , (b) the ionically realistic O’Hara-Rudy (ORd) model  in two dimensions (2D), and (c) the ORd model in an anatomically realistic simulation domain. In each one of these models, we control parameters (see below) in such a way that the ion-channel properties change anisotropically in our simulation domains, thereby inducing an anisotropic spatial variation in the local action potential duration APD . We show that this variation in the APD leads, in all these models, to an anisotropic reduction of the wavelength of the spiral or scroll waves; and this anisotropic reduction of the wavelength paves the way for an instability that precipitates turbulence, the mathematical analog of VF, in these models. 

The Aliev-Panfilov model provides a simplified description of an excitable cardiac cell . It comprises a set of coupled ordinary differential equations (ODEs), for the normalized representations of the transmembrane potential V and the generalized conductance r of the slow, repolarizing current: d V d t = - k V ( V - a ) ( V - 1 ) - V r ; (1) d r d t = [ ϵ + μ 1 r μ 2 + V ] [ - r - k V ( V - b - 1 ) ] ; (2) fast processes are governed by the first term in , whereas, the slow, recovery phase of the AP is determined by the function ϵ + μ 1 r μ 2 + V in . The parameter a represents the threshold of activation and k controls the magnitude of the transmembrane current. We use the standard values for all parameters , except for the parameter k . We write k = g × k o , where g is a multiplication factor and k o is the control value of k . In 2D simulations we introduce a spatial gradient (a linear variation) in the value of k along the vertical direction of the domain. To mimic the electrophysiology of a human ventricular cell, we perform similar studies using a slightly modified version of the ionically-realistic O’Hara-Rudy model (ORd) . Here, the transmembrane potential V is governed by the ODE d V d t = - I ion C m , I i o n = Σ x I x , (3) where I x , the membrane ionic current, for a generic ion channel x , of a cardiac cell, is I x = G x f 1 ( p a c t ) f 2 ( p i n a c t ) ( V m - E x ) , (4) where C m = 1 μ F is the membrane capacitance, f 1 ( p act ) and f 2 ( p inact ) are, respectively, functions of probabilities of activation ( p act ) and inactivation ( p inact ) of the ion channel x , and E x is its Nernst potential. We give a list of all the ionic currents in the ORd model in . We write G i = g × G io , where G io is the original value of the maximal conductance of the ion channel x in the ORd model, and g is a multiplication factor. We model gradients in G i as follows: G i ( y ) = [ g m i n + y ( g m a x - g m i n ) L ] G i o , 0 ≤ y ≤ L ; (5) here, L is the length of the side of the square simulation domain, and g max and g min are, respectively, the maximal and minimal values of g ; we can impose gradients in k in the Aliev-Panfilov model in the same manner. For simplicity, we induce the gradient along one spatial direction only: the vertical axis in 2D; and the apico-basal (apex-to-base) direction in 3D. The spatiotemporal evolution of V in both models is governed by the following reaction-diffusion equation: ∂ V ∂ t + I = ∇ . ( D ∇ V ) , (6) where D is the diffusion tensor, and I = I ion C m and kV ( V − a )( V − 1) + Vr for ORd and Aliev-Panfilov models, respectively. For the numerical implementation of the diffusion term in , we follow Refs. . We construct our anatomically realistic simulation domain with processed human-ventricular data, obtained by using Diffusion Tensor Magnetic Resonance Imaging (DTMRI) . For our 2D isotropic-domain with ORd model, we set D = 0.0012 cm 2 / ms . The temporal and spatial resolutions are set to be δx = 0.02 cm and δt = 0.02 ms, respectively, and all the simulations are performed in a domain with 960 × 960 grid points. For the anatomically-realistic domain, we use a phase-field method for the boundary conditions . The value of diffusion constant along the fiber ( D ∥ ) is set equal to the value of D in the 2D isotropic case (i.e., 0.0012 cm 2 / ms ) and its value perpendicular ( D ⊥ ) to the fiber is 1/4 times D ∥ . The simulation is performed in a cubical domain with 512 3 grid points with the same spatial and temporal resolutions that we use in our 2D simulations. We do not incorporate the intrinsic ionic heterogeneities that are present in real mammalian hearts . In our single-cell simulations, the APD is calculated by measuring the duration over which the cell depolarizes and repolarizes to 90% ( APD 90 ) of its peak transmembrane voltage in the action potential. 

Spiral-wave instability In we show the variation, with the parameter g , of A P D ¯ = A P D / A P D o , where APD o is the control APD value for g = 1. We find that A P D ¯ decreases with increasing g . Changes in the APD at the single-cell level influence electrical-wave dynamics at the tissue level. In particular, such changes affect the rotation frequency ω of reentrant activity (spiral waves). If θ and λ denote, respectively, the conduction velocity and wavelength of a plane electrical wave in tissue, then ω ≃ θ λ , λ ≃ θ × APD . Therefore, if we neglect the effects of curvature  and excitable gap, the spiral-wave frequency ω ≃ 1 A P D . (7) We find, in agreement with this simple, analytical estimate, that ω decreases as the APD increases. We show this in by plotting ω ¯ = ω / ω 0 versus g ; here, ω 0 is the frequency for g = 1. For the parameter a this simple relation between ω and APD is not observed, because change in a affects not only the APD but also other quantities like θ , which has effects on the value of ω . The spiral-wave frequency ω is obtained by simulating a spiral-wave in a homogeneous domain for every value of g . Similarly, in the ionically realistic ORd model, changes in the ion-channel conductances G i alter the APD of the cell and, therefore, the spiral-wave frequency ω . In we present a family of plots to illustrate the variation in A P D ¯ with changes in G i . We find that A P D ¯ decreases with an increase in g for most currents ( I Kr , I Ks , I K 1 , I Na and I NaK ); but it increases for some other currents ( I Ca , I NaCa and I to ). The rate of change of A P D ¯ is most significant when we change G Kr ; by contrast, it is most insensitive to changes in G Na and G to . In we show the variation of ω ¯ with g for different ion channels x . We find that changes in G i , which increase APD , decrease ω and vice versa; this follows from . The sensitivity of ω , with respect to changes in G i , is most for G i = G Kr and least for G i = G to : ω ¯ increases by Δ ω ¯ ≃ 1 . 23 , as g goes from 0.2 to 5; for G to , the same variation in g decreases the value of ω ¯ by Δ ω ¯ ≃ 0 . 04 . We have done many simulations for each G i with different values of Δ ω ¯ to check if a critical value Δ ω ¯ c exists such that above (below) Δ ω ¯ c we see wave breaks (no wave breaks) for all G i s. We find, however, that no such Δ ω ¯ c exists, that is common for all G i s, which is because the stability of the spiral waves depends on the local values of the gradients in APD . We now investigate the effects, on spiral-wave dynamics, of spatial gradients in k , in the 2D Aliev-Panfilov model, and in G i , in the 2D ORd model. A linear gradient in k , in the Aliev-Panfilov model, induces a gradient in ω ¯ (see ); and such a spatial gradient in ω ¯ induces a spiral-wave instability in the low- ω ¯ region. In we demonstrate how a gradient in k ( g max = 1.5 and g min = 0.5) leads to the precipitation of this instability (also see ). Similarly, for each current listed in for the ORd model, we find wave breaks in a medium with a gradient in G i . We illustrate, in , such wave breaks in our 2D simulation domain, with a gradient (∇ G i ) in any G i , for 3 representative currents; we select I Kr , because it has the maximal impact on the single-cell APD , and also on ω in tissue simulations; and we choose I K 1 and I NaCa , because they have moderate and contrary effects on APD and ω . Our results indicate that gradient-induced wave breaks are generic, insofar as they occur in both the simple two-variable (Aliev-Panfilov) and the ionically realistic (ORd) models of cardiac tissue. In , we present power spectra of the time series of V , recorded from a representative point of the simulation domain; these spectra show broad-band backgrounds, which are signatures of chaos, for the gradients ∇ G Kr and ∇ G K 1 ; however, the gradient ∇ G NaCa induces wave breaks while preserving the periodicity of the resultant, reentrant electrical activity, at least at the points from which we have recorded V . The instability in spiral waves occurs because spatial gradients in k (Aliev-Panfilov) or in G i (ORd) induce spatial variations in both A P D ¯ and ω ¯ : In our simulation domain, the local value of ω ¯ ( A P D ¯ ) decreases (increases) from the top to the bottom. In the presence of a single spiral wave (left panel of ), the domain is paced, in effect, at the frequency ω of the spiral, i.e., with a fixed time period T = 1/ ω = APD + DI , where DI is the diastolic interval (the time between the repolarization of one AP and the initiation of the next AP ). Thus, the bottom region, with a long APD , has a short DI and vice versa. The restitution of the conduction velocity θ implies that a small DI leads to a low value of θ and vice versa  (see ). To compensate for this reduction of θ , the spiral wave must reduce its wavelength λ, in the bottom, large- APD (small- DI ) region, so that its rotation frequency ω ≃ θ λ remains unchanged, as shown in (also see ), where the shortening of the spiral arms is indicated by the variation of λ along the spiral arm (λ 2 > λ 1 , in the pseudocolor plot of V m in the top-left panel t = 1.46 s). Clearly, this shortening is anisotropic, because of the uni-directional variation in k or G i ; this anisotropy creates functional heterogeneity in wave propagation, causing a local conduction block, which leads in turn to the spiral-wave instability we have discussed above . The phenomenon of conduction block in a medium with a gradient in ionic properties has been extensively investigated in an earlier study ; here, it is this local conduction block (caused by the anisotropy of the medium) that leads to the break-up of the spiral arms. It should be noted that the stability of the spiral wave depends on the APD difference between the region, where the spiral is initiated, and the top region, where the APD is maxmium; therefore, its stability depends on the location of the spiral-wave initiation along the vertical direction. In the ORd model, we find that gradients in G Kr easily induce instabilities of the spiral for small values of Δ g ≡ g max − g min ≃ 0.5; by contrast, in a medium with gradients in G to , the spiral remains stable for values of Δ g as large as 4.8 (shown in ). This implies that the stability of the spiral depends on the magnitude of the gradient in ω that is induced in the medium. Scroll-wave instability In (also see ), we extend our study to illustrate the onset of scroll-wave instabilities in a 3D, anatomically realistic human-ventricular domain, in the presence of spatial gradients in G Kr . In mammalian hearts, the APD is typically lower in the apical region as compared to that in the basal region . Therefore, we use values of the APD that increase from the apex to the base (and, hence, ω decreases from the apex to base). With g max ( G Kr ) = 6 and Δ g = 4, we observe breakup in a scroll wave that is otherwise stable in the absence of this spatial gradient. We note that the mechanism for the onset of such scroll-wave instabilities is the same as in 2D, and it relies on the gradient-induced anisotropic shortening of the scroll wavelength. For control, we also perform a simulation with small Δ g = 0.1 that does not show scroll-wave instability (see ). 

In we show the variation, with the parameter g , of A P D ¯ = A P D / A P D o , where APD o is the control APD value for g = 1. We find that A P D ¯ decreases with increasing g . Changes in the APD at the single-cell level influence electrical-wave dynamics at the tissue level. In particular, such changes affect the rotation frequency ω of reentrant activity (spiral waves). If θ and λ denote, respectively, the conduction velocity and wavelength of a plane electrical wave in tissue, then ω ≃ θ λ , λ ≃ θ × APD . Therefore, if we neglect the effects of curvature  and excitable gap, the spiral-wave frequency ω ≃ 1 A P D . (7) We find, in agreement with this simple, analytical estimate, that ω decreases as the APD increases. We show this in by plotting ω ¯ = ω / ω 0 versus g ; here, ω 0 is the frequency for g = 1. For the parameter a this simple relation between ω and APD is not observed, because change in a affects not only the APD but also other quantities like θ , which has effects on the value of ω . The spiral-wave frequency ω is obtained by simulating a spiral-wave in a homogeneous domain for every value of g . Similarly, in the ionically realistic ORd model, changes in the ion-channel conductances G i alter the APD of the cell and, therefore, the spiral-wave frequency ω . In we present a family of plots to illustrate the variation in A P D ¯ with changes in G i . We find that A P D ¯ decreases with an increase in g for most currents ( I Kr , I Ks , I K 1 , I Na and I NaK ); but it increases for some other currents ( I Ca , I NaCa and I to ). The rate of change of A P D ¯ is most significant when we change G Kr ; by contrast, it is most insensitive to changes in G Na and G to . In we show the variation of ω ¯ with g for different ion channels x . We find that changes in G i , which increase APD , decrease ω and vice versa; this follows from . The sensitivity of ω , with respect to changes in G i , is most for G i = G Kr and least for G i = G to : ω ¯ increases by Δ ω ¯ ≃ 1 . 23 , as g goes from 0.2 to 5; for G to , the same variation in g decreases the value of ω ¯ by Δ ω ¯ ≃ 0 . 04 . We have done many simulations for each G i with different values of Δ ω ¯ to check if a critical value Δ ω ¯ c exists such that above (below) Δ ω ¯ c we see wave breaks (no wave breaks) for all G i s. We find, however, that no such Δ ω ¯ c exists, that is common for all G i s, which is because the stability of the spiral waves depends on the local values of the gradients in APD . We now investigate the effects, on spiral-wave dynamics, of spatial gradients in k , in the 2D Aliev-Panfilov model, and in G i , in the 2D ORd model. A linear gradient in k , in the Aliev-Panfilov model, induces a gradient in ω ¯ (see ); and such a spatial gradient in ω ¯ induces a spiral-wave instability in the low- ω ¯ region. In we demonstrate how a gradient in k ( g max = 1.5 and g min = 0.5) leads to the precipitation of this instability (also see ). Similarly, for each current listed in for the ORd model, we find wave breaks in a medium with a gradient in G i . We illustrate, in , such wave breaks in our 2D simulation domain, with a gradient (∇ G i ) in any G i , for 3 representative currents; we select I Kr , because it has the maximal impact on the single-cell APD , and also on ω in tissue simulations; and we choose I K 1 and I NaCa , because they have moderate and contrary effects on APD and ω . Our results indicate that gradient-induced wave breaks are generic, insofar as they occur in both the simple two-variable (Aliev-Panfilov) and the ionically realistic (ORd) models of cardiac tissue. In , we present power spectra of the time series of V , recorded from a representative point of the simulation domain; these spectra show broad-band backgrounds, which are signatures of chaos, for the gradients ∇ G Kr and ∇ G K 1 ; however, the gradient ∇ G NaCa induces wave breaks while preserving the periodicity of the resultant, reentrant electrical activity, at least at the points from which we have recorded V . The instability in spiral waves occurs because spatial gradients in k (Aliev-Panfilov) or in G i (ORd) induce spatial variations in both A P D ¯ and ω ¯ : In our simulation domain, the local value of ω ¯ ( A P D ¯ ) decreases (increases) from the top to the bottom. In the presence of a single spiral wave (left panel of ), the domain is paced, in effect, at the frequency ω of the spiral, i.e., with a fixed time period T = 1/ ω = APD + DI , where DI is the diastolic interval (the time between the repolarization of one AP and the initiation of the next AP ). Thus, the bottom region, with a long APD , has a short DI and vice versa. The restitution of the conduction velocity θ implies that a small DI leads to a low value of θ and vice versa  (see ). To compensate for this reduction of θ , the spiral wave must reduce its wavelength λ, in the bottom, large- APD (small- DI ) region, so that its rotation frequency ω ≃ θ λ remains unchanged, as shown in (also see ), where the shortening of the spiral arms is indicated by the variation of λ along the spiral arm (λ 2 > λ 1 , in the pseudocolor plot of V m in the top-left panel t = 1.46 s). Clearly, this shortening is anisotropic, because of the uni-directional variation in k or G i ; this anisotropy creates functional heterogeneity in wave propagation, causing a local conduction block, which leads in turn to the spiral-wave instability we have discussed above . The phenomenon of conduction block in a medium with a gradient in ionic properties has been extensively investigated in an earlier study ; here, it is this local conduction block (caused by the anisotropy of the medium) that leads to the break-up of the spiral arms. It should be noted that the stability of the spiral wave depends on the APD difference between the region, where the spiral is initiated, and the top region, where the APD is maxmium; therefore, its stability depends on the location of the spiral-wave initiation along the vertical direction. In the ORd model, we find that gradients in G Kr easily induce instabilities of the spiral for small values of Δ g ≡ g max − g min ≃ 0.5; by contrast, in a medium with gradients in G to , the spiral remains stable for values of Δ g as large as 4.8 (shown in ). This implies that the stability of the spiral depends on the magnitude of the gradient in ω that is induced in the medium. 

In (also see ), we extend our study to illustrate the onset of scroll-wave instabilities in a 3D, anatomically realistic human-ventricular domain, in the presence of spatial gradients in G Kr . In mammalian hearts, the APD is typically lower in the apical region as compared to that in the basal region . Therefore, we use values of the APD that increase from the apex to the base (and, hence, ω decreases from the apex to base). With g max ( G Kr ) = 6 and Δ g = 4, we observe breakup in a scroll wave that is otherwise stable in the absence of this spatial gradient. We note that the mechanism for the onset of such scroll-wave instabilities is the same as in 2D, and it relies on the gradient-induced anisotropic shortening of the scroll wavelength. For control, we also perform a simulation with small Δ g = 0.1 that does not show scroll-wave instability (see ). 

We have shown that gradients in parameters that affect the APD of the constituent cells induce spatial gradients in the local value of ω . This gradient in the value of ω leads to an anisotropic reduction in the wavelength of the waves, because of the conduction-velocity restitution property of the tissue, and it paves the way for spiral- and scroll-wave instability in the domain. We would like to point out that this instability is not because of the condition of steep APD restitution curves as reported in ref. . We find that the value of slopes of APD restitution curves for all values of g for all G i in are less than one. Therefore, the instability of waves in our study is induced by the anisotropic variation of APD in the medium. This gradient-induced instability is a generic phenomenon because we obtain this instability in the simple Aliev-Panfilov and the detailed ORd model for cardiac tissue. Such an instability should be observable in any excitable medium that has the conduction-velocity-restitution property. We find that the spiral or scroll waves always break up in the low- ω region. This finding is in line with that of the experimental study by Campbell, et al .,  on neonatal-rat-ventricular cell cultures and a computational study by Xie, et al ., , who observe spiral-wave break-up in regions with a large APD . We find that the stability of the spiral is determined by the magnitude of the gradient in ω ; the larger the magnitude of the gradient in the local value of ω , the more likely is the break up of the spiral or scroll wave. By using the ORd model, we find that ω varies most when we change G Kr (as compared to other ion-channel conductances) and, therefore, spiral waves are most unstable in the presence of a gradient of G Kr . By contrast, we find that ω varies most gradually with G to , and hence the spiral wave is most stable in the presence of a gradient in G to (as compared to gradients in other conductances). Earlier studies have investigated the effects of ionic-heterogeneity on spiral-wave dynamics. The existence of regional ionic heterogeneities have been found to initiate spiral waves , attract spiral waves to the heterogeneity , and destabilize spiral waves . The presence of APD gradients in cardiac tissue has been shown to drive spirals towards large- APD (low- ω ) regions  or small- APD regions , called ‘anomalous drift’, by varying model parameters. We have also observed the drift of spiral waves towards the large- APD region (see ) in the initial time before the waves break up. A study by Zimik, et al .,  finds that spatial gradients in ω , induced by gradients in the density of fibroblasts, can precipitate a spiral-wave instability. However, none of these studies provides a clear understanding of the mechanisms underlying the onset of spiral- and scroll-wave instabilities, from a fundamental standpoint. Moreover, none of these studies has carried out a detailed calculation of the pristine effects of each individual major ionic currents, present in a myocyte, on the spiral-wave frequency; nor have they investigated, in a controlled manner, how gradients in ion-channel conductances lead to spiral- or scroll-wave instabilities. Our work makes up for these lacunae and leads to specific predictions that should be tested experimentally. We end our paper by discussing certain limitations of our work. We have shown that large spatial gradients in APD can induce scroll-wave breaks in real hearts via a representative simulation on anatomically realistic heart domain with fiber orientation; however, we have not incorporated other important physiological details of real mammalian hearts, like the intrinsic heterogeneities that exists in them , and the bidomain nature of the tissue . Moreover, in our study we induce heterogeneity in the medium by applying a spatial gradient that extends throughout the domain, but, heterogeneities in real hearts tend to occur in localized regions. However, our results of spiral-wave break at large- APD region should still hold even if the heterogeneites are localized, as has been shown in . 

S1 Video Spiral-wave instability in the Aliev-Panfilov model. Video of pseudocolor plots of transmembrane potential V showing the formation of spiral-wave instability in a medium with gradient in k : g min = 0.5 and g max = 1.5. For the video, we use 10 frames per second with each frame separated from the succeeding frame by 20ms in real time. (AVI) Click here for additional data file. S2 Video Spiral-wave instability in the ORd model. Video pseudocolor plots of transmembrane potential V m showing the formation of spiral-wave instability in a medium with a gradient in G Naca ( g min = 0.2 and g max = 2). For the video, we use 10 frames per second with each frame separated from the succeeding frame by 20ms in real time. (AVI) Click here for additional data file. S3 Video Scroll-wave instability. Video pseudocolor plots of transmembrane potential V m showing the formation of scroll-wave instability in an anatomically realistic model for human ventricles. A linear gradient in G Kr is applied along the apico-basal direction: g min = 2 in the apex and g max = 6 in the base. For the video, we use 10 frames per second with each frame separated from the succeeding frame by 20ms in real time. (AVI) Click here for additional data file. S4 Video Stable scroll-wave. Video pseudocolor plots of transmembrane potential V m showing a stable scroll-wave for small Δg = 0.1 in an anatomically realistic model for human ventricles. A linear gradient in G Kr is applied along the apico-basal direction: g min = 2 in the apex and g max = 2.1 in the base. For the video, we use 10 frames per second with each frame separated from the succeeding frame by 20ms in real time. (AVI) Click here for additional data file. 

