^{1}

^{2}

^{1}

^{1}

^{3}

^{*}

Conceived and designed the experiments: STK RP. Performed the experiments: STK ARN RP. Analyzed the data: STK ARN RP. Contributed reagents/materials/analysis tools: STK ARN RP. Wrote the paper: STK ARN RP.

The authors have declared that no competing interests exist.

Regular electrical activation waves in cardiac tissue lead to the rhythmic contraction and expansion of the heart that ensures blood supply to the whole body. Irregularities in the propagation of these activation waves can result in cardiac arrhythmias, like ventricular tachycardia (VT) and ventricular fibrillation (VF), which are major causes of death in the industrialised world. Indeed there is growing consensus that spiral or scroll waves of electrical activation in cardiac tissue are associated with VT, whereas, when these waves break to yield spiral- or scroll-wave turbulence, VT develops into life-threatening VF: in the absence of medical intervention, this makes the heart incapable of pumping blood and a patient dies in roughly two-and-a-half minutes after the initiation of VF. Thus studies of spiral- and scroll-wave dynamics in cardiac tissue pose important challenges for in vivo and in vitro experimental studies and for in silico numerical studies of mathematical models for cardiac tissue. A major goal here is to develop low-amplitude defibrillation schemes for the elimination of VT and VF, especially in the presence of inhomogeneities that occur commonly in cardiac tissue. We present a detailed and systematic study of spiral- and scroll-wave turbulence and spatiotemporal chaos in four mathematical models for cardiac tissue, namely, the Panfilov, Luo-Rudy phase 1 (LRI), reduced Priebe-Beuckelmann (RPB) models, and the model of ten Tusscher, Noble, Noble, and Panfilov (TNNP). In particular, we use extensive numerical simulations to elucidate the interaction of spiral and scroll waves in these models with conduction and ionic inhomogeneities; we also examine the suppression of spiral- and scroll-wave turbulence by low-amplitude control pulses. Our central qualitative result is that, in all these models, the dynamics of such spiral waves depends very sensitively on such inhomogeneities. We also study two types of control schemes that have been suggested for the control of spiral turbulence, via low amplitude current pulses, in such mathematical models for cardiac tissue; our investigations here are designed to examine the efficacy of such control schemes in the presence of inhomogeneities. We find that a local pulsing scheme does not suppress spiral turbulence in the presence of inhomogeneities; but a scheme that uses control pulses on a spatially extended mesh is more successful in the elimination of spiral turbulence. We discuss the theoretical and experimental implications of our study that have a direct bearing on defibrillation, the control of life-threatening cardiac arrhythmias such as ventricular fibrillation.

Cardiac arrhythmias like ventricular tachycardia (VT) and ventricular fibrillation (VF) are a major cause of death in industrialised countries. Experimental studies over the past decade or so have suggested that VT is associated with an unbroken spiral wave of electrical activation on cardiac tissue but VF with broken spiral waves

The Panfilov model

Our goal is to carry out a comparison of spiral-wave dynamics in these four models of cardiac tissue especially in the presence of different types of inhomogeneities, such as conduction and ionic inhomogeneities; a comparison of such spiral waves in the Priebe Beuckelmann (PB), RPB, TNNP, models and the model of Iyer

Conduction inhomogeneities in cardiac tissue can affect spiral waves in several ways. Experimental studies

ten Tusscher

A detailed numerical and analytical study of the interaction of excitation waves with a piecewise linear obstacle has been carried out in Ref.

Ionic inhomogeneities, formed say by local modifications of the maximal conductance of calcium ion channels, affect the action potential of a cardiac cell; in particular, the action potential duration (APD) and other time scales, such as the extent of the plateau region and the refractory period _{si}) as illustrated by the numerical study of Qu _{si} they first observed a rigidly rotating spiral wave, then one in which the spiral tip meandered quasiperiodically, and eventually chaotically; finally they obtained spiral turbulence with broken spiral waves. Furthermore, the numerical studies of Refs.

We consider spiral-wave dynamics in an otherwise homogeneous medium with a square region in which the conduction or ionic parameters are different from their values in the rest of the simulation domain. We find that such inhomogeneities can have dramatic effects on spiral wave dynamics. We have reported earlier that conduction inhomogeneities can act as anchoring sites for spirals, or lead to the complete elimination of spiral waves, or have no effect on spiral-wave break up; which one of these results is obtained depends on the size and position of the conduction inhomogeneity

This paper is organised as follows: In the Section on “

The Panfilov model _{1}V, for V<V_{1}, f(V) = −C_{2}V+a, for V_{1}≤V≤V_{2}, and f(V) = C_{3}(V−1), for V>V_{2}. The physically appropriate parameters are _{1} = 0.0026, V_{2} = 0.837, C_{1} = 20, C_{2} = 3, C_{3} = 15, a = 0.06 and k = 3. The time scales of the recovery variable are determined by the function ϵ(V,g) that is ϵ_{1} for V<V_{2}, ϵ_{2} for V>V_{2}, and ϵ_{3} for V<V_{1} and g<g_{1}; g_{1} = 1.8, ϵ_{1} = 0.01, ϵ_{2} = 1.0, and ϵ_{3} = 0.3; here we deal with dimensionless quantities. To obtain dimensioned quantities ^{2}/s. In some of our studies we vary ϵ_{1} to mimic the effects of ionic inhomogeneties. Voltages in this model are scaled such that the resting potential is zero.

Even though the Panfilov model is very simple when compared to the LRI, RPB, and TNNP models, which retain details of ion channels, it captures several essential properties of the spatiotemporal evolution of V

The LRI model uses the Hodgkin-Huxley formalism for ion channels in a given cell. It accounts for 6 ionic currents (e.g., Na^{+}, K^{+}, and Ca^{2+}) that flow through voltage-gated ion channels; 9 gate variables regulate the transport of ions across the cell membrane

We also study the Reduced-Priebe-Beuckelmann (RPB) model

The most realistic model we study is the one introduced recently ^{+} current I_{Na}, the L-type Ca^{2+} current I_{CaL}, the transient outward potassium current I_{to}, the slow, potassium, delayed, rectifier current I_{Ks}, the rapid, potassium, delayed, rectifier current I_{Kr}, and the inward, rectifier K^{+} current I_{K1}. These and other currents and the details of the dynamics of calcium ions are given in Section 2 of the

Since this model has many variables, numerical simulations of spiral-wave dynamics in it are considerably harder than in the simpler LRI and RPB models. In both the RPB and TNNP models, we can study human epicardial, endocardial, and M cells by a suitable choice of parameters; for the RPB model we use the parameters for M cells and for the TNNP model we use parameters for epicardial cells (the equilibrium voltage across the cardiac cell in the latter is −86.2 mV).

To integrate the Panfilov model PDEs in d spatial dimensions we use the forward-Euler method in time t, with a time step δt = 0.022, and a finite-difference method in space, with step size δx = 0.5, and five-point and seven-point stencils, respectively, for the Laplacian in d = 2 and d = 3 for spatial grids on square or simple-cubic simulation domains with side L mm, i.e., (2L)^{d} grid points. We use a similar forward-Euler method for the LRI PDEs, with δt = 0.01 ms, and a finite-difference method in space, with δx = 0.0225 cm, and a square simulation domain with side L = 90 mm, i.e., 400×400 grid points. We have checked in representative simulations for the LRI model that a Crank Nicholson scheme yields results in agreement with the numerical scheme described above. The simulation schemes that we use for the RPB and TNNP models are similar to the one we use for the LRI model; for the RPB model we use δt = 0.01 ms, δx = 0.0225 cm, and a 512×512 square simulation domain; for the TNNP case we use δt = 0.02 ms, δx = 0.0225 cm, and a 600×600 square simulation domain (i.e., L = 135 mm).

For all the models that we study, we use Neumann (no-flux) boundary conditions. The initial conditions we use will be specified below. For numerical efficiency, these simulations have been carried out on parallel computers with MPI codes that we have developed for all these models.

As suggested in Ref.

We introduce conduction inhomogeneities, which we also refer to as obstacles, in the simulation domains of the models described above by making the conductivity constant D = 0 in the region of the obstacle. In most of our studies we use square and square-prism obstacles in two and three dimensions, respectively. When we set D = 0 we decouple the cells inside the obstacle from those outside it. Furthermore, we use Neumann (i.e., no-flux) boundary conditions on the boundaries of the obstacle; we have checked in representative cases that, even if we do not impose Neumann boundary conditions on the obstacle boundaries, our results are not changed qualitatively.

We insert ionic inhomogeneities into our simulation domains by changing the values of the maximal conductances of Ca^{2+} channels, in the region of the inhomogeneity, for the LRI, RPB, and TNNP models. Since the Panfilov model does not account explicitly for Ca^{2+} ion channels, we mimic ionic inhomogeneities here by altering the value of the parameter ϵ_{1} in the region of the ionic inhomogeneity. In most of our studies we use square ionic inhomogeneities in two dimensions.

All the models described above can support spiral waves in a homogeneous simulation domain if we use suitable initial conditions. We begin the description of our results with an overview of such homogeneous simulations for the TNNP model; our work here complements that of Ref.

Given the diffusive coupling between cardiac cells, an action potential, generated in one cell, can excite neighboring cells and thus spread out as an expanding wave. However, since the initial excitation is followed by a refractory period, a second wave cannot follow the first one immediately. Each such wave leaves in its wake a nonexcitable region, so the next wave can follow it only at a distance determined by the product of the refractory period and the wave-conduction velocity

Furthermore, the conduction velocities of these waves depend on the curvature of the wavefront

We show below how such spiral waves can be generated, and how they break up subsequently, in representative, two-dimensional simulations for the TNNP model in homogeneous domains. (Similar simulations for Panfilov, LRI and RPB models are given in Section 3 of the

We obtain spiral waves in the TNNP model by injecting a plane wave into the domain via a stimulation current of 150 µA/cm^{2} for 2 ms at the left boundary. As this plane wave moves towards the right boundary and 270 ms after the first stimulus, we apply a second stimulus of 450 µA/cm^{2} along a line behind this wave but parallel to it ^{2}/ms between 304 ms to 524 ms; this yields the fully developed spiral wave. We then reset the conductivity to its original value after 524 ms. At the moment the first plane wave is initiated the currents and gating variables are initialised as follows: the gating variables are given in _{Na}, I_{CaL}, I_{to}, I_{Ks}, I_{Kr}, I_{K1}, I_{NaCa}, I_{NaK}, I_{pCa}, I_{pK}, I_{bNa}, and I_{bCa} (

Pseudocolor plots of the transmembrane potential V at (A) t = 0 s; (B) t = 0.8 s; (C) t = 3.2 s; and (D) t = 4.8 s.

(A) fast Na^{+} current (I_{Na}); (B) L-type Ca^{2+} current (I_{CaL}); (C) transient outward current (I_{to}); (D) slow delayed rectifier current (I_{Ks}); (E) rapid delayed rectifier current (I_{Kr}); (F) inward rectifier current (I_{K1}).

From our studies of spiral waves in the four models described above, we see that many qualitative features of spiral-wave dynamics are the same in the Panfilov, LRI, RPB, and TNNP models. However, there are important differences, some qualitative and the others quantitative. For example, the Panfilov model cannot address directly any questions regarding currents in ion channels since it does not follow their evolution but only considers one slow recovery variable g. The LRI, RPB, and TNNP models do give spatiotemporal information for several ion-channel currents as shown, for a representative case, in _{Na} is substantial only at the beginning of the action potential (_{Na} (_{CaL} is significant in the plateau regime of the action potential (_{CaL} (_{K1} is substantial in the repolarization regime of the action potential (_{K1} to have significant structure along the back of this wave, where repolarization occurs; this expectation is borne out as can be seen from

Negative currents move into the cell and positive currents move out of the cell.

It is useful to contrast pseudocolor plots of V for the Panfilov, LRI, RPB, and TNNP models. The qualitative features of these plots are the same but they differ in detail

Note that the action potentials from the Panfilov and TNNP models fall off more sharply in the repolarization phase than those for the LRI and RPB models.

We have elucidated the effects of conduction inhomogeneities in the Panfilov and LRI models in Ref.

We first examine the dependence of spiral-wave dynamics on the size of an obstacle by fixing its position and changing its size (cf., Ikeda

Furthermore, we find that the final state of the system depends on where the obstacle is placed with respect to the tip of the initial spiral wavefront. Even a small obstacle placed near this tip [e.g., an obstacle with l = 10 mm at (100 mm, 100 mm)] can prevent the spiral from breaking up; but a bigger obstacle, placed far away from the tip [e.g., an obstacle with l = 75 mm at (125 mm, 50 mm)], does not affect the spiral. In Ref.

We show here that this sensitive dependence of spiral-wave dynamics on the position of an obstacle also occurs in the TNNP models. (Similar results for the RPB model are given in Section 4 of the

We have carried out a systematic study of spiral-wave dynamics in the TNNP model in the presence of an obstacle with the initial conditions specified above, namely, one spiral wave [

We start with the initial condition of

This wave leads to periodic temporal evolution as can be seen from plots of (D) the time series of V from a sample of 100000 iterations (after the removal of the first 300000 iterations) taken from the representative point (90 mm, 90 mm) in the square simulation domain of side L = 135 mm, (E) the IBI versus the beat number n (a sample of 400000 iterations) that settles, eventually, to a constant value of ≃118 ms, and (F) the power spectrum of V (from a time series of 200000 iterations after removal of the initial 200000 iterations) that has discrete peaks at the fundamental frequency ω_{f}≃8.5 Hz and its harmonics.

(D) The time series of V from a sample of 200000 iterations recorded from the representative point (90 mm, 90 mm); after t = 3.2 s the system is quiescent and V is −86.2 mV.

Specifically, if we use a square obstacle of side l = 22.5 mm at (54 mm, 22.5 mm), the system is in the state ST as shown in

If we change the position of the obstacle and place it at (45 mm, 31.5 mm), the spiral gets attached to it. For this case the analogs of _{f}, where m is a positive integer and ω_{f} is the fundamental frequency of spiral rotation (ω_{f}≃8.5 Hz here).

We now place the obstacle at (63 mm, 22.5 mm); in this case the spirals get eliminated completely and we get the quiescent state shown via pseudocolor plots of V in

This sensitive dependence of the dynamics of spiral waves on the position of an obstacle has been investigated systematically, especially for the Panfilov model, in our earlier work _{p} = 10 mm. We then carry out a sequence of simulations. In each one of these simulations the square obstacle (slightly larger than the squares into which the simulation domain has been divided) is placed so that its bottom-left corner coincides with the bottom-left corner of one of the small squares; we now study the dynamics of spiral waves, with the initial condition described above, and determine whether the system reaches the ST, RS, or Q state. Our stability diagram, given in Ref.

Similar stability diagrams can be obtained, in principle, for LRI, RPB, and TNNP models; however, as the complexity of the models increases so does the difficulty of obtaining a stability diagram. Though we have not found complete stability diagrams for these detailed ionic models, the representative studies that we have carried out indicate that their stability diagrams are qualitatively similar to that of the Panfilov model, which we have given in Ref. _{CaL} = 0.000044 in _{pCa} = 3.825 is given in Section 4 of the

For each one of these positions of the obstacle we determine the final state of the system; the colors of the small squares, of side l_{p} = 4.5 mm, indicate the final state of the system when the position of the bottom-left corner of the obstacle coincides with that of the small square (red, blue, and green denote ST, RS, and Q, respectively).

Apart from obstacles that arise from inhomogeneities in the inter-cellular coupling, cardiac tissue can contain other types of inhomogeneities that originate from changes in single-cell properties, caused say by changes in the chemical environment or metabolic modifications _{1}. We insert such inhomogeneities in LRI, RPB, and TNNP models by considering spatial variations in conductances of calcium ion channels.

In the Panfilov model (1) _{1} increases the absolute refractory period of the action potential decreases. In turn this decreases the pitch of the spiral wave (cf. _{1} we can investigate spiral-wave dynamics in the presence of time-scale inhomogeneities. If the square simulation domain, of linear size L = 200 mm, is homogeneous, then with ϵ_{1}>0.03 we obtain a single, periodically rotating spiral wave but, as it decreases, for instance, if ϵ_{1}≲0.02, the tip of this rotating spiral starts meandering so that the temporal evolution of the system is quasiperiodic. At even lower values, say at ϵ_{1} = 0.01 that we have used above, we see spatiotemporal chaos. These behaviors are shown in the illustrative pseudocolor plots of V in Section 5 of the

We now introduce a square inhomogeneity in ϵ_{1} in the Panfilov model (all other parameters are uniform over the simulation domain): ϵ_{1} is assigned the value

If, instead,

The pseudocolor plot of V in (A) is at t = 2200 ms. The plots of the IBI associated with time series taken from a point outside the inhomogeneity and that from a point inside the inhomogeneity versus the beat number n are given in (B) and (C) respectively; these show period-5 and period-4 behaviors; the power spectrum associated with (C) is given in (D).

It is natural to ask whether the rich spatiotemporal dynamics of spiral waves in the Panfilov model with inhomogeneities in ϵ_{1}, which lead to inhomogeneities in local times scales since _{si} and G_{k} in the LRI model we can modify the action potential and the time scales associated with it, such as its duration and the extent of the plateau region, which in turn affect spiral-wave dynamics in much the same way as alterations of ϵ_{1} do in the Panfilov model. In particular, we can effect transitions from periodic to quasiperiodic rotating spiral waves or from quasiperiodic rotating spiral waves to broken waves with spatiotemporal chaos by changing these conductances; moreover, inhomogeneities in these conductances lead to spatiotemporal patterns in LRI, RPB, and TNNP models that are reminiscent of those we have described above for the Panfilov model with inhomogeneities in ϵ_{1}. We show this in detail below by investigating the effects of changes of the maximal calcium and potassium conductances, G_{si} and GK, respectively, in the LRI model, of Gsi in the RPB model, and of the maximal conductance GCaL for the L-type calcium current in the TNNP model. Illustrative simulations for the LRI and RPB models with ionic inhomogeneities are given in Section 5 of the

To investigate ionic inhomogeneities in the TNNP model we consider the calcium conductance G_{CaL} that governs the ionic current I_{CaL} (in the _{CaL} from 0.000175, the maximal channel conductance, to 0.00011, then to 0.00005, and finally to 0 [i.e., I_{CaL} channel block]). We now study the TNNP model in a square simulation domain of side 13.5 cm with the initial condition of _{CaL} the spiral wave breaks up because the slope of the APD restitution curve steepens and eventually exceeds 1: This break up is shown in the pseudocolor plots of V, at t = 3.2 s, of _{CaL} = 0.000175, 0.00011, and 0.0005, respectively. In _{CaL} = 0.000175, 0.00011, and 0.00005, respectively. The discrete lines in the power spectrum of _{1}≃8.25 Hz, ω_{2}≃9 Hz, and ω_{3}≃9.5 Hz) and chaotic states, the latter associated with the break up of spiral waves. We get similar results if we increase the plateau Ca^{2+} conductance G_{pCa} instead of changing the L-type Ca^{2+} conductance.

Panels (D), (E), and (F) show the corresponding power spectra of V (from a time series of 200000 iterations after the removal of the initial 80000 iterations) from the representative point (90 mm, 90 mm) in the simulation domain. These plots indicate that, as G_{CaL} decreases, the system goes from a state with a single rotating spiral wave to the spiral-turbulence state.

We now insert a square G_{CaL} inhomogeneity of side 33.75 mm in a square simulation domain of side 135 mm and _{1}ω_{1}+n_{2}ω_{2}+n_{3}ω_{3} with n_{1}, n_{2}, and n_{3} integers and ω_{1}≃4 Hz, ω_{2}≃6.25 Hz, and ω_{3}≃10.5 Hz (inside the inhomogeneity), and ω_{1}≃2.25 Hz, ω_{2}≃4.25 Hz, and ω_{3}≃6.25 Hz (outside the inhomogeneity). Since these frequencies are not related to each other by simple rational numbers we conclude that the spiral wave rotates quasiperiodically both inside and outside the inhomogeneity. In some cases we observe that the inhomogeneity does not have a significant qualitative effect on the dynamics of spiral waves; e.g., when the obstacle is at (45 mm, 45 mm), the position of the spiral tip shifts towards the bottom-left corner of the simulation domain but we still have a state with a single rotating spiral wave whose arms pass through the inhomogeneity. Like conduction inhomogeneities, ionic inhomogeneity can also remove spirals from the medium to leave the system in a quiescent state, e.g., when our G_{CaL} ionic inhomogeneity is at (45 mm, 22.5 mm). (See Fig. 33 in the

Associated plots of (F) the IBI versus the beat number n (a sample of 400000 iterations) and (G) the power spectrum of V (from a sample of 200000 iterations after the removal of the initial 200000 iterations) indicating quasiperiodic temporal evolution [the peaks in the power spectrum can be indexed (see text) in terms of three incommensurate frequencies (2.25, 4.25, and 6.25 Hz)]. Figures (H), (I), and (J) are the analogs of (E), (F), and (I), respectively, when data for

As we have mentioned above, there is growing consensus that the breakup of spiral waves of electrical activation in ventricular tissue leads to ventricular fibrillation (VF). In the usual clinical treatment of VF electrical stimuli are applied to the affected heart. This is believed to reset all irregular waves in the ventricular tissue leaving it ready to receive the regular sinus rhythm

Early attempts at controlling spiral-wave turbulence in models for cardiac tissue focused on applying well-known techniques of controlling low-dimensional chaos, which are based on the principle that, if a system's trajectory in state space comes very close to an unstable fixed point, it stays in its vicinity for a small duration of time. Different methods have been proposed to drive chaotic trajectories towards such an unstable fixed point, so that the system can stay in the neighborhood of the fixed point for the duration of the control stimulus

Biktashev and Holden

Osipov and Collins

Rappel, Fenton, and Karma

To suppress a spatiotemporally chaotic state with broken spiral waves, Sinha, Pande, and Pandit ^{2} smaller blocks by a mesh of line electrodes, and the mesh size is chosen to be small enough that spirals cannot persist for long inside the block of side ^{2} are used on the control mesh; this is much less than in conventional electrical defibrillation which uses pulses of amplitude 1 A/cm^{2}. This control algorithm has been extended to suppress spiral turbulence in the two-dimensional Beeler-Reuter and LRI models _{z} domain, the above scheme works with a slight modification: Instead of applying a pulse for a time _{z}>2 mm: its propagation in the z direction is impeded once the interior of the simulation domain becomes refractory. However, if we use a sequence of short pulses and if

Recently Zhang, _{f} of the forcing are chosen carefully: For example, for the Panfilov model with d = 2 it is shown in Ref. _{f} = 0.82, and

The control scheme of Ref. ^{2}, applied for 20 ms on a mesh that divides our square simulation domain of side 13.5 cm into 16 square cells of side 3.375 cm each, suffices to control spiral turbulence. The pseudocolor plots of V in

We apply a control pulse of amplitude 27.75 µA/cm^{2} for t = 20 ms over a square mesh with each block of size L/K = 33.75 mm, i.e., the simulation domain is divided into 4^{2} square blocks. Pseudocolor plots of V at (A) 0 ms, (B) 24 ms, (C) 80 ms, and (D) 280 ms, after the initiation of the control pulse, show how spiral turbulence is suppressed.

As we mentioned above, the control scheme of Ref. _{z} simulation domain, we can suppress broken scroll waves (the three-dimensional analogs of broken spiral waves) in the entire medium, provided that L_{z}<2 mm. Illustrative simulations are given in Section 6 of the

For L_{z}>2 mm this control scheme for a three-dimensional domain must be modified as follows: We use a control mesh on one face, but, instead of using a single long pulse, we use a series of short pulses, each of duration ^{3} simulation domain, a control mesh of square cells, each of side 16 mm, on the bottom face of the domain,

A series of 32 pulses, each lasting for 0.22 ms, and separated by 220 ms is applied on the bottom of the domain on a mesh; the subsequent evolution of of the scroll waves is shown at 440 ms (B), 1760 ms (C), and 1870 ms (D) by which time we see the last wave moving out of the simulation domain.

In the last Section we have given a short overview of some control schemes that have been used to suppress spiral-wave break up in two- and three-dimensional simulations in some mathematical models for cardiac tissue. Cardiac tissue can have inhomogeneities, such as scar tissue. It is important, therefore, to study whether these control schemes are effective in controlling spiral-wave turbulence in the presence of such inhomogeneities. To the best of our knowledge this has not been investigated systematically so far. We present such an investigation here (for simplicity we restrict ourselves to conduction inhomogeneities). In particular, it is important to ensure that a control scheme does not lead inadvertently to spiral break up in the presence of inhomogeneities.

We begin by studying the control scheme of Zhang _{f} = 0.82]; we use parameters such that, in the absence of this forcing, a single spiral wave would have been attached to the conduction inhomogeneity (

The subsequent evolution of the system in the presence of the forcing is shown in (B), (C), and (D). The control stimulus, of the form _{f} = 0.82, is applied on a small square region of side 0.3 cm in the center of the simulation domain of side L = 25 cm; the square conduction inhomogeneity of side 4 cm is placed at (8.5 cm, 8.5 cm). The target waves break up and contribute to spiral turbulence in the medium because of the obstacle as shown in (B) immediately after the control is applied and in (C) and (D) after 200,000 and 400,000 iterations, respectively; given our time discretization 1 iteration is 0.11 ms.

By contrast, the control scheme proposed in Ref. ^{2} is applied for 20 ms on a mesh that divides our square simulation domain of side 13.5 cm into 16 square cells of side 3.375 cm each. This suffices to control spiral turbulence in the TNNP model, even though there is a square obstacle of side 2.25 cm placed in the simulation domain. (If this obstacle is at (45 mm, 31.5 mm), in the absence of the control pulse, a single rotating spiral wave would have been anchored to this obstacle; if instead the obstacle is at (54 mm, 22.5 mm), as in

The square obstacle is at (54 mm, 22.5 mm) and has side l = 22.5 mm. Without control the spiral turbulence persists as in

We have shown above how the control scheme of Ref. _{z} simulation domain, if L_{z}<2 mm. This scheme works even in the presence of an obstacle as is illustrated for representative cases in ^{3} placed at (80 mm, 60 mm). In the absence of control pulses, scroll-wave turbulence is obtained. We now apply a control pulse of strength 0.48 on a mesh with square cells, each of side 16 mm, on the bottom face of the simulation domain for 748 ms; the evolution of the states of the system after the initiation of the control are depicted at (B) 220 ms, (C) 440 ms, and (D) 880 ms in

We now apply a control pulse of strength 0.48 on a mesh with square cells, each of side 16 mm, on the bottom face of the simulation domain for 748 ms; the evolution of the V iso-surfaces, after the initiation of the control, is depicted at (B) 220 ms, (C) 440 ms, and (D) 880 ms, by which time scroll waves have left the simulation domain and it is completely quiescent.

When L_{z}>2 mm, the control scheme described in the previous paragraph fails just as its counterpart did in the absence of an inhomogeneity. A representative simulation for this case is given in Section 6 of the

We have presented the most extensive numerical study carried out so far of the effects of inhomogeneities on spiral-wave dynamics in mathematical models for cardiac tissue. In particular, we have investigated such dynamics in the Panfilov, LRI, RPB, and TNNP models for homogeneous simulation domains and also in the presence of conduction and ionic inhomogeneities. Furthermore, we have considered two low-amplitude control schemes in detail; these have been designed to eliminate spiral-wave turbulence in these models but have not been tested systematically in the presence of inhomogeneities; we carry out such tests here.

One of the principal results of our studies is the confirmation that spiral-wave and scroll-wave dynamics in mathematical models of cardiac tissue depend very sensitively on the positions of conduction or ionic inhomogeneities in the simulation domain. Our results here extend significantly those presented in Ref.

Our studies have practical implications for experimental investigations of spiral-wave dynamics in cardiac tissue. In particular, the studies of Refs.

Our results, especially those on the elimination of spiral-wave turbulence in the presence of inhomogeneities, should also have important implications for the development of low-amplitude electrical defibrillation schemes, which is a major challenge that lies at the interfaces between nonlinear science, biophysics, and biomedical engineering. One of the lessons of our numerical studies, namely, the sensitive dependence of spiral-wave dynamics on inhomogeneities, implies that low-amplitude defibrillation schemes might well have to be tuned suitably to account for inhomogeneities in cardiac tissue. Furthermore, it would be very interesting to develop the mesh-based control scheme that we have described in the previous section and to see how it might be realised experimentally.

While this paper was being prepared for publication two new suggestions appeared for the elimination of spiral turbulence in models such as the LRI model

As we have emphasized throughout this paper, one of the principal goals of our study is a qualitative one, namely, the elucidation of the sensitive dependence of spiral-wave dynamics on inhomogeneities in mathematical models of cardiac tissue. We have, therefore, carried out extensive simulations of such dynamics in the Panfilov, LRI, RPB, and TNNP models; but we have not, so far, extended our study to bidomain models

In our earlier 3D simulations

Spiral-Wave Turbulence and its Control in the Presence of Inhomogeneities in Four Mathematical Models of Cardiac Tissue

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