Exact Coherent Structures and Dynamics of Cardiac Tissue

Greg Byrne, Christopher Marcotte, and Roman Grigoriev

Abstract— High dimensional spatiotemporal chaos under-pins the dynamics associated with life threatening cardiacarrhythmias. Our research seeks to suppress arrhythmias andrestore normal heart rhythms by using low-energy transfersbetween recurrent unstable solutions embedded within thechaotic dynamics of cardiac tissue. We describe the search forthese unstable solutions in nonlinear reaction diffusion modelsof cardiac tissue and discuss recent efforts to reduce localsymmetries that emerge as a result of spiral wave interactions.

I. INTRODUCTION

Sudden cardiac arrest is the leading cause of death in theindustrialized world. The majority of these deaths are due tocardiac arrhythmic disorders such as ventricular fibrillationscaused by myocardial infractions, or “heart attacks.”

Arrhythmias are characterized by spatially complex, high-dimensional dynamics that occur as excitation waves propa-gate through cardiac tissue. Figure 1 shows an example ofhow ventricular tachycardia (an arrhythmic behavior char-acterized in two dimensions by spiral wave dynamics) candeteriorate into ventricular fibrillation – a dangerous chaoticregime characterized by a loss of spatial coherence in thecontractile dynamics of the heart. While in this state, theheart’s ability to pump blood is severely degraded, and deathusually follows within minutes unless corrective measuresare taken.

Fig. 1: An unstable spiral wave formed during ventricu-lar tachycardia breaks up into multiple interacting spirals.This process leads to the chaotic activity observed duringventricular fibrillation. Simulations are performed using theKarma model defined by Eqs. ( 1, 2) with “no flux” boundaryconditions.

The most common and effective way to terminate ven-tricular fibrillation is by using a defibrillator – an externalor implanted medical device that delivers one or more high

978-1-4799-3969-5/14/$31.00 2014 IEEEThis material is based upon work supported by theNational Science Foundation under Grant No. CMMI-1028133.G.B., C.M., and R.G. are from the Georgia Institute of Technology, USAContact email: gregory.byrne@physics.gatech.edu

energy electrical shocks to forcefully “reset” the excitationdynamics of cardiac tissue. This “brute force” methodologyreestablishes the normal sinus rhythm of the heart, but causesa variety of undesired side effects including severe pain,cardiac tissue damage, and a rapid loss of battery life inimplanted devices [1], [2]. Our research seeks to improvethe defibrillation process by using a novel dynamical systemsapproach that provides a safer, lower energy alternative withfewer negative side effects.

The dynamical systems approach to low-energy controlis based on two equivalent representations of the cardiacdynamics: (1) a physical space representation and (2) astate space representation. In the former, the state of thetissue is described by a vector field w where the vectorw(x, y, t) describes the state at time t of a cardiac cell(cardiomyocyte) located at position (x, y). The latter usesan arbitrary alternative representation of the vector field wthat is convenient for computational purposes, e.g., the valuesof w on a computational mesh or Fourier coefficients of win the spectral representation.

The advantage of the state space representation is that itallows solutions in the physical space to be described interms of steady-, time-periodic and quasi-periodic solutionsthat are known to organize chaotic dynamics in state space.The physical space versions of these solutions are conven-tionally referred to as exact coherent states due to theirsimilarity with the spatiotemporal structures (coherent states)frequently observed in turbulent fluid flow experiments.

These solutions and their dynamical (e.g. heteroclinic)connections provide ideal pathways to guide the system awayfrom the dangerous regions of state space associated withcardiac arrhythmias and back towards safe regions wherenormal rhythms can be restored. The theory based on theseunstable solutions has already enabled impressive progressin understanding space manifold dynamics and other highdimensional chaotic systems such as weakly turbulent fluidflows [3], [4], [5], [6].

Numerical methods originally developed in the context offluid turbulence have enabled us to identify several invari-ant solutions in nonlinear reaction-diffusion models of car-diac tissue. These include equilibria and relative equilibria–traveling wave solutions (e.g. pulse and spiral waves) whichrespect global translational and rotational symmetries inunbounded domains [7], [8] but which become time-periodicsolutions in bounded domains.

We have also identified numerous multi-spiral solutionswhich are not strictly time-periodic because of local symme-tries that form in the presence of weakly interacting spiralwaves. In order to find exact periodic solutions featuringmany interacting spirals, a new approach is needed thatincorporates the dynamics associated with local rather thanglobal symmetries. The reduction of these local symmetries

will provide an important step towards obtaining the sets ofunstable periodic solutions needed to implement an optimallow-energy control.

This work is organized as follows. In Sec. II we describethe computational procedure for locating unstable recurrentsolutions and show examples of equilibria, relative equilibriaand some nearly time-periodic trajectories. In Sec. III weillustrate the effects of local symmetry by constructing multi-spiral solutions from a single-spiral solution. In Sec. IVwe introduce the concept of tiling to partition the domainfor local symmetry reduction. In Sec. V we provide someconclusions.

II. THE SEARCH FOR COHERENT STRUCTURES

Many modern mathematical models of cardiac tissue arealgebraically complex and incorporate tens of variables percell. The tissue dynamics are typically described in termsof coupled nonlinear reaction-diffusion partial differentialequations

∂tw = D∇2w + f(w), (1)

where the vector field w = (u, v) represents the transmem-brane voltage and gating variables (ionic concentrations, stateof ionic channels, etc.) for the cardiac muscle cells makingup the tissue, D is a diagonal matrix of diffusion coefficients,and the nonlinear function f(w) describes the local dynamicsof individual cardiac cells.

These models require substantial computational resourcesto obtain numerical solutions with appropriate spatial andtemporal resolution. When compared to some algebraicallysimpler, lower dimensional models, the added complexityprovides little additional insight into the dynamical mecha-nisms responsible for generating and maintaining complexarrhythmic behaviors.

The two-variable Karma model [9] provides a greatlysimplified description of cardiomyocyte excitation dynamicswhile still reproducing the essential alternans instability thatis responsible for the breakup of single-spiral solutions andthe transition to fibrillation in two or three dimensions.The cellular dynamics of this model are described by thefollowing function

f(w) =

(u2(1− tanh(u− 3))(u∗ − v4)/2− uε((β−1 − v)Θ(u− 1)−Θ(1− u)v)

), (2)

where Θ(·) is the Heaviside step function and (ε, u∗, β) arethe control parameter set. The parameter ε describes the ratioof excitation and relaxation time-scales, u∗ is a phenomeno-logically chosen voltage scale, and β = 1− exp (−Re) con-trols the restitution (and susceptibility of traveling excitationwaves to alternans, or period-doubling instability) through asecondary control parameter Re. We chose the parametersu∗ = 1.5415, ε = 0.01 and Re = 1.273 such that simpletraveling wave solutions such as those shown in Fig. 2(a-b)would break up due to alternans instability.

The equations are solved on an N ×N uniform grid withspacing ∆ = 262 µm which corresponds to the size ofcardiomyocytes. No-flux boundary conditions are used unlessotherwise specified. At each time instant t, the discretizedfields u and v that define the state of the system can bethought of as an element z(t) of a 2N2-dimensional vector

0.000 3.725

(a)

0.000 3.471

(b)

0.000 3.599

(c)

7e−11 7e−04

(d)

Fig. 2: Voltage distributions u(x, y, t) of exact (a-b) andnearly-exact (c) time-recurrent states of the Karma model:(a) relative equilibrium generated by translational symmetry,∂tw − c∂xw = 0 and (b) relative equilibrium generatedby rotational symmetry, ∂tw − ω∂θw = 0, where θ isthe polar angle in the (x, y) plane. (c) A snapshot of anearly recurrent solution found using the method of closereturns. This solution has a period of T = 377.95ms andis computed using periodic boundary conditions. (d) Therelative residual |u(x, y, 0) − u(x, y, T )|/|u(x, y, 0)|. TheL2-norm of the residual is small but non-vanishing and islocalized near three slowly drifting spiral cores.

space M ⊂ R2N2

. Under time evolution, z(t) defines atrajectory in this high-dimensional state space.

The first class of recurrent solutions are the equilibria. Inthe absence of spatial variation, the Laplacian term vanishesand the reaction diffusion equations are reduced to a setof 2N2 identical uncoupled ordinary differential equationswhose solutions represent the dynamics of individual cells.Equilibrium points for the ordinary differential equationscan be obtained by finding the intersections of the null-clines defined by f(w) = 0. Three such solutions existfor the choice of parameters given above: a stable (butexcitable) equilibrium point at the origin w∗

1 = (0, 0), asaddle point at w∗

2 = (0.6547, 0) and an unstable focus atw∗

3 = (1.0073, 0.853). Each of these equilibrium points alsoprovides a steady state solution for the tissue model whenD 6= 0.

Another class of unstable solutions exist due to the Eu-clidean symmetry of the underlying equations. These solu-tions are known as relative equilibria and describe travelingwaves. Two examples are shown in Fig. 2(a) and (b). For theplane traveling wave shown in Fig. 2(a), time evolution isequivalent to a global translation in the horizontal direction.For the spiral traveling wave shown in Fig. 2(b), timeevolution is equivalent to a global rotation about a zero-dimensional core [10] that pins the spiral. In the symmetry-

reduced representation, where continuous translational androtational symmetries are factored out of the dynamics, bothtypes of relative equilibria reduced to an equilibrium [7].

A slightly more complex type of exact coherent structuresare described by unstable time-periodic solutions. Thesesolutions can be found using the method of close returns.This method is based on the observation that the differenceE(t, τ) ≡ ‖w(t) − w(t − T )‖ is small when a state spacetrajectory w(t) shadows an unstable periodic orbit of periodT , where ‖ ·‖ denotes the L2 norm in R2N2

. In contrast, | · |will be used below to denote the absolute value.

t

T

200 400 600 800 1000 1200 1400

50

100

150

200

250

300

E(t

,τ)

0.2

0.4

0.6

0.8

1

Fig. 3: A recurrence plot is used to identify close returns inthe state space M. The black circle in the plot identifies aminimum associated with a close return.

The recurrence plot shown in Fig. 3 illustrates how weidentify nearly recurrent solutions. Once a sufficiently lowminimum of the recurrence function E(t, τ) is identified inthe plot, a matrix-free Newton-Krylov solver (GMRES) isused to refine the nearly time-periodic solution into an exacttime-periodic solution. Computational challenges associatedwith the high dimensionality of the state space M are han-dled by developing heavily parallelized numerical integrators[11] and Krylov-space algorithms that use powerful NVIDIAGPU hardware.

Nearly exact periodic solutions are produced when theNewton-Krylov solver fails to converge. Figure 2(c) showsan example of such a nearly recurrent, multi-spiral solutionwith a period T = 377.95ms and a relative residual of‖w(T ) − w(0)‖/‖w(0)‖ = 5 × 10−4. The convergencefailure can be understood by examining the residual dis-tribution in the spatial domain, shown in Fig. 2(d). Theresidual is concentrated near the wave fronts and cores ofthree specific spiral waves that undergo slow drifts. Thesedrifts are associated with local symmetries.

III. LOCAL SYMMETRIES

Spiral waves interact very weakly when separated by largedistances. This means that neither their relative positions,nor their relative phases are fixed. The ability to apply smalldisplacements to a core or small shifts to its phase (which isequivalent to a spatial rotation of the spiral about its core)relative to the other cores implies that the solutions respectEuclidean symmetry both globally and locally.

We illustrate the local Euclidean symmetries encounteredin the dynamics by constructing two simple multi-spiralsolutions. The first solution is constructed by placing a copyof the single, isolated spiral solution shown in Fig. 2(b) nextto a second copy that has been integrated forward in time

0.000 3.509

0e+00 3e−17

Fig. 4: A two-spiral solution with local symmetry in whichone of the spirals is phase-shifted relative to the other. Thevoltage (top), core trajectories (middle), and the residual(bottom) are shown.

by a quarter of its temporal period (which is equivalent to aspatial rotation of this spiral by π/2). The resulting solutionis shown in Fig. 4.

The cartoon description of the spiral core motion shownin the middle of Fig. 4 shows that there is no net drift.Unlike the single-spiral state (which is properly classified asa relative equilibrium), the two-spiral state is a time-periodicsolution. The two-spiral solution recurs exactly (within thenumerical precision of the simulation) after a period T =125.98ms. In this case the Newton-Krylov solver (whichautomatically reduces global symmetries) succeeds in find-ing the two-spiral solution because each of the spirals ischaracterized by the same type of local symmetry (a phaseshift/rotation).

The stability of a recurrent solution in state space isdetermined from the eigenvalues (stability multipliers) Λkand corresponding eigenvectors of the finite-time jacobianJTp

kj (zp) = ∂z(Tp)k/∂z(t0)j , where J0(zp) = 1, zp is theinitial condition on the state space orbit and Tp is the time tocomplete the orbit. Because of the high dimensionality of thestate space, the leading stability multipliers are determinednumerically using Arnoldi iteration. The stability exponentsλk = µk + iωk are related to the stability multipliers Λk =eTpλk whose magnitude |Λk| = eTpµk determines whethersmall perturbations grow or decay under the dynamics.

The multipliers for the phase shifted, two-spiral solutionare plotted in Fig. 5 along with some of the correspondingeigenmodes for the unstable (expanding) |Λ| > 1, stable(contracting) |Λ| < 1 and marginal |Λ| = 1 directions.The unstable modes in the spectrum are associated withan alternans instability which is responsible for the spiralbreakup shown in Fig. 1. The pair of marginal modesfound in the spectrum indicate that either spiral can undergosmall phase shifts (or rotations) relative to the other withoutdestroying the time-periodic nature of the solution on the

Fig. 5: The leading stability multipliers for the phase shifted,two spiral exact coherent structure shown in Fig. 4. Severaleigenmodes corresponding to the unstable and marginaldirections are also shown. The unstable modes are associatedwith an alternan instability while the two marginal modesindicate that the solution contains a local symmetry in whicheither spiral is free to undergo small phase shifts relative tothe other.

entire domain.A nearly periodic, three-spiral solution with a residual of

3.5 × 10−5 is shown in the top of Fig. 6. This solutionillustrates a local symmetry involving core drift. The bottomof Fig. 6 shows that the residual is localized to the vicinityof the smaller drifting core. The failure of the Newton-Krylov solver to converge to a time-periodic three-spiralsolution in this case is likely due to the difference in thetypes of the local symmetry for each of the three spirals–rotation for the two large spirals vs. a combined rotation andspatial translation for the small spiral. This failure requiresdevelopment of a new approach that identifies regions in thedomain where symmetry reduction can be applied locally.

IV. LOCAL SYMMETRY REDUCTION

Spiral waves naturally divide the domain by forming thinwalls known as shock lines along which the phases ofneighboring spirals are equal. The resulting sub-domains, ortiles, are ideal for local symmetry reduction because theyeach enclose a single-spiral.

Tiles were constructed in [12] for the complex Ginzburg-Landau equation (CGLE) by using the local amplitude of os-cillation. Unlike the CGLE which describes weak nonlinearoscillations, the Karma model is strongly nonlinear, makingit difficult to define, let alone compute the local amplitude.Instead, we use cycle areas in the (u, v) plane to define ameasure equivalent to the oscillation amplitude.

Figure 7(a) plots the instantaneous (u, v) values at eachcell on the computational grid for the three-spiral solutionshown in Fig. 6. Over time, the dynamics of each oscillatortrace out a continuous cycle (with a corresponding local pe-riod) whose area can be computed after each full revolution.Each point in the figure is colored by the area of its cycle. Wefind that the larger cycles (red) correspond to the shock linesbetween spirals, while the smaller cycles (blue) identify the

0.001 3.501

1.02e−15 5.06e−06

Fig. 6: A three-spiral solution with local symmetry in whichthe small core in the center drifts relative to the two largerspirals. The voltage (top), core trajectories (middle), and theresidual (bottom) are shown. The L2-norm of the residual issmall but non-vanishing and is concentrated near the slowlydrifting spiral core in the center of the domain.

(a)

0 1 2 3

0.85

0.9

0.95

1

U

V

0.003

0.124

0.246

0.367

0.488

(b)

Fig. 7: The oscillatory dynamics for each grid point areused to construct tiles on the domain. (a) The three-spiralsolution from Fig. 6 shown in the u−v representation. Eachpoint is colored by the area its cycle traces out during afull revolution. The cycle area is useful in determining thelocation of phase singularities (spiral cores) and shock linesof uniform phase. (b) Three important cycles are extractedfrom (a). Ordered from the smallest to largest area, theycorrespond to points at a spiral core (blue), in the interior ofa tile (lighter red), and on a tile boundary (dark red).

phase singularities that represent the spiral cores. The tilesdefined by the cycle areas for the two-, three-, and ten-spiralsolutions from Figs. 4, 6 and 2(c) are shown in Fig. 8.

Similar representations have been found useful for describ-ing frozen/glassy spiral states in the CGLE [12], [13]. Theauthors show that tiles do not correspond to Voronoi tessel-lation. Instead, the shocks are approximated as segments ofhyperbolae with the two nearest spiral cores as foci. Futurework will determine whether, and under what conditions,the shock lines for the solutions of the Karma model arehyperbolic.

The decomposition of the domain into a set of weaklyinteracting tiles greatly simplifies the dynamical notion ofinvariance. On short time-scales, the pattern and temporalevolution within each tile is defined by a single-spiral so-lution. Because the interactions with neighboring spirals areweak, they can, to leading order, be neglected. This allowseach tile to be treated as its own independent domain wherethe evolution of the spiral wave is affected by the action of a“global” symmetry. The independence of the tiles allows usto symmetry reduce the local (on the tile) solutions beforerecombining them to find global (on the entire domain)solutions that organize spatiotemporally chaotic dynamics.

Symmetries and tile geometries on the domain have im-portant consequences for the globally recombined solutions.On a tile with an unbounded or circular domain, a spiralsolution corresponds to a relative equilibrium. On a generictile of finite size, a spiral solution will correspond to atime-periodic state since the geometry will typically breaksthe continuous rotational symmetry. For stationary tiles (i.e.those that contain spirals with pinned cores), a time-periodicsolution in the entire domain would naturally decomposeinto time-periodic solutions within each of the tiles. Fordrifting tiles (i.e. those whose boundaries deform because ofdrifting cores), the corresponding spirals would be describedby relative periodic solutions. Hence, more generally, a classof multi-spiral solutions in the entire domain can be decom-posed into periodic and relative time-periodic solutions insidethe tiles.

V. CONCLUSIONS

The ability to understand and reduce local symmetriesis essential for classifying and computing unstable time-dependent solutions of cardiac tissue models. These solutionsform a skeleton for the spatiotemporally chaotic dynamicsassociated with cardiac arrhythmias in their state space rep-resentation. A dynamical description of arrhythmic behaviorsbased on transitions between such solutions provides animportant first step towards designing efficient low-energymethods to suppress fibrillation. We have developed a com-putational procedure for finding time-periodic solutions inthe state space representation corresponding to single- andmulti-spiral states with fixed spiral cores in the physicalspace representation.

In its current form, this procedure is unable to findsolutions in which spiral waves drift relative to one another.As a result, local symmetries continue to mask many of theunstable recurrent solutions that are important for implement-ing an effective method of low-energy control. The ability topartition the spatial domain into tiles on which the dynamics

0.004 0.488

(a)

0.004 0.488

(b)

0.001 0.484

(c)

Fig. 8: The cycle area computed for the solutions shown inFig. 4 (top), Fig. 6 (middle) and Fig. 2(c) (bottom). Thedark regions correspond to shock fronts that separate thespirals and divide the domain into tiles where local symmetryreduction can be applied. The blue circles correspond tophase singularities that define the location of the spiral cores.

effectively decouple on short time-scales should provide aframework for a systematic application of local symmetryreduction both in models of cardiac tissue and other systemsdisplaying spatiotemporally chaotic dynamics.

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