16667050

Convection-induced anisotropy in excitable media subject to

solenoidal advective ¯ow ®elds

J.I. Ramos

Dpto. de Leng. y Cienc. de la Comp., E.T.S. Ingenieros Industriales, Universidad de Malaga, Plaza El Ejido, s/n, Room I-325, 29013 Malaga, Spain

Accepted 26June 2000

Abstract

The e€ects of solenoidal velocity ®elds on the propagation of spiral waves in excitable media is studied numerically by means of a time-linearized method. It is shown that the advective ®eld distorts the spiral wave at moderate frequencies, whereas, at large fre-quencies, the average shape of the spiral wave is nearly identical to that in the absence of convection, although its inner and outer parts exhibit spatial oscillations whose frequency increases as that of the velocity ®eld is increased. At low frequencies and high amplitudes of the velocity ®eld, the concentration of the activator and the wave propagation are controlled by the symmetry of the velocity and the number and location of the stagnation points, and the concentration of the activator may exhibit either counter-rotating regions or a layered structure. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction

In many spatially extended systems such as ¯ame propagation [1] and waves in biological media [2], one may observe moving fronts that separate parts of the medium in di€erent states. The stability of such fronts depends on the convective and di€usive transport, because transport provides the coupling between spa-tially separated elements and brings about the front propagation [1]. Of the two kinds of transport, the di€usive one has been more extensively studied than the convective transport. Biktashev et al. [3,4] con-sidered an excitable medium in two dimensions with a cubic nonlinearity given by the FitzHugh±Nagumo system and a shear characterized by a velocity ®eld in thex-direction which is either a linear or a sinusoidal function of the y-coordinate, and showed that the shear can distort and then destroy spiral waves. Such breaks were found to result in a chain reaction of spiral wave births and deaths. The velocity ®elds em-ployed by Biktashev et al. [3] are one-directional and solenoidal, but not irrotational, and they do not satisfy the no-penetration condition at the boundaries of the domain; in fact, these authors used periodicity conditions for the sinusoidal velocity ®eld. Elkin et al. [5] have studied numerically the movement of ex-citation wave breaks, while Biktashev and Holden [6] analyzed the resonant drift of autowave vortices in two-dimensional and the e€ects of boundaries and inhomogeneities. As will be shown in this paper, the activator concentration may exhibit vortical patterns for certain amplitudes and frequencies of the sole-noidal advective ®eld.

Wellner et al. [7] considered the drift of stable, meandering spiral waves in a singly di€usive FitzHugh± Nagumo medium caused by a weak time-independent gradient or convection in the fast-variable equation,

*_{Tel.: +34-95-2131402; fax: +34-95-2132816.} E-mail address:[email protected] (J.I. Ramos).

0960-0779/01/$ - see front matterÓ 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 60 - 0 7 7 9 ( 0 0 ) 0 0 1 5 3 - 3

showed by means of perturbati on methods the equivalence between gradient and convective perturbations, and proposed a semiempirical solution to the drift of spiral waves that depends on the period of rotation and the value of the fast variable at the center of the spiral wave. Biktashev and Holden [8] and Zhang and Holden [9] have explained the hypermeander of spiral waves as a chaotic attractor that leads to a motion of the spiral wave tip analogous to that of a Brownian particle. Rovinsky et al. [10] have analyzed a gener-alized Kuramoto±Sivashinky equation describing the dynamics of perturbations of a planar front in sys-tems with di€erential ¯ows, i.e., the ¯ow velocities of various components may be di€erent, and showed that a periodic pattern of modulation may appear on the front. Kñrn and Menzinger [11] considered a one-dimensional reaction±di€usion equation and a plug ¯ow velocity, and predicted the existence of stationary waves quite di€erent from those associated with the Turing mechanism which requires a fast inhibitor di€usion for the formation of spatially periodic patterns. The stationary waves obtained by Kñrn and Menzinger [11] were also found to be quite di€erent from those associated with di€erential ¯ow instabilities. Andresen et al. [12] considered the formation of stationary periodic patterns in a one-dimensional excitable medium with a constant plug ¯ow and the Brusselator model, and showed that their model may exhibit such patterns even in the case of equal di€usion coecients for certain types of boundary conditions in an open system.

In this paper, a numerical study of excitable media governed by the Belousov±Zhabotinsky (BN) equation in two-dimensional, stationary, convective/advective, solenoidal ¯ow ®elds which satisfy the no-slip condition at the solid walls is presented. The ¯ow ®eld is of the trigonometric type, and the propagation and distortion of spiral waves in excitable media are analyzed as functions of the amplitude and frequency of the velocity ®eld; both small and large velocities are considered. The velocity ®eld is assumed to be the same for both the fast and slow variables, i.e., the same for both the activator and the inhibitor. Fur-thermore, di€erential ¯ow e€ects due to the di€erent di€usive ¯uxes owing to the di€erent (constant) dif-fusivities of the slow and fast variables are considered, and these ¯uxes may induce spatio-temporal patterns. In addition, since the velocity ®elds considered in this paper are not uniform, they introduce anisotropic e€ects which alter the shape and speed of propagation of the spiral wave. This convection-induced anisotropy is used to determine the equivalent anisotropic di€usive tensor which, when employed in excitable media governed by reaction-di€usion equations, would result in the same patterns as those observed in excitable media with ¯ow straining.

2. Governing Equations

The study presented here is based on the BZ reaction which is often modelled by the Oregonator equations [13] whose periodic solutions and global structure have been recently analyzed by Zuohuan et al. [14], and may be written as

ou

ot‡v ruˆ r …Duru† ‡Fu; …1†

ov

ot‡v rvˆ r …Dvrv† ‡Fv; …2†

wheretis time,uandvdenote the concentrations of the activator and the inhibitor, respectively,DuandDv are the di€usivity tensors foruandv, respectively, andvˆ …vx;vy†Tis the velocity ®eld, where the subscripts xandydenote the corresponding axes. Since the interest in this paper lies in the e€ects of the velocity ®eld on wave propagation in excitable media, it is assumed thatDuˆIandDuˆ0:6I, whereIis the unit tensor of second rank, and it is known that, for these di€usivity tensors, the Oregonator model has spiral wave solutions forvˆ0.

The source terms in Eqs. (1) and (2) can be written as

Fuˆ1 u

u2 _fvu q

u‡q

; Fvˆu v; …3†

Eqs. (1) and (2) were solved in the (square) spatial domain Xˆ ‰ Lx=2;Lx=2Š ‰ Ly=2;Ly=2Š, where

LxˆLy ˆ7:5 subject to homogeneous Neumann boundary conditions on all the boundaries and to the following initial condition:

uˆ0; for 0<h<0:5; uˆq…f ‡1†=…f 1†elsewhere; …4†

vˆqf ‡1 f 1‡

h

8pf ; …5†

wherehis the angle with respect to the origin of coordinates measured counterclockwise from the positive x-axis. In the absence of convection, this initial condition results in the formation of a spiral wave which rotates counter-clockwise.

The velocity ®eld employed in this study is solenoidal, i.e.,r vˆ0, satis®es the no-slip condition, i.e., vˆ0 onoX, and has the following components:

vxˆAsin4npx_L cos2npx_L

… 1†n_{; v}

yˆAsin2npx_L sin2 2npy_L ; …6†

whereAis a constant andnis an integer. This velocity ®eld is not irrotational, i.e.,r v6ˆ0, although it is easy to show that the vorticity is nil where sinhyˆ0 and/or 2 sinhx coshy‡sinhy coshx ˆ0 where

hx ˆ2npx=Landhy ˆ2npy=L.

In order to illustrate the anisotropy induced by the solenoidal velocity ®eld, let us consider the following reaction±di€usion equation:

ou

ot ˆ r …Kuru† ‡Fu; …7†

where the di€usivity tensor Ku has componentskij…x;y;t†. An analogous equation can be written for the inhibitor concentration,v.

Eq. (7) can be written as ou

ot‡v ruˆKur2ru‡Fu; …8†

where the components ofvare

vxˆ o_ok_x11

‡o_ok_y21

; vy ˆ o_ok_x12

‡o_ok_y22

: …9†

Therefore, if this advective ®eld is to be solenoidal, then

o2_k11 ox2 ‡

o2

oxoy…k12‡k21† ‡

o2_k22

oy2 ˆ0: …10†

On the other hand, the condition that the di€usivity tensor be positive de®nite requires that kij>0,

k11k22>k12k21. This condition together with Eqs. (9) and (10) provide four equations for the four com-ponents kij if the velocity ®eld is speci®ed; on the other hand, if the di€usivity tensor is speci®ed, its components must satisfy the above inequality, and the velocity ®eld generated by the spatial dependence of the di€usivity tensor is then given by Eq. (9). This implies that the solenoidal and/or irrotational conditions impose further restrictions on the di€usivity tensor. These conditions are Eq. (10) and

o2

oxoy…k11‡k22† ‡

o2_k 21 oy2

o2_k 12

ox2 ˆ0; …11†

respectively.

For diagonal di€usivity tensors, i.e.,kijˆ0 ifi6ˆj, the irrotational condition can be integrated and the result is

k11 k22ˆF…y† ‡G…x†; …12† whereas the solenoidal condition becomes

o2_k 22 ox2 ‡

o2_k 22 oy2 ˆ

d2_G

dx2; …13†

and the no-penetration condition onoX, i.e.,vnˆ0, wherenis the unit normal to the boundary, implies that

ok22 oy x;

L₂y

ˆ0; o_ok_x22

L₂x;y

ˆ d_dG_x

L₂x

: …14†

The no-slip condition requires Eq. (14) and ok22

oy

L₂x;y

ˆ0; o_ok_x22 x;

L₂y

ˆ d_dG_x; …15†

which are satis®ed ifk22…x;y† ˆ G…x†andk11…x;y† ˆF…y†, but these imply thatvˆ0. Therefore, the no-slip, irrotational and solenoidal conditions for Eq. (8), i.e., space-dependent di€usivity tensors, may yield advective ®elds which are nil. As a consequence, there is not, in general, a complete analogy between re-action±di€usion equations in anisotropic media with space-dependent di€usivity tensors and convection± reaction±di€usion equations such as the one analyzed in this paper.

Eqs. (1) and (2) were solved numerically by means of an implicit, time-linearized, second-order accurate (in both space and time) ®nite di€erence method [15]. This method factorizes the elliptic equation that results upon discretization of time at each time level, into two one-dimensional boundary value problems and employs an iterative technique to account for the approximate factorization errors. Computations were performed on a 10001000 point equally spaced mesh and a time step equal to 10 4_{. Computations were} also performed with equally spaced meshes of 200200 and 500500 points and di€erent time steps, in order to insure that the results were independent of both the number of grid points and the time step. In the next section, some sample results obtained with 10001000 point equally spaced meshes and a time step equal to 10 4 _{are presented; however, only 100100 points are illustrated in the graphs.}

3. Presentation of results

In this section, some sample results on the interaction between spiral waves and solenoidal ¯ow ®elds are presented as functions of the amplitude and frequency of the velocity ®eld. For…A;n† ˆ …0:1;1†, a spiral wave which rotates in a counter-clockwise manner is observed, and its tip rotates around the origin of coordinates, i.e., the center of the domain [16]; it should be stated that the spiral wave does meander about the origin of the domain and that such a meandering has also been observed in numerical studies of the FitzHugh±Nagumo system [8,9].

The values ofuandvmonitored as functions of time at two ®xed positions indicate the spiky nature of the time traces ofu which are characterized by large values of the activator concentration in very small intervals; between two successive spikes, the concentration of the activator is very small. The time traces of vare also spiky and are characterized by a high slope whenvincreases and a smaller slope whenvdecreases. The separation between two succesive spikes ofuisT 1:6092 for…A;n† ˆ …0:1;1†. The separation between two succesive spikes ofu are presented in Table 1 for all the cases considered in this paper.

The results for…A;n† ˆ …0:1;1†di€er very little from those corresponding tovˆ0 [16], thus indicating the small in¯uence of the advective ®eld for small values ofA. ForAˆ1, however, the concentrations of both the activator and inhibitor are very sensitive ton, i.e., the frequency of the imposed advective ®eld, as indicated in Fig. 1 which corresponds to…A;n† ˆ …1;3†. This ®gure clearly illustrates that the spiral wave preserves its shape although it is highly distorted by the velocity ®eld. This distorsion decreases as n is increased; in fact, for …A;n† ˆ …1;6†, it was observed that the spiral wave is nearly identical to that for

correspond to those of the velocity ®eld. It is worth noting that no break-up of the spiral wave was observed forAˆ1 regardless of the value ofn, and that the period of the activator concentration decreases asnis increased as shown in Table 1.

The distorsion of the spiral wave was found to increase asAwas increased as illustrated in Fig. 2 which corresponds to…A;n† ˆ …3;2†. This ®gure clearly shows the large straining that the spiral wave undergoes between stagnation points where the velocity gradients are largest. This straining causes the formation of hook-like fronts, spikes in the regions engulfed by the spiral wave, and smoother albeit distorted fronts, whereuis largest. The number of spikes increases but their sharpness decreases asnis increased. In fact, for

…A;n† ˆ …3;7†, the spiral wave is similar to that for …A;n† ˆ …0:1;1†, but its front and back exhibit large undulations whose amplitude decreases asn is increased.

Fig. 1. Concentration of the activatoruat (from left to right, from top to bottom)tˆ50:4;50:6;50:8;51;51:2;51:4;51:6;51:8 and 52 for…A;n† ˆ …1;3†.

Table 1

Separation between the pulses in the activator's concentrationuas a function of…A;n†

…A;n† (1,1) (1,2) (1,3) (3,6)

Period 1.9355 1.7742 1.6667 1.6129

…A;n† (3,1) (3,2) (3,3) (3,6)

Period 4.1398 2.2043 1.8280 1.6129

…A;n† (5,1) (5,2) (5,3) (5,6)

Period 4.1935 4.1935 1.8280 1.6667

…A;n† (7,1) (7,2) (7,3) (7,6)

Similar features to those exhibited in Fig. 2 have also been observed for…A;n† ˆ …5;3†. For still larger values ofA, both the velocity gradient and the straining on the spiral wave which are proportional toA, increase and result in patterns such as those shown in Fig. 3 which corresponds to…A;n† ˆ …7;3†. This ®gure indicates that the front and back of the spiral wave are highly distorted and convoluted, and that small spiral waves are born between the inner and outer boundaries of the largest spiral wave. These smaller spiral waves appear between adjacent stagnation points of the velocity ®eld. For…A;n† ˆ …7;6†, the spiral wave has a shape similar to that for…A;n† ˆ …0:1;1†, but exhibits undulations in its front and back, and the activator concentration is smallest in the engulfed or inner part of the wave. Moreover, the largest con-centration of the activator occurs in a region much thinner than that for…A;n† ˆ …0:1;1†.

For …A;n† ˆ …3;1†;…5;1†;…5;2†and …7;1†, the results were found to be quite di€erent from those dis-cussed previously. For example, the time traces monitored at two di€erent points indicate that the distance between two successive spikes ofuis larger and the inhibitor concentration exhibits a longer decay period for…A;n† ˆ …5;2†than for …A;n† ˆ …0:1;1†. Similar time traces to those for…A;n† ˆ …5;2†have also been observed for…A;n† ˆ …3;1†;…5;1†and…7;1†.

Fig. 4 shows that very thick patterns where the concentration of the activator is large, may be formed for

…A;n† ˆ …3;1†; these patterns rotate counter-clockwise and may form small spiral waves near the middle of the domain. As time evolves, these patterns create two counter-clockwise rotating cells characterized by small concentrations of the activator; the pattern in the upper part of the domain has a slightly larger concentration than the one in the lower part. For times larger than those of Fig. 4, the concentration of the activator is similar to that shown in the lower right corner of that ®gure until the small island which appears in the left upper corner of that ®gure is created. The process illustrated in Fig. 4 is periodic and has a small

Fig. 2. Concentration of the activatoruat (from left to right, from top to bottom)tˆ50:4;50:6;50:8;51;51:2;51:4;51:6;51:8 and 52 for…A;n† ˆ …3;2†.

duration. Similar results have been found for…A;n† ˆ …5;1†and…A;n† ˆ …7;1†as illustrated in Figs. 5 and 6, respectively.

Fig. 5 illustrates the fast formation of cells as well as the clockwise rotation of the high-concentration region, which is followed by a relatively long period of quietness. The bottom frames of this ®gure exhibit the same trends as the corresponding frames of Fig. 4, whereas the frame on the ®rst row and second column of Fig. 4 is almost identical to the re¯ection with respect toyˆ0 of the frame on the ®rst row and third column of Fig. 5.

Fig. 6shows that the activator concentration is characterized by two small counter-rotating cells, one in the upper and the other in the lower part of the domain. This pattern persists for some time and is followed by a rotation of these cells around the center of the domain and a decrease in the activator concentration. This rotation results in a high concentration of the activator in the lower part of the domain which then decreases and the cycle is repeated in a periodic fashion.

For…A;n† ˆ …5;2†, one may observe an almost layered structure in the vertical direction as shown in Fig. 7. This structure is characterized by a higher concentration in the lower left corner of each frame which then moves to the upper right corner and, ®nally, to the lower left corner. The layered structure observed in Fig. 7 illustrates that the linexˆ0 is a special one; moreover, these ®gures exhibit a mirror symmetry about the lineyˆ xfollowed by a re¯ection along the lineyˆx. This symmetry applies to the isocontours, not to the values ofu. The reason for the symmetry and the special signi®cance of the linexˆ0 can be gathered from the velocity ®eld employed in this study. Fornˆ2,vx…0;y† ˆvy…0;y† ˆ0; therefore, the linexˆ0 is a stagnation line and the derivative of the velocity along this line is largest. This means that, along this line, the straining is largest.

Fig. 3. Concentration of the activatoruat (from left to right, from top to bottom)tˆ50:4;50:6;50:8;51;51:2;51:4;51:6;51:8 and 52 for…A;n† ˆ …7;3†.

Alongyˆx; vx ˆvyˆ0 atxˆk…L=4†, wherekˆ 2; 1;0;1;2; therefore, these points are also stag-nation points and correspond to the re-entrant regions observed in Fig. 7. On the other hand, along the line

yˆ x; vxˆvyˆ0 atyˆk…L=4†, wherekˆ 2; 1;0;1;2.

It is worth noting that the results presented in Fig. 7 show four cells in the vertical direction on each side of the linexˆ0 in agreement with the ®ve stagnation points discussed above. In the horizontal direction, one can also observe four cells, two on each side of the linexˆ0, but there is a transition along this line. The di€erences between Figs. 4±6and Fig. 7 are mainly due to the di€erent values ofn. Fornˆ1,vyˆ0 along xˆ0 and yˆ0; however, vx…x;0† ˆ0 and vx…0;y† ˆ2Asin…4py=L†. Therefore, yˆ0 is a line of stagnation points and this explains the appearance of the upper and lower parts in the results presented in Figs. 4±6. The points…x;y† ˆ …0;k…L=4††wherekˆ 2; 1;0;1;2 are also stagnation points. Moreover, for

nˆ1,vx…x;y†is symmetric with respect to they-axis, and antisymmetric with respect to thex-axis, whereas

vy…x;y†is antisymmetric with respect to they-axis and symmetric with respect to thex-axis. The symmetry and antisymmetry of the velocity ®eld can also be observed in the concentration of the activator presented in Figs. 4±6.

The results shown in this paper and others not shown here indicate that the solenoidal velocity ®eld represented by Eq. (6) does not result in wave breakup, although it distorts the spiral wave substantially. This distortion increases as the magnitude of the velocity ®eld increases; at high frequencies, however, the spiral wave exhibits oscillations on its front and back, but it essentially has an average shape which is nearly identical to that observed in the absence of advection. The results presented here also show that the dis-tortion created in excitable media by time-independent, solenoidal velocity ®elds depends on the number and location of the stagnation points and the magnitude and direction of the velocity gradients. They also

Fig. 4. Concentration of the activatoruat (from left to right, from top to bottom)tˆ52;52:2;52:4;52:6;52:8;53;53:2;53:4 and 55.6 for…A;n† ˆ …3;1†.

show that both the amplitude and frequency of the velocity ®eld play a paramount role in determining the periodicity of spiral wave propagation in excitable media.

The periods shown in Table 1 indicate that the period of the spiral wave decreases as the frequency of the convective ®eld increases and tends to an almost constant value, regardless of the amplitude of the velocity. The concentrations of the activator and inhibitor were monitored at two points in order to determine the periodicity of the spiral wave propagation, and the time traces/histories at these monitoring points indicate that the inhibitor concentration may reach values of the order of 0.01 for…A;n† ˆ …3;1†;…5;1†;…5;2†and

…7;1†which are about ®ve times smaller than those for the other cases considered in this study.

Although the term straining caused by the velocity ®eld has been used throughout the paper, it must be pointed out that, for solenoidal velocity ®elds, Eq. (1) may be written as

ou

ot ˆ r …Duru vu† ‡Fu; …16†

and, therefore, the total (convective‡diffusive) ¯ux of the activator is nil wherever Duru vuˆ0. Similarly, the total ¯ux of the inhibitor is nil whereverDvrv vvˆ0. This implies that, for the di€usivity tensors employed in this paper, the ¯uxes ofuandvcannot be nil at the same points, unlessuˆvˆ0.

Although a comparison with the results of Biktashev et al. [3] is not possible because these authors employed the FitzHugh±Nagumo equations and a one-dimensional velocity ®eld in thex-direction which depends on they-coordinate, whereas, in this study, the BZ system is considered and the ¯ow ®eld is two-dimensional and solenoidal and satis®es the no-slip condition at the domain boundaries, it is convenient to indicate that Biktashev et al. [3] observed spiral wave breakup and a chain reaction that resulted in spiral wave births and deaths, whereas the results of this paper show that the spiral wave maintains its integrity although it is highly distorted and convoluted at large amplitudes and frequencies, but it may undergo quite

Fig. 5. Concentration of the activatoruat (from left to right, from top to bottom)tˆ52;52:2;52:4;52:6;52:8;53;53:2;53:4 and 55.6 for…A;n† ˆ …5;1†.

a few changes at high amplitudes and low frequencies. When this occurs, it was observed that the activator concentration exhibited patterns in accord with those of the velocity ®eld. Moreover, our results also in-dicate that the number and location of the stagnation points and the velocity gradients play a paramount role in determining the propagation of spiral waves in excitable media subject to time-independent sole-noidal velocity ®elds which satisfy the no-slip condition at the domain boundaries.

The results presented in this paper also show that a shear ¯ow breaks the spatial re¯ection symmetry of reaction±di€usion systems and, therefore, the angular velocity of the spiral wave in a shear ¯ow depends on the direction of rotation. The results are also in qualitative accord with those of Biktashev et al. [3] who showed that spiral waves rotating against the shear, i.e., counter-rotating spiral waves, do rotate faster than in absence of shear. Moreover, since in an excitable medium, faster waves entrain slower ones, co-rotating spiral waves are entrained by counter-rotating ones and driven away by low-shear regions. As a conse-quence, spiral waves in low-shear regions do not develop because the spiral wave rotation frequency there is lower than that in the high-shear regions [3].

4. Conclusions

The e€ects of a solenoidal advective ®eld which satis®es the no-slip condition on the boundaries of the domain, on wave propagation in excitable media have been studied numerically, and it has been shown that, if the amplitude of the velocity ®eld is small, the shape of spiral wave is nearly identical to that ob-served in the absence of advection. It has also been shown that the advective ®eld distorts the spiral wave decreasing its thickness and creating oscillations on its front and back, albeit the average shape of the spiral

Fig. 6. Concentration of the activatoruat (from left to right, from top to bottom)tˆ54;54:08;54:16;54:24;54:32;54:40;54:48;54:56 and 54.64 for…A;n† ˆ …7;1†.

wave is nearly identical to that observed in the absence of advection at high frequencies, i.e., when the number of stagnation points is large.

The most important e€ects of the solenoidal velocity ®eld have been found for frequencies equal to one or two and amplitudes larger than one. In these cases, the symmetry of the velocity ®eld and the number and location of stagnation points play a paramount role in determining the concentrations and period of both the activator and inhibitor, the concentration of the activator re¯ects the symmetry/asymmetry of the velocity ®eld, thick regions of high concentrations can be found, and the activator concentration may exhibit counter-rotating cells or layered structures. These cases are periodic and multiple spiral waves or layers characterize the excitable medium.

It has also been shown that the total ¯ux of the activator concentration is not nil at the same locations as that of the inhibitor, and that wave propagation in anisotropic excitable media in the absence of advection, may be quite di€erent from wave propagation in isotropic excitable media with convective ¯ow ®elds.

Acknowledgements

The research reported in this paper was supported by Project PB97-1086from the D.G.E.S. of Spain.

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