Spiral breakup in a modified FitzHugh Nagumo model

Transcription

1 Physics Letters A 176 (1993) North-Holland P H YSI CS LETTERS A Spiral breakup in a modified FitzHugh Nagumo model Alexandre Panfilov and Pauline Hogeweg Bioinformatica, University of Utrecht, Padualaan 8, 3584 CH Utrecht, The Netherlands Received 20 January 1993; revised manuscript received 17 March 1993; accepted for publication 22 March 1993 Communicated A.P. Fordy In a modified FitzHugh Nagumo model for excitable tissue a spiral wave is found to break up into an irregular spatial pattern. The main difference between our equations and the standard FitzHugh Nagumo model is that we use two different time constants: one for the relative refractory period and another for the absolute refractory period. Breakup occurs when the relative refractory period is short. The effect is numerically stable at least for a five-fold decrease in the space integration step. 1. Introduction modifying the dynamics of the recovery variable. Spiral waves in excitable media provide an important example of self-organization phenomena in 2. Mathematical model and method of computation spatially distributed biological, physical and chemical systems [1]. Usually, spiral waves occur either For numerical computation we use FitzHugh because of inherent heterogeneity in the excitable Nagumo-type equations with piecewise linear tissue, or because of some special initial conditions Pushchino kinetics [12 13], [2]. However, in some cases, spirals can be generated even in homogeneous systems. Recent studies have demonstrated that a spiral can break up, and ~9e/ôt=V 2e f(e) g, ôg/ôt E(e g)(ke g), (1) spontaneously generate complicated spatio-temporal patterns [3 10].This process of spiral breakup has withf(e) =C 1e when e<e1 f(e)= C2e+a when been observed in some sophisticated equations for e1 v~e~e2 f(e) = C3 (e-.- 1) when e> e2, and ~(e, g) cardiactissue [4 7 and in several cellular automata = ~ when e < e2 ~(e, g) = ~2 when e> e2, and ~( e, g) models [3,8 10]. However, the mechanism in- =~ when e<e1 and g<g1. The parameters detervolved is unknown, but the breakup is believed to be mining the shapeof the functionf(e) are e1 = , associated with the discrete nature ofthe model [8], e2=0.837, C1 =20, C2= 3, C3= 15, a=0.06 and k=3. or in some cases to be connected with the presence With these parameter values the function f(e) is of two inward ionic currents [7]. One way of study- continuous. The shape of the functionf(e) specifies ing the mechanism underlying this effect is to re- fast processes such as the initiation of the action poproduce it in simple minimal partial differential tential. The dynamics of the recovery variable g in equation models of excitable tissue, which allow us (1) is determined by the function ~(e, g). In ~(e, g) to study the influence of discrete effects, pulse shape, the parameter ~ specifies the recovery time condispersion relation, etc., on the process of spiral stant for small values of e and g. This region apbreakup. However, recent computations by Winfree proximately corresponds to the relative refractory show that this effect is unlikely to exist in classical period. Similarly, ~ specifies the recovery time FitzHugh Nagumo equations [11]. Here we show constant for relatively large values of g and interthat it is possible to obtain the spiral breakup in a mediate values of e. This region approximately corsimple two-component model of excitable tissue by responds to the wave front, wave back and to the ab /93/$ Elsevier Science Publishers B.V. All rights reserved. 295

2 Volume 176, number 5 PHYSICS LETTERS A I 7 May I 99~ - solute refractory period. The main difference between time constant for the relative refractory period v~ model (I) and the previous model [I 3] is that model We found that spiral breakup occurred if e~ was less I ) uses two independent constants ~ and e~ for than 5.5. Figure I shows the evolution of a spatial the refractory state. The values of these parameters pattern in a medium with ~ =2.8. In this case the were fixed at fj = 75, ~ = I - = 1.8, and 0.5< wave makes several rotations with pronounced <10. meandering and begins to fragment close to the centre For numerical computations we used the explicit of the spiral (fig. Ic). However, at this moment in Euler method with Neumann boundary conditions, time, there is not enough recovered space and the and the rectangular grid measured from l00~100 wave breaks disappear. Later, at time 1= Ill, a larger elements up to 1000x700 elements. To initiate the fragment of the wave breaks away (fig. le). initifirst spiral we used initial data corresponding to a ating two new spirals. This process continues, and al 2D broken wave front, the break being located in the 1= 183 we see five interacting rotating spatial waves middle of the excitable tissue. (11g. If). In a large excitable medium, the behaviour of the system becomes more complicated. Figure 2 shows 3. Results and discussion the same computation as in fig. I, hut on a grid of l000x 700 elements. The final structure comprises We found that the spiral breakup occurred spon- many wave breaks of various sizes and a complitaneously in the excitable media that have a short- cated spatial distribution of the recovery variable. ened relative refractory period. In our computations This picture evolves in time, new breaks occur conwe fixed the time constant for the absolute refractory tinuously and disappear; generically, however, the period at ~ =75, and we varied the value of the picture remains similar. Note that in this case the I... ~I ~ F a b C r :.~~i ~:.~ j j :~ ~ Fig. I. The spiral breakup in model (1). The pictures are ai time (a) 1=27, (b) 67. (c) 82. (d) 107, (ci III, (f) 183. Numerical integration with space step h, = 0.5 and time step h, = on the grid of 120x 120 elements. The black area represents the excited state of the tissue (e> 0.6), dark grey shows the region where g> 1.8 (close to the absolute refractory state) and intermediate shading from grey to white shows different levels of g, 0< g< 1.8 (estimate for ihe relative refractory period). 296

3 Volume 176, number 5 PHYSICS LETTERS A 17 May 1993 p Fig. 2. The spatial pattern that occurred spontaneously due to the spiral breakup in the excitable medium measuring 1000x 700 elements at the time t= All other settings are the same as in fig. I. process of spiral breakup also started close to the results of our computation. The first major breakup centre of the initial spiral, and spread out over the appeared in the same way as in figs. 3a and le. entire medium. For the breakup of spirals, it is not necessary for We studied the stability of spiral breakup at var- the two nullclines in eq. (1) to be parallel to each ious spatial and time integration steps. Figure 3a other (k=c 2=3); however, the slope of the nullshows the same computation as fig. 1 but with the dine for the slow variable affects the process of spispace step decreased five-fold and the time step de- ral breakup. We found the effect of breakup at k=4.5 creased 25-fold. In both cases the picture of the first and C2=3, but it was less pronounced. The first mamajor breakup is similar. (Compare fig. 3a and fig. jor breakup in this situation occurredat time t= 305, 1 e.) The error in these computations, estimated us- whereas in fig. 1 it occurred at I = 111. The threshold ing the difference between the computed and the sat- of the excitable medium (the parameter a) was also urated value for the velocity of plane wave propa- important for the process of spiral breakup. We did gation, is less than 0.2% for space step h5=0.l and observe the breakup when the threshold was inless than 5% for h~= 0.5. creased three-fold (a = 0.18); however, the first ma- Because the difference in the time constants of the jor breakup in this situation occurred later, at t= 360. absolute and relative refractory periods is important Another parameter that is important for spiral for the effect studied in this paper, we looked for a breakup is relative refractoriness (dependence of ~ possible source of numerical instability: the region of on the variables e and g). As we mentioned above, abrupt change from ~ and ~. We made a compu- increasing the time constant for the relative refractation in which we used linear interpolation from the tory period makes the breakup impossible at ~ value of~ =75 to the value of~ =2.8. In this case 5.5. the excitable tissue spends about 30% of time at the The effect of spiral breakup is not a unique propregion where ~- changes to ~ j. Figure 3b shows erty of model (1) with particular piecewise linear that this modification of recovery did not affect the Pushchino kinetics. Modifying the dynamics of the 297

4 Volume 176, numbers PHYSI( S LETTERS A I Mas 1 ~)Y~ recovery variable ( using two independent constants for relative and absolute refractory periods) svc v~ere -~ able to observe the effect of spiral breakup in ihc FitzHugh Nagumo model, with the standard function 1(e) in the form of a cubic parabola (/( ) -- 20e( c O.l) (e I)). We can see (fig 4) that ihc picture which occurs in this situation is sirnilai it figs. I and 2. I quite Theclear; mechanism however, underlying our case spiral it is connected bieakup is ~cith iiot functional heterogeneity of the tissue s~ithrespect to the refractory period, the heterogeneits. being in duced by the spiral itself. The patch in the refrauoi period (fig. Id) at which the wave latei breaks up can be seen clearly. We presume that this kind of hel erogeneity is associated with sonic kind of bifurca tion in the pulse propagation during high frequenc\ -- forcing. Figure 5 shows the time space course of one (a) dimensional wave propagation during forcing sviilt period i~=24.5 The length of the forcing period v -~ within the range of periods ofa meandering spiral at 0 these parameter values. We see (fig. 5 small osc ~ lations in the velocit~ and duration of the ~vcited _ H ~.. (b) Fig. 3. The first major breakup of a spiral wave ai various inicgralion steps (a) and recovery kinetics (b). (a) Compuiaiion ot eqs. (I) with h~=0.1. h,= , grid 600 x 600 clemenis. (h) Computation with linear interpolation from ihe value of = Fig. 4 1 he spatial pattern thai occurred spontaneously due to thi 75 to lhe saluc of ~ =2.8: (t. e) = when e.- t-~ and spiral breakup in the cscitable medium measuring 500xyOOclc- ~<2l: s(i.gl =r~ +(~ --v )(g. l.5)/om when menisatthetimci_2266 Numcrieal intcgrationwiththefun~and I.5<g<2.l:v(c,g)=~ whenc <e 1andc<it.5.h~-=(IS. tiont( )=20L I) I lit - I thcotherparametersarc =4 h,= grid I20x 12(1 elements: hoih pictures arc at time ~() 049 1) I8S 1 o All other seiiings are the same a~ /=111. in fig I

5 Volume 176, numbers PHYSICS LETTERS A 17 May ~ - - ~ ~,c ~ 1000!Z~~ 100 SPACE 150 Fig. 5. Pulse propagation during periodic forcing with period t~=24.5.h,=0.l, h,= ; gray scale coding is the same as in fig. 1. Space and time are measured in nondimensional units. The dotted line corresponds to moment 1= 1038 in time. state of the pulse, and large oscillations in the re- Acknowledgement fractory period. The important feature of these oscillations is the nonmonotonic dependence on space We are grateful to Professor J.P. Keener, Professor of the edge of the absolute refractory period. In fig. A.T. Winfree and M.C. Boerlijst for discussing im- 5 we can see clearly the maxima and minima in the portant aspects ofthis paper and to Miss S.M. McNab absolute refractory period profile and the excitation for linguistic advice. propagating steadily upward and to the right. This means that if we consider the spatial state ofthis onedimensional excitable medium at some moment of References time 1= 1038, we will find a refractory patch behind the propagating pulse. So, the patch can be consid- [II AT. Winfree, When time breaks down (Princeton Univ. ered as a heterogeneity with respect to refractoriness ~ B 3 (1978) 539. at this moment tn time. We think that such hetero- [3] M. Gerhard, H. Schuster and J.J. Tyson, Science 247 (1990) geneities break up the two-dimensional spiral. The other important feature of this patch heterogene- [41AT. Winfree, J. Theor. Biol. 138 (1989) 353. ity is that it disappears quickly, providing re- [5]A.V.Panfilov and A.V. Holden, Phys. Lett. A 151 (1990) covered space in which the new breaks can survive. [6] 23. A.V. Panfilov and A.V. Holden, mt. J. Bifurcation Chaos I Note that the spiral breakup observed here is sim- (1991) 219. ilar to the onset of the complicated spatio-temporal [71M. Courtemanche and A.T. Winfree, Tnt. J. Bifurcation patterns observed in cellular automata models for a Chaos. 1(1991)431. process of evolution [9] and for spread of excitation [81 H. Ito and L. Glass, Phys. Rev. Lett. 66 (1991) 671. in cardiac tissue [8]. Studies of the models [8,9] 191 MC. Boerlijst and P. Hogeweg, Physica D 48 (1991) 17. [101 M.P. Hassell, H.N. Comins and R.M. May, Nature 353 have also revealed that relative refractory period (1991) 255. plays an important role in the process of spiral [11] AT. Winfree, Chaos 1 (1991) 303. breakup. [121 A.V. Panfilov and AM. Pertsov, DokI. Akad. Nauk SSSR 274(1984) [13}A.V. Panfilov and J.P. Keener, J. Electrocard. Physiol. (1992), in press. 299