mesoscopic modeling of wave propagation in excitable media

PHYSICA EI~EVIER Physica D 79 (1994) 16-40

Mesoscopic modeling of wave propagation in excitable media

Hiroyuki Ito Department of Information and Communication Sciences, Kyoto Sangyo University, Kamigamo, Kita-ku, Kyoto 603. Japan

Received 18 October 1993; revised 21 January 1994; accepted 8 March 1994 Communicated by K. Kaneko

Abstract

A new theoretical model is introduced to abstract the significant factors leading to wave propagation in excitable media. Contrary to the previous theoretical models constructed based on the microscopic physical mechanism, our model is formulated based on experimentally measurable "mesoscopic" characteristics of the media: dispersion relation and curvature relation. Analytic investigations lead to the nonlinear curvature relation which is approximated by a linear relation for a small curvature. Numerical simulations incorporated with the experimental data of the above two characteristics in the Belousov-Zhabotinskii reaction reproduce spiral waves in agreement with experiment. Both stationary rotating spiral waves and a meandering motion of spiral tip are obtained in accord with other experimental and numerical studies. Our results suggest that both the dispersion relation and the curvature relation are important factors leading to those universally observed wave phenomena. The model can be used as a convenient numerical tool to understand the nature of wave propagation in excitable media through extensive numerical experiment.

1. Introduct ion

Spiral waves of excitation are universally observed in a wide variety of excitable media from

chemical reactions [1,2], cardiac tissue [3,4], neural tissue (depression waves in retina [5] and

cerebral cortex [6]), and to slime mold [7]. The

dynamical behaviors of these self-sustained

waves have attracted much attention for many years [8,9]. One of the main interest consists in a basic question "which common characteristics of such diverse systems lead to these universal

behaviors?"

] A part of this work was carried out when the author was staying at Department of Physiology, McGill University, Canada.

Traditionally theoretical studies of spiral

waves have been carried out mainly by using

nonlinear partial differential equations (PDEs) [10-12]. Recently, however, cellular automaton models with multiple states [13-16] and coupled

map lattices [17] have been proposed for faster

numerical computations. The important point is

that all of these models are based on (or inspired

by) the microscopic physical mechanism of gen- eration and propagation of excitation (e.g., molecular reaction and diffusion in chemical reactions, ionic channels in physiological systems, etc.). Therefore the parameters of these models have rather tight connections with the micro-

scopic physical quantities such as a reaction rate constant, an equilibrium constant, chemical con- centration and diffusion coefficient of each mole-

0167-2789/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI I )167 -2789(94 )00091-4

H. lto / Physica D 79 (1994) 16-40 17

cule. The models are used to simulate wave phenomena observed at a macroscopic level of the system such as spiral waves. Although these microscopic models are mathematically well defined, due to the complexity of their mathematical structure, analytical studies are difficult and most studies are carried out by computer simulations except for some studies of perturbation analysis [10-12, 18-20]. Further the forms of the PDEs are different for different systems in an object dependent way reflecting their specific microscopic mechanism. For example, the Hodg- kin-Huxley model [21] and the FitzHugh- Nagumo model [22, 23] were proposed for neural tissue, the Beeler-Reuter model [24] for cardiac tissue, the Oregonator model [25,26] for the Belousov-Zhabotinskii (BZ) reaction, and the Martiel-Goldbeter model for the slime mold [27] z.

In this paper, we introduce a new theoretical model from a different point of view. We need a reasonable motivation for this, because there have already been much studies by the PDEs on this subject. Our motivation is two-fold. First, we introduce a flexible computation tool that can be used to abstract the significant factors leading to universal behaviors observed in general excitable media. Although microscopic mechanisms are different in different media, similar behaviors of spiral waves are observed in a wide variety of excitable media. Therefore, in between the microscopic level and the macroscopic level of the system, there should exist some universal characteristics determining the physics of spiral waves. Since the PDEs have been constructed on a rigid microscopic foundation from the beginning, abstraction at the higher description level is no longer adequate. There- fore we need a new theoretical model that is constructed directly at the intermediate ("mesoscopic") level of the system.

The concept of the mesocopic modeling itself

2 Systematic comparisons among some of these different models were investigated by Winfree [28].

is not so novel. One classical prototype was the study of irregular dynamics in periodic stimulation of spontaneously beating aggregates of cardiac cells by Guevara, Glass and Shrier [29]. The detailed ionic mechanism of the membrane potential in cardiac cell is highly complicated. Currently most computer simulation studies adopt the Beeler-Reuter model [24] with six different ionic currents. Instead of starting from these microscopic mechanisms of the system, they formulated the model based only on a single mesoscopic characteristic of the cell. Although the experimental studies of the detailed ionic mechanism are not so easy, the latency of the action potential after the electric stimulus can be measured systematically. Their model is essentially a simple one dimensional map based on this phase resetting curve. They demonstrated that the irregular dynamics of the action po- tentials appeared in a macroscopic level can be well described by the simple model constructed in a mesoscopic level. The study of the microscopic mechanism leading to the phase resetting curve is also important. However, the existence of this curve itself is enough for our understanding of the essential mechanism of the phenomena. After their observation, similar behaviors have been reported in many other systems [30]. The abstraction of the mathematical structure in the mesoscopic level has facilitated the discovery of the universality in these phenomena.

In this study, we construct a model based on two well-known basic mesoscopic characteristics of excitable media 3. Those are dispersion rela-

3 It might be a delicate quest ion whether one regards these two relations as mesoscopic or not. In this paper , we use a term "mesoscopic" in the following sense. The two relations, which appear by the interaction among microscopic local kinetics of the excitable med ium, govern the local characteristics of the wave propagation. On the other hand, we believe that the nature of the macroscopic wave such as geometry or stability of spiral waves can be s tudied by the consistency conditions of these local characteristics in a global spatio-temporal extent. The emergence of physical event at each level of the system is not trivial f rom the lower level. Therefore our definition of "mesoscopic" is subjective reflecting our concept in model ing the excitable media.

18 H. lto / Physica D 79 (1994) 16-40

tion: the propagation velocity of excitation wave front depends on the recovery time elapsed since the passage of the last excitation, and curvature relation: the normal propagation velocity of the wave front depends on its curvature. However, in principle, the model can be formulated based on any mesoscopic characteristics. If the model successfully simulates wave phenomena in accord with actual experimental data and numerical results of more complex theoretical models, the selected characteristics are proved to be really essential to these phenomena. If not, some important characteristics are still missing in our mesoscopic modeling.

As our second motivation, the present model provides a more interactive connection between experimental data and theoretical studies than the previous approaches. For the reproduction of wave phenomena observed in actual excitable media, the connection of the PDEs with the experimental data is still not so straightforward, because some of the important microscopic parameters remain poorly measured experimentally [31]. On the other hand, the mesoscopic characteristics of the media are more accessible experimentally. The experimental data of dispersion relations are available in the BZ reaction [31-33], cardiac tissue [34,35], neural tissue (retina) [5], and slime mold [36]. The data of curvature relations were also obtained in the BZ reaction [32] and slime mold [37]. Since the present model is based on these experimentally measurable characteristics of the media, we can explicitly incorporate these data to study macroscopic wave phenomena and can compare the results directly with experimental data. In fact, in electrophysiology laboratory, the change in wave phenomena is often discussed with the associated change in the mesoscopic tissue characteristics [35]. We can easily examine this correlation by numerical simulations of our model even if the microscopic mechanism of this change is still unknown [38].

We believe that the present model facilitates further understanding of universality in excitable

media from a different perspective than the traditional approach by the PDEs. Our model has a very flexible structure to examine the significant factor leading to the observed phenomena. There is a serious difficulty in the PDEs for this purpose, because a change in one model parameter usually affects several properties of the media in a complex way. For example, our model can easily realize the special case when the media has a curvature effect but the propagation velocity does not depends on the recovery time (dispersionless media). It is an interesting question whether the instability to meandering motion of the spiral tip would appear even in such a case.

Fig. 1 shows the experimental curve of (a) dispersion relation and (b) curvature relation in the BZ reaction [32]. Fig. la plots the velocity of the plane wave Csz as a function of the recovery time T. The solid line is obtained by a least square fitting of experimental data assuming the functional form

c ( T ) = l / [ a e x p ( - T / / 3 ) + y ] , i fT>-O, (1)

where c~,/3, and y are positive parameters. After the excitation, the medium needs a certain time interval called the refractory period 0 to recover

(a)

lO. 2f .... , 1 o.lo° u 0.08 • Z

0.05

o 30 60 ooo

T (s)

(b)

i Kcr !

/

" \ i l + I

" 2~) ~ 0

K (ram -I) Fig. 1. Two mesoscopic characteristics in the Belousov Zhabotinskii reaction. (a) Dispersion relation: fitted curve of the experimental data in Ref. [32] by Eq. (1). The conduction velocity of plane wave front c is a monotonical ly increasing function of the recovery time T. The excitation cannot occur when T is less than the refractory period 0. (b) Curvature relation: linear relation given by Eq. (2) was assumed in Ref. [32]. The normal velocity of the curved convex wave front N decreases as the curvature K is increased. The propagation cannot occur when K is larger than the critical curvature K .

H. Ito / Physica D 79 (1994) 16-40 19

its ability to conduct excitation. The data were well fitted with the parameters aBz = 21.7 s /mm, /3az = 11.1 s, and 3'Bz = 8.79 s /mm and we esti- mate 0Bz = 15 s from the experimental data. This relation reflects slow conduction velocity for short recovery time due to incomplete recovery of excitability in the medium. On the other hand, the normal propagation velocity of a circular wave N is plotted as a function of its curvature K in Fig. lb. In general, the theoretical curvature relation is given by a nonlinear function of K that is approximated by a linear relation only in a limited region of small curvature. However , in the only available experimental report , Foerster, M/iller and Hess assumed a linear relation over an entire range in their data. In Fig. lb , we plot their linear relation

N = c o - D K , (2)

where c o -- 0.0825 mm/s and D = 2 x 10 -3 mm2/s are the asymptotic wave velocity in the fully recovered medium and the linear slope, respectively. As the curvature is increased, the normal velocity decreases until the critical curvature Kcr above which no outward propagation is possible. They concluded that the linear slope D was close to the diffusion coefficient of autocatalytic species. The linear curvature relation can be derived by the PDEs only in a singular perturbation limit when the dynamics of the excitation variable is much faster than that of the recovery variable [10,12]. As one of the important find- ings in this work, we obtain the exact analytic form of the nonlinear curvature relation that is approximated by a linear relation for a small curvature.

In this paper, we introduce our theoretical model by demonstrating how the model can reproduce the spiral waves in the BZ reaction based on the experimental data of the two mesoscopic characteristics given in Fig. 1. In principle, our model can be used to reproduce the wave phenomena in any excitable media, if both the dispersion relation and the curvature relation are measured experimentally. In this

study, however, we restrict our attention only to the BZ reaction because these two characteristics have been best measured in this system. In Section 2, we introduce our theoretical model. The method of numerical integration of the model is explained applying an event scheduling algorithm. The model is essentially a continuum model both spatially and temporally. Section 3 is used to explain how we can adjust the model parameters to realize the given dispersion relation in Fig. la. The exact analytic form of the curvature relation is obtained in Section 4. Details of the analytic computations are given in Appendices. Using the results in these sections, the model can be adjusted to reproduce the experimental data of both the dispersion relation and the curvature relation. And then, in Section 5, we simulate spiral waves, that is the macroscopic wave phenomena. The reproduction of the spiral waves in the BZ system is examined and the results are compared with the experimental data. The model supports both the stationary rotation and the meandering motion of spiral waves. We discuss the results in Section 6. A part of the present results has been summa- rized in a letter article [39].

2. Model

Our final goal in the modeling is to construct a computer model that satisfies the following re- quirements. The model has a designed computa- tional algorithm and some system parameters. We input the experimental data of the mesoscopic characteristics (dispersion curve and curvature curve) into the model. The computer model automatically adjusts its internal parameters so that it can simulate the wave propagation satisfying the given mesoscopic characteristics. The important point is that the internal computer algorithm may have nothing to do with the microscopic physical mechanism leading to those mesoscopic characteristics. The internal mechanism of the computer model can be anything as

20 H. Ito / Physica D 79 (1994) 16-40

long as it can reproduce those characteristics correctly. This is because we are interested in not the detailed mechanism below the mesoscopic level but the nature of macroscopic wave propagat ion in the minimal system realizing the given mesoscopic characteristics.

We consider a sheet of excitable medium with the size L × L. Although our model is defined on this spatially continuous medium, we introduce a discrete grid for numerical integration by a digital computer . This is exactly the same procedure as the space discretization in numeri-

cal integration of the PDEs. The two-dimensional sheet is divided into a regular square lattice of N × N lattice points with a lattice constant d. At each lattice point there is an excitable element (cell) with identical property. The term "cell" is used here in a technical sense as in cellular au tomata . Each cell is directly connected to all the neighboring cells within a circle of an interaction radius r by a conducting cable. The cell can be excited by incoming pulses from exciting neighboring cells. When a cell gets excited, it emits an excitation pulse which conducts to its neighboring cells through the conducting cable.

We assume that the conduction velocity of the pulse v depends on the recovery time T, that is the time interval from the preceding excitation to the present excitation at the cell which is emit- ting the pulses. The conduction is assumed to be isotropic. The cell gets excited when a sequence

of pulses exceeding a finite threshold number Nth arrives after the refractory duration with a high frequency so that all the Nth'S pulses arrive within a finite t ime interval (window time), t w. The next excitation will be induced at the instant

when Nthth pulse arrives at the cell. The continuum limit of the model should be

taken because the model was originally defined on the continuum medium. This can be realized when the lattice constant d goes to zero keeping the quantities L = Nd, r, t w, and the ratio p = Nth/#nn constant, where # . , is the number of neighboring cells within the interaction range [w(r/d) 2 in the continuum limit]. The cell is more

excitable with smaller p and larger t w. Although a finite d is used in the actual numerical simulations presented in this paper , we are always interested in only the result in the continuum limit. In every simulation, the smallness of d is checked so that the result does not depend on the value of d and the simulations well reproduce the result in the continuum limit. In fact, our choice of d in the reproduction of the BZ spirals

shown in Section 5 was 0.01004 mm, that is even smaller than the space discretization used in the numerical integration of the Oregonator model

[311 (0.0225 mm). The integration of dynamics is pe r formed

based on the event scheduling algorithm [40]. To carry out the numerical simulation for each cell, one must keep track of the time of the last excitation t~a~, and the time of the next excitation tn~xt and store the arrival t ime of previously incoming pulses, {AT ' } ( i = 1 . . . . ), where i represents a temporal index of arriving pulses and smaller number of i corresponds to an earlier arriving pulse. The integration proceeds in the following steps. At any time t, the cell

with the smallest tnext > t is excited. The t ime variable t jumps to t,e×, and the stored data {AT ' } of this cell is reset. For each neighboring cell, the arrival t ime of the pulse from this cell is calculated by using the distance between the two cells and the conduction velocity v (T) , where

T = tnex, - tlast is the recovery t ime of the cell. If the pulse arrives within the refractory period of the neighboring ceil, the pulse is blocked and the arrival t ime is not stored in { A T ' } of the cell. Otherwise the arrival t ime is registered in {ATe}

after a sorting to keep a temporal order. If there

exits the smallest j that satisfies ATj+N," 1 - A T j < t w, the value of t.cxt is replaced by ATj+N,h 1- Finally t~st and t . . . . of the exciting cell are replaced by the present t ime t and ~, respectively. One needs to store only the arrival t ime {AT,.} that satisfies AT" > t - t w. This saves memory area of computers . Although each cell is assumed to take one of the three states (excitable, excited and refractory) at any time, these

H. Ito / Physica D 79 (1994) 16-40 21

variables do not appear explicitly in the above

algorithm. Since the time of excitation t n e x t at each cell can take any real number , the present model is essentially a continuum model not only spatially but also temporally. This proper ty is significantly different from the discrete time steps in cellular au tomata and standard coupled map lattices that makes analytical approaches difficult.

3. D i spers ion re lat ion

A single cell needs to wait for many incoming pulses for its excitation. This leads to a conduction delay of the wave front in the wave propagation. The propagat ion velocity of the macroscopic wave front is smaller than the local

conduction velocity of pulses between the neighboring cells. At first, the functional form of the local conduction velocity of excitation pulses v (T ) should be determined so that the plane wave satisfies the dispersion relation of the BZ reaction CBz(T ).

We consider the periodic passings of plane waves with a t ime interval T in the continuum

sheet of excitable medium. As shown in Appen- dix A, we can explicitly calculate the domain of neighboring cells that contribute to the excitation of the cell. A self-consistency condition between the local conduction of excitation pulses and the

propagat ion of plane wave leads to a scaling relation,

v ( r ) = CBz(r ) /cos plr . (3)

Thus u(T) is given by the relation (1) with

the parameters a =cos(pax)aBz, /3 =/3Bz, Y = cos (p~)yaz , and 0 = 0az to realize the dispersion relation CBz(T ). It should be noted that since any specific functional form is not assumed for CBz(T ) in relation (3), the model can be adjusted to realize any given dispersion relation.

We confirmed the dispersion relation by numerical simulations of the model. A periodic boundary condition is imposed on both opposite

sides of the lattice of 20 × N r cells (i .e. , torus shape). We initiate a uni-directional circulation of plane wave along the y-direction and calculate

the steady state circulation period T O after the initial transient dies out. By changing Ny, we get

the relation between the recovery time T O and the propagat ion velocity of plane wave c s = Zy/ T 0, where Ly- -Nyd . Interestingly we find that the simulation on the regular lattice with a finite d does not reproduce the dispersion relation in

Fig. la. The regularity of the lattice leads to a systematic deviation from the cont inuum limit in which relation (3) was obtained. In fact, we can obtain a different analytical scaling relation f rom

Eq. (3) in the case of the regular lattice with a finite d. This is identical to Eq. (3) only when d

goes to zero. We discover that the randomizat ion of the cell positions [15] is quite effective to reproduce the results in the cont inuum limit even using the system with a finite d. The position of each cell deviates randomly f rom its regular position by &r in x-direction (By in y-direction), where both ~x and 8y are independently Gaus- sian distributed with mean zero and standard deviation o-. However the cells at the edges are fixed on the regular lattice to keep the total size L x and Ly constant. The open squares in Fig. 2a show the results with r = 0.1 mm, p = 0.3714 and

(a) (b)

~ - ~ ~.0

0.081 J

004 / , 0.8 • 0 3'0 60 0.2 0.5

T (s)

Fig. 2. (a) simulations

Dispersion relation reproduced by numerical of the theoretical model (open squares). The

fitted curve based on experimental data in Fig. la is shown by a solid line. (b) The steady conduction velocity of the numerical simulation c s scaled by the theoretical value CBz is plotted as a function of the parameter p• The medium is more excitable with smaller p. In (a) and (b), each data point with an error bar is the result of the average over 10 samples with different random distributions of the cells•

22 H. lto / Physica D 79 (1994) 16-40

o-= 0.3d ( d - ' Nth 42, and #nn ']T(F/M) 2 - - ~ r , : =

113.1). The center of the open square is the average over 10 samples with different random distributions of the cells. The data show an excellent agreement with the dispersion curve Cnz (solid line) and the statistical error is barely visible. The dispersion relation was checked also with different excitability p by changing Nth in a range 30-< N,h--< 49. Now we fix N v = 402 (Ly = 6.7 ram) and plot the steady conduction velocity c s scaled by the theoretical value CBZ as a function of p = N t h / ~ J ~ n n in Fig. 2b. The steady recovery time in the continuum limit T0(--59.57 s) is given by the root of the transcendental equation T 0 = Ly/%z(To). The model reproduces the dispersion relation fairly well even with the coarse grid r/d = 6 except for the cases with very large Nth ~---46. The randomiza- tion of the cell positions is considered to elimi- nate the artificial discreteness effect by the local statistical averaging over many fluctuating quantities.

217 12 -- vFi -_ r/2

× {1 - ~[1 +½e(1 +r/)][1 - ½ e ( 1 - ~ ) ] } ,

2(3r/2 ~21) ( I ~ / 1 + 13_ - 1 ~ tan- i - n n

+ o)[1- ½e(1- - t a n V ( 1 - r / ) [ 1 T ½ e ( 1

(5)

and e = rK is a scaled curvature. Fig. 3 shows the curve N(K, T) obtained by solving this transcendental equation numerically with different values of p [(a) p = 0.3118, and (b) p = 0.4287]. When p > 0 . 2 5 , N(K, T) is a concave down function of K and decreases monotonically from Cnz at K = 0 with a linear slope and ends at Kcr taking N = 0. A perturbation expansion of Eq. (5) in case of small curvature (e << 1) leads to a familiar linear curvature relation

N ( K , T ) ~ c B z ( T ) - D ( T ) K , K<<l /r , (6)

4. Curvature relation

In two dimension, the curved wave front propagates with a velocity that depends on its curvature [10-12]. The normal propagation velocity N(K, T) of the curved convex wave front with a curvature K (>0) can be derived analytically in a similar way as the dispersion relation. After some straightforward computations (see Appendix B), we can get the relation

N(K, T) = (r//cos p~r)CBz(T ) , (4)

where r/ is the root of the transcendental equation,

- l ( 2 r / + e(1 + r /2 ) ) p~r = cos 2(1 + er/)

1 + (er/)2 (11 + 12 + / 3 ) , P > 0.25,

I, = c o s - ' r / - c o s - l ( 2r/+ e(1 + r/z)) 2(1 + er/) '

where D(T) = [sinZp'rr/(3 cos p~r)]rCBz(T) is the linear slope at K = 0. The details of this perturbation analysis are also given in Appendix B. As seen in Fig. 3, the curvature relation is well

(a) + - ~ ~ , - - --+ I I

1 I

£.) -~ ~" 7 7

0 1 ~' ½ £ c r

8

(b) t l l + ~ +

i

2 I

°O i ' 8 o r

8

Fig. 3. Curvature relation: normal velocity of curved convex wave front N, scaled by the velocity of the plane wave CBz, is plotted as a function of the scaled curvature e - rK (solid lines). The two figures show the relations with the different values of p: (a) 0.3118 and (b) 0.4287. The wave front cannot propagate with a curvature greater than the critical curvature ecp The linear curvature relation shown by dashed lines approximates the nonlinear curvature relation in the region of small curvature.

H. Ito / Physica D 79 (1994) 16-40 23

approximated by a linear relation in a wider range when the excitability of the medium is very low with larger p (~<0.5).

The critical curvature Kcr above which the curved wave front fails to propagate can be calculated by the root of another transcendental

equat ion for ecr = rKcr ,

~(1 2 -- ecrP) -- (2 - e~r ) - 1 1 cos ( ~ . )

_ecrV 1 _ 1 2 XScr = 0, (7)

that can be derived f rom Eq. (5) by imposing rl = 0. The solid line in Fig. 4a shows the depen-

dence of the critical radius R c r - - l / g c r on the pa rame te r p in a range 0 . 2 5 < p < 0 . 5 . The critical radius obtained by assuming the linear

curvature relation in Eq. (6), D ( T ) / c B z ( T ) =

[sin2p'rr/(3 cos p l r ) ] r , is also plotted by a dashed line. Since the nonlinear curvature relation is a concave down function, the critical radius obtained by a linear extrapolation by Eq. (6) always leads to an underestimation. As discussed in Appendix B, when the excitability of the med ium is very high with p < 0.25, the nonlinear curvature relation is no longer a monotonically decreasing function of the curvature. It is a two valued function in some range of the curvature. Also in Appendix B, we discuss the effect due to a finiteness of the window time t w, which leads to a minor correction to the critical curvature.

The critical radius Rcr can be confirmed by numerical simulations. At ' t = 0, all the cells within a circle of radius Rstim at the center of the system are excited. The critical radius Rcr is the

smallest value of Rstim that leads to an outward propagat ion of the circular wave. In this simulation, each cell is fully recovered before its excitation (i.e., T--oo). The critical radius aver-

aged over 30 samples with different random distributions of the cells is plotted by open squares in Fig. 4a [r -- 0.1 mm, 0.265 -<p -<0.433

(30 -< Nth -< 49), tr = 0.3d, N = 101 and d = ~r]. The data show a good agreement with the theoretical value by the nonlinear curvature

relation. A systematic deviation is observed in the range of larger p.

The linear slope at K = 0, D in Eq. (6), was also checked by numerical simulations. The circular wave is initiated at t -- 0 by the excitation of the cells at the center within the stimulation

radius Rstim and the distance f rom the center R and the time of excitation t is measured for each cell. The relation between the two quantities is well fitted by the theoretical curve,

D c B z R - D R - R~tim + In - CBZt (8)

CBZ ¢BzRstim -- O

that is obtained by assuming Eq. (6). Fig. 4b

plots the quantity B =/ ) / ( r~ 'Bz ) averaged over 30 samples, where D and C-Bz are the fitted

-.b. b rr

(a)

2.501 ' ' /

125 t j / '

0.00 ~- ~', ...... ~ ," 0.2 0.5

rn

(b) 2.50 ' t / 1.1

1.0 1.25

0.00 l J 0.2 0.5

P P

(c) ,, 0.8 , 0.2 0.5

P Fig. 4. Comparison between the numerical simulation and the theoretical analysis. Dependences of various quantities on the parameter p are examined. (a) Critical radius Re, scaled by the interaction radius r: theoretical curve given by Eq. (7) using the nonlinear curvature relation (solid line), theoretical curve given by the linear curvature relation in Eq. (6) (dashed line), and the data of numerical simulations (open squares). (b) Scaled linear slope of the curvature relation at e = 0, B -~ D/(rCBz): theoretical curve by B = sin2p~r/(3 cos per) (solid line) and the data of numerical simulations (open squares). (c) Asymptotic normal velocity CBz of the numerical simulations scaled by the theoretical value CBz. In (a)-(c), the numerical data with an error bar were obtained by the average over 30 samples with different random distributions of the cells.

(a) (b)

24 H. Ito / Physica D 79 (1994) 16-40

Fig. 5. Numerical s imulations of the propagation of curved wave front. (a) Spread of a circular wave initiated by the forced excitations at the center. Each wave front represents the successive excitations of cells during a time interval of 0 . 4 s separated by 2.4s . L - 1 .68mm ( N = 101) and R~,~,, = 0.1 mm. (b) Collision of two circular waves initiated by the excitations at the distant sites. The successive excited cells during a t ime interval of 0.4 s separated by 1.2 s are overwritten. L = 1 .67ram ( N = 100), R,,~m = 0 . 0 9 m m and the dis tance between the two stimulation sites was 0.67 ram. In both simulations, the parameters were r =0 .1 ram, p = 0.3714 (N .h=42 , # , ° = 1 1 3 . 1 ) , ~ = 0 . 3 d , d = ~ r and tw= 4 .0s .

realistic wave fronts. Fig. 5a shows the spread of a circular wave initiated by the excitations at the center. The successive excited regions during a time interval of 0.4s separated by 2 .4s are overwrit ten to animate the spread of the circular wave. The model well simulates the round curved wave front. Since the front with a larger positive curvature propagates more slowly, the

spatial interval between the successive wave fronts in Fig. 5a increases as being apart f rom

the center. The simultaneous excitations at the two distant sites lead to a collision of the two circular waves (Fig. 5b). The cusp structure created by the collision evolves into a plane wave immediately because of its large curvature. The

shape of the wave front approaches to a circle far away from the center.

5. Simulation of spiral waves

paramete rs a. The numerical data show a good agreement with the theoretical value of B = sinZp'rr/(3 cos p~r) (solid line in Fig. 4b). A small

systematic deviation of the data from the theoretical line is considered to be related with an increase of nonlinearity in the curvature relation

as p is decreased. The fitted velocity ~Bz shown in Fig. 4c is in good accord with the theoretical

velocity of plane wave % z ( T = ~) = 1/Y~z except for the cases with larger p(~<0.5). Both the dispersion relation and the curvature relation were checked by the simulations with the finer grid sizes, both of r / d = 7 and 8. We confirmed that the grid size of r / d = 6 is small enough to reproduce the results in the continuum limit.

Curvature effect is essential in simulations of

4 The fitting by Eq. (8) is only valid in the region of relatively small curvature where the nonlinear curvature relation in Eq. (5) is well approximated by the linear relation (6). There- fore, for each N,, (30-< N,h--< 49), we chose an adequately large st imulat ion radius R,,~m such that the deviation between the nonl inear relation and the linear relation in N(K = 1/ Rs.m)/CBz was as small as 0.02 (this deviation is zero and N(K)/c~z = 1 at K = 0).

In the previous sections, we have confirmed that our model can realize both the dispersion relation and the curvature relation. The analytical results in Eqs. (3) and (6) are used to adjust the parameters of the model to reproduce the given data of the two characteristics. In this section, we simulate the macroscopic wave phenomena, spiral waves, based on the exper imental data of the two mesoscopic characteristics in the BZ reaction. Since the dispersion relation of the BZ reaction % z has been reproduced in Section 3, we need to realize the curvature relation of the experimental data. In Section 4, we showed that the curvature relation is given by a nonlinear function in general. However , in the only available experimental report of the curvature relation, Foerster et al. [32] showed that their data were approximated by a linear relation [Fig. lb]. Therefore in order to realize the exper imental data, we adjust the parameters so that the linear slope at K = 0 in the curvature relation in Eq. (6) is equal to the experimental data (DBz = 2 × 10 3 mm2/s) . We find that the change in the

linear slope D leads to the transition from the

H. lto / Physica D 79 (1994) 16-40 25

stat ionary rotation of spiral waves into non- stat ionary (meandering) spiral waves. These two phenomena are discussed in Sections 5.1 and 5.2, respectively.

5.1. Stationary rotation

In the previous sections, the nature of the wave propagat ion was not influenced by the time

interval t w as long as t w was not too small. However , the proper ty of spiral waves strongly

depends on t w because spiral wave adjusts its geomet ry (rotation period and core size) depending on the excitability of medium. With the

paramete rs r = 0 . 1 m m , p = 0.3714 (Nth = 42, #nn = 113.1), d = l r , o ' = 0 . 3 d , the system of 1.67 m m x 1 .67mm (100 × 100 cells) supports a stat ionary rotating spiral wave with constant rotat ion period T O over a wide range of t w (2.0 s --- t w <- 6.0 s) that we have examined 5. Fig.

6 shows a single snapshot of the spiral wave front (t w = 4.0 s and T O = 23.1 s). The filled circles represent the excited cells during a t ime interval

of 1 s accompanied by the counter-clockwise revolution of the wave front. To initiate the spiral wave, the simulation starts from a broken wave line propagat ing downward. We take the open boundary condition, that is, the cells on the boundary are connected to only the inside cells. Behind the wave front there exists the refractory tail where the cells are still in the refractory state (open circles in Fig. 6). In case of stationary rotation, both the excitation wave front and the bo t tom of the refractory tail revolve with the

5 In the numerical simulations, the simply rotating spiral solution may coexists with the other spiral solutions whose tip travels the closed non-circular line. This might be related with mode-locked meander ing motion. In Figs. 8a-c , we averaged the data only over the samples of stationary spirals. A l though Jahnke and Winfree [41] also reported the mode- locked meander ing mot ion in the simulation of the Oregona to r model , its existence has been confirmed neither in exper iment of the BZ reaction [42] nor in other simulation studies [17].

Fig. 6. Snapshot of propagation of a stat ionary rotating spiral wave. The direction of rotation is counter-clockwise. The filled circles and the open circles represent the excited cells during a t ime interval of 1 s and the cells in the refractory state, respectively. Between the excitation wave front and the bot tom of the refractory tail, there exists an excitable gap (white region) where the cells are excitable. With t w = 4 s and the refractory period 0Bz = 15 s, the steady rotation period T 0 takes 23.1 s. r = 0.1 mm, p = 0.3714 (Nth = 42, # . n = 113.1), tr =0 .3d , L = 1 .67mm ( N = 100) and d -

same speed. Therefore the size of the excitable gap (white region in Fig. 6) between the two boundaries remains constant. Fig. 7a also shows the excited cells during a time interval of i s accompanied by the revolution of the same spiral wave as shown in Fig. 6. The successive excited regions separated by 3.6s are overwritten to animate the revolution of the spiral. The spiral wave fronts shown in Fig. 7b were obtained by the simulation with the same parameters except

(a) (b)

Fig. 7. Stationary rotation of spiral wave. (a) Simulation on random lattice (o- = 0.3d). Each region shows the successive excitation of cells during a t ime interval of 1 s separated by 3.6 s. Same spiral wave as shown in Fig. 6. (b) Simulation on regular lattice (tr = 0).

26 H. lto / Physica D 79 (1994) 16-40

for o-= 0, that is, on the regular lattice. Ran- domization of the cell positions eliminates the anisotropy of the regular lattice. Around the center of the spiral, there exists a self-organized circular core region in which cells cannot be excited because the high curvature of the tip of spiral wave prevents these cells from receiving more than Nth pulses within t w.

We measured the steady rotation period T 0 and the core radius, rq = V~q/W d, after spiral rotation became stationary. The quantity #q is the sum of the number of cells forming the core and that of cells on the tip trace surrounding the core. The position of the tip is determined by the cell that has a non-excited nearest neighboring cell during a reasonably long time interval around its own excitation. Figs. 8a, b, respectively, show rq and T o as a function of t w. The relation between rq and T O is plotted in Fig. 8c. Each data point is the average over many samples (10- -30) . Both the core radius and the rotation period decrease as the excitability of the medium is increased. Tyson and Keener [12] have derived an approximate formula for the relation between T O and a radius r 0,

1 + (4cro/D1) - ~/1 + (8cro/D,)

where c and D 1 are the plane wave velocity and the diffusion coefficient of the excitation (fast)

variable, respectively. This formula can be re- written into our format,

= ( 8~rr~ ) l + ~ b

T,, \CBz~0) 1 + 4~'b - ~/1 + 8~'b '

where (=r~j/r and b = 1/B. The quantity r 0 is the radius where the spiral wave front has zero tangential velocity. This relation is valid only in a singular perturbation limit and for the P D E with non-diffusive recovery (slow) field [19, 43]. In the singular perturbation limit, the spiral wave becomes coreless (rq = 0 ) and r 0 is the only relevant quantity characterizing the geometry of the tip. However the relation between the two q u a n t i t i e s rq and % in the system with the general parameters (not in the singular perturbation limit) is still unclear. The geometry of spiral waves teaches that r 0 is a little larger than rq.

Irrespective of these problems, we think that comparing our results with this formula gives some information on qualitative validity of our model. The relation between r 0 and T 0 by the above formula is shown by a solid line in Fig. 8c. By using this relation and the data in Fig. 8b, we get the dependence of r 0 on t w in Fig. 8a. Our result for stationary spiral waves is qualitatively consistent with the result of the PDEs. However , the detailed quantitative comparison should wait for further studies until the connection of the Keener and Tyson's result with general excitable medium would be clarified.

Z

(a) /~. I ) I

2 "."X ro

0 2 2 6

tw (s)

(b) 4O

~9 30 ,E

20 2 4 6

tw (s)

(c) 4 ~---+- , ,/,

! / / t 2 ,' i

/ / " t 1

20 30 40

TO (s)

Fig. 8. Quant i ta t ive characterizations of stationary rotating spiral waves. Dependences of (a) core radius rq (scaled by the interaction radius r) and (b) rotation period T O on the window time t w. (c) Relation between r and T 0. The radius % given by the T y s o n - K e e n e r formula is also plotted in (a) and by a solid line in (c). In (a)-(c), numerical data with an error bar were obtained by the average over many samples (10 -- 30) with different random distributions of the cells. The parameters were the same as the s imulat ion in Fig. 6 except for t w (2 s <- t~ -< 6 s).

H. Ito I Physica D 79 (1994) 16-40 27

5.2. Meandering

The simulations in Section 5.1 were carried out with the linear slope in the curvature relation D = 6 - - 7 × 10 -3 mm2/s that is larger than the value of the BZ reaction (DBz = 2 x 10 -3 mm2/

s). The model has three parameters, r, p , and t w that should be determined to realize the spiral wave in the BZ reaction. Since the linear slope D in our model depends on CBZ, we need either the rotation period of the spiral wave T O or the asymptotic normal velocity %z(T0) from the experimental data to adjust the value of D. By using T 0 - 1 7 . 3 s [32], we determine r = 0.06024 mm and p = 0.3183 (Nth = 36) to realize DBz. Since D depends on both r and p, it is impossible to determine the unique set of (r, p) only based on the linear curvature relation. However , the difference appears only in the region of a large curvature where the nonlinearity in N(K) is dominant. Here we choose them for a numerical convenience. Contrary to the experimental observation, in the numerical simulations with these parameters, we do not obtain a stationary rotating spiral wave over a range of 1.5 s -< tw -< 6.0 s that we have examined. Fig. 9a shows the excited cells during a time interval of

(a) (b)

Fig. 9. Meandering motion of spiral wave. (a) Each region with a temporally ordered number shows the successive excitation of cells during a time interval of 0.5 s separated by 2.0 s. The size of the box is L = 1.004 mm. (b) Trace of the tip of the spiral wave over 73 s. The tip starts from the open square and ends at the closed square. The size of the box is 0.75 mm. In these simulations, we used r = 0.06024 mm and p = 0.3183 (N~h = 36) to adjust the linear slope in the curvature relation D in Eq. (6) to the experimental data DBz (2 >( 10 -3 mm:/s) , tr = 0.3d, d = ~r and t w = 1.5 s.

0.5 s accompanied by the counter-clockwise revolution of spiral wave (tw = 1.5 s). The successive excited regions separated by 2.0 s are overwritten. The number attached with each snapshot represents the temporal order. Now the tip traces a flower-petal pattern as shown in Fig. 9b and the trace does not close. This behavior known as meandering is a generic property of the model and not a numerical artifact.

The successive movements of the wave front and the refractory tail are animated in Figs. 10a-i. In each figure, the excited cells during a time interval 0.5 s are shown by filled circles. The elapsed time after the event in Fig. 10a (t = 0) is shown below each figure. In Fig. 6, the movements of the wave front and the bot tom of the refractory tail show a concordance to per- form the stationary rotation of spiral wave so that there is no collision between them. How- ever, such a concordance is no longer the case in meandering spiral. Because of the non-stationary revolution of the wave front, the refractory tail temporally shows a spatial inhomogeneity. The wave front cannot reenter as in the stationary rotation because the front collides with the refractory tail (Fig. 10e). The tip suffers a sliding motion along the wall of the refractoriness (Figs. 10e-g) until the refractory tail is finally wiped out after the refractory period 0az = 15 S (Fig. 10h). The non-stationary motion of the wave front is caused by this interaction between the wave front and the refractory tail and again it leads to the temporal inhomogeneity in the refractory tail. Such a circulation repeats forever and the tip traces a flower-petal pattern as seen in Fig. 9b. The meandering motion has been widely observed both experimentally [31,42,44] and numerically [31,45,17,41]. The motion of the tip shown in Fig. 9a is very similar to the experimental data in plate I(e) in Ref. [42] and in Fig. 3 in Ref. [44]. Both the bifurcation mechanism to the meandering motion and the dependence of the system parameters on meandering patterns have been topics of great current interest. Many systematic studies were carried out

28 H. lto / Physica D 79 (1994) 16-40

(a) (b) (c)

t= 0.0 2.25

(d) (e)

6.75 7.5

(g) (h)

4.5

(f)

8.25

( i )

9.0 9.75 10.5 Fig. 10. Interaction between the wave front and the refractory tail in meander ing spiral. Each Figure of (a) - ( i ) shows the successive snapshot of the excited cells (filled circles) during a time interval of 0.5 s starting from the time shown below it. The open circles and the white region represent the cells in the refractory state and the excitable cells, respectively. The time interval be tween the successive snapshots is 2.25 s in (a)-(d) and it is decreased to 0.75 s in (d ) - ( i ) to show the collision between the wave front and the refractory tail more in detail. The parameters were the same as those used in the simulation in Fig. 9.

H. Ito / Physica D 79 (1994) 16-40 29

both experimentally [31,42,44] and theoretically [28,31,45,17,46,41,47,48] by using the PDEs. As we increase t w to make the medium more excitable, the path of the tip approaches to a straight line with a sharp turn around .leading to the movement in the opposite direction. There- fore the tip tends to spend a large amount of time to complete a "flower" pattern with a large number of "petals". Similar behaviors have been reported in the simulations of the PDE [41] and the cellular automata [16].

Although meandering spirals are commonly observed in the BZ reaction, the model predicts a meandering spiral in a parameter region where the experiment shows a stationary one. This unfortunate disagreement suggests a prematurity of the model in a quantitative discussion that should be addressed in the future study. The possible origin of this discrepancy is discussed in the next section. By numerical simulations using the Oregonator model, Tyson and Keener [12] were also unsuccessful in obtaining a stationary rotating spiral. They characterized the meandering spiral by a mean rotation period T0 and an effective core radius ref f within which the meandering tip was confined. They pointed out a quantitative agreement between their results (T 0 = 2 0 s and ref f = 0 . 1 8 m m ) and the experimental data of the stationary spiral in the BZ reaction (T O = 17.3s [32] and a core radius 0.09 mm [2,14]). In our model, by adjusting the final free parameter t w to 1.7 s, the mean rotation period T0 can be about 17.3 s that is very close to the above experimental data. The meander of the tip is now confined to a region of radius reff--0.13mm. Although we obtained better quantitative agreement than the Keener and Tyson's result, the validity of these quantitative comparison is skeptical, because the predic- tions by these two models (appearance of meandering spiral) are already wrong qualitatively. Such a quantitative comparison may be of some value only when the model parameter is close to the onset of the bifurcation to the meandering motion so that the amplitude of meander is still small [19].

6. Discussion

We developed a new theoretical model for wave propagation in excitable media. Although the previous models were constructed based on the microscopic physical mechanism of the excitable media, our model is constructed based on the experimentally measurable mesoscopic characteristics of the media (dispersion relation and curvature relation). A theoretical framework is obtained for adjustment of the model parameters to reproduce the given dispersion relation and curvature relation. We derive an analytical expression of nonlinear curvature relation, which is approximated by a linear relation for a small curvature. The reproduction of spiral waves in the Belousov-Zhabotinskii (BZ) reaction incorporated with the experimental data of the two mesoscopic characteristics leads to an agreement with the experimental data. Both stationary rotating spiral waves and a meandering motion of the spiral tip are observed in accord with other experimental and theoretical studies.

Although our model is based only on the two mesoscopic characteristics of the excitable media and each excitable element can take only one of the two states (excited or non-excited), it reproduces well-known behaviors (stationary rotating spiral waves with a core, meandering motion, etc.) observed in actual excitable media and more complex theoretical models with continuous variables. As shown in Section 5, lowering D in our model leads to the transition from stationary rotation to meandering motion. This property is consistent with the well-established fact that decreasing the linear slope in the curvature relation leads to the transition to meandering [19;28,48]. However the simulation of our model incorporated with experimental data of the two characteristics predicted a meandering spiral in a parameter region where the experiment showed a stationary one. Our model is still premature in a quantitative level. In fact, even if the model is adjusted to realize the experimental data of the two characteristics, we still need to determine the parameter t w characterizing the excitability of

30 H. lto / Physica D 79 (1994) 16-40

the medium. The connection of this a priori

parame te r with the microscopic physical quantities should be investigated in order to make our model more reliable even in a quantitative discussion.

The reproduction of the solution of the PDEs

by a faster computat ional algorithm is not our main motivat ion of the modeling. However it is of value to discuss the efficiency of the present method. Because our event scheduling algorithm traces only the jump to the next occurrence of the excitation event, there is no slow dynamics that makes the numerical integration of the P D E s so stiff. In our algorithm, most computation t ime is spent to register the arrival t ime of excitation pulse at many neighboring cells. One revolution of the spiral wave takes about 13s CPU time on HP workstation 9000 in the simulation shown in Fig. 7a. In the remainder of the discussion, we discuss several different issues

raised by this work.

6.1. Nonlinear curvature relation

In both the BZ reaction [32] and the slime mold [37], Foerster et al. reported that their exper imenta l data of curvature relation were approximated by a linear relation and the linear slope D was close to the diffusion coefficient of autocatalytic species. However we consider these are very special cases within the current theoretical f ramework. First of all, in general, curvature relation N ( K ) is given by a nonlinear function. This is true also in the PDEs [11,49]. Only in a limited range of small curvature, it can be approximated by a linear relation with the slope D. In the PDEs, the linear curvature relation over an entire range can be obtained only in the singular per turbat ion limit. Second, the linear slope D is not always identical to a diffusion coefficient and has a dependence on the recovery t ime even in the PDEs. In our model , D is propor t ional to CBz(T ) and is not a system constant. However the dependence of D on the recovery time is different between our model and the PDEs [11,49]. This is because the

excitation in our model spreads out by a conduction of excitation pulses and the interaction between the elements is of convolution type. This is not the case for the PDEs in which the excitation spreads out by a molecular diffusion. In the PDEs of two components , D is proved to be equal to the diffusion coefficient and be

constant only when both components have the same diffusion coefficient or when the diffusion coefficient of the recovery (slow) variable is zero and the smallness pa ramete r e goes to zero (singular perturbat ion limit) [49]. Both are very special cases not realized in the actual physical systems. Therefore it is a very important problem whether the linear slope of the curvature relation at small curvature depends on T or not in actual media. Since there has been no systematic report on this problem even in the simulations of the PDEs, bet ter characterization of the curvature relation should be carried out urgently both in numerical simulations and experiment. Those nonlinearities are supposed to play an important role in the motion of the spiral tip with a large curvature. They are also important in the estimation of the critical radius of the stimulation electrode that can initiate the circular wave. In cardiac physiology, this prob-

lem has a critical significance [50]. The fact that the linear slope D is proport ional

to CBz in our model is the most significant difference from the PDEs. The dependence of the recovery time T on D is much weak in the PDEs [11,49]. How far does this difference lead to the different global structure in the bifurcation diagram in the pa ramete r space? For example , is the selection of a stationary spiral wave still unique with the given paramete r? Is the bifurcation to meandering still supercritical Hopf type without a mode-locking between the two fre- quencies? However answers to those questions should wait for a further systematic pa ramete r

scanning. Recently Cour temanche [51] revised the pres-

ent model so that the interaction between the elements is of diffusion type. In both models, some system paramete r should depend on the

H. lto / Physica D 79 (1994) 16-40 31

recovery time to realize the dispersion relation. In our model, local conduction velocity of excitation pulse v is recovery dependent. The recovery dependence is included in the threshold for excitation p in his model. In his model also, the curvature relation is nonlinear and the linear slope at the small curvature D does depend on the recovery time.

6.2. Other mesoscopic models

There have been proposed many other mesoscopic modelings of the excitable media. Krinsky [52] studied a system of two coupled excitable cells by his 0-model, which was essentially two coupled maps based on a mesoscopic characteristic, the latency of the action potential after the stimulus. Miller and Rinzel [53] studied the instability of periodic wave trains of impulse in one-dimensional excitable cable of neural tissue. They reduced the system of many wave trains into a simple interacting pulse model based on the dispersion relation obtained by numerical simulations of the PDE (the Hodgkin-Huxley model). Oono [54] proposed the cell dynamical system that skeletonizes the complex physical systems to abstract the essential mathematical structure leading to the universally observed phenomena like the pattern formation.

Gerhardt, Shuster and Tyson [13,14] introduced a multi-state cellular automata. They succeeded in simulating both dispersion relation and curvature relation. The reproduction of spiral waves in the BZ reaction was also carried out by their model. Contrary to our approach, their model was constructed based on the microscopic mechanism of local kinetics in the PDEs. In order to adjust the microscopic model parameters to realize the experimental data of dispersion relation, they need to know the relation between the recovery variable and the plane wave velocity. Since this relation is not experimentally measurable at present, they apply the relation obtained by the PDE (the Oregonator model). Therefore their model is not

self-contained without a direct connection to the experimental data.

Finally Zykov [55] introduced a simple framework so-called "kinematic approach". First he derived a geometrical relation among the local variables describing the wave front, that is, the local normal velocity, tangential velocity and the curvature of the wave front. Next, he assumed the fohn of the dispersion relation and the curvature relation based on the experimental data and the results of the PDEs. The movement of the wave front can be computed by solving the integro-differential equation with special boundary conditions.

6.3. Future problems

In our reproduction of stationary spiral waves in the BZ reaction, we obtained the meandering spiral wave. This discrepancy might be attributed to the incompleteness of the experimental data of the dispersion relation that was applied in our simulation (Fig. la). In the experiment [32], concentric waves were initiated by periodic pac- ings by a sharp electrode. The propagation velocities were measured at the position far away from the center where the curvature was negli- gible. The initiation of waves became impossible because of their initial high curvatures even when the pacing period was still greater than the refractory period. Therefore the short end of the dispersion curve was not obtained correctly in this protocol [41]. The better characterization of the dispersion relation should be carried out, because this is one of the essential characteristics of the excitable media.

In principle the present model can be used to reproduce the wave phenomena in any excitable media, if both the dispersion relation and the curvature relation are measured experimentally. Application to other excitable media, neural tissue, cardiac tissue, and slime mold should be examined to check the validity of the present model. Especially, spiral waves in cardiac tissue has been a main research object in the study of

32 H. Ito / Physica D 79 (1994) 16-40

excitable media [3]. This study is also clinically important because spiral wave of activation is believed to be a cause of some dangerous cardiac disease. However in order to better characterize this system, we should include the effect of inhomogeneity, anisotropy, obstacles, and the existence of pacemaker cells (possibly also three dimensionality). Although these complications would make the model less tractable analytically, the inclusion of these effects into the numerical simulation is much easier in our model than the PDEs.

The connection between our model and the partial differential equations still needs to be investigated to understand the universal mathematical structure of excitable media. In this paper , we simulated the wave phenomena based on the experimental data of the dispersion relation. However , we can apply the dispersion relation obtained by the numerical simulations of the PDEs. It is a challenging problem to see how far our model can reproduce the results of the PDEs. Since the dispersion relation can be obtained easily, we hope to reproduce the wave phenomena in the complex PDEs (e.g., Bee le r - Reuter model) that requires much computation time for their numerical integrations.

Another type of instability, breakup of spiral waves, has been reported in some numerical simulations: cellular automata [13], PDEs [56- 58] and discrete version of the present model [59]. We need a systematic parameter scanning to explore whether such an instability exists in our model or not. Winfree [60] reported that spiral waves with different rotation period can coexist in the system with a non-monotonic dispersion curve. However , finding the system parameters realizing such a special curve is not so easy in the PDEs. Since our model can realize such a dispersion relation quite easily, the study of complex behaviors by non-standard dispersion curve is of theoretical interest.

In cardiac tissue, the variability in the duration of excitation (action potential duration) plays an important role in dynamics of wave propagation

[38]. The action potential duration depends on the recovery time of the cell and this relation is called electrical restitution curve. The inclusion of this another mesoscopic characteristic into our model is straightforward. It is an interesting problem to study how the additional variable changes the dynamics of spiral waves.

Acknowledgements

The author thanks L. Glass, A.T. Winfree, D. Kaplan and M. Courtemanche for valuable dis- cussions. He also thanks T. Kiyomoto for help in computer programmings. This research was sup- ported by grants from the Natural Science and Engineering Research Council of Canada and les Fonds des Recherches en Sant6 du Qu6bec. The support was also from Grant-in-Aid for Encour- agement of Young Scientists and Grant-in-Aid for Scientific Research on Priority Areas by the Ministry of Education, Science and Culture of Japan. The author was a research fellowship of the Heart and Stroke Foundation of Canada.

Appendix A. Derivation of scaling relation

Assume the periodic passing of plane wave fronts with a time interval T in the two dimensional sheet of excitable medium. The Cartesian coordinates (x, y) are introduced as seen in Fig. l l a . The fronts propagate downward perpendicular to the y-axis. Since any point in the medium has a recovery time of T between the successive excitations, the velocity of the wave front satisfies the dispersion relation CBz(T ). The cell at the origin (0 ,0 ) is connected to its neighborings cells within a circle of the interaction radius r. Only the upper-half of this circle is drawn in Fig. l l a . Suppose that the cell at the origin is excited at t = 0 when the wave front just arrives at the x-axis. The neighboring cell at the position (x, y) was excited at

H. Ito / Physica D 79 (1994) 16-40 33

(a ) / / / / / / / / / /

/ ,Y / / / / / / / / / /

' CBZ

- r (0,0) r

(b) Y

Fig. 11. (a) Propagation of plane wave front in the (x, y) plane. Wave front propagates downward perpendicular to the y-axis with the velocity CBz. The cell at the origin (0, 0) can get excited by many incoming excitation pulses from the neighboring cells within the interaction circle of radius r. (b) Domain of the neighboring cells that contribute to the excitation at the origin is shown by the hatched area S. In this case, S is a sector of the interaction circle.

t=- -y /CBz , ~xZ+yZ<--r, y>--O, (A.1)

and the excitation pulse emitted at this instant arrives at the origin when

t = --Y/CBz + ~X 2 + yZ/v , (A.2)

where we introduced the polar coordinates £ = sin ~ and )7 = fi cos ~. Since the RHS of Eq.

(A.4) is a constant, this defines boundaries ~ = _+~b 0 within the semi-circle of the interaction range as seen in Fig. l l b . It can be shown that the excitation pulses emitted from the neighboring cells within the sector S divided by the two boundaries (hatched area in Fig. l l b ) arrive at the origin not later than t = 0 and contribute to the excitation at the origin. The ratio of the number of the neighboring cells within S to # ~, is given by (area of S) / (a rea of full c i rc le)= ~borZ/xrr 2. Since this ratio should be equal to p,

we get th0 = P ~ and the scaling relation

v(T) = CBz/COS p'rr. (A.5)

In the above discussion, the window time t w was assumed to be large enough so that every excitation pulse arrives at the origin within the time interval t w and can contribute to the excitation at the origin. Courtemanche [51] discussed the appearance of unstable wave front propagation when this assumption does not hold. This problem will be discussed in Appendix B. Within the range of t w used in the numerical simulations in the text, t w was always large enough to satisfy this assumption.

where v is the local conduction velocity of the pulse between the cells. After the wave front enters the domain of its interaction range, the cell at the origin receives many incoming pulses from its neighbors. The cell gets excited when it has received as many as P#nn pulses, where # nn is the total number of neighboring cells within the interaction radius. Since this instant was assumed to be t = 0, the pulse that arrives at t = 0 is coming from the cell at the position (£, y-),

-)7/CBz + ~ )72/0 = 0 , (A.3)

which leads to

COSt~=CBz/O , - -½~r<~<½7r , 0</~- -<r ,

(A.4)

Appendix B. Derivation of curvature relation

The basic concept of the computation is same as the derivation of the scaling relation for the plane wave fronts in Appendix A. We assume the periodic passing of the identical curved wave fronts in the sheet of excitable medium as depicted in Fig. 12. The wave front has a positive curvature K - - 1 / R , where R is a radius of curvature. After passing of enough numbers of successive waves with a time interval T, the front propagates with a stationary normal velocity N(K, T). The Cartesian coordinates (x, y) are introduced in this space. The center of curvature of the wave front at (0, R - Nt) is moving down on the y-axis. The cell at the origin (0, 0) is

34 H. lto / Physica D 79 (1994) 16-40

Y A

R ~ i

- r (0,0) r

Fig. 12. Propagation of curved convex wave front in the (x, y) plane. The wave front has a curvature K = 1/R (R: radius of curvature) and propagates downward with a normal velocity N. The cell at the origin is connected to its neighboring cells within the interaction circle of radius r.

excited at t = 0 when the wave front reaches at

this point. The cell is excited by incoming excitation pulses emit ted from its neighboring cells within the interaction radius r. The upper half of this interaction circle is drawn in Fig. 12. In the calculation of the curvature relation, we separate

the two cases depending on the degree of the curvature e, where e = r / R = rK is the scaled curvature. When the curvature is not too large satisfying e-< X/2 (case I) , only the wave front ( lower half) of circular wave has entered the interaction circle when the front arrives at the origin (t = 0) as shown in Fig. 13a. On the other hand, some part of the wave tail (upper half) of circular wave has entered the interaction circle at t = 0 when the curvature is large, X/2 < e (case I I ) as seen in Fig. 14a.

Case h e <-X/2. The neighboring cell at the

position (x, y) is excited at

R_y_V~-Y_x 2 t - N ' ~ x 2 + y 2 < - r ' y > - O '

(B.1)

(a) , ~ ,

I I

I I

- r - R R r

(b) ~Y

I

[ > ×

Fig. 13. Propagation of the curved wave front when the curvature is not too large, Case I: e -< X/2. (a) Only the wave front (lower half) of the circular wave has entered the interaction circle when the front reaches the origin. (b) Domain of the neighboring cells that contr ibute to the excitation at the origin is represented by the area S, that is the union of the two differently hatched regions: S~ (hatched by horizontal lines) and S 2 (vertical lines).

and the excitation pulse emit ted at this instant

arrives at the origin when

R - y - V ~ - x 2 t - N + ~ x 2 + y21v ' (B.2)

where v is the local conduction velocity of the pulses between the cells. The cell at the origin gets excited when it has received as many as P#,1, pulses within the window time t w. For simplicity, at first, we assume t w is large enough and ignore the effect due to the finitness of t w, which will be discussed later. The pulse that arrives at t = 0 originates from the cell at the

position (£, y~,

2

+ ~ / ~ + f 2 / v = 0 , (8 .3 ) N

which leads to

fi - 1 / ~ / , ( B . 4 )

c o s - R + 2 - ¢ 2 sin2

H. Ito / Physica D 79 (1994) 16-40 35

(a) I

- r -R

y I I

R r

(b)

I

Fig. 14. Propagation of the curved wave front when the curvature is large, Case II(a): V~<-e-<2. (a) Part of the wave tail (upper half) of the circular wave has entered the interaction circle when the front reaches the origin. (b) The area S represents the domain of the neighboring cells that contribute to the excitation at the origin. S is subdivided into the two parts: Sl (hatched by horizontal lines) and S 2 (vertical lines). In this case, the heavy dashed lines in the boundary of S z appear because the excitatory influence of the neighboring cells is terminated by the wave tail of the circular wave.

where ~=f i s in~ ,37=f i cosq~ , and ~/=N/v. Since the RHS of Eq. (B.4) is a constant, this relation defines the boundary ~(q~) within the semi-circle of the interaction range shown in Fig. 13b by a thick line. With the given R, the boundary depends on the value of "0 that should be determined by the following self-consistency equation. The excitation pulses emitted from the neighboring cells above this boundary (the domain S: the union of the two differently hatched regions in Fig. 13b), arrive at the origin not later than t = 0 and contribute to the excitation at the origin. Since the number of these neighboring cells is equal to p#~ , , the relation

(area of S) = p "rrr 2 , (B.5)

should be satisfied. Together with this relation and Eq. (B.4), we can obtain the normal velocity N as a function of K, p, and v.

The area of S can be calculated in a straightforward way. For a convenience in computation, S is divided into two regions, $I: a sector ( -¢z -< q~-< Ca, hatched by horizontal lines in Fig. 13b) and $2: the rest of S in the both outsides of S 1 (--¢1 ~ (~ ~ --¢2 and ¢2 -< q~ -< ¢1, hatched by vertical lines), where

c 0 s ¢ 1 = 7 , a t t ~ = 0 , (B.6)

cos ¢2 + (~1 + e e sinZ¢2 - 1)/e = , / , at fi = r .

(B.7)

From Eq. (B.4), the boundary of the domain S 2 can be given by

_1{2r/+ e~6(1 + ~72)]

0<-- ~ = f i / r ~ 1. (B.8)

Simply, we get

(area of S 1) = r2¢2 , (B.9)

and

~(8) 4,~

(area ofS2) = 2 J t~ dt~ J dq~. (B.10) 0 4,2

Substitution of the relation

t ~ - 1 ( 1 + -r/-2 - 1 ) (B.I1) R n 1 + 7 2 - 2 7cost~ '

which is obtained by Eq. (B.4), into Eq. (B.10) and subsequent integrations lead to

(areas of S2) = (11 + 12 + 1 3 ) , (B. 12)

11 = ¢1 -- ¢2 ,

( s i n ¢ l s i n C z )

/2 =2~/ l+7 /2_2T/cos¢1 l+nZ-2ncosO2 '

[ 1 / l + r / 1 \ 13 2 ( i ~ @ 1 ) [ t a n - ~ - f - - ~ _ tan(~¢l) )

_ 1 / 1 + 7 / t a n - ~ l _ - - ~ t a n (1¢z)) ] .

Substituting the explicit forms

36 H. lto / Physica D 79 (1994) 16-40

&l = cos It/, (B.13)

and

x{2-O + e(1 + r/2)~ &2 = cos- ~- - 2 7 i 7 ~ } , (B.14)

which are obtained by Eqs. (B.6) and (B.7) respectively, into the above expressions, we get the results underneath Eq. (5) in the text. Finally, we obtain Eq. (5) by summarizing the relations (B.5), (B.9), and (B.12). The relations

= N / v and v = CBz/COS p~r lead to eq. (4) in the text. The curvature relation is a monotonically decreasing function of the curvature as seen in Fig. 3.

Case H: V ~ < e. Contrary to Case I, some part of the wave tail (upper half) of the circular wave has entered the interaction circle when the wave front arrives at the origin at t = 0 (Fig. 14a). The neighboring cell at the position (x, y) was excited at the time given in Eq. (B.1) and the excitation pulse arrived at the origin at the instant given by Eq. (B.2). The cell can contribute to the excitation at the origin for the time interval of t w after the arrival of the pulse. However , the tail of the circular wave arrives at (x, y) when

R - y + V ~ - x 2 t - U ' (B.15)

and the excitatory influence to the cell at the origin is forced to be termined. After this termination of influence arrives at the origin when

R - y + V ~ - x 2 t - N + ~x2 + y 2 / v , (B.16)

the cell can no longer contribute to the excitation at the origin. By a similar discussion as the derivation of Eq. (B.4), we get the equation

= air / , (B.17) c o s 8 - R - X /R 2 - s i n %

which gives another boundary within the interaction circle. When r/->-Oc(e) = ( - 1 + V~e 2 - 1)/e, Eq. (B.17) has no solution for /~(q~) within the interaction circle. Thus there is only a

single boundary given by Eq. (B.4) and the same discussion as Case I does hold. On the other hand, when r I <r/c(e), the second boundary appears as shown in Fig. 14b. Above the boundary given by a solid line in Fig. 14b, the excitation pulse arrives at the origin no later than t = 0. On the other hand, the second boundaries given by heavy dashed lines separate the regions where the termination of the excitatory influence arrives at the origin also no later than t = 0 . Therefore the excitation at the origin is attributed to the excitatory influences only from the neighboring cells in the domain S in Fig. 14b (the union of the two differently hatched regions). The same self-consistency equation as Eq. (B.5) determines the curvature relation. The form of the curvature relation changes depending on whether V~ < e-< 2 (Case II(a)) or 2 < e (Case II(b))). In Case II(a), the equation for ~b z is now given by

cos 4~2 - (~1 + e 2 sinZ~b2 + 1)/e =~7, a t t ~ = r ,

(B.18)

that is different from Eq. (B.7). However Eqs. (B.14) and (B.8) still give the solution for (])2 and the entire boundary of the domain S 2, respectively. Therefore all the equations (B .9 ) - (B .14) hold also in this case, and the final expression of the curvature relation in Eq. (5) in the text gives a monotonically decreasing function of e even in Case II(a).

The critical curvature given by Eq. (7) in the text takes ecr = 2 when p = 0.25. In this case, the curvature relation has a singularity at e = 2. When p is decreased further below 0.25 to increase the excitability of the medium, the curvature relation has a branch in the region e > 2 and changes its topology (Case II(b)). As seen in Fig. 15, the curve is composed of two branches. In the upper branch above the boundary r I = 1 - 2/e (dashed line in Fig. 15), the same discussion as Case II(a) can be applied. However below this boundary there appears the lower branch in which r/ is an increasing function of e.

H. Ito / Physica D 79 (1994) 16-40 37

COS pTr

0 (~t Y/tt)

0 1 ^ 2 ,I_ 3 ~,cr

Fig. 15. Curvature relation (e, ~7) when the excitability of the medium is very high with p < 0.25, where e = rK and r /= (N/ % z ) cos p~r. The curve is composed of the upper branch above the boundary -,7 = 1 - 2/e (dashed line) and the lower branch ending at (ecr,0). The two branches connect at

(~,,n,)-

In Fig. 16b, the domain S in which cells can contribute to the excitation at the origin has a different geometry from that in Case II(a). Although the boundary of the domain S 2 can be still given by Eq. (B.8), ~b 2 is zero and there is

( a ) , ~'Y

- r -R i R r

( b ) y

Fig. 16. Propagation of the curved wave front when the curvature is very large, Case II(b): 2 < e. (a) The entire part of the wave tail (upper half) of the circular wave has entered the interaction circle when the front reaches the origin. (b) Domain of the neighboring cells that contribute to the excitation at the origin is represented by the area S. The entire boundary of S is given by Eq. (B.8).

no region for S 1. Eqs. (B.5) and (B.10) lead to the transcendental equation

p r r = c o s - ~ / + V~-_ r/2

- 1 1+'0 + 2(3"02-~1) tan 1 ~ + _ ~ ) , (B.19) l - r /

that gives the lower branch of the curvature relation. The lower branch connects to the upper branch at the point (et, ~t) which can be obtained by the relation 7/t = 1 - 2 / e t and Eq. (B.19). By imposing r /= 0 in Eq. (B. 19), the critical curvature can be given by ecr = I /V~. In the case when p < 0.25, the curvature relation is a two- valued function in the range ecr < e < e t and the maximum curvature of the propagating wave front is given by e t.

B. 1. Finiteness effect o f t w

In the above discussion, we ignored the effect due to the finiteness of the window time t w and assumed that the cell at the origin can receive all the incoming excitation pulses within t w. How- ever the finiteness of t w leads to a minor correction to the curvature relation. At first, the case of plane wave is considered. As discussed in Appendix A, the excitation at the origin is caused by many incoming pulses from the neighboring cells in the interaction circle. It can be shown that the origin receives the earliest arriving pulse from the cell at the position (x, y ) = (0, r), that is (p, 4)) = (r, 0), when t I =

- r ( 1 - cos pTr)/CBz. Since the origin receives the P#nnth arriving pulse at t = 0, the time interval 0- t I should be less than t W to be consistent with our assumption. This condition

r CBz ->Tw (1 - cos p~r), (B.20)

restricts the valid range of the plane wave velocity in our discussion. In the dispersion relation given in Fig. la, the minimum velocity is taken at the refractory period O. Therefore, if

38 H. l to / Physica D 79 (1994) 1 6 - 4 0

CBZ(0 ) does not satisfy the above condition, our discussion in Appendix A cannot be valid in some range of the dispersion relation. In the reproduct ion of the dispersion relation by the numerical simulation in Fig. 2, we applied the

paramete rs r = 0.1 m m and t w = 3 s. Since r / t w =

0 . 0 3 3 m m / s , which is smaller than CBz(0= 15 s ) = 0 . 0 6 9 m m / s , the condition Eq. (B.20) is always satisfied with any p.

Also in the case of the curved wave front ( p - > 0 . 2 5 ) , it can be shown that the origin receives the earliest arriving pulse from the cell at (p, ~b) = (r, 0). The pulse arrives at the origin

at t t = - r cos p~(1 /~7 - l ) / C B z , where 77 = N / v .

Now the validity condition is given by

( twCBZ t 1 r/-> 1 + . (B.21)

r cos p-rr /

As the curvature e is increased up to the critical curvature ecr, r /decreases monotonically to zero. There fore there always exists another critical curvature e*, above which the condition (B.21)

2

f

b c~ 1

:i

0 i j l 0.0 .... 0.25 0.5 p~

P

Fig. 17. Dependence of critical radius on the parameter p ( 0 < p < 0 . 5 ) . Ordinate is the critical radius, that is the inverse of the critical curvature, scaled by the interaction radius r. The medium is more excitable with smaller p. Solid line and dashed line represent 1/ecr and l / e* , respectively, e* is the critical curvature corrected by the finiteness effect of the window time t w. The data with an error bar (open circle) are the result of numerical s imulation averaged over 30 samples with different random distributions of the cells. The equat ion for e* is different depending on whether p is greater than p* or not (see text for more in detail). In both the computa t ion of e* and the numerical simulations, we applied t w = 10 s, which leads to p* = 0.2326.

is violated and our derivation of the curvature

relation is no longer valid. The dependence of e* on p is computed and compared with e~, in Fig. 17. In the figure, the inverse of the curvature,

that is the scaled critical radius Rcr / r , is plot ted ( l /~ 'cr" solid line and l /e*: dashed line). We applied the value of t w = 10 s. The value of e* is identical to the critical curvature ecr when the

finiteness effect of t w can be ignored in the case when t w = 2. ec r is given by the root of Eq. (7) in the text when p ->0 .25 and by 1/X/fi when p < 0.25. The critical curvature obtained by the

numerical simulations with tw = 10s is also shown in the figure. The method of simulation was explained in Section 4 and the data in the

range 0.25 < p < 0 . 5 were shown in Fig. 4a. When p < 0.25, the curvature relation is com-

posed of the two branches connected at (e t, 7/t) as seen in Fig. 15. When p* - p <0 .25 , the R H S of Eq. (B.21) is larger than ~t so that the

crossing between the curvature relation and the line defined by Eq. (B.21) at e* occurs in the upper branch. Therefore the same discussion as the case p -> 0.25 can be applied. The value of p* can be obtained by the equality between the RHS of Eq. (B.21) and ~,. With t w = 10 s, we get p* =0.2326. When p < p * , e * locates in the

lower branch. As seen in Fig. 16b the domain S that contributes to the excitation at the origin has a different geometry from that in the previous case and the cell at (p, 6 ) = (r, 0) is no longer included in it. In this case, it can be shown that the earliest arriving pulse contribut- ing to the excitation is coming f rom the upper

edge of the domain S, that is (p, ~b) = (2/re(1 - ~7)],0). This pulse arrives at t x = - 2 R c o s p T r / (CBzr/), which leads to the condition

2r cos pw (B.22) r / -> CBZtw 8 ,

for the validity of our derivation of the curvature relation. Contrary to Eq. (B.21), the RHS of Eq. (B.22) depends on e. The dependence of l / e* on p (<0.25) is plotted by a dashed line in Fig. 17. However , in this figure, it is not so

H. Ito / Physica D 79 (1994) 16-40 39

cer ta in whe the r the numer ica l da ta show a be t te r a g r e e m e n t with ecr or e*.

B.2. Linear curvature relation

T h e l inear curva tu re re la t ion in Eq. (6) in the text can be ob ta ined by the pe r tu rba t ion analysis o f Eq . (5). We assume tha t

r~ = 770 + eA + ~(e2) , (B.23)

whe re r/0 = cos p ~ , and calculate the coefficient A by the pe r tu rba t ion expans ion of Eq. (5). Since the second t e r m in the R H S of Eq. (5) has the d e n o m i n a t o r e 2, I t , I 2 and 13 should be evalu- a ted up to the o rde r of e 3 to get the first o rde r cor rec t ion of Eq. (5) in respec t to e < < 1. Af t e r lengthy compu ta t i ons , we get

( -- ) 2/r/°(3"q2 - 3 - 4 A ) ] ,1= VT2 ° - - = = ~" 8 V' - / - r / 2 'P

3( 1 [_12A2 + e 4 8 ( 1 - r / 2 ) 3/2

+ 18A(1 - r/02)(2"q~ - 1)

2 2 2 ) + ( l - r/0) (14770 + 1)] + ~(E4), /

2

2( r/o [ _ ( 1 _ r/2) + 4A(2 _ r/z)]) - e 4 ( 1 - ' q 2 ) 3'z

3( 1 - ~ 8 ( 1 ~, , ,~ [n~(a - r/~)2 _ 2 A ( 1 - r/2)

- r i o )

+ 4A~(2 + ' 7 2 / ] ) + e ( d ) ,

and

13= \ 2 1 ~ _ ~ 0 2 /

2( ff°-z,3/z [ -4A(3 r /0 z - 5) + e 8 ( l - r / 0 )

- (3"0 2 - 1)(1 _ r/z)])

3( 1 + e 48(1 2,5/2 [(3"0~ - 1)

- r / 0 )

x ( 1 - 2x2,,,.~ 2 r/o) ~,Lr/o + 1)

+ 6 A ( 1 - 2 4 770)(6o0 - 9,70 ~ + 1)

+ 12AZ(r/2 + 5)]) + (~(E4) .

S u m m a t i o n of the th ree quant i t ies leads to

11 + I 2 + I 3 = E3\- r / 0 ( -- .~- ~ ( E 4 ) ,

Afte r the subst i tut ion of the above result into Eq. (5) in the text and the pe r tu rba t i on expansion of the first t e rm in the R H S of eq. (5), we get the final result ,

1 • 2 A = - 3 (sin p-u) .

Subst i tut ions of this result and Eq. (B.23) into Eq. (4) in the text lead to the re la t ion

N(K, T) = CBz(T ) -- D ( T ) K + ~7(K2),

where the l inear coefficient D ( T ) is g iven by

sin 2 pxr D ( T ) 3 cos p'u rCaz(T)"

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mesoscopic modeling of wave propagation in excitable media

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