J. theor. Biol. (1998) 191, 299–308

0022–5193/98/070299+10 $25.00/0/jt970585 7 1998 Academic Press Limited

Intercellular Spiral Waves of Calcium

M W*§ J S†‡

*Department of Mathematics and Statistics, University of Canterbury, Christchurch, NewZealand and †Department of Mathematics, University of Michigan, 2072 East Hall, 525 East

University Avenue, Ann Arbor, MI, 48109-1109, U.S.A.

(Received on 14 July 1997, Accepted in revised form on 5 November 1997)

Intercellular calcium waves have been observed in a large number of cell types, and are known to resultfrom a variety of stimuli, including mechanical or hormonal stimulation. Recently, spiral intercellularwaves of calcium have been observed in slices of hippocampal tissue. We use an existing model to studythe properties of spiral intercellular calcium waves. Although intercellular spiral waves are well knownin the context of cardiac muscle, due to the small value of the calcium diffusion coefficient intercellularcalcium waves have fundamentally different properties. We show that homogenisation techniques givea good estimate for the plane wave speed, but do not describe spiral behaviour well. Using an expressionfor the effective diffusion coefficient we estimate the intercellular calcium permeability in liver. For thebistable equation, we derive an analytic estimate for the value of the intercellular permeability at whichwave propagation fails. In the calcium wave model, we show numerically that the spiral period is firsta decreasing, then an increasing, function of the intercellular permeability. We hypothesise that this isbecause the curvature of the spiral core is unimportant at low permeability, the period beingapproximately set instead by the speed of a plane wave along a line of coupled cells in one dimension.

7 1998 Academic Press Limited

1. Introduction

In the past few years, a great deal of evidence hasaccumulated that intracellular and intercellularcalcium (Ca2+) signalling is one of the crucialmethods of cellular coordination and control.Oscillations in the concentration of free intracellularCa2+ are thought to play an important role in manycellular processes, such as secretion, reproduction andmovement, and in many cells these oscillations areorganised into periodic intracellular waves (Berridge,1993; Rooney & Thomas, 1993).

Calcium waves have also been often observedtravelling between cells, presumably coordinatingcellular responses over much larger regions. Forinstance, mechanically stimulated Ca2+ waves have

been observed propagating through ciliated trachealepithelial cells (Sanderson et al., 1990, Boitano et al.,1992), rat brain glial cells (Charles et al., 1991, 1992,1993), and many other cell types (Sanderson et al.,1994). Furthermore, intercellular Ca2+ waves havebeen observed passing from glial cells to neurons, andvice versa (Charles, 1994; Nedergaard, 1994; Parpuraet al., 1994). Particularly interesting are recentdiscoveries that spontaneous intercellular Ca2+ wavesexist in tissue slice preparations from the hippo-campus (Dani et al., 1992), and in the intact liver(Robb-Gaspers & Thomas, 1995) and thus are not anartifact of the cell culture or the mechanical stimulus.When a hippocampal slice is stimulated by bathapplication of N-methyl-D-aspartate (NMDA), inter-cellular glial Ca2+ waves are immediately observed.Since glial cells do not respond directly to NMDA, itis likely that the glial responses result from the releaseof glutamate by neurons which have been stimulatedby the NMDA. Hippocampal intercellular Ca2+

‡Author to whom correspondence should be addressed.E-mail: jsneyd.math.lsa.umich.edu§Present address: Courant Institute, New York University, NewYork, U.S.A.

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waves are highly structured, with forms similar to thespatio-temporal organisation of wave activity seen inXenopus oocytes. In particular, intercellular Ca2+

waves in hippocampal slices have recently beenobserved to form spontaneous spiral waves (Harris–White et al., 1998. Similar intercellular Ca2+ waves,travelling at similar speeds (7–45 mm s−1), have beenobserved in the intact liver (Robb-Gaspers &Thomas, 1995). Perfusion of a rat liver withvasopressin causes intracellular Ca2+ oscillations thatare similar to those observed in isolated hepatocytes.Most interesting is the observation that the intracellu-lar oscillations are coordinated over large spatialareas, and thus form coherent intercellular waves thatpropagate along the hepatic plates. A pacemakerregion of the liver acts as an initiating center, andthe intercellular waves spread from this region.Each intercellular wave consists of a sequence ofintracellular waves, and the delay in intercellulartransmission decreases with increasing agonist con-centration.

It is widely believed that intercellular Ca2+ wavesare a mechanism by which a group of cells cancommunicate with one another, and coordinate amulticellular response to a local event. An under-standing of the mechanisms underlying intercellularCa2+ waves is of importance, not only for a generalunderstanding of intercellular communication, butalso for the understanding of a wide range of specificprocesses such as mucociliary clearance, woundhealing, mechanical transduction, cell growth, andinformation processing. For example, in ciliatedtracheal epithelial cells the rate of ciliary motionincreases as the intracellular Ca2+ concentrationincreases. Thus, an intercellular Ca2+ wave wouldserve to coordinate the ciliary beating of a number ofadjacent cells, resulting in more efficient mucusclearance. Intercellular Ca2+ waves are also impli-cated in the wound healing response. If a monolayerof epithelial cells is mechanically damaged, theresultant intercellular Ca2+ wave sets up intercellularCa2+ gradients which influence the initiation anddirection of cell migration (Sanderson et al., 1994).Not well understood, but potentially of extremesignificance, are recent findings that intercellularCa2+ waves in glial cells can be communicated toneurons, and thus glial cells could play an activerole in information processing (Charles, 1994;Nedergaard 1994; Parpura et al., 1994; Hassingeret al., 1995). Intercellular Ca2+ waves in glia have alsobeen linked to spreading depression in the brain, inwhich waves of depolarisation spread at a similarspeed to intercellular Ca2+ waves in glia (Leibowitz,1992).

2. The Mechanism of Intercellular Waves

Intracellular calcium signalling has been intensivelystudied in many cell types, and there is generalagreement on the initial steps of the calcium response(Berridge, 1993). When an agonist binds to itsreceptor, it initiates a series of reactions that resultsin the production of a chemical called inositol1,4,5-trisphosphate (IP3). IP3 diffuses through the cellcytoplasm and binds to IP3 receptors located on theendoplasmic reticulum (ER) and the nuclear mem-brane. IP3 receptors are also Ca2+ channels, and whenIP3 binds, they open, releasing large amounts of Ca2+

from the ER. Note that, at steady state, theconcentration of Ca2+ in the ER is high, and thusthere is a steep concentration gradient of Ca2+

between the ER and the cytoplasm. Released Ca2+

then activates the IP3 receptors, leading to the releaseof further Ca2+, in an autocatalytic process calledCa2+-induced Ca2+ release (CICR). Release of Ca2+

through the IP3 receptor is terminated when highcytoplasmic Ca2+ concentrations inactivate thereceptor. Thus, Ca2+ has both a positive and anegative feedback effect on the IP3 receptor. Ca2+

pumps actively remove Ca2+ from the cytoplasm,pumping it back into the ER or out of the cell, untilthe cell returns to steady state. The cytoplasm is anexcitable medium (with respect to Ca2+ release) inmuch the same way that the nerve axon is an excitablemedium with respect to electrical potential.

Such a mechanism has been used by a number ofgroups to explain propagating waves of Ca2+ insidesingle cells, and their organisation into intracellularspiral waves (Lechleiter & Clapham, 1992; Atri et al.,1993; Dupont & Goldbeter, 1994; Sneyd et al.,1995b). Other groups have hypothesised that thecoupling of excitable cells by gap junctions could alsolead to the propagation of intercellular Ca2+ waves ofthe types observed experimentally. For instance, sinceit is known that vasopressin causes the production ofIP3, it has been hypothesised (Robb-Gaspers &Thomas, 1995) that intercellular waves in liver are theresult of the coupling of excitable intracellular Ca2+

dynamics by an intercellular messenger. Further, thedirection of the intercellular waves is not altered bythe direction of the perfusion, suggesting that theintercellular messenger is communicated through gapjunctions. Since glial cells in hippocampal explantsare extensively coupled by gap junctions, a similarhypothesis has been proposed to explain intercellularCa2+ waves in hippocampal explants (Dani et al.,1992).

It is important to note that the intercellular wavesdescribed above are fundamentally different from the

301

mechanically-stimulated waves described and mod-elled by Sanderson and his colleagues (Sandersonet al., 1994; Sneyd et al., 1994, 1995a). Theirhypothesis is that mechanically-stimulated intercellu-lar waves result from the passive diffusion of IP3 fromthe stimulated cell and are not actively propagated.Clearly, such a mechanism could not possibly resultin a spiral wave, and is therefore not applicable to theobserved waves in hippocampal slice explants.

Our goal is to study the properties of intercellularCa2+ waves in a fairly general framework. Our resultsare not tied to the details of any specific model(although we perform most of our simulations usinga particular model), but are applicable to any systemin which intercellular waves are propagated bycoupled excitable systems. Similar systems have beenstudied in detail in the context of cardiac electrophysi-ology. However, we shall show that, because the Ca2+

diffusion coefficient is so small, intercellular Ca2+

waves have some unique properties.

3. The Model

Our model and numerical computations are basedon the model of Atri et al. (1993) that was originallydesigned to model intracellular spiral Ca2+ waves inXenopus oocytes. Here, only the bare essentials of themodel will be presented.

The model equations are

1c1t

=Dc 012c1x2 +

12c1y21+ Jflux − Jpump, (1)

Jpump =gc

kg + c, (2)

Jflux = kfm(p) 0b+(1− b)ck1 + c 1 h, (3)

thdhdt

=k2

2

k22 + c2 − h, (4)

m(p)=p

km + p, (5)

where c=[Ca2+] and p=[IP3].Jpump describes the amount of Ca2+ being pumped

out of the cytoplasm back into the ER or out throughthe plasma membrane. Jflux models the flux of [Ca2+]through the IP3 receptor, and depends on the fact thatCa2+ activates the IP3 receptor quickly, butinactivates it on a slower time-scale. The binding sitesfor IP3 and Ca2+ are assumed to be independent.

m(p) is the fraction of IP3 receptors that have boundIP3, and is an increasing function of p. The term

b+(1− b)c/(k1 + c) controls the activation of theIP3 receptor by Ca2+. It is assumed that Ca2+

instantaneously activates the IP3 receptor. Thevariable h denotes the fraction of IP3 receptors thathave not been inactivated by Ca2+. At steady stateh= k2

2/(k22 + c2), a decreasing function of [Ca2+].

Inactivation by Ca2+ is assumed to act on a slowertime-scale, with time constant th. The constant kf

denotes the Ca2+ flux when all IP3 receptors are openand activated.

Equation (1) only applies within each cell, notacross the entire culture. At the boundaries betweencells it is assumed that the flux across the cellmembrane is proportional to the concentrationdifference across the membrane. For instance,consider a membrane situated at x=0 in one spatialdimension. Then,

Dc1c(0−,t)

1x=Dc

1c(0+,t)1x

=Fc[c(0+,t)− c(0−,t)], (6)

where superscripts − and + denote limits from theleft and right respectively. In our simulations, all cellboundaries are oriented along one of the coordinateaxes, and the equations are discretised with a regulargrid aligned with the coordinate axes. All cellboundaries are assumed to lie between grid points.Thus, difficulties caused by the orientation of theboundary are avoided.

The parameter values were obtained by experimen-tal measurements in a variety of cell types (Atri et al.,1993). Although there is no reason to believe thatthese are the correct values for glial cells, no bettermeasurements are presently available. In any case, aswe pointed out above, our results do not dependqualitatively on the details of the model. In Table 1we give the parameters and their numerical values.

4. Oscillations and Intracellular Waves in the Model

First we note that intercellular spiral waves inhippocampal slices are produced experimentally by abath application of NMDA, which, presumably,raises the concentration of IP3 approximatelyuniformly over the entire tissue slice. Thus, we mayassume that p is fixed, and treat it like a bifurcationparameter. Equivalently, we may use m as thebifurcation parameter. Different fixed values of m willgive different model behaviour. In the absence ofdiffusion, as the value of m increases, stable limitcycles appear via a homoclinic bifurcation (Fig. 1).

When m is less than, but close to, the value at whichthe homoclinic bifurcation occurs, the model withdiffusion exhibits travelling waves (Fig. 2). To give thedesired value of m(p), an initial [IP3] of 0.672 mM is

0.8

0.2

0.4

0.6

0.01000 200 300 400 500 600

Distance along cell ( m)

[Ca

2+]

(

M)

0.5

1.0

1.5

2.0

2.5

3.0

0.00.40 0.45 0.500.350.300.25 0.55

[Ca

2+]

(

M)

Period goeshere

Stable

StableOscillations

Unstable S.S.Stable

S.S.Stable

S.S.

∞

Oscillations

Un

stab

le

Oscilla

tion

s

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T 1The parameter values of the model [taken from Atri et al. (1993)]

Parameter Explanation Value

b Fraction of activated IP3 receptors when [Ca2+]=0 0.111k1 Km for activation of IP3 receptors by Ca2+ 0.7 mMk2 Km for inactivation of IP3 receptors by Ca2+ 0.7 mMDc Diffusion coefficient of Ca2+ 20 (mm)2 s−1

g Maximum rate of pumping of ER Ca2+ pumps 2 mM/skf Ca2+ flux when all IP3 receptors are open and activated 16.2 mM/skg Km of endoplasmic reticulum Ca2+ pumps 0.1 mMkm Km for binding of IP3 to its receptor 0.7 mMth Time constant for inactivation of IP3 receptors 2 s

placed over the whole domain. The [Ca2+] is initiallyat steady state, which, for the numerical values inTable 1, is 0.13 mM. To start the travelling wave,1 mM of Ca2+ is added in the first 0.9 mm of thedomain on the left-hand side. Travelling waves wereproduced when the background [IP3] is approximatelybetween 0.66 mM and 0.673 mM. This narrow range ofvalues for [IP3] is an unattractive feature of the model,but has no qualitative effect on the results presentedhere. Other models are more robust with respect towave generation.

One crucial fact to note about the travelling wavesis that the width of the wave front (10 mm or so) issmall compared with the length of a typical cell(30 mm). Thus, an entire wave front is containedwithin a single cell, and indeed the entire wave,front and back, is contained within only two or threecells.

According to the model, the cell cytoplasm is anexcitable system, with behaviour similar to that of theFitzHugh–Nagumo model, a generic excitable system.

Once the Ca2+ concentration gets above a thresholdvalue, the autocatalytic process of CICR takes over,and leads to the release of a large amount of Ca2+.Eventually, the high Ca2+ concentration shutsoff the Ca2+ flux through the IP3 receptor, andthe Ca2+ concentration returns to steady state.However, although the behaviour of the model issimilar to that of the FitzHugh–Nagumo model, it isinteresting to note that the structure of the modelequations is rather different, as the nullclinesdo not have the generic shape. This is not anissue we shall explore further here, but just note inpassing.

5. Intercellular Plane Waves

Next we consider the propagation of an intercellu-lar plane wave. We solve the model equations,

F. 2. The wave (in a single large cell) shown at two differenttimes. The wave is travelling at about 7 mm S−1. Key: ——time=50 s; – – – time=70 s.

F. 1. [Ca2+ ] versus the bifurcation parameter m. The line madeup of circles shows the maximum and minimum of the [Ca2+]oscillations (filled circles are stable and hollow are unstable).Oscillations occur for m above about 0.29. Waves occur for m lessthan, but close to, 0.29.

6

3

4

5

20.50.0 1.0 1.5 2.0

Fc

Tra

vel

lin

g w

av

e sp

eed

(

m s

–1)

303

together with the internal boundary conditions (6) ona grid of 20 by 20 cells, each of which has a side lengthof 30 mm. As before, the domain is primed with IP3,so [IP3] is constant over the whole domain. Ca2+ isadded in on the left-hand side of the domain, and thisinitiates a wave. However, this time there are cellmembranes for the Ca2+ wave to get through. Thus,the wave travels across a cell, reaches the membraneat the far end, and then pauses until enough Ca2+ hasleaked through to initiate a wave in the neighbouringcell. As Fc decreases, it takes longer for a wave in onecell to initiate a wave in the neighbouring cell, and sothe intercellular wave speed decreases (Fig. 3). WhenFc is below 0.3 mm s−1 the wave ceases to propagateat all.

We note that, technically, an intercellular planewave is not a travelling wave, as it is not translationinvariant in space and time. However, although thismakes it difficult to construct the wave solutionsanalytically, it is still possible to obtain estimates forwhen the wave fails to propagate.

It is interesting to investigate why wave propa-gation fails when Fc is below a certain critical value.When the wave reaches a membrane the front of thewave stops while Ca2+ slowly leaks throughthe membrane. The [Ca2+] on the other side of themembrane must reach the threshold value before thewave will start to propagate in the next cell. Therecould be two reasons why the [Ca2+] doesn’t reach thethreshold. Firstly, the Ca2+ may be being pumpedaway faster than it is leaking through, so the [Ca2+]will never build up sufficiently. Alternatively, the backof the wave could reach the membrane before the[Ca2+] on the other side of the membrane exceeds thethreshold.

5.1.

Since the model of Ca2+ wave propagation is anexcitable system, it is reasonable to expect that someinsight into the model’s behaviour may be gained bythe study of simpler models of excitable systems.Thus, to gain an analytic understanding of propa-gation failure, we study a simpler model, thegeneralised bistable equation,

1c1t

=Dc12c1x2 + f(c), (7)

where f(c) has a generic cubic shape, with roots atc=0, a and 1. It is well known that the generalisedbistable equation has a travelling wave solution when0Q aQ 1/2, and so we restrict ourselves to this case.There has been a considerable amount of work doneon the effects of blocks, or gap junctions, on

reaction-diffusion equations and wave propagation(Pauwelussen, 1981, 1982; Othmer, 1983; Keener,1991; Sneyd & Sherratt, 1997). The argument usedhere is based on the methods presented in Keener(1991).

Since an entire wave front is contained within asingle cell, it is sufficient to suppose we have twosemi-infinite cells, one extending from x=−a tox=0 and the other extending from x=0 to x=a.The discontinuity in the solution at the cell boundaryis given by (6). We look for a standing solution to thebistable equation, under the assumption that astanding solution will preclude the existence of apropagating wave (Fife, 1979; Keener, 1987, 1991;Pauwelussen, 1981, 1982). Although the non-exist-ence of a standing solution does not prove theexistence of a propagating wave solution, numericalsolutions indicate that, at least for the case consideredhere, if a standing solution does not exist, a wavesolution usually does.

The standing solution must satisfy c : 1 asx : −a, and c : 0 as x : a. At steadystate

Dcc0=−f(c), (8)

and so

12

ddx

(c')2 =ddx

G(c), (9)

F. 3. The wave speed versus Fc for the normal simulations (withcell membranes) and the homogenised equations. As Fc decreasesthe plane wave speed decreases, but below Fc =0.3 mm s−1 the cellboundaries prevent wave propagation. The speed asymptoticallyincreases to about 6.5 mm s−1 with increasing Fc. Key: w,homogenisation; q, normal.

. . 304

where

G(c)=−1Dcg

c

0

f(v)dv. (10)

It follows that

12(c')2 =6G(c)−G(1) x$(−a, 0],

G(c), x$[0, a),(11)

where we have used the boundary conditionsat 2a. If we now let c− denote c(0−) and similarly

for c+, we get the boundary conditions atx=0,

z2[G(c−)−G(1)]=Fc

Dc(c− − c+)=z2G(c+). (12)

This can now be solved for c− and c+. Then, requiringc+ Q a (which is necessary in order for the solutionto be valid), we get a relationship between Fc

and a that guarantees the existence of a standingsolution.

As a particular realisation of this, we choosef(c)=−c+H(c− a) to give the piecewise linear

F. 4. How an intercellular spiral is created numerically. The cells are 30 mm across. (a) At time=40 s the initial wave has moved acrossapproximately half the domain. (b) At time=57.2 s, a bolus of Ca2+ (1 mM) is added. (c) At times 65 s the tip of the bolus has curledaround and (d) at time 88 s it has developed into a spiral wave. In this simulation Fc =1.1 mm s−1.

34

22

24

26

28

30

32

100806040200

Fc

Sp

ira

l p

erio

ds

(s)

34

22

24

26

28

30

32

104 6 820

Fc

Sp

ira

l p

erio

ds

(s)

305

bistable equation. In this case, a propagating waveexists only when

Fc

2Fc +zDc q a. (13)

When a=0.1 and Dc =20, a propagating wave existsonly when Fc q 0.56. (Of course, this numericalestimate does not necessarily apply to more complexmodels.)

The analysis above does not take into account theexistence of a wave back, which can lead topropagation failure even when Fc is greater than thecritical value. For, when Fc is close to the criticalvalue, the wave will take a long time to cross themembrane; this will lead to propagation failure ifthe wave back catches up to the membrane before thefront has progressed through. Thus, the analysisabove just gives a lower bound for the critical valueof Fc.

6. The Effective Diffusion Coefficient

One common technique used in the study ofintercellular waves in cardiac tissue is homogenis-ation, in which the details of the cell boundaries arereplaced by an averaged description (Hunter et al.,1975; Keener, 1991; Neu & Krassowska, 1993).Homogenisation smooths out local variations to givean effective diffusion coefficient that ignores vari-ations on the microscopic scale.

Given a line of N cells, each of length L, theeffective diffusion coefficient is defined by assumingthat an analog of Ohm’s law holds. Thus, ifc(0,t)=C0 and c(NL,t)=C1, the effective diffusioncoefficient, De, is defined by

J=De

NL(C0 −C1), (14)

where J is the steady state flux. It is a standard result(see, for example, Weidmann, 1966; Hunter et al.,1975) that, when L is small with respect to thediffusion space constant,

1De

=1De

+1

LFc. (15)

The assumption that the length of each cell is smallcompared with the diffusion space constant is anaccurate one for wave propagation in cardiac tissue,the context in which this theory is often used. Thus,in typical cardiac tissue, the wave front extends overmany cells, the effects of the cell boundaries beingmerely to add small perturbations to the overall waveshape. The opposite is true however for intercellularCa2+ waves. The diffusion coefficient of Ca2+ is only

about 20 mm2 s−1, while each cell is about 30 mm long.Thus, the wave front is completely contained withina single cell, and the wave progresses in anintermittent fashion, stopping at the cell boundaries,and travelling quickly across each cell. There is thusno a priori reason to expect that the wave can bemodelled by the use of an effective diffusioncoefficient.

Nevertheless, numerical computations show thatthe plane intercellular wave speed is closely approxi-mated by that predicted by the effective diffusioncoefficient, as is shown in Fig. 3. At low values of Fc

the homogenised theory cannot of course explainpropagation failure, but at higher values of Fc theagreement is excellent. However, the homogenisedequations do not provide a good description of theintercellular spiral wave.

6.1.

Interestingly, the above result gives us a simple wayof measuring Fc, the intercellular permeability. Sincethe wave speed is proportional to the square root ofthe diffusion coefficient, it follows that sp/si =zDc/zDe, where sp is the plane wave speed, and si is theintercellular wave speed. Thus, using (15), it followsthat

Fc =Dc

L $0sp

si12

−1%. (16)

F. 5. Period of the intercellular spiral versus Fc for cells ofwidth 30 mm and 1 mm and for the homogenised equations. key: w,homogenised equations; q, small cells; *, normal sized cells.

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it is more useful to write this in a different form. Ifthe intercellular wave takes t seconds to travelbetween cells, then

si =Lsp

L+ tsp, (17)

where L is the length of an individual cell. Thus,

Fc =Dc

2tsp +(tsp)2/L. (18)

From the data of Robb–Gaspers & Thomas (1995) weestimate L=25 mm, t=2.5 s, and sp =21 mm s−1,which gives Fc 1 0.1 mm s−1.

7. Intercellular Spiral Waves

It is a relatively simple matter to generate anintercellular spiral, as long as Fc is large enough.Firstly, a wave is set up to propagate across thedomain of cells. When it is approximately in thecentre, a bolus (in the shape of a line) of Ca2+ isplaced directly behind the refractory region of thewave. Because of the refractory period of the medium,the bolus cannot spread in the direction of the wave,and thus goes the other way. As it moves, the ends ofthe bolus start to curl round on themselves thuscreating a spiral (Fig. 4). Even though a wave will

F. 6. The core of the intercellular spiral wave when Fc =0.5 mm s−1. Panels (a–d) are in order of increasing time.

307

propagate for Fc as low as 0.3 mm s−1, a spiral cannotbe formed for such low values of Fc. For Fc =0.5 mms−1 or above the spirals are stable. At Fc =0.4 mm s−1

a spiral can be formed, but it is not stable, and breaksup after a few minutes. At Fc =0.3 mm s−1 a spiralcannot even be formed. It is interesting to note thatthe value of Fc estimated from experiments in theintact liver is close to the value needed to propagatean intercellular wave in the model. This confirms thatthe observed intercellular waves may indeed be theresult of the intercellular coupling of excitable cells, asoriginally proposed.

In Fig. 5 we plot the spiral period versus Fc for threecases: the homogenised equations (using the effectivediffusion coefficient), normal sized cells (side length of30 mm) and very small cells (side length of 1 mm).Firstly, notice that as Fc : a all three lines converge(the period for the small cells is a bit low—this isprobably due to discretisation error as the small cellscreate some numerical difficulties). This convergenceis as expected. As Fc : a, cell membranes will haveno effect on wave propagation. Secondly, thehomogenised equations do a good job at predictingthe spiral period when the cells are small (and, ofcourse, if Fc is not too small). Again, this is asexpected. As the cells get smaller, the assumptionsunderlying the homogenised equations become moreaccurate, and thus the theory works better.

However, the homogenised equations do verypoorly at predicting the period of the spirals over thelarger cells. Not only that, but the spiral periodbehaves in a fundamentally different manner as afunction of Fc. For the other two cases the period isa monotone decreasing function of Fc. This is simplydue to the increase in the effective diffusioncoefficient, which speeds up the waves, thusdecreasing the period. But for the cells of normal size,the period first decreases and then increases as Fc

increases. The reasons for this are not completelyclear, but an educated guess can be made.

The period of the spiral is governed by the speedat which the core rotates. Fc affects the core in twodifferent ways. With decreasing Fc, any intercellularwave, and thus the core, travels slower because of theincreased resistance. However, as Fc decreases thecore also loses its curvature, and consists rather of acombination of straight wave portions, interrupted bythe cell boundaries. As can be seen from Fig. 6, forFc =0.5 mm s−1 the core is essentially made up ofstraight wave fronts propagating around a two by twosquare of cells. Thus, the core of the spiral does nothave a high curvature, as do the spiral cores in ahomogeneous medium, and so curvature effects donot slow down the spiral core, as is the case in a

homogeneous excitable medium (Keener, 1986;Zykov, 1980; Tyson & Keener, 1988). On thecontrary, as the spiral core loses curvature it actuallyspeeds up. In summary, as Fc decreases, there are twocompeting effects; firstly, the decrease in wave speedcaused by the decrease in the effective diffusioncoefficient, and secondly, an increase in wave speedcaused by the loss of curvature. Competition betweenthese two effects causes the non-monotonic behaviourof spiral period as a function of Fc.

We can check this hypothesis in a qualitativefashion by considering the decrease in spiral period atlow values of Fc. As Fc increases from 0.5 mm s−1 upto 2 mm s−1 the plane wave speed increases by a factorof about 5.6/3.1=1.32 and the period goes down bya similar amount of about 34/22=1.55. Hence, itappears that the decrease in spiral period is causedalmost entirely by the increased speed of theintercellular plane wave, and curvature is having verylittle effect. However as Fc increases from about 3 mms−1 the core starts to gain curvature, the increase inthe speed of the core is lessened, and so the periodincreases again.

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