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Spiral wave breakdown in an excitable medium model of cardiac tissue
Chaos, Solitom
& FractaLF Vol 5, Nm 3/4, pp. 635-643, 1995 Copyright 0 1995 Elsevier Science Lfd Printed in Great Britain. All rights reserved 6964~u779p5$9.50 + .cnl
Spiral Wave Breakdown in an Excitable Medium Model of Cardiac Tissue H. ZHANG and N. PATEL The Department
of Physiology
and Centre for Nonlinear
Studies, University
Abstract-Cardiac tissue is modelled by an excitable medium based Meander and the rate-dependence of action potential duration turn medium into a functionally inhomogeneous medium, and cause spiral between the propagating waves leads to a spatio-temporal chaos excitable medium.
of Leeds, Leeds LS2 9JT, UK
on the Noble 1962 equations. a parametrically homogeneous wave breakdown. Interactions that spreads throughout the
1. INTRODUCTION
The rhythmic beating of the heart is triggered by its electric activity, waves that propagate out from the sinoatrial pacemaker and spread through the heart muscle and conducting system. Propagating activity in heart muscle may be idealized as activity in a bounded twoor three-dimensional excitable medium [l-3], whose local properties represent the nonlinear behaviour of excitability, and one of whose kinetic variables (the membrane potential) interacts diffusively. With the development of computing power, mathematical models for a piece of cardiac tissue have evolved from simplified symbolic models [4,5] to biophysically derived equations [6-91. Although these equations are cumbersome in numerical simulations, they allow modelling of specific features of cardiac tissue [lo]. Re-entry of a propagating wave, where excitation passes repeatedly through the same part of the medium, is a general phenomenon observed in excitable media. For cardiac tissue, re-entrant waves have been found experimentally in thin sheets of cardiac tissue [ll-141. Re-entrant waves in cardiac tissue are believed to underly some cardiac arrhythmias [l-4]. Spiral waves in some two-dimensional models for excitation in cardiac tissue [6-81, and for some simple two-dimensional phenomenological models [15] are not stable, but break down. Breakdown of re-entrant waves, producing new re-entrant sources, is believed to be an important mechanism in the genesis of atria1 and ventricular fibrillation. Here, we use the Noble 1962 equations [16] in a partial differential system in R2 to model wave phenomena in a sheet of cardiac tissue. In numerical experiments we started with a parametrically homogeneous medium, with the parameters chosen so that the equations have a stable, spatially uniform equilibrium state. We trace the tip movement of the spiral wave, and record time series of action potentials from different sites in the medium. In order to study the dynamical behaviours of the medium in the temporal and spatial domain, we also measure the period of the wave front arriving the spatially distributed recording sites with time evolution. The spiral wave meanders, and the rate-dependence of action potential duration turns the medium into a medium with inhomogeneous excitability after a transition time. Re-depolarization of an incompletely recovered refractory tail of a propagating wave causes a spiral wave breakdown. Interactions between the propagating 635
H. ZHANG and N. PATEL
636
waves leads to an irregular spatio-temporal pattern that spreads out from the centre of spiral wave throughout the whole excitable medium. 2. NUMERICAL MODEL AND EXPERIMENTS
The Noble 1962 equations are derived from voltage-clamp experiments, and represent an early description of the ionic current mechanism for Purkinje fibre membrane. Even though more mechanistically accurate models exist [17], the Noble 1962 equations reproduce the stiffness and shape changes of membrane action potential for Purkinje fibres and ventricular tissue. Since cardiac tissue is a functional syncytium composed of cardiac cells coupled by low-resistance pathways, it may be considered as a continuous medium represented by a nonlinear parabolic partial differential equation of the reaction-diffusion type, which takes the form: C c = -I tot i- DAE (1) m at where E is the membrane potential, C, is the capacity, I,,, is the total ionic current, D is the diffusion coefficient and A is the Laplacian operator. D and A together present the coupling and communications between cardiac cells. In the model, the time is measured in ms, space in mm, the membrane potential in mV, the ionic current in nA, and the capacity in @. Numerical experiments were performed on a uniform grid of 200 x 200 points representing a medium of 320 x 320 mm. The spatial resolution is about 1.6 mm in both X- and y-direction. We used the explicit Euler integration method to solve the partial differential equation with integration time-step dt = 0.2 ms. The boundary condition is: (nA)El, = 0, where n is the normal unit vector to the boundary I’. The coupling coefficient D is simply taken as 1. To reduce the effect of the grid geometry projection to the numerical solution, we use a 9-point neighbourhood for the Laplacian operator, which takes the form: 2 Ei,j+l + Ei,j-1 + E,-l,j
AE=%+i%=ax2
ay*
+ L Ei+l,j+l 3
+ E,+l,j - 4Ei,j
AX2
3 + Ei+l,j-1 + Ei-l,j+l
+ E;-l,j-1
- 4Ei,j
ww2
In experiments, we use the parameters gNa= 0.132, gkr = 1.0. Other parameters are as in [16]. At these parameters, the Noble 1962 equations have a stable, spatially-uniform equilibrium state. 3. UNSTABLE AND NON-STATIONARY SPIRAL WAVE
In an excitable medium a broken travelling plane wave of excitation will develop into a spiral wave in a 2D medium, and a scroll wave in a 3D medium. A spiral wave is characterized by its rotation period and travelling speed, stability and stationarity, all of which are determined by the properties of the medium, i.e. the local kinetics and the spatial parametric structure of the medium [18]. In a homogeneous medium, a stable stationary spiral wave can follow rigid rotation around a circular core of radius r,,, or can undergo non-stationary rotation. The tip of a non-stationary spiral wave can meander in a complicated pattern [4]. Inhomogeneity of the parametric properties of the medium also can induce the spiral wave meandering and drifting [19]. Spiral wave breakdown will produce new interactive spiral waves.
Spiral wave breakdown in a model of cardiac tissue
631
Our numerical solutions demonstrate the non-stationarity and instability of a spiral wave in a parametrically homogeneous medium, with parameters chosen so that the medium has a spatially uniform, stable equilibrium state. The unstable spiral wave evolves into multiple interacting spiral waves that generate autowave turbulence. Figure 1 shows the evolution of spiral wave breakdown. The spiral wave was established from a broken plane wave, and present snapshots taken at different times. In Fig. 1, (a) is the spiral wave 2800 ms after its initiation, (b) is 720 ms after (a), (c) is 3520 ms after (b), and (d) is 3720 ms after (c). The spiral wave (a) initially breaks down in a region of sharp curvature where the gap between the wave front and the wave back of the preceding wave is narrow (c.f. Fig. l(b)), producing multiple spiral waves. The new spiral waves break down again (Fig. l(c)), produce a spatio-temporal irregular pattern (Fig. l(d)) that spreads outwards throughout the medium. The propagation velocity-curvature relation determines that the more the curvature the slower the propagating speed [18]. At the sharply curved places, the propagating speed is very low, the gap between the propagating wave front and its preceding wave back is narrow. The wave front can re-excite the refractory tail of the preceding wave before it is fully recovered, but the excitation is with short duration and low amplitude of the action potential. The re-excitation of an incompletely recovered cardiac cell produces irregularity of cardiac action potential, and interaction between the propagating wave and its preceding wave. The spiral wave rotates in a non-stationary manner. The tip movement of the spiral wave is illustrated in Fig. 2. Figure 2(a) is the tip path, 2(b) and 2(c) are the time recordings of x-coordinate and y-coordinate of the core position respectively. The tip of the vortex followed an open, not closed, path after a time of transition. After the spiral wave breaks down, there are several core positions. From the time recordings of core position coordinates (x in 2(b), y in 2(c)), spiral wave breakdown appears as a cascade of bifurcations. Interactions between the propagating waves lead to an irregular spatio-temporal pattern. The spatio-temporal activity of the medium can be illustrated by the time series recorded from a different recording sites. To study the spatio-temporal dynamical behaviours of the medium, we recorded time series from different points of the medium and (see Fig. 3) measure the periods for the wave front arriving a specific recording site. The time series and periods of the wave front arriving a specific point are illustrated in Fig. 3 Figure 3(a)-(d) are the time series of the action potential recorded from the coordinates (25, 25), (50, 50), (75, 75) and (100, 100) in the medium, respectively. All the recorded time series are irregular in the sense that the duration and amplitude of the action potential, and the interval between two action potentials are irregular. In Fig. 3(c) and (d) one can see that, even before one action potential is fully recovered, the next excitation occurs with very short duration and low amplitude, resulting from the rate-dependent properties [20]. In this way, the parametrically homogeneous medium evolves into a functionally inhomogeneous medium. The non-stationarity can also be illustrated by the time series of the period of wave fronts arriving a specific recording site in the medium. In an excitable medium, when non-stationary rotation occurs, the tip of the spiral wave meanders (see Fig. 2) the source of the spiral wave changes position from time to time, and the period of the wave front arriving at a specific recording site oscillates. The oscillation of the period for the wave front arriving a specific recording site is an indicator of the non-stationary rotation [21]. If the oscillation is regular, the spiral wave meanders in a cycloid; otherwise, the spiral wave meanders irregularly and the temporal behaviour at a specific point of the medium may be chaotic. In Fig. 3(e), we can see that the oscillation of the period of wave fronts arriving
638
H. ZHANG
and N. PATEL
a
b
h c a zic .E
142 .
8 Y X 15
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time (ms) C
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time (ms) Fig. 2. The path of core movement (a) the core path of the spiral wave. The spiral wave is non-stationary, its tip moves along a open curve. After the spiral wave breaks down, there are multiple core movement regions; (b) the time series of core position. A sequence of bifurcations in the core position time series shows that the spiral wave breaks down to produce a spatio-temporal complex pattern.
position (i, j) = (50, 50) (noted by symbol n) is irregular, which suggests that the temporal behaviour at the recording site may be chaotic. To observe the spatial behaviour of the medium, we measure the periods between the wave fronts arriving at recording sites ((i, j) = (25, 25), (i, j) = (50, 50), (i, j) = (75, 75),
Spiral wave breakdown in a model of cardiac tissue
639
d
Fig. 1. The development of a non-stationary and unstable spiral wave in a two-dimensional homogeneous excitable cardiac: tissue modelled by Noble 1962 equations with size of 320 x 320 mm, forming 200 x 200 grid points with spatial resolution 1.6 mm in both x- and y-directions. The time-step is 0.2 ms. The parameters are EN. = 0.132, gkl = 1.0, at which the Noble 1962 equation is in an equilibrium state. With time evolution, the spiral wave breaks down and evolves into spatial-temporal chaos. (a) A spiral wave formed 2HIOms after initiation of spiral wave; (b) a spiral wave formed 720 ms after (a); (c) a spiral wave formed 3520 ms after (b); (d) a spiral wave formed 3720 ms after (c).
641
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I, 0
I, 1
I, 2
I, 3
I 4
,,, 5
I, 6
I 7
,I,‘, 8
9
10
k-th period Fig. 3. The recordings of action potentials and periods for wave front arriving different recording sites over the medium. All the time series are irregular. (a) The time series of the recordings from points (25, 25); (b) the time series of the recordings from points (50, 50); (c) the time series of the recordings from points (75, 75); (d) the time series of the recordings from points (100, 100); (e) the fluctuating periods for the spiral wave front arriving at different recording points. Abscissa: the ith number of period. Ordinate: the period (ms). 1 at the point (i, j) = (25, 25). * at the point (i, j) = (50, 50). l at point (i, j) = (75, 75). o at the point (i, j) = 100. All periods fluctuate in a bigger region. This is different from an excitable medium supporting a hypermeandering spiral wave where near to the centre point of the core region of the spiral wave the period of wave front fluctuates in a bigger variance than period oscillation far from the centre point of the core region.
(i, j) = (100, 100)). All the periods oscillate irregularly with a similar variance. This is different from the spatio-temporal behaviour of an excitable medium produced by hypermeander of a spiral wave, where near to the source of the spiral wave the period of a wave front oscillates irregularly, with a larger variance than the period oscillations far from
642
H. ZHANG and N. PATEL
the source of the spiral wave. The spatio-temporal chaos associated with the hypermeander is localized, and measures of its irregularity decrease with distance from the vortex centre [22]. However, the spatio-temporal chaos produced by spiral wave breakdown and interactions is not localized, and propagates through the whole medium. 4. CONCLUSION
Cardiac tissue is not a simple excitable medium with a fixed threshold for initiating a travelling wave. Rate-dependent changes in conduction velocity and action potential shape and duration [20] change a homogeneous medium to a functionally inhomogeneous medium. A spatio-temporal irregularity is produced by the self-destruction of a single spiral wave in the Noble 1962 model with parameters such that there is a spatially uniform, stable equilibrium state; earlier studies used an auto-rhythmical medium [6,7]. The self-destruction of a spiral wave results from the interaction between the back of a preceding wave and the front of the subsequent wave. The rate-dependent properties of a cardiac cell [20] and the curvature dependence of conduction speed of a spiral wave [18] reduce the speed of wave propagation. In a region of high curvature, i.e. close to the tip, a wave front re-depolarizes the refractory tail of the preceding wave. Interactions between two successive action potentials can cause fragmentation of the spiral wave in these regions of sharp curvature. The spatio-temporal chaos produced by spiral wave breakdown and interactions is global, and spreads throughout the medium. This differs from the spatio-temporal chaos associated with hypermeander, where the spatio-temporal chaos is localized, and far from the core of the spiral wave, the irregularity decreases [22]. Acknowledgements-The authors would like to thank Dr A. V. Holden for supervision. This work is partly supported by Welcome Trust and MRC UK: SPG 9070859.
REFERENCES 1. J. Jalife (editor), Mathematical approaches to cardiac arrhythmias, Ann. NY Acad. Sci. 591, l-417 (1990). 2. A. V. Holden, M. M. Markus and H. G. Othmer, Nonlinear Wave Processes in Excitable Media, iATd ASI Series 244B. Plenum, New York (1991). 3. A. T. Winfree, The Geometry of Biological Time. Springer, New York (1980); When Time Breaks Down. Princeton University Press, Princeton, NJ (1987). 4. A. T. Winfree, Varieties of spiral wave behaviour: An experimentalist’s approach to the theory of excitable media, Chaos 1, 303-334 (1991). 5. A. V. Holden and H. Zhang, Modelling propagation and re-entry in anisotropic and smoothly heterogeneous cardiac tissue, J. Chem. Sot. Faradav Trans. 89. 2833-2837 (1993). 6. A. V. Panfilov and A. V. Holden, Self-generation of turbulent vortices in a two-dimensional model of cardiac tissue, Phys. Lett. A151,23-26 (1990). 7. A. Panfilov and A. V. Holden, Spatio-temporal chaos in a model of cardiac electric activity, Int. J. Bifurcation & Chaos 1,219-225 (1991). 8. M. Courtemanche and A. T. Winfree, Re-entrant rotating waves in Beeler-Reuter based model of two-dimensional cardiac conduction, Znt. J. Bifurcation & Chaos 1, 431-444 (1991). 9. R. L. Winslow, A. L. Kimball, A. Varghese and D. Nobel, Simulation cardiac sinus and atria1 network dynamics on the connect machine, Physica D64, 281-298 (1993). 10. M. Boyett, A. V. Holden, I. Kodama, R. Suzuki and H. Zhang, Vagal control of sino-atria1 node-experiments and simulations, Chaos, Solitons & Fractals 5(3/4), 425-438 (1995). 11. J. H. Kingma, N. M. van Hemel and K. I. Lie (editors), Afrial Fibrillation, a Treatable Disease? Kluwer,
Dordrecht (1992). 12. M. A. Allessie, M. J. Schalij, C. J. H. Kirchhof, L. Boersma., M. Huyberts and J. Hollen, Ann. NY Acad. Sci. 591, 247 (1990). 13. J. M. Davidenko, A. V. Pertsov, R. Salomonsz, W. Baxter and J. Jalife, Nature Land. 355, 449 (1991). 14. A. M. Pertsov, J. M. Davidenko, R. Salomonsz, W. Baxter and J. Jalife. Circ. Res. 72. 631 (19931. \ , 15. A. V. Panfilov and P. Hogeweg,.Phys. Lea. Al76, 295 (1993).
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16. D. Noble, A modification of the Hodgkin-Huxley equations application to Purkinje fiber action and pacemaker potentials, J. Physiol. U&317-352 (1962). 17. D. Noble, OXSOFT Heart Manual, Version 2.1. 25 Beaumont St., Oxford OX1 2NP, England (1988). 18. J. P. Keener and J. J. Tyson, SIAM Rev. 34, 1 (1992). 19. A. S. Mikhailov, V. S. Zykov and A. V. Davydov, Complex dynamics of spiral waves and motion of curves, Physica Iwo, l-39 (1995). 20. M. R. Boytte and B. R. Jewel& Analysis of the effects of the effects of changes in rate and rhythm upon electrical activity in the heart, Prug. Biophys. Molec. Biol. 31, 1-52 (1981). 21. A. V. Panfilov and A. V. Holden, Computer simulation of re-entry sources in myocardium in two and three dimensions. J. Theor. Biol. 161, 271-285 (1993). 22. H. Zhang and A. V. Holden, Chaotic meandering of spiral waves in the FitzHugh-Nagumo system, Chaos, Solitons & Fractds 5(3/4), 661-670 (1995).
& FractaLF Vol 5, Nm 3/4, pp. 635-643, 1995 Copyright 0 1995 Elsevier Science Lfd Printed in Great Britain. All rights reserved 6964~u779p5$9.50 + .cnl
Spiral Wave Breakdown in an Excitable Medium Model of Cardiac Tissue H. ZHANG and N. PATEL The Department
of Physiology
and Centre for Nonlinear
Studies, University
Abstract-Cardiac tissue is modelled by an excitable medium based Meander and the rate-dependence of action potential duration turn medium into a functionally inhomogeneous medium, and cause spiral between the propagating waves leads to a spatio-temporal chaos excitable medium.
of Leeds, Leeds LS2 9JT, UK
on the Noble 1962 equations. a parametrically homogeneous wave breakdown. Interactions that spreads throughout the
1. INTRODUCTION
The rhythmic beating of the heart is triggered by its electric activity, waves that propagate out from the sinoatrial pacemaker and spread through the heart muscle and conducting system. Propagating activity in heart muscle may be idealized as activity in a bounded twoor three-dimensional excitable medium [l-3], whose local properties represent the nonlinear behaviour of excitability, and one of whose kinetic variables (the membrane potential) interacts diffusively. With the development of computing power, mathematical models for a piece of cardiac tissue have evolved from simplified symbolic models [4,5] to biophysically derived equations [6-91. Although these equations are cumbersome in numerical simulations, they allow modelling of specific features of cardiac tissue [lo]. Re-entry of a propagating wave, where excitation passes repeatedly through the same part of the medium, is a general phenomenon observed in excitable media. For cardiac tissue, re-entrant waves have been found experimentally in thin sheets of cardiac tissue [ll-141. Re-entrant waves in cardiac tissue are believed to underly some cardiac arrhythmias [l-4]. Spiral waves in some two-dimensional models for excitation in cardiac tissue [6-81, and for some simple two-dimensional phenomenological models [15] are not stable, but break down. Breakdown of re-entrant waves, producing new re-entrant sources, is believed to be an important mechanism in the genesis of atria1 and ventricular fibrillation. Here, we use the Noble 1962 equations [16] in a partial differential system in R2 to model wave phenomena in a sheet of cardiac tissue. In numerical experiments we started with a parametrically homogeneous medium, with the parameters chosen so that the equations have a stable, spatially uniform equilibrium state. We trace the tip movement of the spiral wave, and record time series of action potentials from different sites in the medium. In order to study the dynamical behaviours of the medium in the temporal and spatial domain, we also measure the period of the wave front arriving the spatially distributed recording sites with time evolution. The spiral wave meanders, and the rate-dependence of action potential duration turns the medium into a medium with inhomogeneous excitability after a transition time. Re-depolarization of an incompletely recovered refractory tail of a propagating wave causes a spiral wave breakdown. Interactions between the propagating 635
H. ZHANG and N. PATEL
636
waves leads to an irregular spatio-temporal pattern that spreads out from the centre of spiral wave throughout the whole excitable medium. 2. NUMERICAL MODEL AND EXPERIMENTS
The Noble 1962 equations are derived from voltage-clamp experiments, and represent an early description of the ionic current mechanism for Purkinje fibre membrane. Even though more mechanistically accurate models exist [17], the Noble 1962 equations reproduce the stiffness and shape changes of membrane action potential for Purkinje fibres and ventricular tissue. Since cardiac tissue is a functional syncytium composed of cardiac cells coupled by low-resistance pathways, it may be considered as a continuous medium represented by a nonlinear parabolic partial differential equation of the reaction-diffusion type, which takes the form: C c = -I tot i- DAE (1) m at where E is the membrane potential, C, is the capacity, I,,, is the total ionic current, D is the diffusion coefficient and A is the Laplacian operator. D and A together present the coupling and communications between cardiac cells. In the model, the time is measured in ms, space in mm, the membrane potential in mV, the ionic current in nA, and the capacity in @. Numerical experiments were performed on a uniform grid of 200 x 200 points representing a medium of 320 x 320 mm. The spatial resolution is about 1.6 mm in both X- and y-direction. We used the explicit Euler integration method to solve the partial differential equation with integration time-step dt = 0.2 ms. The boundary condition is: (nA)El, = 0, where n is the normal unit vector to the boundary I’. The coupling coefficient D is simply taken as 1. To reduce the effect of the grid geometry projection to the numerical solution, we use a 9-point neighbourhood for the Laplacian operator, which takes the form: 2 Ei,j+l + Ei,j-1 + E,-l,j
AE=%+i%=ax2
ay*
+ L Ei+l,j+l 3
+ E,+l,j - 4Ei,j
AX2
3 + Ei+l,j-1 + Ei-l,j+l
+ E;-l,j-1
- 4Ei,j
ww2
In experiments, we use the parameters gNa= 0.132, gkr = 1.0. Other parameters are as in [16]. At these parameters, the Noble 1962 equations have a stable, spatially-uniform equilibrium state. 3. UNSTABLE AND NON-STATIONARY SPIRAL WAVE
In an excitable medium a broken travelling plane wave of excitation will develop into a spiral wave in a 2D medium, and a scroll wave in a 3D medium. A spiral wave is characterized by its rotation period and travelling speed, stability and stationarity, all of which are determined by the properties of the medium, i.e. the local kinetics and the spatial parametric structure of the medium [18]. In a homogeneous medium, a stable stationary spiral wave can follow rigid rotation around a circular core of radius r,,, or can undergo non-stationary rotation. The tip of a non-stationary spiral wave can meander in a complicated pattern [4]. Inhomogeneity of the parametric properties of the medium also can induce the spiral wave meandering and drifting [19]. Spiral wave breakdown will produce new interactive spiral waves.
Spiral wave breakdown in a model of cardiac tissue
631
Our numerical solutions demonstrate the non-stationarity and instability of a spiral wave in a parametrically homogeneous medium, with parameters chosen so that the medium has a spatially uniform, stable equilibrium state. The unstable spiral wave evolves into multiple interacting spiral waves that generate autowave turbulence. Figure 1 shows the evolution of spiral wave breakdown. The spiral wave was established from a broken plane wave, and present snapshots taken at different times. In Fig. 1, (a) is the spiral wave 2800 ms after its initiation, (b) is 720 ms after (a), (c) is 3520 ms after (b), and (d) is 3720 ms after (c). The spiral wave (a) initially breaks down in a region of sharp curvature where the gap between the wave front and the wave back of the preceding wave is narrow (c.f. Fig. l(b)), producing multiple spiral waves. The new spiral waves break down again (Fig. l(c)), produce a spatio-temporal irregular pattern (Fig. l(d)) that spreads outwards throughout the medium. The propagation velocity-curvature relation determines that the more the curvature the slower the propagating speed [18]. At the sharply curved places, the propagating speed is very low, the gap between the propagating wave front and its preceding wave back is narrow. The wave front can re-excite the refractory tail of the preceding wave before it is fully recovered, but the excitation is with short duration and low amplitude of the action potential. The re-excitation of an incompletely recovered cardiac cell produces irregularity of cardiac action potential, and interaction between the propagating wave and its preceding wave. The spiral wave rotates in a non-stationary manner. The tip movement of the spiral wave is illustrated in Fig. 2. Figure 2(a) is the tip path, 2(b) and 2(c) are the time recordings of x-coordinate and y-coordinate of the core position respectively. The tip of the vortex followed an open, not closed, path after a time of transition. After the spiral wave breaks down, there are several core positions. From the time recordings of core position coordinates (x in 2(b), y in 2(c)), spiral wave breakdown appears as a cascade of bifurcations. Interactions between the propagating waves lead to an irregular spatio-temporal pattern. The spatio-temporal activity of the medium can be illustrated by the time series recorded from a different recording sites. To study the spatio-temporal dynamical behaviours of the medium, we recorded time series from different points of the medium and (see Fig. 3) measure the periods for the wave front arriving a specific recording site. The time series and periods of the wave front arriving a specific point are illustrated in Fig. 3 Figure 3(a)-(d) are the time series of the action potential recorded from the coordinates (25, 25), (50, 50), (75, 75) and (100, 100) in the medium, respectively. All the recorded time series are irregular in the sense that the duration and amplitude of the action potential, and the interval between two action potentials are irregular. In Fig. 3(c) and (d) one can see that, even before one action potential is fully recovered, the next excitation occurs with very short duration and low amplitude, resulting from the rate-dependent properties [20]. In this way, the parametrically homogeneous medium evolves into a functionally inhomogeneous medium. The non-stationarity can also be illustrated by the time series of the period of wave fronts arriving a specific recording site in the medium. In an excitable medium, when non-stationary rotation occurs, the tip of the spiral wave meanders (see Fig. 2) the source of the spiral wave changes position from time to time, and the period of the wave front arriving at a specific recording site oscillates. The oscillation of the period for the wave front arriving a specific recording site is an indicator of the non-stationary rotation [21]. If the oscillation is regular, the spiral wave meanders in a cycloid; otherwise, the spiral wave meanders irregularly and the temporal behaviour at a specific point of the medium may be chaotic. In Fig. 3(e), we can see that the oscillation of the period of wave fronts arriving
638
H. ZHANG
and N. PATEL
a
b
h c a zic .E
142 .
8 Y X 15
10
20
25
30
35
40
l 1d
time (ms) C
5. z
loo-:..- L _ _ ."-. .-.iv.,.
5 'i5
so-
.-
_:' - .....i.'.,. >;- :-...... ;G . .,, .
6
_.-i _..._.... ... : -..r _. __: -.
:.....
E: h
t,l(II
IllI
10
IIII,"'I"'II""l""I"
15
20
25
30
35
40
'10'
time (ms) Fig. 2. The path of core movement (a) the core path of the spiral wave. The spiral wave is non-stationary, its tip moves along a open curve. After the spiral wave breaks down, there are multiple core movement regions; (b) the time series of core position. A sequence of bifurcations in the core position time series shows that the spiral wave breaks down to produce a spatio-temporal complex pattern.
position (i, j) = (50, 50) (noted by symbol n) is irregular, which suggests that the temporal behaviour at the recording site may be chaotic. To observe the spatial behaviour of the medium, we measure the periods between the wave fronts arriving at recording sites ((i, j) = (25, 25), (i, j) = (50, 50), (i, j) = (75, 75),
Spiral wave breakdown in a model of cardiac tissue
639
d
Fig. 1. The development of a non-stationary and unstable spiral wave in a two-dimensional homogeneous excitable cardiac: tissue modelled by Noble 1962 equations with size of 320 x 320 mm, forming 200 x 200 grid points with spatial resolution 1.6 mm in both x- and y-directions. The time-step is 0.2 ms. The parameters are EN. = 0.132, gkl = 1.0, at which the Noble 1962 equation is in an equilibrium state. With time evolution, the spiral wave breaks down and evolves into spatial-temporal chaos. (a) A spiral wave formed 2HIOms after initiation of spiral wave; (b) a spiral wave formed 720 ms after (a); (c) a spiral wave formed 3520 ms after (b); (d) a spiral wave formed 3720 ms after (c).
641
Spiral wave breakdown in a model of cardiac tissue a O-
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-80 0
1000
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01
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I, 1
I, 2
I, 3
I 4
,,, 5
I, 6
I 7
,I,‘, 8
9
10
k-th period Fig. 3. The recordings of action potentials and periods for wave front arriving different recording sites over the medium. All the time series are irregular. (a) The time series of the recordings from points (25, 25); (b) the time series of the recordings from points (50, 50); (c) the time series of the recordings from points (75, 75); (d) the time series of the recordings from points (100, 100); (e) the fluctuating periods for the spiral wave front arriving at different recording points. Abscissa: the ith number of period. Ordinate: the period (ms). 1 at the point (i, j) = (25, 25). * at the point (i, j) = (50, 50). l at point (i, j) = (75, 75). o at the point (i, j) = 100. All periods fluctuate in a bigger region. This is different from an excitable medium supporting a hypermeandering spiral wave where near to the centre point of the core region of the spiral wave the period of wave front fluctuates in a bigger variance than period oscillation far from the centre point of the core region.
(i, j) = (100, 100)). All the periods oscillate irregularly with a similar variance. This is different from the spatio-temporal behaviour of an excitable medium produced by hypermeander of a spiral wave, where near to the source of the spiral wave the period of a wave front oscillates irregularly, with a larger variance than the period oscillations far from
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the source of the spiral wave. The spatio-temporal chaos associated with the hypermeander is localized, and measures of its irregularity decrease with distance from the vortex centre [22]. However, the spatio-temporal chaos produced by spiral wave breakdown and interactions is not localized, and propagates through the whole medium. 4. CONCLUSION
Cardiac tissue is not a simple excitable medium with a fixed threshold for initiating a travelling wave. Rate-dependent changes in conduction velocity and action potential shape and duration [20] change a homogeneous medium to a functionally inhomogeneous medium. A spatio-temporal irregularity is produced by the self-destruction of a single spiral wave in the Noble 1962 model with parameters such that there is a spatially uniform, stable equilibrium state; earlier studies used an auto-rhythmical medium [6,7]. The self-destruction of a spiral wave results from the interaction between the back of a preceding wave and the front of the subsequent wave. The rate-dependent properties of a cardiac cell [20] and the curvature dependence of conduction speed of a spiral wave [18] reduce the speed of wave propagation. In a region of high curvature, i.e. close to the tip, a wave front re-depolarizes the refractory tail of the preceding wave. Interactions between two successive action potentials can cause fragmentation of the spiral wave in these regions of sharp curvature. The spatio-temporal chaos produced by spiral wave breakdown and interactions is global, and spreads throughout the medium. This differs from the spatio-temporal chaos associated with hypermeander, where the spatio-temporal chaos is localized, and far from the core of the spiral wave, the irregularity decreases [22]. Acknowledgements-The authors would like to thank Dr A. V. Holden for supervision. This work is partly supported by Welcome Trust and MRC UK: SPG 9070859.
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