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International Journal of Bifurcation and Chaos, Vol. 14, No. 7 (2004) 2457­2466 c World Scientific Publishing Company

MODEL OF CARDIAC TISSUE AS A CONDUCTIVE SYSTEM WITH INTERACTING PACEMAKERS AND REFRACTORY TIME
ALEXANDER LOSKUTOV , SERGEI RYBALKO and EKATERINA ZHUCHKOVA Physics Faculty, Moscow State University, Leninskie Gory, 119992 Moscow, Russia loskutov@moldyn.phys.msu.ru Received Decemb er 9, 2002; Revised Septemb er 12, 2003
The model of the cardiac tissue as a conductive system with two interacting pacemakers and a refractory time is proposed. In the parametric space of the model the phase locking areas are investigated in detail. The obtained results make possible to predict the behavior of excitable systems with two pacemakers, depending on the type and intensity of their interaction and the initial phase. Comparison of the described phenomena with intrinsic pathologies of cardiac rhythms is given. Keywords : Arrhythmia; phase diagrams; interacting pacemakers; maps.

1. Intro duction
One of the remarkable examples of excitable media is the cardiac tissue. Because the stability of its b ehavior is vitally imp ortant, investigations of processes o ccurring in the cardiac muscle attract a considerable scientific interest. Owing to extraordinary complexity of the system, many alternative mo dels have b een tested. One of them treats the cardiac tissue as an active conductive system. Then, the cardiac rhythms are describ ed on the basis of the dynamical system theory (see e.g. [Courtemanche et al., 1989; Goldb erger, 1990; Bub & Glass, 1994; Glass et al., 2002] and refs. therein). Excitation waves in the cardiac tissue originate in the sinoatrial no de (SA) and spread successively over the right atrium and the left atrium. Then they pass through the atrioventricular no de (AV), bundle of His and Purkinje fib ers, and finally to the walls of the right and left ventricles. The normal rhythm of the heart is determined by the activity of the SA no de which is called the leading pacemaker (a source of concentric excitation waves) or the first order driver of the rhythm. In addition to the SA no de cells, the other parts of the cardiac

conductive system can reveal automaticity. So, the second order driver of the rhythm is lo cated in the AV conjunction. The Purkinje fib ers are the rhythm driver of the third order. Often arrhythmia is the disturbance of the normal heartb eat caused by several propagation failures [Keener, 2000; Lewis & Keener, 2000]. In particular, arrhythmias can b e evoked by the violation of the restitution of the cardiac tissue (so-called the reentry phenomenon) (see e.g. [Panfilov & Keener, 1995; Biktashev & Holden, 1998; Cytrynbaum & Keener, 2002; Glass et al., 2002] and refs. therein). In addition to such pathologies, the disturbances of the cardiac rhythms are induced by the app earance of ectopic excitation sources [Schamorth, 1980; Marriot & Conover, 1983; Zip es & Jalife, 1995; Winfree, 1987; Glass & Mackey, 1988]. Furthermore, a few abnormal sources and spiral waves result in a fibrillation phenomenon (in fact, a spatio-temp oral cardiac chaos [Qu et al., 1997; Qu et al., 2000]). In this case it is necessary to involve the application of a high energy electric sho ck. A numb er of recent clinical studies aim at the improvement of defibrillation proto cols in order
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to minimize failure rates and reduce the amount of energy used and thus damage risk [Zip es & Jalife, 1995]. Recent studies provide a theoretical understanding of the underlying mechanisms of defibrillation [Krinsky & Pumir, 1998; Pumir et al., 1998; Panfilov et al., 2000]. When arrhythmias are presented via interaction of the pacemaker and the ectopic source (i.e. the recipro cal action of nonlinear sources), investigation of such cardiac pathologies can b e subdivided into two large groups: First is based on the continuous time representations (i.e. using systems of ordinary differential equations) [van der Pol & van der Mark, 1928; Keener, 1981; Guevara & Glass, 1982; Hopp ensteadt & Keener, 1982; Keener & Glass, 1984; Keith & Rand, 1984; West et al., 1985; Signorini et al., 1998] (and refs. therein), second -- on the maps representation [Mo e et al., 1977; Jalife et al., 1982; Honerkamp, 1983; Ikeda et al., 1983; Glass & Mackey, 1988; Courtemanche et al., 1989; He et al., 1992; Bub & Glass, 1994; Kremmydas et al., 1996; Ostb orn et al., 2001]. Since arrhythmias are dangerous heart diseases, investigation of such pathologies is of great imp ortance. Moreover, analysis of the complex cardiac rhythms, treated as chaotic phenomena, can give a clue to the problem of controllability of the complex cardiac dynamics [Goldb erger & Rigney, 1988; Goldb erger, 1990; Garfinkel et al., 1992]. In the present investigation, a general mo del of two nonlinear coupled oscillators describing certain typ es of cardiac arrhythmias (AV-blo cks and parasistoles) is elab orated. The mo del o ccurs to b e a universal in the sense that its predictions are not sensitive to the sp ecific form of interactions, i.e. on the phase resp onse curve (PRC). The exp erimentally obtained PRC is approximated by a certain p olynomial function with a plateau. This plateau mo dels the refractory stage when the system do es not resp ond to an external action. Note that the refractory stage plays an imp ortant role in the normal cardiac functioning. For example, the refractoriness extends almost over the p erio d of the cardiac contraction, protecting the myo cardium from premature heartb eats caused by the external p erturbation. The refractoriness provides also the normal sequence of an excitation propagation in the cardiac tissue and the electrical stability of the myo cardium [Marriot & Conover, 1983; Zip es & Jalife, 1995; Winfree, 1987; Glass & Mackey, 1988]. In the prop osed mo del, p ossible areas of phase lo ckings,

caused by the refractoriness, are investigated. We observe the splitting of the resonance tongues and the sup erp osition of the synchronization areas. Using the obtained results we can classify the dynamics of the excitable media with two active pacemakers dep ending on the typ e and intensity of their interaction and the initial phase difference. Moreover, generalizing our approach a theory of excitable media with interacting pacemakers under external actions may b e elab orated. This fact can b e of a great practical imp ortance due to p ossible application in controlling the cardiac rhythms by external stimuli.

2. Heart Tissue as a Dynamical System
The cardiac arrhythmias may b e describ ed sometimes as an interaction of two sp ontaneously oscillating nonlinear sources. Such interaction can b e analyzed in terms of influence of an external p erio dic p erturbation (with a constant amplitude and frequency) on a nonlinear oscillator. In this case it is p ossible to use the well-known circle map [Guevara & Glass, 1982; Glass et al., 1983; Glass & Mackey, 1988; Bub & Glass, 1994; Kremmydas et al., 1996]: x
n+1

= xn + f (xn )

(mo d 1) ,

(1)

where xn is a phase difference of oscillators and the function f (x) determines a change in phase after the action of the stimulus. This function is called a phase resp onse curve (PRC). One of the most imp ortant characteristics of the circle map is a rotation numb er . It is defined as follows: xn - x 0 . = lim n n For stable phase lo cking the rotation numb er is rational. If is irrational then the system b ehavior is quasip erio dic or chaotic. It should b e noted that such a one-dimensional approach imp oses restrictions: We neglect the propagation of the excitation waves on the heart surface. Thus, the violation of the cardiac rhythm due to the absence of pulse co ordinations in the whole myo cardium cannot b e describ ed within this mo del. Analyzing dynamics of the mo del based on the circle map, it is necessary to find a prop er analytical approximation for the exp erimentally obtained PRC. This allows investigation of the most salient features of the system of interest.


A Model of Cardiac Tissue with Interacting Pacemakers
1

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0.8

2.6 mkA 5.0 mkA

0.6

0.4

0.2

h(x)

0

-0.2

-0.4

-0.6

-0.8

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x
Fig. 1. Phase response curves: the experimental curves (dotted line) and their analytical approximation (solid line). The experimentally obtained phase response curves show the duration of perturbed cycle (in %) as a function of the input phase.

as the amplitude parameter a continuously changes. In the context of the circle map theory this means that the transformation (1) changes a top ological degree. Unfortunately, consideration of such maps is a rather complex problem, and a continuous PRC approximation is commonly used. In the present pap er we also accept this restriction. The basic feature of any approximation of the PRC is the dep endence on two physical parameters: on the amplitude of stimulus and the input phase. In the ideal case the other (so-called "internal") parameters can b e reduced. Applying the p olynomial function as an approximation of the PRC, we construct a mo del of two bidirectional ly interacting active oscillators.

3. A Mo del with Mutual Influence of Impulses
Let us consider the system of two nonlinear interacting oscillators (Fig. 2). Supp ose that the pulse of the first oscillator with p erio d T 1 b eats at time tn , and the pulse of the second oscillator (with the p erio d T2 ) b eats at time n . Then, the subsequent moments of the app earance of impulses are defined as tn+1 = tn + T1 , n+1 = n + T2 . Now, taking into account that under the influence of the second impulse the p erio d of the first oscillator changes by the value of 1 ((n - tn )/T1 ) (where the relation in brackets means that this value dep ends only on the phase of the second impulse), we get the expression for tn+1 : tn+1 = tn + T1 + 1 ((n - tn )/T1 ). For further analysis, let us consider the case when the pulses of two oscillators strictly alternate

Exp eriments on the recording of phase shifts have b een carried out for quite a large numb er of variety of systems, but we are interested in the PRC extracted from the real cardiac tissues. In [Weidmann, 1951; Jalife & Mo e, 1976] measurements of the cycle durations of the sp ontaneously b eating Purkinje fib ers after stimulation by short electric current pulses have b een p erformed. The obtained phase resp onse curve is shown in Fig. 1 (dotted lines). Taking into account this exp erimental material, it is p ossible to make the following general conclusions [Weidmann, 1951; Jalife & Mo e, 1976; Glass et al., 1986; Glass & Mackey, 1988]: (1) Dep ending on the phase, the single input can lead to either increasing or decreasing in the p erio d of the p erturb ed cycle. (2) After p erturbation, the rhythm is usually restored (after some transient time) with the same frequency and amplitude, but with shifted phase. (3) At some amplitudes of the stimulus the obvious breaks app ear. Also, it is necessary to take into account that: (i) The assumption of the immediate resetting of the pacemaker rhythm after the action of the external stimulus is a certain kind of idealization; (ii) for an adequate description of the third PRC feature it is necessary to cho ose a parametrical function f (x) = fa (x) which o ccurs to b e discontinuous

tn
D
1

t

1
n+1

T
tn

1

2

D

t
2

n+1

T

2

Fig. 2. Construction of the model of two nonlinearly interacting oscillators.


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A. Loskutov et al.
n+1

each other.1 Then for the second oscillator we obtain: n+1 = n + T2 + 2 ((t relations by T1 , we arrive at the corresp onding values for the phases: n+1 = n + 1 1 (n - n ), T1
n+1

- n )/T2 ). Dividing these

= n +

T2 1 + T1 T1

2

T1 1 n tn + + 1 (n - n ) - T2 T2 T2 T2

.

Here n = tn /T1 is the phase of the first p erturb ed oscillator with resp ect to the unp erturb ed one (with p erio d T1 ), and n = n /T1 is the phase of the second p erturb ed oscillator with resp ect to the same first oscillator with p erio d T1 . Using parameters a = T2 /T1 and 1 /T1 = f1 , 2 /T1 = f2 one can write: n+1 = n + f1 (n - n ), 1 n+1 = n + a + f2 (n + 1 + f1 (n - n ) - n ) , a which after some algebra yields the final expression for the phase difference of the oscillators: x
n+1

= xn + a + f - f1 (xn )

2

1 (1 + f1 (xn ) - xn ) a (mo d 1) , (2)

In nomial tained studies

the present pap er the map (3) with the polyfunction h(x) will b e addressed. The obresults are generalization of our previous [Loskutov, 1994; Loskutov et al., 2002].

where xn = n - n . Obviously, the PRC changes its form dep ending on the amplitude of the external stimulus. In the simplest case this dep endence can b e approximated by the multiplicative relation. Then the phase resp onse curves can b e written as follows: f1 = h(x) , f2 = h(x) , where h(x) is a p erio dic function, h(x + 1) = h(x). Under such assumption, the transformation (2) takes the form: 1 (1 + h(xn ) - xn ) xn+1 = xn + a + h a - h(xn ) (mo d 1) . (3)

4. Phase Diagrams for Unidirectional Coupling of Oscillators
Let us analyze the situation when p ermanent puts act on the nonlinear oscillator, i.e. f 2 (x) or = 0. As an analytical approximation of exp erimental curve in Fig. 1, let us consider the lowing p olynomial function: h(x) = C x
2

in0 the fol(4)

1 - x (1 - x)2 . 2

The normalizing factor C is chosen that the amplitude of h(x) is equal 20 5 (see Fig. 1, solid line). Then count the refractory time we can (3) as 0 xn , (mo d 1),

in such a way to 1, i.e. C = taking into acwrite the map

x

n+1

=

xn + a,

where h(·) is determined by (4). Now let us analyze the dep endence of the system dynamics on the refractory time. We start with the case when the refractory p erio d is trivial, i.e. = 0. The phase lo cking regions in the parametric space (a, ) obtained by numeri1

x + a - h n

xn - 1-

(5) (mo d 1),

,

< xn 1,

cal analysis are shown in Fig. 3(a), where a [1, 2] is chosen. Different colors define the phase lo cking areas with multiplicity N : M , where N cycles of external stimulus corresp ond to M cycles of the nonlinear oscillator. One can see that "tales" of the

The case when the pulses of two oscillators are not intermittent is addressed in [Loskutov et al., 2003].


A Model of Cardiac Tissue with Interacting Pacemakers

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(a) Fig. 3. Phase diagrams of the map (5): (a) = 0; (b) = 0.1.

(b)

(a) Fig. 4.

(b) Phase locking areas of the map (5): (a) = 0.3; (b) = 0.5.

main lo cking regions are slightly split and overlap at large . As it follows from the analysis of the system (5) with = 0.1 [Fig. 3(b)], nonvanishing refractory time leads to the extension of the phase

lo cking areas and enhances splitting and overlap of their tales. In Fig. 4(a) the numerically obtained phase diagram for = 0.3 is shown. In this figure the same


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(a) Fig. 5. Phase diagrams of the map (5): (a) = 0.7; (b) = 0.9.

(b)

N : M stable phase lo ckings as in Fig. 3 are given for comparison. As it follows from the figure, the 2 : 3 phase lo cking area increases with increasing refractory time; simultaneously the 1 : 1 and 1 : 2 areas decrease. The phase lo cking regions for = 0.5 are shown in Fig. 4(b). This phase diagram is qualitatively different from those shown ab ove. The form of 2 : 3 phase lo cking area is stretched and lo oks like an arrow. The forms of 3 : 4 and 3 : 5 regions also resemble arrows for = 0.7 [Fig. 5(a)]. At = 0.9 all phase lo ckings are degenerated into vertical lines. This situation is illustrated in Fig. 5(b). Note that for = 1 (i.e. the system do es not resp ond to the external action) there is no dep endence on the stimulus amplitude .

5. Phase Diagrams for Systems with Bidirectional Interaction
In this section the system (3) at = 0.1 is considered. The analysis is p erformed in ( , a)- and ( , )-parametric spaces.

5.1. Phase locking areas in the ( , a)-space
Assume that the influence of the first oscillator on the second is small enough, for example, = 0.1.

The corresp onding phase diagram for = 0.1 is shown in Fig. 6(a). One can see that the mutual action leads to deformation and splitting in the phase lo cking areas. Note that even for small values of the amplitude of the second stimulus , the main phase lo cking areas overlap. This means that the system dynamics b ecomes multistable: Its limiting stage dep ends on an initial phase difference x 0 . The growth of the refractory time in the mo del with = 0.1 leads to a more deep distortion of the forms of main tongues and disapp earance of the splitting areas. If, however, we increase the influence of the first oscillator up to, e.g. = 0.5, a very complicated structure with much more deep deformation of the main phase lo cking areas [see Fig. 6(b)] will app ear. For example, the 1 : 1 area will degenerate into a narrow strip, whereas the 1 : 2 phase lo cking area will expand due to app earance of long narrow tongues. Numerical analysis shows that the increase of up to approximately 0.5 is accompanied by the expansion of resonance zones. At the same time, the shap e of the phase lo cking zones b ecomes more complex, and their lo cation changes. This leads to the complete mixture of zones: Zones of various multiplicity may b e found in a small neighb orho o d of almost any p oint ( , a). Nevertheless


A Model of Cardiac Tissue with Interacting Pacemakers

2463

(a) Fig. 6.

(b)

Phase locking regions of the system of two bidirectionally interacting oscillators with = 0.1: (a) = 0.1; (b) = 0.5.

at any given value of self-similarly structures are clearly observed. Additionally, we have found as the nonlinearity parameter further grows, the resonance zones shrink and o ccupy a smaller area. In this case, the mixing of the resonance tongues also takes place. Thus, the increase of the interaction of the oscilla-

tors causes the mixing of the initially regular structures in the ( , a)-space.

5.2. Phase locking regions in the ( , )-space
Now we construct the phase diagrams of the interacting oscillators in the space of influence

(a) Fig. 7.

(b)

Phase lockings in the space of stimulus amplitudes ( = 0.1): (a) a = /2; (b) a = 2.


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amplitude, i.e. ( , ). First we consider a = 2 [Fig. 7(a)]. This value of the p erio d ratio implies that for = = 0 the rotation numb er (see Sec. 2) is rational, so that the dynamics of the system is p erio dic with the 1 : 2 phase lo cking. Although for a large nonlinearity there exist phase lo ckings with another multiplicity, the b ehavior of the system is p erio dic, with 1 : 2 phase lo cking, even at large and . Qualitatively different b ehavior is observed at a = /2. In this case the rotation numb er is irrational at zero stimulus amplitudes, and the system exhibits quasip erio dicity or chaoticity. However, as the nonlinearity increases the p erio dic b ehavior b ecomes p ossible [Fig. 7(b)]: For a sufficiently large , the decrease of the area o ccupied by the resonance zones may b e observed. Therefore, at irrational values of a there exists a significant probability of the complex b ehavior of system (3).

typ e of synchronization corresp onding to interaction parameters a, , and . Moreover, the phase pictures indicate that as the nonlinearity increases (i.e. at growing ) the areas with various phase lo ckings start to overlap. The knowledge of such regions, predicted by the present mo del, is necessary to op erate the system dynamics. In particular, removing the system from an undesirable mo de of synchronization to an appropriate state by the external action may b e of crucial imp ortance for applications.

7. Conclusion
In the present study a quite general mo del of two nonlinear interacting impulse oscillatory systems is elab orated. On the basis of this mo del it is p ossible to predict certain typ es of cardiac arrhythmias. The constructed mo del is a universal one, in the sense that its prop erties do not dep end on the chosen interaction typ e, i.e. on the form of phase resp onse curve. Taking into account the refractory time the phase lo cking regions of the p olynomial maps (which describ e a nonlinear oscillator under the p ermanent inputs), are investigated. It is found that the nonvanishing refractory time causes the extension of the phase lo cking areas, significant splitting and overlap of their tails. Moreover, the phase lo cking areas shrink and tend to the vertical lines as the refractory time tends to one. The detailed analysis of the phase diagram of the system with two bidirectionally interacting oscillators in the ( , a)-space shows that b esides the splitting of the central tongues there is an overlap of the main regions of synchronization, which corresp onds to various typ es of cardiac arrhythmias. This bistability is observed even for smal l enough stimulus amplitude. The increase of refractory time leads to the distortion in the forms of main tongues and disapp earance of the splitting areas. For sufficiently large values of the first stimulus a very complicated picture is observed, where the phase lo cking areas are interwoven with each other. Another imp ortant prop erty of the suggested mo del is that phase lo ckings in the space of the stimulus amplitudes are observed. It is found that the interacting oscillators can b e synchronized even if the ratio of their p erio ds is irrational (note, that the probability of this phenomenon is quite small). If the coupling lacks (, 0), however, this would corresp ond only to the complex dynamics (quasip erio dic or chaotic).

6. Applications to Heart Rhythm Pathologies
Let us consider the analogy b etween the obtained results and pathological states of the cardiac tissue. Using the develop ed mo dels, it is p ossible, for example, to describ e the interaction of the sinus and the ectopic pacemakers, the SA (sinoatrial) and AV (atrioventricular) no des and impact of an external p erturbation on the sinus pacemaker. Consider the typ es of arrhythmias which one can predict on the basis of our mo del. If the first pulse oscillator is presented as the SA no de and the second one is considered as the AV no de, then one can conclude that certain stable phase lo ckings corresp ond to cardiac pathologies which are detected in a clinical practice. In this case among various lo ckings one can observe the normal sinus rhythm (1 : 1 phase lo cking). In addition, in the diagrams we can see the classical rhythms of Wenckebach (N : (N - 1) phase lo ckings) and N : 1 AV-blo cks. When the first pulse system is considered as the AV no de and the second one is presented as the SA no de, we obtain the inverted Wenckebach rhythms (that are similar to the direct rhythms but the roles of ventricles and atria change places) which were recorded for some patients. The existence of wide areas of phase lo ckings (see Figs. 3­7) confirms it is p ossible to observe synchronization of two oscillators qualitatively corresp onding to some typ es of cardiac arrhythmias. The phase diagram makes p ossible to determine the


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The obtained results make p ossible to predict the dynamics of oscillatory systems, dep ending on the initial phase difference, on the typ e of interaction and its strength. Moreover, using the ab ove approach one can develop a quite general theory of interacting oscillators under a certain p erio dic p erturbation. In this case the knowledge of multistability areas would b e helpful to stabilize the system dynamics and return the cardiac tissue to the required typ e of b ehavior.

References
Biktashev, V. N. & Holden, A. V. [1998] "Reentrant waves and their elimination in a model of mammalian ventricular tissue," Chaos 8, 48­56. Bub, G. & Glass, L. [1994] "Bifurcations in a continuous circle map: A theory for chaotic cardiac arrhythmia," Int. J. Bifurcation and Chaos 5, 359­371. Courtemanche, M., Glass, L., Belair, J., Scagliotti, D. & Gordon, D. [1989] "A circle map in a human heart," Physica D49, 299­310. Cytrynbaum, E. & Keener, J. P. [2002] "Stability conditions for the traveling pulse: Modifying the restitution hypothesis," Chaos 12, 788­799. Garfinkel, A., Spano, M. L. & Ditto, W. L. [1992] "Controlling cardiac chaos," Science 257, 1230­1235. Glass, L., Guevara, M. R., Shrier, A. & Perez, R. [1983] "Bifurcation and chaos in a periodic stimulated oscillator," Physica D7, 89­101. Glass, L., Guevara, M. R. & Shrier, A. [1986] "Phase resetting of spontaneously beating embryonic ventricular heart cell aggregates," Am. J. Physiol. 251, H1298­H1305. Glass, L. & Mackey, M. [1988] From Clocks to Chaos: The Rhythms of Life (Princeton Univ. Press, Princeton). Glass, L., Nagai, Yo., Hall, K., Tala jie, M. & Nattel, S. [2002] "Predicting the entrainment of re-entrant cardiac waves using phase resetting curves," Phys. Rev. E65, 021908-1­021908-10. Goldberger, A. L. & Rigney, D. R. [1988] "Sudden death is not chaos," Dynamic Patterns in Complex Systems, eds. Kelso, J. A. S., Mandell, A. J. & Schlesinger, M. F. (World Scientific, Singapore), pp. 248­264. Goldberger, A. L. [1990] "Nonlinear dynamics, fractals and chaos: Applications to cardiac electrophysiology," Ann. Biomed. Eng. 18, 195­198. Guevara, M. R. & Glass, L. [1982] "Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias," J. Math. Biol. 14, 1­23. He, Y., Chi, R. & He, N. [1992] "Lyapunov exponents of

the circle map in human hearts," Phys. Lett. A170, 29­32. Honerkamp, J. [1983] "The heart as a system of coupled nonlinear oscillators," J. Math. Biol. 18, 69­88. Hoppensteadt, F. C. & Keener, J. [1982] "Phase locking of biological clocks," J. Math. Biol. 15, 339­349. Ikeda, N., Yoshizawa, S. & Sato, T. [1983] "Difference equation model of ventricular parasystole as an interaction between cardiac pacemakers based on the phase response curve," J. Theor. Biol. 103, 439­465. Jalife, J. & Moe, G. K. [1976] "Effects of electronic potencials on pacemaker activity of canine Purkinje fibers in relation to parasistole," Circ. Res. 39, 801­808. Jalife, J., Antzelevitch, C. & Moe, G. [1982] "The case for modulated parasistole," Pace 5, 911­926. Keener, J. P. [1981] "On cardiac arrhythmias: AV conduction block," J. Math. Biol. 12, 215­225. Keener, J. P. & Glass, L. [1984] "Global bifurcations of a periodically forced oscillator," J. Math. Biol. 21, 175­190. Keener, J. P. [2000] "Propagation of waves in an excitable medium with discrete release sites," SIAM J. Appl. Math. 61, 317­334. Keith, W. L. & Rand, R. H. [1984] "1:1 and 2:1 phase entrainment in a system of two coupled limit cycle oscillators," J. Math. Biol. 20, 133­152. Kremmydas, G. P., Holden, A. V., Bezerianos, A. & Bountis, T. [1996] "Representation of sino-atrial node dynamics by circle maps," Int. J. Bifurcation and Chaos 6, 1799­1805. Krinsky, V. & Pumir, A. [1998] "Models of defibrillation of cardiac tissue," Chaos 8, 188­203. Lewis, T. J. & Keener, J. P. [2000] "Wave-block in excitable media due to regions of depressed excitability," SIAM J. Appl. Math. 61, 293­316. Loskutov, A. [1994] "Nonlinear dynamics and cardiac arrhythmia," Appl. Nonlin. Dyn. 2, 14­25 (Russian). Loskutov, A., Rybalko, S. D. & Zhuchkova, E. A. [2002] "Dynamics of excitable media with two interacting pacemakers," Biophys. 47, 892­901. Loskutov, A., Rybalko, S. D. & Zhuchkova, E. A. [2004] in preparation. Marriot, H. J. L. & Conover, M. M. [1983] Advanced Concepts in Cardiac Arrhythmias (C.V. Mosby, St. Louis). Moe, G. M., Jalife, J., Mueller, W. J. & Moe, B. [1977] "A mathematical model for parasistole and its application to clinical arrhythmias," Circulation 56, 968­979. Ostborn, P., Ohlen, G. & Wohlfart, B. [2001] "Simulated sinoatrial exit blocks explained by circle map analysis," J. Theor. Biol. 211, 219­227. Panfilov, A. V. & Keener, J. P. [1995] "Re-entry in threedimensional FitzHugh­Nagumo medium with rotational anisotropy," Physica D84, 545­552.


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A. Loskutov et al.

Panfilov, A. V., Muller, S. C., Zykov, V. S. & Keener, J. P. [2000] "Elimination of spiral waves in cardiac tissue by multiple electrical shocks," Phys. Rev. E61, 4644­4647. Pumir, A., Romey, G. & Krinsky, V. [1998] "Deexcitation of cardiac cells," Biophys. J. 74, 2850­2861. Qu, Z., Weiss, J. N. & Garfinkel, A. [1997] "Spatiotemporal chaos in a simulated ring of cardiac cells," Phys. Rev. Lett. (USA) 78, 1387­1390. Qu, Z., Weiss, J. N. & Garfinkel, A. [2000] "From local to global spatiotemporal chaos in a cardiac tissue model," Phys. Rev. E61, 727­732. Schamorth, L. [1980] The Disorders of the Cardiac Rhythm (Blackwell, Oxford). Signorini, M. G., Cerutti, S. & di Bernardo, D. D. [1998] "Simulation of heartbeat dynamics: A nonlinear model," Int. J. Bifurcation and Chaos 8, 1725­1731.

van der Pol, B. & van der Mark, J. [1928] "The heartbeat considered as a relaxation oscillation and an electrical model of the heart," Phil. Mag. 6, 763­775. Weidmann, S. [1951] "Effects of current flow on the membrane potential of cardiac muscle," J. Physiol. (London) 115, 227­236. West, B. J., Goldberger, A. L., Rovner, J. & Bhargava, V. [1985] "Nonlinear dynamics of the heartbeat. I. The AV junction: Passive conduit or active oscillator?" Physica D17, 198­206. Winfree, A. T. [1987] When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias (Princeton Univ. Press, Princeton). Zipes, D. P. & Jalife, J. [1995] Cardiac Electrophysiology -- From Cel l to Bed-Side, 2nd edition (W. B. Saunders, Philadelphia).