Nonlinear dynamic behavior of cardiac oscillators

International Elsevier

CARD10

Journal of Cardiology,

211

26 (1990) 211-216

01006

Nonlinear dynamic behavior of cardiac oscillators Qingyie

Tong

Zhejiang (Received

Tong Q, Liu Z. Nonlinear

dynamic

22 August

behavior

and Zhicheng

University. Hangzhou, 1988; revision

of cardiac

accepted

Liu

China 20 September

oscillators.

By studying a mathematical model of the effects of brief current cells, we have simulated the phase transition curves of the oscillators.

1989

Int J Cardiol

1990:26:211-216

impulses on the oscillation of cardiac This has permitted discussions of the

interactions of two coupled sinus oscillators and the periodic stimulation of one ventricular results have been compared with findings from the normal electrocardiogram and applied mechanisms of some arrhythmias. Key words:

Nonlinear

dynamics;

Cardiac

oscillator;

The general classification of arrhythmias assumes that most disturbances of rhythm result from one of two primary abnormalities in the activities

abnormality abnormality

of the heart

Arrhythmias

tion and chaos. Due to some physiological reasons, such as the “noise” intrinsic to the experi-

Introduction

electrial

oscillator. The to explain the

[l]. The first is an

in the initiation and the second an in the propagation of the cardiac

impulse. Using the theory of ectopic pacemakers, investigators have explained qualitatively many

mental

preparations

and

there are discrepancies

environmental

between

“noise”,

the dynamics

pre-

dicted theoretically and those observed experimentally. We know that phase locking, bifurcation and chaos are the intrinsic

properties

of a nonlin-

ear system, and may be expected in a mathematical model. We think it more convenient and precise.

therefore,

to use a nonlinear

mathematical

arrhythmias in clinical practice. But, as a nonlinear oscillator, their complicated dynamic behavior

model of cardiac cells to formulate these behaviors. The model chosen here was formulated by

has rarely been analysed. Glass and his colleagues [2], however, have studied the effects of single and periodic stimulation of spontaneously beating ag-

Beeler and Reuter [3]. It is an equation of the Hodgkin-Huxley type. It includes a few components of ionic currents, but most of the parameters were determined by voltage clamp experiments. A

gregates of cells taken from the ventricle of the embryonic chick heart, observing complicated dynamic

processes,

including

phase locking,

bifurca-

brief introduction dVm/dT=

is as follows:

- (l/Cm)(Zna

+ 1s + Ikl

+ Zk2

+ Ix1 - lex) Correspondence to: Dr. Qingyie Tong, Scientific Instrument Department, Zhejiang University, Hangzhou, China. This work is supported by the grant of Natural Science Foundation. 0167-5273/90/$03.50

0 1990 Elsevier

Science Publishers

dY(i)/dT= i=l

B.V. (Biomedical

...6 Division)

- (P(i)

+ Q(i))

(1) x y(i)

+ P(i)

(2)

212

where V/m = membrane potential (mV), T = time (msec), Cm = membrane capitance (pF/cm’), Ina = excitatory inward sodium currents (PA/ cm2 ), Is = slow inward calcium currents (PA/ cm2 ). I kl = time-independent outward potassium currents (pA/cm2), Zk2 = pacemaker potassium currents ( pA/cm2), 1x1 = background potassium currents (PA/cm’). lex = external stimulating currents (PA/cm’), Y(i) = conductance parameters of those ionic currents, and P(i), Q(i) = rate constants (l/msec). By analysing this model on a computer, we have explored the dynamic mechanisms of two kinds of arrhythmias.

be zero. The phase of the point x(t) is then defined to be t/To (mod 1). Thus, a phase (9 < = < 1) can be assigned to every point on the cycle of the oscillator. We then considered the effect of delivering a brief stimulus when the oscillator is at some phase (old). In general, after the perturbation. the oscillator will lie on a different phase (new), so we get

B=‘d+)

(3)

where + is the old phase, and 8 is the new one. The function g depends on the strength of the stimulus, which is called a phase transition function. If the perturbed cycle of the oscillator is T. we can obtain the expression of g as:

Methods Assuming the period of an oscillator of x(f) to be To, we set the phase of the initial point x(0) to

f9=g(+)

0

=

-(TO-

T)/T0

\-r-----l

(4)

I

I

400

Time(

msec)

‘\.\-

200

mv

Vm

Fig. 1. The results of numerically solving the Hodgkin-Huxley type of equation for cardiac models. A. Spontaneous rhythmic activities of a cell of the sinus node with a period of 916 msec. B. The action potential of a ventricular cell when being stimulated (arrow) at 6 PA/cm’. 10 msec. C. The oscillatory behavior of a ventricular cell with a bias current of 2.4 PA/cm’. Its period is 1220 msec. No stimulus is applied

213

It is convenient to analyse the effect of periodic stimulation of an oscillator using the phase transition function. Suppose the interval between successive stimuli is T, then by calling (i) the phase of the oscillator immediately before the ith stimulus, we obtain

0.2

0.3

0.4

Starting from an initial phase (0) we can utilize our fifth equation to generate a sequence (0) (l)=f((O)),...(N)=f((N1)) and if G(N) +(i)

=9(O)

0.5

0.6

< > +(O)l < i < N 0.7

then the +(O), +(l), . . . +(N - 1) is a cycle of period N. If there are m stimulations in one cycle, we say the cycle of period N corresponds to M: N phase locking. Using equations (1) and (2) with different parameters, we obtained models of the cells of the ventricle and of the sinus node. The results of numerically solving the two kinds of models are shown in Fig. 1. The cell of the sinus node has an autonomous rhythm of 916 msec. The normal ventricular cell does not have rhythmic activity but, under some circumstances, does become an intrinsically rhythmical cell. Fig. 1C shows the result of numerically solving the model of a ventricular cell with a small bias current (2.4 pA/cm2). Its period is 1220 msec. In our work, we obtained first the phase transition curves of the two kinds of oscillators at a brief current stimulus. Secondly, we analysed the coupling effect of two sinus oscillators and the effect of periodic stimulation of an oscillating ventricular cell.

Fig. 2. The action potential of a cell of the sinus node with a stimulus applied at different phases. The stimulus intensity is 2 PA/cm’. 6 msec in Fig. A and 10 PA/cm’. 6 msec in Fig. B. Note that the perturbed cycles are longer or shorter than the original spontaneous cycle dependent on the phase where the stimulus is applied and the intensity of the stimulus.

0.6

0.0

0.2

0.4

0.6

0.8

:O Q,

A

Results When a stimulus is applied at different phases of a spontaneous oscillator, the behavior of the oscillator will be different. In some cases, the effect is to advance the next beat. In other cases, the effect is to delay the next beat. Fig. 2 illustrates the effects of applying two kinds of current at different phases to an oscillating cell of the

B

UJ

c

CD

Fig. 3. Phase transition curves of three cells of the sinus node of different periods (916 msec, 704 msec, 666 msec) with the same stimulus of 2 PA/cm’, 6 msec. Note that the phase transition curves are discontinuous for the reasons discussed in the text.

214 Time

672

672

672

672

640

672

672

672

672

640

I

I

886

918

818

706

886

918

818

706

A

I

I

712

712

712

712

712

712

712

712

721

---------

712

B

Fig. 4. The long-term behavior of two coupled sinus oscillators. Each vertical line represents the occurrence of a marker event. A. The effect of oscillator of period 916 msec coupling that of period 666 msec. They do not act simultaneously but with phase locking of 5 : 4. B. The effect of oscillator of period 916 msec coupling that of period 704 msec. They act simultaneously with a common period of 712 msec.

sinus node. We know the period of the sinus oscillator to be To (Fig. 1). So. by measuring the perturbed cycle T, we can obtain the phase transition curves of the oscillator at these stimuli using equation (3). Fig. 3 shows the phase transition curves of three sinus oscillators with different period cycles of 916 msec, 704 msec, and 666 msec. The stimulus is 2 PA/cm* and has a duration of 6 msec. We can now study the dynamic process of two coupled sinus cells. We regard the depolarization of a cell as a marker event. A marker of one oscillator has an effect on the activity of the other. Suppose that all of the effects between two oscillators can be expressed by a brief current impulse of 2 PA/cm*, 6 msec (of course, the strength of the current impulse is arbitrary as long as it is at the same scale and gives the equation a convergent solution). Using the phase transition curves of the two oscillators with this stimulus, we obtained the long-term behaviors of the two coupled oscillators. The results are shown in Fig. 4. In Fig. 4A, the oscillator with a period of 916 msec couples with that having a period of 666 msec. Because the

difference of the periods between the two oscillators is considerable, their markers do not occur simultaneously. The intervals of the two markers of each oscillator are different, but there is a stable phase locking of 5 : 4. Thus, if the initial conditions of the two oscillators are set at phase zero, the fifth marker of the first oscillator and the fourth marker of the second one will occur at the same time. Fig. 4B shows the behavior of the cell oscillating with a period of 916 msec which couples with the cell oscillating with a period of 704 msec. The difference between the periods is so great that they will emit impulses simultaneously. producing 1: 1 phase locking. The new common period is between the original periods of the two oscillators. Normally, all cells of the sinus node function under the same circumstances and their periods should not be markedly different. The effect of co-operation between then is that the sinus node emits impulses at a single frequency. If, in con-

Fig. 5. with a intensity in Fig. 2, and change

The action potential of an oscillatory ventricular cell stimulus applied at different phases. The stimulus is 2 pA/cm’. 6 msec in Fig. A and 6 pA/cm2, 6 msec B. The behaviors are similar to those illustrated in Fig. increasing the intensity of the stimulus makes the occur earlier and become more abrupt. going from prolonging to shortening the perturbed cycle.

trast, there are abnormal cells, which have distinct and different rhythmic activities from others, then various arrhythmias of the sinus node may occur. As shown in Fig. lC, a ventricular cell will have an autonomous rhythm once it is stimulated with a bias current. The behaviors of such an oscillator when stimulated at two different phases are illustrated in Fig. 5. Note that there are abrupt changes of the perturbed intervals in both circumstances. Before further discussion, we must make an assumption that an incomplete depolarization will not be regarded as a marker of the oscillator since it cannot lead to excitement or systole of the heart. In particular, we think that an incomplete depolarization with a peak below 0 mV belongs to this class. This is why the intervals of the perturbed cycles change abruptly, and in correspondence with the law of “all or nothing”. And, for the same reason, the phase transition curves of ventricular oscillators (as well as sinus oscillators) must be discontinuous, as shown in Figs. 3 and 6. We also find that, as the intensity of stimulus increases, the phase transition curves of the oscillators will change from being monotonic to nonmonotonic. (We say that the function g is monotonic if 0 changes greater when increased.) Now, we can observe the dynamic properties of the ventricular oscillator (its period is 1220 msec) by giving a little stimulus to it with the period of

1 .o

8 0.0 0.6

0.6

0.4

0.4

0.2

0.2

o.o0.0 A

0.2

0.0 0.4

0.6

0.8

1.0 0

300

600

900 Time

1200 (msec)

Fig. 7. Different phase locking regions when stimulating the ventricular oscillator with different periods T. There are four major stable phase locking regions (3: 1. 2 : 1, 3 : 2, 1: 1). The complex dynamic behaviors. including unstable phase locking and chaos. could be observed in the rest zones.

stimulation T from 300 msec to 1220 msec. For some values of T, after a few oscillations, every stimulus will fall at some fixed phases of the oscillator, producing phase locking. For other values of T, the stimulus falls at phases which are either not fixed or unstable (phase locking will change concomitant with change in the initial phase). There exist four major stable regions of phase locking regions (3 : 1, 2: 1, 3 : 2, 1 : 1). The rest zone has complex dynamics including unstable phase locking and chaos as shown in Fig. 7. The results obtained above are similar to the aggregate experiments of Glass et al. [2]

Discussion

:\ 0.0 0

0.2

0.4

0.6

0.0

1.0 0

Fig. 6. The phase transition curves of a ventricular oscillator with two different stimuli of intensities 2 PA/cm’. 6 msec and 10 PA/cm’, 6 msec, respectively. Note that the phase transition curves are discontinuous because an incomplete depolarization cannot emit an impulse capable of leading to one of excitement and systole of the heart. This is in correspondence with the law of “all or nothing”. The phase transition curves are monotonic in Fig. A and nonmonotonic in B because of the increasing intensity of the stimulus.

The study of the nonlinear dynamic behavior of cardiac cells is helpful for understanding the mechanisms of cardiac electrical activities. But solutions of a complete and complex nonliner mathematic model based on ionic currents, such as equations (1) and (2) are not available, because it takes too much time for a computer to calculate the long-term behavior of the interaction between cardiac cells. Furthermore, some simple models of coupling oscillators give only a qualitative conclusion because there is no experimental basis or appropriate bionomic parameters to support the

216

Fig. 8. Electrocardiograms illustrating pathological cardiac rhythms. A. Two kinds of pacemakers within the sinus node compete to control the activities of the heart and form one of the abnormalities of sinus rhythm. B. Many pacemakers within the sinus node compete to control the activities of the heart. The main pacemaker cells shift within the sinus node. C. 2 : 1 ventricular premature beat. Every two normal impulses are followed by a ventricular ectopic impulse. D. 3 : 2 ventricular premature beat. Two ventricular ectopic impulses are initiated for every three normal impulses.

theories. In our work, we have found it effective and simple to analyse the nonlinear properties of oscillators using their curves of phase transition. We can compare our model with four examples of abnormal electrocardiograms (Fig. 8). It is easy to explain their mechanisms using our results. In

the sinus node, there are many cells emitting impulses of different periods. When the differences among them are not too great, all cells of the sinus node will act simultaneously, and the sinus node will produce a regular impulse. Presence of abnormal cells within the sinus node, in contrast, will lead to irregular activity. This is why abnormalities of rhythm occur within the sinus node. In ventricular myocardium, one ectopic pacemaker may produce a ventricular premature beat, which often is of fixed rhythm (phase locking). We can theorize that two or more ectopic pacemakers will lead to ventricular tachycardia or even fibrillation. Using the same method, Chay and Lee [5] have examined the impulse responses of automaticity in Purkinje fibers and analysed their dynamic characteristics. In one word, each autonomously rhythmic cell of the heart plays its role in controlling cardiac systole. The traditional view of the dominant pacemaker of highest rate of discharge controlling all electrical activities of the heart should be supplanted by analysis of the heart in terms of a dynamic oscillator.

References Hoffman BF. Rosen MR. Cellular mechanisms for cardiac arrhythmias. Circ Res 1981:49: l-15. Glass L, Guevara MR, Scrier A. Bifurcation and chaos in a periodically stimulated cardiac oscillator. Physica 1983: 7D: 899101. Beeler GW. Reuter H. Reconstruction of the action potential of ventricular myocardial fibres, J Physiol 1977;268:177-210. Mandell AJ. Dynamical complexity and pathological order in the cardiac monitoring problem. Physica 1987:27D:235242. Chay TR, Lee YS. Impulse responses of automaticity in the Purkinje fiber. J Biophys 1984;45:841-849.