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Nonlinear dynamics of the heartbeat
Physica 17D (1985) 198-206 North-Holland, Amsterdam
NONLINEAR
DYNAMICS
I. THE AV JUNCTION:
Bruce J. WEST L.uJollu Institute, Center Culifornia,
OF THE HEARTBEAT
PASSIVE CONDUIT
for Siudies
of Nonlinear
OR ACTIVE OSCILLATOR?
Dynumics,
(Affiliured
with the Unioersi
Sun Diego),
La Jollu,
USA
Ary L. GOLDBERGER Depurtment
of Medicine,
Sun Diego
Veterans Administrution
Medicul
Center,
Unioersity
of Culiforniu,
Sun Diego,
Culifornru,
USA
Galina ROVNER L.uJollu Institute, Center Culifomiu,
for Studies of Nonlineur
D.ynumics, (AfJiliuted
with the Unirlersity
oj Culiforniu,
Sun Diego),
Lu Jollu,
USA
and Valmik BHARGAVAT Depurtment
of Medicine,
Sun Diego
Veieruns
Administrution
Medicul
Center.
University
of Culijorniu,
Sun Diego,
Culiforniu,
USA
Received 30 November Revised 16 April 1985
1984
Under physiologic conditions, the AV junction is traditionally regarded as a passive conduit for the conduction of impulses from the atria to the ventricles. An alternative view, namely that subsidiary pacemakers play an active role in normal electrophysiologic dynamics during sinus rhythm, has been suggested based on nonlinear models of cardiac oscillators. A central problem has been the development of a simple but explicit mathematical model for coupled nonlinear oscillators relevant both to stable and perturbed cardiac dynamics. We use equations describing an analog electrical circuit with an external d.c. voltage source (V,) and two nonlinear oscillators with intrinsic frequencies in the ratio of 3 : 2, comparable to the SA node and AV junction rates. The oscillators are coupled by means of a resistor. 1 : 1 (SA : AV) phase-locking of the oscillators occurs over a critical range of V,. Externally driving the SA oscillator at increasing rates results in 3 : 2 AV Wenckebach periodicity and a 2 : 1 AV block. These findings appear with no assumptions about conduction time or refractoriness. This dynamical model is consistent with the new interpretation that normal sinus rhythm may represent 1 : 1 coupling of two or more active nonlinear oscillators and also accounts for the appearance of an AV block with critical changes in a single parameter such as the pacing rate.
1. Introduction
The cardiac conduction system is a network of self-excitatory pacemakers, with the sino-atria1 (SA) node, having the highest intrinsic frequency. tAddress for correspondence and reprints: Ary L. Goldberger, M.D., Harvard Medical School, Cardiovascular Division, Beth Israel Hospital, 330 Brookline Avenue, Boston, MA 02215, USA.
Subsidiary pacemakers with slower firing rates are located in the atrioventricular (AV) junction and the His-Purkinje system. Suppression of SA nodal activity will unmask the lower frequency AV junctional “escape” rhythm. Under physiologic conditions, however, the slower pacemaker sites are traditionally regarded as passive conduits for the conduction of the impulse from the atria to the ventricles [l]. This linear concept of cardiac im-
0167-2789/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B.J. West et al./ Nonlinear cardiac oscillators I
pulse conduction is exemplified in the conventional “ladder diagrams” used in clinical electrocardiography to map the spread of potentials from the sinus node through the junction, where there is a delay before the impulse passes on to the ventricular conduction system. This view assumes that with normal conduction the AV junction is passive and does not actively generate impulses or influence the SA node. An alternative viewpoint, namely that the slower pacemakers play a more active role in normal electrophysiologic dynamics, has been suggested based on nonlinear models of cardiac oscillators [2-51. Models of this sort have been motivated in part by the clinical appearance of arrhythmias such as isorhythmic AV dissociation, characterized by the apparent desynchronization of two active oscillators, the SA node and the AV junction. Van der Pol and van der Mark [2] proposed the first analog nonlinear model of the cardiac conduction system which consisted of relaxation oscillators having the qualitative features of the following equation E-(t)-[E”-
V2(t)]~(t)+S2g2V(f)=0,
(1)
where L’(t) is the voltage, &, is the natural frequency, E is an adjustable parameter and where the dots denote differentiation with respect to time. However, the authors [2] did not attempt to solve, or indeed to construct the system of equations associated with the analog circuit. Instead each oscillator was represented by a battery of voltage V, in series with a resistor of ohmic value R, both in series with a parallel combination of a capacitor C and a neon lamp. The relaxation time (T,,) of the oscillator is T,, = RC = 1 s for R = 1 MS2 and C = 1 pf, so that the neon lamp is set to give a short flash once each second. A sequenc,e of three such resistively coupled electrical circuits (corresponding to the sinus node, atria and ventricles) provided an electrical analog of the heart. The recorded pulses, after proper filtering and reshaping, resembled the atria1 (P wave) and ventricular (QRS complex) pulses of the electrocardiogram.
199
Depending on the coupling, this model generated voltage-time traces consistent with sinus rhythm (1 : 1 P wave-QRS relationship) or a variety of patterns mimicking AV heart block. Subsequent models, based on the van der Pol relaxation oscillator, yielded similar results [6-81. However, these simulations did not account for possible bidirectional interaction of pacemaking centers that would allow the SA node to be influenced by, as well as to influence, the AV junction under physiological conditions. Such bidirectional interaction may occur between spatially separated nonlinear oscillators which are resistively coupled. Although it is clear that there are action potentials propagating between these two sites, this style of microscopic description has not proved useful in the present context. Instead, one can construct an equivalent circuit model such as a relaxation oscillator to capture the essential features of this kind of system without necessarily simulating its physical processes. This is the strategy adopted here. This equivalent circuit model does have an implicit microscopic description, the discussion of which we postpone until later. A central problem, therefore, has been the development of a simple but explicit mathematical model (see e.g. Grasman, ref. 9) consisting of only two coupled nonlinear oscillators which might be relevant for describing stable as well as perturbed dynamics of the heartbeat.
2. Methods In the present study, we use the equations describing an analog electrical circuit to represent the dynamic interaction between the faster (SA) and slower (AV) cardiac pacemakers. A computer program was written to simulate this circuit (fig. 1A) consisting of two anharmonic (nonlinear) oscillators. In this model and its predecessors, the activity of all the cells in a given region of the heart is coarse-grained into a single relaxation oscillator. This coarse-graining is intended to
200
B.J. West et al./ Nonlinear cardiac oscillators I
CURRENT
I&)
Fig. 1. A) Analog circuit described by eqs. (2) and (3) with tunnel diodes. B) voltage-current responses curves across diodes 1 and 2 when R in fig. A = 0 (uncoupled system); C) response curves for coupled system (R # 0). The circuit was externally driven (fig. 3) by applying an external voltage pulse train of variable frequency between R, and V,. VL, VCi,I,, IG are defined in text.
simulate the net effect of multicellular depolarization at the pacemaker sites [3, lo]. Each oscillator is composed of an inductor L,, a resistor Rj, and a tunnel diode Dj (j = 1,2) in series. Experiments with a similar circuit have been reported [ll, 121. The experimental results, however, were described using a simple mapping algorithm rather than directly integrating the nonlinear equations of motion as in the present study. The two oscillators are coupled together by the common resistor R and energy is supplied by the battery with voltage V,. I( t the total current supplied the circuit the battery and divided into and t) at the juncture the two branches the circuit. anharmonicity in this system arises from the nonlinear response of each diode to the applied voltage. In a linear device, the response is directly proportional to the input. In figs. 1B and lC, the preselected response of the diode to the applied voltage is shown in the form of a current-voltage curve where the operating characteristics of the diode are depicted by graphing the voltage drop across the diode for each value of the current flowing through the diode. The physiologic similarities of the SA and AV pacemakers are incorporated into this model by
using the same qualitative current response curves for each diode. The intrinsic difference in rate between the two pacemakers (AV/SA = 2/3) is taken into account by the selected values of the “free-running” oscillator periods L/R,. Experiments in dogs have demonstrated that the intrinsic firing rate of the AV junction is about 2/3 that of the SA node [13]. The non-instantaneous transitions between the upper and lower branches of the hysteresis loops in figs. 1B and 1C are modeled by using a hyperbolic tangent function. These hysteresis loops were selected to yield voltage-time traces similar to those reported with the slow channel (calcium dependent) action potential recorded at the SA node and AV junction sites. When the coupling resistor R is zero, the two oscillators will be uncoupled (‘free-running’) and an external voltage V, will initiate current flow along the two closed current-voltage paths depicted in fig. 1B. Each oscillator will complete one cycle in the time interval L,/R,, during which the voltage drop across the first diode will switch from its greater value Vo to its-lesser value V, and back again. The time interval over which the switching occurs is very much shorter than the overall period and gives the appearance of a discontinuous change
B.J. West et al./ Nonlinear cardiac osciliaiors I
201
B.
L
1.
0
1.5 E 5is
1.4-
Is
1.31.2 -
l
b g
I.I-
z
1.0 -
0
I0.0
I
0.1
I1 0.15
I 0.20
I
I 0.25
EXTERNAL
I
I. 0.30
I1 0.35
VOLTAGE
In 0.40
I 0.45
V,
c.
0.6
TIME Fig. 2. A) Voltage pulses as function of time (dimensionless units) for SA (solid line) and AV (dotted line) oscillators with parameter values R = 3.251, V0 =0.32 V, R, = 1.39, L, = 2.112 pH, R, = 1.4 51, L2 = 3.732 PH. The inset indicates current through diode 1 as function of current through diode 2; B) ratio of frequences of SA and AV oscillators as function of external voltage ( VO). Open dots (0) indicate coupled system (R = 3.2 51) Closed dots (0) indicate uncoupled system (R = 0), with all other parameters as in fig. 2A. The solid curve indicates the general dependence on the winding number of VO. However, the actual relation is not continuous (as one can see on close inspection) because the rotation number is constant for a range of voltages when the frequencies are in the ratio of integers. The variability in the uncoupled winding number is due to the variability in the applied voltage in making the transition from I,_ -) To or I, --, IL in the hysteresis loop (see appendix); C) 1 : 1 phase locking for V. = 0.182 V.
in voltage, i.e., the voltage time series appears to be a sequence of pulses (fig. 2). The current through the first diode has the form of rising and falling exponentials, switching direction after transitions to the upper or lower branches of the current-voltage curve. This cycle of conduction in the current-voltage phase space of the diode is an example of a hysteresis loop. The hysteresis loops of the diodes and the resultant voltage-time pulses were designed to simulate the morphology of action potential recordings at the SA node and AV junction sites. Finally, the resistance and induc-
tance determine the intrinsic frequency of each pacemaker. The time rates of change in the current through the two diode branches of the circuit are determined by Kirchhofi’s laws: ~J&)
+ (4
+zU,(r) M*(t)
+ R)&(t)
+ [v,(t)
- v,] = 0,
(2)
+ (R, + W*(t)
+Rz,(t)
+ [v*(t) - V-J = 0.
(3)
202
B.J. West et al./ Nonlineur cardiac oscilhtors
These two equations constitute a coupled feedback system through the I2 dependence of the i, equation and the I, dependence of the i, equation. Thus the two oscillators are linearly coupled with a resistor. However, each oscillator is driven by the difference between the voltage source and that dropped across the diode, introducing the anharmonic effect of the current-voltage response curve. Because the tunnel diodes are hysteretic (nonlinear) devices, as the current in one of them increases, the voltage across it remains nearly the same until the current reaches its value Io, at which time an external voltage source is activated to drive the system to the level V, (fig. 1B) (see appendix). Once on the upper part of the hysteresis loop, the current diminishes, again with little or no change in the voltage until the current reaches its value I,, at which point an external voltage of opposite polarity is activated to drive the system to the lower branch of the hysteresis curve. The cycle then repeats itself. This cycling of the coupled system is depicted in fig. 1C which shows that the sharply angled regions of the uncoupled hysteresis loops have been smoothed out by the means of the coupling. In the related model of Gollub et al. [ll] the transition between the upper and lower branches of the current-voltage loop is instantaneous. To mimic the smooth change from one branch of the curve to the other observed in physiological oscillators, we replaced this discontinuity with a hyperbolic tangent function along with a voltage which linearly increases in magnitude with time at the transition points IG and I, (see appendix). The set of coupled nonlinear rate equations (2) and (3) were numerically integrated on a VAX 11/750 computer using a modified divided difference form of the standard Adams Pete formulas and local extrapolation. The values of the circuit parameters were chosen to be R = 3.2 9, R, = 1.3 9, R,= 1.4 s2, L, = 2.772 /.LH, L,= 3.732 PH and V, varied in the range from 0.16 to 0.45 V. In the appendix we specify the functional form of the nonlinear voltage-current relation depicted in fig. 2 and used in the numerical integration. This
I
particular form was chosen for convenience, but does have the appropriate qualitative features of SA and AV current-voltage loops. Finally, to simulate the effects of driving the right atrium at incrementally higher rates with an external pacemaker, an external voltage of variable frequency was applied to the SA node oscillator branch of the electric circuit.
3. Results Each diode (Dj) cycles through its hysteresis loop at a period determined by the decay time of the oscillator, i.e., L,/( R + Rj). The periodic pulse form of the isolated nonlinear oscillator is now changed due to its interaction with the other nonlinear oscillator. The phase space for the coupled system has the four coordinate axes (I,, I,, V,, V,) corresponding to the four dynamic variables of the composite system. In figs. 1B and lC, we have superimposed the two phase spaces rather than trying to draw the actual four dimensional one. The dynamics of the coupled system can be depicted by the orbits in the reduced phase space (II, 12) for a certain set of system parameter values. A periodic solution to dynamic equations (2) and (3) is a closed curve in the reduced phase space as shown in fig. 2A (inset). Here, for two periods in one oscillator we have three in the other so that the coupled frequencies are in the ratio of three to two. A closed orbit with 2m turns along one direction and of 2n turns in the orthogonal direction indicates a phase locking between the two diodes such that one diode undergoes m cycles and the other n cycles in a constant time interval T for the coupled system. Fig. 2A also shows the time trace of the voltages across diodes 1 and 2 for this case. We observed the 3 : 2 ratio of oscillator frequencies over a broad range of values of V,. However, in the region V, I 0.225 V the frequency ratio of the two oscillators becomes phase locked (1 : 1 coupling) (fig. 2B) at a frequency slower than the intrinsic frequency of the SA node but faster than that of the AV junction. Fig. 2C shows the
B.J. West et al. / Nonlineur curdiuc oscillurors I
203
Finally, for an interval of 1.5 units between consecutive external pulses, every other pulse fails to trigger the AV junction (fig. 3C).
4. Discussion
C.2:l AVK&Xi
0.6 ,_
I
TIME
Fig. 3. A) With parameter values same as in fig. 2C; 1 : 1 phase locking persists when the SA node is driven by an external voltage pulse train with pulse width 0.5 dimensionless time units and period 4.0. B) Driver period reduced to 2.0 with emergence of 3 : 2 Wenckebach periodicity. C) Driver period reduced to 1.5, resulting in a 2 : 1 AV block. Closed brackets denote SA pulse associated with AV response. Open brackets denote SA pulse without AV response (“nonconducted beat”).
outputs of both oscillators in the coupled system with parameter values such that the uncoupled frequencies are in the ratio of three to two. In the coupled system, the SA and AV oscillators are clearly 1: 1 phase-locked due to their dynamic interaction. The effects of driving the two oscillators at incrementally higher rates by an external pacemaker are illustrated in fig. 3. At first, when the driving period is 4.0 units between consecutive pulses and the pulses are 0.5 units wide, the two oscillators remain 1: 1 locked (fig. 3A). As the interval between consecutive pulses is reduced to 2.0 units, the output of the two oscillators is no longer one to one, but three to two. This means that the AV oscillator is failing to respond to one out of three impulses from the SA oscillator (fig. 3B). Decreasing the interval to 1.8 units results in two out of every five pulses from the SA oscillator failing to elicit a response from the AV oscillator.
This new nonlinear model of supraventricular conduction was motivated by several objectives. First, using a minimum number of elements we sought to construct a circuit whose dynamics would be relevant both to physiologic and pathologic cardiac function. In contrast to more complex models proposed in the past [2, 81, the present circuit consists of just two oscillators and a total of only eight components, each with a physiologic analog. The voltage source (V,) corresponds to the metabolic energy supply of the conduction system and oscillators. The resistance (R) provides the coupling between the oscillators. When R = 0,the oscillators revert to their free-running, uncoupled state, analogous to complete AV block. Each pacemaker unit itself consists of three elements: diode (O]), inductor (Li) and resistor ( Ri).We chose values such that the free-running rates of the SA/AV oscillators would be 3/2, close to their physiologic ratio. The present model differs in other important respects from earlier nonlinear representations of the cardiac conduction system. For example, the three-oscillator model of van der Pol and van der Mark [2] which serves as the foundation for cardiac nonlinear dynamics did not specify the relevant set of coupled equations. Similarly, subsequent analog simulations [7, 8, 141 lacked explicit mathematical descriptions. The coupled set of equations provided by Katholi et al. [3] afforded an important advance in this regard. In their model, the SA node and AV junction were represented as van der Pol oscillators with free-running frequencies in the ratio of 3 : 2. An arbitrary time delay was interposed between these pacemakers. This model supported the hypothesis that sinus rhythm corresponds to 1: 1 phase-locking of two coupled relaxation oscillators
204
B.J. West ei al./ Nonlinear cardiac oscillators I
with different intrinsic firing rates. Perturbations leading to AV Wenckebach, however, were not studied in the model. Subsequently, Guevara and Glass [4] observed AV Wenckebach-like periodicity in a phase transition curve model of the response of a single oscillator driven by an external stimulus. Their mathematical model, however, did not address the physiologic problem of having two spontaneously firing pacemakers which are bidirectionally-coupled. Such bidirectional coupling implies that the action potentials propagating between pacemaker sites influence both sites. In the equivalent circuit model, the influence is manifest by means of the resistive coupling. How this influence is physically mediated would require a description not only of the action potentials, but also of the physical-chemical processes generating them. The microscopic mechanism for the bidirectional interaction is not known, but is implicit in the equivalent circuit model presented here. This simple dual-oscillator circuit is based on an explicit set of coupled equations and has several features of potential relevance to cardiac electrical dynamics. First, the model, like that of Katholi et al. [3], demonstrates that two nonlinear oscillators with free-running frequencies of approximately 3 : 2 may become 1: 1 phase locked (coupled) with the intrinsically faster (“SA”) pacemaker leading the slower (“AV”) pacemaker. In addition, the present model demonstrates that abrupt changes (bifurcations) in the phase relationship between the two oscillators occur when the intrinsically faster pacemaker is driven at progressively higher rates. In the present model, over a critical range of frequencies, a distinctive type of periodicity is observed such that the interval between the “SA” and “AV” oscillators becomes progressively longer until one “SA” pulse is not followed by an “AV” pulse (fig. 3B). This cycle then repeats itself, analogous to AV Wenckebach periodicity [14] which is characterized by progressive prolongation of the PR interval until a P-wave is not followed by a QRS complex. These AV Wenckebach cycles, which may be seen under a variety of pathological conditions, are also a feature of normal electro-
physiologic dynamics and can be induced by driving the atria with an electronic pacemaker [15]. In addition, this model demonstrates the phenomenon of 2 : 1 block when the system is driven past a critical point. In this new parameter zone, every other “SA” pulse is apparently uncoupled from an “AV” pulse, resulting in the type of period-doubling (subharmonic) bifurcation reported in a variety of perturbed nonlinear systems [17-221. The findings of both phase-locking and bifurcation-like behavior are particularly noteworthy in this dual oscillator model because they emerge without any special assumptions regarding conduction time between oscillators, refractoriness of either oscillator to repetitive stimulation or the differential effect of one oscillator on the other. The observed dynamics support the contention that the AV junction may be more than a passive conduit for impulses generated by the sinus node, as also suggested by Guevara and Glass [4]. The present model is consistent with the alternative interpretation that normal sinus rhythm corresponds to 1: 1 phase-locking (entrainment) of two or more active oscillators, and does not require complete suppression of the slower pacemaker by the faster one, as do the passive conduit models. It should be emphasized, however, that when two active pacemakers become 1: 1 phase-locked, the intrinsically slower oscillator may be mistaken as a passive element because of its temporal relation to the intrinsically faster oscillator. Furthermore, the model is of interest because it demonstrates marked qualitative changes in system dynan&s, characteristic of AV Wenckebach and 2 : 1 AV block, occurring when a single parameter (driving frequency) is varied over some critical range of values. The present model may also be useful in making predictions about cardiac electrical dynamics. For example, a prediction based on this representation is that the interval between the pulse peaks of the two major cardiac oscillators (analogous to the PR interval of the electrocardiogram) in their 1: 1 phase-locked state will be affected not only by the intrinsic firing rate of the slower (AV) pacemaker
B.J. West et al. / Nonlinear cardiac oscillators I
but by that of the faster (SA) oscillator as well. ‘According to this nonlinear model, selective. slowing of the period of the SA node leads to a decrease in the SA-AV coupling interval. The traditional concept of passive AV conduction predicts no change in this interval if the SA firing rate alone is perturbed. In addition, this new model and subsequent modifications may be helpful in studying a variety of other cardiac phenomena with possible nonlinear mechanisms, such as “electrotonic” coupling in modulated parasystole [23]. The effect on SA-AV coupling relationships induced by changes in the qualitative features of the oscillator hysteresis loops can also be systematically studied. These perturbations may be used to model electrolyte or drug effects which selectively alter one or more phases of the action potential and hence the slope of the corresponding limb of the hysteresis loop. Finally, a multi-oscillator system could be of potential importance in modeling the nonlinear traveling waves which may underlie arrhythmias such as flutter and fibrillation [2, 241.
To numerically integrate the set of coupled rate equations (2) and (3) requires that a functional form for the voltage-current curve in fig. 1 be specified. We denote the voltage current pair by (V,, Z,) where n = 1,2. On the upper (up) branch of the current-voltage curve, we have dZ,,/dt < 0 and for 3 X lob3 I Z, I 22 X 10e3, the voltage is then on the relatively flat portion of the hysteresis loop: (A.1)
When Z, 5 3 x 10v3 = ZLc, the critical value of current at which the transition to the lower branch is initiated. (A.l) is replaced with V, = G,““) + Z-Z,‘).
G,(“*), EZ,?, ZZL*)are defined as fol-
Gi”*)= Vt,+k,AV{l+tanhk,(Z,-I,,)} +Ak,(Z,-Z,), ZZ;“=k,AV{l-
(A-3)
tanhk,(Zo-I,,)-
0” - ZLC)
xK3-&_,) ’ Hi*)=
-k,AV(l+
(A-2)
VL} (A.41
tanhk,(Z,-Z,,)-(YZL}
K-ZLC)
x(z,-z,c)~
(A.5)
Note that a linear voltage (cwZr)is introduced after the transition is initiated to insure it is completed. On the lower (low) branch of the hysteresis loop we have dZ,/dl>O and for 10-3~Z,~2~10-3 the voltage is again in the relatively flat region: I/ = G(i‘+ + R(i) n . n n
(A.61
When Z, 2 21 x low3 = I,, the transition to the upper branch is initiated and (A.6) is replaced with V = G(‘“‘“)+ R(2) n n n 3
Appendix A
V, = G,(U*)+ H,“‘.
The function lows:
205
(~4.7)
where G;“‘“‘= V,+k,AY{l+tanh[k,,(Z,-I,,)]} +Ak,(Z,,-Z,),
(A.81
R(,‘)=k,AV{l+a~~-tanh[k,~,(Z,-I,,]}
(4 -I”,)
x&-Z”C) R’,Z)=k,AV{l-
’ V,+tanh[k,,(Z,-I,,)]}
w-Z”d u”-z”c)’
x
64.9)
(A.lO)
We have selected the parameter values k, = 0.4, k, = 0.45Ak, = 434(0.5 - k,), I, = 10-3, ZG= 24 x 10-3, k,, = 434, (Y= 0.1, v, = 0.05, vo = 0.5, AV= V, - ?‘, = 0.45 and k,, = 868.
206
B.J. West et ul./Nonlineur
References [l] D.M. Vassalle, Circ. Res. 41 (1977) 269. [2] B. van der Pol and J. van der Mark, Phil. Mag. 6 (1928) 763. [3] C.R. Katholi, F. Urthaler, J. Macy Jr. and T.N. James, Comp. Biomed. Res. 10 (1977) 529. [4] M.R. Guevara and L. Glass, J. Math. Biol. 14 (1982) 1. [5] N. Ikeda, Biol. Cybem. 43 (1982) 157. [6] R.P. Grant, Am. J. Med. 20 (1956) 334. [7] D.A. Side& J. Electrocardiol. 9 (1976) 227. (81 D.A. Sideris and SD. Moulopoulos, J. Electrocardiol. 10 (1977) 51. [9] J. Grasman, Bull. Math. Biol. 46 (1984) 407. [lo] A.T. Winfree, J. Theor. Biol. 16 (1967) 15. [ll] J.P. Gollub, T.O. Brunner and B.G. Danly, Science 200 (1978) 48. [12] J.P. Gollub, E.J. Romer and J.G. Socolar, J. Stat. Phys. 23 (1980) 321. [13] F. Urthaler, C. Katholi, J. Macy Jr. and T.N. James, Am.
curdioc oscilkutors I
Heart J. 86 (1973) 189. [14] F.A. Roberge and R.A. Nadeau, Can. J. Physiol. Pharmacol. 47 (1969) 695. [15] M.E. Josephson and SF. Seides, Clinical Cardiac Electrophysiology: Techniques and Interpretations (Lea and Febiger, Philadelphia, 1979), p. 33. (161 R.M. May, Nature 216 (1976) 459. [17] M.J. Feigenbaum, Phys. Lett. 74A (1979) 375. [18] P.S. Linsay, Phys. Rev. Lett. 19 (1981) 1349. [19] M.R. Guevara, L. Glass and A. Shrier, Science 214 (1981) 1350. [20] A.L. Goldberger, R. Shabetai, V. Bhargava, B.J. West and A.J. Mandell, Am. Heart J. 107 (1984) 1297. [21] A.L. Ritzenberg, D.R. Adam and R.J. Cohen, Nature 307 (1984) 159. [22] W.L. Keith and R.H. Rand, J. Math. Biol. 20 (1984) 133. [23] C. Antzelevitch, M.J. Bernstein, H.N. Feldman and G.K. Moe, Circulation 68 (1983) 1101. [24] A.L. Goldberger, V. Bhargava, B.J. West and A.J. Mandell, Chn. Res. 32 (1984) 169A.
NONLINEAR
DYNAMICS
I. THE AV JUNCTION:
Bruce J. WEST L.uJollu Institute, Center Culifornia,
OF THE HEARTBEAT
PASSIVE CONDUIT
for Siudies
of Nonlinear
OR ACTIVE OSCILLATOR?
Dynumics,
(Affiliured
with the Unioersi
Sun Diego),
La Jollu,
USA
Ary L. GOLDBERGER Depurtment
of Medicine,
Sun Diego
Veterans Administrution
Medicul
Center,
Unioersity
of Culiforniu,
Sun Diego,
Culifornru,
USA
Galina ROVNER L.uJollu Institute, Center Culifomiu,
for Studies of Nonlineur
D.ynumics, (AfJiliuted
with the Unirlersity
oj Culiforniu,
Sun Diego),
Lu Jollu,
USA
and Valmik BHARGAVAT Depurtment
of Medicine,
Sun Diego
Veieruns
Administrution
Medicul
Center.
University
of Culijorniu,
Sun Diego,
Culiforniu,
USA
Received 30 November Revised 16 April 1985
1984
Under physiologic conditions, the AV junction is traditionally regarded as a passive conduit for the conduction of impulses from the atria to the ventricles. An alternative view, namely that subsidiary pacemakers play an active role in normal electrophysiologic dynamics during sinus rhythm, has been suggested based on nonlinear models of cardiac oscillators. A central problem has been the development of a simple but explicit mathematical model for coupled nonlinear oscillators relevant both to stable and perturbed cardiac dynamics. We use equations describing an analog electrical circuit with an external d.c. voltage source (V,) and two nonlinear oscillators with intrinsic frequencies in the ratio of 3 : 2, comparable to the SA node and AV junction rates. The oscillators are coupled by means of a resistor. 1 : 1 (SA : AV) phase-locking of the oscillators occurs over a critical range of V,. Externally driving the SA oscillator at increasing rates results in 3 : 2 AV Wenckebach periodicity and a 2 : 1 AV block. These findings appear with no assumptions about conduction time or refractoriness. This dynamical model is consistent with the new interpretation that normal sinus rhythm may represent 1 : 1 coupling of two or more active nonlinear oscillators and also accounts for the appearance of an AV block with critical changes in a single parameter such as the pacing rate.
1. Introduction
The cardiac conduction system is a network of self-excitatory pacemakers, with the sino-atria1 (SA) node, having the highest intrinsic frequency. tAddress for correspondence and reprints: Ary L. Goldberger, M.D., Harvard Medical School, Cardiovascular Division, Beth Israel Hospital, 330 Brookline Avenue, Boston, MA 02215, USA.
Subsidiary pacemakers with slower firing rates are located in the atrioventricular (AV) junction and the His-Purkinje system. Suppression of SA nodal activity will unmask the lower frequency AV junctional “escape” rhythm. Under physiologic conditions, however, the slower pacemaker sites are traditionally regarded as passive conduits for the conduction of the impulse from the atria to the ventricles [l]. This linear concept of cardiac im-
0167-2789/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B.J. West et al./ Nonlinear cardiac oscillators I
pulse conduction is exemplified in the conventional “ladder diagrams” used in clinical electrocardiography to map the spread of potentials from the sinus node through the junction, where there is a delay before the impulse passes on to the ventricular conduction system. This view assumes that with normal conduction the AV junction is passive and does not actively generate impulses or influence the SA node. An alternative viewpoint, namely that the slower pacemakers play a more active role in normal electrophysiologic dynamics, has been suggested based on nonlinear models of cardiac oscillators [2-51. Models of this sort have been motivated in part by the clinical appearance of arrhythmias such as isorhythmic AV dissociation, characterized by the apparent desynchronization of two active oscillators, the SA node and the AV junction. Van der Pol and van der Mark [2] proposed the first analog nonlinear model of the cardiac conduction system which consisted of relaxation oscillators having the qualitative features of the following equation E-(t)-[E”-
V2(t)]~(t)+S2g2V(f)=0,
(1)
where L’(t) is the voltage, &, is the natural frequency, E is an adjustable parameter and where the dots denote differentiation with respect to time. However, the authors [2] did not attempt to solve, or indeed to construct the system of equations associated with the analog circuit. Instead each oscillator was represented by a battery of voltage V, in series with a resistor of ohmic value R, both in series with a parallel combination of a capacitor C and a neon lamp. The relaxation time (T,,) of the oscillator is T,, = RC = 1 s for R = 1 MS2 and C = 1 pf, so that the neon lamp is set to give a short flash once each second. A sequenc,e of three such resistively coupled electrical circuits (corresponding to the sinus node, atria and ventricles) provided an electrical analog of the heart. The recorded pulses, after proper filtering and reshaping, resembled the atria1 (P wave) and ventricular (QRS complex) pulses of the electrocardiogram.
199
Depending on the coupling, this model generated voltage-time traces consistent with sinus rhythm (1 : 1 P wave-QRS relationship) or a variety of patterns mimicking AV heart block. Subsequent models, based on the van der Pol relaxation oscillator, yielded similar results [6-81. However, these simulations did not account for possible bidirectional interaction of pacemaking centers that would allow the SA node to be influenced by, as well as to influence, the AV junction under physiological conditions. Such bidirectional interaction may occur between spatially separated nonlinear oscillators which are resistively coupled. Although it is clear that there are action potentials propagating between these two sites, this style of microscopic description has not proved useful in the present context. Instead, one can construct an equivalent circuit model such as a relaxation oscillator to capture the essential features of this kind of system without necessarily simulating its physical processes. This is the strategy adopted here. This equivalent circuit model does have an implicit microscopic description, the discussion of which we postpone until later. A central problem, therefore, has been the development of a simple but explicit mathematical model (see e.g. Grasman, ref. 9) consisting of only two coupled nonlinear oscillators which might be relevant for describing stable as well as perturbed dynamics of the heartbeat.
2. Methods In the present study, we use the equations describing an analog electrical circuit to represent the dynamic interaction between the faster (SA) and slower (AV) cardiac pacemakers. A computer program was written to simulate this circuit (fig. 1A) consisting of two anharmonic (nonlinear) oscillators. In this model and its predecessors, the activity of all the cells in a given region of the heart is coarse-grained into a single relaxation oscillator. This coarse-graining is intended to
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B.J. West et al./ Nonlinear cardiac oscillators I
CURRENT
I&)
Fig. 1. A) Analog circuit described by eqs. (2) and (3) with tunnel diodes. B) voltage-current responses curves across diodes 1 and 2 when R in fig. A = 0 (uncoupled system); C) response curves for coupled system (R # 0). The circuit was externally driven (fig. 3) by applying an external voltage pulse train of variable frequency between R, and V,. VL, VCi,I,, IG are defined in text.
simulate the net effect of multicellular depolarization at the pacemaker sites [3, lo]. Each oscillator is composed of an inductor L,, a resistor Rj, and a tunnel diode Dj (j = 1,2) in series. Experiments with a similar circuit have been reported [ll, 121. The experimental results, however, were described using a simple mapping algorithm rather than directly integrating the nonlinear equations of motion as in the present study. The two oscillators are coupled together by the common resistor R and energy is supplied by the battery with voltage V,. I( t the total current supplied the circuit the battery and divided into and t) at the juncture the two branches the circuit. anharmonicity in this system arises from the nonlinear response of each diode to the applied voltage. In a linear device, the response is directly proportional to the input. In figs. 1B and lC, the preselected response of the diode to the applied voltage is shown in the form of a current-voltage curve where the operating characteristics of the diode are depicted by graphing the voltage drop across the diode for each value of the current flowing through the diode. The physiologic similarities of the SA and AV pacemakers are incorporated into this model by
using the same qualitative current response curves for each diode. The intrinsic difference in rate between the two pacemakers (AV/SA = 2/3) is taken into account by the selected values of the “free-running” oscillator periods L/R,. Experiments in dogs have demonstrated that the intrinsic firing rate of the AV junction is about 2/3 that of the SA node [13]. The non-instantaneous transitions between the upper and lower branches of the hysteresis loops in figs. 1B and 1C are modeled by using a hyperbolic tangent function. These hysteresis loops were selected to yield voltage-time traces similar to those reported with the slow channel (calcium dependent) action potential recorded at the SA node and AV junction sites. When the coupling resistor R is zero, the two oscillators will be uncoupled (‘free-running’) and an external voltage V, will initiate current flow along the two closed current-voltage paths depicted in fig. 1B. Each oscillator will complete one cycle in the time interval L,/R,, during which the voltage drop across the first diode will switch from its greater value Vo to its-lesser value V, and back again. The time interval over which the switching occurs is very much shorter than the overall period and gives the appearance of a discontinuous change
B.J. West et al./ Nonlinear cardiac osciliaiors I
201
B.
L
1.
0
1.5 E 5is
1.4-
Is
1.31.2 -
l
b g
I.I-
z
1.0 -
0
I0.0
I
0.1
I1 0.15
I 0.20
I
I 0.25
EXTERNAL
I
I. 0.30
I1 0.35
VOLTAGE
In 0.40
I 0.45
V,
c.
0.6
TIME Fig. 2. A) Voltage pulses as function of time (dimensionless units) for SA (solid line) and AV (dotted line) oscillators with parameter values R = 3.251, V0 =0.32 V, R, = 1.39, L, = 2.112 pH, R, = 1.4 51, L2 = 3.732 PH. The inset indicates current through diode 1 as function of current through diode 2; B) ratio of frequences of SA and AV oscillators as function of external voltage ( VO). Open dots (0) indicate coupled system (R = 3.2 51) Closed dots (0) indicate uncoupled system (R = 0), with all other parameters as in fig. 2A. The solid curve indicates the general dependence on the winding number of VO. However, the actual relation is not continuous (as one can see on close inspection) because the rotation number is constant for a range of voltages when the frequencies are in the ratio of integers. The variability in the uncoupled winding number is due to the variability in the applied voltage in making the transition from I,_ -) To or I, --, IL in the hysteresis loop (see appendix); C) 1 : 1 phase locking for V. = 0.182 V.
in voltage, i.e., the voltage time series appears to be a sequence of pulses (fig. 2). The current through the first diode has the form of rising and falling exponentials, switching direction after transitions to the upper or lower branches of the current-voltage curve. This cycle of conduction in the current-voltage phase space of the diode is an example of a hysteresis loop. The hysteresis loops of the diodes and the resultant voltage-time pulses were designed to simulate the morphology of action potential recordings at the SA node and AV junction sites. Finally, the resistance and induc-
tance determine the intrinsic frequency of each pacemaker. The time rates of change in the current through the two diode branches of the circuit are determined by Kirchhofi’s laws: ~J&)
+ (4
+zU,(r) M*(t)
+ R)&(t)
+ [v,(t)
- v,] = 0,
(2)
+ (R, + W*(t)
+Rz,(t)
+ [v*(t) - V-J = 0.
(3)
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B.J. West et al./ Nonlineur cardiac oscilhtors
These two equations constitute a coupled feedback system through the I2 dependence of the i, equation and the I, dependence of the i, equation. Thus the two oscillators are linearly coupled with a resistor. However, each oscillator is driven by the difference between the voltage source and that dropped across the diode, introducing the anharmonic effect of the current-voltage response curve. Because the tunnel diodes are hysteretic (nonlinear) devices, as the current in one of them increases, the voltage across it remains nearly the same until the current reaches its value Io, at which time an external voltage source is activated to drive the system to the level V, (fig. 1B) (see appendix). Once on the upper part of the hysteresis loop, the current diminishes, again with little or no change in the voltage until the current reaches its value I,, at which point an external voltage of opposite polarity is activated to drive the system to the lower branch of the hysteresis curve. The cycle then repeats itself. This cycling of the coupled system is depicted in fig. 1C which shows that the sharply angled regions of the uncoupled hysteresis loops have been smoothed out by the means of the coupling. In the related model of Gollub et al. [ll] the transition between the upper and lower branches of the current-voltage loop is instantaneous. To mimic the smooth change from one branch of the curve to the other observed in physiological oscillators, we replaced this discontinuity with a hyperbolic tangent function along with a voltage which linearly increases in magnitude with time at the transition points IG and I, (see appendix). The set of coupled nonlinear rate equations (2) and (3) were numerically integrated on a VAX 11/750 computer using a modified divided difference form of the standard Adams Pete formulas and local extrapolation. The values of the circuit parameters were chosen to be R = 3.2 9, R, = 1.3 9, R,= 1.4 s2, L, = 2.772 /.LH, L,= 3.732 PH and V, varied in the range from 0.16 to 0.45 V. In the appendix we specify the functional form of the nonlinear voltage-current relation depicted in fig. 2 and used in the numerical integration. This
I
particular form was chosen for convenience, but does have the appropriate qualitative features of SA and AV current-voltage loops. Finally, to simulate the effects of driving the right atrium at incrementally higher rates with an external pacemaker, an external voltage of variable frequency was applied to the SA node oscillator branch of the electric circuit.
3. Results Each diode (Dj) cycles through its hysteresis loop at a period determined by the decay time of the oscillator, i.e., L,/( R + Rj). The periodic pulse form of the isolated nonlinear oscillator is now changed due to its interaction with the other nonlinear oscillator. The phase space for the coupled system has the four coordinate axes (I,, I,, V,, V,) corresponding to the four dynamic variables of the composite system. In figs. 1B and lC, we have superimposed the two phase spaces rather than trying to draw the actual four dimensional one. The dynamics of the coupled system can be depicted by the orbits in the reduced phase space (II, 12) for a certain set of system parameter values. A periodic solution to dynamic equations (2) and (3) is a closed curve in the reduced phase space as shown in fig. 2A (inset). Here, for two periods in one oscillator we have three in the other so that the coupled frequencies are in the ratio of three to two. A closed orbit with 2m turns along one direction and of 2n turns in the orthogonal direction indicates a phase locking between the two diodes such that one diode undergoes m cycles and the other n cycles in a constant time interval T for the coupled system. Fig. 2A also shows the time trace of the voltages across diodes 1 and 2 for this case. We observed the 3 : 2 ratio of oscillator frequencies over a broad range of values of V,. However, in the region V, I 0.225 V the frequency ratio of the two oscillators becomes phase locked (1 : 1 coupling) (fig. 2B) at a frequency slower than the intrinsic frequency of the SA node but faster than that of the AV junction. Fig. 2C shows the
B.J. West et al. / Nonlineur curdiuc oscillurors I
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Finally, for an interval of 1.5 units between consecutive external pulses, every other pulse fails to trigger the AV junction (fig. 3C).
4. Discussion
C.2:l AVK&Xi
0.6 ,_
I
TIME
Fig. 3. A) With parameter values same as in fig. 2C; 1 : 1 phase locking persists when the SA node is driven by an external voltage pulse train with pulse width 0.5 dimensionless time units and period 4.0. B) Driver period reduced to 2.0 with emergence of 3 : 2 Wenckebach periodicity. C) Driver period reduced to 1.5, resulting in a 2 : 1 AV block. Closed brackets denote SA pulse associated with AV response. Open brackets denote SA pulse without AV response (“nonconducted beat”).
outputs of both oscillators in the coupled system with parameter values such that the uncoupled frequencies are in the ratio of three to two. In the coupled system, the SA and AV oscillators are clearly 1: 1 phase-locked due to their dynamic interaction. The effects of driving the two oscillators at incrementally higher rates by an external pacemaker are illustrated in fig. 3. At first, when the driving period is 4.0 units between consecutive pulses and the pulses are 0.5 units wide, the two oscillators remain 1: 1 locked (fig. 3A). As the interval between consecutive pulses is reduced to 2.0 units, the output of the two oscillators is no longer one to one, but three to two. This means that the AV oscillator is failing to respond to one out of three impulses from the SA oscillator (fig. 3B). Decreasing the interval to 1.8 units results in two out of every five pulses from the SA oscillator failing to elicit a response from the AV oscillator.
This new nonlinear model of supraventricular conduction was motivated by several objectives. First, using a minimum number of elements we sought to construct a circuit whose dynamics would be relevant both to physiologic and pathologic cardiac function. In contrast to more complex models proposed in the past [2, 81, the present circuit consists of just two oscillators and a total of only eight components, each with a physiologic analog. The voltage source (V,) corresponds to the metabolic energy supply of the conduction system and oscillators. The resistance (R) provides the coupling between the oscillators. When R = 0,the oscillators revert to their free-running, uncoupled state, analogous to complete AV block. Each pacemaker unit itself consists of three elements: diode (O]), inductor (Li) and resistor ( Ri).We chose values such that the free-running rates of the SA/AV oscillators would be 3/2, close to their physiologic ratio. The present model differs in other important respects from earlier nonlinear representations of the cardiac conduction system. For example, the three-oscillator model of van der Pol and van der Mark [2] which serves as the foundation for cardiac nonlinear dynamics did not specify the relevant set of coupled equations. Similarly, subsequent analog simulations [7, 8, 141 lacked explicit mathematical descriptions. The coupled set of equations provided by Katholi et al. [3] afforded an important advance in this regard. In their model, the SA node and AV junction were represented as van der Pol oscillators with free-running frequencies in the ratio of 3 : 2. An arbitrary time delay was interposed between these pacemakers. This model supported the hypothesis that sinus rhythm corresponds to 1: 1 phase-locking of two coupled relaxation oscillators
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B.J. West ei al./ Nonlinear cardiac oscillators I
with different intrinsic firing rates. Perturbations leading to AV Wenckebach, however, were not studied in the model. Subsequently, Guevara and Glass [4] observed AV Wenckebach-like periodicity in a phase transition curve model of the response of a single oscillator driven by an external stimulus. Their mathematical model, however, did not address the physiologic problem of having two spontaneously firing pacemakers which are bidirectionally-coupled. Such bidirectional coupling implies that the action potentials propagating between pacemaker sites influence both sites. In the equivalent circuit model, the influence is manifest by means of the resistive coupling. How this influence is physically mediated would require a description not only of the action potentials, but also of the physical-chemical processes generating them. The microscopic mechanism for the bidirectional interaction is not known, but is implicit in the equivalent circuit model presented here. This simple dual-oscillator circuit is based on an explicit set of coupled equations and has several features of potential relevance to cardiac electrical dynamics. First, the model, like that of Katholi et al. [3], demonstrates that two nonlinear oscillators with free-running frequencies of approximately 3 : 2 may become 1: 1 phase locked (coupled) with the intrinsically faster (“SA”) pacemaker leading the slower (“AV”) pacemaker. In addition, the present model demonstrates that abrupt changes (bifurcations) in the phase relationship between the two oscillators occur when the intrinsically faster pacemaker is driven at progressively higher rates. In the present model, over a critical range of frequencies, a distinctive type of periodicity is observed such that the interval between the “SA” and “AV” oscillators becomes progressively longer until one “SA” pulse is not followed by an “AV” pulse (fig. 3B). This cycle then repeats itself, analogous to AV Wenckebach periodicity [14] which is characterized by progressive prolongation of the PR interval until a P-wave is not followed by a QRS complex. These AV Wenckebach cycles, which may be seen under a variety of pathological conditions, are also a feature of normal electro-
physiologic dynamics and can be induced by driving the atria with an electronic pacemaker [15]. In addition, this model demonstrates the phenomenon of 2 : 1 block when the system is driven past a critical point. In this new parameter zone, every other “SA” pulse is apparently uncoupled from an “AV” pulse, resulting in the type of period-doubling (subharmonic) bifurcation reported in a variety of perturbed nonlinear systems [17-221. The findings of both phase-locking and bifurcation-like behavior are particularly noteworthy in this dual oscillator model because they emerge without any special assumptions regarding conduction time between oscillators, refractoriness of either oscillator to repetitive stimulation or the differential effect of one oscillator on the other. The observed dynamics support the contention that the AV junction may be more than a passive conduit for impulses generated by the sinus node, as also suggested by Guevara and Glass [4]. The present model is consistent with the alternative interpretation that normal sinus rhythm corresponds to 1: 1 phase-locking (entrainment) of two or more active oscillators, and does not require complete suppression of the slower pacemaker by the faster one, as do the passive conduit models. It should be emphasized, however, that when two active pacemakers become 1: 1 phase-locked, the intrinsically slower oscillator may be mistaken as a passive element because of its temporal relation to the intrinsically faster oscillator. Furthermore, the model is of interest because it demonstrates marked qualitative changes in system dynan&s, characteristic of AV Wenckebach and 2 : 1 AV block, occurring when a single parameter (driving frequency) is varied over some critical range of values. The present model may also be useful in making predictions about cardiac electrical dynamics. For example, a prediction based on this representation is that the interval between the pulse peaks of the two major cardiac oscillators (analogous to the PR interval of the electrocardiogram) in their 1: 1 phase-locked state will be affected not only by the intrinsic firing rate of the slower (AV) pacemaker
B.J. West et al. / Nonlinear cardiac oscillators I
but by that of the faster (SA) oscillator as well. ‘According to this nonlinear model, selective. slowing of the period of the SA node leads to a decrease in the SA-AV coupling interval. The traditional concept of passive AV conduction predicts no change in this interval if the SA firing rate alone is perturbed. In addition, this new model and subsequent modifications may be helpful in studying a variety of other cardiac phenomena with possible nonlinear mechanisms, such as “electrotonic” coupling in modulated parasystole [23]. The effect on SA-AV coupling relationships induced by changes in the qualitative features of the oscillator hysteresis loops can also be systematically studied. These perturbations may be used to model electrolyte or drug effects which selectively alter one or more phases of the action potential and hence the slope of the corresponding limb of the hysteresis loop. Finally, a multi-oscillator system could be of potential importance in modeling the nonlinear traveling waves which may underlie arrhythmias such as flutter and fibrillation [2, 241.
To numerically integrate the set of coupled rate equations (2) and (3) requires that a functional form for the voltage-current curve in fig. 1 be specified. We denote the voltage current pair by (V,, Z,) where n = 1,2. On the upper (up) branch of the current-voltage curve, we have dZ,,/dt < 0 and for 3 X lob3 I Z, I 22 X 10e3, the voltage is then on the relatively flat portion of the hysteresis loop: (A.1)
When Z, 5 3 x 10v3 = ZLc, the critical value of current at which the transition to the lower branch is initiated. (A.l) is replaced with V, = G,““) + Z-Z,‘).
G,(“*), EZ,?, ZZL*)are defined as fol-
Gi”*)= Vt,+k,AV{l+tanhk,(Z,-I,,)} +Ak,(Z,-Z,), ZZ;“=k,AV{l-
(A-3)
tanhk,(Zo-I,,)-
0” - ZLC)
xK3-&_,) ’ Hi*)=
-k,AV(l+
(A-2)
VL} (A.41
tanhk,(Z,-Z,,)-(YZL}
K-ZLC)
x(z,-z,c)~
(A.5)
Note that a linear voltage (cwZr)is introduced after the transition is initiated to insure it is completed. On the lower (low) branch of the hysteresis loop we have dZ,/dl>O and for 10-3~Z,~2~10-3 the voltage is again in the relatively flat region: I/ = G(i‘+ + R(i) n . n n
(A.61
When Z, 2 21 x low3 = I,, the transition to the upper branch is initiated and (A.6) is replaced with V = G(‘“‘“)+ R(2) n n n 3
Appendix A
V, = G,(U*)+ H,“‘.
The function lows:
205
(~4.7)
where G;“‘“‘= V,+k,AY{l+tanh[k,,(Z,-I,,)]} +Ak,(Z,,-Z,),
(A.81
R(,‘)=k,AV{l+a~~-tanh[k,~,(Z,-I,,]}
(4 -I”,)
x&-Z”C) R’,Z)=k,AV{l-
’ V,+tanh[k,,(Z,-I,,)]}
w-Z”d u”-z”c)’
x
64.9)
(A.lO)
We have selected the parameter values k, = 0.4, k, = 0.45Ak, = 434(0.5 - k,), I, = 10-3, ZG= 24 x 10-3, k,, = 434, (Y= 0.1, v, = 0.05, vo = 0.5, AV= V, - ?‘, = 0.45 and k,, = 868.
206
B.J. West et ul./Nonlineur
References [l] D.M. Vassalle, Circ. Res. 41 (1977) 269. [2] B. van der Pol and J. van der Mark, Phil. Mag. 6 (1928) 763. [3] C.R. Katholi, F. Urthaler, J. Macy Jr. and T.N. James, Comp. Biomed. Res. 10 (1977) 529. [4] M.R. Guevara and L. Glass, J. Math. Biol. 14 (1982) 1. [5] N. Ikeda, Biol. Cybem. 43 (1982) 157. [6] R.P. Grant, Am. J. Med. 20 (1956) 334. [7] D.A. Side& J. Electrocardiol. 9 (1976) 227. (81 D.A. Sideris and SD. Moulopoulos, J. Electrocardiol. 10 (1977) 51. [9] J. Grasman, Bull. Math. Biol. 46 (1984) 407. [lo] A.T. Winfree, J. Theor. Biol. 16 (1967) 15. [ll] J.P. Gollub, T.O. Brunner and B.G. Danly, Science 200 (1978) 48. [12] J.P. Gollub, E.J. Romer and J.G. Socolar, J. Stat. Phys. 23 (1980) 321. [13] F. Urthaler, C. Katholi, J. Macy Jr. and T.N. James, Am.
curdioc oscilkutors I
Heart J. 86 (1973) 189. [14] F.A. Roberge and R.A. Nadeau, Can. J. Physiol. Pharmacol. 47 (1969) 695. [15] M.E. Josephson and SF. Seides, Clinical Cardiac Electrophysiology: Techniques and Interpretations (Lea and Febiger, Philadelphia, 1979), p. 33. (161 R.M. May, Nature 216 (1976) 459. [17] M.J. Feigenbaum, Phys. Lett. 74A (1979) 375. [18] P.S. Linsay, Phys. Rev. Lett. 19 (1981) 1349. [19] M.R. Guevara, L. Glass and A. Shrier, Science 214 (1981) 1350. [20] A.L. Goldberger, R. Shabetai, V. Bhargava, B.J. West and A.J. Mandell, Am. Heart J. 107 (1984) 1297. [21] A.L. Ritzenberg, D.R. Adam and R.J. Cohen, Nature 307 (1984) 159. [22] W.L. Keith and R.H. Rand, J. Math. Biol. 20 (1984) 133. [23] C. Antzelevitch, M.J. Bernstein, H.N. Feldman and G.K. Moe, Circulation 68 (1983) 1101. [24] A.L. Goldberger, V. Bhargava, B.J. West and A.J. Mandell, Chn. Res. 32 (1984) 169A.