International Journal of Bifurcation and Chaos, Vol. 13, No. 3 (2003) 743–754c© World Scientific Publishing Company

OBSERVATION OF A PERIOD DOUBLING

BIFURCATION DURING ONSET OF HUMAN

VENTRICULAR FIBRILLATION

MICHAEL SMALL∗

Department of Electronic and Information Engineering,

Hong Kong Polytechnic University, Kowloon, Hong Kong

ensmall@polyu.edu.hk

DEJIN YU and ROBERT G. HARRISONDepartment of Physics, Heriot-Watt University, Edinburgh, UK

Received August 30, 2001; Revised March 20, 2002

We find evidence of a period doubling bifurcation and transition from pseudo-periodic chaos toa stable limit cycle during onset of ventricular fibrillation (VF) in humans. Novel radial basismodeling techniques are applied to time series of spontaneous VF to estimate time dependentchanges in the dynamics. Furthermore, we show that the spectral power content may be usedas a surrogate for the underlying bifurcation parameter. With this spectral power measure wedemonstrate a characteristic transition from pseudo-periodic chaos to a second pseudo-periodicchaotic regime. The methods described in this paper are generic and applicable to any timeseries data.

Keywords : Ventricular fibrillation; period doubling; radial basis modeling.

1. Introduction

Ventricular fibrillation (VF) is a rapidly lethal car-diac arrhythmia and a common cause of death inthe industrialized world. However, the mechanismunderlying onset of VF and successful treatment viaelectrical defibrillation is poorly understood.

Experimental observations in humans andanimals have shown that the cardiac system is in-herently nonlinear and that significant nonlinear de-terministic structure can be extracted from ECGtime series recordings [Small et al., 2000c; Smallet al., 2000b; Ravelli & Antolini, 1992]. Experimen-tal preparations of cardiac cells have been shownto exhibit phase locking, period doubling and on-set of chaos [Guevara et al., 1981; Chialvo & Jalife,1990; Savino et al., 1989; Karagueuzian et al., 1993].

However, these experiments consist of externallydriven isolated cellular preparations. Theoreticalmodels of the atrio-ventricular (AV) junction andperiodically forced cardiac cells predict phase lock-ing and period doubling bifurcation [West et al.,1985]. Period doubling, from period 1 to period 2(known in the medical literature as alternans), canbe occasionally observed in the diseased heart (see,e.g. [Goldberger et al., 1985]) and has recently beenshown to precede ventricular fibrillation in theo-retical models [Hastings et al., 2000]. Cohen andcolleagues have demonstrated period doubling (upto five times the base period) in the canine heart[Ritzenberg et al., 1984] and have more recentlyshown that T-wave alternans is a significant pre-dictor of arrhythmia [Klingenheben et al., 2000].

∗Author for correspondence.

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744 M. Small et al.

Although alternans is indicative of a perioddoubling bifurcation, a period doubling bifurcationroute to chaos may not necessarily manifest itselfas alternans. Alternans is defined as a doubling ofthe underlying period of a time series (see Fig. 3),a period doubling bifurcation occurs when the or-

der of the periodic behavior doubles — typically totwo periods very close to one another, i.e. Alternansis sufficient but not necessary. Many authors havespeculated that a period doubling bifurcation mayprovide a route to chaos during ventricular arrhyth-mia in humans [Goldberger et al., 1988; Hoekstraet al., 1997; Zhang et al., 1999]. However, no con-clusive evidence has been provided. In this paperwe infer the existence of a period doubling bifurca-tion route to chaos prior to the onset of ventriculartachyarrhythmia in humans through the analysis ofnonlinear models of clinical ECG data. Whereasalternans is an easily observable and measurablephenomenon (Fig. 3) we apply nonlinear modelingtechniques to search for more subtle features, notvisually apparent in the original data. A perioddoubling bifurcation is not apparent from a cursoryexamination of the data in Fig. 1. We deduce theexistence of such a bifurcation in a model of thisdata.

To search for time dependent nonlinear dy-namic structure we employ powerful nonlinearradial basis modeling techniques described in [Judd& Mees, 1995] and generalized in [Small & Judd,1998]. These modeling methods have the nec-essary nonlinearity to accurately describe cardiacdynamics (unlike linear or polynomial models)and are amenable to analysis with the tech-niques of nonlinear dynamical systems theory (un-like neural networks or local linear methods).Furthermore, these models are generic and themodeling algorithm is (in general) assumption free.By employing an information theoretic model se-lection criteria this algorithm chooses the simplestmodel that is consistent with the observed dynam-ics. To accurately model bifurcation in cardiacdynamics these existing methods must be modi-fied. By embedding time as a state variable [Smallet al., 2001] we observe a period doubling bifurca-tion and chaotic dynamics during sinus rhythm andventricular tachycardia (VT) immediately preced-ing VF. However, we find that successive runs of themodeling algorithm do not produce quantitativelyidentical results. Therefore, we explicitly calculatean index which we use as a surrogate for the true

(hidden) bifurcation parameter and embed this as astate variable. This technique produces characteris-tic and reproduceable results. We observe a transi-tion from pseudo-periodic chaos (sinus rhythm priorto VT and VF) to a stable limit cycle (during VT)and to (distinct) pseudo-periodic chaos (VF).

By pseudo-periodic chaos we mean the com-monly observed type of chaos in systems such asthe Rossler equations. A periodic orbit becomesbi-periodic and cascades through successive perioddoubling bifurcations; in the Poincare section; tochaos. In the chaotic regime the system exhibitschaotic, approximately cyclic behavior. The mod-eling described in this paper characterizes the tran-sition from the physiological states of sinus rhythmto VT and VF as a period doubling bifurcation.

We must emphasize that applying this mod-eling algorithm to experimental time series datacannot prove the existence of a bifurcation to chaosin the dynamical system underlying the observeddata. However, this information theoretic modelingalgorithm estimates the simplest dynamical system(model) that is consistent with the observed time se-ries. We may then apply computational dynamicalsystems theory to estimate the bifurcation behaviorof that model. From this we are able to show thatthis observed behavior is both consistent with thetime series and also the simplest consistent behav-ior; from the class of models employed.

1.1. Data

We examine four recordings of spontaneous initia-tion and evolution of VF in human subjects. Suchtime series data are limited and difficult to ob-tain. Using a new data collection facility [Smallet al., 2000a] we have been able to record time seriesshowing spontaneous evolution from sinus rhythmto VF, and subsequent treatment. In two of theserecordings VF is preceded by VT. One of the othertwo recordings shows a direct transition from si-nus rhythm to VF. The fourth recording showsbradycardic (slow rhythm) behavior prior to on-set of VF. For each of these four recordings weselected an approximately 90 second (45000 datapoint) sub-sequence covering the transition from si-nus rhythm, through any intermediate rhythms toVF. We down-sampled the time series by takinga four point moving average and sampling everyfourth point, yielding 90 second episodes of approxi-mately 11250 data points. Down sampling was done

Period Doubling Bifurcation During Onset of Human Ventricular Fibrillation 745

22 24 26 28 30 32 34 36

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3

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mV

)

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3

3.5

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G (

mV

)

52 54 56 58 60 62 64 66

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3

3.5

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G (

mV

)

Time (Seconds)

Fig. 1. Spontaneous evolution of VF in a human. The data shown here has been down-sampled to 125 Hz. On eachplot the horizontal axis is time in seconds and the vertical axis is surface (ECG) electrical potential in milli-volts (mV). Thetop plot shows a short section of sinus rhythm, including ectopic beats. The second plot shows initiation and evolution ofVT and the third shows VF. These plots are contiguous and the horizontal axes are equal. The entire data set is shown inFig. 4(a).

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3

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)

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Fig. 2. Spontaneous evolution of VT and bigeminy in a human. The data shown here has been down-sampled to125 Hz. On each plot the horizontal axis is time in seconds and the vertical axis is surface (ECG) electrical potential inmilli-volts (mV). The top plot shows a short section of sinus rhythm and spontaneous evolution of VT. The second plot showsself termination of VT, following this regular QRS complexes (normal heart beats) and ectopic (abnormal) beats alternate.In the third panel sinus rhythm is restored. These plots are contiguous and the horizontal axes are equal.

746 M. Small et al.

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(m

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Fig. 3. Period doubling in human ECG. The data shown here has been down-sampled to 125 Hz. On each plot thehorizontal axis is time in seconds and the vertical axis is surface (ECG) electrical potential in milli-volts (mV). This datashows a spontaneous period doubling in the respiratory rate of a human subject (at around 76 seconds). Prior to this severalectopic (premature) beats are observed (for example, at 57 and 71 seconds). These plots are contiguous and the horizontalaxes are equal.

to reduce the computational load to a manageablelevel. By taking a four point moving average beforedown sampling we reduced the amount of tempo-ral information in the time series but increased thespatial resolution. After down sampling, the first11000 data points were used to construct a cylin-drical basis model.

Figure 1 shows one of the recordings used inthis study. For comparison we have also applied thisanalysis to recordings of other spontaneous evolv-ing physiological rhythms. Figure 2 shows a record-ing of bigeminy (a phenomenon where normal andabnormal beats alternate) and Fig. 3 shows spon-taneous alternans.

1.2. Overview

We apply nonlinear cylindrical basis modelingtechniques to these data in two slightly differentincarnations. In Sec. 2 we describe the nonlinearcylindrical basis modeling methodology we employ.Section 3 describes the application of this method

with no further assumptions about the underlyingbifurcation. In Sec. 4 we assume that visually ap-parent structure in the time series is indicative ofthe underlying bifurcation and estimate a scalartime dependent parameter as a first approximationto the bifurcation parameter.

2. Cylindrical Basis Models andMinimum Description Length

Cylindrical basis models are a generalization of thewell known radial basis models [Powell, 1992]. Let{yt}

Nt=1 be a scalar time series on N observations.

For an embedding window dw, a cylindrical basismodels is a function F : Rdw 7−→ R such that

F (zt) = a0 +k∑

i=1

aiyt−`i

+m∑

j=1

λjφ

(

‖Pj(zt − cj)‖

rj

)

, (1)

Period Doubling Bifurcation During Onset of Human Ventricular Fibrillation 747

where zt = (yt−1, yt−2, . . . , yt−dw) ∈ Rdw , ai, λj ∈

R are the weights, cj ∈ Rdw are the centers, andrj ∈ R+ are the radii. The lags `i satisfy 0 < `i <`i+1 ≤ dw and the functions Pj(·) are orthogonalprojections from Rdw to some subset of the coordi-nate axis. The functions φ are typically some classof C2 functions satisfying

∫

∞

0φ(x)dx < ∞. For the

computations in this paper we use a combination ofGaussian basis functions and Morlet wavelets.

The essential difference between cylindrical ba-sis models and standard radial basis models is theinclusion of the projection functions Pj(·). Thesefunctions allow for the projection onto different, sig-nificant, subspaces of Rdw in different parts of phasespace. This is both useful and intuitive since thecomplexity of most nonlinear dynamical systemsvaries with location in phase space. For example,the Lorenz attractor is basically two dimensional onthe “wings”, but the central separatrix contains athree-dimensional structure [Judd & Mees, 1998].

The function F is used to approximate theevolution operator of a dynamical system by

F (zt) = F (yt−1, yt−2, . . . , yt−dw) = yt . (2)

With the model size (k,m) fixed, our modelingalgorithm selects ai, `i, λj, cj , rj and Pj(·) suchthat the model prediction error

Error =∑

e2t =

N∑

t=dw+1

(F (zt) − yt)2 , (3)

is minimized. The optimal model size is then se-lected by finding the values of k and m such thatthe model description length is minimized. The de-scription length of a model is roughly the amountof compression of the original data that is achievedby describing the model and model prediction er-rors, over specifying the original data values [Judd& Mees, 1995; Small & Judd, 1998; Small et al.,2001]. The idea of minimum description length wasoriginally described in [Rissanen, 1989].

For a given model (1) with error specified by(3) the description length is approximately

N − dw

2

(

1 + ln2πeT e

N − dw

)

+

(

1

2+ ln γ

)

(k + m) −k+m∑

j=1

ln δj (4)

where γ is the constant number of (binary) ordersof magnitude required to cover the range of the data

and δj are the precision with which one must specifyeach of the (k + m) model parameters (the ai andλj terms in (1)). The δj terms are computed asthe solution of (Qδ)j = 1/δj where Q is the secondderivative of (1) with respect to the model param-eters Ai and λj .

3. Assumption Free Modeling

The model described by Eqs. (1) and (2) is timeindependent. The process we are modeling exhibitstime dependent features and these equations are un-likely to provide an adequate model. One possiblesolution is to model each of the parameters ai, `i, λj ,cj , rj and Pj(·) as a function of time. However, thisapproach is somewhat impractical. A more feasibleapproach is to extend the embedding

zt = (yt−1, yt−2, . . . , yt−dw) (5)

to

zt = (yt−1, yt−2, . . . , yt−dw, ξ(t)) (6)

so that time t is explicitly included as a dependentvariable of F (zt) [Small et al., 2001]. The affinetransformation ξ is included so that the time in-dex “stretches” the attractor. If ξ is too large, thenthe embedded points zt will be too sparse in phasespace, if ξ is too small then there will effectively beinsufficient separation. For this data we have foundthat it is sufficient to choose ξ so that the maxi-mum and minimum values of ξ(t) are the same asthe maximum and minimum of yt. In general, amore robust choice may be to define ξ in terms ofthe standard deviation of yt.

The modeling algorithm we apply in this pa-per must minimize a nonlinear function (descrip-tion length) of several variables (ai, `i, λj , cj , rj

and Pj(·)). To solve this problem completely is be-yond our limited computational resources. For thisreason the algorithm we have implemented involvesa stochastic search for a local minimum of descrip-tion length. For many data sets this is sufficient toobtain reproducible and consistent results [Judd &Mees, 1998, 1996; Judd & Small, 2000; Small et al.,1999a; Small et al., 1999b]. We have tested thismodeling algorithm with four systems with knownbifurcation structure and obtained good agree-ment between actual and predicted results [Smallet al., 2001]. This modeling methodology is able

748 M. Small et al.

to accurately uncover bifurcation structure fromtime series simulations of (i) the Rossler equations,(ii) FitzHugh–Nagumo type simulations of ven-tricular arrhythmia, and (iii) nonstationary noiseprocesses as well as from (iv) experimental data ofinfant respiration [Small et al., 2001]. The compu-tational simulations examined in [Small et al., 2001]have well known and easily verifiable dynamics, thisprovides a test of the accuracy of this approach. Wewere able to correctly extract a nonstationary trendfrom an i.i.d. noise simulation [Small et al., 2001]with a slight nonstationarity. When applied to anoisy simulation of the Rossler system undergoinga period doubling bifurcation the modeling algo-rithm was able to correctly identify the qualitativefeatures of the bifurcation from the noisy data (5000observations) [Small et al., 2001].

The model we utilize here is an ordinaryiterative map with a tunable parameter. This bi-furcation parameter has been allowed to vary in theconstruction of the model. By keeping this parame-ter fixed and observing the iterated behavior of themodel we have an estimate of the asymptotic dy-namics of the system for that (fixed) value of thebifurcation parameter. For each such trajectory weestimate the (asymptotic) peak and trough valuesobserved. For a periodic orbit these will be fixed;for period 2 dynamics there will be two distinct am-plitudes; and for chaos the sequence of amplitudeswill be irregular. It is these estimates of the asymp-totic amplitudes that are plotted in Figs. 4 and 6 asa function of the bifurcation parameter. Represen-tative trajectories for various values of the bifurca-tion parameter are depicted in Fig. 5. For example,

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& T

roug

h

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0.2

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Time index (125 Hz)/Bif. param.

Est

. Am

plitu

de

Fig. 4. Period doubling chaos prior to onset of VF. The top panel shows the original time series. The second panelshows the estimated asymptotic peak (red) and trough (green) values for fixed values of the bifurcation parameter. Thebottom plot shows the estimated asymptotic amplitudes (blue). The horizontal axis is time series datum number, the modelbifurcation parameter is an affine transformation of this. Sinus rhythm is characterized as a periodic orbit, prior to initiationof VT this periodic orbit undergoes a period doubling bifurcation and becomes chaotic. At onset of VT the chaotic behaviorbifurcates to pseudo-periodic chaos and period 8 dynamics. At the transition to VF this behavior changes to a stable limitcycle and eventually a stable focus.

Period Doubling Bifurcation During Onset of Human Ventricular Fibrillation 749

100 200 300 400 500 600 700 800 900 1000da

tum

100

0

100 200 300 400 500 600 700 800 900 1000

datu

m 2

000

100 200 300 400 500 600 700 800 900 1000

datu

m 4

000

100 200 300 400 500 600 700 800 900 1000

datu

m 5

000

100 200 300 400 500 600 700 800 900 1000

datu

m 6

000

100 200 300 400 500 600 700 800 900 1000

datu

m 8

000

Fig. 5. Trajectories estimates from a model. For fixed values of the bifurcation parameter, the system is run for 1000iterations. The dynamic behavior for various values of the bifurcation parameter are shown (from top): periodic, period 2,period 3, chaos, period 4 and a fixed point. The chosen fixed values of bifurcation parameter correspond to datum 1000, 2000,4000, 5000, 6000 and 8000 of Fig. 4 (respectively). Trajectories such as these are used to estimate the amplitudes shown inFigs. 4 and 6. Note that while these trajectories do not capture the exact quantitative features of the system (for example,the periodic behavior observed at datum 1000 is unlike sinus rhythm ECG) these features are quantitatively appropriate.

from the representative trajectories shown in Fig. 5we would deduce (from top to bottom): a periodicorbit with only one observable amplitude, period 2,period 3, chaos, period 4, and a fixed point (zeroamplitude).

In Figs. 4 and 6 we provide an estimate, derivedfrom a single application of our modeling scheme tothe data set of Fig. 1. This estimate is representa-tive of our other results. The dynamics presentedin this figure contain all the various complex be-haviors observed in separate runs of the modeling

routine. Some iterations of the modeling routine didnot produce the complete period doubling bifurca-tion illustrated here. However, they all exhibiteda complex bifurcation from pseudo-periodic regime(corresponding to sinus rhythm) to chaos (VT) anda stable periodic orbit or focus (VF). Models fromeach of the four recordings of onset of VF are qual-itatively similar.

We compared this technique and the bifurca-tion behavior estimated from models with tradi-tional frequency domain methods applied directly

750 M. Small et al.

Time (Seconds)

Est

. Am

plit

ude

0 10 20 30 40 50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

0.3

Fig. 6. Period doubling chaos prior to onset of VF. The calculation of Fig. 4 is shown here as a probability densityplot. The figure depicts the estimated distribution of asymptotic oscillation amplitudes as a function of time (in seconds).High likelihood is depicted in bright red and low likelihood in dark blue.

Time (Seconds)

Freq

uenc

y (H

z)

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

Time (Seconds)

Peri

od (

log 2(s

ampl

es)

)

0 10 20 30 40 50 60 70 80 90

2

4

6

8

10

12

(a)

(b)

Fig. 7. Period doubling chaos prior to onset of VF. (a) Fourier spectrogram (256 FFT with an overlap of 128 points)and (b) wavelet analysis of the data used in Figs. 4 and 6. In contrast to the analysis in Figs. 4 and 6 these calculations providelittle additional information compared to the original time series. Most of the change in frequency content that is observablefrom these plots is also observable directly from the original time series. High density is shown in red and low density in blue.

Period Doubling Bifurcation During Onset of Human Ventricular Fibrillation 751

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 110000.2

0.4

0.6

0.8

Bif

. par

am. (

arb.

uni

ts)

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

2.6

2.8

3

Pea

k &

Tro

ugh

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

0.1

0.2

0.3

0.4

Time index (125 Hz)/Bif. param.

Est

. Am

plit

ude

Fig. 8. Bifurcation diagram estimated with surrogate bifurcation parameter. The top plot shows values used forthe bifurcation parameter as a function of the time index (sampled at 125 Hz) for the same data set as Fig. 1. The secondplot shows the asymptotic peak (red) and trough (green) values as a function of the time index and the third plot showsthe estimated amplitude behavior (blue). The complex period doubling bifurcation evident in Fig. 4(b) is absent, but a cleartransition from large scale Rossler type chaos (for sinus rhythm preceding VF) through period 6 behavior (VT) to anotherpseudo-periodic chaotic regime (VF), is apparent.

to the data. Figure 6 is an estimate of the proba-bility distribution of the amplitude of system orbitsfor various values of the bifurcation parameter (timeindex). The data is the same as the lower panelof Fig. 4, expressed as a probability distribution.For comparison we computed a windowed spectro-gram and wavelet transform from the original data(Fig. 7). While the coarse time dependent featuresare evident in these plots, we find very little in-formation not evident in the original time series.The asymptotic time behavior displayed in Fig. 6 isachieved by extrapolating the model dynamics fora model fitted to the data.

Qualitative features of period doubling and on-set of chaos are present in each of the models despitethis complex structure not being obvious in the orig-inal time series. The modeling algorithm is fittingsubtle features of the asymptotic dynamics that

may not be observed directly during the compara-tively rapid change in the system dynamics prior to(and during) onset of ventricular arrhythmia.

Although essential features are consistent be-tween most models, this variation between modelsis somewhat unsatisfactory. The problem is that inaddition to estimating unknown dynamics we arerelying on the modeling algorithm to estimate anunknown bifurcation parameter at each point intime. To overcome this problem we estimated thebifurcation parameter a priori and built a modelbased on this bifurcation parameter.

4. A priori Bifurcation ParameterEstimation

Generically, the time series recordings of evolutionof VF exhibit two or three of four distinct dynamical

752 M. Small et al.

regimes: sinus rhythm, VT, bradycardia and VF.A bifurcation parameter must reflect this. For thedata presented in this paper, we found that theproportion of the power spectral content between4 and 8 Hz performed well. Alternative indicesincluding entropy, complexity and point-wise corre-lation dimension were also considered. However, wefound that spectral power content provided the bestseparation between different physiological rhythms.In our experience spectral power content also per-formed best for real time identification of arrhyth-mia [Small et al., 2000a]. Spectral power capturesmany of the essential features of the time series(Fig. 7). However, this spectral variation alone isinsufficient to deduce the period doubling bifurca-tion and onset of chaos observed in models of thisdata. Figure 8 shows estimates of this quantity forthe same time series as in Fig. 4.

We estimated the power content between 4and 8 Hz for a 2048 point sliding window (beforedown sampling the original recordings) and usedthis quantity as a surrogate for the true, unknown,bifurcation parameter. That is, the embedding (5)or (6) is replaced by

zt = (yt−1, yt−2, . . . , yt−dw, ξ(p(yt))) (7)

where p(yt) is the above mentioned statistic, andξ(·) is an affine scale transformation. The remain-der of the modeling process is identical.

Figure 8 depicts a representative result of thiscalculation for the data illustrated in Fig. 4. Re-peated runs of the modeling algorithm on thesame data produced equivalent results. The re-sults between data sets were similar. Whereas thecalculation estimating both model and bifurcationparameter would typically yield smooth bifurcationfrom a periodic orbit, to chaos (sinus rhythm priorto VT and during VT) and back to a noisy peri-odic orbit (VF) this is not the case with an a priori

estimate of the bifurcation parameter. For smoothand gradual change in the values of this bifurcationparameter the change in the underlying dynamics issimilarly subtle. However, as this parameter is esti-mated directly from the time series the value oftenchanges fairly rapidly (Fig. 8), and this in turn pro-duces sudden changes in the underlying dynamics.Closer examination of small changes in the bifurca-tion parameter do show a period doubling bifurca-tion during the transition between sinus rhythm andtachycardia. The weakness of this approach is thatwe are constraining the time dependent dynamics

of the model to conform to our expectation of thesystem, i.e. ξ(p(yt)).

5. Conclusion

In this paper we have described two alternativetechniques to examine the changing dynamics in ex-perimental time series. Although we consider onlyhuman ECG recordings the method itself is genericand equally applicable to any experimental time se-ries. Estimating the model and bifurcation param-eter simultaneously (Sec. 3) provides a method ofexamining the changing dynamics with no a priori

information. For short, noisy, or experimental datathis method has its limitations. Primarily, the com-putational burden to estimate the optimal model,model parameters, and bifurcation parameter froma single scalar time series is immense. This problemmeant that it was difficult to get fully repeatable re-sults (for a single time series). Rather than obtainthe global minimum of (4) the modeling algorithmwould halt at a locally optimal value.

To reduce the computational effort we providedan alternative approach — estimating the modelbifurcation parameter a priori (Sec. 4). This ap-proach meant that the models produced were moreeasily repeatable (for a single time series) and alsoconsistent (between different time series). The mainproblem with this approach is finding a “reason-able” surrogate for the bifurcation parameter, andthe restriction of the dynamics that this implies.

Estimating the bifurcation parameter andmodel simultaneously meant that the problem ofreproducing the dynamics was reduced to findingthe right nonlinear function from (an affine trans-formation of) the time index to the underlying bi-furcation parameter. By assuming an appropriatevalue of bifurcation parameter first, the existence ofan injective mapping from the surrogate bifurcationparameter ξ(p(yt)) to the underlying bifurcation pa-rameter is not guaranteed (i.e. p(yt) is not necessar-ily a 1-to-1 function of t). However, we hope that a“reasonable” choice of the surrogate parameter willprovide a trivial transformation from ξ(p(yt)) to theunderlying bifurcation parameter.

When applied to recordings of initiation of VFthese modeling techniques provide a description ofthe simplest dynamical system (within the modelclass) consistent with the observed data. Observa-tions derived from these models are possible dynam-ical phenomena consistent with the data. This does

Period Doubling Bifurcation During Onset of Human Ventricular Fibrillation 753

not imply that the data must have been producedby such phenomena, only that such phenomena area simple and compact explanation for the observeddata. Prior to onset of ventricular arrhythmia, sinusrhythm may behave as a noisy dynamical systemexhibiting Rossler type chaos (i.e. chaotic pseudo-periodic orbits). We also find evidence that a perioddoubling bifurcation may precede imminent ventric-ular tachyarrhythmia. In the arrhythmic state thisdynamical system behaves as either Rossler typechaos (Sec. 3) or a stable focus (Sec. 4). We be-lieve that this possible contradiction is due to lim-itations of the modeling process. When modelingboth dynamics and estimating bifurcation param-eter (Sec. 3) the modeling algorithm is unable toextract sufficient information from the VF and VTstates to model them as more than noisy periodicorbits. When the bifurcation parameter is assumeda priori (Sec. 4), and p(yt) is a nonmonotonic func-tion of t (i.e. the value remains the same at differenttimes, leading to a more densely populated phasespace) arrhythmic (VT and VF) behavior can bebetter modeled as pseudo-periodic chaos (Rosslertype chaos).

These results imply that both sinus rhythmand VF may be characterized as chaotic dynamicalsystems. However the underlying dynamics duringsinus rhythm, VT and VF are all fundamentallydifferent. This result is therefore consistent withearlier work estimating dynamical invariants fromtime series [Small et al., 2000c; Small et al. 2000b;Ravelli & Antolini, 1992; Small et al., 1999b; Yuet al., 2000]. The similarity between this result andvarious observations of alternans is significant butnot conclusive. We observe an increase in the order

of periodic behavior in the amplitude of respirationwhereas alternans typically manifests as a doubling

in the underlying respiratory rate (and sometimesobserved directly as two distinct amplitudes). Wefound no evidence for chaotic bifurcations in record-ings of alternans or bigeminy.

We also found that prior to onset of tach-yarrhythmia the pseudo-periodic chaos observed inmodels of these time series becomes strictly pe-riodic and a period doubling bifurcation may beobserved as a mechanism consistent with transi-tion between these states. These results are pre-liminary, and certainly not clinically significant. Amuch larger sample needs to be considered beforedefinitive statements may be made. Nevertheless,observation of period doubling bifurcations prior toonset of VF suggest that the human cardiac system

undergoes a fundamental change in its dynamicalstate prior to onset. Methods such as Lyapunovexponent and point-wise correlation dimension [Yuet al., 2000] estimation may therefore be applied topredict imminent arrhythmia. However, such algo-rithms need to be substantially modified to be bothsensitive and accurate for extremely short time se-ries if they are to be applied as clinical indicatorsof imminent arrhythmia.

Acknowledgments

This research was funded through a Research De-velopment Grant by the Scottish Higher Educa-tion Funding Council (SHEFC), No. RDG/078. M.Small is supported by a Hong Kong PolytechnicUniversity Postdoctoral Fellowship award (No. G-YW55). We would also like to acknowledge theassistance of N. Grubb, K. Fox and the staff ofthe Coronary Care Unit of the Royal Infirmary ofEdinburgh.

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