Complexity in the living: a modelistic approachProceedings of an International Meeting

Rome, February 1997

Recurrent Structuring of Dynamical andSpatial Systems

C.L. WEBBER (1) Jr., J.P. ZBILUT (2)

(1) Dept. of Physiology - Loyola University ChicagoStritch School of Medicine2160 S. First Ave. - Maywood, IL 60153 USAcwebber@luc.edu

(2) Dept. of Molecular Biophysics and Physiology - Rush Medical CollegeRush-Presbyterian-St. Luke’s Medical Center1653 W. Congress Parkway - Chicago, IL, 60612, USAjzbilut@rush.edu

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Complex Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Recurrence Quantification Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Mathematical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Biological Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Bioinformatic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

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Abstract

The complexity and nonlinearity of physiological systems typically defy comprehen-sive and deterministic mathematical modeling, except from a statistical perspective.Living systems are governed by numerous interacting variables (high dimensionalproblem) with drifting parameters (nonstationarity) in the presence of noise (internaland external perturbations). Depending upon the frame of reference, many biologicalsignals can be shown to be discontinuous alternations between deterministic trajecto-ries and stochastic pauses (terminal dynamics). One promising approach for assessingsuch nondeterministic complexity is recurrence quantification analysis (RQA). As re-viewed in this paper, strategies implementing quantification of recurrences have beensuccessful in diagnosing numerous systems in mathematics, biology, and bioinformat-ics. RQA may likewise be applicable to linguistic texts and financial markets. It isconcluded that recurrence quantification analysis is a powerful discriminatory toolwhich, when properly applied to any dynamical system of choice, temporal or spatial,and correctly interpreted, can yield definitive clues regarding the degree of determin-istic structuring characterizing the system, state changes through which the systemis passing, as well as degrees of complexity comprising the system.

1 Complex Systems

1.1 Definition of Complexity

Dynamical and spatial can be characterized by complexity, but a strict definition forcomplexity remains an enigma. For example, by complexity do we mean that: 1)the system is comprised of numerous interacting variables (high dimensional)?; 2) fewvariables feed back on one another in a nonlinear fashion (chaotic)?; 3) additive noisedistorts pristine trajectories rendering them unpredictable (quasi-periodic)?; 4) thedynamic is discontinuous with intervening singularities and stochastic pauses (ter-minal dynamic)?; 5) successive events are absolutely independent (Markovian)?; 6)the system is characterized by large variance (broad distribution)?; or 7) the systemis punctuated by periods of non-stationarity (drifting)? Probably the best answerto each of these seven questions is an unqualified, “Yes, and more!” We concludethat the definition of complexity is at best amorphous. Not fully understanding,but nevertheless acknowledging this conundrum, legitimizes investigators of differ-ent disciplines and varied technologies to contribute significantly to the problem ofcomplexity. Possibly the best definitions of complexity may incorporate scale depen-dency. Thus, one type of complexity local to a larger field of study (e.g. metabolicpathways in biochemistry) may not be fully applicable to a particular sub-systemof interest (e.g. molecular dynamics of hemoglobin). Scientific investigations haveclearly demonstrated that systems are not simplified (reduced) by descending scales,either in time or space. Rather, complexity prevails at every level of observation,macro to micro, fractal or non-fractal [1].

1.2 Terminal Dynamics

The assumption has been made in classical dynamics that time-variant and space-variant systems are continuous. General acceptance of the Lipschitz constraint ofunique solutions has enabled the implementation of differential equations to modelreal-world systems, supposedly. It is interesting that the replacement of time (t) with

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negative time (-t) is mathematically sound, but in stark contradiction to actual expe-rience. Some dissenting voices in physics contend that physical systems are actuallydiscontinuous [2]. In fact, relaxing the Lipschitz requirement provides for branchingsolutions to equations, disallows for the mathematical reversal of equations (-t is ille-gal), gives insight as how noise might steer a dynamic, and provides an explanationfor dynamical uncertainty in future trajectories. Zak et al.[3] discuss the many facetsand implications of nondeterministic systems characterized by terminal dynamics.As these authors point out, discontinuous systems seem ideally suited for describingboth physical and biological systems extant in the real world, and exhibiting complexbehaviors.

In brief, nondeterministic systems are characterized by alternating deterministicand stochastic sequences. During the execution of a deterministic set of rules (tra-jectory) the system is robust against external buffeting, but while resting at its pointof singularity (pause) the system is subject to low-energy inputs and noise factors.With idealized data, discontinuities between deterministic trajectories and stochasticpauses can be readily identified by spikes in the second derivative or discrete wavelets[4]. Real-world noise, however, renders these techniques all but useless, and filteringblurs actual discontinuities. Recurrence plots, on the other hand, seem to be robustagainst noise influences, making it possible to discover evidence for terminal dynamicsin noisy physiological data.

1.3 Pattern Recognition

Depending upon one’s definition of complexity it is possible to devise measures ofcomplexity (standard deviation, fractal dimension, information entropy, etc.). Theremay be some practical utility for such metrics in disambiguating different states ofdynamical systems (time series) or distinguishing various qualities of anatomical tis-sues (spatial series). However, studies on perception have demonstrated the exquisiteability of the human eye (and ear) to discern subtle patterns in otherwise complexand noisy signals [5]. However, pattern-recognition algorithms have in some casesbeen shown to be superior to human perception. In attempting to best characterizedynamical systems, instead of placing the full responsibility on some single metric,the more noble (and less myopic) strategy might be to represent a system’s subtle“wiggles” graphically. It is hypothesized that viewing any complex but organized sys-tem from different frames of reference (altered dimensional perspectives) will revealunique, albeit subtle, patterns arising as meaningful signals above the backgroundnoise.

2 Recurrence Quantification Analysis

2.1 History of Recurrence Plots

Although the concept of recurrences in numerical sequences has precedent in the for-mer mathematical literature, Eckmann, et al. [6] in the physics literature were the first(1987) to graphically represent recurrences in higher dimensional space as recurrenceplots. By the use of embedding procedures and distance matrices, recurrent pointswere plotted at specific coordinates whenever paired i-j vectors were close in N-space(within a ball of radius r). Rhythms not observable in the one-dimensional time seriescould be realized as specific patterns within the constellations of distributed recur-rent points. More importantly, the presence of diagonal line structuring in recurrence

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plots revealed deterministic behavior within the time series, leading to their conjec-ture that the longest line segment in the recurrence plot was inversely proportionalto the largest positive Lyapunov exponent (> 0), a measure of diverging trajectoriesin chaotic time series.

2.2 Quantification of Recurrences

Recurrence plots soon had appeal to investigators studying a variety of time series,particularly biological data [7]. Dissatisfied with relying upon human judgment todiagnose recurrence plots of complex physiological phenomena, however, Webber andZbilut [8–10] introduced the concept of recurrence quantification analysis (RQA). Thetwo primary contributions culminating from this work were: 1) recurrence variablescould be quantitated by pattern recognition algorithms, and 2) recurrence variablescomputed within a short, episodic window could be diagnostic for state changes occur-ring over long time series. Six nonlinear recurrence variable were generated by RQAincluding: mean distance (average vector distance in N-space), %recurrence (per-cent of plot filled with recurrent points); %determinism (percent of recurrent pointsforming diagonal lines with a minimum of at least two adjacent points); entropy(Shannon information entropy of line-segment distribution); divergence (reciprocalof the longest diagonal line segment, 1/Lmax), and trend (measure of the fading ofrecurrent points away from the central line of identity). The intent of the remainderof this paper is to illustrate the utility of RQA in diagnosing many different types oftime series stemming from mathematics, physiology and bioinformatics. Applicabilityof this approach to complex dynamics in even more divergent fields is also conjecturedwith the hope of stimulating new directions for research.

3 Mathematical Systems

3.1 Periodic Signals

Koebbe and Mayer-Kress [11] were careful to point out that periodic signals scoredas large-scale diagonal lines in recurrence plots (< 0). This property is illustrated inFigure 1 for a sine wave before and after the addition of white noise. Upward, paralleldiagonals indicate direct correlations between two rising phases or two falling phasesof the sine wave. Downward, antiparallel diagonals reflect the palindromic sequencingof recurrent points derived from the rising phase and falling phase of the sine wave(Fig.1B). Added noise breaks up the long diagonals, but the basic diagonal structuringis still apparent (Fig. 1D). Typical recurrence plots of square waves, triangle waves,and a relaxation oscillator at various embedding dimensions are available elsewhere[11].

3.2 Henon Attractor

The following set of coupled equations generate iterated maps of points which enmasse form diverse, but stable geometric structures, Henon attractors [12], dependingupon the numeric values assigned to the parameters:

xi+1 = yi + a− (bx2i ) (1)

yi+1 = cxi . (2)

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With a = 1.0, b = 1.054 and c = 0.3, the plot of yi versus xi creates a 16-pointperiodic attractor (Fig. 2A) which can be transformed into the chaotic Henon strangeattractor by simply changing parameter b to 1.4 (Fig. 2D). Each Henon attractor isa two-variable system, whose topological structure can be reproduced by the methodof time delays [13]. This remarkable theorem of Takens is illustrated by plottingthe current x interval (xi) against the next x interval (xi+1 ) which fully reproducesthe periodic and chaotic attractors (Fig. 2B,E). Randomization of xi points, whileleaving the mean and standard deviation unchanged, destroys the phasic associationof sequential points and obliterates both attractor structures (Fig. 2C,F). Recurrenceplots of the Henon x variable yield long diagonals (λ < 0) for the periodic attractor(Fig. 3A) and short diagonals (λ > 0) for the chaotic attractor (Fig. 3B), bothstructures of which degrade with randomization of xi sequencing. When present,the diagonal lines are only represented with upward slopes, indicating the absence ofpalindromic patterns.

3.3 Logistic Difference Equation

Below is the deceptively simple equation originally described by May [14] which pos-sess very interesting bifurcation properties depending upon the numeric value of pa-rameter a ( 0 ≤ a ≤ 4):

xi+1 = axi(1− xi/k) (3)

For example, as reviewed by Glass and Mackey [15] (and with k = 1), xi+1 decays to0 for constant low values of a (0 ≤ a <1 ), xi+1 goes through period doubling period-icities for constant intermediate values of a (1 ≤ a < 3.57), and xi+1 values becomeaperiodic with the exception of a few periodic oscillations (regions of calm withinthe chaotic storm) for constant high values of a (3.57 ≤ a ≤ 4). After seeding theequation and allowing initial transients to die away, recurrence plots of this logistic at-tractor show similar diagonal line structuring as described for the Henon system (notillustrated). However, to evaluate the utility of recurrence analysis in diagnosing thelogistic system under continuous transitions between states, Trulla et al. [16] gener-ated a series of xi points in which parameter a was continuously increased by 0.00001on each iteration over the range from 2.8 to 4.0. The results showed that regions ofbifurcations could be easily identified by abrupt changes in recurrence variables. Inaddition, the 1/Lmax value (RQA divergence variable) correlated positively with thelargest positive Lyapunov exponent ( λ > 0), confirming the conjecture of Eckmannet al. [6]. Finally, the drifting parameter a in this study rendered the logistic equationmore biologically relevant.

4 Biological Systems

4.1 Pulmonary

Recurrence plot variables have been used to describe the state-dependent determinis-tic structuring of breathing patterns in rodents. For example, Webber [8] recorded in-trapleural pressure in the spontaneously breathing, unanesthetized rat and performedrecurrence analysis (embedding dimension = 10) on breath-by-breath intervals dur-ing different physiological states. Compared with the active state (resident intrudermodel), the quiet state had significantly higher %recurrence values (P < 0.0005) andsignificantly lower information entropy values (P < 0.0005). To study the transition

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between states, Webber and Zbilut [10] again recorded intrapleural pressure beforeand immediately after anesthetization with pentobarbital. As expected, the awakeand agitated rat had a very excited breathing pattern which slowed (increased meanperiods) and evened out (decreased standard deviation of periods) as the rat lost con-sciousness. A one-dimensional running average (200-point epoch) through the perioddata suggested that the rat reached a new anesthetized steady state about 10 min-utes following the administration of the drug. A ten-dimensional perspective on thesame data by RQA, however, revealed sustained aperiodic oscillations in %recurrence,%determinism, and entropy up to 40 minutes after the initial injection, indicating thepresence and persistence of a more complicated dynamic than implied by the simplemean data. One working interpretation of the results posited that pentobarbital wasbeing processed by numerous systems, each possessing a unique time constant (ab-sorption by brain and fat, catabolism by the liver, secretion/excretion by the kidney,etc.).

The complexity of breathing is also expressed by the three phases of the respi-ratory cycle including inspiration, post-inspiration, and expiration [17]. As shownin Figure 4, however, these three phases are very difficult to identify in the phrenicneurogram (cat) or its integrated transform. This ambiguity is completely removedby computing the mean distance variable of the phrenic neurogram using recurrenceanalysis which clearly distinguishes both inspiratory and post-inspiratory phases ofthe phrenic discharge (Fig. 4, third panel).

Finally, Webber and Zbilut [4] also employed RQA to identify the discontinu-ous, terminal dynamic characteristics of breathing. Recurrence analysis was applieddirectly to intrapleural pressure traces recorded in the rat as detailed in Figure 5.Recurrent patterns were reminiscent of noisy sine wave patterns (see Fig. 1), includ-ing palindromic structuring, but definitive recurrent rectangles (or horizontal lines)marked off the presence and position of singularities in the dynamic. Singularitieswere found at the transitions from inspiration to expiration and from expiration toinspiration. Due to the presence of noise in the dynamic, the detection of these singu-larities by wavelet analysis or the second differential of the intrapleural pressure datawas precluded.

4.2 Cardiovascular

The healthy heart does not beat like a predictable metronome [15]. Clinical medicinecan take advantage of this fact by hypothesizing that deviations in heart rate variabil-ity (HRV) from normal may be indicative of cardiac pathology [18]. RR intervals inthe electrocardiogram (ECG) can be studied to quantitate variability. For example,in the time domain, HRV is manifested as non-zero coefficients of variation (covar)for sequential RR intervals; in the frequency domain, HRV appears as specific peaksin the fast-Fourier transform (FFT) of RR intervals [19].

As a new approach, Webber and Zbilut [20] applied recurrence analysis to theproblem of HRV and found the model of terminal dynamics to be fully applicable tothe electrical activation of the heart. For example, a standard ECG recorded from ahuman volunteer in Figure 6 shows the alternation between the PQRST complex andthe TP pause. Interestingly. the PT duration of the active complex is remarkablyconstant (covar = 0.014) whereas the TP variability (covar = 0.141) essentially ex-plains the variability of the entire PP cycles (covar = 0.056). As a terminal dynamic,the cardiac system can be viewed as deterministic process during the activation andrepolarization states (PQRST), interspersed by singularities (TP), at which points thesystem awaits reactivation. These stochastic pauses impart a random-walk character

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to the cardiac dynamic [21].Cardiac variability has also been extensively studied in the frequency domain

[19]. Spectral analysis of RR intervals from normal electrocardiograms, for example,typically produce three prominent peaks: high- (HFO), low- (LFO), and very low-frequency (VLFO) oscillations, classically attributed to cardiac inputs from parasym-pathetic, predominantly sympathetic, and possibly renin-angiotensin-aldosterone sys-tems, respectively. Webber and Zbilut [20] hypothesized that the total variability inthe cardiac cycle could not be explained by the parasympathetic component alone.This hypothesis was tested by computing the FFT of 512 PP intervals from each of 14normal and healthy human subjects. Three spectral peaks were realized as expected(Fig. 7A, solid trace). Randomization of the PP interval sequences obliterated allthree peaks (Fig. 7A, dashed line), suggesting dynamical structuring of the nativePP orderings. To test the parasympathetic contribution to the heart-beat variability(e.g. cardiac slowing), the HFO peak was numerically integrated (Fig. 7A, shadedarea) and correlated with the PP standard deviation. As hypothesized, no signif-icant correlation (linear or curvilinear) could be found between HFO area and PPstandard deviation (Fig.7B), suggesting that other physiological variables were alsocontributing to the overall cardiac variability.

Recurrence analysis confirmed that HRV is a higher dimensional phenomenon.For example, the PP intervals of the same 14 subjects were studied by RQA at anembedding dimension of 13 meaning that the PP variable was serving as a surrogatevariable for at least 6 unmeasured variables [22]. Figure 8 shows the correlation ofRQA variables with PP standard deviation, the %recurrence data exhibiting widerscatter (P = 0.068) than the %determinism data (P = 0.002). Importantly, theseresults imply that factors other than parasympathetic inputs contribute to total car-diac variability, but exactly what those other factors are (besides sympathetic andhormonal inputs) requires further experimentation. The point is that recurrence anal-ysis, a nonlinear multi-dimensional tool, can reveal further information regarding thedegree of complexity of the system under investigation. Full recurrence analysis, ofcourse, would require a reexamination of the same system in parameter space (i.e.the behavior of RQA variables during systematized fine tuning of RQA parameters,especially embedding dimension and radius).

4.3 Muscular

The first application of recurrence analysis to muscle dynamics was made by Web-ber et al. [23]. In this study, normal human volunteers were asked to sit quietlywhile holding first a 1.4 kg weight (control) and then a 5.1 kg weight (experimen-tal). Electromyogram (EMG) recordings were digitized (1000 Hz) during 60 secondsof control and throughout the experimental phase until the onset of biceps musclefatigue. The continuous data (1 to 6 minutes in duration among 14 subjects) werewindowed into 1.024-second epochs and overlapped by 75% (0.256 sec delay betweenepochs) for recurrence and spectral analyses. By this means it was possible to studythe evolution of the muscle dynamics through imposed state changes by two differentanalytical tools (linear FFT, nonlinear RQA). As expected, spectral center frequency,%recurrence and %determinism all remained stable throughout the light-load controlperiod. With heavy loading, however, %determinism increased at a faster rate thanspectral center frequency decreased. In fact, strict quantitation demonstrated thatRQA was significantly more sensitive than spectral analysis in detecting state changespresent in contracting, fatiguing muscles. It was concluded that physiological changesoccurring during the fatiguing process are nonlinear and multidimensional. Finally,

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Kankannp, et al. [24] exploited the notion that RQA could be used to establishthe stationarity of EMG signals, legitimizing the submission of data to FFT analysiswithout necessitating initial detrending (numerical distorting) of the data.

4.4 Ion channels

The gating characteristics of single ion channels can be studied by patch-clamp record-ings from individual cells or membrane fragments [25]. Quantitative analysis of gatingdata in the first dimension includes distributions of open- and closed-time intervals,open- and closed-time probabilities (Po and Pc), and fractal-time structuring [26]Rothberg et al. [27] have been successful in using two-dimensional distributions ofion channel dwell times, open and closed, to select the best multi-state models mim-icking the kinetics of simulated ion channels.

Although it is commonly assumed that ion channel gating events are independent,permitting stochastic modeling of the Markovian type, Webber and Zbilut (unpub-lished) mused whether there might be deterministic structuring in ion channel gates,generating information remotely analogous to a Morse Code signal. For example, asystem might be broadcasting more information than can be captured in the meanopen or closed times (average number of dots or dashes) which constitutes too severedata filtering and compression. To evaluate this issue, dynamical gating data wereobtained from a colleague who was studying sodium channels in membrane vesiclesprepared from rat brains [28]. The data were digitized at 2000 Hz, low-pass filteredat 200 Hz, and idealized to integers (0 for closed, 1 for open) by setting a numericalthreshold at 50% of the open-channel ionic current. This is illustrated in Figure 9Awhere Po is 0.540. The duration of sequential gating cycles was then computed as thetime from one open state to the next (Fig. 9B). Recurrence analysis proceeded on thistime series over a range of embedding dimensions (1 to 30) before after randomization(Fig. 9C). Interestingly, both native and randomized sequences of ion channel datayielded essentially identical %determinism values over the range of embedding dimen-sions studied, as did %recurrence and entropy computations (not plotted). Whereasthese negative results seem compatible with Markovian requirements of event inde-pendence, they may rather stem from relatively low data sampling rates. We hypoth-esize that ion channel gatings are better modeled as terminal dynamics in which theopening-closing mechanisms are deterministic in nature, but where the dwell timesin the open or closed state are stochastic (latched at one of two singularities). Totest this notion, experiments are planned whereby ion channel data are digitized andfiltered at 10-times higher frequencies (20,000 Hz, 2,000 Hz, respectively). Still, theinherent molecular “programs” to open or close channel proteins may be magnitudesof order faster than the dwell times themselves, rendering experimental verificationvery difficult.

5 Bioinformatic Systems

5.1 Biological Codes

Unlike ion channel gating events, the spatially organized text strings of DNA (4-letteralphabet) and biologically-relevant amino acids (20-letter alphabet) possess knownand meaningful information as delineated by investigators in the fields genetics andbiochemistry. Since these codes direct the primary, secondary, tertiary and quater-nary structures of proteins, one important quest for the new field of bioinformatics

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is to find correlations between encoded information and protein structure using amulti-disciplinary approach [29]. Such knowledge might be very useful, for example,in extracting meaningful information from the human genome for intelligent drugdesign [30]. Peng et al. [31] collapsed DNA strings into purines and pyrmadinesand reported different fractal structuring in exon versus intron regions. Webber andZbilut [32] encoded 9770 DNA bases of the human immunodeficiency virus (HIV)into the first four integers (A=1, C=2, G=3, T=4) and evaluated the data by recur-rence analysis before and after randomization. The results demonstrated significant%determinism and information entropy within a 1000-point epoch window slidingdown the DNA bases. Further work is needed in this field, but it is significant to re-alize that recurrence analysis is applicable for any ordered series, spatial or temporal.The applicability of RQA to several different bioinformatic systems is now addressed.

5.2 InsP3 Receptor

The complete primary nucleotide sequence of inositol 1,4,5-trisphosphate receptorfrom the rat brain was reported by Mignery et al. [33]. The expression of this in-tracellular receptor (2734- or 2749- amino acid sequence) is activated by InsP3, asecond messenger, resulting in the release of calcium ions from intracellular stores.To test for deterministic structuring in the nucleotide sequence, recurrence analysiswas performed on the native and randomized DNA sequences of InsP3 (9852 bases).Recurrence variables were computed within windows of 1000 bases, shifting one basebetween epochs. As shown in Figure 10, randomization of the IsnP3 sequence de-creased both %determinism and information entropy throughout the entire sequence.Since the recurrence analysis was restricted to the first embedding dimension, recur-rence plots of the InsP3 sequence (not shown) resembled diagonal plots generated bysoftware from the Genetics Programming Group (GPG, Milwaukee, WI). Still, RQAhas the advantage of detecting subtle deterministic structures within DNA exons thatmay go unnoticed by visual examination of recurrence plots.

To examine the InsP3 coding sequence in higher dimensions, base distances be-tween identical nucleotides were computed, resulting in four series: 2687 A intervals,2323 C intervals, 2497 G intervals, and 2341 T intervals. Distances were also com-puted on the randomized base sequence, resulting in four control series. RQA wasperformed on the native and randomized intervals over embedding dimensions 1 to10. Figure 11 summarizes the results in terms of recurrence variable %determinism,although very similar plots were found for information entropy. All four plots assumeflattened-U shapes with the minimum %determinism usually occurring at embed-ding dimension 4 or 5. Randomization of the data decreased %determinism morefor nucleotide intervals A and T than intervals C and G. These initial results againunderscore the presence of deterministic structuring in the IsnP3 coding sequence,but deeper interpretation and comprehension must await further study.

5.3 Alpha-Spectrin

The cytoskeleton of the human erythrocyte has a complex anatomical organizationwhich includes a twisted-pair of fibrous protein chains, α- and β-spectrin dimers,just beneath the membrane surface [34]. These proteins impart functional robust-ness to the red blood cells, allowing them to contort in shape, without damage, asthey are forcefully ejected by the heart, propelled through the circulatory system,and squeezed through tortuous capillary pathways. The known primary amino acidsequences of α-spectrin (2429 residues) and β-spectrin (2137 residues) are believed

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%REC %DET ENT DIV TREND

α-spectrinnative 7.091 14.126 0.432 142.9 -0.106random 7.091 13.837 0.393 250 -0.029difference 0 0.289 0.039 107.1 0.077

β-spectrin

native 6.743 13.562 0.417 142.9 -0.437random 6.743 12.915 0.379 250 0.005difference 0 0.647 0.038 107.1 0.442

Table 1: Recurrence quantification analysis of spectrin

to be organized into 20 (α-spectrin) or 18 (β-spectrin) repeating segments, each con-sisting of 106 amino acids. Cartoons of the molecular anatomy of spectrin moleculespicture inflexible domains (106 amino-acid segments) that are serially linked at flexpoints, analogous to coupled box cars in a freight train. One glance at spectrin’s pre-sumed structure would seem to justify a vocabulary shift from “repeating segments”to “recurring segments,” implicating the use of recurrence analysis to validate theenvisaged structure. To do this, the amino acid sequences of α- and β-spectrin wereencoded with integers 1 through 20 and subjected to recurrence analysis at the ortho-graphic level (embedding dimension = 1), only scoring recurrences for exact residuematches (radius = 0). Surprisingly, not only could no 106-amino acid rhythms bedetected in the data but, as shown in Table 1, randomization of the α- and β-spectrinsequences resulted in only minimal reductions in %determinism and information en-tropy. Even the longest recurrent line segments (7 for both α- and β-spectrin), wereonly truncated by 1 to 2 points following randomization, but this gave large increasesin the divergence variable defined as 1000/Lmax (Table 1). The protein sequenceswere then transformed into random walk patterns by assigning each of the 20 aminoacids a unique 18-degree sector on a 360-degree circle. This analysis demonstratedthat the sequence data could be described by random walks of the Brownian-motionvariety (i.e. integrated white noise, 1/f2 process), leading to the interpretation thatdeterministic structuring within α- and β-spectrin, at least in the first dimension, issubtly present but buried within noise. No numerical evidence for the supposed 106amino-acid repeats (recurrences) could be found.

Protein structure-function studies have been performed by substituting physicalproperties of amino acids for the string positions of those amino acids [35]. Thisapproach permits amino acid sequences to be examined from a higher dimensionalperspective. In our case, we substituted hydrophobicity values of Kyte-Doolittle [36]for each amino acid and computed recurrence variables over embedding dimensions 1to 10. Randomized sequences served as controls.

The results of Figure 12 reveal that RQA variables %recurrence, %determinismand entropy generally increase with increases in embedding dimension. Randomiza-tion of the hydrophobicity sequences reduces %recurrence and entropy at embeddingdimensions 3 or 6 and higher, respectively. However, randomization has no apparenteffect on %determinism. Thus randomization of -spectrin at the higher embeddingdimensions decreases the number of recurrent points within and outside of diagonalline structures (no change in %determinism) while altering the distribution of linesegments (decrease in entropy). This explains why entropy, in some cases, can bea more sensitive measure of subtle pattern shifts than %determinism. Results for

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-spectrin were not as robust. It is concluded that the deterministic structuring of-spectrin is definitely a high-dimensional problem possibly relating to specific proteinfoldings, turns, helices, and sheets. Further study is required to elucidate the 106amino-acid repeats by RQA.

5.4 Bacteriophage Lambda

Li [37] reviews the many statistical approaches taken to detect long-range correlationswithin DNA sequences which are attributed to multiple scalings of different character-istic lengths within the same genome. For example, the non-random structure of theentire genome of bacteriophage lambda (48502 nucleotides) has been demonstratedby Karlin and Brendel [38]. These investigators showed that the random-walk plots ofthe native sequence versus the shuffled sequence were dramatically distinct, the effectbeing attributed to the mosaic character of DNA (patches of different composition).Given that global patterns within DNA sequences are real (and interesting), probablythe more important structuring occurs at the local level which must be known to pin-point genes, promoters, open reading frames, introns, etc. . If the quest is to discoverorganizing principles of the genome, and it is [37], recurrence analysis may be anotherrobust tool to digest long DNA sequences (global) within episodic windows (local).

To illustrate this approach, we, too, examined the entire coding region of bacterio-phage lambda, a characteristic prokaryote lacking introns. Recurrence variables werecomputed within a 1000-base window, and recomputed throughout the genome byrepetitively shifting the window forward by one base. As shown in Figure 13, RQAvariables %recurrence and %determinism share similar and tortuous topographieswithin local regions along the genome. This structuring is destroyed by randomiza-tion, an important statistical control verifying deterministic patterning in the nativesequence. Recurrence analysis is too new to give a meaningful interpretation of thepeaks and valleys in the bacteriophage lambda plots, but we posit that the “trans-lation” of spatial fluctuations in recurrence variables constitutes an important futuregoal for genomic research..

5.5 Human Serum Albumin Gene

Another long-standing goal of genomic research is to distinguish coding regions (ex-ons) from non-coding regions (introns) within long-strand DNA sequences of eukary-otes [37]. Buldyrev et al. [39] for example, describe how DNA exons and intronspossess different long-range correlative properties, but the resolution is not sufficientto detect borders between these one-dimensional regions. To see if any topographicaldifferences could be detected by RQA, Webber and Zbilut (unpublished data) exam-ined the 19002 DNA bases of the human serum albumin gene. Like beads on a string,the 15 exons are represented in Figure 14 as black dots, scattered unevenly betweenthe surrounding 16 introns. Although it is tempting to describe this eukaryotic DNAcode as nondeterministic (alternating coding and non-coding regions), %recurrenceand %determinism topographies reveal no obvious correlations with either the exonsor introns. From such results it might rather be speculated that the introns as wellas the exons provide structural or functional roles for DNA, albeit not necessarilyprotein coding.

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6 Other Systems

6.1 Financial Markets

There is a large and growing interest in applying nonlinear analyses to financial andeconomic data [40]. In this light, nonlinear forecasting is becoming important fordiscerning trends in the market [41], but one which may prove as difficult as weatherprediction. It was natural for Webber and Zbilut (unpublished data) to apply RQA tofinancial data, specifically the Dow Jones Industrial Averages (DJIA) as summarizedin Figure 15. The time series data represent the DJIA closing prices from January1, 1970 through April 1, 1997 as acquired from a historical data base accompanyingcommercial financial software (Omega Research, Miami, FL). RQA digestion of thisdata (embedding dimension = 5, epoch = 200 days) resulted in complicated fluctua-tions in %determinism which could be destroyed by prior randomization of the DJIAseries (control). Careful scrutiny of this RQA variable reveals an interesting finding.Before the severe fall in the market by more than 500 index points on Monday, Oc-tober 19, 1987 (Fig. 15A), the %determinism oscillates aperiodically around a gentlyrising mean (Fig. 15B). Following the market crash of 1987, however, %determinismshows a dramatic slope increase (Fig. 15B). Interpretation of such results is not easy,but one conjecture is that increases in %determinism since 1987 reflect the success ofimposed safeguards on financial markets. One such safeguard prohibited automaticbuying and selling by computers whenever the Dow Jones index changed by morethan 50 units in a single day. When this threshold was exceeded, buying and sellinghad to resort to manual methods (e.g. telephone), an effective strategy for slowingthe rate of transactions. In a sense, the degrees of freedom in the market were de-creased, resulting in a more confined and slowed dynamic and, hence, an elevated%determinism from the RQA perspective. Although the DJIA data might easily befit with a simple polynomial equation, effects of the restraining regulations would stillgo unnoticed (insensitive methodology). Some think that the stock market follows arandom walk process, random in the sense that it is not possible for experts to con-sistently beat the average market prices [42], but recurrence analysis may be able todelineate important trends in this very complex system. Examination of the data fromdifferent dimensional perspectives (various embeddings), might yield more importantclues. But buyer beware, to the extent that financial markets are nondeterministic(terminal dynamic), the results will be complicated, the interpretations strained.

6.2 Linguistics

For pedagogical purposes only, Webber and Zbilut [32] used the children’s poem“Green Eggs and Ham” by Dr. Seuss [43] to illustrate the concept of recurrencewithin a string of data. This 812-word poem is constructed with a limited vocabularyof 50 words, which means that words must recur throughout the text. Recurrenceplots in this case were generated by assigning a unique integer (1, 2, 3, ... 26) toeach letter in the 26-letter English alphabet, case insensitive (Aa, Bb, Cc, ... Zz),scoring recurrences at positions of exact matches. The alliteration and rhythm inthe poem could thus be captured by diagonal banding in the recurrence plot and byoscillations in RQA variables (%recurrence, %determinism, etc.). Recurrence plots atthe word level for this poem are shown in Figure 16 before and after randomizationof the word order. As at the orthographic level, the native sequence of words presentswith strong diagonal line structuring (Fig. 12A) which is homogenized (Fig. 12B) byprior shuffling of the words into a nonsensical sequence. RQA results of this poem

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%REC %DET ENT DIV TREND

letter string

native 7.309 35.547 2.055 7.246 -0.087random 7.309 14.012 0.408 200 0.143difference 0 18.535 1.647 192.8 0.097

word string

native 4.827 45.901 2.137 21.739 -2.029random 4.827 8.833 0.284 200 0.870difference 0 37.068 1.853 178.3 2.899

Table 2: Rqa of dr. Seuss’ poem

at the letter and word levels are summarized in Table 2. Since randomization ofletter and word sequences greatly changes RQA variables (%determinism, entropy,divergence), it is concluded that Dr. Seuss’ poem possesses deterministic structure.Randomization cannot affect %recurrence (letter matches letter, word matches word,independent of string position) and randomization does not affect the trend variable(native and randomized strings are both stationary). Of course it is impossible toextract the linguistic meaning of the poem from the recurrence variables, but thesequences of letters and words nevertheless form logical units that can be dissectedmathematically by RQA.

Finally, Webber et al. [44] discuss how RQA might be useful in studying differenttexts, including ancient Hebrew manuscripts (22-letter alphabet). As explained for the“Green Eggs and Ham” poem, recurrence analysis was extended from the orthographiclevel (number of symbols limited by size of alphabet) to the word level (number ofsymbols limited by size of vocabulary). Our impression is that complex texts arebetter distinguished from children’s literature at the word level, not the spelling level.So literary studies could be launched to examine textual data using RQA on nativeand randomized series. It may be that subtle shifts in style between different authorscould be discerned by careful RQA scrutiny, but this challenge remains a serious (andlaborious) project for linguistic research.

7 Conclusions

Recurrence plots were originally devised by Eckmann et al. [6] to detect hiddenrhythms in data derived from complex mathematical systems. Webber and Zbilut[8–10] realized that recurrence analysis was ideally suited for physiological data char-acterized by nonstationarity and noise. They introduced the computation of specificrecurrence variables and showed how RQA could be used to detect dynamical statechanges occurring within physiological systems. Hopefully, RQA is a welcome additionto the armamentaria of other time-series analyses, contributing a higher-dimensionalperspective and nonlinear rendering. This has special significance for biological sys-tems which are notoriously nonlinear, high dimensional (many operating variables),and noisy [45]. Due to technicalities in setting recurrence parameters, RQA remainsan uncalibrated technique. Successful implementation requires a thorough under-standing of one’s system under study as rendered in recurrence parameter space.However, one excellent “calibration” procedure is to study a system both before andafter randomization using identical parameter settings, as exampled throughout this

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paper. Differences in %determinism values, for example, can be attributed to thepresence of structure in the native temporal dynamic or original spatial sequence.

To conclude, the growing impression is that RQA is applicable to numerous dy-namical systems, irrespective of whether they be time-varying (e.g. heart beat inter-vals)or space-varying (e.g. DNA sequences). An understanding of recurrence analysisheightens one’s awareness to the shared properties and commonality of various dy-namical systems in seemingly unrelated disciplines. For this reason, divergent fields ofstudy might be unified by appropriate shifts in vocabulary. Still, the scope across dis-ciplines is far too broad for any one investigator or team to master, but the door hasbeen opened for serious investigation of unique data sets from the RQA perspective.

Acknowledgments

Appreciation and thanks are extended to Dr. Samuel Cukierman for providing ionchannel data, to Dr. Leslie Fung for providing coding sequences of α- and β-spectrin,and to Omega Research, Inc. for providing a complementary copy of their software,“Wall Street Analyst Deluxe,” from which the 27-year historical data of Dow JonesIndustrial Averages were obtained in ASCII format.

Recurrence Software Availability

The most recent RQA software is available for downloading from the author’s URLaddress 1. These C programs currently run under the DOS environment and havebeen extensively tested and debugged for accuracy. RQA implementation is ratherstraightforward, but RQA interpretation is under careful scrutiny by many investiga-tors worldwide.

1http://homepages.luc.edu/˜cwebber/

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Legends to the figures

Fig. 1 Recurrence plots of sine wavesA: Idealized 1-Hz sine wave (100 Hz digitization). B: Recurrence plot of idealized

sine wave. C: Noisy 1-Hz sine wave (100 Hz digitization). D: Recurrence plot of noisysine wave. Additive noise was uniformly distributed with a maximum magnitude of10% peak-to-peak sine wave amplitude. RQA parameters: Euclidean norm; embed-ding dimension = 1; delay = 1; radius = 0 (B) or 1 (D); line = 2; epoch = 500 points.

Fig. 2 Henon mapsA: Periodic attractor. B: Reconstruction of periodic attractor by method of time

delays. C: Destruction of periodic attractor by randomization of x variable. D:Chaotic attractor. E: Reconstruction of chaotic attractor by method of time delays.E: Destruction of chaotic attractor by randomization of x variable. Epoch = 999points per plot.

Fig. 3 Recurrence plots of Henon x variableA: Periodic attractor. B: Chaotic attractor. RQA parameters: Euclidean norm;

embedding dimension = 3; delay = 1; radius = 0 (A) or 1 (B); line = 2; epoch = 200points.

Fig. 4 Three-phase respiration in the feline phrenic dischargeVariables (top to bottom): phrenic electroneurogram (500 Hz digitization); inte-

grated phrenic activity (100 ms time constant); RQA mean vector distance variable;transistor-transistor-logic (TTL) pulse triggered from the integrated phrenic. Threeof the four variables are expressed in relative units (ru). RQA parameters: Euclideannorm; embedding dimension = 10; delay = 1; radius = 1; line = 2; epoch = 400points; #epochs = 5001; epoch shift = 1 point.

Fig. 5 Terminal dynamics of rodent breathingA: Intrapleural pressure recording contaminated by cardiac motion (physiological

noise) within the thoracic space (200 Hz digitization). B: Recurrence plot of intrapleu-ral pressure on same time axis identifying regions of singularity. RQA parameters:minimum norm; embedding dimension = 10; delay = 1; radius = 1; line = 2; epoch= 480 points.

Fig. 6 Terminal dynamics of normal human electrocardiogram (1000Hz digitization)

A: Cardiac activation (PQRS waves) and repolarization (T wave) define PT inter-vals which alternate with TP pauses between beats (bars). B: Variability in cardiacPT, TP and PP intervals for 512 cardiac cycles (PP = PT + TP).

Fig. 7 Frequency peaks in PP intervals of the normal human electro-

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cardiogramA: High frequency oscillation (HFO, parasympathetic inputs), low frequency oscil-

lations (LFO, sympathetic inputs), and very low frequency peak (VLFO, other inputs)computed by spectral analysis of cardiac PP intervals. B: Correlation of HFO areawith the standard deviation of PP intervals in 14 normal subjects (r = correlationcoefficient for 13 points, excluding single outlier - open circle). Fast Fourier transform(FFT): 512-point FFT; linear detrended; DC component removed; Hanning window(10% cosine tapering of side lobes); spectral curves smoothed with low-pass digitalfilter.

Fig. 8 High dimensionality of human cardiac dynamicsCorrelation of RQA %recurrence (A) and %determinism (B) with the standard

deviation of PP intervals in 14 normal subjects (r = correlation coefficient). RQAparameters: Euclidean norm; embedding dimension = 13; delay = 1; radius = 0.2;line = 3; epoch = 500 points.

Fig.9 Stochasticity of ion-channel gatingA: Open (current > T) and closing (current <= T) transitions of a sodium channel

reconstituted in an artificial membrane. B: Sequence of gating cycle durations fromone open state to the next (n = 918 cycles). C: Recurrence analysis of gate cycle du-rations in their native sequence (N) and randomized sequence (R). RQA parameters:Euclidean norm; embedding dimension = 1 to 30; delay = 1; radius = 2; line = 2;epoch = 889 points; #epochs = 30.

Fig.10 Recurrence analysis of inositol 1,4,5-trisphosphate receptor nu-cleotides from the rat brain (n = 9852 nucleotides)

RQA %determinism (A) and RQA information entropy (B) computed within a1000-point window sliding down the entire DNA code before (N) and after (R) ran-domization of the nucleotide sequence. RQA parameters: Euclidean norm; embeddingdimension = 1; delay = 1; radius = 0.0; line = 2; epoch = 1000 points; #epochs =8853; epoch shift = 1 point.

Fig.11 Recurrence analysis of inter-base distances of inositol 1,4,5-triphosphate receptor nucleotides from the rat brain at different embed-ding dimensions

A: 2687 base-A intervals. B: 2323 base C intervals. C: 2497 base G intervals. D:2341 base T intervals. Computation are shown before (N) and after (R) randomizationof the nucleotide intervals. RQA parameters: Euclidean norm; embedding dimension= 1 to 10; delay = 1; radius = 1; line = 2; epoch = 2658 (A), 2294 (C), 2468 (G), or2312 (T) points; #epochs = 10 per base interval.

Fig.12 Recurrence analysis of -spectrin amino acids encoded for hy-drophobicity (n = 2429 residues)

RQA %recurrence (A), %determinism (B) and information entropy (C) as func-tions of embedding dimension for native (N) and randomized (R) amino-acid se-quences. RQA parameters: maximum norm; embedding dimension = 1 to 10; delay

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= 1; radius = 1; line = 2; epoch = 2420 points; #epochs = 10 N and 10 R.

Fig.13 Recurrence analysis of the entire genome of bacteriophagelambda (n = 48502 nucleotides)

RQA %recurrence (A) and %determinism (B) computed within a 1000-point win-dow sliding down the entire DNA code before (N) and after (R) randomization of thenucleotide sequence. RQA parameters: Euclidean norm; embedding dimension = 1;delay = 1; radius = 0.0; line = 2; epoch = 1000 points; #epochs = 47503; epoch shift= 1 point.

Fig.14 Recurrence analysis of the human albumin gene (n = 19002nucleotides)

RQA %recurrence (A) and %determinism (B) computed within a 1000-point win-dow sliding down the entire DNA code. Centers of coding exon regions (C) are markedby dots and vertical bands which alternate with non-coding intron regions. RQA pa-rameters: Euclidean norm; embedding dimension = 1; delay = 1; radius = 0; line =2; epoch = 1000 points; #epochs = 18003; epoch shift = 1 point.

Fig.15 Recurrence analysis of the Dow Jones Industrial AverageA: Index data for 7085 trading days. B: RQA %determinism computed within a

365-day epoch and embedded at 5 days before (N) and after (R) randomization ofthe financial data. Correlation coefficients are specified prior to and after the marketcrash of 1987, and for the entire randomized data (r = correlation coefficient). RQAparameters: Euclidean norm; embedding dimension = 5; delay = 1; radius = 0.1; line= 2; epoch = 200 points; #epochs = 6886; epoch shift = 1point.

Fig.16 Recurrence plots of Dr. Seuss’ poem ”Green Eggs and Ham.”The fifty vocabulary words of this 812-word poem were sequentially encoded into

integers 1 to 50 from which recurrence plots were generated before (A) and afterrandomization of the word order. RQA parameters: Euclidean norm; embeddingdimension = 1; delay = 1; radius = 1; line = 2; epoch = 812 points; #epochs = 1.

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Figures:

Fig. 1 Fig. 2

Fig. 3 Fig. 4

Fig. 5 Fig. 6

Fig. 7 Fig. 8

Fig. 9 Fig. 10

Fig. 11 Fig. 12

Fig. 13 Fig. 14

Fig. 15 Fig. 16