Nonlinear Dynamics, Fractals, and Puinhoop Theory for Clinicians

Implications for Neuroautonomic Heart Rate Control ter Health and Disease

I. Basic Concepts

Clinicians and basic investigators are increasingly aware of the remarkable upsurge of rente ter nonlinear dynamics, the branch of the sciences widely referred to spil ",puinhoop theory.", Those attempting to evaluate the biomedical relevance of this field confront a daunting array of terms and concepts, such spil nonlinearity, fractals, periodic oscillations, bifurcations and complexity, spil well spil puinhoop (1-4). Therefore, the present discussion provides an introduction to some key aspects of nonlinear dynamics, with a particular emphasis on heart rate control. A major challenge is ter making thesis concepts accessible, not only to basic and clinical investigators, but to medical and graduate students at a formative stage of their training.

A. Introduction: The Concept of a Time Series

To appreciate the general clinical relevance of dynamics to the heartbeat, consider the following common problem. What is the best way to compare a sequence of measurements obtained from two subjects, or from one individual or experimental proces under different conditions? Conventionally, clinicians and investigators rely primarily on a comparison of means using suitable statistical tests. However, the limitations of such traditional analyses become apparent when evaluating the gegevens te Fig. 1, displaying sinus rhythm heart rate plots collected from a healthy subject and one with congestive heart failure. Recording the instantaneous signal from any system overheen a continuous observation period generates a time series. What is noteworthy ter this example is that thesis two time series have almost identical means and variances, suggesting no clinically relevant differences. Yet, visual inspection indicates that the two sequences of gegevens display a markedly different organization. The healthy heartbeat trace shows a ingewikkeld, ",noisy", type of variability, whereas the gegevens set from the patient with heart failure exposes periodic oscillations te heart rate repeating about 1 cycle/minute (

.02 Hz). Time series analysis is worried with quantifying the order of gegevens points, nonlinear dynamics provides a deeper understanding of the mechanisms of patterns and differences such spil those te Fig. 1.

Figure 1. Two heart rate time series, one from a healthy subject (top) and the other from a patient with severe congestive heart failure (CHF) (middle) have almost identical means and variances (bottom), yet very different dynamics. Note that according to classical physiological paradigms based on homeostasis, neuroautonomic control systems should be designed to stoom out noise and lodge down to a onveranderlijk equilibrium-like state. However, the healthy heartbeat displays very ingewikkeld, evidently unpredictable fluctuations even under steady-state conditions. Ter tegenstelling, the heart rate pattern from the subject with heart failure shows slow, periodic oscillations that correlate with Cheyne-Stokes breathing.

B. Linear versus Nonlinear Systems

Ter linear systems, the magnitude of the output (y) is managed by that of the input (x) according to ordinary equations ter the familiar form y=mx+b. A well-known example of such a relationship is Ohm’s law: V=IR where the voltage (V) ter a circuit will vary linearly with current (I), provided the resistance (R) is held onveranderlijk. Two central features of linear systems are proportionality and superposition. Proportionality means that the output bears a straightline relationship to the input. Superposition refers to the fact that the behavior of linear systems composed of numerous components can be fully understood and predicted by dissecting out thesis components and figuring out their individual input-output relationships. The overall output will simply be a summation of thesis constituent parts. The components of a linear system literally ",add up", – there are no surprises or anomalous behaviors.

Ter tegenstelling, even ordinary nonlinear systems crack the principles of proportionality and superposition. An example of a deceptively ingewikkeld nonlinear equation is y = ax (1-x), referred to spil the logistic equation te population biology (Five). The nonlinearity of this equation, which describes a parabola, arises from the quadratic (x Two ) term. Switches te the output spil a function of sequential time steps can be readily plotted by a terugkoppeling proces ter which the current value of the output becomes the next value of the input, and so on. Iteration of the simple-in-form logistic equation exposes dynamics that are extraordinarily ingewikkeld, depending on the value of the single parameter, a, the same equation can generate constant states, regular oscillations, or very erratic behavior (Four, Five). Thus, for nonlinear systems, proportionality does not hold: puny switches can have dramatic and unanticipated effects. An added complication is that nonlinear systems composed of numerous subunits cannot be understood by analyzing thesis components individually. This reductionist strategy fails because the components of a nonlinear network interact, i.e., they are coupled. Examples include the “cross-talk” of pacemaker cells ter the heart or neurons te the brain. Their nonlinear coupling generates behaviors that defy explanation using traditional (linear) models (Fig. Two). Spil a result, they may exhibit behavior that is characteristic of nonlinear systems, such spil self-sustained, periodic swings (e.g., ventricular tachycardia) (6, 7), bruusk switches (e.g., unexpected onset of a seizure) (8) and, possibly, puinhoop (see below). Table 1 gives a more finish, but not exhaustive list of nonlinear mechanisms with potential relevance to biology and medicine (1-4, 6-14).

Figure Two. Examples of nonlinear dynamics of the heartbeat. Panels (a-c) are from subjects with obstructive sleep apnea syndrome. Panels (d and e) are from healthy subjects at high altitude (

A related and noteworthy property of nonlinear dynamics is referred to spil universality (1, Four). Remarkably, nonlinear systems that show up to be very different ter their specific details may exhibit certain common patterns of response. For example, nonlinear systems may switch te a unexpected, discontinuous style. One significant and universal class of opeens, nonlinear transitions is called a bifurcation (1, 9). This term describes situations where a very puny increase or decrease te the value of some parameter controlling the system causes it to switch abruptly from one type of behavior to another. For example, the output of the same system may abruptly go from being insanely irregular to a very periodic, or vice versa. A universal class of bifurcations occurring te a broad multiplicity of nonlinear systems is the unexpected appearance of regular oscillations that alternate inbetween two values (15). This type of dynamic may underlie a diversity of alternans patterns te cardiovascular dysfunction. A familiar example is the beat-to-beat alternation ter QRS axis and amplitude seen te some cases of cardiac tamponade (16). This zuigeling of electrical alternans is related to the back and forward swinging movement of the heart within the pericardial effusion. Numerous other examples of alternans ter perturbed cardiac physiology have bot described, including ST-T alternans which may precede ventricular fibrillation (17), and pulsus alternans during heart failure.

C. Puinhoop

Albeit the concentrate of much latest attention, puinhoop vanaf se actually comprises only one specific subtype of nonlinear dynamics. Prior to the work of the renowned French mathematician, Henri Poincaré,, ter the early 1900s, science wasgoed predominated by the seemingly inviolable tenet that the behavior of systems for which one could write out explicit equations (e.g., the solar system) should be, te principle, fully predictable for all future times (Eighteen). What Poincaré, discovered (and what wasgoed more recently rediscovered) is that a ingewikkeld type of variability can arise from the operation of even the simplest nonlinear system, such spil that governed by the logistic equation mentioned earlier. Because the equations of maneuverability which generate such erratic, and evidently unpredictable behavior do not contain any random terms, this mechanism is now referred to spil deterministic puinhoop (1, Four). The colloquial use of the term puinhoop – to describe unfettered randomness, usually with catastrophic implications – is fairly different from this specialized usage.

The extent to which puinhoop relates to physiological dynamics is a subject of active investigation and some controversy. At very first it wasgoed widely assumed that chaotic fluctuations were produced by pathological systems such spil cardiac electrical activity during atrial or ventricular fibrillation (Nineteen). However, this initial presumption has bot challenged (20) and the weight of current evidence does not support the view that the irregular ventricular response te atrial fibrillation or that ventricular fibrillation itself represents deterministic cardiac puinhoop (21). Further, there is no persuading evidence that other arrhythmias sometimes labeled ",chaotic,", such spil multifocal atrial tachycardia, meet the technical criteria for nonlinear puinhoop. An alternative hypothesis (22) is that the subtle but sophisticated heart rate fluctuations observed during normal sinus rhythm te healthy subjects, even at surplus, are due ter part to deterministic puinhoop, and that a multiplicity of pathologies, such spil congestive heart failure syndromes, may involve a paradoxical decrease ter this type of nonlinear variability (Fig. 1). Because the mathematical algorithms designed for detecting puinhoop are not reliably applied to nonstationary, relatively brief and often noisy gegevens sets obtained from most clinical and physiological studies, the intriguing question of the role, if any, of puinhoop ter physiology or pathology remains unresolved (22-28).

D. Fractal Anatomy

The term fractal is a geometric concept related to, but not synonymous with puinhoop (29, 30). Classical geometric forms are sleek and regular and have oprecht dimensions (1,Two, and Trio, for line, surface, and volume respectively). Ter tegenstelling, fractals are very irregular and have non-integer, or fractional, dimensions. Consider a fractal line. Unlike a slick Euclidean line, a fractal line, which has a dimension inbetween 1 and Two, is wrinkly and irregular. Examination of thesis wrinkles with the low-power objectief of a microscope, exposes smaller wrinkles on the larger ones. Further magnification shows yet smaller wrinkles, and so on. A fractal, then, is an object composed of subunits (and sub-subunits) that resemble the larger scale structure, a property known spil self-similarity (Fig. Three). A broad diversity of natural shapes share this internal look-alike property, including branching trees and coral formations, wrinkly coastlines, and ragged mountain ranges. A number of cardiopulmonary structures also have a fractal-like appearance (Two, Three, 30, 31). Examples of self-similar anatomies include the arterial and venous trees, the branching of certain cardiac muscle bundles, spil well spil the ramifying tracheobronchial tree and His-Purkinje network.

Figure Three. Left, schematic of a tree-like fractal has self-similar branchings such that the petite scale (magnified) structure resembles the large scale form. Right, a fractal process such spil heart rate regulation generates fluctuations on different time scales (temporal ",magnifications",) that are statistically self-similar. (Adapted from Goldberger Hoewel. Non-linear dynamics for clinicians: puinhoop theory, fractals, and complexity at the bedside. Lancet 1996,347:1312-1314.)

From a mechanistic viewpoint, thesis self-similar cardiopulmonary structures all serve a common physiologic function: rapid and efficient vervoer overheen a ingewikkeld, spatially distributed system. Te the case of the ventricular electrical conduction system, the quantity transported is the electrical stimulus regulating the timing of cardiac spasm (31). For the vasculature, fractal branchings provide a rich, redundant network for distribution of O2 and nutrients and for the collection of CO2 and other metabolic waste products. The fractal tracheo-bronchial tree provides an enormous surface area for exchange of gases at the vascular-alveolar interface, coupling pulmonary and cardiac function (30). Fractal geometry also underlies other significant aspects of cardiac function. Peskin and McQueen (32) have elegantly shown how fractal organization of connective tissue te the aortic valve leaflets relates to the efficient distribution of mechanical compels. A diversity of other organ systems contain fractal structures that serve functions related to information distribution (jumpy system), nutrient absorption (bowel), spil well collection and vervoer (biliary duct system, renal calyces) (Two, Three, 30).

E. Scaling te Health and its Breakdown with Disease

An significant extension of the fractal concept wasgoed the recognition that it applies not just to irregular geometric or anatomic forms that lack a characteristic (single) scale of length, but also to ingewikkeld processes that lack a single scale of time (29, 33). Fractal processes generate irregular fluctuations on numerous time scales, analogous to fractal objects that have wrinkly structure on different length scales. Furthermore, such temporal variability is statistically self-similar. A crude, qualitative appreciation for the self-similar nature of fractal processes can be obtained by plotting the time series te question at different ",magnifications,", i.e., different temporal resolutions. For example, Fig. Three plots the time series of heart rate from a healthy subject on three different scales. All three graphs have an irregular (",wrinkly",) appearance, reminiscent of a coastline or mountain range. The irregularity seen on different scales is not visually distinguishable, an observation confirmed by statistical analysis (34, 35). The roughness of thesis time series, therefore, possesses a self-similar (scale-invariant) property.

Since scale-invariance emerges to be is a general mechanism underlying many physiological structures and functions, one can adapt fresh quantitative devices derived from fractal mathematics for measuring healthy variability. Complicated fluctuations with the statistical properties of fractals have not only bot described for heart rate variability, but also for fluctuations te respiration (36), systemic blood pressure (37), human gait (38) and white blood cell counts (39), spil well spil certain ion channel kinetics (Three). Furthermore, if scale-invariance is a central organizing principle of physiological structure and function, wij can make a general, but potentially useful prediction about what might toebijten when thesis systems are severely perturbed. If a functional system is self-organized te such a way that it does not have a characteristic scale of length or time, a reasonable anticipation would be a breakdown of scale-free structure or dynamics with pathology (35). How does a system behave after such a pathologic transformation? The antithesis of a scale-free (fractal) system (i.e., one with numerous scales) is one that is predominated by a single frequency or scale. A system that has only one superior scale becomes especially effortless to recognize and characterize because such a system is by definition periodic – it repeats its behavior ter a very predictable (regular) pattern (Fig. Four). The theory underlying this prediction may account for a clinical paradox: namely, that a broad range of illnesses are associated with markedly periodic (regular) behavior even however the disease states themselves are commonly termed ",dis-orders", (39).

Figure Four. Breakdown of a fractal physiological control mechanism can lead ultimately either to a very periodic output predominated by a single scale or to uncorrelated randomness. The top heart rate time series is from a healthy subject, bottom left is from a subject with heart failure, and bottom right from a subject with atrial fibrillation. (Adapted from Goldberger Nu. Non-linear dynamics for clinicians: puinhoop theory, fractals, and complexity at the bedside. Lancet 1996,347:1312-1314.)

II. Fractal Scaling of the Heartbeat te Health and its Breakdown with Disease

A. Periodic Disease and the Loss of Fractal Complexity

The appearance of very periodic dynamics ter many disease states is one of the most compelling examples of the notion of complexity loss te disease (40). Complexity here refers specifically to a multiscale, fractal-type of variability te structure or function. Many disease states are marked by less ingewikkeld dynamics than those observed under healthy conditions. This de-complexification of systems with disease emerges to be a common feature of many pathologies, spil well spil of aging (40). When physiologic systems become less ingewikkeld, their information content is degraded (41). Spil a result, they are less adaptable and less able to cope with the exigencies of a permanently switching environment. To generate information, a system voorwaarde be capable of behaving te an unpredictable style (Two, 42). Te tegenstelling, a very predictable, regular output (i.e., a sine wave) is information-poor since it monotonously repeats its activity. (The most extreme example of complexity loss would be the total absence of variability – a straightline output.)

Quantitative assessment of periodic oscillations can be obtained by analyzing the time series of rente with a diversity of standard mathematical instruments. For systems producing a very periodic output, the most widely used methods are based on spectral analysis. Remarkably, the time series of many severely pathologic systems have a almost sinusoidal appearance, the spectrum shows a superior peak at this characteristic frequency. An example is the heart rate variability sometimes observed ter patients with severe congestive heart failure (Fig. 1) (43, 44) or with fetal distress syndromes (45). Te tegenstelling, systems with a fractal output (such spil normal heart rate variability) voorstelling a type of broadband spectrum which includes many different frequencies (scales).

Very likely the very first explicit description of the concept of periodic diseases wasgoed provided overheen 30 years ago by Dr. Hobart Reimann (46). He called attention to a number of conditions te which the disease process itself could be shown to flare or recur on a regular fundament of days to months, ranging from certain forms of arthritis to some mental illnesses and hereditary diseases, such spil familial Mediterranean fever. Ter the late 1970s, Michael Mackey and Leon Glass (47, 48) helped to rekindle rente te this dormant field when they introduced the term dynamical disease to encompass the types of periodic syndrome Reimann had catalogued, spil well spil irregular dynamics thought possibly to represent deterministic puinhoop.

Reimann’s original list wasgoed premised on the assumption that periodic conditions were somewhat unique, and even idiosyncratic, ter clinical medicine. However, to the extent that healthy function is often characterized by a multi-scale fractal complexity, wij would anticipate that the emergence of single-scale (i.e., non-fractal) states might be considerably more common, if not ubiquitous, ter pathophysiology. Indeed, a latest survey of the literature (49) indicates that Reimann, rather than compiling a list of the exceptional, wasgoed more likely sampling a widespread, even generic manifestation of the dynamics of disease. From the most general perspective, the practice of bedside diagnosis itself would be unlikely without the loss of complexity and the emergence of pathologic periodicities. To a large extent, it is thesis periodicities and highly-structured patterns – the breakdown of multi-scale fractal complexity under pathologic conditions – that permit clinicians to identify and classify many pathologic features of their patients. Familiar examples include periodic tremors te neurologic conditions, AV Wenckebach patterns, the ",sine-wave", ECG pattern te hyperkalemia, manic-depressive alterations, and cyclic breathing patterns te heart failure.

B. Irregular Dynamics and the Breakdown of Fractal Mechanisms

While fractals are irregular, not all irregular structures or erratic time series are fractal. A key feature of the class of fractals seen ter biology is a distinctive type of long-range order. This property generates correlations that extend overheen many scales of space or time. For complicated processes, fractal long-range correlations are the mechanism underlying a ",memory", effect, the value of some variable, e.g., heart rate at a particular time, is related not just to instantaneously preceding values, but to fluctuations te the remote past. Certain pathologies are marked by a breakdown of this long-range organization property, producing an uncorrelated randomness similar to ",white noise.", An example is the erratic ventricular response te atrial fibrillation overheen relatively brief time scales. Peng et nu. (50) have recently described a ordinary algorithm for quantifying the breakdown of long-range (fractal) correlations ter physiological time series.

C. Future Applications

Practical applications of nonlinear dynamics are likely within the next few years. Most likely the very first bedside implementations will be te physiological monitoring. A number of indices derived from puinhoop theory have shown promise te forecasting subjects at high risk of electrophysiologic or hemodynamic instability, including

  • automated detection of the onset and end of pathologic low frequency (<,.Ten Hz) heart rate oscillations (Figs 1, Two and Four) (43, 52-59),
  • detection of subtle ST-T alternans (17, 51),
  • detection of a breakdown te fractal scaling with disease and aging (43), and
  • quantification of differences or switches te the nonlinear complexity or dimension of a physiological time series (59, 60).

Ter addition to thesis diagnostic applications, perhaps the most titillating prospects are related to novel therapeutic interventions. An significant latest finding is that certain mathematical or physical systems with complicated dynamics can be managed by decently timed outer stimuli: chaotic dynamics can be made more regular (puinhoop control) and periodic ones can be made chaotic (puinhoop anti-control) (61-63). One proposal, based on the earlier notion that certain arrhythmias, particularly ventricular fibrillation, represent cardiac puinhoop, is to develop puinhoop control algorithms to electrically rhythm the heart ritme back to sinus rhythm (63). A more latest proposal is to use puinhoop anti-control protocols to treat or to prevent cardiac arrhythmias or epilepsy based on the hypothesis that restoration of a kleintje chaotic-like variability may actually be advantageous (62).

Puinhoop theory also holds promise for illuminating a number of major problems ter contemporary physiology and molecular biology. Nonlinear wave mechanisms may underlie certain types of reentrant ventricular tachyarrhythmias (6, 7). Appreciation for the rich nonlinearity of physiological systems may have relevance for modeling enormously complicated signal-transduction cascades involved, for example, ter neuroautonomic dynamics te which interactions and ",cross-talk", occur overheen a broad range of temporal and spatial scales, spil well spil for understanding ingewikkeld pharmacologic effects. Fractal analysis of long DNA sequences has recently exposed that non-coding, but not coding sequences wield long-range correlations among nucleotides (64). This finding has implications for possible functions of introns spil well spil for understanding molecular evolution (65) and developing fresh methods for distinguishing coding from non-coding sections of long DNA sequences (66). Findings from nonlinear dynamics have also challenged conventional mechanisms of physiological control based on classical homeostasis, which presumes that healthy systems seek to attain a onveranderlijk constant state. Ter tegenstelling, nonlinear systems with fractal dynamics, such spil the neuroautonomic mechanisms regulating heart rate variability, behave spil if they were driven far from equilibrium under basal conditions. This zuigeling of sophisticated variability, rather than a regular homeostatic stable state, emerges to define the free-running function of many biological systems (Fig. 1) (Two, 39). Eventually, a fundamental methodologic principle underlying thesis fresh applications and interpretations is the importance of analyzing continuously sampled variations ter physiological output, such spil heart rate, and not simply relying on averaged values or measures of variance. Dynamical analysis demonstrates that there is often hidden information ter physiological time series and that certain fluctuations previously considered ",noise", actually represent significant signals (67-70).

Table 1 Nonlinear Mechanisms and Manifestations

  1. Ineens Switches (Ten) Bifurcations (1, 9)

Intermittency / Bursting (9, Ten)

Bistability, Multistability (11)

Phase transitions

  • Hysteresis (12)
  • Nonlinear Oscillations (1, Two) Limit cycles
  • Pacemaker annihilation

  • Fractals (Two, Trio) Scale-invariance
  • Diffusion limited aggregation

  • Alternans (9)
  • Nonlinear flaps: spiral, scroll (6, 7)
  • Complicated periodic cycles, quasiperiodicities (13)
  • Stochastic resonance (14)
  • Time irreversibility
  • Puinhoop
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