Proceedings of the American Thoracic Society Email Content Delivery
HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS
QUICK SEARCH:   [advanced]


     

The Proceedings of the American Thoracic Society 3:467-471 (2006)
© 2006 The American Thoracic Society
doi: 10.1513/pats.200603-028MS
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Similar articles in this journal
Right arrow Similar articles in PubMed
Right arrow Alert me to new issues of the journal
Right arrow Download to citation manager
Right arrow reprints & permissions
Citing Articles
Right arrow Citing Articles via HighWire
Right arrow Citing Articles via Google Scholar
Google Scholar
Right arrow Articles by Goldberger, A. L.
Right arrow Search for Related Content
PubMed
Right arrow PubMed Citation
Right arrow Articles by Goldberger, A. L.

Giles F. Filley Lecture. Complex Systems

Ary L. Goldberger

Cardiology Division, and Margret and H.A. Rey Institute for Nonlinear Dynamics in Medicine, Department of Medicine, Beth Israel Deaconess Medical Center, Boston, Massachusetts

Correspondence and requests for reprints should be addressed to Ary L. Goldberger, M.D., Cardiology Division, Beth Israel Deaconess Medical Center, 330 Brookline Avenue, GZ-435, Boston, MA 02215. E-mail: agoldber{at}bidmc.harvard.edu

ABSTRACT

Physiologic systems in health and disease display an extraordinary range of temporal behaviors and structural patterns that defy understanding based on linear constructs, reductionist strategies, and classical homeostasis. Application of concepts and computational tools derived from the contemporary study of complex systems, including nonlinear dynamics, fractals and "chaos theory," is having an increasing impact on biology and medicine. This presentation provides a brief overview of an emerging area of biomedical research, including recent applications to cardiopulmonary medicine and chronic obstructive lung disease.

Key Words: chaos theory • chronic obstructive pulmonary disease • fractal • nonlinear dynamics • sleep apnea

Investigators and clinicians attempting to evaluate the potential applications of complex systems analysis are faced with unfamiliar terms and concepts, including "nonlinear dynamics," "fractals," "periodic oscillations," "bifurcations," as well as "chaos" (1–4). This presentation attempts to provide a brief overview of some key aspects of complex systems, with an emphasis on cardiopulmonary applications.

BASIC CONCEPTS OF COMPLEX SYSTEMS

The relevance of nonlinear dynamics to bedside cardiopulmonary medicine is illustrated in Figure 1, which presents 15-min heart rate time series from two subjects, both in sinus rhythm. One is from a healthy individual; the other from a patient during episodes of severe obstructive sleep apnea. The two datasets have nearly identical mean values and variances and, therefore, escape statistical distinction based on conventional comparisons. However, visual inspection of the raw time series (Figure 2) reveals dramatic differences in the temporal structure of the original data. The time series from the healthy subject reveals a complex pattern of nonstationary fluctuations. In contrast, the heart rate dataset from the subjects with sleep apnea shows a much more predictable pattern with a characteristic time scale defined by prominent, low-frequency oscillations at about 0.03 Hz. Both the complex behavior in the healthy case and the sustained oscillations in the pathologic one suggest the presence of nonlinear mechanisms.


Figure 1
View larger version (6K):
[in this window]
[in a new window]
  		Figure 1. Conventional statistical summary of heart rate dynamics in two subjects over 15 min showing nearly identical mean values and SDs (error bars). One subject is healthy; the other was having episodes of severe obstructive sleep apnea. bpm = beats per minute.

 

Figure 2
View larger version (18K):
[in this window]
[in a new window]
  		Figure 2. Original ("raw") heart rate time series from the two subjects (one healthy, one during obstructive sleep apnea) whose data are summarized in Figure 1. Note the marked differences in the underlying dynamics despite the essentially identical mean values and SDs. The healthy dynamics (top) show the more complex pattern of variability, in contrast to the relatively periodic temporal structure of the sleep apnea dataset (bottom).

 
LINEAR VERSUS NONLINEAR MECHANISMS

Two key features of linear systems are proportionality and superposition. Proportionality indicates that the output bears a straight-line relationship to the input. Superposition describes the fact that the output of linear systems composed of multiple components can be fully understood and predicted by the reductionist strategy of parsing out these components and defining their individual input–output relationships. The overall system behavior will be a summation of its constituent parts. The components of a linear system literally "add up."

In contrast, even the simplest nonlinear systems contravene the principles of proportionality and superposition. A well-studied example of a deceptively complex nonlinear equation is y = ax (1 – x), referred to as the logistic equation in population biology (5). The nonlinearity of this equation, which describes a parabola, arises from the x2 term. Changes in the output as a function of sequential time increments can be tracked by a feedback procedure in which the current value of the output becomes the next value of the input, and so on. Iteration of the apparently simple logistic equation reveals dynamics that are extraordinarily complex. Depending only on the value of the parameter, a, the same equation can generate a single steady state, sustained periodic oscillations, or highly erratic fluctuations (5, 6). Thus, for nonlinear systems, proportionality does not hold. Small changes can have dramatic and unanticipated consequences, sometimes referred to as the "butterfly effect."

A related complication is that nonlinear systems composed of multiple subunits cannot be understood by analyzing these components individually and treating the global system in a modularized way (1, 4). The superposition paradigm fails because the components of a nonlinear system interact. Examples include the "cross-talk" of pacemaker cells in the heart and of neurons in the respiratory control center of the brain (7). Their nonlinear coupling generates behaviors that cannot be explained using traditional models. Instead, these systems exhibit behaviors that are markers of nonlinear mechanisms, such as self-sustained oscillations (e.g., periodic cardiopulmonary waves in central and obstructive sleep apnea syndromes; Figure 2, bottom), abrupt changes (e.g., sudden onset of an apneic episode or cardiac arrhythmia), and irregular dynamics referred to as "deterministic chaos" (1, 5, 6). In the parlance of complex systems, the term "emergence" is used to refer to the often abrupt appearance of qualitatively unanticipated global patterns and behaviors resulting from local interactions. Table 1 gives a more complete, but by no means exhaustive, inventory of nonlinear mechanisms and phenomena with potential relevance to biology and medicine.


View this table:
[in this window]
[in a new window]
  		TABLE 1. NONLINEAR/COMPLEXITY-RELATED MECHANISMS AND PHENOMENA IN PHYSIOLOGY: A SHORT LIST

 
A fundamental underpinning of nonlinear dynamics is referred to as universality (1, 3, 4). Surprisingly, nonlinear systems that differ in their specific details may exhibit certain common patterns of response. For instance, nonlinear systems may change in a sudden, discontinuous fashion. One universal class of abrupt, nonlinear transitions is called a bifurcation (1, 3, 4, 6). This term describes situations where a minuscule change in the value of some parameter causes the system to change abruptly from one type of behavior to another. A universal class of bifurcations occurring in a wide variety of nonlinear systems is the sudden appearance of regular oscillations that consist of alternation between two values. This type of ABABAB dynamic may underlie the variety of alternans patterns observed in pathologic cardiovascular electrical and mechanical syndromes, including those seen before sudden cardiac arrest from ventricular tachyarrhythmia (T or U wave alternans), with congestive heart failure (pulsus alternans), and with pericardial tamponade (electrical alternans associated with the swinging heart phenomenon) (4, 8). Another example of universality in complex systems is related to fractals (2, 4, 9, 10).

FRACTAL GEOMETRIES AND DYNAMICS

The concept of a fractal is most often associated with irregular objects that display a property called self-similarity or scale-invariance (2, 4, 10). Fractal forms are composed of subunits (and sub-subunits, etc.) that resemble the structure of the macroscopic object (Figure 3, left). In an idealized model, this internal "look-alike" property holds on all scales. The real world imposes upper and lower bounds over which such scale-free behavior applies. Many irregular structures in nature, such as branching trees, crenellated coastlines, and craggy mountain surfaces, are fractal (2). A number of complex anatomic structures also display fractal-like geometry (4, 9, 10), including branching networks such as the His-Purkinje conduction system (11), and the tracheobronchial tree (10–18). These self-similar cardiopulmonary structures subserve at least one fundamental physiologic function: rapid and efficient transport over multiscale, spatially distributed networks. Related to this function, the fractal tracheobronchial tree may also support chaotic-like mixing of gases in the distal airways (19) and avalanche-like cascades in pulmonary inflation (20). With aging and pathology, fractal anatomic structures may show degradation in their structural complexity. Examples include loss of dendritic arbor in aging neurons, vascular "pruning" in primary pulmonary hypertension (21), and, possibly, alterations in distal airway architecture with chronic obstructive lung disease (22).


Figure 3
View larger version (19K):
[in this window]
[in a new window]
  		Figure 3. Schematic representations of self-similar structures and self-similar fluctuations. The treelike, spatial fractal (left) has self-similar branchings, such that the small-scale structure resembles the large-scale form. A fractal temporal process, such as healthy heart rate regulation (right), generates fluctuations on different time scales that are statistically self-similar. Adapted by permission from Reference 4.

 
The fractal concept can be applied quantitatively not only to irregular geometric forms that lack a characteristic (single) length scale but also to complex processes that lack a single time scale (2, 4, 9, 10). Instead, fractal processes generate irregular fluctuations across multiple time scales, analogous to scale-invariant objects that have a branching or wrinkly structure across multiple scales of length. A qualitative approximation for the self-similar nature of fractal processes, such as the healthy heartbeat, can be obtained by plotting their fluctuations at different temporal resolutions (Figure 3, right). An important methodologic challenge is how to detect and quantify the fractal scaling and related correlation properties of physiologic time series, which are typically not only irregular but also nonstationary since their statistical properties change with time. To help deal with the ubiquitous "complication" of nonstationarity in biological signals, Peng and colleagues (23) have introduced a fractal analysis method termed "detrended fluctuation analysis" (DFA), which has been widely applied to cardiopulmonary and other physiologic time series.

Systematic analyses using the DFA technique and a variety of complementary methods have confirmed that the healthy human heartbeat, even at rest, displays fractal (long-range) correlation properties and that this organization breaks down with aging and disease (10, 24). More recent studies (25, 26) have also disclosed a fractal, multiscale organization of breathing dynamics under healthy conditions. The presence of fractal dynamics in spontaneous cardiopulmonary fluctuations has implications for understanding and modeling physiologic regulation. Traditional (homeostatic) models of control that predict a constant, "steady state" output under physiologic conditions do not account for the observed fractal dynamics (27). Analysis of scaling behavior in a number of life-threatening cardiovascular pathologies, including heart failure and after extensive myocardial infarction, indicates significant alterations in short- and long-range fractal heartbeat correlation properties (10, 28). The effects of chronic obstructive lung disease and other pulmonary pathologies on nonlinear and fractal-related breathing dynamics remain to be investigated.

FRACTAL PHYSIOLOGY: IMPLICATIONS FOR THE DYNAMICS OF HEALTH AND DISEASE

A defining feature of healthy function is adaptability, the capacity to respond to often unpredictable stimuli. Physiologic plasticity requires a broad range of integrated multiscale outputs. Fractal physiology, exemplified by long-range correlations in heartbeat and breathing dynamics, may be adaptive for at least two reasons (10): (1) long-range correlations serve as an organizing mechanism for highly complex processes that generate fluctuations across a wide range of time scales and (2) the absence of a characteristic scale may inhibit the emergence of very periodic behaviors that greatly narrow system responsiveness. This hypothesis is supported by findings from life-threatening conditions, such as chronic heart failure, where the breakdown of fractal correlations is often accompanied by the emergence of a dominant mode (e.g., the Cheyne-Stokes frequency). Abrupt transitions to strongly periodic dynamics are observed in many other pathologies, including at high altitudes and with obstructive sleep apnea (29, 30) (Figure 2), sudden cardiac death, epilepsy, and fetal distress syndromes, to name but a few (4).

The paradoxical appearance of highly ordered dynamics with pathologic states ("disorders") exemplifies the concept of complexity loss (decomplexification) in aging and disease (4, 10, 31). Physiologic stability appears to relate in part to complex patterns of variability that incorporate long-range correlations, together with distinct classes of nonlinear interactions (10, 24). The opposite of a fractal (scale-free) system (Figure 2, top) is one dominated by a characteristic frequency (Figure 2, bottom). Physiologic systems evincing only one (or a few) dominant scale(s) become especially easy to recognize clinically because they stereotypically repeat their behavior in a highly predictable fashion (31, 32).

SPECIAL NEEDS FOR OPEN DATABASES AND SOFTWARE TOOLS

A major obstacle to the dynamic analysis of cardiopulmonary signals has been the unavailability of large, well-characterized, open-access databases and open-source computational tools necessary to promote multidisciplinary and collaborative efforts to find "hidden information" in such complex recordings. In September 1999, under the auspices of the National Center for Research Resources of the National Institutes of Health, our colleagues and we inaugurated the "Research Resource for Complex Physiologic Signals" (33, 34). The "PhysioNet" resource has three interdependent components: (1) "PhysioBank" is a large and expanding collection of well-characterized digital recordings of physiologic signals. Currently available databases include multiparameter cardiopulmonary, neural, and other biomedical signals from subjects with a variety of conditions with major public health implications, such as life-threatening cardiac arrhythmias, congestive heart failure, sleep apnea, neurologic disorders such as Parkinson's disease, and aging, as well as recordings from healthy subjects. (2) "PhysioToolkit" is a library of open-source software for physiologic signal processing and analysis, using both classical techniques and novel methods based on statistical physics and nonlinear dynamics (including fractal and information/entropy-based analyses). (3) "PhysioNet" (from which the resource derives its name) is the online forum for dissemination and exchange of recorded biomedical signals and open-source software for analyzing these complex signals.

FUTURE DIRECTIONS

One promising application of dynamic analysis involves strategies to restore complex biological variability (Figures 2–4GoGo), including fractal fluctuations, to cardiopulmonary systems (35, 36). Initial results using artificial ventilation in experimental animals and clinical settings suggest the possibility of improving physiologic function with "noisy" versus "metronomic" parameter settings (35–42). The use of dynamic assays to uncover basic and clinical information encoded in time series, such as those in Figures 2 and 4, also promises to provide new, readily implemented diagnostic tests for prevalent conditions such as sleep-disordered breathing (30, 41). The extent to which dynamic measures and complexity-informed models and interventions will enhance diagnostic capabilities and therapeutic options in chronic obstructive lung disease is an intriguing area for future study (19, 22, 42, 43).


Figure 4
View larger version (16K):
[in this window]
[in a new window]
  		Figure 4. Illustrative interbreath interval (IBI) time series (obtained at rest by inductance plethysmography) for (a) a healthy young male adult and (b) a healthy elderly male subject. Both time series show an irregular pattern of spontaneous breathing with considerable variability (ordinate) in IBIs over 1,000 consecutive breathing cycles (abscissa). A fractal analysis of the correlation properties of the two datasets is presented in (c) on a log-log plot using the detrended fluctuation analysis (DFA) technique. Dashed lines give the least-square fits for the data from both subjects. The slope of these fitted lines in the DFA exponent, {alpha}. The power-law fits are consistent with fractal dynamics. Of note, the slope for the elderly subject (closer to a value of 0.5 indicative of an uncorrelated process) is consistent with an age-related degradation of the long-range fractal organization of respiratory control (see Reference 25). Adapted by permission from Reference 25.

 
FOOTNOTES

Supported by the National Institutes of Health/National Center for Research Resources (P41-RR13622; A.L.G.), the G. Harold and Leila Y. Mathers Charitable Foundation, the James S. McDonnell Foundation, and the Defense Advanced Research Projects Agency.

Conflict of Interest Statement: A.L.G. does not have a financial relationship with a commercial entity that has an interest in the subject of this manuscript.

(Received in original form March 15, 2006; accepted in final form April 7, 2006)

REFERENCES

  1. Glass L, Mackey MC. From clocks to chaos: the rhythms of life. Princeton, NJ: Princeton University Press; 1988.
  2. Mandelbrot BB. The fractal geometry of nature. New York: WH Freeman; 1982.
  3. Beuter A, Glass L, Mackey M, Titcombe MS. Nonlinear dynamics in physiology and medicine. New York: Springer-Verlag; 2003.
  4. Goldberger AL. Non-linear dynamics for clinicians: chaos theory, fractals, and complexity at the bedside. Lancet 1996;347:1312–1314.[CrossRef][Medline]
  5. May RM. Simple mathematical models with very complicated dynamics. Nature 1976;261:459–467.[CrossRef][Medline]
  6. Kaplan DT, Glass L. Understanding nonlinear dynamics. New York: Springer-Verlag; 1995.
  7. Paydarfar D, Buerke DM. Dysarrhythmias of the respiratory oscillator. Chaos 1995;5:18–29.[Medline]
  8. Rigney DR, Goldberger AL. Non-linear mechanics of the heart's swinging during pericardial effusion. Am J Physiol 1989;257:H1292–H1305.[Medline]
  9. Bassingthwaighte JB, Liebovitch LS, West BJ. Fractal physiology. Oxford, UK: University Press; 1994.
  10. Goldberger AL, Amaral LAN, Hausdorff JM, Ivanov PCh, Peng CK, Stanley HE. Fractal dynamics in physiology: alterations with disease and aging. Proc Natl Acad Sci USA 2002;99:2466–2472.[Abstract/Free Full Text]
  11. Goldberger AL, Bhargava V, West BJ, Mandell AJ. On a mechanism of cardiac electrical stability: the fractal hypothesis. Biophys J 1985;48:525–528.[Medline]
  12. Nelson TR, West BJ, Goldberger AL. The fractal lung: universal and species-related scaling patterns. Experientia 1990;46:251–254.[CrossRef][Medline]
  13. Weibel ER. Fractal geometry: a design principle for living organisms. Am J Physiol 1991;261:L361–L369.[Medline]
  14. Glenny RW, Robertson HT. Fractal properties of pulmonary blood flow: characterization of spatial heterogeneity. J Appl Physiol 1990;69:532–545.[Abstract/Free Full Text]
  15. Glenny RW, Robertson HT. Fractal modeling of pulmonary blood flow heterogeneity. J Appl Physiol 1991;70:1024–1030.[Abstract/Free Full Text]
  16. McNamee JE. Fractal perspectives in pulmonary physiology. J Appl Physiol 1991;71:1–8.[Abstract/Free Full Text]
  17. Robertson HT, Altemeier WA, Glenny RW. Physiological implications of the fractal distribution of ventilation and perfusion in the lung. Ann Biomed Eng 2000;28:1028–1031.[CrossRef][Medline]
  18. Kitaoka H, Suki B. Branching design of the bronchial free based on a diameter-flow relationship. J Appl Physiol 1997;82:968–976.[Abstract/Free Full Text]
  19. Tsuda A, Rogers RA, Hydon PE, Butler JP. Chaotic mixing deep in the lung. Proc Natl Acad Sci USA 2002;23:10173–10178.
  20. Suki B, Barabasi AL, Hantos Z, Petak F, Stanley HE. Avalanches and power-law behavior in lung inflation. Nature 1994;368:615–618.[CrossRef][Medline]
  21. Boxt LM, Katz J, Liebovitch L, Jones R, Essen PD, Reid L. Fractal analysis of pulmonary arteries: the fractal dimension is lower in pulmonary hypertension. J Thorac Imaging 1994;9:8–13.[Medline]
  22. Mitsunobu F, Ashida K, Hosaki Y, Tsugeno H, Okamoto M, Nishida K, Takata S, Yokoi T, Mishima M, Tanizaki Y. Complexity of terminal airspace geometry assessed by computed tomography in asthma. Am J Respir Crit Care Med 2003;167:411–417.[Abstract/Free Full Text]
  23. Peng CK, Havlin S, Stanley HE, Goldberger AL. Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos 1995;5:82–87.[CrossRef][Medline]
  24. Ivanov PCh, Amaral LAN, Goldberger AL, Havlin S, Rosenblum MG, Struzik Z, Stanley HE. Multifractality in human heartbeat dynamics. Nature 1999;399:461–465.[CrossRef][Medline]
  25. Peng CK, Mietus JE, Liu Y, Lee C, Hausdorff JM, Stanley HE, Goldberger AL. Quantifying fractal dynamics of human respiration: age and gender effects. Ann Biomed Eng 2002;30:683–692.[CrossRef][Medline]
  26. Fadel PJ, Barman SM, Phillips SW, Gebber GL. Fractal fluctuations in human respiration. J Appl Physiol 2004;97:2056–2064.[Abstract/Free Full Text]
  27. Amaral LAN, Diaz-Guilera A, Moreira A, Goldberger AL, Lipsitz LA. Emergence of complex dynamics in a simple model of signaling networks. Proc Natl Acad Sci USA 2004;101:15551–15555.[Abstract/Free Full Text]
  28. Huikuri HV, Mäkikallio TH, Peng C-K, Goldberger AL, Hintze U, Møller M, for the DIAMOND Study Group. Fractal correlation properties of R-R interval dynamics and mortality in patients with depressed left ventricular function after an acute myocardial infarction. Circulation 2000;101:47–53.[Abstract/Free Full Text]
  29. Lipsitz LA, Hashimoto F, Lubowsky LP, Mietus J, Moody GB, Appenzeller O, Goldberger AL. Heart rate and respiratory rhythm dynamics on ascent to high altitude. Br Heart J 1995;74:390–396.[Abstract/Free Full Text]
  30. Mietus JE, Peng CK, Ivanov PCh, Goldberger AL. Detection of obstructive sleep apnea from cardiac interbeat interval time series. Comput Cardiol 2000;27:753–756.
  31. Goldberger AL. Fractal variability versus pathologic periodicity: complexity loss and stereotypy in disease. Perspect Biol Med 1997;40:543–561.[Medline]
  32. Belair J, Glass L, an der Heiden U, Milton J. Dynamical disease: mathematical analysis of human illness. New York: American Institute of Physics Press; 1995.
  33. Goldberger AL, Amaral LAN, Glass L, Hausdorff JM, Ivanov PCh, Mark RG, Mietus JE, Moody GB, Peng C-K, Stanley HE. PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation 2000;101:e215–e220.[Abstract/Free Full Text]
  34. Moody GB, Mark RG, Goldberger AL. PhysioNet: a web-based resource for the study of physiologic signals. IEEE Eng Med Biol 2001;20:70–75.[CrossRef][Medline]
  35. Suki B, Alencar AM, Sujeer MK, Lutchen KR, Collins JJ, Andrade JS Jr, Ingenito EP, Zapperi S, Stanley HE. Life-support system benefits from noise. Nature 1998;393:127–128.[Medline]
  36. Graham MR, Warrian RK, Girling LG, Doiron L, Lefevre GR, Cheany M, Math M, Mutch WAC. Fractal or biologically variable delivery of cardioplegic solution prevents diastolic dysfunction after cardiopulmonary bypass. J Thorac Cardiovasc Surg 2002;123:63–71.[Abstract/Free Full Text]
  37. Goldberger AL. Heartbeats, hormones, and health: is variability the spice of life? Am J Respir Crit Care Med 2001;163:1289–1290.[Free Full Text]
  38. Mutch WAC, Alan C, Harms S, Lefevre GR, Graham MR, Girling LG, Kowalski SE. Biologically variable ventilation increases arterial oxygenation over that seen with positive end-expiratory pressure alone in a porcine model of acute respiratory distress syndrome. Crit Care Med 2000;28:2457–2464.[CrossRef][Medline]
  39. Mutch WAC, Harms S, Ruth Graham M, Kowalski SE, Girling LG, Lefevre GR. Biologically variable or naturally noisy mechanical ventilation recruits atelectatic lung. Am J Respir Crit Care Med 2000;162:319–323.[Abstract/Free Full Text]
  40. Boker A, Graham MR, Walley KR, McManus BM, Girling LG, Walker E, Lefevre GR, Mutch WAC. Improved arterial oxygenation with biologically variable or fractal ventilation using low tidal volumes in a porcine model of acute respiratory distress syndrome. Am J Respir Crit Care Med 2002;165:456–462.[Abstract/Free Full Text]
  41. Thomas RJ, Mietus JE, Peng CK, Goldberger AL. An electrocardiogram-based technique to assess cardiopulmonary coupling during sleep. Sleep 2005;28:1135–1143.[Medline]
  42. Suki B. Fluctuations and power laws in pulmonary physiology. Am J Respir Crit Care Med 2002;166:133–137.[Free Full Text]
  43. Venegas J, Winkler T, Musch G, Vidal Melo MF, Layfield D, Tgavalekos N, Fischman AJ, Callahan RJ, Bellani G, Harris RS. Self-organized patchiness in asthma as a prelude to catastrophic shifts. Nature 2005;434:777–782.[CrossRef][Medline]



This article has been cited by other articles:


Home page
 J. Neurol. Neurosurg. PsychiatryHome page
M P Pender, P A Csurhes, A Lenarczyk, C M M Pfluger, and S R Burrows
Decreased T cell reactivity to Epstein-Barr virus infected lymphoblastoid cell lines in multiple sclerosis
J. Neurol. Neurosurg. Psychiatry, May 1, 2009; 80(5): 498 - 505.
[Abstract] [Full Text] [PDF]

Home page
 Cardiovasc ResHome page
K. Hu, F. A.J.L. Scheer, R. M. Buijs, and S. A. Shea
The circadian pacemaker generates similar circadian rhythms in the fractal structure of heart rate in humans and rats
Cardiovasc Res, October 1, 2008; 80(1): 62 - 68.
[Abstract] [Full Text] [PDF]

Home page
 Evid Based Complement Alternat MedHome page
E. H. Logue
Geffen Faculty Highlight Concerns Linking CAIM and Conventional Researchers at UCLA Symposium
Evid. Based Complement. Altern. Med., August 13, 2008; (2008) nen051v2.
[Abstract] [Full Text] [PDF]

Home page
 Am. J. Public HealthHome page
K. Resnicow and S. E. Page
Embracing Chaos and Complexity: A Quantum Change for Public Health
Am J Public Health, August 1, 2008; 98(8): 1382 - 1389.
[Abstract] [Full Text] [PDF]

This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Similar articles in this journal
Right arrow Similar articles in PubMed
Right arrow Alert me to new issues of the journal
Right arrow Download to citation manager
Right arrow reprints & permissions
Citing Articles
Right arrow Citing Articles via HighWire
Right arrow Citing Articles via Google Scholar
Google Scholar
Right arrow Articles by Goldberger, A. L.
Right arrow Search for Related Content
PubMed
Right arrow PubMed Citation
Right arrow Articles by Goldberger, A. L.

HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS