Deterministic Chaos and Rhythms of Life

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Deterministic Chaos and Rhythms of Life

• Dr. Thomas Caraco

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Outline

• 1. Ecological Complexity
• Background
• Population Dynamics Route to Chaos
• Evolution to Edge of Chaos?
• 2. Dynamic Complexity Human Physiology
• Human Heart Beat, Gait
• (Dis)-Order, Chaos and Disease

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Chaos Research Significance

• Biological, Physical and Social Sciences
• Systems with Nonlinear Dynamics
• Generator of Chaos and Complexity
• New Perspective on Law of Causality
• Very Similar Cause ?Very Similar Effect?
• Practical Applications
• Control Dynamical Systems

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Intl Sun-Earth Explorer-3/ICE
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Deductive Modeling

• Specify Detailed Assumptions
• Essence of Question, Avoid Nonessential
• Apply Math (Deduction)
• Results Follow from Assumptions
• Theory
• Predictions, Hypotheses for Test

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• One of the principle objects of theoretical
  research in any department of knowledge is to
  find the point of view from which the subject
  appears in its greatest simplicity.
• J. W. Gibbs, 1881
• Poetry leaves other (nonessential) data out of
  the relational field narrows the focus.
• J. Johnson, 2004

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Inductive Modeling

• Observing, Manipulating
• Statistical Inference
• Interpret
• Hypotheses for Test

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• Chaos Theory New?
• Poincarè (1892-94) Dynamic Tangles
• Lorenz (1963) Aperiodic Complexity
• Sarkovskii (1964) Windows of Order
• 1974 1985 Dynamical Renaissance

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Chaos Ecological Significance

• Population Regulation before 1975
• Physical Factors ? Random Fluctuations
• Density Dependence Stabilizing
• New Perspective on Density Dependence
• Constancy to Chaotic Complexity
• Understand Mix Nonlinear, Random Dynamics

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Simple Model, Complex Dynamics

• R.M. May
• Logistic Map
• General Paradigm
• for
• Emergence of Chaos
• Metric Universality

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Annual Life Cycle

• Population Density
• x(t) Map to x(t1)
• Individual Reproduction
• Density-Dependent

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Nonlinear Map x(t1) r x(t) x(t)2

• Increase Fecundity r

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Behavior of Map Dynamics

• 1
• Equilibrium Node
• Any Initial Density
• ? Same Equilibrium

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Dynamics

• r 3.3
• Bifurcation
• Equilibrium 2-Cycle
• Periodic Dynamics
• Time Symmetry

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Dynamics

• r 3.56
• Bifurcation
• Equilibrium 4-Cycle
• Increased Complexity

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Bifurcation Cascade

• Period-Doubling Route to Chaos
• Infinite Number of Bifurcations
• Feigenbaum Point
• r 3.56994456
• Chaos Non-Equilibrium in Nature

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Deterministic Chaos

• Bounded
• Close to Extinction
• Aperiodic
• No State Repeats!
• Not Random!
• Correlations
• Sensitive Dependence
• Initial Conditions

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Bifurcation Diagram

• Route to Chaos
• Periodic Windows
• Universality
• Strange
• Attractor

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Fractal Behavior

• Self-Similarity
• Scale Invariance
• Repeating Geometry
• Signature of Chaos

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• Metric Universalities
• Feigenbaum Ratios
• Periodic Window
• Sequence
• Quantitative Identicality

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Strange AttractorCantor Set
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Real Populations Chaotic?

• Within Populations
• Favor Faster Growth
• Complex Dynamics, Fluctuations
• ? Extinction
• Among Populations
• Dynamic Stability ? Persistence
• Evolve to Edge of Chaos?

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Real Populations

• Remove Random Error
• Reconstruct Map
• Test for Divergence
• Lack of Data Require Lengthy Records
• Costantino et al. 1997. Science 275389-391.
• Ellner Turchin. 1995. Amer. Naturalist
  145343-375.
• Olsen Schaffer. 1990. Science 249499-504.

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Childhood Disease
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Lessons from Simple Nonlinearities

• Universalities Stability ? Complexity
• Equilibrium ? Non-Equilibrium
• Small Parameter Changes
• ? Qualitative Change in Behavior
• Chaos Small Change in State
• ? Quantitative Divergence of Systems

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Lessons from Simple Nonlinearities

• Chaos Emergence of Fractal Order
• Break Symmetry of Past Future
• Non-Random Behavior, Correlations
• Ecological Complexity ? Loss Predictability

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Fractal Physiology

• Diseases were explained in terms of
  disharmony and imbalance the goal of medicine
  was to restore balance.
• V. Ng (1990), Madness in Chinese Culture
• Compelling examples of chaotic dynamics are
  found in periodic stimulation of biological
  oscillators.
• D. Kaplan L. Glass, 1995

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Human Heart EKG
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Fractal Dynamics of Human Heart Rate

• Classical Paradigm
• Equilibrium ?
• Homeostasis
• Average Rates
• Novel
• Hidden Variability

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Fractal Process (Inter-beat Interval thru Time)

• Self-Similarity
• Sub-unit Statistically Identical to Whole
• Scaling Between Time Windows
• ? Self-Similarity Parameter (DFA)
• http//reylab.bidmc.harvard/tutorial/DFA
• http//www.physionet.org/challenge

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Scaling Rate Variability

• Periodicity 0
• Random (Uncorrelated Noise) ? 0.5
• Fractal (Power Law Behavior,
• Long Range Correlations) 0.5
• Random Walk ? 1

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Complexity of Heart Beat Dynamics

• Fractal-Type Variability Inter-beat Interval
• No Characteristic Time Scale
• Generates Long-Range Organization
• Order in Chaotic Signal

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Heart Rate Healthy SubjectInter-beat Interval
Fractal
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Heart Rate Healthy, Disease, Aging
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CHF Patients

• Clinical Utility
• Complexity Loss

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Meditation Heart Rate ?
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Fractal Dynamics of Human Walking
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Human Gait Fractal
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Walking Rate Stride Dynamics
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Gait in Aging Disease
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Huntingtons Disease

• Low Severity (Score10)
• ? Fractal Gait
• Severe (Score
• ? Periodic Gait

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Heart Dynamics

• Health
• Fractal Over 1000s Heartbeats
• Persistently Chaotic Sleep-Wake Cycle
• Cardiovascular Disease
• Loss of Complexity
• Random, Periodic Heart Rate
• Complexity Can Predict Survival

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Gait Dynamics

• Health
• Fractal Over 1000s Strides
• Persistent Across Pace
• Neurodegenerative Disease
• Loss of Complexity
• Random Gait
• Complexity May Predict Injury

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Loss of Fractal Complexity

• EEG Epilepsy
• Respiration Sleep Apnea
• White Cell Count Myelogenous Leukemia
• Blood Pressure Kidney Function

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Fractal Physiological Rates

• How?
• Complex Regulation
• Mechanisms Effective Different Time Scales
• Information Content
• Why? Adaptive Significance?
• Inhibits Mode-Locking Response Scale
• Maintains Organisms Functional Plasticity
• Respond at Multiple Scales of Time

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Sudden Cardiac Death

• Kills ½ Million Annually in U.S.
• Ventricular Fibrillation
• Uncoordinated Shivering Multiple Modes
• Myocardial Infarction ? Fibrillation
• ¼ Male SCD, Ages 20-64, No Infarction

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• EKG Faint Current, Bodys Saline
• Cardiac Cycle Phases
• Spatial Temporal Organization
• Local Current, Local Voltage
• Voltage Spatial Diffusion, Couples Locations
• Different Location, Different Phase

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Electrical Heartbeat

• Time t, Spatial Location x
• V(t, x) Membrane Voltage
• g(t, x) Ion channel conductance
• K Diffusion coefficient

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Time Space Isochrones
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Singularity Time Breaks Down
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Phase Singularities

• Rotors in Excitable Media
• Myocardium
• Excitable, Biological Oscillators
• Has Phase Singularities, Clock Breaks

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Singularities Re-entrants
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Attractors Electrical Activity

• Normal Oscillation
• Almost Periodic, Functional
• Space-Time Chaos (Turbulence)
• Re-entrant Waves
• Rotor(s) Induced by Singularity
• Fibrillation, Dysfunctional
• Alternate Attractors, Fibrillation cartoon

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Citations

• http//reylab.bidmc.harvard/tutorial/DFA
• Keener, J.P. 2002. Heart attacks can give you
  mathematics.
• http//www.math.utah.edu/keener/lectures/Arrhyth
  mias
• Goldberger, A.L. 1999. Nonlinear dynamics,
  fractals, and chaos theory implications for
  neuroautonomic heart rate control in health and
  disease. In Bolis, C.L. Licinio, J. (eds) The
  Autonomic Nervous System. World Health
  Organization, Geneva, Switzerland.
• Alligood, K.T., Sauer, T.D. Yorke, J.A. 1997.
  Chaos An Introduction to Dynamical Systems.
  Springer, New York, NY.
• Kaplan, D. Glass, L. 1995. Understanding
  Nonlinear Dynamics. Springer, New York, NY.
• http//www.economist.com/displayStory.cfm?Story_ID
  324885

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• The danger already exists that the
  mathematicians have made a covenant with the
  devil to darken the spirit and to confine man in
  the bonds of Hell.
• St. Augustine

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• People who wish to analyze nature without using
  mathematics must settle for a reduced
  understanding.
• Richard Feynman