1. Deterministic Chaos and Rhythms of Life
Dr. Thomas Caraco

2. Ecological Complexity Background
Outline
Ecological Complexity
Background
Discrete-Time Self-Regulation
Overcompensation
Population Dynamics: Route to Chaos
Evolution to Edge of Chaos?

3. 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?

4. Chaos: Ecological Significance
Population Regulation before 1975
Physical Factors  Random Fluctuations
Density Dependence Stabilizing
New Perspective on Density Dependence
Constancy to Chaotic Complexity

5. Simple Model, Complex Dynamics
Robert M. May
Logistic Map in Ecology
General Paradigm for Emergence of Chaos
Challenge: Distinguish Deterministic Chaos from Stochastic Flux

6. Discrete-Time Logistic Growth
Density-Dependence: Self-Regulation = Intraspecific Competition Assume: Individual’s Contribution to Next Generation Declines Linearly with This Generation’s Density

7. Discrete-Time Logistic Growth
Time t, Population density 𝑥 𝑡 0 < 𝑥 𝑡 <1 Occupied fraction of environment Individual contribution to next generation 𝑥 𝑡+1 𝑥 𝑡

8. Annual Life Cycle Population Density: x(t) Map to x(t+1)
Individual Reproduction
Density-Dependent

9. Logistic: x(t+1) =  {x(t) – [x(t)]2}
Universality: “All” single-peaked maps show same increase in dynamic complexity!

10. Nonlinear Map: x(t+1) =  {x(t) – [x(t)]2}
Increase Fecundity 

11. Behavior of Map: Dynamics
1 <  < 3
Equilibrium Node
Any Initial Density
 Same Equilibrium
r = 2.9

12. Dynamics  = 3.3 Bifurcation: Equilibrium 2-Cycle Periodic Dynamics
Time Symmetry
r = 3.3

13. Dynamics  = 3.56 Bifurcation: Increased Complexity
Equilibrium 4-Cycle
Increased Complexity
r = 3.56

14. Bifurcation Cascade Period-Doubling Route to Chaos
Infinite Number of Bifurcations
Feigenbaum Point
 = …
Chaos

15. Deterministic Chaos Bounded Aperiodic No State Repeats! Not Random!
Close to Extinction
Aperiodic
No State Repeats!
Not Random!
Correlations
Sensitive Dependence
Initial Conditions

16. Bifurcation Diagram “Route to Chaos” Periodic Windows Universality
Strange
Attractor

17. Fractal Behavior Self-Similarity Scale Invariance Repeating Geometry
Signature of Chaos

18. Real Populations Chaotic?
Within Populations
Favor Faster Growth
Complex Dynamics, Fluctuations
 Extinction
Among Populations
Dynamic Stability  Persistence
Evolve to Edge of Chaos?

19. Lack of Data; Require Lengthy Records
Real Populations
Remove Random “Error”
“Reconstruct Map”
Test for Divergence
Lack of Data; Require Lengthy Records
Costantino et al Science 275:
Ellner & Turchin Amer. Naturalist 145:
Olsen & Schaffer Science 249:

20. Chaos in lab?

21. Chaos in nature?

22. Small mammal: latitude and dynamics

23. Childhood Disease

24. 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
Bifurcations Sensitivity Initial Conditions

25. Lessons from Simple Nonlinearities 2
Chaos: Emergence of Fractal Order
Break Symmetry of Past & Future
Non-Random Behavior, Correlations
Ecological Complexity  Loss Predictability
Bifurcations Sensitivity Initial Conditions
STOP

26. 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

27. Human Heart & EKG

28. Fractal Dynamics of Human Heart Rate
Classical Paradigm
Equilibrium 
Homeostasis
“Average” Rates
Novel
Hidden Variability

29. Fractal Process (Inter-beat Interval thru Time)
Self-Similarity
Sub-unit Statistically Identical to Whole
Scaling Between Time Windows
 : Self-Similarity Parameter (DFA)

31. Scaling Rate Variability
Periodicity <  < 0.5
Random (Uncorrelated Noise)  = 0.5
Fractal (Power Law Behavior,
Long Range Correlations) <   1
Random Walk  > 1

32. Complexity of Heart Beat Dynamics
Fractal-Type Variability: Inter-beat Interval
No Characteristic Time Scale
Generates Long-Range Organization
Order in Chaotic Signal

33. Heart Rate: Healthy Subject Inter-beat Interval Fractal

34. Heart Rate: Healthy, Disease, Aging

35. CHF Patients
Clinical Utility
Complexity Loss

36. Meditation: Heart Rate ?

37. Fractal Dynamics of Human Walking

38. Human Gait Fractal

39. Walking Rate & Stride Dynamics

40. Gait in Aging & Disease

41. Huntington’s Disease Low Severity (Score>10)  Fractal Gait
Severe (Score < 4)
 Periodic Gait

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

43. Gait Dynamics Health: Fractal Over 1000’s Strides
Persistent Across Pace
Neurodegenerative Disease:
Loss of Complexity
Random Gait
Complexity May Predict Injury

44. Loss of Fractal Complexity
EEG: Epilepsy
Respiration: Sleep Apnea
White Cell Count: Myelogenous Leukemia
Blood Pressure: Kidney Function

45. Fractal Physiological Rates
How?
Complex Regulation
Mechanisms Effective Different Time Scales
Information Content
Why? Adaptive Significance?
Inhibits “Mode-Locking”: Response Scale
Maintains Organism’s Functional Plasticity
Respond at Multiple Scales of Time

46. 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

47. EKG: Faint Current, Body’s Saline
Cardiac Cycle: Phases
Spatial & Temporal Organization
Local Current, Local Voltage
Voltage: Spatial Diffusion, Couples Locations
Different Location, Different Phase

48. Electrical Heartbeat Time t, Spatial Location x
V(t, x) Membrane Voltage
g(t, x) Ion channel conductance
K Diffusion coefficient

49. Time & Space: Isochrones

50. Singularity: Time Breaks Down

51. Phase Singularities Rotors in Excitable Media Myocardium
Excitable, Biological Oscillators
Has Phase Singularities, “Clock” Breaks

52. Singularities & “Re-entrants”

53. 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

54. Citations http://reylab.bidmc.harvard/tutorial/DFA
Keener, J.P Heart attacks can give you mathematics.
Goldberger, A.L 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 Chaos: An Introduction to Dynamical Systems. Springer, New York, NY.
Kaplan, D. & Glass, L Understanding Nonlinear Dynamics. Springer, New York, NY.

55. “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

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