Dynamics and time series: theory and applications

Stefano Marmi

Scuola Normale Superiore

Lecture 6, Feb 17, 2009

• Lecture 1: An introduction to dynamical systems and to time series. Periodic and

quasiperiodic motions. (Tue Jan 13, 2 pm - 4 pm Aula Bianchi)

• Lecture 2: Ergodicity. Uniform distribution of orbits. Return times. Kac inequality

Mixing (Thu Jan 15, 2 pm - 4 pm Aula Dini)

• Lecture 3: Kolmogorov-Sinai entropy. Randomness and deterministic chaos. (Tue Jan

27, 2 pm - 4 pm Aula Bianchi)

• Lecture 4: Time series analysis and embedology. (Thu Jan 29, 2 pm - 4 pm Dini)

• Lecture 5: Fractals and multifractals. (Thu Feb 12, 2 pm - 4 pm Dini)

• Lecture 6: The rhythms of life. (Tue Feb 17, 2 pm - 4

pm Bianchi)

• Lecture 7: Financial time series. (Thu Feb 19, 2 pm - 4 pm Dini)

• Lecture 8: The efficient markets hypothesis. (Tue Mar 3, 2 pm - 4 pm Bianchi)

• Lecture 9: A random walk down Wall Street. (Thu Mar 19, 2 pm - 4 pm Dini)

• Lecture 10: A non-random walk down Wall Street. (Tue Mar 24, 2 pm – 4 pm

Bianchi)

• Seminar I: Waiting times, recurrence times ergodicity and quasiperiodicdynamics (D.H. Kim, Suwon, Korea; Thu Jan 22, 2 pm - 4 pm Aula Dini)

• Seminar II: Symbolization of dynamics. Recurrence rates and entropy (S. Galatolo, Università di Pisa; Tue Feb 10, 2 pm - 4 pm Aula Bianchi)

• Seminar III: Heart Rate Variability: a statistical physics point of view (A. Facchini, Università di Siena; Tue Feb 24, 2 pm - 4 pm Aula Bianchi )

• Seminar IV: Study of a population model: the Yoccoz-Birkeland model (D. Papini, Università di Siena; Thu Feb 26, 2 pm - 4 pm Aula Dini)

• Seminar V: Scaling laws in economics (G. Bottazzi, Scuola Superiore Sant'Anna Pisa; Tue Mar 17, 2 pm - 4 pm Aula Bianchi)

• Seminar VI: Complexity, sequence distance and heart rate variability (M. Degli Esposti, Università di Bologna; Thu Mar 26, 2 pm - 4 pm Aula Dini )

• Seminar VII: Forecasting (M. Lippi, Università di Roma; late april, TBA)

Today’s bibliography:

L. Glass and M.C. Mackey: “From Clocks to Chaos.

The Rhythms of Life” Princeton University Press

(1988)

Fractal dynamics in physiology: Alterations with disease and

aging

Goldberger AL, Amaral LAN, Hausdorff JM, Ivanov PC,

Peng CK, Stanley HE

PNAS USA 2002;99:2466-2472.

• Physiological rhythms are central to life. Some are maintained

throughout life without interruptions.

• Rhythms interact with one another and with exernal

environment

• Variations of rhythms outside normal limits, or appearance of

new ones, is associated with disease

Homeostasis

• homeostasis: is the property of a system, either open or closed,

that regulates its internal environment so as to maintain a

stable, constant condition

• Human body: relative constancy of

• Claude Bernard (1813–78), French physiologist: showed the

role of the pancreas in digestion, the method of regulation of

body temperature, and the function of nerves supplying the

internal organs.

• W.B. Cannon (1871-1945) : Showed that physiological

variables like blood sugar, blood gases, electrolytes,

osmolarity, blood pressure, pH are relatively constant.

• Homeostasis can be associated to stable steady states

Arterial and mean arterial pressure responses to a quick mild

hemorrage in a dog anesthetized with sodium pentobarbital

(Figure 1.1, Glass Mackey From Clocks to Chaos, p.4)

Physiological variables when measured with sufficient accuracy will never be absolutely constant or exactly periodic

There are always fluctuations. Sometimes they are due to the environment, as shown here with blood insulin and glucose levels

B=breakfast, L=lunch, SK=snack, D=dinner, E=exercise

(Figure 1.3, Glass Mackey From Clocks to Chaos, p.5)

The human heart

http://www.nhlbi.nih.gov/health/dci/Diseases/hhw/hhw_anatomy.html

The hearbeatThe frequency of the cardiac cycle is the heart rate. Every single

'beat' of the heart involves three major stages: atrial systole,

ventricular systole and complete cardiac diastole. The

term diastole is synonymous with relaxation of a muscle. During

diastole, the atria and ventricles of your heart relax and begin to

fill with blood. At the end of diastole, your heart’s atria contract

(an event called atrial systole) and pump blood into the ventricles.

The atria then begin to relax. Next, your heart’s ventricles contract

(an event called ventricular systole) and pump blood out of your

heart.

Throughout the cardiac cycle, the blood pressure increases and

decreases. The cardiac cycle is coordinated by a series of

electrical impulses that are produced by specialized heart cells

found within the sinoatrial node and the atrioventricular node.

Your heart uses the four valves to ensure your blood flows only in one direction.

Blood without oxygen from the two vena cavae fill your heart’s right atrium.

The atrium contracts (atrial systole). The tricuspid valve located between the right

atrium and ventricle opens for a short time and then shuts. This allows blood to

enter into the right ventricle without flowing back into the right atrium.

When your heart’s right ventricle fills with blood, it contracts (ventricular systole).

The pulmonary valve located between your right ventricle and pulmonary artery

opens and closes quickly. This allows blood to enter into your pulmonary artery

without flowing back into the right ventricle. Blood travels through the pulmonary

arteries to your lungs to pick up oxygen.

Oxygen-rich blood returns from the lungs to your heart’s left atrium through the

pulmonary veins. As your heartís left atrium fills with blood, it contracts. This

event also is called atrial systole. The mitral valve located between the left atrium

and left ventricle opens and closes quickly. This allows blood to pass from the left

atrium into the left ventricle without flowing back into the left atrium.

As the left ventricle fills with blood, it contracts. This event also is called

ventricular systole. The aortic valve located between the left ventricle and aorta

opens and closes quickly. This allows blood to flow into the aorta. The aorta is the

main artery that carries blood from your heart to the rest of your body. The aortic

valve closes quickly to prevent blood from flowing back into the left ventricle,

which is already filling up with new blood.http://www.nhlbi.nih.gov/health/dci/Diseases/hhw/hhw_pumping.html

Electrocardiogram

Pictures taken from Wikipedia

The heart cycle and ECG

Picture taken from Wikipedia

Picture taken fromWikipedia

Your Heart’s Electrical System.swf

http://www.nhlbi.nih.gov/health/dci/Diseases/hhw/hhw_electrical.html

Heart Contraction and Blood Flow.swf

http://www.nhlbi.nih.gov/health/dci/Diseases/hhw/hhw_pumping.html

Heart rate vs embryonic age

At 21 days

after conception,

the human heart

begins beating at 70

to 80 beats per

minute and

accelerates linearly

for the first month

of beating

From:WIKIPEDIA

Heart rate from childhood to senescence

http://reylab.bidmc.harvard.edu/pubs/1999/circulation-1999-100-393.pdf

Stable periodic behaviour: limitcycles and ECG

G. Israel:Balthasar van der Pol e il primo modello matematico del battito cardiaco. http://giorgio.israel.googlepages.com/Art76.pdf

http://www.scholarpedia.org/article/Van_der_Pol_oscillator

,

Lienard transformation

Time series analysis ofphysiological signals

Physiological signals are characterized by extreme variability both inhealthy and pathological conditions. Complexity, erratic behaviour,chaoticity are typical terms used in the description of manyphysiological time series.

Quantifying these properties and turning the variability analysis fromqualitative to quantitative are important goals of the analysis of time-series and could have relevant clinical impact.

From ECG to heart rate variability time series

• Example of ECG signal

• The time interval between

two consecutive R-wave

peaks (R-R interval) varies in

time

• The time series given by the

sequence of the durations of

the R-R intervals is called

heart rate variability (HRV)

Healthy? Statistical vs. dynamical tools for diagnosis

• The HRV plots of an healthy

patient show a very different

dynamics from those of a sick

patient but the traditional

statistical measures (mean and

variance) are almost the same.

• www.physionet.org

Time series and self-similarity

http://physionet.org/tutorials/fmnc/img25.png

A cardiac inter-heartbeat interval (inverse of heart rate) time series is shown in (A) and

a randomized control is shown in (B). Successive magnifications of the sub-sets

show that both time series are self-similar with a trivial exponent , albeit the

patterns are very different in (A) and (B).

www.physionet.org

Healthy ?

www.physionet.org

Healthy or pathological?

(Adapted from Goldberger AL. Non-linear dynamics for clinicians: chaos theory, fractals, and complexity at the bedside. Lancet 1996;347:1312-1314.)

Here is what can happen as the heart goes out of its normal healthy state. At the bottom left :

heart failure. The signal is approximately periodic with small amplitude fluctuations. At the

bottom right: atrial fibrillation. The heart rate is very erratic with quick accelerations and

decelerations, with no particular pattern.

Scientific American, February 1990http://reylab.bidmc.harvard.edu/pubs/1990/sa-1990-262-42.pdf

Healthy or not?

The breakdown of long-range power law correlations may lead to any of three

dynamical states: (i) a random walk (``brown noise'') as observed in low frequency

heart rate fluctuations in certain cases of severe heart failure; (ii) highly periodic

oscillations, as also observed in Cheyne-Stokes pathophysiology in heart failure, as well

as with sleep apnea (Fig. 1c), and (iii) completely uncorrelated behavior (white noise),

perhaps exemplified by the short-term heart rate dynamics during atrial fibrillation.

www.physionet.org

Correlation between diseaseseverity and fractal scalingexponent

www.physionet.org

Distribution ofR-R intervals

www.physionet.org

Stochastic or chaotic?

• An important goal of time-series analysisis to determine, given a times series (e.g. HRV) if the underlying dynamics (the heart) is:

– Intrinsically random

– Generated by a deterministic nonlinearchaotic system which generates a randomoutput

– A mix of the two (stochastic perturbations ofdeterministic dynamics)

Nonlinear dynamics and chaos in cardiology?

The normal heart rhythm in humans is set by a small group of cells called the

sinoatrial node. Although over short time intervals, the normal heart rate often

appears to be regular, when the heart rate is measured over extended periods of

• time, it shows significant fluctuations. There are a number of factors that affect

these fluctuations: changes of activity or mental state, presence of drugs,

presence of artificial pace- makers, occurrence of cardiac arrhythmias that

might mask the sinoatrial rhythm or make it difficult to measure.

Following the widespread recognition of the possibility of deterministic chaos in the

early 1980s, considerable attention has been focused on the possibility that heart rate

variability might reflect deterministic chaos in the physiological control system

regulating the heart rate. A large number of papers related to the analysis of heart

rate variability have been published in Chaos and elsewhere. However, there is still

considerable debate about how to characterize fluctuations in the heart rate and the

significance of those fluctuations. There has not been a forum in which these

disagreements can be aired. Accordingly, Chaos invites submissions that address

one or more of the following questions:

• Is the normal heart rate chaotic?

• If the normal heart rate is not chaotic, is there some more appropriate term to

characterize the fluctuations e.g., scaling, fractal, multifractal?

• How does the analysis of heart rate variability elucidate the

underlying mechanisms controlling the heart rate?

• Do any analyses of heart rate variability provide clinical information that can

be useful in medical assessment e.g., in helping to assess the risk of sudden

cardiac death. If so, please indicate what additional clinical studies would be useful

for measures of heart rate variability to be more broadly accepted by the medical

community.

EEG: electroencephalography

We use EEG to measure the constant flux of electrical activity generated by

cells in the head

Brain “waves” occur in the EEG record due to changes across time in voltages

between two different regions of the brain, as recorded from different

regions several centimeters apart on the scalp

The two fundamental aspects of brain activity, as measured in the EEG are the

frequency and amplitude of the waves

Frequency typically varies from less than 1 to more than 25 Hz

Amplitude usually varies between 10 and 200 microvolts when measured

from the scalp

Generally, the higher the frequency, the lower the amplitude

Alpha: from 8 Hz to

12 Hz. Closing the

eyes, relaxation;

"posterior dominant

rhythm" Beta: from

12 Hz to about 30 Hz.

Active, busy, anxious

thinking,

concentration; most

evident frontally.

Gamma: 26–100 Hz,

cognitive or motor

function of neuron

networks. Delta: up

to 3 Hz. Adults

in slow wave sleep

and also seen

normally in babies.

Theta: 4 Hz to 7 Hz.

Young children.

Meditatative and

creative states

Source:wikipedia

• Human EEG associated with different

stages of sleep and wakefulness. (a)

Relaxed wakefulness (eyes shut) shows

rhythmic 8–12-Hz alpha waves. (b) Stage 1

non-REM sleep shows mixed frequencies,

especially 3–7-Hz theta waves. (c) Stage 2

non-REM sleep shows 12–14-Hz sleep

spindles (bursts of activity) .(d) Delta sleep

shows large-amplitude (>75 μV) 0.5–2-Hz

delta waves. (e) REM sleep shows low-

amplitude, mixed frequencies with

sawtooth waves.

• From: McGraw-Hill Encyclopedia of

Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc.

Soft excitation: supercriticalHopf bifurcation

• A mathematically natural way of turning on/off oscillations is

via parameter-dependent dynamics

dr/dt = r(c-r2)

dφ/dt = 2π

figure 5.3 Glass Mackey From

Clocks to Chaos, p. 85

Hard excitation: subcritical Hopfbifurcation

Here as a parameter increases

there suddenly arises a

stable large-amplitude

oscillation

dr/dt=r(c+2r2-r4)

dφ/dt=2π

figure 5.8 Glass Mackey From

Clocks to Chaos, p. 91

Visual hystheresis

Hysteresis in

perception: start from

the left upper corner,

follow the first row and

then proceed the same

way along the lower

row. You will notice a

jump from the

perception of a man's

face to that of a woman.

Do the same

observations in the

reverse direction: the

transition from woman

to men's face occurs at a

later point than before.

http://www.scholarpedia.org/article/Self-organization_of_brain_function

Dynamical diseases?

Glass-Mackey : irregular physiological rhythms might be associated

with deterministic chaos. Many diseases characterized by abnormal

rhythmicity could be associated with qualitative changes in

dynamics in mathematical models of physiological systems…the

terms “dynamical disease” is appropriate to describe medical

disorders characterized by abnormal dynamics associated with

abnormalities in the control systems generating and regulating the

physiological rhythms. The irregular chaotic rhythms might

correspond either to disease or to normal physiological fluctuation.

Mathematics may help in classification of diseases and development

of new therapies.

L. Glass: Dynamical disease – The impact of nonlinear dynamics and chaos on cardiology and

medicine. In Chaos and Complexity, Proceedings of the Vth Rencontre de Blois (1995)

Dynamical diseases?

Three routes to (dynamical) disease

The signature of a dynamical disease is a marked change in the

dynamics of some variable:

(constant→periodic) variables that are constant or undergoing

small amplitude random fluctuations can develop large

amplitude oscillations that may be more or less regular

(periodic to new period) new periodicities can arise in an already

periodic process

(periodic to constant or to chaotic) periodic processes can

disappear and be replaced by relatively constant dynamics or

by aperiodic (chaotic) dynamics

H.A. Reimann (1963) Periodicdiseases

From www.cnd.mcgill.ca/courses_mackey/lecture_3.ppt

Temperature oscillations in Hodgkin’s disease

From www.cnd.mcgill.ca/courses_mackey/lecture_3.ppt

Periodic hematological diseases

Periodic leukemiaFrom www.cnd.mcgill.ca/courses_mackey/lecture_3.ppt

70 days

54 days

83 days

43 days

Delay-differential equations

Delay-differential equations (DDEs) often arise in biological

systems where time lags naturally occur.

In particular, in hematology several processes are controlled

through feedback loops and these feedbacks are generally

operative only after a certain time, thus introducing a delay in

the system feedback.

DDE with constant delays

dx/dt=f(x(t),x(t-τ1), x(t-τ2),…, x(t-τn))

To obtain a solution for t > 0, one needs to specify a history

function on [−τ, 0], where τ=max(τ1 , τ2 ,…, τn ). Therefore

DDEs are often viewed as infinite-dimensional systems.

Is epilepsy a dynamical disease?

Addictions as dynamical diseases?

Several authors have suggested that chaos theory, the study of nonlinear dynamics

and the application of the knowledge gained to natural and social phenomena, might

yield insight into substance-related disorders. In this article, we examine the

dynamics of substance abuse by fitting a nonlinear model to a time series of the

amount of alcohol, which an adult male with a diagnosis of substance abuse

consumed on a daily basis. The nonlinear model shows a statistically superior fit

when compared to a linear model. We then use the model to explore a question that is

pertinent to the treatment of substance abuse, whether controlled drinking or

abstinence is a preferred strategy for maintaining sobriety.