1. Phase synchronization in coupled nonlinear oscillators
Limei Xu
Department of Physics
Boston University
Collaborators: Plamen. Ch. Ivanov, Zhi Chen, Kun Hu
H. Eugene Stanley
It is such an pleasure to be here! Thanks, Lazlso for giving me the opportunity. Thanks all of you for coming to my talk, It is an honor to speak my work to all of you.
Today I will talk about the Phase synchronization in coupled nonlinear oscillators
The work is collobrated with Zhi Chen, P. Ch. Ivanov, Kun Hu, and my adviosr H. E. Stanley
Z. Chen et al., Phys. Rev. E 73, (2006)
L. Xu et al., Phys. Rev. E ( R ) 73, (2006)

2. Why study synchronization?
Synchronization is often present in physiological systems
Example:
In human heart coupled cells fire synchronously, producing a periodic rhythm governing the contractions of the heart
Alteration of synchronization may provide important
physiological information:
Parkinson’s disease, synchronously firing of many neurons results in tremor
Synchronization is a fundamental phenomenon, and it plays an important role in various fields of science and engineering. It is an indication of coupling between systems. This phenomenon was first observed by Hugens in 17 centrery in the system of two coupled pendelum clock. When the rhythm of one of the pendelum is changed, the second one adjusts it rhythm due to the coupling consequently. They will oscillate together.
Very often it is found in living systems, being observed on a level of single cells, physiological subsystems, organisms, and even on the level of populations. Sometimes this phenomenon is effential for a normal functioning of a system. For example, for the performance of the pacemaker inside heart: many coupled of cells inside heart tend to coordinate and fire synchronously, which produce a macroscopic rhythm than controls the heart contraction, respiration, etc.
Alteration of the synchronization may provide important physiology information. For example, the onset of synchronization leads to a severe pathlogy, in the case of the Parkinson’s disease, synchronization of many neurons in the central nervous systems results in tremor.
In general, the signals from living systems can be treated as the outputs of self-sustained oscillators, and synchronization indicates the adjustment their rhythms due to coupling interations. In the following, I give the defination of self-sutained oscillator, and types of sel-sustained oscillator we may encounter in analyzing physioligical signals.
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interactions of locomotor and respiratory central controllers likely play an important role in regulating respiratory pattern during locomotion in birds
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PD: The primary symptoms are the results of excessive muscle contraction, normally caused by the insufficient formation and action of dopamine, which is produced in the dopaminergic neurons of the brain.
in the case of Parkinson’s disease, when locking of many neurons in the central nervous system results in the tremor activity

3. Concept of synchronization
Huygens clock (1657)
two identical pendulums + common bar
Phenomenon:
anti-phase synchronized α x
Synchronization of two self-sustained oscillators due to coupling
Self-sustained oscillators exhibit “regular” rhythm
driven by internal source
How to describe the motion of self-sustained oscillator?
The first observation of synchronization dates back to more than 300 hundred years ago by Huygens when he tried to design a marinetime clock.
I schetch the system here. It consists of three parts: two identical pendulum, and a common metal bar.
He found that two pendulum was always anti-phase synchronized. Why this happened?
As we know that they share the same bar on the top, and they are coupled! The energy can transfer from one oscillator to the other through the bar. Just because of the coupling,one can possible observe synchronization.
On the other hand, by detecting sychronization, we can measure the coupling between different systems.
Very often, we encounter the situation of detecting coupling from two time series
Very often, we encounter to extract information from time series,
If we can extract information from time series, we can detect the coupling
between systems by synchronization approach
two identical pendulem share the same metal bar, in the sense that they are coupled, he found that the two clock was always anti-phased. Each of the pendulum is a self-sustained oscillator, rhythm is driven by internal source sqt(l/g). Biological signals can be considered as the output of a self-sustained oscillators. Next, I will discuss how to describe the motion of a self-sustained oscillator
Pendulum is a simple example of self-sustained oscillators. Self-sustained is in the sense that oscillator is driven by internal source. It continues to oscillate without external forcing. It is free to perturbation. For example, When you try to perturbate the system, it can adjust itself quickly to its own rhythm after a short time.
How to describe the motion of the oscillator?
Huygens pendulum clock: l=9 inchesx2.54cm=24.83cm, m=0.5lb

4. Linear self-sustained oscillators
x(t) : time series of a self-sustained oscillator
x(t) 1 φ 2
y(t) 3
Description of the motion:
One variable is not enough!
x(t) + i y(t)
Aeiφ(t) .
Phase:
tanΦ(t) =y/x
(angular speed Φ(t)= constant)
Amplitude:
A2=x2+y2
(constant)
Output signals of biological systems are more complex=>nonlinear oscillators

5. Nonlinear Oscillators: signal of human postural control
X1(Red): back-forth sway X2(Black): left-right sway
X1(t)
X2(t)
time
Characteristics:
both amplitude and rhythm are time dependent
extraction of amplitude and phase is nontrivial

6. Extract phase and amplitude from signal
for arbitrary time series x(t), analog to the case of linear oscillator
time
x(t)
y(t)
x(t)
A(t)
Φ(t)
time dependent amplitude of the original signal is preserved
for time series of postural control: x1(t), x2(t)
x1(t) + і y1(t)
A1(t)eiφ1(t)
x2(t) + і y2(t)
A2(t)eiφ2(t)
How to detect coupling between two time series?

7. Coupled nonlinear oscillators
time series of human postural control a)
Phase A2
Phase difference
time A1
Amplitude: not synchronized synchronized: A2/A1  constant
Phase: synchronized
Phase information is important to detect coupling between systems!
A. Piously et al., Intl. J. Bifurcation and Chaos, Vol. 10, 2291 (2000)

8. Characterization of Phase Synchronization
ΔΦmod2π
Strong
synchronization
distribution P(ΔΦ)
Weak No
no coupling
Phase difference ΔΦ
strong coupling
weak coupling
no coupling uniform distribution
weak coupling broad distribution
strong couplingsharp distribution
Synchronization is a measure of the strength of coupling

9. Detect of correlation between blood flow velocity in the brain and blood pressure in the limbs for healthy and stroke subjects

10. Background: cerebral autoregulation
Mean blood pressure
Mean cerebral blood flow
50mm Hg
150mm Hg
This control may be impaired with disease, e.g., stroke.
Cerebral autoregulation controls cerebral blood flow velocity (BFV) in the brain in response to blood pressure (BP) in the limbs.
How it works for normal people and unhealthy people
Cerebral autoregulation is an autonomic function to keep constant blood flow velocity (BFV) in the brain even when blood pressure (BP) changes.

11. Application to Cerebral Autoregulation
Health subject
Stroke subject
t (s)
Note: bring the constant of band-pass filtering: low frequency band (trend)
Characteristics: (1) regular rhythm 1Hz (driven by heart beat)
(2) nonstationarity
Question: Is there any difference in the correlations between two signals for healthy subject and unhealthy subjects?

12. Amplitude cross-correlation between BP and BFV
Healthy subject
Stroke subject
Amplitude cross-correlation
Time lag
Time lag
Amplitude cross correlation:
C()=<A(t)BP A(t+ )BFV>/<A(0)BP A(0 )BFV>
Simple approach does not work well 

13. Phase synchronization between BFV and BP
Healthy subject
Healthy subject
 (t)BP- (t)BFV
Stroke subject
stroke subject
time (s)
Healthy subject: weak synchronization
Stroke subject: strong synchronization
A change in blood pressure, how long its effects on BFV will last?

14. Cross-correlation of phases between BFV and BP
C()=<(t)BP (t- )BFV>/<(0)BP (0 )BFV>
C()
Healthy subject
Stroke subject
time lag 
correlation strength at time 0: C0
time correlation length 0: time lag at which correlation drops by 80%
Note: example of sudden get up from the bed
Health subject: short rangedweak synchronization  Cerebral autoregulation works
Stroke subject: long ranged strong synchronization Cerebral autoregulation impaired

15. Diagnosis for stroke? 11 healthy 12 stroke subjects
time correlation length 0
stroke subjects: the instantaneous response of BFV to the change
of BP is more pronounced and lasts longer compared to healthy subjects

16. Effect of band-pass filtering on phase synchronization
phase synchronization approach:
sensitive to the choice of band-pass filtering
excessive band-pass filtering spurious detection
filtering:
Low frequency range 0-0.2Hz
nonstationarity (trend)
High frequency:
f>10Hz, same results
power spectrum
Zhi’s power spectrum Hz,
f (Hz)
Our detection of the synchronization between BP and BFV is valid

17. Conclusion
Phase synchronization is a useful concept to quantify complex behavior in coupled nonlinear systems.
Useful approach to detect the coupling between systems
Phase synchronization has practical applications to physiological data and is useful to understand mechanisms of physiologic regulation.

18. Effect of band-pass filtering for the phase synchronization approach
MEG Signal for Parkinson’s patient
Power spectrum for Parkinson’ patient
time
More than one peak
broad power spectrum

19. Effect of band-pass filtering on external noise on coupled output of oscillators
High noise hides synchronization
filtering recovers true behavior
Increasing Noise strength
narrowing down band-width
Band-pass filtering is effective to reduce external noise in coupled systems!

20. Effect of band-pass filtering on phase synchronization
Remaining band-width
Band-pass filtering generate spurious synchronization
Recipes for band-pass filtering
Phase synchronization is very sensitive to the choice of band-width,
it is effective to reduce external noise
excess of band-pass generating spurious detection of phase synchronization
Therefore, many trials has to be done before getting consistent results.