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, 031915 (2006) L. Xu et al., Phys. Rev. E ( R ) 73, 065201 (2006)

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. Cardiorespiratory endurance is the ability of the body's circulatory and respiratory systems to supply fuel during sustained physical activity (USDHHS, 1996 as adapted from Corbin & Lindsey, 1994). To improve your cardiorespiratory endurance, try activities that keep your heart rate elevated at a safe level for a sustained length of time such as walking, swimming, or bicycling. The activity you choose does not have to be strenuous to improve your cardiorespiratory endurance. Start slowly with an activity you enjoy, and gradually work up to a more intense pace. interactions of locomotor and respiratory central controllers likely play an important role in regulating respiratory pattern during locomotion in birds According to the Epilepsy Foundation of America, epilepsy is a physical condition that occurs when there is a sudden, brief change in how the brain works. When brain cells are not working properly, a person's consciousness, movement, or actions may be altered for a short time. These physical changes are called epileptic seizures. Epilepsy is therefore sometimes called a seizure disorder. Epilepsy affects people in all nations and of all races. Some people can experience a seizure and not have epilepsy. For example, many young children have convulsions from fevers. These febrile convulsions are one type of seizure. Other types of seizures not classified as epilepsy include those caused by an imbalance of body fluids or chemicals or by alcohol or drug withdrawal. A single seizure does not mean that the person has epilepsy. 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

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

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

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

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?

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)

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

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

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.

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?

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 

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?

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

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

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 0.2-10Hz, f (Hz) Our detection of the synchronization between BP and BFV is valid

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.

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

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!

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.