My favourite tool for synch is the Kuramoto order parameter (http://en.wikipedia.org/wiki/Kuramoto_model).

The main advantage of Kuramoto order parameter is in fact its flexibility: it is capable to capture the "degree of synchronization" for rotators, i.e. oscillators that can be mapped onto a circle. In principle it works for infinite coupled oscillators, I have never tried it with just two, but I think it can give some indications. Or perhaps if you have an extended system, you could try to sample the system in several, not just two, points. In any case, it is a relatively simple tool , I think it is worth trying.

Dear Pawel, coincidentally I'm working on a very similar problem. See my question here in RG:

www.researchgate.net/post/Which_correlation_coefficient_is_better_to_use_Spearman_or_Pearson

In my case, I´m trying to use Pearson correlation coeficient to quantify how much two variables oscillatory with respect the time are in-phase or out-of-phase... So, perhaps the answers given to my question may also be useful to you.

Thanks Walter - I had a look at the question. However, its not exactly what I'm looking for. Imagine, in your problem, that you compute your correlation using sliding window and obtain a plot of time evolution of correlation coefficient. If it wouldn't be a flat line, what could you obtain from specific shape of obtained function?

Yes Pawel, I understood your problem...

Well, in such case I think that your correlation coefficient would oscillate along time between a positive value (when the two oscillatory systems are in-phase) and a negative value (when the two systems are in counter-phase). The frequency of the observed oscillations on the correlation coefficient is exactly the beat frequency due to the coupling of the two systems...

Dear Pawel,

From a theoretical point of view, synchronization cannot be applied stricto sens for the case of non-stationary signals. Certainly, synchronization is an state achieved asymptotically, something that cannot be considered in the case of non-stationary signals. However, it is possible to address the matter of coordinated activity/phase locking and to apply the framework defined by tools as the transfer entropy and methodologies based on the sliding window technique. In addition, it could be useful to go through the works of Rainer Dalhaus about partial phase synchronization and locally stationary processes.

Cheers,

David

Dear Pawel,

You question is one of my main research interests. This is an obvious challenged when one deals with interacting biomedical systems.

You might want to check my recent publication on the topic:

'Inference of time-evolving coupled dynamical systems in the presence of noise.'

https://www.researchgate.net/publication/231816443_Inference_of_time-evolving_coupled_dynamical_systems_in_the_presence_of_noise?ev=prf_pub

(or its long version https://www.researchgate.net/publication/230899403_Dynamical_Bayesian_Inference_of_Time-evolving_Interactions_From_a_Pairof_Coupled_Oscillators_to_Networks_of_Oscillators?ev=prf_pub)

Under time-evolving conditions you can study e.g. coupling strength/directionality, coupling function and synchronization. In other words you can describe the structural connectivity and the functional relationship that define the interactions, which if time-evolving can lead to sync transitions. Furthermore, you can describe also the effect from the dynamical noise and determine if the phase-slips are noise-induce.

Notes on practical implementation and Matlab codes can be found in

https://www.researchgate.net/publication/236589349_A_Tutorial_on_Time-Evolving_Dynamical_Bayesian_Inference?ev=prf_pub

Hope this helped

Article Inference of Time-Evolving Coupled Dynamical Systems in the ...

Article Dynamical Bayesian Inference of Time-evolving Interactions: ...

Article A tutorial on time-evolving dynamical Bayesian inference

Dear Pawel,

On the one hand, you can regard your two mentioned systems as a whole dynamical system with a time-varying parameter. Thus you can analyze the new's stability, i.e. the two mentioned systems' synchronization.

On the other hand, if your two mentioned systems have similar topplogy structure, then you can regard them as two dynamical systems with time-varying parameters. Thus you can analyze their synchronization by means of stability control technology.

Essentially the two aspects are the same.

All the best,

Baogui

Since you mentioned a brain as example, I assume that your problem (or system) is biological. Biology always try to reach some form of homoeostasis. When you couple another system and some form of synchronization is formed, the homoeostasis is shifted but still maintained. You can investigate your problem from this point of view by looking at the effects on homoeostasis at each time you add - couple another system and at another location.

My favourite tool for synch is the Kuramoto order parameter (http://en.wikipedia.org/wiki/Kuramoto_model).

The main advantage of Kuramoto order parameter is in fact its flexibility: it is capable to capture the "degree of synchronization" for rotators, i.e. oscillators that can be mapped onto a circle. In principle it works for infinite coupled oscillators, I have never tried it with just two, but I think it can give some indications. Or perhaps if you have an extended system, you could try to sample the system in several, not just two, points. In any case, it is a relatively simple tool , I think it is worth trying.

This question about synchronization is VERY fundamental, but the answer (Pearson correlation coefficient, etc) is not. The idea will work probably only for “two interacting systems with weak coupling” (two oscillators), but not for “spatially extended” systems (brain, chemical systems with oscillations). Two points on the surface have nothing in common with two simple oscillators. What results do you expect in case of global oscillations, irregular oscillations, spiral waves? It must be remembered that many of the processes affected by noise, which sometimes is the source of synchronization.

I worked in this area many years ago, for example:

1) Global synchronization via homogeneous nucleation in oscillating surface reactions O Kortluke, VN Kuzovkov, W von Niessen , Physical Review Letters 01/1999; 83(15):3089-3092.

2) Internal spatiotemporal stochastic resonance in the presence of weak noise. O Kortlüke, V N Kuzovkov, W von Niessen , Physical Review E 10/2002; 66(3 Pt 2A):036139.

3) Synchronization of surface reactions via Turing-like structures. R Salazar, A P J Jansen, V N Kuzovkov , Physical Review E 04/2004; 69(3 Pt 1):031604

@ David

True about synchronization reached asymptotically but I’m sure you also see benefit of quantifying this transient, partial and “imperfect” synchronization. Especially in real-world applications. I’m working in area of cardiac electrophysiology, and it’s basic characteristic is “non-stationary” and “short duration” (signals collected during experiments are usually shorter than one would wish). Thanks for tip about Rainer Dalhaus work.

@ Tomislav

Brilliant paper (“Inference of Time-Evolving Coupled Dynamical Systems in the Presence of Noise”), congrats! Directionality is example of things I’m looking for – it gives you insight about underlying dynamical structure. I started to read but will take time to wrap my head around this concept. Just one question, you applied this method to HR-respiration signals. How long signals did you use? In this example, what would be the shortest recording you would consider as appropriate to apply this method to?

@ Baogui Xin

Thank you for comment. I’m aware of such generalization we can make. E.g. having 2 coupled oscillators we can combine their 2D phase spaces and have 4D phase space in which we have whole system evolving as a point. But this is not giving us any insight into dynamical structure of our systems and their interaction.

@ Adam

Yes, my problem is biological, and more specifically “cardiac-electrophysiological”. Your comment is interesting in light of oscillations in our body that happen on different time scales (from biochemical, through heart beat, breathing, circadian and beyond). We may say that almost at every-time scale we have processes with lower/higher frequency which make our oscillation of interest non-stationary. So assuming homeostasis in analysis of oscillations is bit tricky...

@ Giovanni

Thanks for tip about Kuramoto model – for sure it’s well researched!

@ Vladimir

Thank you for suggestion of papers – will have a look.

>> What results do you expect in case of global oscillations, irregular oscillations, spiral waves?

I would rather turn this question around – I have results (in terms of time evolving synchronization measures) and I want to find out what is happening in the system (so e.g. as you say, global oscillations, irregular oscillations, spiral waves etc.).

>> It must be remembered that many of the processes affected by noise, which sometimes is the source of synchronization.

Issue of confounding effect of noise on synchronization seems to be addressed very well by Tomislav Stankovski (see few posts above).

Thank you Pawel, I am glad someone is dealing with similar problems from other perspectives.

Well for the paper we used 12min long signals, and we have never studied very short signals. But the methods is evaluated in time windows which are much shorter. (Also for the particular dynamical Bayesian inference other factors apply e.g. if one has informative priors and if it propagates the information between the windows, then the size of the windows can be reduced significantly;...)

However, in papers about cardio-respiratory interactions there is a kind of consensus that one should consider at least 5 to 10 cycles of the slower oscillation - in this case the respiration. You can find such details for example in:

https://www.researchgate.net/publication/43161158_Nonlinear_dynamics_of_cardiovascular_ageing?ev=prf_pub

Article Nonlinear dynamics of cardiovascular aging

Cross-Recurrence quantifies the degree of synchronization of two time series both globally and locally (general amount of synchronization, hot-spots of high synchronic behavior), it can be used in both normal and 'joint-recurrence' mode and has no limitation of lack of stationarity and/or linearity...see

http://www.recurrence-plot.tk/

for both documentation and downloading software

Dear Pawel,

Look at the properties of autowaves, they have self-synchronization. Perhaps the highest frequency is this parameter since these waves not seen the interference.

How about fourier transform of time domain or in other words frequency domain and let say one sensor output in y axis and other in e axis.

@ Tomislav

Thanks – that’s very important it works in case of short data segments. And very nice tutorial how to implement the method! Good for you – usually it’s extremely confusing to reproduce more complex method from the paper.

@ Alessandro

Thank you for link. I had some experience with recurrence plots. I find your website very useful, especially list of parameters quantifying diagrams. Surely I will employ those measures.

@ Martin

Sorry, I couldn’t follow you. What parameter you mean and what you mean by “waves not seen the interface”?

@ Adam

Assuming sliding window would be used, yes, that would be useful to check whether frequencies are synchronizing. That possibly could tell us if coupling between systems is affecting “base” frequencies.

Dear Pawel,

The autowaves do not possess the property of interference. The wave with the highest frequency is established throughout the whole system. This type of waves are found in brain, heart, at nerve impulse transmission, and also in the clock with a pendulum.

Some time ago I had a desire to deal with molecular systems in which such waves occur, but this is not a simple job.

Dear Pawel,

May be tis review paper may be useful?

http://xxx.lanl.gov/abs/math-ph/0510044

Michel

@ Marin

Ok, I don't use term "autowaves" normally so was bit confused. I did some modeling of spiral waves so I'm familiar little bit with their properties. Actually one of my research interests is the synchronizing property of spiral waves in heterogeneous media (in heterogeneous case, "fastest is the winner" is not always straightforward).

@ Michel

Thank you for link to paper.

Dear Pawel,

I added some Supplementary Resources (Media files = DivX movies)

to my two articles:

1) https://www.researchgate.net/publication/233996517_Kinetic_oscillations_in_the_catalytic_CO_oxidation_on_Pt_single_crystal_surfaces_Theory_and_simulation?ev=prf_pub

pt100 L1024 y20 d0100 k00 v10 kap90.avi

2)

https://www.researchgate.net/publication/233996490_Simulation_of_kinetic_oscillations_in_surface_reactions_on_reconstructing_surfaces?ev=prf_pub

pt110 L1024 y494 d250 k10 v10.avi

pt110 L2048 y500 d250 k10 v10.avi

I see the main problem is that the method of analyzing the synchronization process for spatially extended systems that you are looking for, should work for any oscillatory systems, but until now all the ideas are not universal.

Please watch these movies and imagine that you are unable to see this structure formation. You will have only the information given by your method. Will you be able to distinguish between these three examples?

Article Kinetic oscillations in the catalytic CO oxidation on Pt sin...

Article Simulation of kinetic oscillations in surface reactions on r...

Dear Vladimir,

Those patterns actually resemble fibrillation/ectopic tachycardia seen in models of cardiac wave propagation. Regarding distinguishing the patterns: I think that if you would compute phase synchronization (or maybe even simple correlation would do – period looks stable) between distant points (e.g. corners of the system), you would observe higher synchronization in case of this global “point source” scenario (pt110_L1024_y494_d250_k10_v10) and lower for this coexistence-spiral wave breakup in pt110_L2048_y500_d250_k10_v10.

Dear Pawel,

I work in human movement field, analyzing time series of stride intervals/tap intervals or whatever, with non-linear, fractal tools. Especially one recent hypothesis suggest that two interacting non-linear systems will tend to match their "complexities", i.e. their time series dynamics. This issue is tightly related to the stochastic resonance approach also.

From my point of view, synchronization does not need to be "local" or "event based". Indeed, biological systems are non-linear, but can synchronize in many ways. Also, I do not agree that biological systems tend to "homeostatis", which is by definition a state with as less fluctuations as possible. BUT biological fluctuations are not random, and the long-range correlations in such time series contain valuable informations about the system state (for example, it allow distinguish healthy vs pathological populations).

The paper below is an attempt to apply a specific tool for the analysis of synchronization of two (biological) time series, with a look at synchronization over many time scales (window of observation range from 10 intervals up to 200...).

We still try to define more precisely what informations these methods can give, but I think this might help you at least for a "conceptual" point of view, if not "technical" :)

https://www.researchgate.net/publication/257873043_Strong_anticipation_and_long-range_cross-correlation_Application_of_detrended_cross-correlation_analysis_to_human_behavioral_data?ev=prf_pub

Article Strong anticipation and long-range cross-correlation: Applic...

Dear Pawel,

Your idea ("phase synchronization... between distant points (e.g. corners of the system)") is understandable, but not enough. The problem is that you will see quantitative differences, but not quality. The correlation functions between the (only) two points on the surface ("maybe even simple correlation ") are too primitive. Oscillating systems do have not only characteristic time scale, but also the characteristic spatial scale. One should be able to see both.

It may be useful to work with more general method: set of correlation functions F(r,t), where r - length between points, 0

Dear Vladimir,

In general you are right. But in case of movies from your paper, I think simple cross-correlation would be enough to see a difference. Because in case of this "bursting" activity (waves starting at one point activating whole system) activation times at distant points will be most likely in good sync. Those examples you present are really different.

At the end of my "synchronization" deliberations, I decided to use mean phase coherence as a single number describing degree of synchronization between signal from two distant points (http://www.sciencedirect.com/science/article/pii/S0167278900000877). I think it would work nicely also in your study.

Dear Vivien,

Thank you for your paper – looks interesting and I’m sure it will be a good read. I especially like this concept of “matching complexities” – it is somewhat intuitive that complex systems with their great flexibility and adaptability will have qualitatively new ways of synchronizing. So I’m looking forward to your work extracting those mechanisms.

Regarding homeostasis – I’m not sure its possible not to agree with it. It is observation and a name for it. It is clearly visible in many examples and its difficult to question it. My PhD supervisor had interesting approach – he was talking about “dynamical homeostasis”. Which would mean that organism not only tends to keep some variables at the same level, but also complex oscillatory processes at relatively similar trajectories. But on the other hand we have organism adaptability which requires breaking homeostasis to adapt to environment. So I guess it’s, as almost always in biology, an equilibrium point between two extremes.

Dear pawel,

thanks for your comments.

Here is a nice paper from Bruce West (2010), where the author claim for considering complex (biological) systems as based on allometric control, rather than homeostatic control. The arguments are very similar to those you evoked about 'dynamical homeostasis", so I hope you'll enjoy reading it :)

http://onlinelibrary.wiley.com/store/10.1111/j.1365-2753.2010.01459.x/asset/j.1365-2753.2010.01459.x.pdf?v=1&t=hpe2qlpf&s=f46c807996fa57b8c67dc0b61f0af438f8b4206a

Hi Vivien,

Thaks for link - but I can't access the pdf there - "forbidden access". What is the paper title?

West (2010) Homeostasis and Gauss statistics: barriers to understanding natural variability. Journal of Evaluation in Clinical Practice

Maybe this link:

http://onlinelibrary.wiley.com/doi/10.1111/j.1365-2753.2010.01459.x/full

@ Vivien,

Thanks! This one worked!