Dear Pawel,

Ronny Bartsch et al. utilize ad-hoc statistical tools to quantify different cardio-respiratory synchronization regimes, as discussed here:

http://www.pnas.org/content/109/26/10181.full

I hope this is helpful.

Best, Justus

I don't think those are ad-hoc so much as they are some close cousins of the very same circular statistics that are used by any number of motor-control researchers like Peter Beek, Scott Kelso and Didier Delignieres and their colleagues.

One good way to figure out an appropriate time window may be to submit each series to an autoregression measure (autocorrelation, linear redundancy, and if you want the linearity-blind measure, perhaps average mutual information) and pick the time lag where the autocorrelation *first* drops below the significance level--or if you're using AMI, pick the time lag where you reach the first local minimum. Any or all of these would give you a well-reasoned feel for when the changes in variable X at times t are no longer trivially predictable from changes in variable X at times t-lag, and that's where you'd hope that the coherence with the other variable Y were important.

It seems in order to determine how long a meaningful period is, it would be helpful to explain what types of constraints you set on the 'quality' of synchronization.

In any case, it makes sense to define synchrony only over time periods at least longer than a number of oscillation periods (probably you don't see sync otherwise anyway)...

Window= 2^n

I recommend using wavelet coherency and phase:

https://www.researchgate.net/publication/230744210_Assessing_the_relationship_between_solar_activity_and_some_large_scale_climatic_phenomena?ev=prf_pub

http://paos.colorado.edu/research/wavelets/interdecadal.pdf

Article Assessing the relationship between solar activity and some l...

It all depends on the smalllest meaningful frequency in the signal. your periods of observation should last something like 10 cycles at least. Significance dependends on the method used for estimation and on how much noise is present.

Hi, this paper (Sun J, Hong X, and Tong S. Phase synchronization analysis of EEG signals: an evaluation based on surrogate tests, IEEE Trans Biomed Eng, 2012, 59: 2254-2263.) may give you some useful information.

Thank you very much for this important question. These topics need to be adressed and discussed since "connectivity" analysis play an important role in different fields of science.

From my point of view there is no general solution for the best time window in coherence/synchronisation etc. analysis.

Some thoughts:

Although the aimed frequency is important to determine how long the time window for an appropriate analysis should be, there are other variables that have to be taken into account as well:

How many datapoints do you have for each cycle i.e. what was the sampling rate?

What is the nature of the underlying oscillators?

Can oscillators only "switch into the same phase" once at each cycle or can they synchronize or desynchronize at every time point?

If the latter holds for your data and if you additionally have a much higher sampling rate than the analysed freuqency has you might determine the phase-synchronization (or coherence, or non-linear/linear coherence) for each sampling point (e.g. 1000 Hz sampling rate for looking at 10 Hz oscillators). Then you might even get a valid statistic for time windows below one cycle length. However, the interpretation of these data is then up to you.

On the other side the length of the time window also depends on the number of expected events your time series synchronize. If this is rather uncommon and only takes place let´s say every 100 cycles then longer time windows should be chosen while shorter windows could yield valid results for time series that "synchronize" at every one or two cycles or even within one cycle.

In order to make sure that the apparent coherency was not just a chance alignment of similar oscillations, you would need to verify that it occurs over a window longer than the beat period:

W > 1 / |f1-f2|

where W is the minimum window length and f1 and f2 are the frequencies in the two timeseries.

@ Justus

Thanks for link to the paper – its very interesting, but unfortunately no answer to my question there. As I understand authors just assumed arbitrary window length to identify synchronization segment (supported by few equations but still arbitrary).

@ Damian

Thanks for suggestion – sounds very reasonable. Would you use this approach on phases of both signals or on time series of intervals between consecutive cycles?

@ Erik

I already solved the issue of “quality” of synchronization: I'm using Rayleigh’s test for circular uniformity (of difference between phases of both signals) and set p level as e.g. 0.01.

I’m not sure what you mean by “ define synchrony only over time periods at least longer than a number of oscillation periods”. Could you please clarify?

@ Victor

Thanks for links.

You wrote “Window= 2^n”

What is n? And why “2^n”?

@ Junfeng

Thanks for link – will check it later since I have no access to IEEE.

@ Sebastian

Thanks for interesting remarks. Answering your questions:

>> What is the nature of the underlying oscillators?

These are electrograms recorded inside the atria during atrial fibrillation.

>> How many datapoints do you have for each cycle i.e. what was the sampling rate?

Sampling rate is usually 1000 Hz and dominant frequency of electrical activity is between 5-9 Hz.

>> Can oscillators only "switch into the same phase" once at each cycle or can they synchronize or desynchronize at every time point?

There is rather a gradual change of frequency (but sometimes quite rapid) and from time to time, naturally, there is a window with similar frequency in electrograms recorded at different sites.

>> If the latter holds for your data and if you additionally have a much higher sampling rate than the analysed frequency has you might determine the phase-synchronization (or coherence, or non-linear/linear coherence) for each sampling point (e.g. 1000 Hz sampling rate for looking at 10 Hz oscillators). Then you might even get a valid statistic for time windows below one cycle length. However, the interpretation of these data is then up to you.

I tried this last week and concluded it wouldn't be a valid approach in case of my signals, since statistical significance is a function of the number of data points. So when I have 200 sampling points per cycle I always get statistically significant coherence (in case of window length holding few cycles). I think I will work on intervals between consecutive cycles.

@ Adam

Thanks for suggestion. Could you please explain your approach?

@Pawel

Do you apply any filter to separate different frequency bands or do you look at a "global" coherence measure without any assumption about the frequency bins?

As far as I did get it from your description, you are "free" to define a threshold where synchrony is "true" by yourself. A total phase syncrhonization in biological systems seems not plausible. Do you want to compare synchronization before and after e.g. an intervention or between deiofferent groups? Perhaps the window length does not play that important role if you just have enough cycels to be analysed? If you try out different different window lengths, your results will probably merge into the same findings? This will more or less depend on the measure for apply fopr assessment of synchronisation I guess.

I attached a Matlab File that served for assessment of phase connectivity values between two time series (of electroencephalogram, filtered for frequency ranges). The output provides an average value for each segment, resulting in a time series (with the time resolution of one segment length) of connectivity values between the two channels. There is only poor documentation of the code and I am not sure if this was the working version of it. However you might want to have a look.

Good luck with the analysis and keep us up to date what will be the "best" window for your data and why!

Resonance in physics, chemistry and biology - http://eng.ikar.udm.ru/sb/sb22e.htm

http://eng.ikar.udm.ru/files/zip/sb22e.zip

chapter 3There is worked out a method allowing finding the areas of dynamic stability of unstable states of complex multi-component non-linear systems of the physical nature in the conditions of resonance and without in the analytical form with the necessary degree of accuracy

Dear Pawel,

I can recommend to first decompose your raw signal in separate frequencies by convolving it with wavelets that are of an integer length of the frequency under observation, i.e. exact 3 cycles long. The length of this wavelet will then differ in number of samples for each frequency. By sliding such a wavelet across your raw signal, you get a time-resolved estimate of amplitude and phase for each frequency per time bin. Next, you can investigate phase coherence between the convolved signals over longer time periods, depending on the time structure of your paradigm.

Most of these steps are well defined in the FieldTrip MATLAB toolbox, which is available for free at: http://fieldtrip.fcdonders.nl/download

Good luck

@ Sebastian,

Many thanks for suggestions and Matlab code! I had a go with it – but since it is based on peak detection it is not ideal for atrial fibrillation signals. There is a lot of noise there with quasi-periodic deflections related with passing wave (see attached image). But I figured out reconstruction of the electrograms phase so this step luckily is already behind me (and it wasn’t easy – there is a lot of variability in electrograms morphology).

>> Do you apply any filter to separate different frequency bands or do you look at a "global" coherence measure without any assumption about the frequency bins?

I’m looking at “global” coherence, without applying filters. During Atrial Fibrillation it’s not a great issue, usually there is dominant frequency (e.g. 6Hz) plus variability of interest within +-4 Hz.

>> Do you want to compare synchronization before and after e.g. an intervention or between deiofferent groups?

That is a part of analysis – but that’s straightforward: calculating a measure and applying statistical tests to check for significance.

>> Perhaps the window length does not play that important role if you just have enough cycels to be analysed?

I agree that this most likely be the case, but, as you know, it’s not enough for most of reviewers ;)

>> If you try out different different window lengths, your results will probably merge into the same findings?

Good point. I think I will use approach to find optimal window length while comparing between groups (before/after etc.). But, identification of periods of coherence is window length dependent – and then results wouldn’t merge...

Thanks again and will keep you updated!

@ Valentin

It’s not exactly what I’m looking for. Two systems don’t have to be dynamically stable to be coherent.

@ Marijn

Thanks for suggestion – I have seen few papers using wavelet coherence and I think it’s awesome method. I myself am using Hilbert transform to get phase and then Mean Phase Coherence ( Sum_t( e^(i*(phase_of_first_signal(t)- phase_of_second_signal(t))). It works well and also allows you to get sort of “spectrum of coherence”. My problem is: you find in this spectrum “blobs” corresponding with periods of high coherence. Is there a systematic way (or even a point of doing that) to find minimum length of such “blob” denoting meaningful coherence event?

Hi Pawel,

If you are not really interested in the phase of the coherency per se, you might try shifting one of your signals in time in small steps, and compute the coherence "fall-off" over time, keeping analysis windows etc. the same. To get at what is meaningful coherence in your data, you could try to estimate the baseline or random coherence present in your data (probably substantial if it is on going oscillatory process in contrast to evoked oscillations) by generating a reference distribution of expected coherence values made up by taking data that are chosen randomly over time.

Hi Marijn

Thanks for input. I like both ideas. Coherence "fall-off" was also suggested by Damian Kelty-Stephen in a first post. Will have a go with this approach.

The empirical idea is to make the relation coh=fn(W) in order to look for the plateau. If the plateau is consistent with observation - you can treat it as true.

I have used similar approach to find the window width for the estimation of dimensional complexity of the high-dimensional signal:

Modifications of the Takens-Ellner algorithm for medium-and high-dimensional signals

KP Michalak - Physical Review E, 2011 - APS

Greetings

K.M.

I don't know how so late is to participate in this discussion, but I would like to share you this paper that I found because a similar problem that I have with EEG registers. Although it doesn't deal with ECG, probably you could follow this method to find that magic time window

Thank you Krzysztof and Alexander for contributions. Both approaches make sense. The search for plateau in f-n of window size seems more straightforward and easy to implement. But on the other hand shuffling provides means to check if coherence is meaningful...

I have V2V measurments at 5.9GHz. I have filtered my data to remove the shadowing . How can I determine the optimum length of the time window to detect a good correlation?