.

you might find some inspiration from the two links below (related to synchrony scores between trains of impulses from spiking neurons ; it will not include any notion of cyclicity, however)

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

http://www.scholarpedia.org/article/SPIKE-distance

.

Thank you for your answer. I already read some paper regarding spike distance, but i operate with 100 trees that act as oscillators, so it is a bit hard to operate in this manner. I guess.

.

"100 trees acting as oscillators" ???

could you be more specific ? just curious ...

.

Each tree fruits with a specific period/cycle. So it can be model as an individual biologic oscillator. The fruition can be modeled as a frequency. But generally trees fruit together / at the same  - they tend to synchronize their fruition frequency. That is why I consider them biologic oscillators :-)

I had to attack that once. I tried using the coefficient of variation of the combined spike activity.  The idea is that if the oscillators are completely desynchronized, the c.v. would be towards zero because the overall oscillation would be averaged out.  If the oscillators were completely synchronized, then the c.v. would be highest.   For my purposes, which was to show that a circuit affected synchrony, it worked well enough.

This is an interesting approach. Once I used the cv of voronoi areas for assessing the degree of uniformity in space, so I am familiar somehow with this. Thank you - it might work.

Could you show us how is your database like? can each tree be consider as an observation and the measures  are 

1) The etymological meaning of the word synchronous which derives from the greek works χρόνος (that is time) and σύν (that is same, common), suggests that synchronization stands for two or more phenomena which occurs at the same time or recurring with some kind of periodicity.

A more informative definition of synchronization: "We consider synchronization as an adjustment of rhythms of oscillating objects due to their weak interaction." 

Measure of synchronization among oscillating objects results from the phase-locking analysis of their signals [that means answering to the question:  how much their trajectories in the phase-plane do diverge ? ]. The analysis of phase locking is a sensible way of detecting synchronization since it captures the covariations in the relative timing of the oscillations while taking apart amplitude fluctuation. However, the method should be used with caution in signals that have broad band spectra or whenever a phase cannot be defined unambiguously.

2) From original signal to analytic signal: Gabor’s method for detecting instantaneous parameters of the signal

If we consider a smoother signal resembling a sine wave with slowly varying frequency and amplitude, the phase of such a narrow-band signal can be obtained by means of the analytic signal concept originally introduced by Gabor. To implement it, one has to construct from a scalar signal s(t) a complex process

z(t) = s(t) + isH(t)= A(t)eiφ(t)

where sH(t) is the Hilbert transform of s(t). The instantaneous phase φ(t) and instantaneous amplitude A(t) of the signal are thus uniquely determined from that equation.

Thus, the analytic signal z(t) has a real part s(t) which is the original data and an imaginary part sH(t) which contains the Hilbert transform. The Hilbert transformed series has the same amplitude and frequency content as the original real data and includes phase information that depends on the phase of the original data [Rosemblum M.G. et al., “Analyzing synchronization phenomena from bivariate data by means of the Hilbert transform”, in Nonlinear Analysis of Physiological data, 91-99, 1998 ].

3) Index of phase locking

The general approach to the detection of phase synchrony relies on computing a correlation statistic from the data. 

The aim is to test whether the phase difference between the signals is bounded, that is: 

|nφ1(t) - mφ2(t)|

Thank you for you answer, but it's a little too much for me :-)

I have to slowly digest all of that and I don't know if I can adapt this to fruition periodicity.  Anyway, thank you and I will give it a try.

Golomb et al. 1997, Heller et al. 1995 and Kjaer et al. 1994 have performed several statistical analyses of synchrony phase and otherwise. You will find a detail list with explanation of different measures of synchrony in the first article. However, it seems advisable that coherence cannot be equated with synchrony as pointed out in the last link. Were it I, have used Hilbert-Huang transform algorithm as few rhythms are so immediate and overlapping that it is difficult to examine them in time window.

http://www.dauwels.com/files/233.pdf

http://arrow.dit.ie/cgi/viewcontent.cgi?article=1114&context=dmccon

http://graduatestudents.ucmerced.edu/aloan/Vitae_files/Frame-Differencing%20Methods%20for%20Measuring%20Bodily%20Synchrony%20in%20Conversation.pdf

http://www.neuralsystemsandcircuits.com/content/pdf/2042-1001-2-5.pdf

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3314884/pdf/nihms-364897.pdf

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

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

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

http://www.nature.com/nature/journal/v421/n6921/full/nature01285.html

I am sorry about the incomplete answer, I have wirtten it from my phone an d I do not know what happened.

Can each tree be considered as a differenent observation? Are the measureaments taken at the same time interval ? If so, I would recommend Functional Data Analysis. Its techniques are implemented in R in the fda package (parametric version) and fda.usc (non-parametric version). I have done some works with fda, and I can help you with the computation.

Keep in touch!

Gigi