
DAMOCO: Data Analysis with Models Of Coupled OscillatorsMATLAB Toolbox for multivariate time series analysisBjörn Kralemann, Michael Rosenblum, Arkady Pikovsky 
This toolbox is a collection of functions for bivariate data analysis, based on the coupled oscillator approach,
developed in our publications. With the help of this toolbox you can:
transform phase estimates, or protophases, of scalar signals into true phases
reconstruct models of coupled oscillators from data
compute synchronization and directionality indices
A description of the toolbox and of how to use it can be found in this manual.
Please mail us if you encounter any problem in using the toolbox! Any questions/suggestions are highly welcome.
Please cite our publications if you use this software!
You can download the whole toolbox as an archive here or as separate files from the list of functions below.
The current version 1.0 of the toolbox contains the following functions:
Function 
What it does 
Version 
General purpose functions 

Auxiliary functions which checks the input data 
17.01.2011 

Instantaneous protophase from a scalar time series, using the Hilbert transform 
17.01.2011 

n:m synchronization index 
17.01.2011 

Maximal n:m synchronization index for a given range of n, m 
17.01.2011 

Directionality index from norms of the coupling functions 
17.01.2011 

Directionality index from partial derivatives of the coupling functions 
17.01.2011 

Univariate transformation 

Onedimensional Fourierbased protophase → phase transformation 
17.01.2011 

Bivariate transformation 

Highlevel function which performs twodimensional protophase → phase transformation and computes the coupling functions, frequencies, directionality index. Fourierbased technique. 
17.01.2011 

Highlevel function which performs twodimensional protophase → phase transformation and computes the coupling functions, frequencies, directionality index. Iteration technique. 
17.01.2011 

Given two (proto)phases, the function yields two coupling functions via fitting a Fourier series 
17.01.2011 

Similar to co_fexp2.m, but only one coupling function is computed 
17.01.2011 

Using the output of co_fexp2.m , this function computes the bivariate protophase → phase transformation function 
17.01.2011 

Using the output of co_fbsolv, this function performs the bivariate protophase → phase transformation 
17.01.2011 

Norm of the coupling function, given by the Fourier coefficients 
17.01.2011 

Using the output of co_fexp2.m or, this function computes the bivariate protophase → phase transformation function by iteration technique; it also returns frequencies, true coupling functions, and their norms. 
17.01.2011 

Using the output of co_itersolv, this function performs the bivariate protophase → phase transformation 
17.01.2011 

Additional functions 

Plot of the coupling function 
17.01.2011 

Plot of two coupling functions in the same window 
17.01.2011 

Plot of the Fourier coefficients of the coupling function 
17.01.2011 

Plot of the Fourier coefficients of two coupling functions 
17.01.2011 

Correlation between two coupling functions (Fourierbased) 
17.01.2011 

Correlation between two coupling functions, given on a grid 
17.01.2011 
The toolbox is illustrated by the following examples:
Example 
Function 
Sample data 
Output 
Two coupled van der Pol oscillators 

Highlevel Fourierbased function 

Fourierbased technique, stepbystep with plots of intermediate results 
same as above 

Highlevel function, iteration technique 

Iteration technique, stepbystep with plots of intermediate results 
same as above 
This site is still under construction and (we hope) will be improved soon!
The toolbox is also under development. What is coming:
efficient algorithms based on kernel density estimation;
functions for analysis of three coupled oscillators.
B. Kralemann, L. Cimponeriu, M.G. Rosenblum, A.S.
Pikovsky, and R. Mrowka,
Phase dynamics of coupled
oscillators reconstructed from data,
Physical Review E, 77,
p. 066205, 2008, pdf file (2
Mb)
B. Kralemann, L. Cimponeriu, M.G. Rosenblum, A.S.
Pikovsky, and R. Mrowka
Uncovering interaction of coupled
oscillators from data,
Physical Review E, 76, p.
055201, 2007, pdf file (280 Kb)
M.G. Rosenblum and A.S.
Pikovsky,
Detecting direction of coupling in interacting
oscillators,
Physical
Review E, 64, p. 045202, 2001, pdf
file (218 Kb)