
DAMOCO: Data Analysis with Models Of Coupled OscillatorsMATLAB Toolbox for multivariate time series analysisBjörn Kralemann, Michael Rosenblum, Arkady
Pikovsky
Version
2.0 (2014)

Last modification 05.06.18
This toolbox is a collection of functions for multivariate data analysis,
based on the coupled oscillator approach, developed in our publications.
With the help of this toolbox you can:
· compute protophases (initial phase estimates) from time series by means of the Hilbert Transform
· transform protophases into true phases
· reconstruct phase dynamics 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. Sorry, the manual for Version 2 is not ready yet.
You can download the whole toolbox as an archive here
or as separate files from the list of functions below.
Main functions are illustrated by examples.
Please mail us if you encounter any bug or problem in using
the toolbox! Any questions/suggestions are highly welcome.
Please cite our publications if you use this
software.
DAMOCO Version 1.0 can be found here.
Brief
illustration to the theory
o Coupled
oscillators approach: main ideas and assumptions
o Phase
dynamics of coupled oscillators
o Synchronization
and its quantification
o Direction
of coupling and its quantification
o Protophase →
phase transformation: why do we need it?
The
current version 2.0 of the toolbox contains the following functions:
Function 
What it does 
Version 
Protophase computation 

Protophase from a scalar time series, using the Hilbert transform 
27.02.2014 

Protophase from a scalar time series, using the
length of the trajectory in the state space 
10.04.2014 

Protophase from a scalar time series via markers
(minima, maxima, zerocrossings) 
06.05.2014 

Protophase from a scalar time series, via the
average cycle 
18.05.2014 

Univariate protophase → phase transformation 

Protophase → phase transformation plus computation of the transformation function 
02.03.2014 

Protophase → phase
transformation with optimization (recommended) 
02.03.2014 

Synchronization analysis 

n:m synchronization index 
17.01.2011 

Maximal n:m synchronization index for a given range of n, m 
27.02.2014 

n:m:p triplet synchronization index 
06.03.2014 

Maximal n:m:p synchronization index for a given range of n, m, p 
06.03.2014 

Two interacting oscillators: coupling functions 

Fourierbased technique, coupling functions for both
oscillators 
04.03.2014 

Fourierbased technique, coupling function for one
oscillator only 
04.03.2014 

Kernel estimation, coupling functions for both
oscillators 
03.03.2014 

Kernel estimation, coupling function for one
oscillator only 
03.03.2014 

Analysis of twodimensional coupling functions 

Norm and constant term (frequency) of the coupling function, given by the Fourier coefficients 
05.03.2014 

Norm and constant term (frequency) of the coupling function, given on the grid 
04.03.2014 

Correlation between two coupling functions (Fourierbased) 
06.03.2014 

Maximal correlation between two coupling functions
(Fourierbased) 
06.03.2014 

Correlation between two coupling functions, given on a grid 
06.03.2014 

Maximal correlation between two coupling functions,
given on a grid 
06.03.2014 

Correlation and difference measure for two coupling
functions on a grid 
02.03.2014 

Directionality index from norms of the coupling functions 
26.02.2014 

Directionality index from partial derivatives of the coupling functions 
26.02.2014 

Phase Response Curve via decomposition of the
coupling function 
28.02.2014 

More than two interacting oscillators 

Three coupled oscillators, coupling functions via a
Fourierbased technique 
06.03.2014 

Coupling structure of the triplet 
12.03.2014 

Triplet analysis of a network with N>3
oscillators 
12.03.2014 

Plot of the triplet coupling structure 
05.04.2014 

Additional functions 

Derivative of the phase via the SavitzkyGolay
filter 
06.03.2014 

Derivatives of two phases via the SavitzkyGolay filter (for bivariate
analysis) 
06.03.2014 

Derivatives of three phases via the SavitzkyGolay filter (for triplet analysis) 
06.03.2014 

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 

Auxiliary function which checks the input data 
17.01.2011 


Publications

Reference 
What it is about 
1 
M. Rosenblum and A. Pikovsky 
Notion of directionality in coupled oscillatory systems, directionality index via partial derivatives 
2,3 
B. Kralemann, L. Cimponeriu, M. Rosenblum, A.
Pikovsky, and R. Mrowka Phase
dynamics of coupled oscillators reconstructed from data, 
Notion of protophases, phase to protophase transformation, protophaase
via the length of the trajectory, recovery of autonomous frequencies 
4 
B. Kralemann, A. Pikovsky, and M. Rosenblum,
Reconstructing
phase dynamics of oscillator networks,
Chaos, 21, p. 025104, 2011 
Reconstruction of small networks, coupling functions for
three coupled oscillators, directionality index via partial norms, recovery
of autonomous frequencies 
5 
B. Kralemann, M. Frühwirth, A. Pikovsky, M. Rosenblum,
T. Kenner,
J. Schaefer, and M. Moser In vivo
cardiac phase response curve elucidates human respiratory heart rate
variability, Nature Communications, 4, 2418, 2013 
Protophase via average cycle, phase response
curve from the coupling function 
6 
B. Kralemann, A. Pikovsky, and M. Rosenblum, Detecting
triplet locking by triplet synchronization indices, Phys. Rev. E, 87, p. 052904, 2013. 
Computation of the triplet index, quantification of the
triplet locking from data 
7 
B. Kralemann, A. Pikovsky, and M. Rosenblum, Reconstructing effective phase
connectivity of oscillator networks from observations, New Journal of Physics, 16, p. 085013, 2014 
Extension of the approach from Ref. [4]; here triplet
analysis of the Noscillator network (N>3) is used instead of traditional pairwise
analysis. Improved differentiation between structurally existing and
nonexisting links. 



The
toolbox is illustrated by the following examples: new
examples are coming
Example 
Function 
Sample data 
Output 
Two coupled van der Pol oscillators 

Highlevel Fourierbased function 



Fourierbased technique, stepbystep with plots of intermediate results 



Highlevel function, iteration technique 



Iteration technique, stepbystep with plots of intermediate results 



Related links
HERMES: Integrated Toolbox to Characterize Functional
and Effective Brain Connectivity 
G. Niso, R. Bruña,
E. Pereda, R. Gutiérrez, R.
Bajo, F.
Maestú,
and F. delPozo. Neuroinform (2013) 11:405–434 
Trentool: an open source toolbox to estimate
neural directed interactions with transfer entropy 
M. Wibral, R. Vicente, V. Priesemann, and M. Lindner. BMC Neuroscience, 12 (Suppl 1): P200, 2011. 
L.
Barnett, A.K. Seth. J. Neurosci Methods, 223:5068, 2014 


