# The Problem

Many complex natural phenomena can be modeled as networks of coupled
oscillators. Examples can be drawn from the physical, chemical, and
biological world. Phase variables in these coupled systems show
characteristic pairwise phase dependencies, shown in the figure below.
Pairwise phase concentrations originate from direct

Phase dependencies for human ECoG recordings (a) filtered in the theta (6 ± 1.2 Hz) frequency band (b). The pairwise phase distribution of two neighboring electrodes (c) shows strong phase dependencies: the empirical distribution is highly concentrated in the difference of the phases, while the marginals are flat (e,f).

# Phase Coupling Concentration vs. Coupling

Phase coupling estimation recovers the true coupling from measurements in situations where phase correlations alone produce false inferences about network coupling.

**True Coupling** (first column): depicts the network
coupling used to simulate a system of oscillators;
**Pairwise Distribution** (second column): the empirical
distribution of the phase difference between oscillators A and
B; **Phase Concentration** (third column): inferred
network coupling using pairwise phase concentration
alone; **Phase Coupling Estimation** (fourth column):
inferred network coupling using our estimation technique. Each row
presents a different situation in which phase correlations provide
incorrect inference about oscillator interactions. **(a)
spurious coupling**: phase correlations in- dicate coupling
between A and B when the true coupling and coupling estimated using
our model is 0. **(b) missing coupling**: phase
correlations indicate a lack of coupling between A and B when there is
an interaction between A and B, which is correctly inferred using our
estimation technique. **(c) incorrect offset**: phase
correlations indicate that oscillator A leads oscillator B, when the
true relationship, and relationship inferred by our technique, is that
the interaction between oscillator A and oscillator B is a phase lag.

This is an example of a network of eight oscillators, in which many pairs of oscillators are coupled with unit coupling strength. The couplings are indicated by links in (a). Network effects lead to phase concentrations of varying strength between all pairs of phase variables as indicated by the red points in (b). Phase Coupling Estimation recovers the true coupling parameters as indicated by the green points in (b).

# Source Code

Matlab and Python code for Phase Coupling Estimation can be obtained here.

# References

Cadieu CF, Koepsell K (2010) Phase coupling estimation from
multivariate phase statistics. **Neural Computation**
22(12), pp. 3107 - 3126.
[pdf]

Cadieu, C., K. Koepsell (2009) Phase Coupling Estimation from Multivariate Phase Statistics. June 2009. arXiv:0906.3844v1 [nlin.AO]. [abstract]

Cadieu, C., K. Koepsell. (2008) A Multivariate Phase Distribution and its Estimation. September 2008. arXiv:0809.4291v2 [q-bio.NC]. [abstract]