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 couplings as well as from indirect network effects. Direct experimental measurements mix these two effects and in order to effectively determine neural interactions we must separate these two contributions. We solve this problem by estimating the unique maximum entropy distribution that gives rise to the observed pariwise phase dependencies.

ECoG phase dependencies

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.

simple phase dependencies 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.

complex phase dependencies 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]