Synchronization of pulse-coupled biological oscillators
SIAM Journal on Applied Mathematics
Lending direction to neural networks
Neural Networks
Sync: The Emerging Science of Spontaneous Order
Sync: The Emerging Science of Spontaneous Order
Estimation of Non-Normalized Statistical Models by Score Matching
The Journal of Machine Learning Research
Some extensions of score matching
Computational Statistics & Data Analysis
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Coupled oscillators are prevalent throughout the physical world. Dynamical system formulations of weakly coupled oscillator systems have proven effective at capturing the properties of real-world systems and are compelling models of neural systems. However, these formulations usually deal with the forward problem: simulating a system from known coupling parameters. Here we provide a solution to the inverse problem: determining the coupling parameters from measurements. Starting from the dynamic equations of a system of symmetrically coupled phase oscillators, given by a nonlinear Langevin equation, we derive the corresponding equilibrium distribution. This formulation leads us to the maximum entropy distribution that captures pairwise phase relationships. To solve the inverse problem for this distribution, we derive a closed-form solution for estimating the phase coupling parameters from observed phase statistics. Through simulations, we show that the algorithm performs well in high dimensions (d = 100) and in cases with limited data (as few as 100 samples per dimension). In addition, we derive a regularized solution to the estimation and show that the resulting procedure improves performance when only a limited amount of data is available. Because the distribution serves as the unique maximum entropy solution for pairwise phase statistics, phase coupling estimation can be broadly applied in any situation where phase measurements are made. Under the physical interpretation, the model may be used for inferring coupling relationships within cortical networks.