Kuramoto model

Model incredibly useful for describing synchronization, in the case of a large set of couple oscillators. While motivated for chemical/biological oscilliations, has found use in neuroscience and more specifically Computational Neuroscience for plasible modeling. Biomimetic model of corticostriatal micro-assemblies discovers new neural code(Pathak et al., 2024) used this for their Neural Circuit based approach.

ωi​, and each is also equally coupled to all the other oscillators.

Despite this being a fully Nonlinear model, an exact solution can be found as the limit number of oscillators $N\to \infty$. OR one can obtain steady state solutions of the order parameter.

$$\frac{d{\theta }_i}{dt}={\omega }_i+\frac{1}{N}\sum _{j=1}^N{K}_{ij}\sin ({\theta }_j-{\theta }_i),i=1\dots N,$$

This represents a system of $N$ limit-cycle oscillators, which have phases ${\theta }_i$ and a coupling constant of $K$.

Order Parameter

we can define the order parameters $r$ and $\Psi$ as:

$$r{e}^{i\Psi }=\frac{1}{N}\sum _{j=1}^N{e}^{i{\theta }_j}$$

Bibliography

• Pathak, A., Brincat, S. L., Organtzidis, H., Strey, H. H., Senneff, S., Antzoulatos, E. G., Mujica-Parodi, L. R., Miller, E. K., & Granger, R. (2024). Biomimetic Model of Corticostriatal Micro-Assemblies Discovers New Neural Code (p. 2023.11.06.565902). bioRxiv. https://doi.org/10.1101/2023.11.06.565902