Generative models of cortical oscillations: neurobiological implications of the kuramoto model

Abstract

Understanding the fundamental mechanisms governing fluctuating oscillations in large-scale cortical circuits is a crucial prelude to a proper knowledge of their role in both adaptive and pathological cortical processes. Neuroscience research in this area has much to gain from understanding the Kuramoto model, a mathematical model that speaks to the very nature of coupled oscillating processes, and which has elucidated the core mechanisms of a range of biological and physical phenomena. In this paper, we provide a brief introduction to the Kuramoto model in its original, rather abstract, form and then focus on modifications that increase its neurobiological plausibility by incorporating topological properties of local cortical connectivity. The extended model elicits elaborate spatial patterns of synchronous oscillations that exhibit persistent dynamical instabilities reminiscent of cortical activity. We review how the Kuramoto model may be recast from an ordinary differential equation to a population level description using the nonlinear Fokker-Planck equation. We argue that such formulations are able to provide a mechanistic and unifying explanation of oscillatory phenomena in the human cortex, such as fluctuating beta oscillations, and their relationship to basic computational processes including multistability, criticality, and information capacity.

Keywords: Fokker–Planck equation; Kuramoto model; cortical oscillations; neural synchrony.

Figure 1

Simulation results for the conventional globally-coupled Kuramoto model (N = 1024) under conditions of weak (K/N = 1), moderate (K/N = 6), and strong (K/N = 12) coupling. Top row (A–C) shows the final phases (in polar form on a unit circle) of the individual oscillators for each condition at t = 10 s. Middle row (D–F) shows the evolution of the oscillator phases during the final 5 s of each corresponding simulation. For clarity, only the first 64 of the 1024 oscillators are shown. (G) Shows the effect of coupling strength (K/N = 0–14) on the phase coherence (r_∞) of 1024 oscillators at t = 10 s. (H) Shows the Gaussian distribution of natural oscillator frequencies used in these simulations.

Figure 2

Spatial patterns of phase locking in the 1D Kuramoto model (n = 128) at convergence (t = 10 s) under conditions of global versus local synaptic kernels. Top row shows the spatial pattern of phase locking adopted by the oscillators (left panel) when coupled using a cosine-with-distance kernel which extends to infinity (as shown in the top-right panel). The bottom-left panel shows the spatial pattern of phase locking evoked by a local kernel corresponding to the fourth derivative of a Gaussian (as shown in the bottom-right panel). Initial conditions were identical in both cases and natural frequencies were normally distributed as per Figure 1.

Figure 3

Spatial patterns of phase-locking in the 2D Kuramoto model (128 × 128) using a local kernel corresponding to the fourth derivative of the Laplacian of the Gaussian. As before, the natural oscillator frequencies were normally distributed (M = 0 Hz, SD = 0.5 Hz).

Figure 4

(A–C) Show the second order phase interaction employed by Hansel and colleagues with β = 0. 25 and three values of R. A saddle-node bifurcation in the fixed points associated with this function occurs as the second peak in this curve crosses 0 (left to right).

Figure 5

Spatiotemporal dynamics in systems of oscillators coupled through a local spatial kernel and a second order PIF, equation (13). Columns left to right depict results for increasing phase offset β between the two modes. Top row (A–C) shows representative oscillator states using the same color scheme as Figure 3. Bottom row (D–F) shows the coupling tension F as defined by equation (12) where blue denotes F = 0.

Figure 6

As with Figure 5 with the exception of a larger spatial kernel whose outer extent is upgoing (phase advancing) hence engendering large coherent fronts and spiral waves.