Frontiers in Human Neuroscience (Nov 2010)
Generative models of cortical oscillations: from Kuramoto to the nonlinear Fokker–Planck equation
Abstract
Understanding the fundamental mechanisms governing fluctuating oscilla- tions in large-scale cortical circuits is a crucial prelude to a proper knowl- edge of their role in both adaptive and pathological cortical processes. Neu- roscience research in this area has much to gain from understanding the Kuromoto 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 Kuromoto model in its original, rather abstract, form and then focus on modifications that increase its neurobiological plau- sibility by incorporating topological properties of local cortical connectivity. The extended model elicits elaborate spatial patterns of synchronous oscil- lations that exhibit persistent dynamical instabilities reminiscent of cortical activity. We review how the Kuramoto model may be recast from an ordi- nary differential equation to a population level description using the nonlin- ear 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 relation- ship to basic computational processes including multistability, criticality and information capacity.
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