However, “physiologically precise” models can quickly become highly parameterized, making systematic explorations of their dynamics an increasing challenge. The cortex has a very detailed cytoarchitectural and physiological make-up and, clearly, this detail is crucial to its many specific functions. Whereas the latter, detailed models allow the study of precise mechanisms to explain specific empirical observations, the former, more abstracted approach seeks to elucidate fundamental mechanisms that may underpin a variety of apparently diverse neurophysiological phenomena. Models hence range from relatively simple algebraic forms, through increasing complexity, to very detailed networks of multi-compartment neurons connected via specific synaptic maps. We believe that unraveling the laws governing fluctuations in large-scale cortical oscillations is a necessary precursor to understanding their role in adaptive and pathological cortical functions.Ĭomputational studies adopt a variety of abstractions in order to deal with complex dynamical systems like the brain. Activity in this field has almost exclusively been devoted toward empirical research, although related advances in computational neuroscience can provide important insights into the fundamental mechanisms of oscillatory activity in neuronal systems. Whilst neurophysiological data attest to the role of high frequency oscillations, there is also tremendous interest in slow frequency (below 0.1 Hz) activity in resting state networks, as evident in functional neuroimaging data ( Biswal et al., 2005). Fluctuations in beta amplitude also appear in spontaneous cortical activity ( Freyer et al., 2009) but are greatly muted in a number of pathological conditions such as Parkinson’s disease ( Eusebio and Brown, 2009). This occurs across a range of spatial and temporal scales, from percept-related changes in gamma oscillations recorded invasively within neuronal microcircuits (e.g., Bressler and Freeman, 1980), to motor-related modulations in cortical beta oscillations observable in extracranial recordings (e.g., Boonstra et al., 2007 Houweling et al., 2010). Over the last few decades, extensive neurophysiological research has established the intimate association between adaptive perceptual and behavioral processes and fluctuating oscillatory activity in the cortex. 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. 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. The extended model elicits elaborate spatial patterns of synchronous oscillations that exhibit persistent dynamical instabilities reminiscent of cortical activity.
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. 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. 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. 5 Research Institute MOVE, VU University Amsterdam, Amsterdam, Netherlands.4 Royal Brisbane and Women’s Hospital, Brisbane, QLD, Australia.3 Queensland Institute of Medical Research, Brisbane, QLD, Australia.2 The Black Dog Institute, Prince of Wales Hospital, Sydney, NSW, Australia.1 School of Psychiatry, University of New South Wales, Sydney, NSW, Australia.Stewart Heitmann 1,2 and Andreas Daffertshofer 5
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