Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review

Abstract

Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott-Antonsen and Watanabe-Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.

Phase-dependence of response curves to deep brain stimulation and their relationship: from essential tremor patient data to a Wilson–Cowan model

“…This applies in particular for the linearisation of the WC model, another popular neuroscience model very different in essence from coupled oscillator models. In the thermodynamic limit and under certain assumptions about the distribution of oscillator frequencies, the Kuramoto model can be reduced to a two-dimensional system [43,44]. Our results are applicable to the linearisation of a fully desynchronised reduced Kuramoto model observed through X 1 = ρ cos θ where r = ρe iθ is the order parameter (ρ is the modulus and θ the angle in the complex plane).…”

“…If we pick values for β, E * and I * , the remaining parameters can be obtained by equating (44) and (45), and by re-arranging Eqs. (42) and (43).…”

“…The amplitude response curve hARC (1) is obtained by only scaling the derivative of hPRC (1) by -X 0 1 as a 2 = b 1 = 0 and F = 1. Note that WC parameters for which the system's Jacobian at the fixed point is J circ cannot be found as the second diagonal term cannot be 0, at least in the version of the WC model used in this work (see equation (44) in Appendix E).…”

Weak Chimeras in Modular Electrochemical Oscillator Networks

“…For example, other electrochemical systems and the BZ reaction can generate rich variety of clusters, in particular, when the sign of the coupling strength can also be varied (e.g., excitatory and inhibitory coupling) [31,39]. The weak chimera state could contribute to exploring chimeras in robust biological systems, e.g., circadian clocks [57], and dynamical diseases [58]; see also [59] for a recent review. Along these lines, we showed that an integrate-and-fire neuron model, with refractory period can generate bistability between cluster states in globally coupled populations [56].…”

Optimal Closed-loop Deep Brain Stimulation with Multi-Contact Electrodes

“…2, is determined by both the amplitude and phase of the system, the expectation is that stimulation will lead to a change in both these quantities, which we refer to as the instantaneous amplitude and phase response of the system. To obtain analytical expressions for these quantities, we can consider an infinite system of oscillators satisfying the ansatz of Ott and Antonsen [26,27]. In our previous work [14], we showed that for a general nPRC given by Equation (12) and assuming the natural frequencies are Lorentzian distributed with centre ω 0 and width γ, the instantaneous change in the order parameter can be written as…”