Synchronization in complex networks of phase oscillators: A survey <s> Canonical Coupled Oscillator Model <s> Abstract The impressive variety of biological rhythms leaves no doubt that autonomously periodic processes contribute to the coordination of life-processes. The question here raised is, “What modes of temporal organization—if any—could result from weak interactions in a population of innately oscillatory devices (e.g. electronic oscillators, secretory cells, spontaneously active neurons, or individual animals)?” By mathematical analysis, electronic experiments, and digital computer simulation, answers are found as functions of the nature of the periodic processes and the kind of interactions involved. For populations of “generalized relaxation oscillators”, threshold conditions are discovered for mutual synchronization in any of a variety of modes. It is proposed that self-entraining communities of this sort may exist within individual metazoan animals and plants as the basis of the observed diurnal coordination of their physiological process. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical Coupled Oscillator Model <s> Abstract We study a pair of weakly coupled van der Pol oscillators and investigate the bifurcations of phase-locked periodic motions which occur as the coupling coefficients are varied. Perturbation methods are used and their relation to the topological structure of solutions in the four dimensional phase space is discussed. While the problem is formulated for general linear coupling, the case of detuning plus diffusive coupling via displacement and velocity is discussed in more detail. It is shown that up to four phase-locked periodic motions can exist in this case. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical Coupled Oscillator Model <s> 1 Introduction.- 2 Bifurcations in Neuron Dynamics.- 3 Neural Networks.- 4 Introduction to Canonical Models.- 5 Local Analysis of WCNNs.- 6 Local Analysis of Singularly Perturbed WCNNs.- 7 Local Analysis of Weakly Connected Maps.- 8 Saddle-Node on a Limit Cycle.- 9 Weakly Connected Oscillators.- 10 Multiple Andronov-Hopf Bifurcation.- 11 Multiple Cusp Bifurcation.- 12 Quasi-Static Bifurcations.- 13 Synaptic Organizations of the Brain.- References. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical Coupled Oscillator Model <s> We undertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli. Recalling and extending results from the literature, we compute phase response curves (PRCs) valid near bifurcations to periodic firing for Hindmarsh-Rose, HodgkinHuxley, FitzHugh-Nagumo, and Morris-Lecar models, encompassing the four generic (codimension one) bifurcations. Phase density equations are then used to analyze the role of the bifurcation, and the resulting PRC, in responses to stimuli. In particular, we explore the interplay among stimulus duration, baseline firing frequency, and population-level response patterns. We interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical Coupled Oscillator Model <s> This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition. In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics. This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience. It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology. "Dynamical Systems in Neuroscience" presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons. It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties. The book introduces dynamical systems starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems. Each chapter proceeds from the simple to the complex, and provides sample problems at the end. The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians. Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines. Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum - or taught by math or physics department in a way that is suitable for students of biology. This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical Coupled Oscillator Model <s> Synchronization is crucial to wireless sensor networks. Recently a pulse-coupled synchronization strategy that emulates biological pulse-coupled agents has been used to achieve this goal. We propose to optimize the phase response function such that synchronization rate is maximized. Since the synchronization rate is increased independently of transmission power, energy consumption is reduced, hence extending the life of battery-powered sensor networks. Comparison with existing phase response functions confirms the effectiveness of the method. <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Canonical Coupled Oscillator Model <s> Oscillator models are central to the study of system properties such as entrainment or synchronization. Due to their nonlinear nature, few system-theoretic tools exist to analyze those models. The paper develops a sensitivity analysis for phase response curves, a fundamental one-dimensional phase reduction of oscillator models. The proposed theoretical and numerical robustness analysis tools are illustrated on several models arising in the systems biology of cellular rhythms. <s> BIB007
In the preceding subsections we have seen how the coupled-oscillator model (1) appears naturally in various applications. At first glance, diffusively-coupled phase oscillator models of the general form (4) appear to be quite specific. We now illustrate how the general phase oscillator network model (4) (and its particular instance (1)) can be obtained as a local canonical model of weakly coupled and periodic limit-cycle oscillators BIB003 . Our presentation is informal, we schematically follow the approaches developed in the computational neuroscience community, and we refer to the textbooks BIB003 BIB005 , the tutorials   , and the pioneering papers BIB001 Kopell, 1984, 1991) for details. Consider a limit-cycle oscillator modeled as a dynamical system with state x ∈ R m and nonlinear dynamicṡ x = f (x). Assume that this system admits a locally exponentially stable periodic orbit S ⊂ R m with period T > 0. By a local change of variables, any trajectory in a neighborhood of S can be characterized by a phase variable ϕ ∈ S 1 with dynamicsφ = Ω, where Ω = 2π/T . Now consider a weakly-forced oscillator of the forṁ where ε > 0 is sufficiently small and g(t) is a timedependent external forcing term. For sufficiently small forcing εg(t), the attractive limit cycle S persists, and the local phase dynamics are obtained aṡ where ϕ → Q(ϕ) is the infinitesimal phase response curve (iPRC) and we dropped higher order terms O(ε 2 ). The iPRC is a linear response function that associates to each point on the periodic orbit S (parameterized by the phase ϕ) the phase shift induced by the input εg(t). Now consider n such limit-cycle oscillators. Let x i ∈ R m be the state of oscillator i with limit cycle S i ⊂ R m and period T i > 0. We assume that the oscillators are weakly coupled with interconnected dynamics given bẏ Here, g ij is the coupling function for the oscillators i, j ∈ {1, . . . , n}. The coupling functions may be continuous or impulsive or take the value zero if oscillators i and j are not interacting. The weak coupling in (13) can be identified with the weak forcing in (12), and a transformation to phase coordinates results in the local phase model where Ω i = 2π/T i . By changing variables as θ i (t) = ϕ i (t) − Ω i t, we arrive at the coupled phase dynamicṡ An averaging analysis applied to the θ-dynamics yieldṡ where the averaged coupling functions h ij are given by and ω i = h ii (0). Notice that the averaged coupling functions h ij are 2π-periodic and the coupling in (14) is diffusive. In the slow time scale τ = εt, the averaged dynamics (14) equal the phase oscillator network (4). This analysis justifies calling the phase oscillator network (4) a local canonical model for weakly coupled limit-cycle oscillators. It also explains the widespread adoption of phase oscillator models as phenomenological model in synchronization studies. If the interaction among the oscillators is anti-symmetric, then all functions h ij are odd, and a first-order Fourier series expansion yields h ij (·) ≈ a ij sin(·) as first harmonic with coefficient a ij . In this case, the dynamics (14) in the slow time scale τ = εt reduce to the coupled oscillator model (1). As a prototypical example, for two van der Pol oscillators (with parameters in the quasi-harmonic limit) coupled through a resistor, the above procedure results exactly in the coupled oscillator model (1), see BIB002 . In general, the coupling functions h ij depend on the iPRC and may not be sinusoidal. Hence, the iPRC serves as a natural analysis  BIB007 BIB004 and design BIB006 tool for general limit-cycle oscillator networks. X r e iψ = 0 