We provide a complete analysis of the Kuramoto model of coupled nonlinear oscillators with uncertain natural frequencies and arbitrary interconnection topology. Our work extends and supersedes existing, partial results for the case of an all-to-all connected network. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value all the oscillators synchronize, resulting in convergence of all phase di#erences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity.

To Larry Sirovich on the occasion of his 70th birthday ABSTRACT We study a class of permutation-symmetric globally-coupled, phase oscillator networks on N-dimensional tori. We focus on the effects of symmetry and of the forms of the coupling functions, derived from underlying Hodgkin-Huxley type neuron models, on the existence, stability, and degeneracy of phase-locked solutions in which subgroups of oscillators share common phases. We also estimate domains of attraction for the completely synchronized state. Implications for stochastically forced networks and ones with random natural frequencies are discussed and illustrated numerically. We indicate an application to modeling the brain structure locus coeruleus: an organ involved in cognitive control. 1 Introduction and

In this dissertation I develop both general results on the dynamics of neural oscil-lators and integrators and specific applications of these results to brain areas involved in simple cognitive tasks. The scientific motivation is broad: neural networks inside our brains are able to adapt to changing information processing demands by exercising cognitive control, for example focussing on salient components of noisy sensory inputs when making specific decisions based on these inputs, but relaxing this focus when other needs become prominent. But what free variables or parameters can account for the observed adaptability? And does this adaptation occur optimally, with respect to simple economic metrics and physiological limitations? Here I address these questions via reduced models of neurons and populations near bifurcations, which characterize the dynamics of a brainstem nucleus involved in adaptive cognitive control, and via variational problems arising from neural signal processing, which clarify the role of this nucleus, and other dynamical mechanisms in decision tasks. First, I study and apply nonlinear oscillator dynamics. I develop and extend phase

Arrays of identical limit-cycle oscillators have been used to model a wide variety of patternforming systems, such as neural networks, convecting fluids, laser arrays and coupled biochemical oscillators. These systems are known to exhibit rich collective behavior, from synchrony and traveling waves to spatiotemporal chaos and incoherence. Recently, Kuramoto and his colleagues reported a strange new mode of organization — here called the chimera state — in which coherence and incoherence exist side by side in the same system of oscillators. Such states have never been seen in systems with either local or global coupling; they are apparently peculiar to the intermediate case of nonlocal coupling. Here we give an exact solution for the chimera state, for a one-dimensional ring of phase oscillators coupled nonlocally by a cosine kernel. The analysis reveals that the chimera is born in a continuous bifurcation from a spatially modulated drift state, and dies in a saddle-node collision with an unstable version of itself.

Abstract not found

A modified Kuramoto model of synchronization in a finite discrete system of locally coupled oscillators is studied. The model consists of N oscillators with random natural frequencies arranged on a ring. It is shown analytically and numerically that finite-size systems may have many different synchronized stable solutions which are characterised by different values of the winding number. The lower bound for the critical coupling kc is given, as well as an algorithm for its exact calculation. It is shown that in general phase-locking does not lead to phase coherence in 1D. PACS numbers: 05.45.Xt 1.