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Spiral waves are a basic feature of excitable systems. Although such waves have been observed in a variety of biological systems, they have not been observed in the mammalian cortex during neuronal activity. Here, we report stable rotating spiral waves in rat neocortical slices visualized by voltage-sensitive dye imaging. Tissue from the occipital cortex (visual) was sectioned parallel to cortical lamina to preserve horizontal connections in layers III-V (500-m-thick, ϳ4 ϫ 6 mm 2 ). In such tangential slices, excitation waves propagated in two dimensions during cholinergic oscillations. Spiral waves occurred spontaneously and alternated with plane, ring, and irregular waves. The rotation rate of the spirals was ϳ10 turns per second, and the rotation was linked to the oscillations in a one-cycle-one-rotation manner. A small (Ͻ128 m) phase singularity occurred at the center of the spirals, about which were observed oscillations of widely distributed phases. The phase singularity drifted slowly across the tissue (ϳ1 mm/10 turns). We introduced a computational model of a cortical layer that predicted and replicated many of the features of our experimental findings. We speculate that rotating spiral waves may provide a spatial framework to organize cortical oscillations.

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We examine the existence and stability of spatially localized "bumps" of neuronal activity in a network of spiking neurons. Bumps have been proposed in mechanisms of visual orientation tuning, the rat head direction system, and working memory. We show that a bump solution can exist in a spiking network provided the neurons re asynchronously within the bump. We consider a parameter regime where the bump solution is bistable with an all-off state and can be initiated with a transient excitatory stimulus. We show that the activity pro le matches that of a corresponding population rate model. The bump in a spiking network can lose stability through partial synchronization to either a traveling wave or the all-off state. This can occur if the synaptic timescale is too fast through a dynamical effect or if a transient excitatory pulse is applied to the network. A bump can thus be activated and deactivated with excitatory inputs that may have physiological relevance.

Chimera states in networks of coupled oscillators occur when some fraction of the oscillators synchronise with one another, while the remaining oscillators are incoherent. Several groups have studied chimerae in networks of identical oscillators, but here we study these states in heterogeneous models for which the natural frequencies of the oscillators are chosen from a distribution. For a model consisting of two subnetworks we obtain exact results by reduction to a finite set of differential equations, and for a network of oscillators in a ring we generalise known results. We find that heterogeneity can destroy chimerae, destroy all states except chimerae, or destabilise chimerae in Hopf bifurcations, depending on the form of the heterogeneity.Synchronisation of interacting oscillators is a problem of fundamental importance, with applications from Josephson junction circuits to neuroscience [19,24,22,27]. Since oscillators are unlikely to be identical, the effects of heterogeneity on their collective behaviour is of interest. One well-studied system of heterogeneous phase oscillators is the Kuramoto model [4, 9, 23], for which there is global coupling. Generalisations of this model with nonlocal coupling [3, 21, 2, 10, 16], or several populations of oscillators [1], have shown interesting types of behaviour referred to as "chimera" states in which some oscillators are synchronised with one another while the remainder are incoherent. Even though the effects of heterogeneity on synchronisation have been emphasised in the past [4,13,9,23], all chimerae have so far been studied in networks of identical oscillators. This raises the obvious question: do chimerae exist in networks of nonidentical oscillators? Here we address the question analytically, first using recent results to exactly derive a finite set of differential equations governing the dynamics of chimerae in two coupled networks of heterogeneous phase oscillators, and then using a similar idea to extend the results of Abrams and Strogatz [3] for chimera states in a ring of coupled oscillators.