No abstract

We investigate the bifurcation structure of stationary localised patterns of the two-dimensional SwiftHohenberg equation on an infinitely long cylinder and on the plane. On cylinders, we find localised roll, square and stripe patches that exhibit snaking and non-snaking behaviour on the same bifurcation branch. Some of these patterns snake between four saddle-node limits: recent analytical results predict then the existence of a rich bifurcation structure to asymmetric solutions, and we trace out these branches and the PDE spectra along these branches. On the plane, we study the bifurcation structure of fully localised roll structures, which are often referred to as worms. In all the above cases, we use geometric ideas and spatial-dynamics techniques to explain the phenomena we encounter.

Oscillatory activity robustly correlates with task demands during many cognitive tasks. However, not only are the network mechanisms underlying the generation of these rhythms poorly understood, but it is also still unknown to what extent they may play a functional role, as opposed to being a mere epiphenomenon. Here we study the mechanisms underlying the influence of oscillatory drive on network dynamics related to cognitive processing in simple working memory (WM), and memory recall tasks. Specifically, we investigate how the frequency of oscillatory input interacts with the intrinsic dynamics in networks of recurrently coupled spiking neurons to cause changes of state: the neuronal correlates of the corresponding cognitive process. We find that slow oscillations, in the delta and theta band, are effective in activating network states associated with memory recall. On the other hand, faster oscillations, in the beta range, can serve to clear memory states by resonantly driving transient bouts of spike synchrony which destabilize the activity. We leverage a recently derived set of exact mean-field equations for networks of quadratic integrate-and-fire neurons to systematically study the bifurcation structure in the periodically forced spiking network. Interestingly, we find that the oscillatory signals which are most effective in allowing flexible switching between network states are not smooth, pure sinusoids, but rather burst-like, with a sharp onset. We show that such periodic bursts themselves readily arise spontaneously in networks of excitatory and inhibitory neurons, and that the burst frequency can be tuned via changes in tonic drive. Finally, we show that oscillations in the gamma range can actually stabilize WM states which otherwise would not persist.

a b s t r a c tThe multistability of composite thin structures has shown potential for morphing applications. The present work combines a Ritz model with path-following algorithms to study bistable plates' behaviour. Classic low-order Ritz models predict stable shapes' geometry with reasonable accuracy. However, they may fail when modelling other aspects of the elastic structural behaviour. A refined higher-order model is here presented. In order to improve the inherently poor conditioning properties of Ritz approximations of slender structures, a non-dimensional version of Classical Plate Lamination Theory with von Kármán nonlinear strains is developed and presented. In the current approach, we continue numerical solutions in parameter space, that is, we path-follow equilibrium configurations as the control parameter varies, find stable and unstable configurations and identify bifurcations. The numerics are carried out using a set of in-house MATLAB Ò routines for numerical continuation. The increased degrees of freedom within the model are shown to accurately reflect buckling loads and provide useful insight into the relative importance of different aspects of nonlinear behaviour. Finally, the complex, experimentally observed snap-through geometry is captured analytically for the first time. Results are validated against finite elements analysis throughout the course of the paper.