Autapse plays an important role in regulating the electric activity of neuron by feedbacking time-delayed current on the membrane of neuron. Autapses are considered in a local area of regular network of neurons to investigate the development of spatiotemporal pattern, and emergence of spiral wave is observed while it fails to grow up and occupy the network completely. It is found that spiral wa...

Slow brain rhythms are attributed to near-simultaneous (synchronous) changes in activity in neuron populations in the brain. Because they are slow and widespread, synchronous rhythms have not been considered crucial for information processing in the waking state. Here we adapted methods from turbulence physics to analyze δ-band (1-4 Hz) rhythms in local field potential (LFP) activity, in multie...

Emergence is easy to exhibit, but very hard to formally explain. This paper deals with square sand grains moving around on nicely stacked columns in one dimension (the physical sandpile is two dimensional, but the support of sand columns is one dimensional). The Kadanoff sandpile model is a discrete dynamical system describing the evolution of finitely many sand grains falling from an hourglass...

Averaging-normalization, applied to weakly nonlinear wave equations provides a tool for identification of slow manifolds in these infinite-dimensional systems. After discussing the general procedure we demonstrate its effectiveness for a Rayleigh wave equation to find lowdimensional invariant manifolds.

Persistence and bifurcations of Lyapunov manifolds can be studied by a combination of averaging-normalization and numerical bifurcation methods. This can be extended to infinite-dimensional cases when using suitable averaging theorems. The theory is applied to the case of a parametrically excited wave equation. We find fast dynamics in a finite, resonant part of the spectrum and slow dynamics e...

The Hokusai woodcut entitled The great wave off Kanagawa has been interpreted as an unusually large storm wave, likely to be classed as a rogue wave, and possibly generated from nonlinear wave dynamics (J. H. E. Cartwright and H. Nakamura, Notes Rec. R. Soc.63, 119-135 (2009)). In this paper, we present a complementary discussion of this hypothesis, discussing in particular how linear and nonli...

The ability of cells to coordinately migrate in groups is crucial to enable them to travel long distances during embryonic development, wound healing and tumorigenesis, but the fundamental mechanisms underlying intercellular coordination during collective cell migration remain elusive despite considerable research efforts. A novel analytical framework is introduced here to explicitly detect and...

A regular network of neurons is constructed by using the Morris-Lecar (ML) neuron with the ion channels being considered, and the potential mechnism of the formation of a spiral wave is investigated in detail. Several spiral waves are initiated by blocking the target wave with artificial defects and/or partial blocking (poisoning) in ion channels. Furthermore, possible conditions for spiral wav...

We analyze the size limits of coupled map lattices with diffusive coupling at the crossover of low-dimensional to high-dimensional chaos. We investigate the existence of standing-wave-type periodic patterns, within the low-dimensional limit, in addition to the stable synchronous chaotic states depending upon the initial conditions. Further, we bring out a controlling mechanism to explain the em...