154 research outputs found
Storing cycles in Hopfield-type networks with pseudoinverse learning rule: admissibility and network topology
Cyclic patterns of neuronal activity are ubiquitous in animal nervous
systems, and partially responsible for generating and controlling rhythmic
movements such as locomotion, respiration, swallowing and so on. Clarifying the
role of the network connectivities for generating cyclic patterns is
fundamental for understanding the generation of rhythmic movements. In this
paper, the storage of binary cycles in neural networks is investigated. We call
a cycle admissible if a connectivity matrix satisfying the cycle's
transition conditions exists, and construct it using the pseudoinverse learning
rule. Our main focus is on the structural features of admissible cycles and
corresponding network topology. We show that is admissible if and only
if its discrete Fourier transform contains exactly nonzero
columns. Based on the decomposition of the rows of into loops, where a
loop is the set of all cyclic permutations of a row, cycles are classified as
simple cycles, separable or inseparable composite cycles. Simple cycles contain
rows from one loop only, and the network topology is a feedforward chain with
feedback to one neuron if the loop-vectors in are cyclic permutations
of each other. Composite cycles contain rows from at least two disjoint loops,
and the neurons corresponding to the rows in from the same loop are
identified with a cluster. Networks constructed from separable composite cycles
decompose into completely isolated clusters. For inseparable composite cycles
at least two clusters are connected, and the cluster-connectivity is related to
the intersections of the spaces spanned by the loop-vectors of the clusters.
Simulations showing successfully retrieved cycles in continuous-time
Hopfield-type networks and in networks of spiking neurons are presented.Comment: 48 pages, 3 figure
Nonlinear Interaction of Transversal Modes in a CO2 Laser
We show the possibility of achieving experimentally a Takens-Bogdanov
bifurcation for the nonlinear interaction of two transverse modes ()
in a laser. The system has a basic O(2) symmetry which is perturbed by
some symmetry-breaking effects that still preserve the symmetry. The
pattern dynamics near this codimension two bifurcation under such symmetries is
described. This dynamics changes drastically when the laser properties are
modified.Comment: 16 pages, 0 figure
Reduced dynamics and symmetric solutions for globally coupled weakly dissipative oscillators
This is a preprint of an article whose final and definitive form has been published in DYNAMICAL SYSTEMS © 2005 copyright Taylor & Francis; DYNAMICAL SYSTEMS is available online at: http://www.informaworld.com/openurl?genre=article&issn=1468-9367&volume=20&issue=3&spage=333Systems of coupled oscillators may exhibit spontaneous dynamical formation of attracting synchronized clusters with broken symmetry; this can be helpful in modelling various physical processes. Analytical computation of the stability of synchronized cluster states is usually impossible for arbitrary nonlinear oscillators. In this paper we examine a particular class of strongly nonlinear oscillators that are analytically tractable. We examine the effect of isochronicity (a turning point in the dependence of period on energy) of periodic oscillators on clustered states of globally coupled oscillator networks. We extend previous work on networks of weakly dissipative globally coupled nonlinear Hamiltonian oscillators to give conditions for the existence and stability of certain clustered periodic states under the assumption that dissipation and coupling are small and of similar order. This is verified by numerical simulations on an example system of oscillators that are weakly dissipative perturbations of a planar Hamiltonian oscillator with a quartic potential. Finally we use the reduced phase-energy model derived from the weakly dissipative case to motivate a new class of phase-energy models that can be usefully employed for understanding effects such as clustering and torus breakup in more general coupled oscillator systems. We see that the property of isochronicity usefully generalizes to such systems, and we examine some examples of their attracting dynamics
Lagrangian Coherent Structures in tropical cyclone intensification
Atmos. Chem. Phys., 12 5483-5507The article of record as published may be located at http://dx.doi.org/10.5194/acpd-11-1-201
Pattern formation in annular convection
This study of spatio-temporal pattern formation in an annulus is motivated by
two physical problems on vastly different scales. The first is atmospheric
convection in the equatorial plane between the warm surface of the Earth and
the cold tropopause, modeled by the two dimensional Boussinesq equations. The
second is annular electroconvection in a thin semetic film, where experiments
reveal the birth of convection-like vortices in the plane as the electric field
intensity is increased. This is modeled by two dimensional Navier-Stokes
equations coupled with a simplified version of Maxwell's equations. The two
models share fundamental mathematical properties and satisfy the prerequisites
for application of O(2)-equivariant bifurcation theory. We show this can give
predictions of interesting dynamics, including stationary and spatio-temporal
patterns
Coupled Mean Flow-Amplitude Equations for Nearly Inviscid Parametrically Driven Surface Waves
Nearly inviscid parametrically excited surface gravity-capillary waves in two-dimensional periodic domains of finite depth and both small and large aspect ratio are considered. Coupled equations describing the evolution of the amplitudes of resonant left- and right-traveling waves and their interaction with a mean flow in the bulk are derived, and the conditions for their validity established. In general the mean flow consists of an inviscid part together with a viscous streaming flow driven by a tangential stress due to an oscillating viscous boundary layer near the free surface and a tangential velocity due to a bottom boundary layer. These forcing mechanisms are important even in the limit of vanishing viscosity, and provide boundary conditions for the Navier-Stokes equation satisfied by the mean flow in the bulk. The streaming flow is responsible for several instabilities leading to pattern drift
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