scholarly journals Pattern formation in a 2-population homogenized neuronal network model

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Karina Kolodina ◽  
John Wyller ◽  
Anna Oleynik ◽  
Mads Peter Sørensen
Keyword(s):  
Pattern Formation ◽  
Firing Rate ◽  
Evolution Equations ◽  
Spatial Dimension ◽  
Wave Number ◽  
Forming Process ◽  
Invariant Model ◽  
Neural Field Model ◽  
Starting Point ◽  
Rate Functions

AbstractWe study pattern formation in a 2-population homogenized neural field model of the Hopfield type in one spatial dimension with periodic microstructure. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. It is shown that the nonlocal synaptic interactions promote a finite band width instability. The stability method relies on a sequence of wave-number dependent invariants of $2\times 2$ 2 × 2 -stability matrices representing the sequence of Fourier-transformed linearized evolution equations for the perturbation imposed on the homogeneous background. The generic picture of the instability structure consists of a finite set of well-separated gain bands. In the shallow firing rate regime the nonlinear development of the instability is determined by means of the translational invariant model with connectivity kernels replaced with the corresponding period averaged connectivity functions. In the steep firing rate regime the pattern formation process depends sensitively on the spatial localization of the connectivity kernels: For strongly localized kernels this process is determined by the translational invariant model with period averaged connectivity kernels, whereas in the complementary regime of weak and moderate localization requires the homogenized model as a starting point for the analysis. We follow the development of the instability numerically into the nonlinear regime for both steep and shallow firing rate functions when the connectivity kernels are modeled by means of an exponentially decaying function. We also study the pattern forming process numerically as a function of the heterogeneity parameters in four different regimes ranging from the weakly modulated case to the strongly heterogeneous case. For the weakly modulated regime, we observe that stable spatial oscillations are formed in the steep firing rate regime, whereas we get spatiotemporal oscillations in the shallow regime of the firing rate functions.

scholarly journals Raman active modes of single-wall boron nitride nanotubes inside carbon nanotubes

2018 ◽  
Vol 3 (2) ◽  
Author(s):  
A.M. Nassir ◽  
Ah Rahmani ◽  
M. Boutahir ◽  
B. Fakrach ◽  
H. Chadli ◽  
...  
Keyword(s):  
Boron Nitride ◽  
Inelastic Neutron Scattering ◽  
Large Diameter ◽  
Chemical Properties ◽  
Spatial Dimension ◽  
Inelastic Neutron ◽  
Boron Nitride Nanotubes ◽  
Hybrid Nanostructures ◽  
Wave Number ◽  
Physical And Chemical

The structure of boron–nitride nanotubes (BNNTs) is very similar to that of CNTs, and they exhibit many similar physical and chemical properties. In particular, a single walled boron nitride nanotube (BNNT) and a single walled carbon nanotube (CNT) have been reported. The spectral moment’s method (SMM) was shown to be a powerful tool for determining vibrational spectra (infrared absorption, Raman scattering and inelastic neutron-scattering spectra) of harmonic systems. This method can be applied to very large systems, whatever the type of atomic forces, the spatial dimension, and structure of the material. The calculations of vibrational properties of BNNT@CNT double-walled hybrid nanostructures are performed in the framework of the force constants model, using the spectral moment's method (SMM). A Lennard–Jones potential is used to describe the van der Waals in-teractions between inner and outer tubes in hybrid systems. The calculation of the BNNT@CNT Raman active modes as a function of the diameter and chirality of the inner and outer tubes allows us to derive the diameter dependence of the wave number of the breathing-like modes, intermediate-like modes and tangential-like modes in a large diameter range. These predictions are useful to interpret the experimental data.

scholarly journals An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc

1986 ◽  
Vol 406 (1830) ◽  
pp. 93-106 ◽  
Keyword(s):  
Boundary Layer ◽  
Nonlinear Problems ◽  
Wave Number ◽  
Viscous Effects ◽  
Rotating Disc ◽  
Expansion Procedure ◽  
Asymptotic Investigation ◽  
Starting Point ◽  
Consistent Manner ◽  
High Reynolds

The paper investigates high-Reynolds-number stationary instabilities in the boundary layer on a rotating disc. The investigation demonstrates that, in addition to the inviscid mode found by Gregory, Stuart & Walker ( Phil. Trans. R. Soc. Lond. A 248, 155 (1955)) at high Reynolds numbers, there is a stationary short-wavelength mode. This mode has its structure fixed by a balance between viscous and Coriolis forces and cannot be described by an inviscid theory. The asymptotic structure of the wave-number and orientation of this mode is obtained, and a similar analysis is given for the inviscid mode. The expansion procedure provides the capacity of taking non-parallel effects into account in a self-consistent manner. The inviscid solution of Gregory et al . is modified to take account of viscous effects. The expansion procedure used is again capable of taking non-parallel effects into account. The results obtained suggest why the inviscid approach of Gregory et al . should give a good approximation to the experimentally measured orientation of the vortices. The results also explain partly why the inviscid analysis should not give such a good approximation to the wavenumber of the vortices. The asymptotic analysis of both modes provides a starting point for the corresponding nonlinear problems.

FIFTH-ORDER EVOLUTION EQUATION OF GRAVITY–CAPILLARY WAVES

2017 ◽  
Vol 59 (1) ◽  
pp. 103-114
Author(s):  
DIPANKAR CHOWDHURY ◽  
SUMA DEBSARMA
Keyword(s):  
Evolution Equation ◽  
Evolution Equations ◽  
Capillary Wave ◽  
Wave Train ◽  
Critical Value ◽  
Capillary Waves ◽  
Wave Number ◽  
Region Of Stability ◽  
Fifth Order ◽  
Nonlinear Gravity

We extend the evolution equation for weak nonlinear gravity–capillary waves by including fifth-order nonlinear terms. Stability properties of a uniform Stokes gravity–capillary wave train is studied using the evolution equation obtained here. The region of stability in the perturbed wave-number plane determined by the fifth-order evolution equation is compared with that determined by third- and fourth-order evolution equations. We find that if the wave number of longitudinal perturbations exceeds a certain critical value, a uniform gravity–capillary wave train becomes unstable. This critical value increases as the wave steepness increases.

A general theory for the dynamics of thin viscous sheets

2002 ◽  
Vol 457 ◽  
pp. 255-283 ◽  
Author(s):  
N. M. RIBE
Keyword(s):  
Evolution Equations ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Extensional Flow ◽  
Constant Thickness ◽  
Scaling Analysis ◽  
Two Dimensional ◽  
Principal Curvatures ◽  
Kinematic Evolution ◽  
Starting Point

A model for the deformation of thin viscous sheets of arbitrary shape subject to arbitrary loading is presented. The starting point is a scaling analysis based on an analytical solution of the Stokes equations for the flow in a shallow (nearly planar) sheet with constant thickness T0 and principal curvatures k1 and k2, loaded by an harmonic normal stress with wavenumbers q1 and q2 in the directions of principal curvature. Two distinct types of deformation can occur: an ‘inextensional’ (bending) mode when [mid ]L3(k1q22 + k2q21)[mid ] [Lt ] ε, and a ‘membrane’ (stretching) mode when [mid ]L3(k1q22 + k2q21)[mid ] [Gt ] ε, where L ≡ (q21 + q22)−1/2 and ε = T0/L [Lt ] 1. The scales revealed by the shallow-sheet solution together with asympotic expansions in powers of ε are used to reduce the three-dimensional equations for the flow in the sheet to a set of equivalent two-dimensional equations, valid in both the inextensional and membrane limits, for the velocity U of the sheet midsurface. Finally, kinematic evolution equations for the sheet shape (metric and curvature tensors) and thickness are derived. Illustrative numerical solutions of the equations are presented for a variety of buoyancy-driven deformations that exhibit buckling instabilities. A collapsing hemispherical dome with radius L deforms initially in a compressional membrane mode, except in bending boundary layers of width ∼ (εL)1/2 near a clamped equatorial edge, and is unstable to a buckling mode which propagates into the dome from that edge. Buckling instabilities are suppressed by the extensional flow in a sagging inverted dome (pendant drop), which consequently evolves entirely in the membrane mode. A two-dimensional viscous jet falling onto a rigid plate exhibits steady periodic folding, the frequency of which varies with the jet height and extrusion rate in a way similar to that observed experimentally.

scholarly journals Effect of Capillarity on Fourth Order Nonlinear Evolution Equation for Two Stokes Wave Trains in Deep Water in the Presence of Air Flowing Over Water

2015 ◽  
Vol 20 (2) ◽  
pp. 267-282
Author(s):  
A.K. Dhar ◽  
J. Mondal
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Water Waves ◽  
Nonlinear Evolution ◽  
Wave Train ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Stokes Wave ◽  
Instability Wave ◽  
Starting Point

Abstract Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves, are derived for deep water surface capillary gravity waves in the presence of second waves in which air is blowing over water. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. A stability analysis is made for a uniform wave train in the presence of a second wave train. Graphs are plotted for the maximum growth rate of instability wave number at marginal stability and wave number separation of fastest growing sideband component against wave steepness. Significant improvements are noticed from the results obtained from the two coupled third order nonlinear Schrödinger equations.

The general structure of integrable evolution equations

1979 ◽  
Vol 365 (1722) ◽  
pp. 283-311 ◽  
Keyword(s):  
Evolution Equations ◽  
Entire Functions ◽  
General Structure ◽  
Spatial Dimension ◽  
Nonlinear Ordinary Differential Equation ◽  
Singular Perturbation Theory ◽  
Field Gradients ◽  
Scattering Transform ◽  
Integrable Evolution

This paper presents some new results in connection with the structure of integrable evolution equations. It is found that the most general integrable evolution equations in one spatial dimension which is solvable using the inverse scattering transform (i. s. t.) associated with the n th order eigenvalue problem V x = ( ξR 0 + P ( x , t )) V has the simple and elegant form G ( D R , t ) P t – F ( D R , t ) x [ R 0 , P ] = Ω ( D R , t ) [ C , P ], where G , F and Ω are entire functions of an integro-differential operatos D R and the bracket refers to the commutator. The list provided by this form is not exhaustive but contains most of the known integrable equations and many new ones of both mathematical and physical significance. The simple structure allows the identification in a straightforward manner of the equation in this class which is closest to a given equation of interest. The x dependent coefficients enable the inclusion of the effects of field gradients. Furthermore when the partial derivative with respect to t is zero, the remaining equation class contains many nonlinear ordinary differential equation of importance, such as the Painleve equations of the second and third kind. The properties of the scattering matrix A( ξ , t ) corresponding to the potential P( x , t ) are investigated and in particular the time evolution of A ( ξ , t ) is found to be G ( ξ , t ) A t + F ( ξ , t ) A ξ = Ω ( ξ , t )[ C , A ], The rôle of the diagonal entries and the principal corner minors in providing the Hamiltonian structure and constants of the motion is discussed. The central rôle that certain quadratic products of the eigenfunctions play in the theory is briefly described and the necessary groundwork from a singular perturbation theory is given when n = 2 or 3.

scholarly journals Core and Periphery of Information Society: Significance of Geospatial Technologies

2013 ◽  
Vol 32 (2) ◽  
pp. 39-49
Author(s):  
Piotr A. Werner ◽  
Tomasz Opach
Keyword(s):  
Information Society ◽  
Information Technologies ◽  
Communication Technologies ◽  
Spatial Dimension ◽  
Global Information ◽  
Geospatial Technologies ◽  
Development Level ◽  
Starting Point ◽  
Global Information Society ◽  
Is Development

Abstract The paper attempts to identify important factors significant for global information society development and to determine the significance of geospatial (geo-information) technologies. The starting point is international measures of the development level of information & communication technologies (ICT) and information society (IS). The relevance of the particular factors was defined using the general segmentation of the milieu, taking into account social, technological, economic, environmental, political, legal and ethical factors and also estimating the global spatial dimension of ICT and IS development. The diagnosis serves as the context of considerations concerning the contribution of geographers and cartographers to IS.

Study on the Surface Quality of Fender NC Incremental Forming

2011 ◽  
Vol 335-336 ◽  
pp. 523-526
Author(s):  
Liu Ru Zhou
Keyword(s):  
Sheet Metal ◽  
Sheet Metal Forming ◽  
Metal Forming ◽  
Process Time ◽  
Forming Process ◽  
Forming Technology ◽  
Starting Point ◽  
Incremental Sheet Metal Forming ◽  
Metal Deformation ◽  
Improvement Method

The NC incremental sheet metal forming technology is a flexible forming technology without dedicated forming dies. The forming locus of the forming tool can be adjusted by correcting the numerical model of the product. Because the deformation of sheet metal only occurs around the tool head and the deformed region is subjected to stretch deformation, the deformed region of sheet metal thins, and surface area increases. Sheet metal forming stepwise is to lead to the whole sheet metal deformation. The principle of NC incremental sheet metal forming and the forming process of the fender are introduced. The effect of process parameters on forming is analysed. The improvement method of the forming quality is suggested. The groove is created in the starting point of tool moving when the starting point of tool moving locus at all layers is identical. The groove can be eliminated when the starting point of tool moving locus at all layers is different. The feed pitch p increase, the process time decrease, production rate and surface degree of roughness increase. In general, the feed pitch is 0.25mm.

Single-point incremental forming of sheet metals: Experimental study and numerical simulation

2016 ◽  
Vol 231 (2) ◽  
pp. 301-312 ◽  
Author(s):  
Matteo Benedetti ◽  
Vigilio Fontanari ◽  
Bernardo Monelli ◽  
Marco Tassan
Keyword(s):  
Numerical Simulation ◽  
Finite Element ◽  
Incremental Forming ◽  
Single Point ◽  
Integration Scheme ◽  
Al Alloy ◽  
Single Point Incremental Forming ◽  
Sheet Metals ◽  
Forming Process ◽  
Starting Point

In this article, the single-point incremental forming of sheet metals made of micro-alloyed steel and Al alloy is investigated by combining the results of numerical simulation and experimental characterization, performed during the process, as well as on the final product. A finite element model was developed to perform the process simulation, based on an explicit dynamic time integration scheme. The finite element outcomes were validated by comparison with experimental results. In particular, forming forces during the process, as well as the final shape and strain distribution on the finished component, were measured. The obtained results showed the capability of the finite element modelling to predict the material deformation process. This can be considered as a starting point for the reliable definition of the single-point incremental forming process parameters, thus avoiding expensive trial-and-error approaches, based on extensive experimental campaigns, with beneficial effects on production time.

scholarly journals On a direct construction of inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension

2004 ◽  
Vol 2004 (58) ◽  
pp. 3117-3128
Author(s):  
H. H. Chen ◽  
J. E. Lin
Keyword(s):  
Inverse Scattering ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Spatial Dimension ◽  
Nonlinear Evolution Equations ◽  
Scattering Problems ◽  
Direct Construction ◽  
Inverse Scattering Problems ◽  
Petviashvili Equation ◽  
The One

We present a method to construct inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension. The temporal component is the adjoint of the linearized equation and the spatial component is a partial differential equation with respect to the spatial variables. Although this idea has been known for the one-spatial dimension for some time, it is the first time that this method is presented for the case of the higher-spatial dimension. We present this method in detail for the Veselov-Novikov equation and the Kadomtsev-Petviashvili equation.

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