Approximations to the time evolution of an Izhikevich neuron

2014 ◽  
Vol 25 (10) ◽  
pp. 1450052 ◽  
Author(s):  
August Romeo ◽  
Hans Supèr
Keyword(s):  
Simple Model ◽  
Power Series ◽  
Small Parameter ◽  
Time Evolution ◽  
Scale Effects ◽  
Spiking Neuron ◽  
Elementary Functions ◽  
Constant Input ◽  
Parameter Expansion

Possible ways of obtaining information about the solutions of Izhikevich's "simple model" for a spiking neuron are considered. The method of power series in time is reviewed. From a different viewpoint, in the case of constant input and weak recovery scale effects, advantage is taken of a small-parameter expansion. The obtained approximations can be expressed in terms of elementary functions.

The effect of barotropic instability on the nonlinear evolution of a Rossby-wave critical layer

1989 ◽  
Vol 207 ◽  
pp. 231-266 ◽  
Author(s):  
Peter H. Haynes
Keyword(s):  
Simple Model ◽  
Numerical Model ◽  
Time Evolution ◽  
Rossby Wave ◽  
Nonlinear Evolution ◽  
Basic Flow ◽  
Critical Layer ◽  
Barotropic Instability ◽  
Long Wavelength ◽  
Subsequent Time

A study of the flow within the critical layer of a forced Rossby-wave is made, using a high-resolution numerical model. The possibility of growth of disturbances through barotropic instability and the extent to which these disturbances modify the subsequent time evolution is of particular interest. The flow is characterized by a parameter μ, equal to the cross-stream lengthscale divided by a downstream wavelength. In the long-wavelength case, μ [Lt ] 1, where there is a clear conceptual division between the instability and the basic flow, the results of the simulation confirm the importance of the growing and saturating disturbances in rearranging the vorticity within the critical layer. When the wavelength is not so long, the distinction between the instability and the straightforward time evolution of the basic flow is less clear. Nonetheless for μ < 0.25 the ultimate evolution is still sensitive to the details of the initial perturbations and in this sense the flow may be regarded as being unstable. The time-integrated absorptivity of the critical layer may be considerably increased by the effects of the instability, sometimes to three or four times that predicted by the Stewartson-Warn-Warn solution. The nature of the flow, at least during the period in which the dynamics are essentially inviscid, does not seem to change when higher harmonics to the forced wave are resonant. The behaviour seen in Béland's (1976) numerical model is re-examined in the light of these findings. A simple model of the redistribution of vorticity by the unstable disturbances is formulated, and its predictions are shown to agree well with the numerical simulations.

Approximate symmetry analysis of nonlinear Rayleigh-wave equation

2018 ◽  
Vol 15 (04) ◽  
pp. 1850055
Author(s):  
Saeede Rashidi ◽  
S. Reza Hejazi ◽  
Elham Dastranj
Keyword(s):  
Wave Equation ◽  
Power Series ◽  
Small Parameter ◽  
Conservation Laws ◽  
Rayleigh Wave ◽  
Symmetry Analysis ◽  
Approximate Symmetry ◽  
Series Method ◽  
Power Series Method ◽  
New Exact Solutions

In this paper, the Lie approximate symmetry analysis is applied to investigate the new exact solutions of the Rayleigh-wave equation. The power series method is employed to solve some of the obtained reduced ordinary differential equations with a small parameter. We yield the new analytical solutions with small parameter which is effectively obtained by the proposed method. The concept of nonlinear self-adjointness is used to construct the conservation laws for Rayleigh-wave equation. It is shown that this equation is approximately nonlinearly self-adjoint and therefore desired conservation laws can be found using appropriate formal Lagrangians.

Localisation of Carriers in Porous Silicon

1996 ◽  
Vol 452 ◽  
Author(s):  
I. Mihalcescu ◽  
J. C. Vial ◽  
R. Romestain
Keyword(s):  
Simple Model ◽  
Porous Silicon ◽  
Temperature Variation ◽  
Time Evolution ◽  
Decay Time ◽  
Additional Assumption ◽  
Shape Evolution ◽  
Photoluminescence Intensity ◽  
Photoluminescence Decay ◽  
Strong Localization

AbstractWe analyze the intensity and decay time evolution of the porous silicon luminescence upon anodic oxidation, aging, chemiral thinning and temperature variation. Strong analogies are pointed out for the photoluminescence intensity as well as for the photoluminescence decay shape evolution. They are interpreted by the variation of the extension of the carrier wavefunction induced by the modification of potential barrier efficiencies. No additional assumption such as hopping of carriers was necessary to explain the decay shapes well fitted by stretched exponential. On the contrary our observations and our simple model are in favor of a strong localization of carriers. Some experimental results are revisited within the frame of this model.

ASYMPTOTIC ANALYSIS OF A MODEL KINETIC EQUATION

1995 ◽  
Vol 05 (07) ◽  
pp. 867-885 ◽  
Author(s):  
JANUSZ R. MIKA ◽  
JACEK BANASIAK
Keyword(s):  
Exact Solution ◽  
Asymptotic Expansion ◽  
Kinetic Equation ◽  
Simple Model ◽  
Asymptotic Analysis ◽  
Small Parameter ◽  
Singularly Perturbed ◽  
Model Kinetic Equation ◽  
Linear Kinetic

For a simple model of a linear kinetic equation the exact solution is expanded in terms of a small parameter whose presence makes the equation, singularly perturbed. Various asymptotic expansion methods are analyzed and it is shown that the compressed method, which is related to the Chapman-Enskog asymptotic procedure, is the most accurate. This holds when the technique of time rescaling is applied to overcome the difficulties with the application of the standard asymptotic procedure.

ON THREE INTEGRALS ARISING IN THE THEORY OF ANISOTROPIC CRITICALITY

1996 ◽  
Vol 10 (09) ◽  
pp. 377-383
Author(s):  
YONKO T. MILLEV
Keyword(s):  
Power Series ◽  
Coupling Constant ◽  
Elementary Functions ◽  
Reduced Dimensionality ◽  
Series Representations ◽  
Anisotropic Dispersion ◽  
By Products ◽  
Double Power Series

Three integrals encountered in earlier studies of anisotropic criticality (anisotropic dispersion laws characterised by the coupling constant f) are solved in terms of elementary functions. The nature of the singularity for the case f→1 which corresponds to an effectively reduced dimensionality has been revealed. Double power-series representations for the integrals are also found and some mathematical implications and by-products are discussed. The advance relates to a prospective full-scope analysis of dipolar criticality.

A SIMPLE MODEL FOR STOCKS MARKETS

2002 ◽  
Vol 13 (05) ◽  
pp. 639-644 ◽  
Author(s):  
JUAN R. SANCHEZ
Keyword(s):  
Time Series ◽  
Simple Model ◽  
Time Evolution ◽  
Stock Markets ◽  
Stock Price ◽  
New Model ◽  
Simulated Time ◽  
Simulated Time Series

A new model for stock markets using integer values for each stock price is presented. In contrast with previously reported models, the variables used in the model are not of binary type, but of more general integer type. It is shown how the behavior of the noise and fundamentalists traders can be taken into account simultaneously in the time evolution of each stock price. The simulated time series generated by the model is analyzed in different ways order to compare parameters with those of real markets.

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