The Spike Order of the Winnerless Competition (WLC) Model and Its Application to the Inhibition Neural System

Liu, Shenquan; Tao, Fan; Lu, Qishao

doi:10.1515/ijnsns.2005.6.2.133

Cited by 15 publications

(10 citation statements)

References 9 publications

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“…It is well known that the Burger-Huxley equation describes many phenomena such as busting oscillation [9], interspike [24], population genetics [1], bifurcation and chaos [33]. In the past few decades, various analytical methods were proposed to solve the Burger-Huxley equation.…”

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confidence: 99%

A robust adaptive grid method for singularly perturbed Burger-Huxley equations

Liu¹,

Liang²,

Zhang³

et al. 2020

era

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In this paper, an adaptive grid method is proposed to solve onedimensional unsteady singularly perturbed Burger-Huxley equation with appropriate initial and boundary conditions. Firstly, we use the classical backward-Euler scheme on a uniform mesh to approximate time derivative. The resulting nonlinear singularly perturbed semi-discrete problem is linearized by using Newton-Raphson-Kantorovich approximation method which is quadratically convergent. Then, an upwind finite difference scheme on an adaptive nonuniform grid is used for space derivative. The nonuniform grid is generated by equidistribution of a positive monitor function, which is similar to the arclength function. It is shown that the presented adaptive grid method is first order uniform convergent in the time and spatial directions, respectively. Finally, numerical results are given to validate the theoretical results.

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confidence: 99%

A robust adaptive grid method for singularly perturbed Burger-Huxley equations

Liu¹,

Liang²,

Zhang³

et al. 2020

era

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“…It was the first model to correctly describe the ionic basis of excitation. Many fascinating phenomena such as bursting oscillator (Duan and Lu, 2006), interspike (Liu et al, 2005), bifurcation and chaos (Zhang et al, 2006), are also associated with this equation. (Wang et al, 1990;Ismail et al, 2004).…”

Section: Particular Casesmentioning

confidence: 99%

Singularly perturbed Burger-Huxley equation: Analytical solution through iteration

Kamboj¹,

Sharma²

2018

Int. J. Eng. Sci. Tech

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The work presented considers the initial boundary value problem for nonlinear singularly perturbed time dependent Burger-Huxley equation. The equation contains two terms with nonlinearities, the cubic term and the advection term. Generally, the severe difficulties of two types encounter in solving this problem. The first one comes from these nonlinearity terms and the second is due to the perturbation parameter in the diffusion term. When the perturbation parameter approaches zero, the problem exhibit boundary layers and most of conventional methods fails to capture this effect. A relatively different analytical technique called variational iteration method (VIM), is used to solve this equation. This method is able to obtain rapidly convergent successive approximations of exact solution without any restrictive approximations or the transformations that may change the physical behaviour of the problem. Numerical examples are studied to demonstrate the significance and effect of various parameters involved in the equation.

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“…where g (y) = / (Ay + Ax *+«,)-/ (AX * +«,) and e m (y) = e m (y + x*)-* m (x*) , which implies #(θ) = 0 and e m (o) = Q.The first equation in system(3)or(6)is the continuous component of the impulsive neural networks (3) or (6). and called the impulse-free neural networks[13][14][15][16][17][18][19][20].2 Main resultsWe now state our main result.Theorem 1. Let e m (y) = m y,m = \,2,-.…”

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confidence: 94%