Spatially localized solutions of the Hammerstein equation with sigmoid type of nonlinearity

Abstract

We study the existence of fixed points to a parameterized Hammertstain operator H β , β ∈ (0, ∞], with sigmoid type of nonlinearity. The parameter β < ∞ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case β = ∞ corresponds to a discontinuous unit step function. We prove that spatially localized solutions to the fixed point problem for large β exist and can be approximated by the fixed points of H∞. These results are of a high importance in biological applications where one often approximates the smooth sigmoid by discontinuous unit step function. Moreover, in order to achieve even better approximation than a solution of the limit problem, we employ the iterative method that has several advantages compared to other existing methods. For example, this method can be used to construct non-isolated homoclinic orbit of a Hamiltionian system of equations. We illustrate the results and advantages of the numerical method for stationary versions of the FitzHugh-Nagumo reaction-diffusion equation and a neural field model.2000 Mathematics Subject Classification. 47H30, 47N60, 47G10, 45L05, 45J05, 92B20.

Neural field equations with neuron‐dependent Heaviside‐type activation function and spatial‐dependent delay

“…The main investigation framework in our research was a Volterra–Hammerstein integral inclusion obtained from the former neural field equation by convexification of the discontinuous activation function. The generality of this approach allowed us to omit the standard Assumption 1 used in the investigations 11,12,17,18 of neural field models involving Heaviside‐type activation functions. The latter fact, in turn, allows to study the so‐called sliding modes in the neural field equations.…”

“…This assumption played a crucial role in the proof of solvability of an initial value problem for (), 12 as well as in the investigations 11,17,18 of particular types of solutions to ().…”

О Непрерывных И Разрывных Моделях Нейронных Полей

“…В [6] также был предложен численный метод построения пространственно локализованного стационарного симметричного решения U λ для непрерывной функции активации f = f λ , λ ∈ (0, 1], использующий известное решение U 0 уравнения (1), отвечающее функции активации f типа Хевисайда.…”

“…В [6] получены условия существования симметричных стационарных локализованных решений уравнения (1) в случае Ω = R и сигмоидальной функции активации, а также предложена схема аппроксимации таких решений соответствующими решениями для случая функции активации типа Хевисайда. Приведем данные результаты.…”

Regularization of Ill-Posed Point Neuron Models

“…The limit process , using different techniques, is studied in [ 18 , 19 ] for the stationary solutions of neural field equations. It has also been observed [ 20 ] for the Wilson–Cowan model that this transition is a subtle matter: Using a steep sigmoid firing rate function instead of the Heaviside mapping can lead to significant changes in a Hopf bifurcation point.…”