We develop two iteration schemes for construction of localized stationary solutions (bumps) of a one-population Wilson-Cowan model with a smoothed Heaviside firing rate function. The first scheme is based on the fixed point formulation of the stationary Wilson-Cowan model. The second one is formulated in terms of the excitation width of a bump. Using the theory of monotone operators in ordered Banach spaces we justify convergence of both iteration schemes.

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We give a necessary and sufficient condition on a positive compact operator T for the existence of a singular trace (i.e. a trace vanishing on the finite rank operators) which takes a finite non-zero value on T. This generalizes previous results by Dixmier and Varga. We also give an explicit description of these traces and associated ergodic states on l (N) using tools of non standard analysis in an essential way.

a b s t r a c tWe study the existence and continuous dependence of stationary solutions of the onepopulation Wilson-Cowan model on the steepness of the firing rate functions. We investigate the properties of the nonlinear nonlocal operators which arise when formulating the stationary one-population Wilson-Cowan model as a fixed point problem. The theory is used to study the existence and continuous dependence of localized stationary solutions of this model on the steepness of the firing rate functions. The present work generalizes and complements previously obtained results as we relax on the assumptions that the firing rate functions are given by smoothed Heaviside functions.

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The problem of structural ambiguity or “sloppiness” of a mathematical model is here studied by multivariate metamodeling techniques. If a given model is “sloppy”, it means that a number of different parameter combinations—“a neutral parameter set”—can give more or less the same model behavior and thus equally good fit to data. This paper presents a way to characterize the structure of such sloppiness. The model used for illustration is a nonlinear dynamic model of reaction kinetics—a simple version of the S‐system model. When fitted to time series data by various nonlinear curve fitting methods, an unexpected problem was discovered: For every time series, a large neutral parameter set was observed. Each of these sets was analyzed by principal component analysis and found to have clear, but nonlinear, subspace structure. The neutral parameter sets were found for many different time series data, and the global sloppiness structure of the model was characterized. This global sloppiness structure of the model allowed us to find strong correlations between parameters, and on this basis to simplify the original model. A method to reduce the ambiguity in kinetic model parameter estimates based on combining several time series is suggested. Copyright © 2014 John Wiley & Sons, Ltd.

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.

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