We review the derivation of the homogenized one-population Amari equation by means of the two-scale convergence technique of Nguetseng in the case of periodic microvariation in the connectivity function. A key point in this derivation is Visintin's theorem for two-scale convergence of convolution integrals. We construct single bump solutions of the resulting homogenized equation using a pinning function technique for the case where the solutions are independent of the local variable and the firing rate function is modelled as a unit step function. The parameter measuring the degree of heterogeneity plays the role of a control parameter. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. A framework for analysing the stability of these structures is formulated. This framework is based on spectral theory for Hilbert-Schmidt integral operators and it deforms to the standard Evans function approach for the translational invariant case in the limit of no heterogeneity. The upper and lower bounds of the growth/decay rates of the perturbations imposed on the bump states can be expressed in terms of the operator norm of the actual Hilbert-Schmidt operator. Intervals for which the pinning function is increasing correspond to unstable bumps, while complementary intervals where the pinning function decreases correspond to stable bumps, just as in the translational invariant case. Examples showing the properties of the bumps are discussed in detail when the connectivity kernels 3 Deceased

We study existence and stability of 2-bump solutions of the one-population homogenized Wilson-Cowan model, where the heterogeneity is built in the connectivity functions by assuming periodic modulations in both the synaptic footprint and in the spatial scale. The existence analysis reveals that the generic picture consists of two bumps states for each admissible threshold value for the case when the solutions are independent of the local variable and the firing rate function is modeled as a Heaviside function. A framework for analyzing the stability of 2-bumps is formulated, based on spectral theory for Fredholm integral operators. The stability method deforms to the standard Evans function approach for the translationally invariant case in the limit of no heterogeneity, in a way analogous to the single bump case for the homogenized model. Numerical study of the stability problem reveals that both the broad and narrow bumps are unstable just as in the translationally invariant case when the connectivity function is modeled by means of a wizard hat function. For the damped oscillating connectivity kernel, we give a concrete example of a 2-bump solution which is stable for all admissible values of the heterogeneity parameter.

We study numerically single bump solutions of a homogenized Amari equation with periodic microvariation. Two attempts are made to detect single bumps that depend on the microvariable. The first attempt which is based on a pinning function technique is applicable in the Heaviside limit of the firing rate function. In the second attempt, we develop a numerical scheme which combines the two-scale convergence theory and an iteration procedure for the corresponding heterogeneous Amari equation. The numerical simulations in both attempts indicate the nonexistence of single bump solutions that depend on the microvariable. Motivated by this result, we finally develop a fixed point iteration scheme for the construction of single bump solutions that are independent of the microvariable when the firing rate function is given by a sigmoidal firing rate function.

Ved Institutt for matematikk og statistikk UiT – Norges arktiske universitet ble Brukerkurset i matematikk høsten 2020 lagt om til et prosjektbasert emne og ny eksamensform med avsluttende muntlig gruppeeksamen. Et av målene for vår studie er å kartlegge ferdigheter og forståelse i algebra i begynnelsen av semesteret og hvordan studentene videre møter de algebraiske aktivitetene i prosjektene. I denne artikkelen rapporterer vi på resultater fra en pre- og en posttest blant studentene i emnet høst 2020 og drøfter algebra i de obligatoriske prosjektene.