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## Effects of memristor-based coupling in the ensemble of FitzHugh-Nagumo elements

We consider a system of two identical FitzHugh-Nagumo elements with electrical and memristor-based couplings and study the effect of these couplings on the nature of neuron-like spiking regimes that were previously observed in such systems with only chemical excitatory couplings.

We study the peculiarities of spiral attractors in the Rosenzweig-MacArthur model, that relates to the life-science systems and describes dynamics in a food chain prey-predator-superpredator. It is well-known that spiral attractors having a \teacup" geometry are typical for this model at certain values of parameters for which the system can be considered as slow-fast system. We show that these attractors appear due to the Shilnikov scenario, the first step in which is associated with a supercritical Andronov{Hopf bifurcation and the last step leads to the appearance of a homoclinic attractor containing a homoclinic loop to a saddle-focus equilibrium with two-dimension unstable manifold. It is shown that the homoclinic spiral attractors together with the slow{fast behavior give rise to a new type of bursting activity in this system. Intervals of fast oscillations for such type of bursting alternate with slow motions of two types: small amplitude oscillations near a saddle-focus equilibrium and motions near a stable slow manifold of a fast subsystem. We demonstrate that such type of bursting activity can be either chaotic or regular.

Stochastic resonance (SR) is said to be observed when the presence of noise in a nonlinear system enables an output signal from the system to better represent some feature of an input signal than it does in the absence of noise. The effect has been observed in models of individual neurons, and in experiments performed on real neural systems. Despite the ubiquity of biophysical sources of stochastic noise in the nervous system, however, it has not yet been established whether neuronal computation mechanisms involved in performance of specific functions such as perception or learning might exploit such noise as an integral component, such that removal of the noise would diminish performance of these functions. In this paper we revisit the methods used to demonstrate stochastic resonance in models of single neurons. This includes a previously unreported observation in a multicompartmental model of a CA1-pyramidal cell. We also discuss, as a contrast to these classical studies, a form of 'stochastic facilitation', known as inverse stochastic resonance. We draw on the reviewed examples to argue why new approaches to studying 'stochastic facilitation' in neural systems need to be developed. © 2015 Informa UK Ltd. All rights reserved: reproduction in whole or part not permitted.

This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a three-dimensional map, which in the general case has no smooth invariant measure. It is shown that homoclinic strange attractors of discrete spiral type (discrete Shilnikov type attractors) arise in this model for certain parameters. From the viewpoint of physical motions, the trace of the contact point of a Chaplygin top on a plane is studied for the case where the phase trajectory sweeps out a discrete spiral attractor. Using the analysis of the trajectory of this trace, a conclusion is drawn about the influence of “strangeness” of the attractor on the motion pattern of the top.

This research is aimed to create a general approach to developing methods and algorithms designed for defining and providing memristor-based artificial neural network (ANNM) components operation accuracy in control and communication systems. Here we list factors which affect the operation accuracy of memristors usied as the synapses in the ANNM. We have created Simscape models of memristor neurons and ANNM for investigating the influence of these factors on the ANNM accuracy. This article presents the results of the initial phase of our research.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.