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## Interaction between PFC neural networks ultra-slow fluctuations and brain oscillations

The **aim** of our work was to study the influence of the different brain rhythms (i.e. theta, beta, gamma ranges with frequencies from 5 Hz to 80 Hz) on the ultra slow oscillations (USOs with frequency of 0.5 Hz and below), where high and low activity states alternate. The USOs is usually observed within neural activity in the human brain and in the prefrontal cortex in particular during rest. The USOs are considered to be generated by the local cortical circuitry together with pulse-like inputs and neuronal noise. Structure of the USOs shows specific statistics and their characteristics has been connected with cognitive abilities, such as working memory performance and capacity. In our study we used the previously constructed computational model describing activity of a cortical circuit consisting of the populations of pyramidal cells and interneurons. This model was developed to mimic global input impinging on the local PFC circuit from other cortical areas or subcortical structures. The studied the model dynamics numerically. We found that frequency increase deferentially lengthens the up states and therefore increases stability of self-sustained activity with oscillations in the gamma band. We argue that such effects would be beneficial to information processing and transfer in cortical networks with hierarchical inhibition.

Despite extensive research on face recognition, only a few studies have examined the integration of perceptual features with semantic, biographical, and episodic information. In order to address this issue, we used repetitive transcranial magnetic stimulation (rTMS) to target the left inferior frontal gyrus (IFG) and the left occipital face area (OFA) during a face recognition task. rTMS was delivered during the encoding of "context" faces (i.e., linked to an occupation, e.g., "lawyer") and "no-context" faces (i.e., linked to a nonword pattern, e.g., "xxxx"). Subjects were then asked to perform a recognition memory task. Accuracy at retrieval showed a mild decrease after left OFA stimulation, whereas rTMS over the left IFG drastically compromised memory performance selectively for no-context faces. On the other hand, absence of rTMS interference on context faces might be due either to the fact that pairing an occupation to a face makes the memory trace stronger, therefore less susceptible to rTMS interference, or to a different functional specificity of the left IFG subregions.

Neuronal oscillations have been shown to be associated with perceptual, motor and cognitive brain operations. While complex spatio-temporal dynamics are a hallmark of neuronal oscillations, they also represent a formidable challenge for the proper extraction and quantification of oscillatory activity with non-invasive recording techniques such as EEG and MEG. In order to facilitate the study of neuronal oscillations we present a general-purpose pre-processing approach, which can be applied for a wide range of analyses including but not restricted to inverse modeling and multivariate single-trial classification. The idea is to use dimensionality reduction with spatio-spectral decomposition (SSD) instead of the commonly and almost exclusively used principal component analysis (PCA). The key advantage of SSD lies in selecting components explaining oscillations-related variance instead of just any variance as in the case of PCA. For the validation of SSD pre-processing we performed extensive simulations with different inverse modeling algorithms and signal-to-noise ratios. In all these simulations SSD invariably outperformed PCA often by a large margin. Moreover, using a database of multichannel EEG recordings from 80 subjects we show that pre-processing with SSD significantly increases the performance of single-trial classification of imagined movements, compared to the classification with PCA pre-processing or without any dimensionality reduction. Our simulations and analysis of real EEG experiments show that, while not being supervised, the SSD algorithm is capable of extracting components primarily relating to the signal of interest often using as little as 20% of the data variance, instead of > 90% variance as in case of PCA. Given its ease of use, absence of supervision, and capability to efficiently reduce the dimensionality of multivariate EEG/MEG data, we advocate the application of SSD pre-processing for the analysis of spontaneous and induced neuronal oscillations in normal subjects and patients.

Most of us use numbers daily for counting, estimating quantities or formal mathematics, yet despite their importance our understanding of the brain correlates of these processes is still evolving. A neurofunctional model of mental arithmetic, proposed more than a decade ago, stimulated a substantial body of research in this area. Using quantitative meta-analyses of fMRI studies we identified brain regions concordant among studies that used number and calculation tasks. These tasks elicited activity in a set of common regions such as the inferior parietal lobule; however, the regions in which they differed were most notable, such as distinct areas of prefrontal cortices for specific arithmetic operations. Given the current knowledge, we propose an updated topographical brain atlas of mental arithmetic with improved interpretative power.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

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.