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## Contribution of synchronized GABAergic neurons to dopaminergic neuron firing and bursting

In the ventral tegmental area (VTA), interactions between dopamine (DA) and gamma-aminobutyric acid (GABA) neurons are critical for regulating DA neuron activity and thus DA efflux. To provide a mechanistic explanation of how GABA neurons influence DA neuron firing, we developed a circuit model of the VTA. The model is based on feed-forward inhibition and recreates canonical features of the VTA neurons. Simulations revealed that gamma-aminobutyric acid (GABA) receptor (GABAR) stimulation can differentially influence the firing pattern of the DA neuron, depending on the level of synchronization among GABA neurons. Asynchronous activity of GABA neurons provides a constant level of inhibition to the DA neuron and, when removed, produces a classical disinhibition burst. In contrast, when GABA neurons are synchronized by common synaptic input, their influence evokes additional spikes in the DA neuron, resulting in increased measures of firing and bursting. Distinct from previous mechanisms, the increases were not based on lowered firing rate of the GABA neurons or weaker hyperpolarization by the GABAR synaptic current. This phenomenon was induced by GABAmediated hyperpolarization of the DA neuron that leads to decreases in intracellular calcium (Ca2+) concentration, thus reducing the Ca2+ dependent potassium (K+) current. In this way, the GABA-mediated hyperpolarization replaces Ca2+-dependent K+ current; however, this inhibition is pulsatile, which allows the DA neuron to fire during the rhythmic pauses in inhibition. Our results emphasize the importance of inhibition in the VTA, which has been discussed in many studies, and suggest a novel mechanism whereby computations can occur locally.

Dopaminergic (DA) neurons display two modes of firing: low-frequency tonic and high-frequency bursts. The high frequency firing within the bursts is attributed to NMDA, but not AMPA receptor activation. In our models of the DA neuron, both biophysical and abstract, the NMDA receptor current can significantly increase their firing frequency, whereas the AMPA receptor current is not able to evoke high-frequency activity and usually suppresses firing. However, both currents are produced by glutamate receptors and, consequently, are often co-activated. Here we consider combined influence of AMPA and NMDA synaptic input in the models of the DA neuron. Different types of neuronal activity (resting state, low frequency, or high frequency firing) are observed depending on the conductance of the AMPAR and NMDAR currents. In two models, biophysical and reduced, we show that the firing frequency increases more effectively if both receptors are co-activated for certain parameter values. In particular, in the more quantitative biophysical model, the maximal frequency is 40% greater than that with NMDAR alone. The dynamical mechanism of such frequency growth is explained in the framework of phase space evolution using the reduced model. In short, both the AMPAR and NMDAR currents flatten the voltage nullcline, providing the frequency increase, whereas only NMDA prevents complete unfolding of the nullcline, providing robust firing. Thus, we confirm a major role of the NMDAR in generating high-frequency firing and conclude that AMPAR activation further significantly increases the frequency.

The combined effect of synaptic NMDA, AMPA, and GABA currents on the neuron model with response differentiation has been considered. It has been shown that the GABA and NMDA currents can compensate the effects of each other, whereas the AMPA current not only leads to the suppression of oscillations but also significantly amplifies the high-frequency activity of the neuron induced by the NMDA current. Two bifurcation scenarios underlying these effects have been revealed. It has been predicted which scenario takes place under the combined influence of all three currents.

Midbrain ventral segmental area (VTA) dopaminergic neurons send numerous projections to cortical and sub-cortical areas, and diffusely release dopamine (DA) to their targets. DA neurons display a range of activity modes that vary in frequency and degree of burst firing. Importantly, DA neuronal bursting is associated with a significantly greater degree of DA release than an equivalent tonic activity pattern. Here, we introduce a single compartmental, conductance-based computational model for DA cell activity that captures the behavior of DA neuronal dynamics and examine the multiple factors that underlie DA firing modes: the strength of the SK conductance, the amount of drive, and GABA inhibition. Our results suggest that neurons with low SK conductance fire in a fast firing mode, are correlated with burst firing, and require higher levels of applied current before undergoing depolarization block. We go on to consider the role of GABAergic inhibition on an ensemble of dynamical classes of DA neurons and find that strong GABA inhibition suppresses burst firing. Our studies suggest differences in the distribution of the SK conductance and GABA inhibition levels may indicate subclasses of DA neurons within the VTA. We further identify, that by considering alternate potassium dynamics, the dynamics display burst patterns that terminate via depolarization block, akin to those observed in vivo in VTA DA neurons and in substantia nigra pars compacta (SNc) DA cell preparations under apamin application. In addition, we consider the generation of transient burst firing events that are NMDA-initiated or elicited by a sudden decrease of GABA inhibition, that is, disinhibition. © 2015 Oster, Faure and Gutkin.

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.