We consider a generalized nonlocal Ginzburg–Landau equation with periodic boundary conditions. For the corresponding initial-boundary value problem we prove the existence of a solution for all ...positive values of the evolution variable. We study the existence and properties of invariant manifolds. We extract a class of invariant manifolds the union of which forms a global attractor. We describe the structure of the attractor and find the Euclidean dimension of its components. In the metric of the space of initial conditions, we also study the Lyapunov stability and orbital stability of solutions that belong in the global attractor.
We consider a periodic boundary value problem for the complex Ginzburg–Landau integro-differential equation. It can be interpreted as a generalized version of the equation known in some physical ...applications as the “nonlocal Ginzburg–Landau equation.” Two natural cases of this equation are studied. One of them can be included in the class of abstract parabolic equations. In this case, the global solvability of the initial–boundary value problem is shown and, most importantly, the existence of a finite-dimensional global attractor is proved. The question of the structure of such an attractor is studied, and the Euclidean dimension of its components is indicated. The analysis of these questions is based on the possibility of obtaining explicit formulas for all solutions of the corresponding initial–boundary value problems. A weakly dissipative version of the equation is also considered, for which the existence of an infinite-dimensional global attractor is shown.
A new algorithm for long-term continuous computer recording and analysis of motor activity of a group of zebrafish in the home tank has been developed. The movements of a group of
Danio rerio
during ...the entire light period and for several days are recorded at a frequency of 1 frame/sec in the form of short (15 min) files. Then these files are analyzed by the unique DanioStudo software, which, using a threshold algorithm and appropriate masks, calculates for each frame the sum of pixels associated with fish (the sum of fish silhouettes), and for two consecutive frames, the sum of altered pixels (the sum of altered fish silhouettes). The following indexes are calculated: the rate of sum of silhouettes alteration as the ratio of the sum of altered silhouettes to the sum of silhouettes (1) and the time spent in the selected area of the home tank as the ratio of the sum of silhouettes in this area to the sum of silhouettes in the entire tank (2). The mean rate of silhouette alteration correlates to the length of the path travelled by the fish and, therefore, serves as a correct measure of the motor activity of a group of fish. Using these algorithms, completely new data were obtained: it was shown that the motor activity of fish remains constant throughout the entire light period, but depends on the size of the home tank. The proposed approach, together with the DanioStudio software, can be effective in studying the dynamics of changes in the behavior of fish under long-term exposure to short daylight, drugs and toxic substances.
We study two rather similar evolutionary partial differential equations. One of them was obtained by Sivashinsky and the other by Kuramoto. The Kuramoto version was taken as the basic version of the ...equation that became known as the Kuramoto–Sivashinsky equation. We supplement each version of the Kuramoto–Sivashinsky equation with natural boundary conditions and, for the proposed boundary-value problems, study local bifurcations arising near a homogeneous equilibrium when they change stability. The analysis is based on the methods of the theory of dynamical systems with an infinite-dimensional phase space, namely, the methods of integral manifolds and normal forms. For all boundary-value problems, asymptotic formulas are obtained for solutions that form integral manifolds. We also point out boundary conditions under which the dynamics of solutions of the corresponding boundary-value problems of the two versions of the Kuramoto–Sivashinsky equation are significantly different.
We consider the periodic boundary value problem for the convective Cahn–Hilliard equation in the case where the unknown function depends on two spatial variables. We obtain sufficient conditions for ...the existence and stability of two- and three-dimensional invariant manifolds. We deduce asymptotic formulas for the solutions. We show that the three-dimensional invariant manifold is a one-parameter family of two-dimensional invariant tori containing tori with ergodic and resonance dynamics simultaneously.
We consider the periodic boundary value problem for two variants of a weakly dissipative complex Ginzburg–Landau equation. In the first case, we study a variant of such an equation that contains the ...cubic and quintic nonlinear terms. We study the problem of local bifurcations of traveling periodic waves under stability changes. We show that a countable set of two-dimensional invariant tori arises as a result of such bifurcations. Both types of bifurcations are possible in the considered formulation of the problem, soft (postcritical) and hard (subcritical) ones, depending on the choice of the coefficients in the equation. We obtain asymptotic formulas for the solutions forming the invariant tori. We also study the periodic boundary value problem for the equation that is called the nonlocal Ginzburg–Landau equation in physics. We show that the boundary value problem in the considered variant has an infinite-dimensional global attractor. We present the solutions forming such an attractor.
We studied the effect of reduced tryptophan hydroxylase (TPH) activity and short daylight exposure on the behavior and the 5-HT system of the brain in
D. rerio
. Male and female
D. rerio
were exposed ...for 30 days to standard (12:12 h light:dark) and short (4:20 h light:dark) photoperiods in the presence or absence of TPH inhibitor (
p
-chlorophenylalanine, pCPA, 5 mg/liter). On day 31, the fish behavior in the “novel tank diving” test, their sex and body weight were determined, and the levels of pCPA, 5-HT, and its metabolite 5-HIAA were measured by HPLC; the levels of the key genes encoding metabolism enzymes (
Tph1a
,
Tph1b
,
Tph2
, and
Mao
) and receptors of 5-HT (
Htr1aa
,
Htr2aa
) were assessed by real-time PCR with reverse transcription. The short daylight exposure caused masculinization of females, reduced body weight, and motor activity in the “novel tank diving” test, but did not affect the 5-HT system of the brain. Long-term pCPA treatment had no effect on sex and body weight, significantly reduced the 5-HIAA level, but increased
Tph1a
and
Tph2
gene expression in the brain. No effects of the interaction of short daylight and pCPA exposure on the sex, body weight, behavior, and 5-HT system of the brain were found. Thus, a moderate decrease in TPH activity cannot potentiate the negative effects of short daylight exposure on the sex, body weight, behavior, and 5-HT system of
D. rerio
.
Our aim is to examine a periodic boundary value problem for the Cahn–Hilliard–Oono equation. This equation appeared as one of the possible modifications of the well-known Cahn–Hilliard equation. This ...modification is designed to assume additional factors in modeling physical and physicochemical processes. The local bifurcations that arise when the stability is changed by spatially homogeneous solutions of the corresponding boundary value problem are studied for all possible variants of stability change, including cases of codimension 2. A special version of the equation is also considered, which leads to the possibility of bifurcations of two-dimensional local attractors formed by Lyapunov unstable solutions that are periodic in the evolutionary variable. The analysis of the problem is based on the use and development of such methods of the theory of infinite-dimensional dynamical systems as the methods of integral (invariant) manifolds and normal forms.
We consider the Cahn–Hilliard equation in the case where its solution depends on two spatial variables, with homogeneous Dirichlet and Neumann boundary conditions, and also periodic boundary ...conditions. For these three boundary value problems, we study the problem of local bifurcations arising when changing stability by spatially homogeneous equilibrium states. We show that the nature of bifurcations that lead to spatially inhomogeneous solutions is strongly related to the choice of boundary conditions. In the case of homogeneous Dirichlet boundary conditions, spatially inhomogeneous equilibrium states occur in a neighborhood of a homogeneous equilibrium state, depending on both spatial variables. An alternative scenario is realized in analyzing the Neumann problem and the periodic boundary value problem. In these, as a result of bifurcations, invariant manifolds formed by spatially inhomogeneous solutions occur. The dimension of these manifolds ranges from 1 to 3. In analyzing three boundary value problems, we use methods of infinite-dimensional dynamical system theory and asymptotic methods. Using the integral manifold method together with the techniques of normal form theory allows us to analyze the stability of bifurcating invariant manifolds and also to derive asymptotic formulas for spatially inhomogeneous solutions forming these manifolds.
•Hindlimb unloading increased the expression of dopanimergic genes in striatum.•Hippocampus Bcl-xl expression was similar for hindlimb unloading and spaceflight.•Hindlimb unloading did not alter the ...expression of MAOA and 5-HT2A receptor genes.
The study of spaceflight effects on the brain is technically complex concern; complicated by the problem of applying an adequate ground model. The most-widely used experimental model to study the effect of microgravity is the tail-suspension hindlimb unloading model; however, its compliance with the effect of actual spaceflight on the brain is still unclear. We evaluated the effect of one month hindlimb unloading on the expression of genes related to the brain neuroplasticity—brain neutotrophic factors (Gdnf, Cdnf), apoptotic factors (Bcl-xl, Bax), serotonin- and dopaminergic systems (5-HT2A, Maoa, Maob, Th, D1r, Comt), and compared the results with the data obtained on mice that spent one month in spaceflight on Russian biosatellite Bion-M1. No effect of hindlimb unloading was observed on the expression of most genes, which were considered as risk neurogenes for long–term actual spaceflight. The opposite effect of hindlimb unloading and spaceflight was found on the level of mRNA of D1 dopamine receptor and catechol-O-methyltransferase in the striatum. At the same time, the expression of Maob in the midbrain decreased, and the expression of Bcl-xl genes increased in the hippocampus, which corresponds to the effect of spaceflight. However, the hindlimb unloading model failed to reproduce the majority of effects of long-term spaceflight on serotonin-, dopaminergic systems and some apoptotic factors.
