Elevated left and reduced right orbitomedial prefrontal fractional anisotropy in adults with bipolar disorder revealed by tract-based spatial statistics

Context - Diffusion tensor imaging (DTI) studies in adults with bipolar disorder (BD) indicate altered white matter (WM) in the orbitomedial prefrontal cortex (OMPFC), potentially underlying abnormal prefrontal corticolimbic connectivity and mood dysregulation in BD. Objective - To use tract-based spatial statistics (TBSS) to examine WM skeleton (ie, the most compact whole-brain WM) in subjects with BD vs healthy control subjects. Design - Cross-sectional, case-control, whole-brain DTI using TBSS. Setting - University research institute. Participants - Fifty-six individuals, 31 having a DSM-IV diagnosis of BD type I (mean age, 35.9 years [age range, 24-52 years]) and 25 controls (mean age, 29.5 years [age range, 19-52 years]). Main Outcome Measures - Fractional anisotropy (FA) longitudinal and radial diffusivities in subjects with BD vs controls (covarying for age) and their relationships with clinical and demographic variables. Results - Subjects with BD vs controls had significantly greater FA (t > 3.0, P = .05 corrected) in the left uncinate fasciculus (reduced radial diffusivity distally and increased longitudinal diffusivity centrally), left optic radiation (increased longitudinal diffusivity), and right anterothalamic radiation (no significant diffusivity change). Subjects with BD vs controls had significantly reduced FA (t > 3.0, P = .05 corrected) in the right uncinate fasciculus (greater radial diffusivity). Among subjects with BD, significant negative correlations (P < .01) were found between age and FA in bilateral uncinate fasciculi and in the right anterothalamic radiation, as well as between medication load and FA in the left optic radiation. Decreased FA (P < .01) was observed in the left optic radiation and in the right anterothalamic radiation among subjects with BD taking vs those not taking mood stabilizers, as well as in the left optic radiation among depressed vs remitted subjects with BD. Subjects having BD with vs without lifetime alcohol or other drug abuse had significantly decreased FA in the left uncinate fasciculus. Conclusions - To our knowledge, this is the first study to use TBSS to examine WM in subjects with BD. Subjects with BD vs controls showed greater WM FA in the left OMPFC that diminished with age and with alcohol or other drug abuse, as well as reduced WM FA in the right OMPFC. Mood stabilizers and depressed episode reduced WM FA in left-sided sensory visual processing regions among subjects with BD. Abnormal right vs left asymmetry in FA in OMPFC WM among subjects with BD, likely reflecting increased proportions of left-sided longitudinally aligned and right-sided obliquely aligned myelinated fibers, may represent a biologic mechanism for mood dysregulation in BD.

Abnormal amygdala-prefrontal effective connectivity to happy faces differentiates bipolar from major depression

Background - Bipolar disorder is frequently misdiagnosed as major depressive disorder, delaying appropriate treatment and worsening outcome for many bipolar individuals. Emotion dysregulation is a core feature of bipolar disorder. Measures of dysfunction in neural systems supporting emotion regulation might therefore help discriminate bipolar from major depressive disorder. Methods - Thirty-one depressed individuals—15 bipolar depressed (BD) and 16 major depressed (MDD), DSM-IV diagnostic criteria, ages 18–55 years, matched for age, age of illness onset, illness duration, and depression severity—and 16 age- and gender-matched healthy control subjects performed two event-related paradigms: labeling the emotional intensity of happy and sad faces, respectively. We employed dynamic causal modeling to examine significant among-group alterations in effective connectivity (EC) between right- and left-sided neural regions supporting emotion regulation: amygdala and orbitomedial prefrontal cortex (OMPFC). Results - During classification of happy faces, we found profound and asymmetrical differences in EC between the OMPFC and amygdala. Left-sided differences involved top-down connections and discriminated between depressed and control subjects. Furthermore, greater medication load was associated with an amelioration of this abnormal top-down EC. Conversely, on the right side the abnormality was in bottom-up EC that was specific to bipolar disorder. These effects replicated when we considered only female subjects. Conclusions - Abnormal, left-sided, top-down OMPFC–amygdala and right-sided, bottom-up, amygdala–OMPFC EC during happy labeling distinguish BD and MDD, suggesting different pathophysiological mechanisms associated with the two types of depression.

An integral equation method for a mixed initial boundary value problem for unsteady Stokes system in a doubly-connected domain

We present a novel numerical method for a mixed initial boundary value problem for the unsteady Stokes system in a planar doubly-connected domain. Using a Laguerre transformation the unsteady problem is reduced to a system of boundary value problems for the Stokes resolvent equations. Employing a modied potential approach we obtain a system of boundary integral equations with various singularities and we use a trigonometric quadrature method for their numerical solution. Numerical examples are presented showing that accurate approximations can be obtained with low computational cost.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant

We study the Cauchy problem for the Laplace equation in a quadrant (quarter-plane) containing a bounded inclusion. Given the values of the solution and its derivative on the edges of the quadrant the solution is reconstructed on the boundary of the inclusion. This is achieved using an alternating iterative method where at each iteration step mixed boundary value problems are being solved. A numerical method is also proposed and investigated for the direct mixed problems reducing these to integral equations over the inclusion. Numerical examples verify the efficiency of the proposed scheme.

Solvability and asymptotics of the heat equation with mixed variable lateral conditions and applications in the opening of the exocytotic fusion pore in cells

We investigate a mixed problem with variable lateral conditions for the heat equation that arises in modelling exocytosis, i.e. the opening of a cell boundary in specific biological species for the release of certain molecules to the exterior of the cell. The Dirichlet condition is imposed on a surface patch of the boundary and this patch is occupying a larger part of the boundary as time increases modelling where the cell is opening (the fusion pore), and on the remaining part, a zero Neumann condition is imposed (no molecules can cross this boundary). Uniform concentration is assumed at the initial time. We introduce a weak formulation of this problem and show that there is a unique weak solution. Moreover, we give an asymptotic expansion for the behaviour of the solution near the opening point and for small values in time. We also give an integral equation for the numerical construction of the leading term in this expansion.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions

We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.

Recovering boundary data in planar heat conduction using boundary integral equation method

We consider a Cauchy problem for the heat equation, where the temperature field is to be reconstructed from the temperature and heat flux given on a part of the boundary of the solution domain. We employ a Landweber type method proposed in [2], where a sequence of mixed well-posed problems are solved at each iteration step to obtain a stable approximation to the original Cauchy problem. We develop an efficient boundary integral equation method for the numerical solution of these mixed problems, based on the method of Rothe. Numerical examples are presented both with exact and noisy data, showing the efficiency and stability of the proposed procedure and approximations.

Seminar 1 impurity in the tomonaga-luttinger model:a functional integral approach

This chapter explains a functional integral approach about impurity in the Tomonaga–Luttinger model. The Tomonaga–Luttinger model of one-dimensional (1D) strongly correlates electrons gives a striking example of non-Fermi-liquid behavior. For simplicity, the chapter considers only a single-mode Tomonaga–Luttinger model, with one species of right- and left-moving electrons, thus, omitting spin indices and considering eventually the simplest linearized model of a single-valley parabolic electron band. The standard operator bosonization is one of the most elegant methods developed in theoretical physics. The main advantage of the bosonization, either in standard or functional form, is that including the quadric electron–electron interaction does not substantially change the free action. The chapter demonstrates the way to develop the formalism of bosonization based on the functional integral representation of observable quantities within the Keldysh formalism.

Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

We consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert's method [24] is employed, which generates a Galerkin-type procedure for the numerical solution via rewriting the boundary integrals over the unit sphere and expanding the densities in terms of spherical harmonics. Numerical results are included as well.