Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations

This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies.

Gaussian process approximations of stochastic differential equations

Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modeling. Current solution methods are limited in their representation of the posterior process in the presence of data. In this work, we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximate solution outperforms standard Gaussian process regression for non-Gaussian Markov processes.

Differential patterns of activity and functional connectivity in emotion processing neural circuitry to angry and happy faces in adolescents with and without suicide attempt

Background - Neural substrates of emotion dysregulation in adolescent suicide attempters remain unexamined. Method - We used functional magnetic resonance imaging to measure neural activity to neutral, mild or intense (i.e. 0%, 50% or 100% intensity) emotion face morphs in two separate emotion-processing runs (angry and happy) in three adolescent groups: (1) history of suicide attempt and depression (ATT, n = 14); (2) history of depression alone (NAT, n = 15); and (3) healthy controls (HC, n = 15). Post-hoc analyses were conducted on interactions from 3 group × 3 condition (intensities) whole-brain analyses (p < 0.05, corrected) for each emotion run. Results - To 50% intensity angry faces, ATT showed significantly greater activity than NAT in anterior cingulate gyral–dorsolateral prefrontal cortical attentional control circuitry, primary sensory and temporal cortices; and significantly greater activity than HC in the primary sensory cortex, while NAT had significantly lower activity than HC in the anterior cingulate gyrus and ventromedial prefrontal cortex. To neutral faces during the angry emotion-processing run, ATT had significantly lower activity than NAT in the fusiform gyrus. ATT also showed significantly lower activity than HC to 100% intensity happy faces in the primary sensory cortex, and to neutral faces in the happy run in the anterior cingulate and left medial frontal gyri (all p < 0.006,corrected). Psychophysiological interaction analyses revealed significantly reduced anterior cingulate gyral–insula functional connectivity to 50% intensity angry faces in ATT v. NAT or HC. Conclusions - Elevated activity in attention control circuitry, and reduced anterior cingulate gyral–insula functional connectivity, to 50% intensity angry faces in ATT than other groups suggest that ATT may show inefficient recruitment of attentional control neural circuitry when regulating attention to mild intensity angry faces, which may represent a potential biological marker for suicide risk.

A variational method and approximations of a Cauchy problem for elliptic equations

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H½(?1 ? ?3) is assumed available on a larger part of the boundary ? of the bounded domain O than the boundary portion ?1 on which the Neumann data is prescribed, is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.

Some aspects of the solution of non-linear differential equations

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Alternative exon usage in the single CPT1 gene of Drosophila generates functional diversity in the kinetic properties of the enzyme:differential expression of alternatively spliced variants in drosophila tissues

The Drosophila melanogaster genome contains only one CPT1 gene (Jackson, V. N., Cameron, J. M., Zammit, V. A., and Price, N. T. (1999) Biochem. J. 341, 483-489). We have now extended our original observation to all insect genomes that have been sequenced, suggesting that a single CPT1 gene is a universal feature of insect genomes. We hypothesized that insects may be able to generate kinetically distinct variants by alternative splicing of their single CPT1 gene. Analysis of the insect genomes revealed that (a) the single CPT1 gene in each and every insect genome contains two alternative exons and (ii) in all cases, the putative alternative splicing site occurs within a small region corresponding to 21 amino acid residues that are known to be essential for the binding of substrates and of malonyl-CoA in mammalian CPT1A.Weperformed PCR analyses of mRNA from different Drosophila tissues; both of the anticipated splice variants of CPT1mRNAwere found to be expressed in all of the tissues tested (both in larvae and adults), with the expression level for one of the splice variants being significantly different between flight muscle and the fat body of adult Drosophila. Heterologous expression of the full-length cDNAs corresponding to the two putative variants of Drosophila CPT1 in the yeast Pichia pastoris revealed two important differences between the properties of the two variants: (i) their affinity (K 0.5) for one of the substrates, palmitoyl-CoA, differed by 5-fold, and (ii) the sensitivity to inhibition by malonyl-CoA at fixed, higher palmitoyl-CoA concentrations was 2-fold different and associated with different kinetics of inhibition. These data indicate that alternative splicing that specifically affects a structurally crucial region of the protein is an important mechanism through which functional diversity of CPT1 kinetics is generated from the single gene that occurs in insects. © 2010 by The American Society for Biochemistry and Molecular Biology, Inc.

The V-ATPase in Paramecium:functional specialization by multiple gene isoforms

The vacuolar H(+)-ATPase (V-ATPase), a multisubunit, adenosine triphosphate (ATP)-driven proton pump, is essential for numerous cellular processes in all eukaryotes investigated so far. While structure and catalytic mechanism are similar to the evolutionarily related F-type ATPases, the V-ATPase's main function is to establish an electrochemical proton potential across membranes using ATP hydrolysis. The holoenzyme is formed by two subcomplexes, the transmembraneous V(0) and the cytoplasmic V(1) complexes. Sequencing of the whole genome of the ciliate Paramecium tetraurelia enabled the identification of virtually all the genes encoding V-ATPase subunits in this organism and the studying of the localization of the enzyme and roles in membrane trafficking and osmoregulation. Surprisingly, the number of V-ATPase genes in this free-living protozoan is strikingly higher than in any other species previously studied. Especially abundant are V(0)-a-subunits with as many as 17 encoding genes. This abundance creates the possibility of forming a large number of different V-ATPase holoenzymes by combination and has functional consequences by differential targeting to various organelles.

Functional characterization of GABAergic pallidopallidal and striatopallidal synapses in the rat globus pallidus in vitro

As a central integrator of basal ganglia function, the external segment of the globus pallidus (GP) plays a critical role in the control of voluntary movement. Driven by intrinsic mechanisms and excitatory glutamatergic inputs from the subthalamic nucleus, GP neurons receive GABAergic inhibitory input from the striatum (Str-GP) and from local collaterals of neighbouring pallidal neurons (GP-GP). Here we provide electrophysiological evidence for functional differences between these two inhibitory inputs. The basic synaptic characteristics of GP-GP and Str-GP GABAergic synapses were studied using whole-cell recordings with paired-pulse and train stimulation protocols and variance-mean (VM) analysis. We found (i) IPSC kinetics are consistent with local collaterals innervating the soma and proximal dendrites of GP neurons whereas striatal inputs innervate more distal regions. (ii) Compared to GP-GP synapses Str-GP synapses have a greater paired-pulse ratio, indicative of a lower probability of release. This was confirmed using VM analysis. (iii) In response to 20 and 50 Hz train stimulation, GP-GP synapses are weakly facilitatory in 1 mm external calcium and depressant in 2.4 mm calcium. This is in contrast to Str-GP synapses which display facilitation under both conditions. This is the first quantitative study comparing the properties of GP-GP and Str-GP synapses. The results are consistent with the differential location of these inhibitory synapses and subtle differences in their release probability which underpin stable GP-GP responses and robust short-term facilitation of Str-GP responses. These fundamental differences may provide the physiological basis for functional specialization.

Optimization of chemical plant simulation using double collocation

A method has been constructed for the solution of a wide range of chemical plant simulation models including differential equations and optimization. Double orthogonal collocation on finite elements is applied to convert the model into an NLP problem that is solved either by the VF 13AD package based on successive quadratic programming, or by the GRG2 package, based on the generalized reduced gradient method. This approach is termed simultaneous optimization and solution strategy. The objective functional can contain integral terms. The state and control variables can have time delays. Equalities and inequalities containing state and control variables can be included into the model as well as algebraic equations and inequalities. The maximum number of independent variables is 2. Problems containing 3 independent variables can be transformed into problems having 2 independent variables using finite differencing. The maximum number of NLP variables and constraints is 1500. The method is also suitable for solving ordinary and partial differential equations. The state functions are approximated by a linear combination of Lagrange interpolation polynomials. The control function can either be approximated by a linear combination of Lagrange interpolation polynomials or by a piecewise constant function over finite elements. The number of internal collocation points can vary by finite elements. The residual error is evaluated at arbitrarily chosen equidistant grid-points, thus enabling the user to check the accuracy of the solution between collocation points, where the solution is exact. The solution functions can be tabulated. There is an option to use control vector parameterization to solve optimization problems containing initial value ordinary differential equations. When there are many differential equations or the upper integration limit should be selected optimally then this approach should be used. The portability of the package has been addressed converting the package from V AX FORTRAN 77 into IBM PC FORTRAN 77 and into SUN SPARC 2000 FORTRAN 77. Computer runs have shown that the method can reproduce optimization problems published in the literature. The GRG2 and the VF I 3AD packages, integrated into the optimization package, proved to be robust and reliable. The package contains an executive module, a module performing control vector parameterization and 2 nonlinear problem solver modules, GRG2 and VF I 3AD. There is a stand-alone module that converts the differential-algebraic optimization problem into a nonlinear programming problem.

A multimodal investigation of dynamic face perception using functional magnetic resonance imaging and magnetoencephalography

Motion is an important aspect of face perception that has been largely neglected to date. Many of the established findings are based on studies that use static facial images, which do not reflect the unique temporal dynamics available from seeing a moving face. In the present thesis a set of naturalistic dynamic facial emotional expressions was purposely created and used to investigate the neural structures involved in the perception of dynamic facial expressions of emotion, with both functional Magnetic Resonance Imaging (fMRI) and Magnetoencephalography (MEG). Through fMRI and connectivity analysis, a dynamic face perception network was identified, which is demonstrated to extend the distributed neural system for face perception (Haxby et al.,2000). Measures of effective connectivity between these regions revealed that dynamic facial stimuli were associated with specific increases in connectivity between early visual regions, such as inferior occipital gyri and superior temporal sulci, along with coupling between superior temporal sulci and amygdalae, as well as with inferior frontal gyri. MEG and Synthetic Aperture Magnetometry (SAM) were used to examine the spatiotemporal profile of neurophysiological activity within this dynamic face perception network. SAM analysis revealed a number of regions showing differential activation to dynamic versus static faces in the distributed face network, characterised by decreases in cortical oscillatory power in the beta band, which were spatially coincident with those regions that were previously identified with fMRI. These findings support the presence of a distributed network of cortical regions that mediate the perception of dynamic facial expressions, with the fMRI data providing information on the spatial co-ordinates paralleled by the MEG data, which indicate the temporal dynamics within this network. This integrated multimodal approach offers both excellent spatial and temporal resolution, thereby providing an opportunity to explore dynamic brain activity and connectivity during face processing.

Quasi-separation of the biharmonic partial differential equation

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

Complexity and chirality indices for molecular informatics::differential geometry approach

Novel molecular complexity measures are designed based on the quantum molecular kinematics. The Hamiltonian matrix constructed in a quasi-topological approximation describes the temporal evolution of the modelled electronic system and determined the time derivatives for the dynamic quantities. This allows to define the average quantum kinematic characteristics closely related to the curvatures of the electron paths, particularly, the torsion reflecting the chirality of the dynamic system. A special attention has been given to the computational scheme for this chirality measure. The calculations on realistic molecular systems demonstrate reasonable behaviour of the proposed molecular complexity indices.

Free energy functional expansion as the generalized approach to the equation of state of dense fluids

A version of the thermodynamic perturbation theory based on a scaling transformation of the partition function has been applied to the statistical derivation of the equation of state in a highpressure region. Two modifications of the equations of state have been obtained on the basis of the free energy functional perturbation series. The comparative analysis of the experimental PV T- data on the isothermal compression for the supercritical fluids of inert gases has been carried out. © 2012.

Computation of rare transitions in the barotropic quasi-geostrophic equations

We investigate the theoretical and numerical computation of rare transitions in simple geophysical turbulent models. We consider the barotropic quasi-geostrophic and two-dimensional Navier–Stokes equations in regimes where bistability between two coexisting large-scale attractors exist. By means of large deviations and instanton theory with the use of an Onsager–Machlup path integral formalism for the transition probability, we show how one can directly compute the most probable transition path between two coexisting attractors analytically in an equilibrium (Langevin) framework and numerically otherWe adapt a class of numerical optimization algorithms known as minimum action methods to simple geophysical turbulent models. We show that by numerically minimizing an appropriate action functional in a large deviation limit, one can predict the most likely transition path for a rare transition between two states. By considering examples where theoretical predictions can be made, we show that the minimum action method successfully predicts the most likely transition path. Finally, we discuss the application and extension of such numerical optimization schemes to the computation of rare transitions observed in direct numerical simulations and experiments and to other, more complex, turbulent systems.

Differential activation in temporal, cingulate and prefrontal cortices during response inhibition in bipolar disorder

Background: Increased impulsivity and aberrant response inhibition have been observed in bipolar disorder (BD). This study examined the functional abnormalities and underlying neural processes during response inhibition in BD, and its relationship to impulsivity. Methods: We assessed impulsivity using the Barratt Impulsiveness Scale (BIS) and, using functional magnetic resonance imaging (fMRI), measured neural activity in response to an Affective Go-NoGo Task, consisting of emotional facial stimuli (fear, happy, anger faces) and non-emotional control stimuli (neutral female and male faces) in euthymic BD (n=23) and healthy individuals (HI; n=25). Results: BD patients were significantly more impulsive, yet did not differ from HI on accuracy or reaction time on the emotional go/no-go task. Comparing neural patterns of activation when processing emotional Go versus emotional NoGo trials yielded increased activation in BD within temporal and cingulate cortices and within prefrontal-cortical regions in HI. Furthermore, higher BIS scores for BD were associated with slower reaction times, and indicative of compensatory cognitive strategies to counter increased impulsivity. Conclusions: These findings illustrate cognition-emotion interference in BD and the observed differences in neural activation indicate potentially altered emotion modulation. Increased activation in brain regions previously shown in emotion regulation and response inhibition tasks could represent a disease-specific marker for BD