A weak trapezoidal method for a class of stochastic differential equations

We present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions. This class of equations arises naturally in the study of population processes and chemical reaction kinetics. We show that the method constructs paths that are second order accurate in the weak sense. The method is simpler than many second order methods in that it neither requires the construction of iterated It̂o integrals nor the evaluation of any derivatives. The method consists of two steps. In the first an explicit Euler step is used to take a fractional step. The resulting fractional point is then combined with the initial point to obtain a higher order, trapezoidal like, approximation. The higher order of accuracy stems from the fact that both the drift and the quadratic variation of the underlying SDE are approximated to second order. © 2011 International Press

Propagation of Fluctuations in Biochemical Systems, II: Nonlinear Chains

We consider biochemical reaction chains and investigate how random external fluctuations, as characterized by variance and coefficient of variation, propagate down the chains. We perform such a study under the assumption that the number of molecules is high enough so that the behavior of the concentrations of the system is well approximated by differential equations. We conclude that the variances and coefficients of variation of the fluxes will decrease as one moves down the chain and, through an example, show that there is no corresponding result for the variances of the chemical species. We also prove that the fluctuations of the fluxes as characterized by their time averages decrease down reaction chains. The results presented give insight into how biochemical reaction systems are buffered against external perturbations solely by their underlying graphical structure and point out the benefits of studying the out-of-equilibrium dynamics of systems

Propagation of fluctuations in biochemical systems, I: Linear SSC networks

We investigate the propagation of random fluctuations through biochemical networks in which the number of molecules of each species is large enough so that the concentrations are well modeled by differential equations. We study the effect of network topology on the emergent properties of the reaction system by characterizing the behavior of variance as fluctuations propagate down chains and studying the effect of side chains and feedback loops. We also investigate the asymptotic behavior of the system as one reaction becomes fast relative to the others. © 2007 Springer Science+Business Media, Inc

Investigating the Spatiotemporal Summation of Perimetric Stimuli in Dry Age-Related Macular Degeneration

Purpose: To measure achromatic spatial, temporal, and spatiotemporal summation in dry age-related macular degeneration (AMD) compared to healthy controls under conditions of photopic gaze-contingent perimetry. Methods: Twenty participants with dry AMD (mean age, 74.6 years) and 20 healthy controls (mean age, 67.8 years) performed custom, gaze-contingent perimetry tests. An area-modulation test generated localized estimates of Ricco’s area (RA) at 2.5° and 5° eccentricities along the 0°, 90°, 180°, and 270° meridians. Contrast thresholds were measured at the same test locations for stimuli of six durations (3.7–190.4 ms) with a Goldmann III stimulus (GIII, 0.43°) and RA-scaled stimuli. The upper limit (critical duration) of complete temporal summation (using the GIII stimulus) and spatiotempo-ral summation (using the RA stimuli) was estimated using iterative two-phase regression analysis. Results: Median (interquartile range [IQR]) RA estimates were significantly larger in AMD participants (2.5°: 0.21 [0.09–0.41] deg2; 5°: 0.32 [0.15–0.65 deg2 ]) compared to healthy controls (2.5°: 0.08 [0.05–0.13] deg2; 5°: 0.15 [0.08–0.22] deg2 ) at all test locations (all P 0.05). Conclusions: Spatial summation is altered in dry AMD, without commensurate changes in temporal summation. Translational Relevance: The sensitivity of perimetry to AMD may be improved by utilizing stimuli that probe alterations in spatial summation in the disease

Magnetic Properties of a Superconductor with no Inversion Symmetry

We study the magnetic properties of a superconductor in a crystal without $z \to -z$ symmetry, in particular how the lack of this symmetry exhibits itself. We show that, though the penetration depth itself shows no such effect, for suitable orientation of magnetic field, there is a magnetic field discontinuity at the interface which shows this absence of symmetry. The magnetic field profile of a vortex in the $x-y$ plane is shown to be identical to that of an ordinary anisotropic superconductor except for a shift in the $-z$ direction by ${\tilde \kappa} \lambda_x$ (see errata). For a vortex along $z$, there is an induced magnetization along the radial direction.Comment: J. Low Temp. Physics, 140, 67 (2005); with Errat

Doses of neighborhood nature: the benefits for mental health of living with nature

This is the author accepted manuscript. The final version is available from OUP via the DOI in this record.Experiences of nature provide many mental health benefits, particularly for people living in urban areas. The natural characteristics of city residents’ neighborhoods are likely to be critical determinants of the daily nature dose that they receive, however which characteristics are important remains unclear. One possibility is that the greatest benefits are provided by characteristics that are most visible during the day and so most likely to be experienced by people. We demonstrate that of five neighborhood nature characteristics tested, vegetation cover and afternoon bird abundances were positively associated with a lower prevalence of depression, anxiety and stress. Further, dose-response modelling shows a threshold response where the population prevalence of mental health issues is significantly lower beyond minimum limits of neighborhood vegetation cover (depression >20% cover, anxiety >30% cover, stress >20% cover). Our findings demonstrate quantifiable associations of mental health with the characteristics of nearby nature that people actually experience

Markov chain aggregation and its application to rule-based modelling

Rule-based modelling allows to represent molecular interactions in a compact and natural way. The underlying molecular dynamics, by the laws of stochastic chemical kinetics, behaves as a continuous-time Markov chain. However, this Markov chain enumerates all possible reaction mixtures, rendering the analysis of the chain computationally demanding and often prohibitive in practice. We here describe how it is possible to efficiently find a smaller, aggregate chain, which preserves certain properties of the original one. Formal methods and lumpability notions are used to define algorithms for automated and efficient construction of such smaller chains (without ever constructing the original ones). We here illustrate the method on an example and we discuss the applicability of the method in the context of modelling large signalling pathways