Katherine A. Bold

Princeton University, Princeton, New Jersey, United States

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Publications (3)2.37 Total impact

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    ABSTRACT: We propose and illustrate an approach to coarse-graining the dynamics of evolving networks (networks whose connectivity changes dynamically). The approach is based on the equation-free framework: short bursts of detailed network evolution simulations are coupled with lifting and restriction operators that translate between actual network realizations and their (appropriately chosen) coarse observables. This framework is used here to accelerate temporal simulations (through coarse projective integration), and to implement coarsegrained fixed point algorithms (through matrix-free Newton-Krylov GMRES). The approach is illustrated through a simple network evolution example, for which analytical approximations to the coarse-grained dynamics can be independently obtained, so as to validate the computational results. The scope and applicability of the approach, as well as the issue of selection of good coarse observables are discussed.
    02/2012;
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    ABSTRACT: We propose a computational approach to modeling the collective dynamics of populations of coupled, heterogeneous biological oscillators. We consider the synchronization of yeast glycolytic oscillators coupled by the membrane exchange of an intracellular metabolite; the heterogeneity consists of a single random parameter, which accounts for glucose influx into each cell. In contrast to Monte Carlo simulations, distributions of intracellular species of these yeast cells are represented by a few leading order generalized Polynomial Chaos (gPC) coefficients, thus reducing the dynamics of an ensemble of oscillators to dynamics of their (typically significantly fewer) representative gPC coefficients. Equation-free (EF) methods are employed to efficiently evolve this coarse description in time and compute the coarse-grained stationary state and/or limit cycle solutions, circumventing the derivation of explicit, closed-form evolution equations. Coarse projective integration and fixed-point algorithms are used to compute collective oscillatory solutions for the cell population and quantify their stability. These techniques are extended to the special case of a "rogue" oscillator; a cell sufficiently different from the rest "escapes" the bulk synchronized behavior and oscillates with a markedly different amplitude. The approach holds promise for accelerating the computer-assisted analysis of detailed models of coupled heterogeneous cell or agent populations.
    Journal of Mathematical Biology 10/2007; 55(3):331-52. · 2.37 Impact Factor
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    ABSTRACT: We propose a computational approach to modeling the collective dynamics of populations of coupled heterogeneous biological oscillators. In contrast to Monte Carlo simulation, this approach utilizes generalized Polynomial Chaos (gPC) to represent random properties of the population, thus reducing the dynamics of ensembles of oscillators to dynamics of their (typically significantly fewer) representative gPC coefficients. Equation-Free (EF) methods are employed to efficiently evolve these gPC coefficients in time and compute their coarse-grained stationary state and/or limit cycle solutions, circumventing the derivation of explicit, closed-form evolution equations. Ensemble realizations of the oscillators and their statistics can be readily reconstructed from these gPC coefficients. We apply this methodology to the synchronization of yeast glycolytic oscillators coupled by the membrane exchange of an intracellular metabolite. The heterogeneity consists of a single random parameter, which accounts for glucose influx into a cell, with a Gaussian distribution over the population. Coarse projective integration is used to accelerate the evolution of the population statistics in time. Coarse fixed-point algorithms in conjunction with a Poincar\'e return map are used to compute oscillatory solutions for the cell population and to quantify their stability.
    12/2006;