Whether single-molecule trajectories, observed experimentally or in molecular simulations, can be described using simple models such as biased diffusion is a subject of considerable debate. Memory effects and anomalous diffusion have been reported in a number of studies, but directly inferring such effects from trajectories, especially given limited temporal and/or spatial resolution, has been a challenge. Recently, we proposed that this can be achieved with information-theoretical analysis of trajectories, which is based on the general observation that non-Markov effects make trajectories more predictable and, thus, more “compressible” by lossless compression algorithms. Toy models where discrete molecular states evolve in time were shown to be amenable to such analysis, but its application to continuous trajectories presents a challenge: the trajectories need to be digitized first, and digitization itself introduces non-Markov effects that depend on the specifics of how trajectories are sampled. Here we develop a milestoning-based method for information-theoretical analysis of continuous trajectories and show its utility in application to Markov and non-Markov models and to trajectories obtained from molecular simulations.
Abstract. The Gegenbauer reconstruction method, first proposed by Gottlieb et.al. in 1992, has been considered a useful technique for re-expanding finite series polynomial approximations while simultaneously avoiding Gibbs artifacts. Since its introduction many studies have analyzed the method's strengths and weaknesses as well as suggesting several applications. However, until recently no attempts were made to optimize the reconstruction parameters, whose careful selection can make the difference between spectral accuracies and divergent error bounds. In this paper we propose asymptotic analysis as a method for locating the optimal Gegenbauer reconstruction parameters. Such parameters are useful to applications of this reconstruction method that either seek to bound the number of Gegenbauer expansion coefficients or to control compression ratios. We then illustrate the effectiveness of our approach with the results from some numerical experiments.
We compare six numerical integrators' performance when simulating a regular spiking cortical neuron model whose 74-compartments are equipped with eleven membrane ion channels and Calcium dynamics. Four methods are explicit and two are implicit; three are finite difference PDE methods, two are Runge-Kutta methods, and one an exponential time differencing method. Three methods are first-, two commonly considered second-, and one commonly considered fourth-order. Derivations show, and simulation data confirms, that Hodgkin-Huxley type cable equations render multiple order explicit RK methods as first-order methods. Illustrations compare accuracy, stability, variations of action potential phase and waveform statistics. Explicit methods were found unsuited for our model given their inability to control spiking waveform consistency up to 10 microseconds less than the step size for onset of instability. While the backward-time central space method performed satisfactorily as a first order method for step sizes up to 80 microseconds, performance of the Hines-Crank-Nicolson method, our only true second order method, was unmatched for step sizes of 1-100 microseconds.
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