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Statistical mechanics for networks of real neurons

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arXiv 2024年

作者： Meshulam, Leenoy Bialek, William Center for Computational Neuroscience Department of Applied Mathematics University of Washington SeattleWA98195 United States Joseph Henry Laboratories of Physics and Lewis Sigler Institute for Integrative Genomics Princeton University PrincetonNJ08544 United States

Perceptions and actions, thoughts and memories result from coordinated activity in hundreds or even thousands of neurons in the brain. It is an old dream of the physics community to provide a statistical mechanics description for these and other emergent phenomena of life. These aspirations appear in a new light because of developments in our ability to measure the electrical activity of the brain, sampling thousands of individual neurons simultaneously over hours or days. We review the progress that has been made in bringing theory and experiment together, focusing on maximum entropy methods and a phenomenological renormalization group. These approaches have uncovered new, quantitatively reproducible collective behaviors in networks of real neurons, and provide examples of rich parameter–free predictions that agree in detail with experiment. © 2024, CC BY.

关键词： Neurons

A PRIORI AND A POSTERIORI ERROR ESTIMATES FOR DISCONTINUOUS GALERKIN TIME-DISCRETE METHODS VIA MAXIMAL REGULARITY

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arXiv 2024年

作者： Akrivis, Georgios Larsson, Stig Department of Computer Science and Engineering University of Ioannina Ioannina451 10 Greece Institute of Applied and Computational Mathematics FORTH Crete Heraklion700 13 Greece Department of Mathematical Sciences Chalmers University of Technology University of Göteborg Göteborg Sweden

The maximal regularity property of discontinuous Galerkin methods for linear parabolic equations is used together with variational techniques to establish a priori and a posteriori error estimates of optimal order under optimal regularity assumptions. The analysis is set in the maximal regularity framework of UMD Banach spaces. Similar results were proved in an earlier work, based on the consistency analysis of Radau IIA methods. The present error analysis, which is based on variational techniques, is of independent interest, but the main motivation is that it extends to nonlinear parabolic equations;in contrast to the earlier work. Both autonomous and nonautonomous linear equations are *** Codes 65M12, 65M15 © 2024, CC BY.

关键词： Galerkin methods

Optimal local approximation spaces for parabolic problems

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arXiv 2020年

作者： Schleuss, Julia Smetana, Kathrin Institute for Computational and Applied Mathematics University of Münster Einsteinstr. 62 Münster48149 Germany Department of Mathematical Sciences Stevens Institute of Technology 1 Castle Point Terrace HobokenNJ07030 United States

We propose local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed in multiscale and domain decomposition methods. The diffusion coefficient can be arbitrarily rough in space and time. To construct local approximation spaces we consider a compact transfer operator that acts on the space of local solutions and covers the full time dimension. The optimal local spaces are then given by the left singular vectors of the transfer operator. To prove compactness of the latter we combine a suitable parabolic Caccioppoli inequality with the compactness theorem of Aubin-Lions. In contrast to the elliptic setting [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406] we need an additional regularity result to combine the two results. Furthermore, we employ the generalized finite element method to couple local spaces and construct an approximation of the global solution. Since our approach yields reduced space-time bases, the computation of the global approximation does not require a time stepping method and is thus computationally efficient. Moreover, we derive rigorous local and global a priori error bounds. In detail, we bound the global approximation error in a graph norm by the local errors in the L2(H1)-norm, noting that the space the transfer operator maps to is equipped with this norm. Numerical experiments demonstrate an exponential decay of the singular values of the transfer operator and the local and global approximation errors for problems with high contrast or multiscale structure regarding space and *** Codes 65M12, 65M15, 65M55, 65M60 Copyright © 2020, The Authors. All rights reserved.

关键词： Vector spaces

On the small mass limit of quantum brownian motion with inhomogeneous damping and diffusion

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arXiv 2017年

作者： Lim, Soon Hoe Wehr, Jan Lampo, Aniello García-March, Miguel Ángel Lewenstein, Maciej Department of Mathematics University of Arizona TucsonAZ85721 United States Program in Applied Mathematics University of Arizona TucsonAZ85721 United States 08860 Spain ICREA - Institució Catalana de Recerca i Estudis Avançats Lluis Companys 23 BarcelonaE-08010 Spain

We study the small mass limit (or: the Smoluchowski-Kramers limit) of a class of quantum Brownian motions with inhomogeneous damping and diffusion. For Ohmic bath spectral density with a Lorentz-Drude cutoff, we derive the Heisenberg-Langevin equations for the particle's observables using a quantum stochastic calculus approach. We set the mass of the particle to equal m = m0ǫ, the reduced Planck constant to equal ~ = ǫ and the cutoff frequency to equal Λ = EΛ/ǫ, where m0 and EΛ are positive constants, so that the particle's de Broglie wavelength and the largest energy scale of the bath are fixed as ǫ → 0. We study the limit as ǫ → 0 of the rescaled model and derive a limiting equation for the (slow) particle's position variable. We find that the limiting equation contains several drift correction terms, the quantum noise-induced drifts, including terms of purely quantum nature, with no classical counterparts. Copyright © 2017, The Authors. All rights reserved.

关键词： Damping

Near-field simulations of pellet ablation for disruptions mitigation in tokamaks

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arXiv 2020年

作者： Bosviel, Nicolas Parks, Paul B. Samulyak, Roman Department of Applied Mathematics and Statistics Stony Brook University Stony BrookNY United States General Atomics San DiegoCA United States Computational Science Initiative Brookhaven National Laboratory UptownNY United States

Detailed numerical studies of the ablation of a single neon pellet in the plasma disruption mitigation parameter space have been performed. Simulations were carried out using FronTier, a hydrodynamic and low magnetic Reynolds number MHD code with explicit tracking of material interfaces. FronTier’s physics models resolve the pellet surface ablation and the formation of a dense, cold cloud of ablated material, the deposition of energy from hot plasma electrons passing through the ablation cloud, expansion of the ablation cloud along magnetic field lines and the radiation losses. A local thermodynamic equilibrium model based on Saha equations has been used to resolve atomic processes in the cloud and Redlich-Kwong corrections to the ideal gas equation of state for cold and dense gases have been used near the pellet surface. The FronTier pellet code is the next generation of the code described in [R. Samulyak, T. Lu, P. Parks, Nuclear Fusion, (47) 2007, 103–118]. It has been validated against the semi-analytic improved Neutral Gas Shielding model in the 1D spherically symmetric approximation. Main results include quantification of the influence of atomic processes and Redlich-Kwong corrections on the pellet ablation in spherically symmetric approximation and verification of analytic scaling laws in a broad range of pellet and plasma parameters. Using axially symmetric MHD simulations, properties of ablation channels and the reduction of pellet ablation rates in magnetic fields of increasing strength have been studied. While the main emphasis has been given to neon pellets for the plasma disruption mitigation, selected results on deuterium fueling pellets have also been presented. Copyright © 2020, The Authors. All rights reserved.

关键词： Neon

PDE model for multi-patch epidemic models with migration and infection-age dependent infectivity

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arXiv 2023年

作者： Pang, Guodong Pardoux, Étienne Department of Computational Applied Mathematics and Operations Research George R. Brown School of Engineering Rice University HoustonTX77005 United States Aix-Marseille Université CNRS I2M Marseille France

We study a stochastic epidemic model with multiple patches (locations), where individuals in each patch are categorized into three compartments, Susceptible, Infected and Recovered/Removed, and may migrate from one patch to another in any of the compartments. Each individual is associated with a random infectivity function which dictates the force of infection depending upon the age of infection (elapsed time since infection). We prove a functional law of large number for the epidemic evolution dynamics including the aggregate infectivity process, the numbers of susceptible and recovered individuals as well as the number of infected individuals at each time that have been infected for a certain amount of time. From the limits, we derive a PDE model for the density of the number of infected individuals with respect to the infection age, which is a system of linear PDE equations with a boundary condition that is determined by a set of integral equations. Copyright © 2023, The Authors. All rights reserved.

关键词： Integral equations

Model for dynamical coherence in thin films of self-propelled microorganisms

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Physical Review E 2007年第4期75卷 040901(R)-040901(R)页

作者： Igor S. Aranson Andrey Sokolov John O. Kessler Raymond E. Goldstein Argonne National Laboratory 9700 South Cass Avenue Argonne Illinois 60439 USA Illinois Institute of Technology 3101 South Dearborn Street Chicago Illinois 60616 USA Department of Physics University of Arizona Tucson Arizona 85721 USA Program in Applied Mathematics University of Arizona Tucson Arizona 85721 USA BIO5 Institute University of Arizona Tucson Arizona 85721 USA Department of Applied Mathematics and Theoretical Physics University of Cambridge Wilberforce Road Cambridge CB3 0WA United Kingdom

Concentrated bacterial suspensions spontaneously develop transient spatiotemporal patterns of coherent locomotion whose correlation lengths greatly exceed the size of individual organisms. Continuum models have indicated that a state of uniform swimming order is linearly unstable at finite wavelengths, but have not addressed the nonlinear dynamics of the coherent state, with its biological implications for mixing, transport, and intercellular communication. We investigate a specific model incorporating hydrodynamic interactions in thin-film geometries and show by numerical studies that it displays large scale persistently recurring vortices, as actually observed.

关键词： ORDER DIFFUSION CELLS

Free time optimal control problems with time delays

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Free time optimal control problems with time delays

IEEE Annual Conference on Decision and Control

作者： Andrea Boccia Paola Falugi Helmut Maurer Richard B. Vinter Department of Electrical and Electronic Engineering Imperial College London Exhibition Road London SW7 2BT UK Institute of Computational and Applied Mathematics University of Munster Einsteinstrasse 62 D-48149 Munster Germany

ISBN: (纸本)9781467357159

Solutions to optimal control problems for retarded systems, on a fixed time interval, satisfy a form of the Maximum Principle, in which the co-state equation is an advanced differential equation. In this paper we present an extension of this well-known necessary condition of optimality, to cover situations in which the data is non-smooth, and the final time is free. The fact that the end-time is a choice variable is accommodated by an extra transversality condition. A traditional approach to deriving this extra condition is to reduce the free end-time problem to a fixed end-time problem by a parameterized change of the time variable. This approach is problematic for time delay problems because it introduces a parameter dependent time-delay that is not readily amenable to analysis;to avoid this difficulty we instead base our analysis on direct perturbation of the end-time. Formulae are derived for the gradient of the minimum cost as a function of the end-time. It is shown how these formulae can be exploited to construct two-stage algorithms for the computation of solutions to optimal retarded control problems with free-time, in which a sequence of fixed time problems are solved by means of Guinn's transformation, and the end-time is adjusted according to a rule based on the earlier derived gradient formulae for the minimum cost function. Numerical examples are presented.

关键词： Time Delays Optimal Control Maximum Principle Free Time Problems maximum principle Optimal control formula unoccupied time time variable

Fast decisions reflect biases, slow decisions do not

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arXiv 2023年

作者： Linn, Samantha Lawley, Sean D. Karamched, Bhargav R. Kilpatrick, Zachary P. Josić, Krešimir Department of Mathematics University of Utah Salt Lake CityUT United States Department of Mathematics Florida State University TallahasseeFL32306 United States Institute of Molecular Biophysics Florida State University TallahasseeFL32306 United States Program in Neuroscience Florida State University TallahasseeFL32306 United States Department of Applied Mathematics University of Colorado Boulder BoulderCO80309 United States Department of Mathematics University of Houston HoustonTX77004 United States Department of Biology and Biochemistry University of Houston HoustonTX77004 United States

Decisions are often made by heterogeneous groups of individuals, each with distinct initial biases and access to information of different quality. We show that in large groups of independent agents who accumulate evidence the first to decide are those with the strongest initial biases. Their decisions align with their initial bias, regardless of the underlying truth. In contrast, agents who decide last make decisions as if they were initially unbiased, and hence make better choices. We obtain asymptotic expressions in the large population limit that quantify how agents’ initial inclinations shape early decisions. Our analysis shows how bias, information quality, and decision order interact in non-trivial ways to determine the reliability of decisions in a group. © 2023, CC BY.

关键词： Reliability analysis