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Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics

Aaditya V. Rangan, David Cai, Louis Tao
2007 Journal of Computational Physics  
The practicability and efficiency of our numerical methods for solving the moment equations of the kinetic theory are illustrated with numerical examples.  ...  It is further demonstrated that the moment equations derived from the kinetic theory of neuronal network dynamics can very well capture the coarsegrained dynamical properties of integrate-and-fire neuronal  ...  Acknowledgments The work was performed under the support of NSF Grant DMS-0506396 for A.V.R., D.C. and L.T., NSF Grant DMS-0211655 for A.V.R. and D.C, and a Sloan Fellowship for D.C.  ... 
doi:10.1016/ fatcat:owawy5slyve47pgsgjgs3bf4yy

Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train

Aaditya V. Rangan, Gregor Kovačič, David Cai
2008 Physical Review E  
numerical evidence for their importance in correctly describing the coarse-grained dynamics of the underlying neuronal network.  ...  We present a kinetic theory for all-to-all coupled networks of identical, linear, integrate-and-fire, excitatory point neurons in which a fast and a slow excitatory conductance are driven by the same spike  ...  DMS-0506287 and gratefully acknowledges the hospitality of the Courant Institute of Mathematical Sciences and Center for Neural Science.  ... 
doi:10.1103/physreve.77.041915 pmid:18517664 fatcat:in3ofybnkradzglsggl335ksm4

An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex

D. Cai, L. Tao, M. Shelley, D. W. McLaughlin
2004 Proceedings of the National Academy of Sciences of the United States of America  
A coarse-grained representation of neuronal network dynamics is developed in terms of kinetic equations, which are derived by a moment closure, directly from the original large-scale integrate-andfire  ...  Finally, I&F networks and kinetic theory are used to discuss orientation selectivity of complex cells for "ring-model" architectures that characterize changes in the response of neurons located from near  ...  We thank Bob Shapley for many discussions throughout the entire course of this work, Dan Tranchina for introducing us to pdf representations of neuronal networks and his comments on this work, and Fernand  ... 
doi:10.1073/pnas.0401906101 pmid:15131268 pmcid:PMC419679 fatcat:y34i3t7srbhonnrplyo2sikdsa

The role of fluctuations in coarse-grained descriptions of neuronal networks

David Cai, Gregor Kovačič, David W. McLaughlin, Aaditya V. Rangan, Maxim S. Shkarayev, Louis Tao
2012 Communications in Mathematical Sciences  
We do not describe the numerical method for solving the kinetic equations here, but instead refer the reader to our work in [98] .  ...  In general, the kinetic equations must be solved numerically.  ...  Conclusions and Discussion Kinetic theory provides an accurate and efficient population-based coarse-graining method for describing neuronal network dynamics [31, 32, 98] .  ... 
doi:10.4310/cms.2012.v10.n1.a14 fatcat:co2g4zc7jza23eu7favisjkka4

Beyond mean field theory: statistical field theory for neural networks

Michael A Buice, Carson C Chow
2013 Journal of Statistical Mechanics: Theory and Experiment  
Mean field theories have been a stalwart for studying the dynamics of networks of coupled neurons. They are convenient because they are relatively simple and possible to analyze.  ...  Statistical field theory methods, in particular the Doi-Peliti-Janssen formalism, are particularly useful in this regard.  ...  Acknowledgments This research was supported by the Intramural Research Program of NIH/NIDDK.  ... 
doi:10.1088/1742-5468/2013/03/p03003 pmid:25243014 pmcid:PMC4169078 fatcat:m6vgzazz7zadpl5emkql6t4qrq

Dynamic Finite Size Effects in Spiking Neural Networks

Michael A. Buice, Carson C. Chow, Bard Ermentrout
2013 PLoS Computational Biology  
The network dynamics are fully characterized by a neuron population density that obeys a conservation law analogous to the Klimontovich equation in the kinetic theory of plasmas.  ...  We investigate the dynamics of a deterministic finite-sized network of synaptically coupled spiking neurons and present a formalism for computing the network statistics in a perturbative expansion.  ...  We adapt the methods of the kinetic theory as applied to gas and plasma dynamics to create a probabilistic description of the network dynamics [45, 46] .  ... 
doi:10.1371/journal.pcbi.1002872 pmid:23359258 pmcid:PMC3554590 fatcat:7fuak4tbgvdzjpgbn3fv3ofulq

Kinetic theory for neuronal network dynamics

David Cai, David W. McLaughlin, Aaditya V. Rangan, Louis Tao
2006 Communications in Mathematical Sciences  
The dimension reduction in our theory is achieved via novel moment closures.  ...  We also describe the limiting forms of our kinetic theory in various limits, such as the limit of mean-driven dynamics and the limit of infinitely fast conductances.  ...  While we may devise ever more efficient numerical methods for simulations of dynamics of large-scale neuronal networks [1] [2] [3] [4] [5] , basic computational constraints will eventually limit our simulation  ... 
doi:10.4310/cms.2006.v4.n1.a4 fatcat:2gve5bx2a5hhxmjxaor2woa4q4

A kinetic theory approach to capturing interneuronal correlation: the feed-forward case

Chin-Yueh Liu, Duane Q. Nykamp
2008 Journal of Computational Neuroscience  
We present an approach for using kinetic theory to capture first and second order statistics of neuronal activity.  ...  We implement a kinetic theory representation of a simple feed-forward network and demonstrate that the kinetic theory model captures key aspects of the emergence and propagation of correlations in the  ...  We thank Dan Tranchina, Brent Doiron, Michael Buice, Carson Chow, and Hide Câteau for helpful discussions.  ... 
doi:10.1007/s10827-008-0116-4 pmid:18987967 fatcat:tpmjppox25hnxcwl6vl4lwpwtu

An embedded network approach for scale-up of fluctuation-driven systems with preservation of spike information

D. Cai, L. Tao, D. W. McLaughlin
2004 Proceedings of the National Academy of Sciences of the United States of America  
We use a newly developed kinetic theory for the description of the coarse-grained background, in combination with a Poisson spike reconstruction procedure to ensure that our method applies to the fluctuation-driven  ...  To address computational "scale-up" issues in modeling large regions of the cortex, many coarse-graining procedures have been invoked to obtain effective descriptions of neuronal network dynamics.  ...  We showed that this kinetic theory is dynamically accurate and numerically efficient, even when the original point-neuron network is fluctuation-dominant.  ... 
doi:10.1073/pnas.0404062101 pmid:15381777 pmcid:PMC521148 fatcat:k53e7be5brd7rh4e6eracqqw6e

Mean-Field and Kinetic Descriptions of Neural Differential Equations [article]

M. Herty, T. Trimborn, G. Visconti
2021 arXiv   pre-print
Since typically neural networks process a very large amount of data, it is convenient to formulate them within the mean-field and kinetic theory.  ...  Furthermore, a modification of the microscopic dynamics, inspired by stochastic residual neural networks, leads to a Fokker-Planck formulation of the network, in which the concept of network training is  ...  -1 and under Germany's Excellence Strategy EXC-2023 Internet of Production 390621612.  ... 
arXiv:2001.04294v4 fatcat:kkd4zfrrwndspa4wkvzwiyxi5i

Page 9002 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
Summary: “Extended thermodynamics of 14 moments is used in solving a boundary value problem for one-dimensional heat conduction in a rarefied gas.  ...  Summary: “Optimal prediction methods compensate for a lack of resolution in the numerical solution of complex problems through the use of prior statistical information.  ... 

Solving the linear transport equation by a deep neural network approach

Zheng Chen, Liu Liu, Lin Mu
2021 Discrete and Continuous Dynamical Systems. Series S  
We demonstrate the accuracy and effectiveness of the proposed DNN method in numerical experiments.  ...  While the interest of using DNN to study partial differential equations is arising, here we adapt it to study kinetic models, in particular the linear transport model.  ...  The deep learning method is a new approach for solving partial differential equations (PDEs) that can resolve some difficulties appeared in traditional numerical methods [52] , such as the expensive computational  ... 
doi:10.3934/dcdss.2021070 fatcat:po2notrupjalpp7j757xp62ffm

Critical Analysis of Dimension Reduction by a Moment Closure Method in a Population Density Approach to Neural Network Modeling

Cheng Ly, Daniel Tranchina
2007 Neural Computation  
In the moment closure method, one approximates the conditional kth centered moment of excitatory conductance given voltage by the corresponding unconditioned moment.  ...  Efficiency of the PDF method is lost as the underlying neuron model is made more realistic and the number of state variables increases.  ...  We thank David Cai and Larry Sirovich for introducing us to moment closure methods in statistical physics and to David Cai for teaching us about the maximum entropy method in particular.  ... 
doi:10.1162/neco.2007.19.8.2032 pmid:17571938 fatcat:gq34mbonjfeobmnv6otlecuili

Firing rate equations require a spike synchrony mechanism to correctly describe fast oscillations in inhibitory networks

Federico Devalle, Alex Roxin, Ernest Montbrió, Sonja Gruen
2017 PLoS Computational Biology  
Here we investigate an exact low-dimensional description for a network of heterogeneous canonical type-I inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous  ...  Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band.  ...  As a consequence of that, the Ott-Antonsen theory becomes a unique method for deriving exact firing rate equations for ensembles of heterogeneous spiking neurons -see also [92] [93] [94] for recent alternative  ... 
doi:10.1371/journal.pcbi.1005881 pmid:29287081 pmcid:PMC5764488 fatcat:vijgcimekjdcdaw3fgwtudo6nu

Page 1277 of Mathematical Reviews Vol. , Issue 96c [page]

1996 Mathematical Reviews  
and train- ing of a recurrent neural network model for a class of nonlinear dynamic systems (415-418).  ...  Toshinori Oaku, Algorithmic methods in the boundary value prob- lem for systems of linear partial differential equations with regular ’ singularities (76-95); E.  ... 
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