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A numerical solver for a nonlinear Fokker–Planck equation representation of neuronal network dynamics

María J. Cáceres, José A. Carrillo, Louis Tao
2011 Journal of Computational Physics  
In this work, we propose a deterministic numerical scheme for a Fokker-Planck model of an excitatory-only network.  ...  A diffusion approximation then led to a nonlinear Fokker-Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances.  ...  In this work, we propose an efficient numerical scheme for the simulation of a nonlinear Fokker-Planck equation representation for neuronal network dynamics.  ... 
doi:10.1016/ fatcat:xqc3it67rrgkbn5uuksdfgwchq

Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: Comparison and implementation

Moritz Augustin, Josef Ladenbauer, Fabian Baumann, Klaus Obermayer, Ralf Haefner
2017 PLoS Computational Biology  
decomposition of the Fokker-Planck operator, the other is based on a cascade of two linear filters and a nonlinearity, which are determined from the Fokker-Planck equation and semi-analytically approximated  ...  When exposed to fluctuating inputs sparsely coupled populations of these model neurons exhibit stochastic collective dynamics that can be effectively characterized using the Fokker-Planck equation. [..  ...  Additionally we thank Srdjan Ostojic for providing his insights and code related to the eigenvalue problem of the Fokker-Planck operator. We would finally like to acknowledge that Douglas J.  ... 
doi:10.1371/journal.pcbi.1005545 pmid:28644841 pmcid:PMC5507472 fatcat:vkxwdfwk7jhodgupfjeualcv74

Deterministic particle flows for constraining stochastic nonlinear systems [article]

Dimitra Maoutsa, Manfred Opper
2021 arXiv   pre-print
Here, we propose a generally applicable and practically feasible non-iterative methodology for obtaining optimal dynamical interventions for diffusive nonlinear systems.  ...  Existing methods for identifying the necessary dynamical adjustments resort either to space discretising solutions of ensuing partial differential equations, or to iterative stochastic path sampling schemes  ...  We thank Sebastian Reich for insightful discussions during the early development of this work. This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG)-SFB1294/ 1-318763901.  ... 
arXiv:2112.05735v1 fatcat:syidouw7yzdejj27pboif4nofa

Neural Galerkin Scheme with Active Learning for High-Dimensional Evolution Equations [article]

Joan Bruna and Benjamin Peherstorfer and Eric Vanden-Eijnden
2022 arXiv   pre-print
fail, especially when features of the solutions evolve locally such as in high-dimensional wave propagation problems and interacting particle systems described by Fokker-Planck and kinetic equations.  ...  Our finding is that the active form of gathering training data of the proposed Neural Galerkin schemes is key for numerically realizing the expressive power of networks in high dimensions.  ...  Given a nonlinear parametric representation (e.g. via a DNN) of the solution of an initial value problem for a time-dependent PDE, we show how to derive a nonlinear evolution equation for the parameters  ... 
arXiv:2203.01360v3 fatcat:xhmep6kt5ndctmyf2gdl47dtcy

Linear noise approximation for oscillations in a stochastic inhibitory network with delay

Grégory Dumont, Georg Northoff, André Longtin
2014 Physical Review E  
The analysis first leads to a nonlinear delay-differential equation (DDE) with multiplicative noise for the mean activity.  ...  Our analytical result is in good agreement with the power spectrum obtained via numerical simulations of the full network dynamics, for the large range of parameters where both the intrinsic stochasticity  ...  This equation can then be further analyzed via the Fokker-Planck equation.  ... 
doi:10.1103/physreve.90.012702 pmid:25122330 fatcat:wxzaxwx52zccxaifvs4p6cnkoa

Integration of Continuous-Time Dynamics in a Spiking Neural Network Simulator

Jan Hahne, David Dahmen, Jannis Schuecker, Andreas Frommer, Matthias Bolten, Moritz Helias, Markus Diesmann
2017 Frontiers in Neuroinformatics  
Contemporary modeling approaches to the dynamics of neural networks consider two main classes of models: biologically grounded spiking neurons and functionally inspired rate-based units.  ...  models in a spiking network simulator.  ...  Partly supported by Helmholtz Portfolio Supercomputing and Modeling for the Human Brain (SMHB), the Initiative and Networking Fund of the Helmholtz Association, and the Helmholtz young investigator group  ... 
doi:10.3389/fninf.2017.00034 pmid:28596730 pmcid:PMC5442232 fatcat:qa7vy55vrncjxkuq54kavx3tkm

Common-input models for multiple neural spike-train data

Jayant E. Kulkarni, Liam Paninski
2007 Network  
Appendix: Deriving the forward-EKS equations We derive the equations of the forward-EKS step, which computes the expectation and variance of the latent processÑ(t), conditioned on the observations made  ...  Yu, and the members of the Center for Theoretical Neuroscience at Columbia University, for many helpful conversations.  ...  Figure 2 . 2 Comparing the Fokker-Planck approach with forward-EKS: In these simulations we numerically solve the Fokker-Planck Equations 10-12, for the case of a single neuron with a simple sinusoidal  ... 
doi:10.1080/09548980701625173 pmid:17943613 fatcat:viel2zv3dbch7h4mmoup7nesra

Estimating Parameters of Generalized Integrate-and-Fire Neurons from the Maximum Likelihood of Spike Trains

Yi Dong, Stefan Mihalas, Alexander Russell, Ralph Etienne-Cummings, Ernst Niebur
2011 Neural Computation  
In this study, we show that although convexity of the negative log-likelihood function is not guaranteed for this model, the minimum of this function yields a good estimate for the model parameters, in  ...  Furthermore, we show that a nonlinear function minimization method (r-algorithm with space dilation) usually reaches the global minimum.  ...  Acknowledgments This work was supported by NIH grants NEI R01EY016281 and NINDS 5R01NS40596 and the Office of Naval Research, grant N000141010278.  ... 
doi:10.1162/neco_a_00196 pmid:21851282 pmcid:PMC3513351 fatcat:cbph3v262bbf7jllt6lutow7hy

Delays and Oscillations in Networks of Spiking Neurons: A Two-Timescale Analysis

Dotan Di Castro, Ron Meir, Irad Yavneh
2009 Neural Computation  
Motivated by these phenomena, we study networks of excitatory and inhibitory integrate and fire neurons within a Fokker-Planck delay partial differential equation formalism, and establish explicit conditions  ...  By employing a two time scale analysis, the full partial differential equation is replaced in this limit by a discrete time iterative map, leading to a relatively simple dynamic interpretation.  ...  Acknowledgment The work of RM was partially supported by a Converging Technologies grant from the Israel Science Foundation.  ... 
doi:10.1162/neco.2008.03-08-723 pmid:19018702 fatcat:achktri7v5bejfcz6362ms53ii

A structure preserving numerical scheme for Fokker-Planck equations of structured neural networks with learning rules [article]

Qing He, Jingwei Hu, Zhennan Zhou
2021 arXiv   pre-print
In this work, we are concerned with a Fokker-Planck equation related to the nonlinear noisy leaky integrate-and-fire model for biological neural networks which are structured by the synaptic weights and  ...  To handle the endowed flux-shift structure and the multi-scale dynamics in a unified framework, we propose a numerical scheme for this equation that is mass conservative, unconditionally positivity preserving  ...  of H(w, t) numerical scheme for the Fokker-Planck equation(1.8) and there exists a strictly positive (every entry is positive) n×1 vector v * > 0 such that M v * > 0, then M is a non-singular M-matrix  ... 
arXiv:2109.04667v1 fatcat:a2xxf3s4ejgnbcnujztedji7ce

Action-Amplitude Approach to Controlled Entropic Self-Organization

Vladimir Ivancevic, Darryn Reid, Jason Scholz
2014 Entropy  
Motivated by the notion of perceptual error, as a core concept of the perceptual control theory, we propose an action-amplitude model for controlled entropic self-organization (CESO).  ...  view of functional compositions and commutative diagrams; (iii) a local geometric view of the Kähler-Ricci flow and time-evolution of entropic action; and (iv) a computational view using various path-integral  ...  Author Contributions Jason Scholz conceived of the problem of modelling military C2 in terms of a Perceptual Control Theory construction, concerning the perceptual errors of each agent, and that minimizing  ... 
doi:10.3390/e16052699 fatcat:kv3s7qfgtfew5les465yavgaci

Multi-Dimensional, Mesoscopic Monte Carlo Simulations of Inhomogeneous Reaction-Drift-Diffusion Systems on Graphics-Processing Units

Matthias Vigelius, Bernd Meyer, Grant Lythe
2012 PLoS ONE  
We demonstrate the validity and applicability of our algorithm with a comprehensive suite of standard test problems that also serve to quantify the numerical accuracy of the method.  ...  For many biological applications, a macroscopic (deterministic) treatment of reaction-drift-diffusion systems is insufficient.  ...  Fokker-Planck equation [54, 55] .  ... 
doi:10.1371/journal.pone.0033384 pmid:22506001 pmcid:PMC3323590 fatcat:d275tuccmrcqfpnzx6nwxa5mty

Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next [article]

Salvatore Cuomo, Vincenzo Schiano di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, Francesco Piccialli
2022 arXiv   pre-print
Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself.  ...  Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method  ...  Then they use a top-down Fokker-Planck model of diffusive development over Waddington-type landscapes, with a PINN learning such landscapes by fitting the PDFs to the Fokker-Planck equation.  ... 
arXiv:2201.05624v3 fatcat:elmdoax7ongblim3cbvkj2pdxi

Multi-Scale Spatio-Temporal Modeling: Lifelines of Microorganisms in Bioreactors and Tracking Molecules in Cells [chapter]

Alexei Lapin, Michael Klann, Matthias Reuss
2010 Biosystems Engineering II  
Creating a homogenous bag of molecules, a well-mixed system, the dynamic behavior of which is modeled with a set of ordinary differential equations is normally not valid.  ...  Agent-based models are rigorous tools for simulating the interactions of individual entities, such as organisms or molecules within cells and assessing their effects on the dynamic behavior of the system  ...  Acknowledgements The authors acknowledge support of the Deutsche Forschungsgemeinschaft (DFG) within the collaborative research center "Sonderforschungsbereich 412" and the Ministry of Science, Research  ... 
doi:10.1007/10_2009_53 pmid:20140659 fatcat:ywqd4jknerhajodlo6fukdtiw4

Uncertainty Propagation for General Stochastic Hybrid Systems on Compact Lie Groups [article]

Weixin Wang, Taeyoung Lee
2022 arXiv   pre-print
A computational framework is proposed to solve the Fokker-Planck (FP) equation that describes the time evolution of the probability density function for the state of GSHS.  ...  The FP equation is split into two parts: the partial differential operator corresponding to the continuous dynamics, and the integral operator arising from the discrete dynamics.  ...  Uncertainty propagation involves advecting a probability density along the flow of a dynamical system according to the Fokker-Planck (FP) equation.  ... 
arXiv:2203.02548v1 fatcat:kt3rqdeh2vax7kijorhzvet5zm
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