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Stochastic neural field theory and the systemsize expansion
 SIAM J. Appl. Math
, 2009
"... Abstract. We analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons. The state of the network is specified by the fraction of active or spiking neurons in each population, and tra ..."
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Abstract. We analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons. The state of the network is specified by the fraction of active or spiking neurons in each population, and transition rates are chosen so that in the thermodynamic or deterministic limit (N →∞)we recover standard activitybased or voltagebased rate models. We derive the lowest order corrections to these rate equations for large but finite N using two different approximation schemes, one based on the Van Kampen systemsize expansion and the other based on path integral methods. Both methods yield the same series expansion of the moment equations, which at O(1/N) can be truncated to form a closed system of equations for the first and secondorder moments. Taking a continuum limit of the moment equations while keeping the system size N fixed generates a system of integrodifferential equations for the mean and covariance of the corresponding stochastic neural field model. We also show how the path integral approach can be used to study large deviation or rare event statistics underlying escape from the basin of attraction of a stable fixed point of the meanfield dynamics; such an analysis is not possible using the systemsize expansion since the latter cannot accurately determine exponentially small transitions. Key words. neural field theory, master equations, stochastic processes, systemsize expansion, path integrals
Stochastic representations of ion channel kinetics and exact stochastic simulation of neuronal dynamics
, 2014
"... In this paper we provide two representations for stochastic ion channel kinetics, and compare the performance of exact simulation strategies with different, commonly used, approximate strategies. The first representation we present is a random time change representation, popularized by Thomas Kurtz, ..."
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In this paper we provide two representations for stochastic ion channel kinetics, and compare the performance of exact simulation strategies with different, commonly used, approximate strategies. The first representation we present is a random time change representation, popularized by Thomas Kurtz, with the second being analogous to a “Gillespie” representation. Stochastic models of ion channel kinetics typically consist of an ordinary differential equation, governing the voltage, coupled with a stochastic jump process, governing the number of open ion channels. The processes are coupled because the parameters of the ODE for the voltage depend upon the number of open ion channels, and the propensity for the opening and closing of the channels depends explicitly upon the timevarying voltage. Exact stochastic algorithms are provided for the different representations, which are preferable to either (a) fixed time step or (b) piecewise constant propensity algorithms, which still appear in the literature. As examples, we provide versions of the exact algorithms for the MorrisLecar conductance based model, and detail the error induced, both in a weak and a strong sense, by the use of approximate algorithms on this model. We include readytouse implementations of the random time change algorithm in both XPP and Matlab. Finally, through the consideration of parametric sensitivity analysis, we show how the representations presented here are useful in the development of further computational methods. The general representations and simulation strategies provided here are known in other parts of the sciences, but less so in the present setting.
Metastability in a Stochastic Neural Network Modeled as a Velocity Jump Markov Process
, 2014
"... Brain dynamics is noisy at the single cell level...but often observe coherent states at the macroscopic level noisy spike trains coherent waves and oscillations at network level SINGLE CELL RECORDINGS Single cell recordings in vivo suggest that individual cortical neurons are noisy with interspike ..."
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Brain dynamics is noisy at the single cell level...but often observe coherent states at the macroscopic level noisy spike trains coherent waves and oscillations at network level SINGLE CELL RECORDINGS Single cell recordings in vivo suggest that individual cortical neurons are noisy with interspike intervals (ISIs) close to Poisson (Softy and Koch 1993) The Journal of Neuroscience, January 1993, 13(l) 335 neocortical units have a very high degree of irregularity, with C, ranging between 0.5 and 1.O. We attempt to understand the origin ofthese values by two different theoretical methods: modified integrateandfire models, and simulations ofdetailed compartmental models of cortical pyramidal cells. Our analysis reveals a strong contradiction between the large observed interspike variability at high firing rates and the much smaller values pre
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, 2011
"... Dynamics of networks of excitatory and inhibitory neurons in ..."
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Dynamics of networks of excitatory and inhibitory neurons in
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, 2011
"... doi: 10.3389/fncom.2011.00059 Emergent properties of interacting populations of spiking neurons ..."
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doi: 10.3389/fncom.2011.00059 Emergent properties of interacting populations of spiking neurons
Onedimensional population density approaches to recurrently coupled networks of neurons with noise
 In preparation 2014
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Learning and Coding Correlations in Stochastic Network States
"... From the scientific side I would like to thank, first of all, my thesis supervisor Alain Destexhe. During these past four years, he has been an excellent mentor, always available and enthousiastic when I needed his help. It has been extremly fulfulling working with him and learning from him how to l ..."
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From the scientific side I would like to thank, first of all, my thesis supervisor Alain Destexhe. During these past four years, he has been an excellent mentor, always available and enthousiastic when I needed his help. It has been extremly fulfulling working with him and learning from him how to lead research projects from the idea proposal to the publication (and further). He trusted me from the begining and let me develop collaborations within the UNIC with a systematic supportive attitude. I am also thankful for letting me attend to summer schools and international conferences that were essential in my formation. His ability to do science in a constant good mood while being dedicated to his personal life has always impressed me and I consider Alain as a role model for my career. I hope we will go on working together in the future. I also thank the lab director Yves Frégnac for his trust and many precious advices he gave me all along my thesis work. I learned a lot from his experience and his broad scientific knowledge. It has been a pleasure collaborating with him and his team on many different projects. I would like to thank as well Daniel Shulz and Thierry Bal for the great collaborations we had and their constantly encouraging attitude.
9 9 IS R N IN R IA /R R 8 4 9 5 F R E
, 2014
"... Asymptotic description of neural networks with correlated synaptic weights Olivier Faugeras, James Maclaurinha l0 ..."
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Asymptotic description of neural networks with correlated synaptic weights Olivier Faugeras, James Maclaurinha l0
SPATIALLY STRUCTURED WAVES AND OSCILLATIONS IN NEURONAL NETWORKS WITH SYNAPTIC DEPRESSION AND ADAPTATION
, 2010
"... UMI Number: 3404426 ..."
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"... Stochastic synchronization of neuronal populations ..."
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