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Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks.

Bressloff PC - J Math Neurosci (2015)

Bottom Line: The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations.In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs.Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural master equation.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112 USA.

ABSTRACT
We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two stochastic variables, a continuous synaptic variable and a discrete activity variable. The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates depend on the synaptic variables. We assume a separation of time scales between fast spiking dynamics with time constant [Formula: see text] and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter [Formula: see text], which can be used to develop various asymptotic expansions of the corresponding path-integral representation of the stochastic dynamics. First, we derive a variational principle for maximum-likelihood paths of escape from a metastable state (large deviations in the small noise limit [Formula: see text]). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects of intrinsic noise on pattern formation in a spatially structured hybrid network. In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural master equation.

No MeSH data available.


Related in: MedlinePlus

Stochastic pattern formation in a scalar neural network. a Plot of Fourier transformed weight distribution as a function of wavenumber k for various values of gain : . b Sketch of corresponding power spectra , showing the peak in the spectrum at the critical wavenumber  for . Parameter values are , , , ,
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Fig2: Stochastic pattern formation in a scalar neural network. a Plot of Fourier transformed weight distribution as a function of wavenumber k for various values of gain : . b Sketch of corresponding power spectra , showing the peak in the spectrum at the critical wavenumber for . Parameter values are , , , ,

Mentions: Spectral theory can now be used to determine the effects of noise on pattern formation. First, Fourier transforming the Langevin equation (5.6) with respect to time gives\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda(k,\varOmega)\widehat{\varPhi}(k,\varOmega)=\widehat {B}_{0}(k)\widehat{\xi }(k,\varOmega) $$\end{document}Λ(k,Ω)Φˆ(k,Ω)=Bˆ0(k)ξˆ(k,Ω) with\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda(k,\varOmega)=-i\varOmega-\widehat{J}_{0}(k) $$\end{document}Λ(k,Ω)=−iΩ−Jˆ0(k) and\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigl\langle \widehat{\xi}(k,\varOmega)\bigr\rangle =0, \quad\quad\bigl\langle \widehat {\xi }(k,\varOmega) \widehat{\xi}\bigl(k',\varOmega'\bigr) \bigr\rangle =\delta_{k,-k'}\delta \bigl(\varOmega+\varOmega' \bigr). $$\end{document}〈ξˆ(k,Ω)〉=0,〈ξˆ(k,Ω)ξˆ(k′,Ω′)〉=δk,−k′δ(Ω+Ω′). It follows that\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \bigl\langle \widehat{\varPhi}(k,\varOmega)\widehat{\varPhi} \bigl(k',\varOmega '\bigr)\bigr\rangle =& \biggl\langle \biggl[\frac{\widehat{B}_{0}(k)}{\varLambda(k,\varOmega )}\widehat {\xi}(k,\varOmega) \biggr] \biggl[ \frac{\widehat{B}_{0}(k')}{\varLambda (k',\varOmega ')}\widehat{\xi}\bigl(k',\varOmega' \bigr) \biggr] \biggr\rangle \\ =&\delta_{k,-k'}\delta\bigl(\varOmega+\varOmega'\bigr) \frac{/B_{0}(k)/^{2}}{/\varLambda (k,\varOmega)/^{2}}. \end{aligned}$$ \end{document}〈Φˆ(k,Ω)Φˆ(k′,Ω′)〉=〈[Bˆ0(k)Λ(k,Ω)ξˆ(k,Ω)][Bˆ0(k′)Λ(k′,Ω′)ξˆ(k′,Ω′)]〉=δk,−k′δ(Ω+Ω′)/B0(k)/2/Λ(k,Ω)/2. Defining the power spectrum by\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigl\langle \widehat{\varPhi}(k,\varOmega) \widehat{\varPhi} \bigl(k',\varOmega '\bigr)\bigr\rangle =S(k,\varOmega) \delta_{k,-k'}\delta\bigl(\varOmega+\varOmega'\bigr), $$\end{document}〈Φˆ(k,Ω)Φˆ(k′,Ω′)〉=S(k,Ω)δk,−k′δ(Ω+Ω′), we deduce that5.10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ S(k,\varOmega)= \frac{/B_{0}(k)/^{2}}{/\varLambda(k,\varOmega)/^{2}}. $$\end{document}S(k,Ω)=/B0(k)/2/Λ(k,Ω)/2. From the deterministic theory, we know that the system undergoes a Turing instability (stationary patterns) rather than a Turing–Hopf instability (oscillatory patterns) so we can set and determine conditions under which has a peak at a non-zero, finite value of k, which is an indication of a stochastic pattern. Substituting the explicit expression for and , we have5.11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ S(k,0)=\frac{2\widehat{w}(k)^{2}F(u_{0})}{[-1+\widehat {w}(k)F'(u_{0})]^{2}}=\frac{2F(u_{0})}{F'(u_{0})^{2}}\bigl[1+\lambda(k)^{-1} \bigr]^{2}. $$\end{document}S(k,0)=2wˆ(k)2F(u0)[−1+wˆ(k)F′(u0)]2=2F(u0)F′(u0)2[1+λ(k)−1]2. Suppose that so the system is below the deterministic critical point for a Turing instability. Clearly becomes singular as , consistent with the fixed point becoming unstable. The main new result is that has a peak at the critical wavenumber for all μ, . This follows from the fact that for all k in the subcritical regime with . Hence, will have a peak at provided that5.12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0< \bigl/\lambda(k_{c})\bigr/\equiv1- \mu\widehat{w}(k_{c})<1 \quad\implies\quad\mu<\mu_{c}. $$\end{document}0</λ(kc)/≡1−μwˆ(kc)<1⇒μ<μc. This is illustrated in Fig. 2. Fig. 2


Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks.

Bressloff PC - J Math Neurosci (2015)

Stochastic pattern formation in a scalar neural network. a Plot of Fourier transformed weight distribution as a function of wavenumber k for various values of gain : . b Sketch of corresponding power spectra , showing the peak in the spectrum at the critical wavenumber  for . Parameter values are , , , ,
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4385107&req=5

Fig2: Stochastic pattern formation in a scalar neural network. a Plot of Fourier transformed weight distribution as a function of wavenumber k for various values of gain : . b Sketch of corresponding power spectra , showing the peak in the spectrum at the critical wavenumber for . Parameter values are , , , ,
Mentions: Spectral theory can now be used to determine the effects of noise on pattern formation. First, Fourier transforming the Langevin equation (5.6) with respect to time gives\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda(k,\varOmega)\widehat{\varPhi}(k,\varOmega)=\widehat {B}_{0}(k)\widehat{\xi }(k,\varOmega) $$\end{document}Λ(k,Ω)Φˆ(k,Ω)=Bˆ0(k)ξˆ(k,Ω) with\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda(k,\varOmega)=-i\varOmega-\widehat{J}_{0}(k) $$\end{document}Λ(k,Ω)=−iΩ−Jˆ0(k) and\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigl\langle \widehat{\xi}(k,\varOmega)\bigr\rangle =0, \quad\quad\bigl\langle \widehat {\xi }(k,\varOmega) \widehat{\xi}\bigl(k',\varOmega'\bigr) \bigr\rangle =\delta_{k,-k'}\delta \bigl(\varOmega+\varOmega' \bigr). $$\end{document}〈ξˆ(k,Ω)〉=0,〈ξˆ(k,Ω)ξˆ(k′,Ω′)〉=δk,−k′δ(Ω+Ω′). It follows that\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \bigl\langle \widehat{\varPhi}(k,\varOmega)\widehat{\varPhi} \bigl(k',\varOmega '\bigr)\bigr\rangle =& \biggl\langle \biggl[\frac{\widehat{B}_{0}(k)}{\varLambda(k,\varOmega )}\widehat {\xi}(k,\varOmega) \biggr] \biggl[ \frac{\widehat{B}_{0}(k')}{\varLambda (k',\varOmega ')}\widehat{\xi}\bigl(k',\varOmega' \bigr) \biggr] \biggr\rangle \\ =&\delta_{k,-k'}\delta\bigl(\varOmega+\varOmega'\bigr) \frac{/B_{0}(k)/^{2}}{/\varLambda (k,\varOmega)/^{2}}. \end{aligned}$$ \end{document}〈Φˆ(k,Ω)Φˆ(k′,Ω′)〉=〈[Bˆ0(k)Λ(k,Ω)ξˆ(k,Ω)][Bˆ0(k′)Λ(k′,Ω′)ξˆ(k′,Ω′)]〉=δk,−k′δ(Ω+Ω′)/B0(k)/2/Λ(k,Ω)/2. Defining the power spectrum by\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigl\langle \widehat{\varPhi}(k,\varOmega) \widehat{\varPhi} \bigl(k',\varOmega '\bigr)\bigr\rangle =S(k,\varOmega) \delta_{k,-k'}\delta\bigl(\varOmega+\varOmega'\bigr), $$\end{document}〈Φˆ(k,Ω)Φˆ(k′,Ω′)〉=S(k,Ω)δk,−k′δ(Ω+Ω′), we deduce that5.10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ S(k,\varOmega)= \frac{/B_{0}(k)/^{2}}{/\varLambda(k,\varOmega)/^{2}}. $$\end{document}S(k,Ω)=/B0(k)/2/Λ(k,Ω)/2. From the deterministic theory, we know that the system undergoes a Turing instability (stationary patterns) rather than a Turing–Hopf instability (oscillatory patterns) so we can set and determine conditions under which has a peak at a non-zero, finite value of k, which is an indication of a stochastic pattern. Substituting the explicit expression for and , we have5.11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ S(k,0)=\frac{2\widehat{w}(k)^{2}F(u_{0})}{[-1+\widehat {w}(k)F'(u_{0})]^{2}}=\frac{2F(u_{0})}{F'(u_{0})^{2}}\bigl[1+\lambda(k)^{-1} \bigr]^{2}. $$\end{document}S(k,0)=2wˆ(k)2F(u0)[−1+wˆ(k)F′(u0)]2=2F(u0)F′(u0)2[1+λ(k)−1]2. Suppose that so the system is below the deterministic critical point for a Turing instability. Clearly becomes singular as , consistent with the fixed point becoming unstable. The main new result is that has a peak at the critical wavenumber for all μ, . This follows from the fact that for all k in the subcritical regime with . Hence, will have a peak at provided that5.12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0< \bigl/\lambda(k_{c})\bigr/\equiv1- \mu\widehat{w}(k_{c})<1 \quad\implies\quad\mu<\mu_{c}. $$\end{document}0</λ(kc)/≡1−μwˆ(kc)<1⇒μ<μc. This is illustrated in Fig. 2. Fig. 2

Bottom Line: The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations.In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs.Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural master equation.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112 USA.

ABSTRACT
We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two stochastic variables, a continuous synaptic variable and a discrete activity variable. The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates depend on the synaptic variables. We assume a separation of time scales between fast spiking dynamics with time constant [Formula: see text] and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter [Formula: see text], which can be used to develop various asymptotic expansions of the corresponding path-integral representation of the stochastic dynamics. First, we derive a variational principle for maximum-likelihood paths of escape from a metastable state (large deviations in the small noise limit [Formula: see text]). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects of intrinsic noise on pattern formation in a spatially structured hybrid network. In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural master equation.

No MeSH data available.


Related in: MedlinePlus