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Path integral methods for stochastic differential equations.

Chow CC, Buice MA - J Math Neurosci (2015)

Bottom Line: Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult.Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs.The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder.

View Article: PubMed Central - PubMed

Affiliation: Mathematical Biology Section, Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, MD 20892 USA.

ABSTRACT
Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs. The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder.

No MeSH data available.


Bold edges represent the sum of all tree level diagrams contributing to that moment. (Top) The mean . (Bottom) Linear response
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Fig3: Bold edges represent the sum of all tree level diagrams contributing to that moment. (Top) The mean . (Bottom) Linear response

Mentions: The loop expansion implies that for each order of D in the expansion, all diagrams with the same number of loops must be included. In some cases, this could be an infinite number of diagrams. However, one can still write down an expression for the expansion because it is possible to write down the sum of all of these graphs as a set of self-consistent equations. For example, consider the expansion of the mean for action (20) for the case where (i.e. no noise term). The expansion will consist of the sum of all tree level diagrams. From Eq. (23), we see that it begins with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bigl\langle x(t) \bigr\rangle = yG(t,t_{0}) + b y^{2} \int G(t,t_{1}) G(t_{1},t_{0})^{2}\,dt_{1} +\cdots. $$\end{document}〈x(t)〉=yG(t,t0)+by2∫G(t,t1)G(t1,t0)2dt1+⋯. In fact, this expansion will be the perturbative expansion for the solution of the ordinary differential equation obtained by discarding the stochastic driving term. Hence, the sum of all tree level diagrams for the mean must satisfy 27\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{d}{dt} \bigl\langle x(t) \bigr\rangle _{\mathrm{tree}} = -a \bigl\langle x(t) \bigr\rangle _{\mathrm{tree}}+ b \bigl\langle x(t) \bigr\rangle _{\mathrm{tree}}^{2} +y\delta(t-t_{0}). $$\end{document}ddt〈x(t)〉tree=−a〈x(t)〉tree+b〈x(t)〉tree2+yδ(t−t0). Similarly, the sum of the tree level diagrams for the linear response, , is the solution of the linearization of (27) around the mean solution with a unit initial condition at , i.e. the propagator equation 28\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{d}{dt} G_{\mathrm{tree}}\bigl(t,t'\bigr)= -a G_{\mathrm{tree}}\bigl(t,t'\bigr)+ 2b \bigl\langle x(t) \bigr\rangle _{\mathrm{tree}}G_{\mathrm{tree}}\bigl(t,t'\bigr) + \delta \bigl(t - t'\bigr). $$\end{document}ddtGtree(t,t′)=−aGtree(t,t′)+2b〈x(t)〉treeGtree(t,t′)+δ(t−t′). The semiclassical approximation amounts to a small noise perturbation around the solution to this equation. We can represent the sum of the tree level diagrams graphically by using bold edges, which we call “classical” edges, as in Fig. 3. We can then use the classical edges within the loop expansion to compute semiclassical approximations to the moments of the solution to the SDE. Fig. 3


Path integral methods for stochastic differential equations.

Chow CC, Buice MA - J Math Neurosci (2015)

Bold edges represent the sum of all tree level diagrams contributing to that moment. (Top) The mean . (Bottom) Linear response
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4385267&req=5

Fig3: Bold edges represent the sum of all tree level diagrams contributing to that moment. (Top) The mean . (Bottom) Linear response
Mentions: The loop expansion implies that for each order of D in the expansion, all diagrams with the same number of loops must be included. In some cases, this could be an infinite number of diagrams. However, one can still write down an expression for the expansion because it is possible to write down the sum of all of these graphs as a set of self-consistent equations. For example, consider the expansion of the mean for action (20) for the case where (i.e. no noise term). The expansion will consist of the sum of all tree level diagrams. From Eq. (23), we see that it begins with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bigl\langle x(t) \bigr\rangle = yG(t,t_{0}) + b y^{2} \int G(t,t_{1}) G(t_{1},t_{0})^{2}\,dt_{1} +\cdots. $$\end{document}〈x(t)〉=yG(t,t0)+by2∫G(t,t1)G(t1,t0)2dt1+⋯. In fact, this expansion will be the perturbative expansion for the solution of the ordinary differential equation obtained by discarding the stochastic driving term. Hence, the sum of all tree level diagrams for the mean must satisfy 27\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{d}{dt} \bigl\langle x(t) \bigr\rangle _{\mathrm{tree}} = -a \bigl\langle x(t) \bigr\rangle _{\mathrm{tree}}+ b \bigl\langle x(t) \bigr\rangle _{\mathrm{tree}}^{2} +y\delta(t-t_{0}). $$\end{document}ddt〈x(t)〉tree=−a〈x(t)〉tree+b〈x(t)〉tree2+yδ(t−t0). Similarly, the sum of the tree level diagrams for the linear response, , is the solution of the linearization of (27) around the mean solution with a unit initial condition at , i.e. the propagator equation 28\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{d}{dt} G_{\mathrm{tree}}\bigl(t,t'\bigr)= -a G_{\mathrm{tree}}\bigl(t,t'\bigr)+ 2b \bigl\langle x(t) \bigr\rangle _{\mathrm{tree}}G_{\mathrm{tree}}\bigl(t,t'\bigr) + \delta \bigl(t - t'\bigr). $$\end{document}ddtGtree(t,t′)=−aGtree(t,t′)+2b〈x(t)〉treeGtree(t,t′)+δ(t−t′). The semiclassical approximation amounts to a small noise perturbation around the solution to this equation. We can represent the sum of the tree level diagrams graphically by using bold edges, which we call “classical” edges, as in Fig. 3. We can then use the classical edges within the loop expansion to compute semiclassical approximations to the moments of the solution to the SDE. Fig. 3

Bottom Line: Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult.Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs.The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder.

View Article: PubMed Central - PubMed

Affiliation: Mathematical Biology Section, Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, MD 20892 USA.

ABSTRACT
Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs. The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder.

No MeSH data available.