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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.


Vertex for multiplicative noise with  in the action (20). This vertex replaces the one in Fig. 1d
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Fig4: Vertex for multiplicative noise with in the action (20). This vertex replaces the one in Fig. 1d

Mentions: Now consider the one-loop corrections when in action (20). First consider the linear response, . For simplicity, we will assume the initial condition . In this case, the vertex in Fig. 1d now appears as in Fig. 4. The linear response will be given by the sum of all diagrams with one entering edge and one exiting edge. At tree level, there is only one such graph, equal to , given in (22). At one-loop order, we can combine the vertices in Figs. 1b and 1d to get the second graph shown in Fig. 5 to obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \bigl\langle x(t) \tilde{x}\bigl(t'\bigr) \bigr\rangle =& G \bigl(t,t'\bigr) + bD \int \,dt_{1} \,dt_{2} G(t,t_{2}) G(t_{2}, t_{1})^{2}G \bigl(t_{2}, t'\bigr) \\ =& e^{-a(t-t')}H\bigl(t - t'\bigr) \biggl[ 1+ bD \biggl( \frac{t - t'}{a} + \frac {1}{a^{2}} \bigl(e^{-a(t-t')} - 1 \bigr) \biggr) \biggr]. \end{aligned}$$ \end{document}〈x(t)x˜(t′)〉=G(t,t′)+bD∫dt1dt2G(t,t2)G(t2,t1)2G(t2,t′)=e−a(t−t′)H(t−t′)[1+bD(t−t′a+1a2(e−a(t−t′)−1))]. This loop correction arises because of two types of vertices. There are vertices that we call “branching” (as in Fig. 4), which have more exiting edges then entering edges. The opposite case occurs for those vertices which we call “aggregating.” Noise terms in the SDE produce branching vertices. As can be seen from the structure of the Feynman diagrams, all moments can be computed exactly when the deterministic part of the SDE is linear because it only involves convolving the propagator (i.e. Green’s function) of the deterministic part of the SDE with the driving noise term, as in the case of the OU process above. On the other hand, nonlinearities give rise to aggregating vertices. Fig. 4


Path integral methods for stochastic differential equations.

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

Vertex for multiplicative noise with  in the action (20). This vertex replaces the one in Fig. 1d
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig4: Vertex for multiplicative noise with in the action (20). This vertex replaces the one in Fig. 1d
Mentions: Now consider the one-loop corrections when in action (20). First consider the linear response, . For simplicity, we will assume the initial condition . In this case, the vertex in Fig. 1d now appears as in Fig. 4. The linear response will be given by the sum of all diagrams with one entering edge and one exiting edge. At tree level, there is only one such graph, equal to , given in (22). At one-loop order, we can combine the vertices in Figs. 1b and 1d to get the second graph shown in Fig. 5 to obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \bigl\langle x(t) \tilde{x}\bigl(t'\bigr) \bigr\rangle =& G \bigl(t,t'\bigr) + bD \int \,dt_{1} \,dt_{2} G(t,t_{2}) G(t_{2}, t_{1})^{2}G \bigl(t_{2}, t'\bigr) \\ =& e^{-a(t-t')}H\bigl(t - t'\bigr) \biggl[ 1+ bD \biggl( \frac{t - t'}{a} + \frac {1}{a^{2}} \bigl(e^{-a(t-t')} - 1 \bigr) \biggr) \biggr]. \end{aligned}$$ \end{document}〈x(t)x˜(t′)〉=G(t,t′)+bD∫dt1dt2G(t,t2)G(t2,t1)2G(t2,t′)=e−a(t−t′)H(t−t′)[1+bD(t−t′a+1a2(e−a(t−t′)−1))]. This loop correction arises because of two types of vertices. There are vertices that we call “branching” (as in Fig. 4), which have more exiting edges then entering edges. The opposite case occurs for those vertices which we call “aggregating.” Noise terms in the SDE produce branching vertices. As can be seen from the structure of the Feynman diagrams, all moments can be computed exactly when the deterministic part of the SDE is linear because it only involves convolving the propagator (i.e. Green’s function) of the deterministic part of the SDE with the driving noise term, as in the case of the OU process above. On the other hand, nonlinearities give rise to aggregating vertices. Fig. 4

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.