Path integral methods for stochastic differential equations.
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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.
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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. |
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Mentions: Hence, terms at the th order of the expansion for the moment are given by directed Feynman graphs with N final endpoint vertices, M initial endpoint vertices, and interior vertices with edges joining all vertices in all possible ways. The sum of the terms associated with these graphs is the value of the moment to th order. Figure 1 shows the vertices applicable to action (20) with . Arrows indicate the flow of time, from right to left. These components are combined into diagrams for the respective moments. Figure 2 shows three diagrams in the sum for the mean and second moment of . The entire expansion for any given moment can be expressed by constructing the Feynman diagrams for each term. Each Feynman diagram represents an integral involving the coefficients of a vertex and propagators. The construction of these integrals from the diagram is encapsulated in the Feynman rules: Fig. 1 |
View Article: PubMed Central - PubMed
Affiliation: Mathematical Biology Section, Laboratory of Biological Modeling, NIDDK, NIH, Bethesda, MD 20892 USA.
No MeSH data available.