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# Path Integral Methods for Stochastic Differential Equations

- Carson C. Chow
^{1}Email author and - Michael A. Buice
^{1}

**5**:8

https://doi.org/10.1186/s13408-015-0018-5

© Chow and Buice; licensee Springer. 2015

**Received:**12 February 2015**Accepted:**13 February 2015**Published:**24 March 2015

## 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.

## 1 Introduction

In mathematical neuroscience, stochastic differential equations (SDE) have been utilized to model stochastic phenomena that range in scale from molecular transport in neurons, to neuronal firing, to networks of coupled neurons, to cognitive phenomena such as decision making [1]. Generally these SDEs are impossible to solve in closed form and must be tackled approximately using methods that include eigenfunction decompositions, WKB expansions, and variational methods in the Langevin or Fokker–Planck formalisms [2–4]. Often knowing what method to use is not obvious and their application can be unwieldy, especially in higher dimensions. Here we demonstrate how methods adapted from statistical field theory can provide a unifying framework to produce perturbative approximations to the moments of SDEs [5–13]. Any stochastic and even deterministic system can be expressed in terms of a path integral for which asymptotic methods can be systematically applied. Often of interest are the moments of \(x(t)\) or the probability density function \(p(x,t)\). Path integral methods provide a convenient tool to compute quantities such as moments and transition probabilities perturbatively. They also make renormalization group methods available when perturbation theory breaks down. These methods have been recently applied at the level of networks and to more general stochastic processes [14–25].

Although Wiener introduced path integrals to study stochastic processes, these methods are not commonly used nor familiar to much of the neuroscience or applied mathematics community. There are many textbooks on path integrals but most are geared towards quantum field theory or statistical mechanics [26–28]. Here we give a pedagogical review of these methods specifically applied to SDEs. In particular, we show how to apply the response function method [29, 30], which is particularly convenient to compute desired quantities such as moments.

The main goal of this review is to present methods to compute actual quantities. Thus, mathematical rigor will be dispensed for convenience. This review will be elementary and self-contained. In Sect. 2, we cover moment generating functionals, which expand the definition of generating functions to cover distributions of functions, such as the trajectory of a stochastic process. We continue in Sect. 3 by constructing functional integrals appropriate for the study of SDEs, using the Ornstein–Uhlenbeck process as an example. Section 4 introduces the concept of Feynman diagrams as a tool for carrying out perturbative expansions and introduces the “loop expansion,” a tool for constructing semiclassical approximations. We carry out a perturbative calculation explicitly for the stochastically forced FitzHugh–Nagumo equation in Sect. 5. Finally, Sect. 6 provides the connection between SDEs and equations for the density \(p(x,t)\) such as Fokker–Planck equations.

## 2 Moment Generating Functionals

The strategy of path integral methods is to derive a generating function or functional for the moments and response functions for SDEs. The generating functional will be an infinite dimensional generalization for the familiar generating function for a single random variable. In this section we review moment generating functions and show how they can be generalized to functional distributions.

*x*. The moments of the PDF are given by

*J*is a complex parameter, with

*a*and variance

*G*, is

*x*is then given by

*n*-dimensional vector \(x=\{x_{1},x_{2},\ldots,x_{n}\}\) to become a generating functional that maps the

*n*-dimensional vector \(J=\{J_{1},J_{2},\dots,J_{n}\}\) to a real number with the form

*α*th eigenvalue and orthonormal eigenvector of \(G^{-1}\), respectively, i.e.

*x*and

*J*in terms of the eigenvectors with

*d*is

*n*segments of length

*h*so that \(T=nh\) and \(x(jh)=x_{j}\), \(j\in\{0,1,\dots,n\}\). We then take the limit of \(n\rightarrow\infty\) and \(h\rightarrow0\) preserving \(T= n h\). We similarly identify \(J_{j}\rightarrow J(t)\) and \(G_{jk}\rightarrow G(s,t)\) and obtain

*t*with

*φ*-4”) theory.

The analogy between stochastic systems and quantum theory, where path integrals are commonly used, is seen by transforming the time coordinates in the path integrals via \(t \rightarrow\sqrt{-1}t\). When the field *φ* is a function of a single variable *t*, then this would be analogous to single particle quantum mechanics where the quantum amplitude can be expressed in terms of a path integral over a configuration variable \(\phi(t)\). When the field is a function of two or more variables \(\varphi(\vec{r},t)\), then this is analogous to quantum field theory, where the quantum amplitude is expressed as a path integral over the quantum field \(\varphi(\vec{r},t)\).

## 3 Application to SDE

*x*at time

*t*. We show how to generalize to other stochastic processes later. Functions

*f*and

*g*are assumed to obey all the properties required for an Ito SDE to be well posed [31]. In particular, the stochastic increment \(dW_{t}\) does not depend on \(f(x_{t},t)\) or \(g(x_{t},t)\) (i.e. \(x_{t}\) is adapted to the filtration generated by the noise). The choice between Ito and Stratonovich conventions amounts to a choice of the measure for the path integrals, which will be manifested in a condition on the linear response or “propagator” that we introduce below.

*h*is given by

*x*and

*w*without indices to denote the vectors \(x = (x_{1}, \dots, x_{N})\) and \(w = (w_{0}, w_{1}, \dots, w_{N-1})\). Formally, the PDF for the vector

*x*conditioned on

*w*and

*y*can be written as

*y*and \(t_{0}\). The moment generating functional for \(x(t)\) and \(\tilde{x}(t)\) is then given by

^{1}Owing to the definition \(i k_{j} \rightarrow\tilde{x}(t)\) the integrals over \(\tilde{x}(t)\) are along the

*imaginary*axis, which is why no explicit

*i*appears in the action above.

### 3.1 Ornstein–Uhlenbeck Process

^{2}Although the generating functional (11) differs from those introduced in Sect. 2 because it is complex and has two source functions

*J*and \(\tilde{J}\), it still obeys Wick’s theorem.

*F*to denote expectation values with respect to the free action. From the action of (11), it is clear the nonzero free moments must have equal numbers of \(x(t)\) and \(\tilde{x}(t)\) due to Wick’s theorem, which applies here for contractions between \(x(t)\) and \(\tilde{x}(t)\). For example, one of the fourth moments is given by

*m*! combinations. Thus (12) is

*G*. Hence, the mean is

## 4 Perturbative Methods and Feynman Diagrams

If the SDE is nonlinear then the generating functional cannot be computed exactly as in the linear case. However, propagators and moments can be computed perturbatively. The method we use is an infinite dimensional generalization of Laplace’s method for finite dimensional integrals [33]. In fact, the method was used to compute the generating functional for the Ornstein–Uhlenbeck process. The only difference is that for nonlinear SDEs the resulting asymptotic series is not generally summable.

There are two types of expansions depending on whether the nonlinearity is small or the noise source is small. The small nonlinearity expansion is called a weak coupling expansion and the small noise expansion is called a semiclassical, WKB, or loop expansion.

### 4.1 Weak Coupling Expansion

*b*can be of any sign. For example, \(p=0\) corresponds to standard additive noise (as in the OU process), while \(p=1\) gives multiplicative noise with variance proportional to

*x*. The action for this equation is

*x*, the only surviving terms in the expansion are those with equal numbers of

*x*and \(\tilde{x}\) factors. Also, because of the Ito condition, \(\langle x(t)\tilde{x}(t)\rangle=0\), these pairings must come from

*different*terms in the expansion, e.g. the only term surviving from the first line is the very first term, regardless of the value of

*p*. All other terms come from the quadratic and higher terms in the expansion. For simplicity in the remainder of this example we limit ourselves to \(p=0\). Hence, the expansion includes terms of the form

*n*! ways of combining terms at order

*n*and terms with

*m*repeats are divided by a factor of

*m*!.

### 4.2 Diagrammatic Expansion

As can be seen in this example, the terms in the perturbation series become rapidly unwieldy. However, a convenient means to keep track of the terms is to use Feynman diagrams, which are graphs with edges connected by vertices that represents each term in the expansion of a moment. The edges and vertices represent terms (i.e. interactions) in the action and hence SDE, which are combined according to a set of rules that reproduces the perturbation expansion shown above. These are directed graphs (unlike the Feynman diagrams usually used for equilibrium statistical mechanics or particle physics). The flow of each graph, which represents the flow of time, is directed from right to left, points to the left being considered to be at times after points to the right. The vertices represent points in time and separate into two groups: *endpoint* vertices and *interior* vertices. The moment \(\langle\prod_{j=1}^{N} x(t_{j}) \prod_{k=1}^{M} \tilde{x}(t_{k}) \rangle\) is represented by diagrams with *N*
*final* endpoint vertices which represent the times \(t_{j}\) and *M*
*initial* endpoint vertices which represent the times \(t_{k}\). Interior vertices are determined from terms in the action.

*n*and

*m*cannot both be ≤1 (those terms are part of the free action). (Nonpolynomial functions in the action are expanded in a Taylor series to obtain this form.) There is a vertex type associated with each \(V_{nm}\). The moment \(\langle\prod_{j=1}^{N} x(t_{j}) \prod_{k=1}^{M} \tilde{x}(t_{k}) \rangle \) is given by a perturbative expansion of free action moments that are proportional to \(\langle\prod_{j=1}^{N} x(t_{j}) \prod_{k=1}^{M} \tilde {x}(t_{k}) V(N_{v}) \rangle_{\mathrm{F}}\) where \(V(N_{v})\) represents a product of \(N_{v}\) vertices. Each term in this expansion corresponds to a graph with \(N_{v}\) interior vertices. We label the

*k*th vertex with time \(t_{k}\). As indicated in equation (25), there is an integration over each such interior time point, over the interval \((t_{0}, \infty)\). The interaction \(V_{nm}\) produces vertices with

*n*edges to the left of the vertex (towards increasing time) and

*m*edges to the right of the vertex (towards decreasing times). Edges between vertices represent propagators that arise from an application of Wick’s theorem and thus every \(\tilde {x}(t')\) must be joined by a factor of \(x(t)\)

*in the future*, i.e. \(t>t'\), because \(G(t,t') \propto H(t-t')\). Also, since \(H(0) = 0\) by the Ito condition, each edge must connect two

*different*vertices. All edges must be connected, a vertex for the interaction \(V_{nm}\) must connect to

*n*edges on the left and

*m*edges on the right.

*N*final endpoint vertices,

*M*initial endpoint vertices, and \(N_{v}\) 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 \(N_{v}\)th order. Figure 1 shows the vertices applicable to action (20) with \(p=0\). 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 \(x(t)\). 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:

(A) For each vertex interaction \(V_{nm}\) in the diagram, include a factor of \(-\frac{v_{nm}}{n!}\). The minus sign enters because the action appears in the path integral with a minus sign.

(B) If the vertex type, \(V_{nm}\) appears *k* times in the diagram, include a factor of \(\frac{1}{k!}\).

(C) For each edge between times *t* and \(t'\), there is a factor of \(G(t,t')\).

(D) For *n* distinct ways of connecting edges to vertices that yield the same diagram, i.e. the same topology, there is an overall factor of *n*. This is the combinatoric factor from the number of different Wick contractions that yield the same diagram.

(E) Integrate over the times *t* of each interior vertex over the domain \((t_{0}, \infty)\).

The diagrammatic expansion is particularly useful if the series can be truncated so that only a few diagrams need to be computed. The weak coupling expansion is straightforward. Suppose one or more of the vertices is associated with a small parameter *α*. Each appearance of that particular vertex diagram contributes a factor of *α* and the expansion can be continued to any order in *α*. The expansion is also generally valid over all time if \(G(t,t')\) decays exponentially for large \(t-t'\) but can break down if we are near a critical point and \(G(t,t')\) obeys a power law. We consider the semiclassical expansion in the next section where the small parameter is the noise strength.

Comparing these rules with the diagrams in Fig. 2, one can see the terms in the expansions in equations (23) and (24), with the exception of the middle diagram in Fig. 2b. An examination of Fig. 2a shows that this middle diagram consists of two copies of the first diagram of the mean. Topologically, the diagrams have two forms. There are connected graphs and disconnected graphs. The disconnected graphs represent terms that can be completely factored into a product of moments of lower order (cf. the middle diagram in Fig. 2b). Cumulants consist only of connected graphs since the products of lower ordered moments are subtracted by definition. The connected diagrams in Fig. 2 lead to the expressions (23) and (24). In the expansion (21), the terms that do not include the source factors *J* and \(\tilde{J}\) only contribute to the normalization \(Z[0,0]\) and do not affect moments because of (18). In quantum field theory, these terms are called vacuum graphs and consist of closed graphs, i.e. they have no initial or trailing edges. In the cases we consider, all of these terms are 0 if we set \(Z[0,0] = 1\).

### 4.3 Semiclassical Expansion

*D*, while

*f*and

*g*are of order one. Now, rescale the action with the transformation \(\tilde{x} \rightarrow\tilde{x}/D\) to obtain

*D*corresponds to a semiclassical approximation. In both quantum mechanics and stochastic analysis this is also known as a WKB expansion. According to the Feynman rules for such an action, each diagram gains a factor of

*D*for each edge (internal or external) and a factor of \(1/D\) for each vertex. Let

*E*be the number of external edges,

*I*the number of internal edges, and

*V*the number of vertices. Then each connected graph now has a factor \(D^{I+E-V}\). It can be shown via induction that the number of closed loops

*L*in a given connected graph must satisfy \(L = I - V + 1\) [26]. To see this, note that for diagrams without loops any two vertices must be connected by at most one internal edge since more than one edge would form a closed loop. Since the diagrams are connected we must have \(V = I + 1\) when \(L = 0\). Adding an internal edge between any two vertices increases the number of loops by precisely one. Thus we see that the total factor for each diagram may be written \(D^{E + L - 1}\). Since the number of external edges is fixed for a given cumulant, the order of the expansion scales with the number of loops.

We can organize the diagrammatic expansion in terms of the number of loops in the graphs. Not surprisingly, the semiclassical expansion is also called the loop expansion. For example, as seen in Fig. 2a the graph for the mean has one external edge and thus to lowest order (graph with no loop), there are no factors of *D*, while one loop corresponds to the order *D* term. The second cumulant or variance has two external edges and thus the lowest order tree level term is order *D* as seen in Fig. 2b. Loop diagrams arise because of nonlinearities in the SDE that couple to moments of the driving noise source. The middle graph in Fig. 2a describes the coupling of the variance to the mean through the nonlinear \(x^{2}\) term. This produces a single-loop diagram which is of order *D*, compared to the order 1 “tree” level mean graph. Compare this factor of *D* to that from the tree level diagram for the variance, which is order *D*. This same construction holds for higher nonlinearities and higher moments for general theories. The loop expansion is thus a series organized around the magnitude of the coupling of higher moments to lower moments.

*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 \(D=0\) (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

*D*for the mean (one loop) and covariance (tree level).

*effective action*, which is beyond the scope of this review. We refer the interested reader to [26].

*D*, the mean is given by the second diagram in Fig. 2a, which immediately gives (29) and (31). Likewise, the variance will be given by the diagram in Fig. 1d leading to (30) and (32).

## 5 FitzHugh–Nagumo Model

*v*is the neuron potential,

*w*is a recovery variable,

*a*,

*b*, and

*c*are positive parameters,

*D*is the noise amplitude, and initial conditions are \(v_{0}\), \(w_{0}\). We will consider a semiclassical or WKB expansion in terms of

*D*. We wish to compute the means, variances, and covariance for

*v*and

*w*as a loop expansion in

*D*.

*v*and

*w*. As both equations in the system must be satisfied simultaneously, we write

*v*is \(\langle v \rangle= V + \langle\nu\rangle\) and the mean of

*w*is \(\langle w \rangle= W+ \langle\omega\rangle\). The diagrams for \(\langle\nu\rangle\), and \(\langle\omega\rangle\) are shown in Fig. 7. \(\langle\nu\rangle\) is given by joining the two vertex diagrams in Figs. 6b and 6c to obtain Fig. 7a, which is topologically equivalent to the middle diagram in Fig. 2a:

*v*is \(\langle\nu(t)\nu(t')\rangle_{C}\) in Fig. 8a is found by using Fig. 6c adjoined to two \(G_{\nu}^{v}\) propagators:

*w*is \(\langle\omega(t)\omega(t')\rangle_{C}\) in Fig. 8b is also given by Fig. 6c but adjoined to two \(G_{\omega}^{v}\) propagators:

*v*and

*w*is \(\langle\nu(t)\omega (t')\rangle_{C}\) in Fig. 8c is given by Fig. 6c adjoined to the \(G_{\nu}^{v}\) and \(G_{\omega}^{v}\) propagators:

To evaluate these expressions we first solve the deterministic equations (37) to obtain \(V(t)\) and \(W(t)\). We then use \(V(t)\) in (39)–(42) and solve for the four propagators, which go into the expressions for the moments. When the solutions of (37) are fixed points, then we can find closed-form solutions for all the equations. Otherwise, we may need to solve some of the equations numerically. However, instead of having to average over many samples of the noise distribution, we only need to solve a small set of equations once to obtain the moments.

## 6 Connection to Fokker–Planck Equation

In stochastic systems, one is often interested in the PDF \(p(x,t)\), which gives the probability density of position *x* at time *t*. This is in contrast with the probability density functional \(P[x(t)]\) which is the probability density of all possible functions or paths \(x(t)\). Previous sections have been devoted to computing the moments of \(P[x(t)]\), which provide the moments of \(p(x,t)\) as well. In this section we leverage knowledge of the moments of \(p(x,t)\) to determine an equation it must satisfy. In simple cases, this equation is a Fokker–Planck equation for \(p(x,t)\).

*J*integral runs along the imaginary axis. This can be rewritten as

*t*. Taylor expanding the exponential gives

*t*gives

*jump moments*are defined by

*y*is the initial condition, \(\tilde{z} = \tilde{x}\) and use the action \(S[z(t)+y, \tilde {z}(t)]\). This shift in

*x*removes the initial condition term. This means we can calculate the

*n*th jump moment by using this shifted action to compute the sum of all graphs with no initial edges and

*n*final edges (as in Fig. 1d for \(n=2\)).

*n*and function \(h(x)\). This will produce a nonzero \(D_{n}\). The PDF for this kind of process will not in general be describable by a Fokker–Planck equation, but we will need the full Kramers–Moyal expansion. If we wished to provide an initial distribution for \(x(t_{0})\) instead of specifying a single point, we could likewise add similar terms to the action. In fact, the completely general initial condition term is given by

*t*. It can be expanded as

## 7 Further Reading

The methods we introduced can be generalized to higher dimensional systems including networks of coupled oscillators or neurons [16, 17, 19, 21–24]. The reader interested in this approach is encouraged to explore the extensive literature on path integrals and field theory. Bressloff [13] covers the connection between the path integral approach and large deviation theory. The reader should be aware that most of the references listed will concentrate on applications and formulations appropriate for equilibrium statistical mechanics and particle physics, which means that they will not explicitly discuss the response function approach we have demonstrated here. For application driven examinations of path integration there is Kleinert [11], Schulman [34], Kardar [27] and Tauber [12]. More mathematically rigorous treatments can be found in Simon [35] and Glimm and Jaffe [36]. For the reader seeking more familiarity with concepts of stochastic calculus such as Ito or Stratonovich integration there are applied approaches [3] and rigorous treatments [37] as well. Zinn-Justin [26] covers a wide array of topics of interest in quantum field theory from statistical mechanics to particle physics. Despite the exceptionally terse and dense presentation, the elementary material in this volume is recommended to those new to the concept of path integrals. Note that Zinn-Justin covers SDEs in a somewhat different manner from that presented here (the Onsager–Machlup integral is derived; although see Chaps. 16 and 17), as does Kleinert. We should also point out the parallel between the form of the action for exponential decay (i.e. \(D = 0\) in the OU process) and the holomorphic representation of the harmonic oscillator presented in [26]. The response function formalism was introduced by Martin et al. [29]. Closely related path integral formalisms have been introduced via the work of Doi [5, 6] and Peliti [7] which have been used in the analysis of reaction–diffusion system [8–10, 30]. Uses of path integrals in neuroscience have appeared in [14, 15, 17–21, 23, 25].

This derivation is, strictly speaking, incorrect because the delta functional fixes the value of \(\dot{x}(t)\), not \(x(t)\). It works because the Jacobian under a change of variables from \(\dot{x}(t)\) to \(x(t)\) is 1.

## Declarations

### Acknowledgements

This research was supported by the Intramural Research Program of the NIH, NIDDK.

**Open Access** This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

## Authors’ Affiliations

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