Large Deviations for Nonlocal Stochastic Neural Fields
© C. Kuehn, M.G. Riedler; licensee Springer. 2014
Received: 22 February 2013
Accepted: 10 June 2013
Published: 17 April 2014
We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a Q-Wiener process, and focus on noise-induced transitions and escape. We argue that proving a sharp Kramers’ law for neural fields poses substantial difficulties, but that one may transfer techniques from stochastic partial differential equations to establish a large deviation principle (LDP). Then we demonstrate that an efficient finite-dimensional approximation of the stochastic neural field equation can be achieved using a Galerkin method and that the resulting finite-dimensional rate function for the LDP can have a multiscale structure in certain cases. These results form the starting point for an efficient practical computation of the LDP. Our approach also provides the technical basis for further rigorous study of noise-induced transitions in neural fields based on Galerkin approximations.
Mathematics Subject Classification (2000):60F10, 60H15, 65M60, 92C20.
Starting from the classical works of Wilson/Cowan  and Amari , there has been considerable interest in the analysis of spatiotemporal dynamics of mesoscale models of neural activity. Continuum models for neural fields often take the form of nonlinear integro-differential equations where the integral term can be viewed as a nonlocal interaction term; see  for a derivation of neural field models. Stationary states, traveling waves, and pattern formation for neural fields have been studied extensively; see, e.g., [20, 29] or the recent review by Bressloff  and references therein.
In this paper, we are going to study a stochastic neural field model. There are several motivations for our approach. In general, it is well known that intra and interneuron  dynamics are subject to fluctuations. Many meso or macroscale continuum models have stochastic perturbations due to finite-size effects [38, 61]. Therefore, there is certainly a genuine need to develop new techniques to analyze random neural systems . For stochastic neural fields, there is also the direct motivation to understand the relation between noise and short-term working memory  as well as noise-induced phenomena  in perceptual bistability . Although an eventual goal is to match results from stochastic neural fields to actual cortex data , we shall not attempt such a comparison here. However, the techniques we develop could have the potential to make it easier to understand the relation between models and experiments; see Sect. 10 for a more detailed discussion.
There is a relatively small amount of fairly recent work on stochastic neural fields, which we briefly review here. Brackley and Turner  study a neural field with a gain function, which has a random firing threshold. Fluctuating gain functions are also considered by Coombes et al. . Bressloff and Webber  analyze a stochastic neural field equation with multiplicative noise while Bressloff and Wilkinson  study the influence of extrinsic noise on neural fields. In all these works, the focus is on the statistics of traveling waves such as front diffusion and the effects of noise on the wave speed. Hutt et al.  study the influence of external fluctuations on Turing bifurcation in neural fields. Kilpatrick and Ermentrout  are interested in stationary bump solutions. They observe numerically a noise-induced passage to extinction as well as noise-induced switching of bump solutions and conjecture that “a Kramers’ escape rate calculation” [, p. 16] could be applied to stochastic neural fields, but they do not carry out this calculation. In particular, the question is whether one can give a precise estimate of the mean transition time between metastable states for stochastic neural field equations; for a precise statement of the classical Kramers’ law; see Sect. 5, Eq. (32). However, to the best of our knowledge, there seems to be no general Kramers’ law or large deviation principle (LDP) calculation available for continuum neural field models although large deviations have been of recent interest in neuroscience applications [13, 33]. It is one of the main goals of this paper to provide the basic steps toward a general theory.
Although Kramers’ law  and LDPs [26, 34] are well understood for finite-dimensional stochastic differential equations (SDEs), the work for infinite-dimensional evolution equations is much more recent. In particular, it has been shown very recently that one may extend Kramers’ law to certain stochastic partial differential equations (SPDEs) [4, 6, 7] driven by space-time white noise. The work of Berglund and Gentz  provides a quite general strategy how to “lift” a finite-dimensional Kramers’ law to the SPDE setting using a Galerkin approximation due to Blömker and Jentzen . Since the transfer of PDE techniques to neural fields has been very successful, either directly  or indirectly [14, 21], one may conjecture that the same strategy also works for SPDEs and stochastic neural fields.
for a trace-class operator Q, nonlinear gain function f, and an interaction kernel w; the technical details and definitions are provided in Sect. 2. Observe that (1) is a relatively general formulation of a nonlocal neural field. Hence, we expect that the techniques developed in this paper carry over to much wider classes of neural fields beyond (1) such as activity-based models.
Remark 1.1 To avoid confusion, we alert readers familiar with neural fields that the nonlinear gain function f in (1) is sometimes also called a “rate function.” However, we reserve “rate function” for a functional, to be denoted later by I, arising in the context of an LDP as this convention is standard in the context of LDPs.
Our main goal in the study of (1) is to provide estimates on the mean first passage times between metastable states. In particular, we develop the basic analytical tools to approximate equation (1) as well as its rate function using a finite-dimensional Galerkin approximation. By making the rate function as explicit as possible, we do not only provide a starting point for further analytical work, but also provide a framework for efficient numerical methods to analyze metastable states.
The paper is structured as follows: The motivation for (1) is given in Sect. 3 where a formal calculation shows that a space-time white noise perturbation of the gain function in a deterministic neural field leads to (1). In Sect. 4, we briefly describe important features of the deterministic dynamics for (1) where . In particular, we collect several examples from the literature where the classical Kramers’ stability configuration of bistable stationary states separated by an unstable state occurs for Amari-type neural fields. In Sect. 5, we introduce the notation for Kramers’ law and LDPs and state the main theorem on finite-dimensional rate functions. In Sect. 6, we argue that a direct approach to Kramers’ law via “lifting” for (1) is likely to fail. Although the Amari model has a hidden energy-type structure, we have not been able to generalize the gradient-structure approach for SPDEs to the stochastic Amari model. This raises doubt whether a Kramers’ escape rate calculation can actually be carried out, i.e., whether one may express the prefactor of the mean first-passage in the bistable case explicitly. Based on these considerations, we restrict ourselves to just derive an LDP. In Sect. 7, the LDP is established by a direct transfer of a result known for SPDEs. The disadvantage of this approach is that the resulting rate function is difficult to calculate, analytically or numerically, in practice. Therefore, we establish in Sect. 8 the convergence of a suitable Galerkin approximation for (1). Using this approximation, one may apply results about the LDP for SDEs, which we carry out in Sect. 9. In this context, we also notice that the trace-class noise can induce a multiscale structure of the rate function in certain cases. The last two observations lead to a tractable finite-dimensional approximation of the LDP and hence also an associated finite-dimensional approximation for first-exit time problems. We conclude the paper in Sect. 10 with implications of our work and remarks about future problems.
2 Amari-Type Models
for , a small parameter , and , where ℬ is bounded and closed. In (2) the solution U models the averaged electrical potential generated by neurons at location x in an area of the brain ℬ. Neural field equations of the form (2) are called Amari-type equations or a rate-based neural field models. The equation is driven by an adapted space-time stochastic process on a filtered probability space . The precise definition of the process W will be given below.
The parameter is the decay rate for the potential, is a kernel that models the connectivity of neurons at location x to neurons at location y. Positive values of w model excitatory connections and negative values model inhibitory connections. The gain function relates the potential of neurons to inputs into other neurons. Typically, the gain functions are chosen sigmoidal, for example, (up to affine transformations of the argument) or . These examples of gain functions are bounded, infinitely often differentiable with bounded derivatives. However, throughout the paper, we only make the standing assumption that
(H1) the gain function f is globally Lipschitz continuous on ℝ.
Throughout the paper, we assume that
(H2) the kernel w is such that K is a compact, self-adjoint operator on .
We note that an integral operator is self-adjoint if and only if the kernel is symmetric, i.e., for all . A sufficient condition for the compactness of K is, e.g., in which case the operator is called a Hilbert–Schmidt operator. Since ℬ is bounded, the continuity of the kernel w on implies the compactness of K considered an integral operator on .
where W is an -valued stochastic process. Interpreting the original equation in this form, we now give a definition of the noise process assuming that
(H3) W is a Q-Wiener process on , where the covariance operator Q is a nonnegative, symmetric trace class operator on .
where are a sequence of independent scalar Wiener processes (cf. [, Proposition 2.1.10]). The series (5) converges in the mean-square on . Furthermore, a straightforward adaptation of the proof of [, Proposition 2.1.10] shows that convergence in the mean-square also holds in the space for every if for all i (corresponding to nonzero eigenvalues) and .
The solution possesses a modification in and from now on we always identify the solution (6) with its continuous modification. It is worthwhile to note that for cylindrical Wiener processes—and thus in particular space-time white noise—there does not exist a solution to (4). This contrasts with other well-studied infinite-dimensional stochastic evolution equations, e.g., the stochastic heat equation. Due to the representation of the solution (6), it follows that a solution can only be as spatially regular as the stochastic convolution . In the present case, the semigroup generated by the linear operator is not smoothing in contrast to, e.g., the semigroup generated by the Laplacian in the heat equation. Thus, the stochastic convolution is only as smooth as the noise which for space-time white noise is not even a well-defined function. To be more specific, for cylindrical Wiener noise, the series representation of the stochastic convolution (cf. see Eq. (8) below) does not converge in a suitable probabilistic sense.
We next aim to strengthen the spatial regularity of the solution (6), which will be required later on. According to [, Theorem 7.10] the solution (6) is a continuous process taking values in the Banach space if the initial condition satisfies , the linear part in the drift of (4) generates a strongly continuous semigroup on , the nonlinear term KF is globally Lipschitz continuous on , and finally, if the stochastic convolution is a continuous process taking values in . It is easily seen that the first conditions are satisfied and sufficient conditions for the latter property are given in the following lemma.
possesses a modification with γ-Hölder continuous paths in for all .
Now, the Kolmogorov–Centsov theorem implies the statement of the lemma. □
We present an example to illustrate the type of noise we are generally interested in. Further motivation is provided in Sect. 3.
3 Gain Function Perturbation
Recall that, by assumption (H2), the integral operator K defined by the kernel w is a self-adjoint compact operator. Thus, the spectral theorem implies that K possess only real eigenvalues , , and the corresponding eigenfunctions form an orthonormal basis of . If additionally we assume that
(H4) K is a Hilbert–Schmidt operator on , that is, ,
is a trace-class Wiener process on the Hilbert space . Note, when comparing to (5) here the coefficients may be negative, however, as is also a Wiener process this slight inconsistency can be neglected.
for a and a . The condition (14) and the first part of (15) are easily checked but for the second part of (15) usually theoretical results on the speed of decay of the eigenvalues have to be obtained. We note that (15) is certainly satisfied with if K is a trace class operator and the eigenfunctions are pointwise bounded independently of i; see, e.g., Example 2.1.
4 Deterministic Dynamics
The linear stability condition is equivalent to where . The stability analysis can be reduced to the understanding of the operator ℒ. However, this is a highly nontrivial problem as the behavior depends upon ℬ, , , and .
An LDP and Kramers’ law are of particular interest in the case of bistability. Therefore, we point out that there are many situations where (16) does have three stationary solutions: , which are stable and which is unstable. The following three examples make this claim more precise.
Suppose there are precisely three solutions for given by with . If , and then (20) has an unstable stationary solution between the two stable stationary solutions.
where is the Heaviside function and b, a, A, and are parameters. Depending on parameter values, one may obtain three constant stationary solutions exhibiting bistability as expected from Example 4.1. However, there are also parameter values so that three stationary pulses exhibiting bistability exist.
Note that the choice is not essential to obtain two deterministically-stable stationary states and one deterministically-unstable stationary state . The important aspect is that certain algebraic equations, such as and in Example 4.1, have the correct number of solutions. Furthermore, one has to make sure that the sign of the nonlinearity f is chosen correctly to obtain the desired deterministic stability results for the stationary solutions. Hence, we expect that a similar situation also holds for bounded domains; see also .
yield three stationary solutions , and . The solutions are stable and satisfy and . The solution is unstable.
Although we only focus on stationary solutions, it is important to remark that the techniques developed here could—in principle—also be applied to traveling waves for . The existence and stability of traveling waves for (16) has been investigated for many different situations; see, e.g. [12, 14, 21, 29], and references therein. However, it seems reasonable to restrict ourselves here to the stationary case as even for this simpler case an LDP and Kramers’ law are not yet well understood.
5 Large Deviations and Kramers’ Law
Furthermore, we are going to assume that the diffusion matrix is positive definite.
where denotes the usual Euclidean norm in . The formula (32) is also known as Kramers’ law  or Arrhenius–Eyring–Kramers’ law [2, 31, 45]. Note that the key differences with the general LDP (29) for the first-exit problem are that (32) yields a precise prefactor for the exponential transition time and uses the explicit form of the good rate function for gradient systems. It is interesting to note that a rigorous proof of (32) has only been obtained quite recently [9, 10].
6 Gradient Structures in Infinite Dimensions
The Gâteaux derivative is equal to the Fréchet derivative by a standard continuity result [, p. 47]. Hence, (36) shows that the stationary solutions of (35) are critical points of the gradient functional V. Since the gradient structure of the deterministic PDE (35) is a key structure to obtain a Kramers’-type estimate for the SPDE (33), we would like to check whether there is an analogue available for the deterministic Amari model (16).
There is still a space-time dependent nonlinear prefactor in (42) for the deterministic system, so the system is not an exact gradient flow for a potential.
Applying the change-of-variable for the stochastic Amari model (2) requires an Itô-type formula so that(43)
Even if we would just assume—without any immediate physical motivation—that the noise term in (43) is purely additive , there is a problem to apply Kramers’ law since we do not have a structure like in (22) with as is a Q-Wiener process defined in (5) and driving space-time white noise in (4) is particularly excluded due to the nonexistence of a solution.
Based on these observations, an immediate approach to generalize a sharp Kramers’ formula to neural fields seems unlikely. Hence we try to understand an LDP for the stochastic Amari-type model (2).
7 Direct Approach to an LDP
A general direct approach for the derivation of an LDP for infinite-dimensional stochastic evolution equations is presented in  and further results have been obtained for certain additional classes of SPDEs [17–19, 57]. The results in  are valid for semilinear equations with suitable Lipschitz assumptions on the nonlinearity and with solutions taking values in . We state the available results applied to continuous solutions of the Amari equation (4) assuming that the conditions of Lemma 2.1 are satisfied.
where this quasipotential relates to the minimal energy necessary to move the control system (44) started at the equilibrium state to z.
Theorem 7.1 ([, Theorem 12.18])
Following further the exposition in [, Sect. 12] explicit formulae for the rate function I are only available in the special case of the drift possessing gradient structure and space-time white noise. As we have argued above, this structure is particularly not satisfied for neural field equations. Hence, the same observations as presented at the end of the last section prevent a further direct analytic approach to the LDP. Therefore, we try to understand the LDP problem for a discretized approximate finite-dimensional version of the neural field equation.
8 Galerkin Approximation
respectively. The following theorem establishes the almost sure convergence of the Galerkin approximations to the solution of (4). Therefore, we may be able to infer properties of the behavior of paths of the solution from the path behavior of the Galerkin approximations. We have deferred the proof of the theorem to the Appendix.
9 Approximating the LDP
Furthermore, for arbitrary functions , which are used in the formulation of the rate function, we use the notation to denote the projection onto the first N Galerkin coefficients. Theorem 5.1 immediately implies the following:
where and . Therefore, the next result just implies that the Galerkin approximation is consistent for the LDP.
Proposition 9.2 For each we have .
by orthonormality of the basis in . □
Hence, we may work with the finite-dimensional Galerkin system and its LDP for computational purposes. However, the truncation N may still be very large. We are going to show, using a formal analysis for a certain case, that there is an intrinsic multiscale structure of the rate function. We assume that we are in the special case considered in Sect. 3 where K and Q have the same eigenfunctions and the corresponding eigenvalues are given by and , respectively.
and observe that the result is independent of N. □
Since and , the last integral is uniformly bounded over by . □
If or as , then we can conclude that , i.e.,(59)
If as , then and the first term dominates the asymptotics. But for all N so that the rate function only has a finite infimum if as . This implies again that (59) holds for the case of a finite infimum.
so that for bounded nonlinearity f, which is represented in the terms in (60), the higher-order modes should really just be governed by .
Hence, Propositions 9.1–9.2 and the multiscale nature of the problem induced by the trace-class noise suggests a procedure how to approximate the rate function and the associated LDP in practice. In particular, we may compute the eigenvalues and eigenfunctions of K and Q up to a sufficiently large given order . This yields an explicit representation of the Galerkin system and the associated rate function. Then one may apply any finite-dimensional technique to understand the rate function. One may even find a better truncation order based on the knowledge that the minimizer of the rate function must have components that decay (almost) exponentially in time for orders bigger than N.
In this paper, we have discussed several steps toward a better understanding of noise-induced transitions in continuum neural fields. Although we have provided the main basic elements via the LDP and finite-dimensional approximations, there are still several very interesting open problems.
We have demonstrated that a sharp Kramers’ rate calculation for neural fields with trace-class noise is very challenging as the techniques for white-noise gradient-structure SPDEs cannot be applied directly. However, we have seen in Sect. 4 that the deterministic dynamics for neural fields frequently exhibits a classical bistable structure with a saddle-state between stable equilibria. This suggests that there should be a Kramers’ law with exponential scaling in the noise intensity as well as a precisely computable pre-factor. It is interesting to ask how this pre-factor depends on the eigenvalues of the trace-class operator Q defining the Q-Wiener process. We expect that new technical tools are needed to answer this question.
From the viewpoint of experimental data, the exponential scaling for the LDP is relevant as it shows that noise-induced transitions have exponential interarrival times. This leads to the possibility that working memory as well as perceptual bistability could be governed by a Poisson process. However, the same phenomena could also be governed by a slowly varying variable, i.e., by an adaptive neural field ; the “fast” activity variable U in the Amari model is augmented by one or more “slow” variables. In this context, the required assumptions on the equilibrium structure in Sect. 4 and the noise in Sect. 3 is not necessary to produce a bistable switch and the fast variable U can, e.g., just have a single deterministically unstable equilibrium and bistable, nonrandom switching between metastable states may occur. Of course, there is also the possibility that an intermediate regime between noise-induced and deterministic escape is relevant .
It is interesting to note that the same problem arises generically across many natural sciences in the study of critical transitions (or “tipping points”) [48, 59]. The question which escape mechanism from a metastable state matches the data is often discussed very controversially and we shall not aim to provide a discussion here. However, our main goal to make the LDP and its associated rate functional as explicit as possible should definitely help to simplify comparison between models and experiment. For example, a parameter study or data assimilation for the finite-dimensional Galerkin system considered in Theorem 8.1 and the associated rate function in Proposition 9.1 are often easier than working directly with the abstract solutions of the stochastic Amari model in .
To study the parameter dependence is an interesting open question, which we aim to address in future work. In particular, the next step is to use the Galerkin approximations in Sect. 8 and the associated LDP in Sect. 9 for numerical purposes . Recent work for SPDEs  suggests that a spectral method can also be efficient for stochastic neural fields. Results on numerical continuation and jump heights for SDEs  can also be immediately transferred to the spectral approximation, which would allow for studies of bifurcations and associated noise-induced phenomena.
One may also ask how far the technical assumptions we make in this paper can be weakened. It is not clear which parts of the global Lipschitz assumptions may be replaced by local assumptions or removed altogether. Similar remarks apply to the multiscale nature of the problem induced by the decay of the eigenvalues of Q. How far this observation can be exploited to derive more efficient analytical as well as numerical techniques remains to be investigated.
On a more abstract level, it would certainly be desirable to extend our basic framework to other topics that have been considered already for deterministic neural fields. A generalization to activity based models with nonlinearity seems possible. Furthermore, it may be highly desirable to go beyond stationary solutions and investigate noise-induced switching and transitions for traveling waves and patterns.
Appendix: Convergence of the Galerkin Approximation
It clearly holds that and the convergence holds by assumption.
Next, as argued above is a.s. finite and the compactness of the operator K implies for , see [, Lemma 12.1.4].
Finally, the third error term vanishes if the Galerkin approximations of the Ornstein–Uhlenbeck process O converge almost surely in the spaces and , respectively. This convergence is proven in Lemma A.1 below.
The proof is completed. □
The following lemma contains the convergence of the Galerkin approximation of the Ornstein–Uhlenbeck process necessary for proving Theorem 8.1.
Remark A.1 Assumptions on the speed of convergence of the series and and readily yield a rate of convergence for the Galerkin approximation due to the definition of the constants in the proof of the lemma.
Proof of Lemma A.1 As in the proof Theorem 8.1 the unspecified norm denotes either the norm in or in and estimates are valid in both cases. We fix , and a with . Throughout the proof denotes a constant that changes from line to line, but depends only on the fixed parameters T, p, ρ, α and the domain .
In order to estimate the p th mean of the process , we proceed separately for the two cases and .
where the final upper bound decreases to zero for by assumption.
where the right-hand side decreases to zero for .
The proof is completed. □
We note that the boundedness assumption on the domain ℬ in this study is only necessary when dealing with results in the space as is the appropriate space for the LDP results. All other results in this paper which only deal with the space , e.g., existence of solutions and convergence of the Galerkin approximation, are also valid for unbounded spatial domains.
Hence, the upper bound, which is finite due to (7), is in independent of x and converges to zero for . Moreover, we further find that implies .
For a centered Gaussian random variable Z, it holds for all .
CK would like to thank the European Commission (EC/REA) for support by a Marie-Curie International Reintegration Grant and the Austrian Academy of Sciences (ÖAW) for support via an APART fellowship. We also would like to thank two anonymous referees whose comments helped to improve the manuscript.
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