Laws of Large Numbers and Langevin Approximations for Stochastic Neural Field Equations

1.1 The Macroscopic Limit

Neural field equations are usually classified into two types: rate-based and activity-based models. The prototype of the former is the Wilson–Cowan equation; see Eq. (1.1), which we also restate below, and the Amari equation, see Eq. (3.7) in Sect. 3, is the prototype of the latter. Besides being of a different structure, due to their derivation, the variable they describe has a completely different interpretation. In rate-based models, the variable describes the average rate of activity at a certain location and time, roughly corresponding to the fraction of active neurons at a certain infinitesimal area. In activity-based models, the macroscopic variable is an average electrical potential produced by neurons at a certain location. For a concise physical derivation that leads to these models, we refer to [5]. In the following, we consider rate-based equations, in particular, the classical Wilson–Cowan equation, to discuss the type of limit theorems we are able to obtain. We remark that the results are essentially analogous for activity based models.

and it is bounded. From an interpretive point-of-view, it is reasonable and consistent to stipulate that f is bounded by one—being a fraction—as well as being monotone. The latter property corresponds to the fact that higher input results in higher activity. In specific models, f is often chosen to be a sigmoidal function, e.g., $f(z)={(1+{\mathrm{e}}^{-({\beta }_{1}z+{\beta }_2)})}^{-1}$ in [6] or $f(z)=(tanh({\beta }_{1}z+{\beta }_2)+1)/2$ in [3], which both satisfy $f\in [0,1]$. Moreover, the most common choices of f are even infinitely often differentiable with bounded derivatives, which already implies the Lipschitz condition (1.4).

The Wilson–Cowan equation (1.3) is well-posed in the strong sense as an integral equation in $L^2(D)$ under the above conditions. That is, Eq. (1.3) possesses a unique, continuously differentiable global solution ν to every initial condition $\nu (0)={\nu }_0\in L^2(D)$, i.e., $\nu \in C^1([0,T],L^2(D))$ for all $T>0$, which depends continuously on the initial condition. Furthermore, if the initial condition satisfies ${\nu }_0(x)\in [0,{\parallel f\parallel }_0]$ almost everywhere in D, then it holds for all $t>0$ that $\nu (t,x)\in (0,{\parallel f\parallel }_0)$ for almost all $x\in D$. For a brief derivation of these results, we refer to Appendix A where we also state a result about higher spatial regularity of the solution: Let $\alpha \in \mathbb{N}$ be such that $\alpha >d/2$. If now ${\nu }_0\in H^{\alpha }(D)$ and if f is at least α-times differentiable with bounded derivatives and the weights and the input function satisfy $w\in H^{\alpha }(D\times D)$ and $I\in C({\mathbb{R}}_{+},H^{\alpha }(D))$, then the equation is well-posed in $H^{\alpha }(D)$, i.e., for all $T>0$ in $\nu \in C^1([0,T],H^{\alpha }(D))$. In particular, this implies that the solution ν is jointly continuous on ${\mathbb{R}}_{+}\times D$.

1.2 Master Equation Formulations of Neural Network Models

For the microscopic model, we concentrate on a variation of the model considered in [5, 6], which is already an improvement on a model introduced in [11]. We extend the model including variations among neuron populations and foremost time-dependent inputs. We chose this model over the master equation formulations in [8, 9] as it provides a more direct connection of the microscopic and macroscopic models; see also the discussion in Sect. 3. We describe the main ingredients of the model beginning with the simpler, time-independent model as prevalent in the literature. Subsequently, in Sect. 1.3 the final, time-dependent model is defined.

We denote by P the number of neuron populations in the model. Further, we assume that the k th neuron population consists of identical neurons which can either be in one of two possible states, active, i.e., emitting action potentials, and inactive, i.e., quiescent or not emitting action potentials. Transitions between states occur instantaneously and at random times. For all $k=1,\dots ,P$, the random variables ${\Theta }_t^k$ denote the number of active neurons at time t. An integer $l(k)$ is used to characterise the population size. This number $l(k)$ can be interpreted as the number of neurons in the k th population, at least for sufficiently large values. However, this is not accurate in the literal sense as it is possible with positive probability for populations to contain more than $l(k)$ active neurons. Nevertheless, a posteriori the interpretation can be salvaged from the obtained limit theorems.^b It is a corollary of these that the probability of more then $l(k)$ neurons being active for some time becomes arbitrarily small for large enough $l(k)$. Hence, for physiological reasonable neuron numbers the probability in these models of observing ‘non-physiological’ trajectories in the interpretation becomes ever smaller.

for a non-negative function $f:\mathbb{R}\to \mathbb{R}$, which obviously corresponds to the gain function f in the Wilson–Cowan equation (1.3). We remark that here f is not the rate of activation of one neuron. In this model, the activation rate of a population is not proportional to the number of inactive neurons but it is proportional to $l(k)$, which stands for the total number of neurons in the population. In [5], this rate is thus interpreted as the rate with which a neuron becomes or remains active.

which is endowed with the boundary conditions $\mathbb{P}[\theta ,t]=0$ if $\theta \notin {\mathbb{N}}_0^P$. In (1.6), the variable $\mathbb{P}[\theta ,t]$ denotes the probability that the process ${\Theta }_t$ is in state θ at time t. Finally, the definition is completed with stating an initial law ℒ, the distribution of ${\Theta }_0$, i.e., providing an initial value for the ODE system (1.6).

The importance of the generator lies in the fact that it fully characterises a Markov process and that convergence of Markov processes is strongly connected to the convergence of their generators; see [15].

1.3 Including External Time-Dependent Input

where the jump intensity λ is given by the sum of all individual time-dependent rates analogously to (1.8). Finally, the post jump value is given by a Markov kernel $\mu ((\theta ,t),\cdot )\times {\delta }_t$ as there clearly do not occur jumps in the progression of time and μ is the obvious time-dependent modification of (1.10).

2.1 A Law of Large Numbers

The first law of large numbers takes the following form. Note that the assumptions imply that the number of neuron populations diverges.

Theorem 2.1 (Law of large numbers)

Remark 2.1 The norm of the uniform convergence ${sup}_{t\in [0,T]}{\parallel \cdot \parallel }_{L^2(D)}$, which we used in Theorem 2.1 is a very strong norm on the space of $L^2(D)$-valued càdlàg functions on $[0,T]$. Hence, due to continuous embeddings, the result immediately extends to weaker norms, e.g., the norms $L^p((0,T),L^2(D))$ for all $1\le p\le \mathrm{\infty }$. Also, for the state space, weaker spatial norms can be chosen, e.g., $L^p(D)$ with $1\le p\le 2$ or any norm on the duals $H^{-\alpha }(D)$ of Sobolev spaces with $\alpha >0$. If weaker norms for the state space are considered, it is possible to relax the conditions of Theorem 2.1 by sharpening some estimates in the proof of the theorem. The results in the following corollary cover the whole range of $\alpha \ge 0$ and splits it into sections with weakening conditions. In particular note that after passing to weaker norms, the convergence does not necessitate that the neuron numbers per population diverge. However, regarding the divergence of the neuron populations, this condition (${\delta }_{+}(n)\to 0$) cannot be relaxed.

Remark 2.2 We believe that fruitful and illustrative comparisons of these convergence results and their conditions to the results in Kotelenez [17, 18], and particularly, Blount [4] can be made. Here, we just mention that the latter author conjectured the conditions (2.13) to be optimal for the convergence, but was not able to prove this result in his model of chemical reactions with diffusions for the region $\alpha \in (0,d/2]$. For our model, we could achieve these rates.

2.2 A Martingale Central Limit Theorem

Here, the process ${(M_t^n)}_{t\ge 0}$ is a Hilbert space-valued, square-integrable, càdlàg martingale using (2.15) as its definition. We have used this representation of the process ${\nu }^n$ in the proof of Theorem 2.2; see Sect. 4.3. We note that the Bochner integral in (2.15) is a.s. well defined due to bounded second moments of the integrand; see (4.7) in the proof of Theorem 2.1. Now an heuristic argument to obtain the convergence to the solution of the Wilson–Cowan equation is the following: The initial conditions converge, the martingale term $M^n$ converges to zero and the integral term in the right-hand side of (2.15) converges to the right-hand side in the Wilson–Cowan equation (1.3). Hence, the ‘solution’ ${\nu }^n$ of (2.15) converges to the solution ν of the Wilson–Cowan equation (1.3). Now interpreting Eq. (2.15) as a stochastic evolution equation, which is driven by the martingale ${(M_t^n)}_{t\ge 0}$ sheds light on the importance of the study of this term. Because, from this point of view, the martingale part in the decomposition (2.15) contains all the stochasticity inherent in the system. Then the idea for deriving a Langevin or linear noise approximation is to find a stochastic non-trivial limit (in distribution) for the sequence of martingales and substituting heuristically this limiting martingale into the stochastic evolution equation. Then it is expected that this new and much less complex process behaves similarly to the process ${({\nu }_t^n)}_{t\ge 0}$ for sufficiently large n. Deriving a suitable limit for ${(M_t^n)}_{t\ge 0}$ is what we set to do next. The result can be found in Theorem 2.3 below and takes the form of a central limit theorem.

which in turn implies convergence in probability and convergence in distribution to the zero limit.

Furthermore, in contrast to Euclidean spaces norms on infinite-dimensional spaces are usually not equivalent. In Corollary 2.1, we exploited this fact as it allowed us to obtain convergence results under less restrictive conditions by changing to strictly weaker norms. In the formulation and proof of central limit theorems, the change to weaker norms even becomes an essential ingredient. It is often observed in the literature, see, e.g., [4, 17, 18] that central limit theorems cannot be proven in the strongest norm for which the law of large numbers holds, e.g., $L^2(D)$ in the present setting, but only in a strictly weaker norm. Here, this norm is the norm in the dual of an appropriate Sobolev space. Hence, from now on, we consider for all $n\in \mathbb{N}$ the processes ${({\nu }_t^n)}_{t\ge 0}$ and the martingales ${(M_t^n)}_{t\ge 0}$ as taking values in the space $H^{-\alpha }(D)$ for an $\alpha >d$, where d is the dimension of the spatial domain D, using the embedding of $L^2(D)$ into $H^{-\alpha }(D)$. The technical significance of the restriction $\alpha >d$ is that these are the indices such that there exists an embedding $H^{\alpha }(D)$ into a $H^{{\alpha }_1}(D)$ with $d/2<{\alpha }_1<\alpha$, which is of Hilbert–Schmidt type^c due to Maurin’s theorem and $H^{{\alpha }_1}(D)$ is embedded into $C(\overline{D})$ due to the Sobolev embedding theorem. These two properties are essential for the proof of the central limit theorem and their occurrence will be made clear subsequently.

Summing these inequalities for all $j\in \mathbb{N}$, we find that the resulting right-hand side is finite as due to Maurin’s theorem the embedding of $H^{\alpha }(D)$ into $L^2(D)$ is of Hilbert–Schmidt type. Moreover, their trace is even bounded independently of t.

Clearly, the resulting bilinear form ${〈C(t)\cdot ,\cdot 〉}_{H^{\alpha }(D)}$ inherits the properties of the bilinear form (2.17). Moreover, due to the positivity of the integrands, it follows that ${〈C(t)\varphi ,\varphi 〉}_{H^{\alpha }(D)}$ is increasing in t for all $\varphi \in H^{\alpha }(D)$. Hence, the family of operators $C(t)$, $t\ge 0$, satisfies the above conditions (i)–(iii), and thus uniquely defines an $H^{-\alpha }(D)$-valued diffusion process.

We are now able to state the martingale central limit theorem. The proof of the theorem is deferred to Sect. 4.4.

Theorem 2.3 (Martingale central limit theorem)

converges weakly on the space of $H^{-\alpha }(D)$-valued càdlàg function to the $H^{-\alpha }(D)$-valued diffusion process defined by the covariance operator $C(t)$ given by (2.18).

Remark 2.3 In connection with the results of Theorem 2.3, two questions may arise. First, in what sense is there uniqueness of the re-scaling sequence, and hence of the limiting diffusion? That is, does a different scaling also produce a (non-trivial) limit, or, rephrased, is the proposed scaling the correct one to look at? Secondly, the theorem deals with the norms for the range of $\alpha >d$ in the Hilbert scale, what can be said about convergence in the stronger norms corresponding to the range of $\alpha \in [0,d]$? Does there exist a limit? We conclude this section addressing these two issues.

Regarding the first question, it is immediately obvious that the re-scaling sequence $\frac{{\ell }_{-}(n)}{v_{+}(n)}$, which we denote by ${\rho }_n$ in the following, is not a unique sequence yielding a non-trivial limit. Re-scaling the martingales $M^n$ by any sequence of the form $\sqrt{c{\rho }_n}$ yields a convergent martingale sequence. However, the limiting diffusion differs only in a covariance operator, which is also re-scaled by c, and hence the limit is essentially the same process with either ‘stretched’ or ‘shrinked’ variability. However, the asymptotic behaviour of the re-scaling sequences, which allow for a non-trivial weak limit is unique. In general, by considering different re-scaling sequences ${\rho }_n^{\ast }$, we obtain three possibilities for the convergence of the sequence $\sqrt{{\rho }_n^{\ast }}{M}^n$. If ${\rho }_n^{\ast }$ is of the same speed of convergence as ${\rho }_n$, i.e., for ${\rho }_n^{\ast }=\mathcal{O}({\rho }_n)$, the thus re-scaled sequence converges again to a diffusion process for which the covariance operator is proportional to (2.18). This is then just a re-scaling by a sequence (asymptotically) proportional to ${\rho }_n$ as discussed above. Secondly, if the convergence is slower, i.e., ${\rho }_n^{\ast }=o({\rho }_n)$, then the same methods as in the law of large numbers show that the sequence converges to zero uniformly on compacts in probability, hence also convergence in distribution to the degenerate zero process follows. Thus, one only obtains the trivial limit. Finally, if we rescale by a sequence that diverges faster, i.e., ${\rho }_n=o({\rho }_n^{\ast })$, we can show that there does not exist a limit. This follows from general necessary conditions for the preservation of weak limits under transformation, which presuppose that $\sqrt{{\rho }_n^{\ast }/{\rho }_n}M$ has to converge in distribution in order for $\sqrt{{\rho }_n^{\ast }}{M}_n$ possessing a limit in distribution; see Theorem 2 in [29]. As the sequence ${\rho }_n^{\ast }/{\rho }_n$ diverges, this is clearly not possible to hold.

Unfortunately, an answer to the second question is not possible in this clarity, when considering non-trivial limits. Essentially, we can only say that the currently used methods do not allow for any conclusion on convergence. The limitations are the following: The central problem is that for the parameter range $\alpha \in [0,d]$ the current method does not provide tightness of the re-scaled martingale sequence, hence we cannot infer that the sequence possesses a convergent subsequence. However, if tightness can be established in a different way then for the range $\alpha \in (max\{1,d/2\},d]$, the limit has to be the diffusion process defined by the operator (2.18) as follows from the characterisation of any limit in the proof of the theorem. Here, the lower bound of $max\{1,d/2\}$ results, on the one hand, from our estimation technique, which necessitates $\alpha \ge 1$, and on the other hand, from the definition of the limiting diffusion. Recall that the covariance operator is only of trace class for $\alpha >d/2$. Hence, for $\alpha \in [0,d/2]$, we can no longer infer that the limiting diffusion even exists.

2.3 The Mean-Field Langevin Equation

Note that the derivation of the above equations was only formal, hence we have to address the existence and uniqueness of solutions and the proper setting for these equations. This is left for future work. It is an ongoing discussion and probably undecidable as lacking a criterion of approximation quality which—if any at all—is the correct diffusion approximation to use. First of all note that for both versions the noise term vanishes for $n\to \mathrm{\infty }$, and thus both have the Wilson–Cowan equation as their limit. And also, neither of them approximates even the first moment of the microscopic models exactly. This means that for neither we have that the mean solves the Wilson–Cowan equation, which would be only the case if f were linear. However, they are close to the mean of the discrete process. We discuss this aspect in Appendix B.

Furthermore, we already observe in the central limit theorem, and thus also in the linear noise and Langevin approximation that the covariance (2.18) or the drift and the structure of the diffusion terms in (2.21) and (2.22), respectively, are independent of objects resulting from the microscopic models. They are defined purely in terms of the macroscopic limit. This observation supports the conjecture that these approximations are independent from possible different microscopic models converging to the same deterministic limit. Analogous statements hold also for derivations from the van Kampen system size expansion [5] and in related limit theorems for reaction diffusion models [4, 17, 18]. The only object reminiscent of the microscopic models in the continuous approximations is the re-scaling sequence ${ϵ}_n$. However, the re-scaling is proportional to the square root of ${\ell }_{-}(n)/v_{+}(n)$, i.e., the number of neurons per area divided by the size of the area, which is just the local density of particles. Therefore, in the approximations, the noise scales inversely to the square root of neuron density in this model, which interpreted in this way can also be considered a macroscopic fixed parameter and chosen independently of the approximating sequence.