Laws of Large Numbers and Langevin Approximations for Stochastic Neural Field Equations
© M.G. Riedler, E. Buckwar; licensee Springer 2013
Received: 4 July 2012
Accepted: 14 January 2013
Published: 23 January 2013
In this study, we consider limit theorems for microscopic stochastic models of neural fields. We show that the Wilson–Cowan equation can be obtained as the limit in uniform convergence on compacts in probability for a sequence of microscopic models when the number of neuron populations distributed in space and the number of neurons per population tend to infinity. This result also allows to obtain limits for qualitatively different stochastic convergence concepts, e.g., convergence in the mean. Further, we present a central limit theorem for the martingale part of the microscopic models which, suitably re-scaled, converges to a centred Gaussian process with independent increments. These two results provide the basis for presenting the neural field Langevin equation, a stochastic differential equation taking values in a Hilbert space, which is the infinite-dimensional analogue of the chemical Langevin equation in the present setting. On a technical level, we apply recently developed law of large numbers and central limit theorems for piecewise deterministic processes taking values in Hilbert spaces to a master equation formulation of stochastic neuronal network models. These theorems are valid for processes taking values in Hilbert spaces, and by this are able to incorporate spatial structures of the underlying model.
Mathematics Subject Classification (2000):60F05, 60J25, 60J75, 92C20.
Often one wants to study deterministic equations such as Eq. (1.1) in order to obtain results on the ‘behaviour in the mean’ of an intrinsically stochastic system. Thus, we first discuss limit theorems of the law of large numbers type for the limit of infinitely many particles. These theorems connect the trajectories of the stochastic particle models to the deterministic solution of mean field equations, and hence provide a justification studying Eq. (1.1) in order to infer on the behaviour of the stochastic system.
Secondly, we aim to characterise the internal noise structure of the complex discrete stochastic models as in the limit of large numbers of neurons the noise is expected to be close to a simpler stochastic process. Ultimately, this yields a stochastic neural field model in terms of a stochastic evolution equation conceptually analogous to the Chemical Langevin Equation. The Chemical Langevin Equation is widely used in the study of chemical reactions networks for which the stochastic effects cannot be neglected but a numerical or analytical study of the exact discrete model is not possible due to its inherent complexity.
In this study, we understand as a microscopic model a description as a stochastic process, usually a Markov chain model, also called a master equation formulation (cf. [3, 5, 8, 9, 22] containing various master equation formulations of neural dynamics). In contrast, a macroscopic model is a deterministic evolution equation such as (1.1). Deterministic mean field equations have been used widely and for a long time to model and analyse large scale behaviour of the brain. In their original deterministic form, they are successfully used to model geometric visual hallucinations, orientation tuning in the visual cortex and wave propagation in cortical slices to mention only a few applications. We refer to  for a recent review and an extensive list of references. The derivation of these equations is based on a number of arguments from statistical physics and for a long time a justification from microscopic models has not been available. The interest in deriving mean field equations from stochastic microscopic model has been revived recently as it contains the possibility to derive deterministic ‘corrections’ to the mean field equations, also called second-order approximations. These corrections might account for the inherent stochasticity, and thus incorporate so called finite size effects. This has been achieved by either applying a path-integral approach to the master equation [8, 9] or by a van Kampen system-size expansion of the master equation . In more detail, the author in the latter reference proposes a particular master equation for a finite number of neuron populations and derives the Wilson–Cowan equation as the first-order approximation to the mean via employing the van Kampen system size expansion and then taking the continuum limit for a continuum of populations. In keeping also the second-order terms, a ‘stochastic’ version of the mean field equation is also presented in the sense of coupling the first moment equation to an equation for the second moments.
However, the van Kampen system size expansion does not give a precise mathematical connection, as it neither quantifies the type of convergence (quality of the limit), states conditions when the convergence is valid nor does it allow to characterise the speed of convergence. Furthermore, particular care has to be taken in systems possessing multiple fixed points of the macroscopic equation, and we refer to  for a discussion of this aspect in the neural field setting. The limited applicability of the van Kampen system size expansion was already well known to Sect. 10 in van Kampen . In parallel to the work of van Kampen, T. Kurtz derived precise limit theorems connecting sequences of continuous time Markov chains to solutions of systems of ordinary differential equations; see the seminal studies [19, 20] or the monograph . Limit theorems of that type are usually called the fluid limit, thermodynamic limit, or hydrodynamic limit; for a review, see, e.g., .
As is thoroughly discussed in  establishing the connection between master equation models and mean field equations involves two limit procedures. First, a limit which takes the number of particles, in this case neurons per considered population, to infinity (thermodynamic limit), and a second which gives the mean field by taking the number of populations to infinity (continuum limit). In this ‘double limit’, the theorems by Kurtz describe the connection of taking the number of neurons per population to infinity yielding a system of ordinary differential equation, one for each population. Then the extension from finite to infinite dimensional state space is obtained by a continuum limit. This procedure corresponds to the approach in . Thus, taking the double limit step by step raises the question what happens if we first take the spatial limit and then the fluid limit, thus reversing the order of the limit procedures, or in the case of taking the limits simultaneously. Recently, in an extension to the work of Kurtz, one of the present authors and co-authors established limit theorems that achieve this double limit , thus being able to connect directly finite population master equation formulations to spatio-temporal limit systems, e.g., partial differential equation or integro-differential equations such as the Wilson–Cowan equation (1.1). In a general framework, these limit theorems were derived for Piecewise Deterministic Markov Processes on Hilbert spaces, which in addition to the jump evolution also allow for a coupled deterministic continuous evolution. This generality was motivated by applications to neuron membrane models consisting of microscopic models of the ion channels coupled to a deterministic equation for the transmembrane potential. We find that this generality is also advantageous for the present situation of a pure jump model as it allows to include time-dependent inputs. In this study, we employ these theorems to achieve the aims (A) and (B) focussing on the example of the deterministic limit given by the Wilson–Cowan equation (1.1).
Finally, we state what this study does not contain, which in particular distinguishes the present study from [5, 8, 9] beyond mathematical technique. Presently, the aim is not to derive moment equations, i.e., a deterministic set of equations that approximate the moments of the Markovian particle model, but rather processes (deterministic or stochastic) to which a sequence of microscopic models converges under suitable conditions in a probabilistic way. This means that a microscopic model, which is close to the limit—presently corresponding to a large number of neurons in a large number of populations—can be assumed to be close to the limiting processes in structure and pathwise dynamics as indicated by the quality of the stochastic limit. Hence, the present work is conceptually—though neither in technique nor results—close to  wherein using a propagation to chaos approach in the vicinity of neural field equations the author also derives in a mathematically precise way a limiting process to finite particle models. However, it is an obvious consequence that the convergence of the models necessarily implies a close resemblance of their moment equations. This provides the connection to [5, 8, 9], which we briefly comment on in Appendix B.
As a guide, we close this introduction with an outline of the subsequent sections and some general remarks on the notation employed in this study. In Sects. 1.1 to 1.3, we first discuss the two types of mean field models in more detail, on the one hand, the Wilson–Cowan equation as the macroscopic limit and, on the other hand, a master equation formulation of a stochastic neural field. The main results of the paper are found in Sect. 2. There we set up the sequence of microscopic models and state conditions for convergence. Limit theorems of the law of large numbers type are presented in Theorem 2.1 and Theorem 2.2 in Sect. 2.1. The first is a classical weak law of large numbers providing uniform convergence on compacts in probability and the second convergence in the mean uniformly over the whole positive time axis. Next, a central limit theorem for the martingale part of the microscopic models is presented in Sect. 2.2 characterising the internal fluctuations of the model to be of a diffusive nature in the limit. This part of the study is concluded in Sect. 2.3 by presenting the Langevin approximations that arise as a result of the preceding limit theorems. The proofs of the theorems in Sect. 2 are deferred to Sect. 4. The study is concluded in Sect. 3 with a discussion of the implications of the presented results and an extension of these limit theorems to different master equation formulations or mean field equations.
Norms in Hilbert spaces are denoted by , is used to denote the supremum norm of real functions, i.e., for we have , and denotes either the absolute value for scalars or the Lebesgue measure for measurable subsets of Euclidean space. Finally, we use to denote the set of integers including zero.
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 . 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., in  or in , which both satisfy . 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 under the above conditions. That is, Eq. (1.3) possesses a unique, continuously differentiable global solution ν to every initial condition , i.e., for all , which depends continuously on the initial condition. Furthermore, if the initial condition satisfies almost everywhere in D, then it holds for all that for almost all . 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 be such that . If now and if f is at least α-times differentiable with bounded derivatives and the weights and the input function satisfy and , then the equation is well-posed in , i.e., for all in . In particular, this implies that the solution ν is jointly continuous on .
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 . 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 , the random variables denote the number of active neurons at time t. An integer is used to characterise the population size. This number 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 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 neurons being active for some time becomes arbitrarily small for large enough . 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 , 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 , which stands for the total number of neurons in the population. In , this rate is thus interpreted as the rate with which a neuron becomes or remains active.
which is endowed with the boundary conditions if . In (1.6), the variable denotes the probability that the process is in state θ at time t. Finally, the definition is completed with stating an initial law ℒ, the distribution of , 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 .
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 as there clearly do not occur jumps in the progression of time and μ is the obvious time-dependent modification of (1.10).
2 A Precise Formulation of the Limit Theorems
In this section, we present the precise formulations of the limit theorems. To this end, we first define a suitable sequence of microscopic models, which gives the connection between the defining objects of the Wilson–Cowan equation (1.3) and the microscopic models discussed in Sect. 1.2. Thus, , , denotes a sequence of microscopic PDMP neural field models of the type as defined in Sect. 1.3. Each process is defined on a filtered probability space , which satisfies the usual conditions. Hence, the defining objects for the jump models are now dependent on an additional index n. That is denotes the number of neuron populations in the n th model, is the number of neurons in the k th population of the n th model and analogously we use the notations and and . However, we note from the beginning that the decay rate is independent of n and τ is the time constant in the Wilson–Cowan equation (1.3). In the following paragraphs, we discuss the connection of the defining components of this sequence of microscopic models to the components of the macroscopic limit.
Connection to the Spatial Domain D A key step of connecting the microscopic models to the solution of Eq. (1.3) is that we need to put the individual neuron populations into relation to the spatial domain D the solution of (1.3) lives on. To this end, we assume that each population is located within a sub-domain of D and that the sub-domains of the individual populations are non-overlapping. Hence, for each , we obtain a collection of non-overlapping sub-sets of D denoted by . We assume that each subdomain is measurable and convex. The convexity of the sub-domains is a technical condition that allows us to apply Poincaré’s inequality, cf. (4.1). We do not think that this condition is too restrictive as most reasonable partition domains, e.g., cubes, triangles, are convex. Furthermore, for all reasonable domains D, e.g., all Jordan measurable domains, a sequence of convex partitions can be found such that additionally the conditions imposed in the limit theorems below are also satisfied. One may think of obtaining the collection by partitioning the domain into convex sub-domains and confining each neuron population to one sub-domain. However, it is not required that the union of the sets in amounts to the full domain D nor that the partitions consists of refinements. Necessary conditions on the limiting behaviour of the sub-domains are very strongly connected to the convergence of initial conditions of the models, which is a condition in the limit theorems; see below. For the sake of terminological simplicity, we refer to simply as the partitions.
where , , is a sequence of functions converging uniformly to f, then all limit theorems remain valid. The proof can be carried out as presented adding and subtracting the appropriate term where the additional difference term vanishes due to for . Hence, any microscopic model with gain rates of such a form reduces to the same Wilson–Cowan equation in the limit. Clearly, the same applies analogously to the decay rate τ, the weights w, and the input I.
Connection to the Input Current I
for all .
Clearly, each is a measurable map into . For the composition of with the stochastic process , we also use the abbreviation , and hence the resulting stochastic process is an adapted càdlàg process taking values in . This process thus states the activity at a location as the fraction of active neurons in the population, which is located around this location.
Connection of the Initial Conditions
Next, assuming that partitions fill the whole domain D for , i.e., , and that the maximal diameter of the sets decreases to zero, i.e., , it is easy to see using the Poincaré inequality (4.1) that the above definition of the initial condition implies that and for all . Then (2.8) holds trivially as the initial condition is deterministic and converges. A simple non-degenerate sequence of initial conditions is obtained by choosing random initial conditions with the above value as their mean and sufficiently fast decreasing fluctuations. Furthermore, a sequence of partitions, which satisfy the above conditions also exists for a large class of reasonable domains D. Assume that D is Jordan measurable, i.e., a bounded domain such that the boundary is a Lebesgue null set, and let be the smallest grid of cubes with edge length covering D. We define to be the set of all cubes, which are fully in D. As D is Jordan measurable, these partitions fill up D from inside and . For a more detailed discussion of these aspects, we refer to .
In the remainder of this section, we now collect the main results of this article. We start with the law of large numbers, which establishes the connection to the deterministic mean field equation, and then proceed to central limit theorems which provide the basis for a Langevin approximation. The proofs of the results are deferred to Sect. 4.
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 , which we used in Theorem 2.1 is a very strong norm on the space of -valued càdlàg functions on . Hence, due to continuous embeddings, the result immediately extends to weaker norms, e.g., the norms for all . Also, for the state space, weaker spatial norms can be chosen, e.g., with or any norm on the duals of Sobolev spaces with . 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 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 () 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  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 . For our model, we could achieve these rates.
2.1.1 Infinite-Time Convergence
In the law of large numbers, Theorem 2.1, and its Corollary 2.1 we have presented results of convergence over finite time intervals. Employing a different technique, we are also able to derive a convergence result over the whole positive time axis motivated by a similar result in . The proof of the following theorem is deferred to Sect. 4.3. Restricted to finite time intervals, the subsequent result is strictly weaker than Theorem 2.1. However, the result is important when one wants to analyse the mean long time behaviour of the stochastic model via a bifurcation analysis of the deterministic limit as (2.14) suggests that is close to for all times for sufficiently large n.
2.2 A Martingale Central Limit Theorem
Here, the process is a Hilbert space-valued, square-integrable, càdlàg martingale using (2.15) as its definition. We have used this representation of the process 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 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’ 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 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 for sufficiently large n. Deriving a suitable limit for 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., 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 the processes and the martingales as taking values in the space for an , where d is the dimension of the spatial domain D, using the embedding of into . The technical significance of the restriction is that these are the indices such that there exists an embedding into a with , which is of Hilbert–Schmidt typec due to Maurin’s theorem and is embedded into 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.
- (i)each is symmetric and positive, i.e.,
- (ii)each is of trace class, i.e., for one (and thus every) orthonormal basis , , in it holds that(2.16)
and the family , , is continuously increasing in t in the sense that the map is continuous and increasing for all .
Summing these inequalities for all , we find that the resulting right-hand side is finite as due to Maurin’s theorem the embedding of into is of Hilbert–Schmidt type. Moreover, their trace is even bounded independently of t.
Clearly, the resulting bilinear form inherits the properties of the bilinear form (2.17). Moreover, due to the positivity of the integrands, it follows that is increasing in t for all . Hence, the family of operators , , satisfies the above conditions (i)–(iii), and thus uniquely defines an -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 -valued càdlàg function to the -valued diffusion process defined by the covariance operator 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 in the Hilbert scale, what can be said about convergence in the stronger norms corresponding to the range of ? 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 , which we denote by in the following, is not a unique sequence yielding a non-trivial limit. Re-scaling the martingales by any sequence of the form 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 , we obtain three possibilities for the convergence of the sequence . If is of the same speed of convergence as , i.e., for , 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 as discussed above. Secondly, if the convergence is slower, i.e., , 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., , 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 has to converge in distribution in order for possessing a limit in distribution; see Theorem 2 in . As the sequence 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 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 , 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 results, on the one hand, from our estimation technique, which necessitates , and on the other hand, from the definition of the limiting diffusion. Recall that the covariance operator is only of trace class for . Hence, for , 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 , 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  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 . However, the re-scaling is proportional to the square root of , 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.
3 Discussion and Extensions
In this article, we have presented limit theorems that connect finite, discrete microscopic models of neural activity to the Wilson–Cowan neural field equation. The results state qualitative connections between the models formulated as precise probabilistic convergence concepts. Thus, the results strengthen the connection derived in a heuristic way from the van Kampen system size expansion.
A general limitation of mathematically precise approaches to approximations, cf. also the propagation to chaos limit theorems in , is that the microscopic models are usually defined via the limit. In other words, the limit has to be known a priori, and we look for models which converge to this limit. Thus, in contrast to the van Kampen system size expansion, the presented results are not a step-by-step modelling procedure in the sense that, via a constructive limiting procedure, a microscopic model yields a deterministic or stochastic approximation. Hence, it might be objected that the presented method can only be used a posteriori in order to justify a macroscopic model from a constructed microscopic model and that somehow one has to ‘guess’ the correct limit in advance. Several remarks can be made to answer this objection.
First, this observation is certainly true, but not necessarily a drawback. On the contrary, when both microscopic and macroscopic models are available, then it is rather important to know how these are connected and qualitatively and quantitatively characterise this connection. Concerning neural field models, this precise connection was simply not available so far for the well-established Wilson–Cowan model. Furthermore, when starting from a stochastic microscopic description working through proving the conditions for convergence for given microscopic models, one obtains very strong hints on the structure of a possible deterministic limit. Therefore, our results can also ease the procedure of ‘guessing the correct limit’.
Secondly, often a phenomenological, deterministic model, which is an approximation to an inherently probabilistic process is derived from ad-hoc heuristic arguments. Given that the model has proved useful, one often aims to derive a justification from first principles and/or a stochastic version, which keeps the features of the deterministic model, but also accounts for the formerly neglected fluctuations. A standard, though somewhat simple approach to obtain stochastic versions consists of adding (small) noise to the deterministic equations. This article, provides a second approach which consists of finding microscopic models, which converge to the deterministic limit to obtain a stochastic correction via a central limit argument.
Thirdly and finally, the method also provides an argument for new equations, i.e., the Langevin and linear noise approximations, which can be used to study the stochastic fluctuations in the model. Furthermore, in contrast to previous studies, we do not provide deterministic moment equations but stochastic processes, which can be, e.g., via Monte Carlo simulations, studied concerning a large number of pathwise properties and dynamics beyond first and second moments.
We now conclude this article commenting on the feasibility of our approach connecting microscopic Markov models to deterministic macroscopic equations when dealing with different master equation formulations that appear in the literature. Additionally, the following discussions also relate the model (1.6) considered in this article to other master equation formulations. We conjecture that the analogous results as presented for the Wilson–Cowan equation (1.3) in Sect. 2 also hold for these variations of the master equations. This should be possible to achieve by an adaptation of the methods of proof presented although we have not performed the computations in detail.
3.1 A Variation of the Master Equation Formulation
with boundary conditions if as stated in . The advantage of the effective spike model interpretation over the interpretation as neurons per population is that the unbounded state space of the model is justified. In principle, there can be an arbitrary number of spikes emitted in the past still active. However, a disadvantage of the master equation (3.2) is that for taking the limit it lacks a parameter corresponding to the system size providing a natural small parameter in the van Kampen system size expansion. This explains the shift in the interpretation of the master equation in the study  following , and subsequently in  to the interpretation we presented in Sect. 1.2, which provides the system-size parameters .
On the level of Markov jump processes, the master equation (3.2) obviously describes dynamics similar to the master equation (1.6) only replacing the activation rate in (1.6) by which is independent of the parameter . Thus, the model (3.2) can be understood as resulting from (1.6) after a limit procedure taking has been applied and the firing rate functions are connected via the formal limit . A qualitative interpretation of this limit procedure connecting the two types of models is given in . This observation motivated the model in  stepping back one limit procedure, and thus providing the correct framework for the derivation of limit theorems.
such that the higher order terms are uniformly bounded and vanish in the limit , and where the weights and inputs are defined as in (2.4) and (2.6). Property (3.3) closely resembles condition (3.1) and trivially holds for linear f with .
3.2 Bounded State Space Master Equations
We have already stated when introducing the microscopic model in Sect. 1.2 that the interpretation of the parameter as the number of neurons in the k th population is not literally correct. The state space of the process is unbounded, hence arbitrarily many neurons can be active, and thus each population contains arbitrarily many neurons. In order to overcome this interpretation problem, it was supposed to consider the master equation only on a bounded state space. That is, the k th population consists of neurons, and almost surely. Such master equations are simply obtained by setting the transition rates for transition of from to zero.
replacing in (1.6). The van Kampen system size expansion was then applied to this bounded state space master equation, tacitly neglecting possible difficulties, which might arise due to the discontinuity of (3.6) considered as a function on . However, for the present, mathematically precise limit convergence results considering bounded state space as originally suggested in  are problematic. The discontinuous activation rate (3.6) causes the machinery developed in , which depends on Lipschitz-type estimates to break down. However, we strongly expect that also in this case the law of large numbers with the deterministic limit given by the Wilson–Cowan equation (1.3) holds. Furthermore, also the Langevin approximations should agree with the equations discussed in Sect. 2.3. However, we have not yet been able to prove such a theorem. We further conjecture that the results in this article can be used to prove the convergence for the bounded state space model by a domination argument. Heuristically, it seems clear that a bounded process should be dominated by a process that possesses the same dynamics inside the state space of the bounded process, but can stray out from that bounded domain. Hence, as the limit of the potentially larger process lies within the domain where the two processes agree also the dominated process should converge to the same limit. Mathematically, this line of argument relies on non-trivial estimates between occupation measures of high-dimensional Markov processes. This is work in progress.
3.3 Activity Based Neural Field Model
We conjecture that also for this type of equations a phenomenological microscopic model can be constructed with a suitable adaptation of the activation rates and that limit theorems analogous to the results in Sect. 2.1 hold. Then also a Langevin equation for this model can be obtained and used for further analysis.
4 Proofs of the Main Results
Moreover, the constant in the right-hand side of (4.1) is the optimal constant depending only on the diameter of the domain D, cf. [1, 23]. Whenever we omit to denote the spatial domain for definition of norms or inner products in or Sobolev spaces , then it is to be interpreted as the norm over the whole domain D. If the norm is taken only over a subset , then this is always indicated unexceptionally.
Note that for corresponds to the generator of applied to the function .
Here, we also used the assumption on the initial condition.
4.1 Proof of Theorem 2.1 (Law of Large Numbers)
In order to prove the law of large numbers, Theorem 2.1, we apply the law of large numbers for Hilbert space valued PDMPs, see Theorem 4.1 in , to the sequence of homogeneous PDMPs . For the application of this theorem, recall that the first, piecewise constant, vector-valued component of this process counts the number of active neurons in each sub-population and the second, deterministic component states time. The process is the usual ‘space-time process’, i.e., homogeneous Markov process which is obtained via a state-space extension to obtain a homogeneous Markov process from the inhomogeneous process . The continuous component satisfies the simple ODE , , and thus the full process is a PDMP. In the terminology of , the sequence of coordinate functions on the different state spaces of the PDMPs into a common Hilbert space is given by the maps (2.7) with the common Hilbert space . Thus, in order to infer convergence in probability (2.10) from Theorem 4.1 in , it is sufficient to validate the following conditions:
- (a)In order to prove condition (4.7), we write the integral with respect to the discrete probability measure as a sum. This yields(4.10)
- (b)The Lipschitz condition (4.8) of the Nemytzkii operators is a straightforward consequence of the Lipschitz continuity (1.4) of the gain function f as
- (c)Finally, we prove the convergence of the generators (4.9). To this end, we employ the characterisation of the norm in by for all , and thus consider first the scalar product of elements with and the difference inside the norm in (4.9). On the one hand, we obtain using definition (4.5) that(4.11)
The upper bound in (4.15) is of order and, therefore, converges to zero for due to assumption (2.9). Hence, condition (4.9) is satisfied. The proof of the convergence in probability (2.10) is completed.
It is now easy to extend this result to the convergence in the r th mean. First of all, the convergence in probability (2.10) implies for all the convergence in probability of the random variables to zero. As convergence in the mean of real valued random variables is equivalent to convergence in probability and uniform integrability it remains to prove the latter for the families , .
We first consider the case , and establish a uniform bound on the second moments . Then the de la Vallée–Poussin theorem, cf. App., Proposition 2.2 in , implies that the random variables , , are uniformly integrable.
can thus be bounded from above by some constant independent of k and n. The proof of Theorem 2.1 is completed.
4.2 Proof of Corollary 2.1 (Corollary to the Law of Large Numbers)
- (a)In the case , i.e., , we used in (4.10) that . For general , we use the representation
- (b)Next, we have to establish that the Nemytzkii operator F on is also Lipschitz continuous with respect to the norms , , i.e., for all there exists a constant such that(4.16)
Hence, taking the supremum on both sides of this inequality over all , we obtain the Lipschitz condition (4.16) with , where is the constant resulting from the continuous embedding of into and the Lipschitz constant L of f satisfies .
4.3 Proof of Theorem 2.2 (Infinite Time Convergence)
- (a)We first present an alternative representation for the jump processes and the solution ν of the Wilson–Cowan equation (1.3). Using the generator of the PDMP , we obtain that the components satisfy(4.17)
and we obtain the final, third term in the right-hand side of (4.19). This completes the proof that (4.19) solves Eq. (4.17).
- (b)As due to Jensen’s inequality , it makes sense to calculate the second moment of the stochastic integral in the right-hand side. For the norm in , we use , and thus obtain using the linearity of the inner product