# A Stochastic Version of the Jansen and Rit Neural Mass Model: Analysis and Numerics

- Markus Ableidinger
^{1}, - Evelyn Buckwar
^{1}and - Harald Hinterleitner
^{1}Email author

**7**:8

https://doi.org/10.1186/s13408-017-0046-4

© The Author(s) 2017

**Received: **26 September 2016

**Accepted: **23 May 2017

**Published: **8 August 2017

## Abstract

Neural mass models provide a useful framework for modelling mesoscopic neural dynamics and in this article we consider the Jansen and Rit neural mass model (JR-NMM). We formulate a stochastic version of it which arises by incorporating random input and has the structure of a damped stochastic Hamiltonian system with nonlinear displacement. We then investigate path properties and moment bounds of the model. Moreover, we study the asymptotic behaviour of the model and provide long-time stability results by establishing the geometric ergodicity of the system, which means that the system—independently of the initial values—always converges to an invariant measure. In the last part, we simulate the stochastic JR-NMM by an efficient numerical scheme based on a splitting approach which preserves the qualitative behaviour of the solution.

## Keywords

## 1 Introduction

Neural mass models have been studied as models describing coarse grained activity of large populations of neurons [1–7] since the 1970s. They have successfully been used to fit neuroimaging data, understanding EEG rhythms [8] or epileptic brain dynamics [9], and are now also a major building block in the Virtual Brain [10]. For a summary on their history, applications and an outlook on their future possible use, we refer to [11]. In general, neural mass models can be derived as a mean-field limit from microscopic models [12] and involve just a few state variables such as average membrane potentials and average population firing rates.

In this article, we focus on the Jansen and Rit neural mass model (JR-NMM) [13], which has been introduced as a model in the context of human electroencephalography (EEG) rhythms and visual evoked potentials [14]. It dates back to the work of Lopes da Silva and Van Rotterdam [3, 5, 15]. The JR-NMM is a biologically motivated convolution-based model of a neuronal population involving two subpopulations, i.e. excitatory and inhibitory interneurons forming feedback loops, which can describe background activity, alpha activity, sporadic and also rhythmic epileptic activity.

The original JR-NMM is formulated as a set of three coupled second-order nonlinear ordinary differential equations (ODEs), i.e. these constitute a system of coupled nonlinear oscillators, often rewritten as the six-dimensional system of first-order equations. After introducing this system in Sect. 2, we rewrite the system in the format of classical mechanics, that is, as a damped Hamiltonian system with a nonlinear displacement. Furthermore, in most of the literature, the JR-NMM includes a term representing extrinsic input or background noise, which essentially is done by declaring that input function to be a stochastic process. Mathematically this implies that the solution process of the ODE system then also is a stochastic process inheriting the analytical properties of the input process and requiring some framework of stochastic analysis for its mathematical treatment. In Sect. 3 we discuss options for such a framework and in this article we choose to formulate a stochastic JR-NMM as a stochastic differential equation (SDE) with additive noise, in particular a stochastic damped Hamiltonian system with a nonlinear term. Systems of SDEs of this or similar form are well studied in the molecular dynamics literature, where they are often called Langevin equations.^{1} In this article we provide a range of results employing various techniques available in the framework of stochastic analysis developed for SDEs: In Sect. 4 we establish basic properties of the SDE such as moment bounds and bounds on the path behaviour. Section 5 augments existing analysis of the dynamics of the deterministic JR-NMM, in particular we consider stochastic versions of equilibrium solutions, i.e. invariant measures, as well as the long-time behaviour of solutions of the SDE with respect to this invariant measure. These results may be interpreted as starting points for studies of phenomenological stochastic bifurcations or noise-induced transitions. Finally, in Sect. 6, we present efficient numerical methods designed for stochastic Hamiltonian problems and show that these numerical methods, which represent discrete stochastic systems for any fixed step-size, respect the properties previously established for the SDE system (subject to mild conditions on the step-size). Thus the resulting numerical methods will not only be quite efficient for future computational studies with the stochastic JR-NMM, they also provide *reliable* computational results.

## 2 Description of the Original Jansen and Rit Neural Mass Model

*output signal*\(y(t):=x_{1}(t)-x_{2}(t)\) describes the average membrane potential of the main family, i.e. the principal neurons of the JR-NMM (see [18, 19, 21]). The function \(p:[0,T]\to\mathbb{R}\) describes the external input which may originate both from external sources or the activity of neighbouring neural populations. We will discuss the mathematical modelling of

*p*in more detail at the end of this section. The sigmoid function \(\operatorname {Sigm}:\mathbb{R}\to[0,\nu_{\max}],\nu_{\max}>0\) (as suggested in [4]) is given by

*A*,

*B*,

*a*,

*b*, \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\), \(\nu_{\max}\),

*r*, \(v_{0}\) and typical values for these parameters, taken from [13, 19], are given in Table 1. The parameters

*A*,

*B*,

*a*and

*b*model basic features of postsynaptic potentials. In particular,

*A*and

*B*denote the excitatory and inhibitory synaptic gain, respectively, and \(a^{-1}\) and \(b^{-1}\) are corresponding time constants. The connectivity constants \(C_{i}\) for \(i\in \lbrace1,2,3,4\rbrace\), modelling the interactions between the main population and interneurons, are assumed to be proportional to a single parameter

*C*which characterises the average number of synapses between populations (see [13]). The solution behaviour of System (2) depends sensitively on the values of the parameters (we refer to the bifurcation analyses in [18, 19, 24]). Especially, changes in the connectivity constants \(C_{i}\) can result in drastic changes of the solution path.

Parameter | Description | Typical value |
---|---|---|

| Average excitatory synaptic gain | 3.25 mV |

| Average inhibitory synaptic gain | 22 mV |

\(a^{-1}\) | Time constant of excitatory postsynaptic potential | 10 ms |

\(b^{-1}\) | Time constant of inhibitory postsynaptic potential | 20 ms |

| Average number of synapses between the populations | 135 |

\(C_{1}\) | Avg. no. of syn. established by principal neurons on excitatory interneurons | C |

\(C_{2}\) | Avg. no. of syn. established by excitatory interneurons on principal neurons | 0.8 C |

\(C_{3}\) | Avg. no. of syn. established by principal neurons on inhibitory interneurons | 0.25 C |

\(C_{4}\) | Avg. no. of syn. established by inhibitory interneurons on principal neurons | 0.25 C |

\(\nu_{\max}\) | Maximum firing rate of the neural populations (max. of sigmoid fct.) | 5 s |

\(v_{0}\) | Value for which 50% of the maximum firing rate is attained | 6 mV |

| Slope of the sigmoid function at \(v_{0}\) | 0.56 mV |

*damped Hamiltonian system with nonlinear displacement*,

*Hamiltonian part*with Hamiltonian function \(H:\mathbb{R}^{6}\to\mathbb{R}^{+}_{0}\),

*damping part*with damping matrix \(\varGamma=\operatorname {diag}[a,a,b]\in\mathbb{R}^{3\times3}\), and a

*nonlinear part*given by the function \(G:[0,T]\times\mathbb{R}^{3}\to\mathbb{R}^{3}\), with

- (i)Random Ordinary Differential Equation (RODE) framework: RODEs are pathwise ODEs involving a stochastic process in their right-hand side, i.e. for a sufficiently smooth function \(f:\mathbb {R}^{m}\times\mathbb{R}^{d} \rightarrow\mathbb{R}^{d}\) and an
*m*-dimensional stochastic process \(\xi(t)\), a*d*-dimensional system of RODEs is given bywith an appropriate initial value. One may then choose the stochastic input process for example as a Wiener process or a coloured noise process, these processes exist in the classical sense and have almost surely continuous paths. In this framework standard deterministic analysis for e.g. guaranteeing existence and uniqueness of solutions can be applied pathwise; see for example [26], Chap. 1. However, the solution of this equation inherits the smoothness properties of the driving stochastic process \(\xi(t)\), independent of the smoothness of the function$$\dot{x}(t)= f\bigl(\xi(t),x(t)\bigr) , $$*f*. Analysis of properties and dynamics of solutions of RODEs may be performed pathwise by standard analysis techniques, bearing in mind that the low smoothness of the solutions limits the applicability of many classical results, such as Taylor’s theorem. We further refer to [27] for relevant results concerning random dynamical systems. Another consequence concerns the numerical treatment: as the order of convergence of classical numerical schemes for ODEs is determined by the smoothness of the solution of that ODE, when such schemes are applied pathwise to RODEs, they usually converge with a lower order than their expected one. In particular, they converge with order at most \(1/2\) when the input process is chosen as the Wiener process or a coloured noise process, as their paths are only Hölder continuous of order less than \(1/2\). We refer to [28] and its references for further information on numerical methods specifically designed for RODEs. - (ii)
Stochastic Differential Equation framework: If one were to choose the stochastic input process in an RODE as above as a Gaussian white noise process, one would need to deal with the fact that such a process exists only in the sense of

*generalised stochastic processes*; see [29], Sect. 3.2, or [30], Appendix I. In particular, Gaussian white noise is usually interpreted as the (generalised) derivative of the Wiener process, which itself is almost surely nowhere differentiable in the classical sense. It is much more convenient to work in the classical stochastic analysis framework designed to deal with differential equations ‘subject to (white) noise’ and interpret Systems (1), (2) or (3) as a stochastic differential equation; see also [29], Sect. 4.1. A considerable amount of results concerning analysis, dynamics, numerics, statistics, etc. of SDEs is available and for stochastic numerics we refer for example to [31], which also treats SDEs driven by coloured noise.

## 3 Jansen and Rit Neural Mass Model as a Damped Stochastic Hamiltonian System with Nonlinear Displacement

Systems of the type (5), typically called Langevin equations, have received considerable attention in the literature of molecular dynamics (see [16] for an overview). In particular, the long-time properties of such systems have been intensively studied in [34–36]. We employ these techniques in Sect. 5 to study the long-time behaviour of System (5).

We briefly discuss the existence of a solution of Eq. (5). As the sigmoid function Sigm is globally Lipschitz continuous, the existence and pathwise uniqueness of an \(\mathcal {F}_{t}\)-adapted solution is a standard result; see e.g. in [29], Theorem 6.2.2. In particular, *Q* is continuously differentiable. In the current context, it makes sense to assume that the functions \(\mu_{i}\) and \(\sigma_{i}\) are smooth and bounded which we will do in the following.

*C*can be reproduced. The numerical values for the parameters are given in Table 1. For the deterministic part of the external inputs we set \(\mu_{3}=\mu_{5}=0\) and \(\mu_{4}=220\), for the diffusion components we set \(\sigma_{3}=\sigma _{5}=10\) and \(\sigma_{4}=1{,}000\) such that ‘weak noise’ is acting on the components \(X_{3}\) and \(X_{5}\); \(X_{4}\) receives a stronger noise input. As in the original paper [14] we see (noisy)

*α*-rhythm-like behaviour as well as spiking behaviour for varying connectivity constants

*C*. In Fig. 2 we provide an illustration of changes in the system behaviour induced by including noise with plots of the phase portrait of the output signal for the case \(C=135\) and \(C=270\). The top two pictures show simulations of

*y*of System (2), i.e. without noise, where the solution curves quickly converge towards a limit cycle. The bottom two pictures show a path of

*Y*of System (5) and in particular for \(C=135\), the behaviour of the path is markedly different from the deterministic case.

## 4 Moment Bounds and Path Behaviour

We have already mentioned in Sect. 2 that the solution paths of Eq. (1) take values in a bounded set. It is natural to ask in which sense this behaviour transfers to the stochastic setting. We answer this question via a twofold strategy. On the one hand we will study the time evolution of the moments of the solution, which describes the average behaviour of all solution paths. On the other hand we will study the behaviour on the level of single paths and estimate the probability that a specific path exceeds a given threshold. Before we study these qualitative properties of Eq. (5) we provide a convolution-based representation for the *Q*-component of Eq. (5) which simplifies the corresponding calculations considerably.

### 4.1 Convolution-Based Representation of the JR-NMM

*M*is a block matrix with diagonal submatrices. Hence, we can calculate an explicit expression for the matrix exponential,

### Theorem 4.1

*The component*

*Q*

*of the unique solution of Eq*. (5)

*solves for*\(t\in[0,T]\)

*the integral equation*

*We call Eq*. (8)

*the convolution*-

*based representation of*

*Q*

*in Eq*. (5).

### Proof

*M*and \(\mathrm {e}^{Mt}\) we obtain

*N*only depends on

*Q*the equation for

*Q*is given by Eq. (8). □

### Remark 1

*P*, however, this formula depends on

*Q*. Indeed, for \(t\in[0,T]\),

### Remark 2

System (1) has originally been deduced by using convolutions of impulse response functions with functions of the output from other subpopulations within the neural mass (see [13, 18, 37, 38]). These response functions have the same shape as the kernel *κ* in Eq. (8), which thus can be interpreted as the stochastic version of this kernel representation.

### 4.2 Moment Bounds

*Q*; analogous results can be derived for

*P*. In the remainder of this section we will perform various componentwise calculations and estimations. For ease and consistency of notation we define the following: Let \(x,y\in\mathbb{R}^{n}\), then \(x\leq_{\odot}y\) denotes \(x_{i}\leq y_{i}\) for all \(1\leq i\leq n\). Furthermore, for \(U,V\in\mathbb{R}^{n\times k}\) we denote the

*Hadamard product*of

*U*and

*V*as \(U\odot V\), which is defined as the elementwise product (see [39, 40]) such that each element of the \(n\times k\) matrix \(U\odot V\) is given as

### Theorem 4.2

*Let*\(\mu_{i}:[0,T]\rightarrow \mathbb {R}^{+}\)

*for*\(i\in\{3,4,5\}\)

*be nonnegative functions bounded by*\(\mu_{i,\max}\in\mathbb{R}^{+}\),

*respectively*,

*and*\(C_{G}:=(Aa(\mu_{3,\max}+\nu_{\max}), Aa(\mu_{4,\max}+C_{2}\nu_{\max}),Bb(\mu _{5,\max}+C_{4}\nu_{\max}))^{T}\).

*Then*\(\mathbb{E}[Q(t)]\)

*is bounded in each component by*

### Proof

*κ*yield the desired estimates. □

Obviously, the bounds provided by Theorem 4.2 also hold for the deterministic equation (3) which justifies our claim in the introduction.

### Remark 3

_{3}. In particular,

Similar calculations can be done for the second moments of the components of \(Q(t)\). We obtain the following result.

### Theorem 4.3

*Let the assumptions of Theorem*4.2

*hold and assume*\(\varSigma (t)\)

*to be a constant matrix*, \(\varSigma\in \mathbb {R}^{3\times3}\).

*We define for*\(x=(x_{1},x_{2},x_{3})^{T}\in \mathbb {R}^{3}\)

*the function*\(\mathbf{1}_{\odot }^{+}(x):=(\mathbf{1}^{+}(x_{1}),\mathbf{1}^{+}(x_{2}),\mathbf{1}^{+}(x_{3}))^{T}\),

*where*\(\mathbf{1}^{+}\)

*denotes the indicator function of the set*\(\mathbb {R}^{+}\).

*Using the functions*

*u*,

*v*

*and*

*w*

*from Theorem*4.2,

*a bound for the second moment of each component of*\(Q(t)\)

*reads*

*In particular*,

### Proof

*t*in the following. By using the Cauchy–Schwarz inequality we bound

*C*. The results for the second moment \(\mathbb{E}[X^{2}_{1}(t)]\) are essentially the same; see Fig. 4. Similar results can be obtained for \(X_{2}\) and \(X_{3}\). The numerical approximations of the expectation (blue curves) stay well within the theoretical bounds (red curves), whereas single trajectories (purple curves) of course may exceed the bounds of the average. Note that, for \(C=68,135\) and 675, the approximations of \(\mathbb{E}[X_{1}(t)]\) rapidly converge towards fixed values for growing

*t*. The same behaviour can be observed for \(C=270\) on larger time scales. We will give a theoretical explanation for this phenomenon in Sect. 5 when we study the long-time behaviour of Eq. (5).

### 4.3 Pathwise Bounds

Theorem 4.2 states that on average the solution of Eq. (5) stays in some bounded set. However, the theorem gives no information for single solution paths, which can in principle attain arbitrarily large values with positive probability; see Lemma A.2 in the Appendix. In this section we want to quantify the probability of such large values. The following theorem provides an upper bound on the escape probability of the components of *Q*, i.e. the probability that for \(i\in\lbrace0,1,2\rbrace\) the solution \(X_{i}\) is larger than a given threshold \(x_{i}^{th}\in \mathbb {R}^{+}\).

### Theorem 4.4

*Let the assumptions of Theorem*4.3

*hold*.

*For fixed*\(t\in [0,T]\)

*we define a Gaussian random vector*\(Y(t)=(Y_{0}(t),Y_{1}(t),Y_{2}(t))\)

*with*

*where its components*\(Y_{i}(t)\)

*are independent*.

*Let*\(F_{Y_{i}(t)}\)

*denote the cumulative distribution function of*\(Y_{i}(t)\).

*Then the probability that the components*\(X_{i}(t)\)

*for*\(i\in\lbrace0,1,2\rbrace\)

*exceed the given thresholds*\(x^{th}_{i}\in \mathbb{R}^{+}\)

*is bounded by*

### Proof

*G*and again integrating

*κ*, we immediately see that each path of

*Q*is bounded by the stochastic process

*Y*defined by

### Remark 4

Theorem 4.4 can, for example, be used for calibration of the noise parameters in *Σ*. Let \(\varSigma=\operatorname {diag}[\sigma_{3},\sigma_{4},\sigma_{5}]\). Suppose we want to choose \(\sigma_{3}\) such that the corresponding component \(X_{0}(t)\) stays below some given threshold \(x_{0}^{th}\) with high probability *α*. Then a suitable choice of \(\sigma_{3}\) is implicitly given by \(F_{Y_{0}(t)} (x_{0}^{th} )=\alpha\).

## 5 Long-Time Behaviour and Stationary Solutions

A further property of interest of an SDE concerns the asymptotic behaviour of solution trajectories. The classical approach in ODE theory for analysing the long-time behaviour of ODE systems is to study the stability of equilibrium solutions and limit cycles. Even in the simplest case of constant input \(p(t)=p\in\mathbb{R}\), the deterministic equation (3) can possess several equilibrium solutions (both stable and unstable) as well as limit cycles with typically nontrivial basins of attraction (again we refer to the bifurcation analyses in [18, 19, 24]). Thus, the choice of the initial value can have large impact on the long-time behaviour of the solution curves of Eq. (3). From a practical point of view, this fact may be problematic, as it is not at all obvious how to estimate the initial value of the *P*-component.

In this section, we will analyse a stochastic counterpart of equilibrium solutions, more precisely invariant measures, and study the long-time asymptotics of Eq. (5). Our main tool is the theory of ergodic Markov processes, for convenience of the reader we recapitulate the basic definitions.

*t*, when it started in the point \(x\in \mathbb{R}^{6}\) at time \(t=0\), is given by

*invariant*if

- (i)
Does Eq. (5) have an invariant measure?

- (ii)
Is the invariant measure unique?

- (iii)
Do quantities of the type \(\mathbb {E}[h(X(t))]\) converge towards stationary values for a suitable class of functions \(h:\mathbb {R}^{6}\to \mathbb {R}\) and any initial value \((Q_{0},P_{0})\)?

*G*does not depend on

*t*and we simply write \(G(Q)\).

### 5.1 Existence of Invariant Measures and Geometric Ergodicity

The existence of invariant measures for Eq. (5) can be established by finding a suitable Lyapunov function. Heuristically speaking, the existence of a Lyapunov function ensures both that the solution trajectories stay in some bounded domain (except for some rare excursions) and in the case of excursions, the trajectories return to the bounded set. The following lemma shows that a perturbed version of the Hamiltonian *H* in Eq. (5) can act as a Lyapunov function (see [42]).

### Lemma 5.1

*Assume*\(a,b>0\)

*and let for*\(n\in\mathbb{N}\)

*Then*\(V_{n}\)

*is a Lyapunov function for Eq*. (5)

*in the following sense*:

- (i)
\(V_{n}\geq1\)

*and*\(V_{n}\rightarrow\infty\)*for*\(\Vert (Q,P)^{T}\Vert _{\mathbb{R}^{6}}\rightarrow\infty\), - (ii)\(\exists\alpha_{n}<0\), \(\beta_{n}>0\)
*such that*$$\begin{aligned} \mathcal{L}V_{n}\leq\alpha_{n} V_{n}+ \beta_{n}, \end{aligned}$$*where*\(\mathcal{L}\)*denotes the generator of Eq*. (5),$$\mathcal{L}:=P^{T}\nabla_{Q}+ \bigl[-Q^{T} \varGamma^{2}-2P^{T}\varGamma +G(Q)^{T} \bigr] \nabla_{P}+\frac{1}{2}\sum_{i=3}^{5} \sigma_{i}^{2}\frac {\partial^{2}}{\partial X_{i}^{2}}. $$*Here*, \(\nabla_{Q}\)*and*\(\nabla_{P}\)*denote the gradient with respect to the**Q**and**P**component*,*respectively*.

### Proof

*Γ*at the intersection of the

*i*th row and

*j*th column. Note that

Lemma 5.1 has two immediate consequences. First, applying Itô’s formula on \(V_{n}\) we obtain the following bounds (see [34]).

### Corollary 5.2

*Let Assumption*(10)

*hold and*\(s,t\in[0,T]\)

*with*\(t\ge s\).

*Then*

*In particular*,

Second, the existence of a Lyapunov function ensures the existence of an invariant measure (see e.g. [43], Corollary 1.11).

### Corollary 5.3

*Let Assumption* (10) *hold and let*
\(X(t)\)
*denote the solution of Eq*. (5). *Then there exists an invariant measure*
η
*of*
\(X(t)\).

Lemma 5.1 does not give any information on the uniqueness of the invariant measure. If we further assume that the three Wiener processes \(W_{i}\) act on all components of *P*, i.e. \(\sigma_{i}>0\) for \(i\in\lbrace3,4,5\rbrace\), we can establish the uniqueness of the invariant measure. Furthermore, the Markov process *X* fulfills the property of geometric ergodicity in the sense of [34]. We give a modification of the result in [34], Theorem 3.2, including the nonlinear function *G*.

### Theorem 5.4

*Let*\(\sigma_{i}>0\)

*for all*\(i\in\lbrace3,4,5\rbrace\).

*The Markov process*\(X(t)\)

*defined by Eq*. (5)

*has a unique invariant measure*η

*on*\(\mathbb{R}^{6}\).

*Furthermore*,

*let*

*Then for any*\(n\in\mathbb{N}\)

*and any initial value*\(X(0)=(Q_{0},P_{0})\)

*there exist positive constants*\(C_{n}\), \(\lambda_{n}\)

*such that*

### Proof

The proof is the same as in [34]. The Lyapunov condition has been established in Lemma 5.1, the corresponding results for the necessary smoothness of the transition probabilities and the irreducibility of the Markov process are given in the Appendix in Lemma A.2 and A.1. Both lemmas rely on the assumption that \(\sigma_{i}>0\) for \(i\in\lbrace3,4,5\rbrace\). □

Theorem 5.4 has two implications for the numerical simulation of Eq. (5). First, the actual choice of the initial value is insignificant as the impact of the initial value on the distribution of \(X(t)\) diminishes exponentially fast for growing *t* and an appropriate approximation of the system behaviour should be obtained with any choice of \((Q_{0},P_{0})\) provided that the system is simulated on a large enough time horizon. Second, due to the correspondence of the time averages and “space averages” of ergodic systems (see [41], Theorem 3.2.4.), one can estimate quantities of the type \(\mathbb{E}[h(X(t))]\) (for *t* sufficiently large) by computing the time average of a single path of \(X(t)\) on a large time horizon instead of using Monte Carlo estimation which requires one to compute a large number of paths of \(X(t)\). Of course, both aspects hold only true if the numerical method reproduces the geometric ergodicity of the original system (see Sect. 6).

*Y*for varying coupling parameter

*C*. We observe the change from unimodal densities (for \(C=68,135\)) to multi-modal densities \(C=270\) and to the peak-like structure \(C=675\). This behaviour can be interpreted as a phenomenological stochastic bifurcation as discussed, for example, in [46] or a noise-induced transition (see [47]).

## 6 Numerical Simulation

In order to obtain an approximation of Eq. (5) which accurately reproduces the qualitative behaviour, it is highly important to construct numerical integrators which on the one hand fulfil the properties of Eq. (5) derived in Sects. 4 and 5, and on the other hand are computationally efficient such that large ensembles of trajectories can be calculated in reasonable time.

- (i)
The methods preserve the moment bounds proposed in Theorems 4.2 and 4.3. Furthermore, for \(\varSigma =\mathbb{O}_{3}\), the numerical method preserves the bounds of the exact solution.

- (ii)
The Markov process generated by the numerical method is geometrically ergodic and fulfils a Lyapunov condition under very mild step-size restrictions.

### 6.1 Splitting Integrators for the JR-NMM

For convenience of the reader we provide a brief introduction to splitting integrators. Further details can be found, for example, in the classical monograph [49], Chapter II, for the deterministic case and [35, 50] for stochastic Langevin-type equations.

*f*. The goal is to choose \(f^{[j]}\) in such a way that the subsystems

*P*-component, and third, a stochastic part which does only arise in the

*P*-component. Therefore,

*Ornstein–Uhlenbeck integrator*. In the second case, the stochastic subsystem defines a Wiener process with drift and we denote the splitting integrator as

*Wiener integrator*.

In the following let \(0=t_{0}< \cdots< t_{N}=T\) with \(N\in \mathbb {N}\) be an equidistant partition of \([0,T]\) with step-size Δ*t*.

### 6.2 Ornstein–Uhlenbeck Integrator

*t*. In either case, we obtain

### 6.3 Wiener Integrator

### 6.4 Order of Convergence and Strang Splitting

As the noise in Eq. (5) is additive, standard integrators such as the Euler–Maruyama method converge with mean-square order one. The same holds true for the splitting integrators constructed above.

### Theorem 6.1

*Let*\(0=t_{0}< \cdots< t_{N}=T\)

*be an equidistant partition of*\([0,T]\)

*with step*-

*size*Δ

*t*,

*and let*\(X^{\mathrm{ou}}(t_{i})\)

*and*\(X^{\mathrm{w}}(t_{i})\)

*denote the numerical solutions defined by Eq*. (19)

*and*(23)

*at time point*\(t_{i}\)

*starting at initial value*\((Q_{0},P_{0})\in\mathbb{R}^{6}\).

*Then the one*-

*step methods defined in Eq*. (19)

*and*(23)

*are of mean*-

*square order one*,

*i*.

*e*.

*there exist constants*\(C_{1},C_{2}>0\)

*such that for sufficiently small*Δ

*t*

*the inequalities*

*hold for all time points*\(t_{i}\).

### Proof

The result can be proved in the same way as in [50], Lemma 2.1.. □

*correct*(as not only observed from both methods with small step-sizes, but also based on our analysis of the method’s properties) results can still be produced with much larger step-sizes than those required for the Euler–Maruyama method. The other important feature of the proposed method is its reliability. Figure 8 shows several plots of the phase portrait of one single path of the output

*Y*, with the splitting method and the Euler–Maruyama method and different step-sizes. It can be observed that the phase portrait obtained with the latter method changes markedly with increasing step-size. These phase portraits have been computed for the coupling parameter \(C=135\), initial value \(X(0)=0_{6}\), \(\sigma_{3}=\sigma_{5}=1\), \(\sigma_{4}=200\) and \(\Delta t\in\{ 10^{-4},10^{-3},2\cdot10^{-3}\}\). Figure 9 corresponds to the upper right plot in Fig. 6, which itself can be interpreted as a computational study of a phenomenological stochastic bifurcation for varying coupling parameter

*C*. It shows the densities of the invariant measure of

*Y*for \(C=135\), \(\sigma_{3}=\sigma_{5}=10\) and \(\sigma_{4}=10^{3}\) and compares the Strang splitting scheme with the Euler–Maruyama method over the time-step-sizes \(\Delta t\in\{10^{-3},2\cdot10^{-3},5\cdot10^{-3}\}\). The Euler–Maruyama method with moderately small step-sizes would report a change from a unimodal to a bimodal density for the parameter \(C=135\), whereas the correct value of

*C*for this change to happen should be much larger.

### 6.5 Moment Bounds and Geometric Ergodicity

The following two lemmas represent the properties presented in Sect. 4 for the numerical approximation schemes defined by Eq. (19) and Eq. (23). Let \(X^{\text{ou}}=(Q^{\mathrm{ou}},P^{\mathrm{ou}})\) and \(X^{\mathrm{w}}=(Q^{\mathrm{w}},P^{\mathrm{w}})\) denote the numerical solutions defined by Eq. (19) and Eq. (23), respectively. We start with proving analogous bounds to those in Theorem 4.2 for the expected value of \(Q^{\mathrm{ou}}\) and \(Q^{\mathrm{w}}\). It is well known already in the deterministic setting that the Euler scheme does not preserve such properties, see [49], Chap. 1, in the stochastic case negative results for the Euler–Maruyama method for (simple) stochastic oscillators have been observed in [48]. Note that the following two lemmas also hold when commuting the compositions in Eq. (19) and Eq. (23).

### Lemma 6.2

*Let*\(\mu_{j}:[0,T]\rightarrow \mathbb {R}^{+}\)

*for*\(j\in\{3,4,5\}\)

*be non*-

*negative functions bounded by*\(\mu_{j,\max}\in\mathbb{R}^{+}\),

*respectively*.

*Then for*\(i\in\lbrace0,\dots,N\rbrace\), \(\mathbb{E}[Q^{\mathrm{w}}(t_{i})]\) (

*and also*\(\mathbb{E}[Q^{\mathrm{ou}}(t_{i})]\))

*is bounded in each component by*

### Proof

*Q*-component reads

*t*. To prove the upper bound it remains to show that

*t*. □

### Remark 5

*κ*is evaluated at the midpoints \((t_{k}+t_{k+1})/2\).

### Remark 6

The last point we discuss in this article is the geometric ergodicity of the discrete Markov processes \(X^{\mathrm{ou}}\) and \(X^{\mathrm{w}}\) defined by Eq. (19) and (23). In analogy to Sect. 5 we assume that Assumption (10) holds. Due to the global Lipschitz continuity of the coefficients of Eq. (5), one would expect that standard numerical methods such as the Euler–Maruyama method are again geometrically ergodic for small enough step-sizes Δ*t* (see [34], Theorem 7.3). The advantage of our proposed splitting integrators is that we can directly prove a discrete analog of Lemma 5.1, i.e. a discrete Lyapunov condition for the same Lyapunov function under very mild restrictions on Δ*t*. We formulate the result for the Wiener integrator, the Ornstein–Uhlenbeck integrator can be treated analogously.

### Lemma 6.3

*Let*\(0< t_{0}\dots< t_{N}=T\)

*be an equidistant partition of*\([0,T]\)

*with step*-

*size*\(\Delta t<1/(2\Vert \varGamma \Vert _{L^{\infty}})\)

*and let*\(X^{\mathrm{w}}\)

*denote the numerical solutions defined by Eq*. (23).

*Then the functional*\(V(X):=V_{1}(Q,P)\)

*defined in Lemma*5.1

*is a Lyapunov function for*\(X^{\mathrm{w}}\),

*i*.

*e*.

*there exist constants*\(\alpha\in(0,1)\)

*and*\(\beta\ge0\)

*such that*

### Proof

*t*there exists a choice \(\tilde{\epsilon}^{*}\) such that

*ϵ*and

*ϵ̂*can be chosen arbitrarily small, therefore the corresponding terms can be neglected. Now let \(\Delta t <1/(2a)\), then Eq. (26a) and (26b) are fulfilled for

- (i)
Smoothness of the transition probability densities: Due to Assumption (10) the transition probability of two (or more) consecutive steps \(\psi^{\mathrm{w}}_{\Delta t}\circ\psi^{\mathrm{w}}_{\Delta t}\) of our integrator has a smooth density.

- (ii)Irreducibility: As in the time-continuous case in Sect. 5 we have to establish a reachability condition, i.e. the numerical method starting at \(x\in\mathbb{R}^{6}\) can reach any \(y\in \mathbb{R}^{6}\) after a fixed number of steps. For our splitting method, two consecutive steps are sufficient to reach any point
*y*by suitably choosing the vectors \(\xi(\Delta t)\) such thatIn fact, Eq. (27) is a six-dimensional system of equations with six degrees of freedom (three Gaussian random variables for each step \(\psi^{\mathrm{w}}_{\Delta t}\)) which can always be solved under Assumption 10.$$\begin{aligned} y= \bigl(\psi^{\mathrm{w}}_{\Delta t}\circ\psi^{\mathrm{w}}_{\Delta t} \bigr) (x). \end{aligned}$$(27)

*t*. Furthermore, as \(X^{\mathrm{ou}}\) and \(X^{\mathrm{w}}\) converge towards

*X*in the mean-square sense, \(\boldsymbol {\eta }^{\mathrm{ou}}_{\Delta t}\) and \(\boldsymbol {\eta }^{\mathrm{w}}_{\Delta t}\) are convergent approximations of the original invariant measure η (see [53], Theorem 3.3, for details). Thus, our numerical approximations of the marginal densities in Sect. 5 (see Fig. 6) are supported by the theory.

## 7 Summary and Conclusions

We proposed a version of the original JR-NMM incorporating random input, as a stochastic Hamiltonian system with nonlinear displacement, and discussed a range of properties based on results available in the framework of stochastic analysis, in particular properties such as moment bounds and the existence of invariant measures. The latter represent a step towards analysing the dynamical properties of a stochastic formulation of the JR-NMM. Furthermore, we presented an efficient numerical scheme based on a splitting approach which preserves the qualitative behaviour of the solution of the system. We have also discussed the advantages of applying such a scheme designed according to the obtained features of the stochastic JR-NMM for future computational studies in contrast to applying other numerical methods such as the Euler–Maruyama scheme. By a suitable introduction of noise our results can be generalised to both the extension of the JR-NMM to multiple populations [37, 54–57] and the extension to multiple areas, e.g. the 2-column model in [13] or the multi-area neural mass model in [56].

The term ’Langevin equation’ goes back to Langevin’s model of Brownian motion (1908) describing the velocity of a Brownian particle in a fluid by what is now called an Ornstein–Uhlenbeck process. However, for example, in the molecular dynamics literature the term refers to a class of models describing the interaction of heavy particles with light ones [16], whereas in some of the physics literature it may also refer to SDEs in general as in [17].

## Declarations

### Acknowledgements

This work was supported by the Austrian Science Fund (FWF) P26314.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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