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A stochastic hierarchical model for the evolution of low grade gliomas is proposed. Starting with the description of cell motion using a piecewise diffusion Markov process (PDifMP) at the cellular level, we derive an equation for the density of the transition probability of this Markov process based on the generalised Fokker–Planck equation. Then, a macroscopic model is derived via parabolic limit and Hilbert expansions in the moment equations. After setting up the model, we perform several numerical tests to study the role of the local characteristics and the extended generator of the PDifMP in the process of tumour progression. The main aim focuses on understanding how the variations of the jump rate function of this process at the microscopic scale and the diffusion coefficient at the macroscopic scale are related to the diffusive behaviour of the glioma cells and to the onset of malignancy, i.e., the transition from low-grade to high-grade gliomas.

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms., 38 pages, 10 figures

In this article we construct weak Runge–Kutta Munthe-Kaas methods for a finite-dimensional version of the stochastic Landau–Lifshitz equation (LL-equation). We formulate a Lie group framework for the stochastic LL-equation and derive regularity conditions for the corresponding SDE system on the Lie algebra. Using this formulation we define weak Munthe-Kaas methods based on weak stochastic Runge–Kutta methods (SRK methods) and provide sufficient conditions such that the Munthe-Kaas methods inherit the convergence order of the underlying SRK method. The constructed methods are fully explicit and preserve the norm constraint of the LL-equation exactly. Numerical simulations are provided to illustrate the convergence order as well as the long time behaviour of the proposed methods.

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.

We are concerned with the numerical solution of a class integro-differential equations, known as Neural Field Equations, which describe the large-scale dynamics of spatially structured networks of neurons. These equations have many applications in Neuroscience and Robotics. We describe a numerical method for the approximation of solutions in the two-dimensional case, including a space-dependent delay in the integrand function. Compared with known algorithms for this type of equation we propose a scheme with higher accuracy in the time discretisation. Since computational efficiency is a key issue in this type of calculations, we use a new method for reducing the complexity of the algorithm. The convergence issues are discussed in detail and a number of numerical examples is presented, which illustrate the performance of the method., Comment: 25 pages, 1 figure

We consider embedding deterministic Runge-Kutta methods with high order into weak order stochastic Runge-Kutta (SRK) methods for non-commutative stochastic differential equations (SDEs). As a result, we have obtained weak second order SRK methods which have good properties with respect to not only practical errors but also mean square stability. In our stability analysis, as well as a scalar test equation with complex-valued parameters, we have used a multi-dimensional non-commutative test SDE. The performance of our new schemes will be shown through comparisons with an efficient and optimal weak second order scheme proposed by Debrabant and Rosler (Appl. Numer. Math. 59:582–594, 2009).

Electromagnetic descriptor models are models which lead to differential algebraic equations (DAEs). Some of these models mostly arise from electric circuit and power networks. The most frequently used modeling technique in the electric network design is the modified nodal analysis (MNA) which leads to differential algebraic equations in descriptor form. DAEs are known to be very difficult to solve numerically due to the sensitivity of their solutions to perturbations. We use the tractability index to measure this sensitivity since it can be computed numerically. Simulation of DAEs is a very difficult task especially for those with index greater than one. To solve higher-index DAEs, one needs to use multistep methods such as Backward difference formulas (BDFs). In this paper, we present an easier method of solving DAEs numerically using special projectors. This is done by first splitting the DAE system into differential and algebraic parts. We then use the existing numerical integration methods to approximate the solutions of the differential part and the solutions of the algebraic parts are computed explicitly. The desired solution of the DAE system is obtained by taking the linear combination of the solutions of the differential and algebraic parts. Our method is robust and efficient, and can be used on both small and very large systems.

The stability of equilibrium solutions of a deterministic linear system of delay differential equations can be investigated by studying the characteristic equation. For stochastic delay differential equations stability analysis is usually based on Lyapunov functional or Razumikhin type results, or Linear Matrix Inequality techniques. In [7] the authors proposed a technique based on the vectorisation of matrices and the Kronecker product to transform the mean-square stability problem of a system of linear stochastic differential equations into a stability problem for a system of deterministic linear differential equations. In this paper we extend this method to the case of stochastic delay differential equations, providing sufficient and necessary conditions for the stability of the equilibrium. We apply our results to a neuron model perturbed by multiplicative noise. We study the stochastic stability properties of the equilibrium of this system and then compare them with the same equilibrium in the deterministic case. Finally the theoretical results are illustrated by numerical simulations.

This paper studies the asymptotic growth and decay properties of solutions of the stochastic pantograph equation with multiplicative noise. We give sufficient conditions on the parameters for solutions to grow at a polynomial rate in $p$-th mean and in the almost sure sense. Under stronger conditions the solutions decay to zero with a polynomial rate in $p$-th mean and in the almost sure sense. When polynomial bounds cannot be achieved, we show for a different set of parameters that exponential growth bounds of solutions in $p$-th mean and an almost sure sense can be obtained. Analogous results are established for pantograph equations with several delays, and for general finite dimensional equations., 29 pages, to appear Electronic Journal of Qualitative Theory of Differential Equations, Proc. 10th Coll. Qualitative Theory of Diff. Equ. (July 1--4, 2015, Szeged, Hungary)

We are concerned with a linear mean-square stability analysis of numerical methods applied to systems of stochastic differential equations (SDEs) and, in particular, consider the @q-Maruyama and the @q-Milstein method in this context. We propose an approach, based on the vectorisation of matrices and the Kronecker product, that allows us to deal efficiently with the matrix expressions arising in this analysis and that provides the explicit structure of the stability matrices in the general case of linear systems of SDEs. For a set of simple test SDE systems, incorporating different noise structures but only a few parameters, we apply the general results and provide visual and numerical comparisons of the stability properties of the two methods.

An asymptotic stability analysis of numerical methods used for simulating stochastic differential equations with additive noise is presented. The initial part of the paper is intended to provide a clear definition and discussion of stability concepts for additive noise equation derived from the principles of stability analysis based on the theory of random dynamical systems. The numerical stability analysis presented in the second part of the paper is based on the semi-linear test equation dX(t) = (AX(t) + f(X(t))) dt + σ dW(t), the drift of which satisfies a contractive one-sided Lipschitz condition, such that the test equation allows for a pathwise stable stationary solution. The θ-Maruyama method as well as linear implicit and two exponential Euler schemes are analysed for this class of test equations in terms of the existence of a pathwise stable stationary solution. The latter methods are specifically developed for semi-linear problems as they arise from spatial approximations of stochastic partial differential equations.

We give a short overview of stochastic problems occurring in neuroscience, of typical questions arising in these problems, and of techniques available for their analysis. These notes are part of the proceedings of the mini-workshop "Dynamics of Stochastic Systems and their Approximation" held in Oberwolfach, August 21-27, 2011.

In this paper, we obtain exact non-exponential rates of growth and decay of solutions of linear functional differential equations with unbounded delay. As a by-product, exponential asymptotic stability is ruled out for asymptotically stable solutions of these equations. Equations with maximum functionals on the right hand side, as well as perturbed equations, are considered. We also give examples showing how the rate of growth or decay of solutions depends on the rate of growth of the unbounded delay. The results established here are obtained by constructing carefully functions which satisfy upper and lower differential inequalities and one aim of the paper is to elucidate this constructive comparison technique.

In this article the numerical approximation of solutions of Ito stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their 𝕃 p -consistency, numerical 𝕃 p -stability and 𝕃 p -convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency.

In the present paper we are concerned with a numerical algorithm for the approximation of the two-dimensional neural field equation with delay. We consider three numerical examples that have been analysed before by other authors and are directly connected with real world applications. The main purposes are 1) to test the performance of the mentioned algorithm, by comparing the numerical results with those obtained by other authors; 2) to analyse with more detail the properties of the solutions and take conclusions about their physical meaning., This article is closely related to previous publication in Arxiv: http://arxiv.org/abs/1508.07484, 2015

Neural Field Equations (NFEs) are integrodifferential equations which describe the electric potential field and the interaction between neurons, in certain regions of the brain. They are becoming increasingly important for the interpretation of EEG, fMRi and optical imaging data. In the present article we describe a new efficient algorithm for the numerical simulation of two-dimensional neural fields with delays. The main features of this method are discussed and its performance is illustrated by some numerical examples.

We consider the effect of random variation in the material parameters in a model for machine tool vibrations, specifically regenerative chatter. We show that fluctuations in these parameters appear as both multiplicative and additive noise in the model. We focus on the effect of additive noise in amplifying small vibrations which appear in subcritical regimes. Coherence resonance is demonstrated through computations, and is proposed as a route for transitions to larger vibrations. The dynamics also exhibit scaling laws observed in the analysis of general stochastic delay differential models.

We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution. As in deterministic numerical analysis we use a linear time-invariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams–Bashforth- and Adams–Moulton-methods, the Milne–Simpson method and the BDF method.

We consider the problem of strong approximations of the solution of Ito stochastic functional differential equations (SFDEs). We develop a general framework for the strong convergence of drift-implicit one-step schemes to the solution of SFDEs. Several examples illustrate the applicability of the framework.

In this article a class of nonlinear Abel-type integral equations is studied. These equations have at least the trivial solution, but we are interested in a continuous, positive solution. We show that there exists a unique positive solution in an order interval of a cone in a Banach space.

Using an approach that has its origins in work of Halanay, we consider stability in mean square of numerical solutions obtained from the θ-Maruyama discretization of a test stochastic delay differential equation [Formula: see text] interpreted in the Itô sense, where W(t) denotes a Wiener process. We focus on demonstrating that we may use techniques advanced in a recent report by Baker and Buckwar to obtain criteria for asymptotic and exponential stability, in mean square, for the solutions of the recurrence [Formula: see text]

We consider linear multi-step methods for stochastic differential equations and present a theorem ensuring their numerical stability and strong convergence. We use this to study the properties of two-step schemes for stochastic differential equations with small noise. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)