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Wilhelm Stannat

Publications and source records attributed to Wilhelm Stannat.

52 records · Page 3Linked to original sources
arXiv2016

Finite-Size Effects on Traveling Wave Solutions to Neural Field Equations

Neural field equations are used to describe the spatiotemporal evolution of the activity in a network of synaptically coupled populations of neurons in the continuum limit. Their heuristic derivation involves two approximation steps. Under the assumption that each population in the network is large, the activity is described in terms of a population average. The discrete network is then approximated by a continuum. In this article we make the two approximation steps explicit. Extending a model by Bressloff and Newby, we describe the evolution of the activity in a discrete network of finite populations by a Markov chain. In order to determine finite-size effects - deviations from the mean field limit due to the finite size of the populations in the network - we analyze the fluctuations of this Markov chain and set up an approximating system of diffusion processes. We show that a well-posed stochastic neural field equation with a noise term accounting for finite-size effects on traveling wave solutions is obtained as the strong continuum limit.

math.PR↗
arXiv2016

$L^2$-Stability of Traveling Wave Solutions to Nonlocal Evolution Equations

Stability of the traveling wave solution to a general class of one-dimensional nonlocal evolution equations is studied in $L^2$-spaces, thereby providing an alternative approach to the usual spectral analysis with respect to the supremum norm. We prove that the linearization around the traveling wave solution satisfies a Lyapunov-type stability condition in a weighted space $L^2(ρ)$ for a naturally associated density $ρ$. The result can be applied to obtain stability of the traveling wave solution under stochastic perturbations of additive or multiplicative type. For small wave speeds, we also prove an alternative Lyapunov-type stability condition in $L^2(\mathfrak{m})$, where $\mathfrak{m}$ is the symmetrizing density for the traveling wave operator, which allows to derive a long-term stochastic stability result.

math.PR↗
arXiv2016

Reliability of signal transmission in stochastic nerve axon equations

We introduce a method for computing probabilities for spontaneous activity and propagation fail- ure of the action potential in spatially extended, conductance-based neuronal models subject to channel noise, based on statistical properties of the membrane potential. We compare different estimators with respect to the quality of detection, computational costs and robustness and propose the integral of the membrane potential along the axon as an appropriate estimator to detect both spontaneous activity and propagation failure. Performing a model reduction we achieve a simplified analytical expression based on the linearization at the resting potential (resp. the traveling action potential). This allows to approximate the probabilities for spontaneous activity and propagation failure in terms of (classical) hitting probabilities of one-dimensional linear stochastic differential equations. The quality of the approximation with respect to the noise amplitude is discussed and illustrated with numerical results for the spatially extended Hodgkin-Huxley.

q-bio.NC↗
arXiv2015

On Average Risk-sensitive Markov Control Processes

We introduce the Lyapunov approach to optimal control problems of average risk-sensitive Markov control processes with general risk maps. Motivated by applications in particular to behavioral economics, we consider possibly non-convex risk maps, modeling behavior with mixed risk preference. We introduce classical objective functions to the risk-sensitive setting and we are in particular interested in optimizing the average risk in the infinite-time horizon for Markov Control Processes on general, possibly non-compact, state spaces allowing also unbounded cost. Existence and uniqueness of an optimal control is obtained with a fixed point theorem applied to the nonlinear map modeling the risk-sensitive expected total cost. The necessary contraction is obtained in a suitable chosen seminorm under a new set of conditions: 1) Lyapunov-type conditions on both risk maps and cost functions that control the growth of iterations, and 2) Doeblin-type conditions, known for Markov chains, generalized to nonlinear mappings. In the particular case of the entropic risk map, the above conditions can be replaced by the existence of a Lyapunov function, a local Doeblin-type condition for the underlying Markov chain, and a growth condition on the cost functions.

math.OC↗
arXiv2015

Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction

We consider a system of $N$ disordered mean-field interacting diffusions within spatial constraints: each particle $θ_i$ is attached to one site $x_i$ of a periodic lattice and the interaction between particles $θ_i$ and $θ_j$ decreases as $| x_i-x_j|^{-α}$ for $α\in[0,1)$. In a previous work, it was shown that the empirical measure of the particles converges in large population to the solution of a nonlinear partial differential equation of McKean-Vlasov type. The purpose of the present paper is to study the fluctuations associated to this convergence. We exhibit in particular a phase transition in the scaling and in the nature of the fluctuations: when $α\in[0,\frac{1}{2})$, the fluctuations are Gaussian, governed by a linear SPDE, with scaling $\sqrt{N}$ whereas the fluctuations are deterministic with scaling $N^{1-α}$ in the case $α\in(\frac{1}{2},1)$.

math.PR↗
arXiv2014

Stochastic partial differential equations: a rough path view

We discuss regular and weak solutions to rough partial differential equations (RPDEs), thereby providing a (rough path-)wise view on important classes of SPDEs. In contrast to many previous works on RPDEs, our definition gives honest meaning to RPDEs as integral equation, based on which we are able to obtain existence, uniqueness and stability results. The case of weak "rough" forward equations, may be seen as robustification of the (measure-valued) Zakai equation in the rough path sense. Feynman-Kac representation for RPDEs, in formal analogy to similar classical results in SPDE theory, play an important role.

math.PR↗
arXiv2014

Lattice Approximation for Stochastic Reaction Diffusion Equations with One-Sided Lipschitz Condition

We consider strong convergence of the finite differences approximation in space for stochastic reaction diffusion equations with multiplicative noise under a one-sided Lipschitz condition only. We derive convergence with an implicit rate depending on the regularity of the exact solution. This can be made explicit if the variational solution has more than its canonical spatial regularity. As an application, spatially extended FitzHugh-Nagumo systems with noise are considered.

math.PR↗
arXiv2014

Mean field limit for disordered diffusions with singular interactions

Motivated by considerations from neuroscience (macroscopic behavior of large ensembles of interacting neurons), we consider a population of mean field interacting diffusions in $\mathbf {R}^m$ in the presence of a random environment and with spatial extension: each diffusion is attached to one site of the lattice $\mathbf {Z}^d$, and the interaction between two diffusions is attenuated by a spatial weight that depends on their positions. For a general class of singular weights (including the case already considered in the physical literature when interactions obey to a power-law of parameter $0<α<d$), we address the convergence as $N\to\infty$ of the empirical measure of the diffusions to the solution of a deterministic McKean-Vlasov equation and prove well-posedness of this equation, even in the degenerate case without noise. We provide also precise estimates of the speed of this convergence, in terms of an appropriate weighted Wasserstein distance, exhibiting in particular nontrivial fluctuations in the power-law case when $\frac{d}{2}\leqα<d$. Our framework covers the case of polynomially bounded monotone dynamics that are especially encountered in the main models of neural oscillators.

math.PR↗
arXiv2014

Front Propagation in Stochastic Neural Fields: A Rigorous Mathematical Framework

We develop a complete and rigorous mathematical framework for the analysis of stochastic neural field equations under the influence of spatially extended additive noise. By comparing a solution to a fixed deterministic front profile it is possible to realise the difference as strong solution to an $L^2(\mathbb{R})$-valued SDE. A multiscale analysis of this process then allows us to obtain rigorous stability results. Here a new representation formula for stochastic convolutions in the semigroup approach to linear function-valued SDE with adapted random drift is applied. Additionally, we introduce a dynamic phase-adaption process of gradient type.

math.PR↗
arXiv2014

Stability of travelling waves in stochastic bistable reaction-diffusion equations

We extend the result on the stability of travelling waves for stochastic Nagumo equations in [St] to general bistable reaction-diffusion equations with both additive and multiplicative noise, using a variational approach based on functional inequalities. Our analysis yields explicit estimates on the rate of stability that can be shown in special examples to be optimal. [St] Stannat, W., Stability of travelling waves in stochastic Nagumo equations, arXiv:1301.6378, 2013.

math.PR↗
arXiv2014

Analysis and Approximation of Stochastic Nerve Axon Equations

We consider spatially extended conductance based neuronal models with noise described by a stochastic reaction diffusion equation with additive noise coupled to a control variable with multiplicative noise but no diffusion. We only assume a local Lipschitz condition on the nonlinearities together with a certain physiologically reasonable monotonicity to derive crucial $L^\infty$-bounds for the solution. These play an essential role in both the proof of existence and uniqueness of solutions as well as the error analysis of the finite difference approximation in space. We derive explicit error estimates, in particular a pathwise convergence rate of $\sqrt{\frac{1}{n}}-$ and a strong convergence rate of $\frac1n$ in special cases. As applications, the Hodgkin-Huxley and FitzHugh-Nagumo systems with noise are considered.

math.PR↗
arXiv2014

Risk-sensitive Markov control processes

We introduce a general framework for measuring risk in the context of Markov control processes with risk maps on general Borel spaces that generalize known concepts of risk measures in mathematical finance, operations research and behavioral economics. Within the framework, applying weighted norm spaces to incorporate also unbounded costs, we study two types of infinite-horizon risk-sensitive criteria, discounted total risk and average risk, and solve the associated optimization problems by dynamic programming. For the discounted case, we propose a new discount scheme, which is different from the conventional form but consistent with the existing literature, while for the average risk criterion, we state Lyapunov-like stability conditions that generalize known conditions for Markov chains to ensure the existence of solutions to the optimality equation.

math.OC↗
arXiv2013

Stability of travelling waves in stochastic Nagumo equations

Stability of travelling waves for the Nagumo equation on the whole line is proven using a new approach via functional inequalities and an implicitely defined phase adaption. The approach can be carried over to obtain pathwise stability of travelling wave solutions in the case of the stochastic Nagumo equation as well. The noise term considered is of multiplicative type with variance proportional to the distance of the solution to the orbit of the travelling wave solutions.

math.PR↗
arXiv2013

Stability of the optimal filter in a hidden Markov model with multiplicative noise

We consider a hidden Markov model with multiplicative noise emerging from studies of software reliability. We show the stability of the optimal filter with respect to general initial conditions in the total variation- and $L^p$-norm and deduce explicit rates. Remarkably, stability turns out to be independent of the ergodic behavior of the signal.

math.PR↗
arXiv2010

Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications

We study regularity properties for invariant measures of semilinear diffusions in a separable Hilbert space. Based on a pathwise estimate for the underlying stochastic convolution, we prove a priori estimates on such invariant measures. As an application, we combine such estimates with a new technique to prove the $L^1$-uniqueness of the induced Kolmogorov operator, defined on a space of cylindrical functions. Finally, examples of stochastic Burgers equations and thin-film growth models are given to illustrate our abstract result.

math.PR↗