Prof. Dr. Wilhelm Stannat

Projects as a project leader

• CH-AP17

  The mathematical analysis of interacting stochastic oscillators

  Prof. Dr. Wilhelm Stannat

  Project heads: Prof. Dr. Wilhelm Stannat
  Project members: -
  Duration: 01.11.2011 - 30.04.2016
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  State-the-art, own contribution: Rigorous mathematical models for spatialy extended neurons and neural systems under the influence of noise will be developed and analysed using the mathematical theory of stochastic evolution equations, in particular stochastic partial differential equations (see [6]). We will take into account thermal noise modelling local exterior forces acting on a couple of adjacent neurons but also parametric noise modelling uncertainties in the parameters. The impact of noise on the whole system will then be analyzed rigorously, to quantify, e.g., the probability for the propagation failure of an action potential. There are only few applications of the mathematical theory of stochastic evolution equations to neural systems subject to noise (see [1,2,8,10]). In particular, the recent developments of the theory based on the semigroup approach for mild solutions and the analysis of the associated Kolmogorov operator (see [9]) has so far only been applied to stochastic FitzHugh Nagumo systems in [4,5].
  
  Cited references:

  • [1] Albeverio S, Cebulla C (2007) Synchronizability of Stochastic Network Ensembles in a Model of Interacting Dynamical Units. Physica A Stat. Mech. Appl. 386, 503-512.
  • [2] Albeverio S, Cebulla C (2008) Synchronizability of a Stochastic Version of FitzHugh-Nagumo Type Neural Oscillator Networks, Preprint, SFB 611, Bonn.
  • [3] Blömker D (2007) Amplitude Equations for Stochastic Partial Differential Equations, World Scientific, New Jersey.
  • [4] Bonaccorsi S, Marinelli C, Ziglio G (2008) Stochastic FitzHugh-Nagumo equations on networks with impulsive noise, EJP 13, 1362-1379.
  • [5] Bonaccorsi S, Mastrogiacomo E (2007) Analysis of the stochastic FitzHugh-Nagumo system, Technical Report UTM 719, Mathematics, Trento, arXiv:0801.2325.
  • [6] Da Prato G, Zabczyk, J (1992) Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge.
  • [7] Es-Sarhir A Stannat W (2008) Invariant measures for Semilinear SPDE's with local Lipschitz Drift Coefficients and applications, Journal of Evolution Equations 8, 129-154.
  • [8] Kallianpur G, Wolpert R (1984) Infinite dimensional stochastic differential equation models for spatially distributed neurons, Appl. Math. Optim. 12, 125-172.

  
  
  https://www.bccn-berlin.de/Research/Projects_II/Branch_A/A11/

• CH-AP18

  Numerical analysis and simulation of cooperative phenomena in interacting stochastic oscillators

  Prof. Dr. Wilhelm Stannat

  Project heads: Prof. Dr. Wilhelm Stannat
  Project members: -
  Duration: 01.05.2013 - 30.04.2016
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  State-the-art, own contribution: In contrast to the case of deterministic reaction diffusion systems there are only few publications on the numerical analysis of stochastic reaction diffusion systems arising in neuroscience, like e.g. stochastic FitzHugh Nagumo systems (see [2,6,7]). From the neuroscience perspective in particular the numerical analysis of stochastic reaction diffusion systems exhibiting various spatial patterns based e.g. on partial synchronization are of interest (see [8] and references therein). From the mathematical viewpoint a major difficulty comes from the fact that the coefficients of the systems typically only satisfy a one-sided Lipschitz condition that cannot be controlled easily if perturbed with stochastic forcing terms. Recent results in the numerical analysis of stochastic differential equations with non-Lipschitz coefficients show that their numerical approximation has to be carried out with additional care (see [3]) in order to validate simulation results. We will be therefore interested in development and rigorous mathematical analysis of the numerical approximation of stochastic reaction diffusion systems in the excitable regime. There is a considerable amount of work in the physics literature on the influence of noise in excitable reacton diffusion systems (see [5] for a survey). On the other hand there is only a limited quantitative understanding of the influence of the stochastic forcing terms on the various effects like wave speed or nucleation of wave patterns. Certainly, spatial correlation of the noise terms will play a crucial role, which will be studied also systematically within this project.
  
  Cited references:

  • [1] Dahlem M A, Graf R,Strong A J, Dreier J P, Dahlem Y A, Sieber M, Hanke W, Podoll K, Schöll E (2010) Two-dimensional wave patterns of spreading depolarization: Retracting, re-entrant, and stationary waves, Physica D 239, 889-903.
  • [2] DeVille R E L, Vanden-Eijnden E (2007) Wavetrain response of an excitable medium to local stochastic forcing, Nonlinearity 20, 51-74.
  • [3] Hutzenthaler M, Jentzen A, Kloeden P E (2011) Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467, 1563–1576.
  • [4] Laing C, Lord G J (Eds.), Stochastic Methods in Neuroscience (2010) Oxford University Press, Oxford.
  • [5] Lindner B, Garcia-Ojalvo J, Neiman A, Schimansky-Geier L (2004), Effects of noise in excitable systems, Physics Reports 392, 321-424.
  • [6] Shardlow T (2005) Numerical simulation of stochastic PDEs for excitable media, J. Comput. Appl. Math. 175, 429-446.
  • [7] Shardlow T (2004) Nucleation of waves in excitable media, Multiscale Model. Simul. 3, 151-167.
  • [8] Tass P (1999) Phase Resetting in Medicine and Biology - Stochastic Modelling and Data Analys, Springer, Berlin.

  
  
  https://www.bccn-berlin.de/Research/Projects_II/Branch_A/A12/

• CH-AP27

  Application of rough path theory for filtering and numerical integration methods

  Prof. Dr. Peter Karl Friz / Prof. Dr. Wilhelm Stannat

  Project heads: Prof. Dr. Peter Karl Friz / Prof. Dr. Wilhelm Stannat
  Project members: -
  Duration: 01.11.2011 - 31.10.2014
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  In 1998 T. Lyons (Oxford) suggested a new approach for the robust pathwise solution of stochastic differential equations which is nowadays known as the rough path analysis. Based on this approach a new class of numerical algorithms for the solution of stochastic differential equations have been developed. Recently, the rough path approach has been successfully extended also to stochastic partial differential equations. In stochastic filtering, the (unnormalized) conditional distribution of a Markovian signal observed with additive noise is called the optimal filter and it can be described as the solution of a stochastic partial differential equation which is called the Zakai equation. In the proposed project we want to apply the rough path analysis to a robust pathwise solution of the Zakai equation in order to construct robust versions of the optimal filter. Subsequently, we want to apply known algorithms based on the rough path approach to the numerical approximation of these robust estimators and further investigate their properties.
  
  http://www.dfg-spp1324.de/abstracts.php?lang=de#8