Project heads:
Prof. Dr. Wilhelm Stannat
Project members:

Duration: 01.11.2011  30.04.2016
Status:
completed
Located at:
Technische Universität Berlin
Description
Statetheart, 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, 503512.

[2] Albeverio S, Cebulla C (2008) Synchronizability of a Stochastic Version of FitzHughNagumo 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 FitzHughNagumo equations on networks with impulsive noise, EJP 13, 13621379.

[5] Bonaccorsi S, Mastrogiacomo E (2007) Analysis of the stochastic FitzHughNagumo 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] EsSarhir A Stannat W (2008) Invariant measures for Semilinear SPDE's with local Lipschitz Drift Coefficients and applications, Journal of Evolution Equations 8, 129154.

[8] Kallianpur G, Wolpert R (1984) Infinite dimensional stochastic differential equation models for spatially distributed neurons, Appl. Math. Optim. 12, 125172.
https://www.bccnberlin.de/Research/Projects_II/Branch_A/A11/