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"Stochastic neural field equations" Wilhelm Stannat (Video)
Mini-course
Neural field equations are used to describe the spatiotemporal evolution of the average activity in a network of synaptically coupled populations of neurons in the continuum limit. This deterministic description is only accurate in the infinite population limit and the actual finite size of the populations causes deviations from the mean field behavior.
We will first rigorously derive stochastic neural field equations with noise terms accounting for these finite size effects. These equations are identified by describing the evolution of the activity in the finite-size populations by Markov chains and then determining the limit of their fluctuations. We then introduce a complete mathematical framework for the analysis of the resulting stochastic neural field equations. As first steps of a multiscale analysis, a geometrically motivated decomposition of the stochastic evolution into a randomly moving wave front and fluctuations is derived. A random ordinary differential equation describing the velocity of the moving wave front can be deduced and the fluctuations around the wave front turn out to be non-Gaussian, even if the driving noise term is a Gaussian process.
The presented geometric approach is in principle applicable to describe the statistics of any macroscopic profile driven by spatially extended noise, like, e.g., wave fronts and pulses in general stochastic reaction diffusion systems. The talk is partially based on joint work with Jennifer Krueger and Eva Lang.
References
J. Krueger, W. Stannat, Front Propagation in Stochastic Neural Fields : A rigorous mathematical framework, SIAM J. Appl. Dyn. Syst., Vol. 13, 1293-1310, 2014.
E. Lang, W. Stannat, Finite-Size effects on traveling wave solutions to neural field equations, 2014, submitted
W. Stannat, Stability of travelling waves in stochastic bistable reaction-diffusion equations, arXiv:1404.3853
Video
The presented geometric approach is in principle applicable to describe the statistics of any macroscopic profile driven by spatially extended noise, like, e.g., wave fronts and pulses in general stochastic reaction diffusion systems. The talk is partially based on joint work with Jennifer Krueger and Eva Lang.
References
J. Krueger, W. Stannat, Front Propagation in Stochastic Neural Fields : A rigorous mathematical framework, SIAM J. Appl. Dyn. Syst., Vol. 13, 1293-1310, 2014.
E. Lang, W. Stannat, Finite-Size effects on traveling wave solutions to neural field equations, 2014, submitted
W. Stannat, Stability of travelling waves in stochastic bistable reaction-diffusion equations, arXiv:1404.3853
Video
Dates
Created on December 11, 2014