Implicit equation-free methods applied on noisy slow-fast systems

Slow-fast systems consist of slow macroscopic and fast microscopic dynamics. By using equation-free methods, one can do a complete bifurcation analysis of these slow macroscopic variables, even if direct equations are only known for the microscopic dynamics.
In the case of deterministic behaviour, a theoretical framework was already given by Kevrekidis et al. [1]. By direct simulation, we can find the stable macroscopic equilibria. For the unstable ones, we apply an implicit equation-free method [2] from which we can numerically gain a time derivative and hence the macroscopic variables’ derivative. With the help of a Newton method, we can find the unstable equilibria in phase space.
Looking on explicit applications in natural and life sciences, we often have noisy microscopic data. As we do not have any knowledge about the underlying macroscopic dynamics and its noise, we need to do further analysis before we can apply statistical tools.
On the example of a neural network for odour distinction in the brain, we want to demonstrate the method’s capability and the so far existing challenges for noisy microscopic dynamics.

References
[1] I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg,
and C. Theodoropoulos. Commun. Math. Sci., 1(4): 715-762, 2003.
[2] C. Marschler, J. Sieber, P. G. Hjorth, and J. Starke. Traffic and Granular
Flow ’13: 423-439, Springer-Verlag, Heidelberg, New York, 2015.

invited by Theresa Lange