The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations

This article deals with the approximation of a stochastic partial differential equation (SPDE) via amplitude equations. We consider an SPDE with a cubic nonlinearity perturbed by a general multiplicative noise that preserves the constant trivial solution and we study the dynamics around it for the deterministic equation being close to a bifurcation. Based on the separation of time-scales close to a change of stability, we rigorously derive an amplitude equation describing the dynamics of the bifurcating pattern. This allows us to approximate the original infinite dimensional dynamics by a simpler effective dynamics associated with the solution of the amplitude equation. To illustrate the abstract result we apply it to a simple one-dimensional stochastic Ginzburg-Landau equation

Dimension Reduction of Neural Models Across Multiple Spatio-temporal Scales

In general, reducing the dimensionality of a complex model is a natural first step to gaining insight into the system. In this dissertation, we reduce the dimensions of models at three different scales: first at the scale of microscopic single-neurons, second at the scale of macroscopic infinite neurons, and third at an in-between spatial scale of finite neural populations. Each model also exhibits a separation of timescales, making them amenable to the method of multiple timescales, which is the primary dimension-reduction tool of this dissertation. In the first case, the method of multiple timescales reduces the dynamics of two coupled n-dimensional neurons into one scalar differential equation representing the slow timescale phase-locking properties of the oscillators as a function of an exogenous slowly varying parameter. This result extends the classic theory of weakly coupled oscillators. In the second case, the method reduces the many spatio-temporal \yp{dynamics of} ``bump'' solutions of a neural field model into its scalar coordinates, which are much easier to analyze analytically. This result generalizes existing studies on neural field spatio-temporal dynamics to the case of a smooth firing rate function and general even kernel. In the third case, we reduce the dimension of the oscillators at the spiking level -- similar to the first case -- but with additional slowly varying synaptic variables. This result generalizes existing studies that use scalar oscillators and the Ott-Antonsen ansatz to reduce the dimensionality and determine the synchronization properties of large neural populations