A simple theory of stochastic dynamics near stable fixed points

The development of a rigorous understanding of the effect of stochastic fluctuations on the dynamics of diverse physical systems is of broad practical significance. We consider systems of many variables that can be described by nonlinear SDEs driven by additive filtered noise, and revisit the analytical theory of stochastic oscillations near the stable fixed points of their dynamics. Specifically, we introduce a simple approach to deriving analytical expressions for the noise power spectrum and coherency spectrum (and their corresponding correlation functions) in terms of rational polynomial functions and identify compact formulas for certain polynomial coefficients. This is achieved through linearization about the fixed point and systematic application of Itô calculus, without resorting to the computation of the formal Fourier transform of the white noise process. Then, we demonstrate that we can identify model parameters by fitting our solution to an experimental (synthetic) noise spectrum and/or coherence data. We illustrate this approach for two distinct types of dynamical systems: a 5-D nonlinear toy model with additive Gaussian white noise, and Wilson-Cowan type cortical population excitatory-inhibitory neural models with exponentially low-pass filtered Poisson noise.