Parameter estimation from an Ornstein-Uhlenbeck process with measurement noise with applications in neuroscience*

This work aims to investigate the impact of noise on parameter fitting for an Ornstein-Uhlenbeck process, focusing on the effects of multiplicative and thermal noise on the accuracy of signal separation. To address these issues, we propose algorithms and methods that can effectively distinguish between thermal and multiplicative noise and improve the precision of parameter estimation for optimal data analysis. Specifically, we explore the impact of both multiplicative and thermal noise on the obfuscation of the actual signal and propose methods to resolve them. Firstly, we present an algorithm that can effectively separate thermal noise with comparable performance to Hamilton Monte Carlo (HMC) but with significantly improved speed. Subsequently, we analyze multiplicative noise and demonstrate that HMC is insufficient for isolating thermal and multiplicative noise. However, we show that, with additional knowledge of the ratio between thermal and multiplicative noise, we can accurately distinguish between the two types of noise when provided with a sufficiently large sampling rate or an amplitude of multiplicative noise smaller than thermal noise.

This finding results in a situation that initially seems counterintuitive. When multiplicative noise dominates the noise spectrum, we can successfully estimate the parameters for such systems after adding additional white noise to shift the noise balance.

We apply our methods to the analysis of resting-state functional magnetic resonance imaging (rsfMRI), which measures neural activation correlations between brain regions.  Here, we combine our measurements of fMRI noise (thermal and multiplicative) using our BrainDancer Dynamic fMRI phantom with the above-described time-series analysis method to improve the estimation of correlation coefficients by simultaneously modeling the noise and signal (OU dynamics).  We show that correlation coefficients that are estimated by our methods more accurately describe the "ground truth" correlation coefficients when applied to human rsfMRI measurements.