This paper proposes a new modeling framework for estimating single-trial event-related potentials (ERPs). Existing studies based on state-space approach use discrete-time random-walk models. We propose to use continuous-time partially observed diffusion process which is more natural and appropriate to describe the continuous dynamics underlying ERPs, discretely observed in noise as single-trials. Moreover, the flexibility of the continuous-time model being specified and analyzed independently of observation intervals, enables a more efficient handling of irregularly or variably sampled ERPs than its discrete-time counterpart which is fixed to a particular interval. We consider the Ornstein–Uhlenbeck (OU) process for the inter-trial parameter dynamics and further propose a nonlinear process of Cox, Ingersoll & Ross (CIR) with a heavy-tailed density to better capture the abrupt changes. We also incorporate a single-trial trend component using the mean-reversion variant, and a stochastic volatility noise process. The proposed method is applied to analysis of auditory brainstem responses (ABRs). Simulation shows that both diffusions give satisfactory tracking performance, particularly of the abrupt ERP parameter variations by the CIR process. Evaluation on real ABR data across different subjects, stimulus intensities and hearing conditions demonstrates the superiority of our method in extracting the latent single-trial dynamics with significantly improved SNR, and in detecting the wave V which is critical for diagnosis of hearing loss. Estimation results on data with variable sampling frequencies and missing single-trials show that the continuous-time diffusion model can capture more accurately the inter-trial dynamics between varying observation intervals, compared to the discrete-time model. 
Modeling and estimation of single-trial event-related potentials using partially observed diffusion processes