Figure 3: Kalman filtering for online estimation in SISO applications.

(a) Standard Kalman filter. Prediction error (e) is used to correct the estimate of state at each time step. (b) Example open-loop (OL) prediction of neuronal response (red) to step input of light (blue) using 5^thorder GLDS fit to noise-driven data, compared to PSTH smoothed with 25 ms Gaussian window (black) and the trial-averaged response estimated using the standard implementation of the Kalman filter (5^th-order GLDS) (purple). (c) Example single-trial Kalman filter estimate (purple) along with corresponding spike raster (grey). (d) Parameter-adaptive Kalman filter. In addition to estimating the state of the system, this approach jointly re-estimates a state disturbance (μ) at each time step. (e) Same as (b) but trial-averaged estimate of firing rate using the parameter-adaptive Kalman filter. (f) Same as (c) except single-trial estimate using parameter-adaptive Kalman filter. (g) Trial-averaged disturbance on the first state estimated using parameter-adaptive Kalman filter. (h) Population average squared-bias in estimation calculated between the single-trial spiking responses and the OL prediction of a 5^th-order GLDS, the standard Kalman filter using the 5^th-order GLDS, and the parameter-adaptive Kalman filter (aKF) using a 1^st- or 5^th-order GLDS. Black and grey data points correspond to error associated with using individually-fit models vs. a single population average fit model, respectively. Error bars represent bootstrapped 95% confidence intervals about the mean (n=48 neurons, 17 recordings, 9 animals).