Denoising neural data with state-space smoothing: method and application

Abstract

Neural data are inevitably contaminated by noise. When such noisy data are subjected to statistical analysis, misleading conclusions can be reached. Here we attempt to address this problem by applying a state-space smoothing method, based on the combined use of the Kalman filter theory and the Expectation-Maximization algorithm, to denoise two datasets of local field potentials recorded from monkeys performing a visuomotor task. For the first dataset, it was found that the analysis of the high gamma band (60-90 Hz) neural activity in the prefrontal cortex is highly susceptible to the effect of noise, and denoising leads to markedly improved results that were physiologically interpretable. For the second dataset, Granger causality between primary motor and primary somatosensory cortices was not consistent across two monkeys and the effect of noise was suspected. After denoising, the discrepancy between the two subjects was significantly reduced.

FIG 1

Top panel: schematic of electrode placement for monkey TI. Data from three prefrontal recording sites marked O, L, and M, referred to as PF1, PF2, and PF3 in the text, are further analyzed. Middle panel: schematic of electrode placement for monkey LU. Data from two recording sites marked K and L, which correspond to S1 and M1, are further analyzed. Bottom panel: schematic of electrode placement for monkey GE. Data from two recording sites marked I and J, which correspond to S1 and M1, are further analyzed.

FIG 2

The power and coherence spectra of a noisy bivariate AR(2) process (example 1). The top and middle panels show the power spectra of channels z₁ and z₂, respectively. The solid lines depict the exact power spectra; the dotted lines indicate the power spectra of the noisy data; the dashed lines indicate the power spectra of the denoised signals; and the filled squares indicate the spectra for noise. The bottom panel represents the coherence spectra between z₁ and z₂. Here, the solid line indicates exact coherence spectrum; the dotted and dashed lines are the coherence spectra of noisy data and denoised signal, respectively. Error bars for the coherence spectra were computed using a bootstrap procedure. See text for more details.

FIG 3

Granger causality spectra for data from a bivariate AR(1) process (example 2). The solid lines, dotted lines, dashed lines represent true Granger causality spectra from data without added noise, spectra from data with added noise, and spectra from the denoised data, respectively. Error bars were computed using a bootstrap procedure. Note that in the top panel the noise-free spectrum and the denoised spectrum nearly coincide.

FIG 4

Granger causality spectra for data from a bivariate AR(1) process (example 3). The amount of noise added to the driver variable z₂ is relatively small while the amount of noise added to the driven variable z₁ is relatively large. The solid lines, dotted lines, dashed lines represent true Granger causality spectra from data without added noise, spectra from data with added noise, and spectra from the denoised data, respectively. Error bars were computed using a bootstrap procedure. Note that all three spectra in the top panel nearly coincide.

FIG 5

The power spectra of the prestimulus data (-90 to 20 ms) at site PF3 for monkey TI. The dotted line, dashed line, and squares represent the power spectrum of the noisy data, the power spectrum of the denoised data, and the power spectrum of the removed noise, respectively. Error bars were computed using a bootstrap procedure.

FIG 6

One realization of the removed noise as a function of time at site PF3 is shown in the top panel. The autocorrelation function is shown as a stem plot in the bottom panel. The horizontal dashed lines indicate the 95% confidence intervals for the process to be a white noise process.

FIG 7

The coherence spectrum of the pre-stimulus data for prefrontal site pair PF1-PF2 of monkey TI. The dotted line and the dashed line represent the coherence of the noisy data and of the denoised data. Error bars were computed using a bootstrap procedure. Inset: The ratios of high gamma band coherence of denoised signal to that of noisy data are shown for all three site pairs with the plotted pair shaded.

FIG 8

Scatter plot showing positive correlation between pre-stimulus prefrontal network coherence in the high gamma band (60 to 90 Hz) and RT for noisy data in monkey TI. Least-squares fit is superimposed. Inset: Spearman rank correlation coefficients between high gamma coherence and RT for all network site pairs are shown with the value for the plotted pair shaded.

FIG 9

Scatter plot showing negative correlation between pre-stimulus prefrontal network coherence in the high gamma band and RT for denoised signal in monkey TI. A Least squares fit is superimposed. Inset: Spearman rank correlation coefficients between high gamma coherence and RT for all network site pairs are shown with the value for the plotted pair shaded.

FIG 10

Granger causality spectra between sites S1 and M1 for monkey LU. The dashed lines and solid lines represent spectra for noisy data and denoised data respectively. Error bars were computed using a bootstrap procedure.

FIG 11

Granger causality spectra between sites S1 and M1 for monkey GE. The dashed lines and solid lines represent spectra for noisy data and denoised data respectively. Error bars were computed using a bootstrap procedure.

FIG 12

Ratio between S1→M1 and M1→S1 Granger causality in beta band for noisy data for monkeys LU and GE. Error bars were computed using a bootstrap procedure.

FIG 13

Ratio between S1→M1 and M1→S1 Granger causality in beta band for denoised data for monkeys LU and GE. Error bars were computed using a bootstrap procedure.