Using a quadratic parameter sinusoid model to characterize the structure of EEG sleep spindles

Abstract

Sleep spindles are essentially non-stationary signals that display time and frequency-varying characteristics within their envelope, which makes it difficult to accurately identify its instantaneous frequency and amplitude. To allow a better parameterization of the structure of spindle, we propose modeling spindles using a Quadratic Parameter Sinusoid (QPS). The QPS is well suited to model spindle activity as it utilizes a quadratic representation to capture the inherent duration and frequency variations within spindles. The effectiveness of our proposed model and estimation technique was quantitatively evaluated in parameter determination experiments using simulated spindle-like signals and real spindles in the presence of background EEG. We used the QPS parameters to predict the energy and frequency of spindles with a mean accuracy of 92.34 and 97.73% respectively. We also show that the QPS parameters provide a quantification of the amplitude and frequency variations occurring within sleep spindles that can be observed visually and related to their characteristic "waxing and waning" shape. We analyze the variations in the parameters values to present how they can be used to understand the inter- and intra-participant variations in spindle structure. Finally, we present a comparison of the QPS parameters of spindles and non-spindles, which shows a substantial difference in parameter values between the two classes.

Keywords: sleep spindle morphology; sleep spindle structure; sleep spindles; sleep spindles model; sleep stages.

Figure 1

(A) Band passed EEG spindle and its envelope (B) Non-sinusoidal variation of the spindle frequency with time.

Figure 2

(A) Raw spindle from MASS-C1/SS2 EEG recording (B) Band-passed version of the EEG spindle (C) QPS spindle generated using the parameters of the band–passed version of the spindle.

Figure 3

Change in simulated QPS spindle with (A) b = 0, c = −20 (B) b = 5, c = −30 (C) b = 10, c = −40 (D) e = 50, f = 0 (E) e = 70, f = 20 (F) e = 90, f = 40.

Figure 4

Simulated QPS spindle with white Gaussian noise and the predicted QPS spindle using estimated parameters.

Figure 5

GOF measures calculated over a range of SNR values for five simulated spindles (A) Sum of Squared Error (B) R-Squared Error (C) Adjusted R-Squared Error (D) Root Mean Squared Error.

Figure 6

GOF measures calculated over a range of SNR values for a single spindle but with different initial values for NLLS (A) Sum of Squared Error (B) R-Squared Error (C) Adjusted R-Squared Error (D) Root Mean Squared Error.

Figure 7

(A) Simulated QPS spindle along with the spindle with added delta components (B) Simulated QPS spindle along with the predicted QPS spindle from the noisy spindle with added delta components.

Figure 8

Boxplot depicting spindle parameter values in the presence of random delta components.

Figure 9

Box plot depicting percentage error in model energy and frequency using data from MASS-C1/SS2 participants.

Figure 10

Box plot depicting the comparison of spindle parameter values using different scorers for parameters (A)a(B)b(C)c(D)d(E)e and (F)f using data from MASS-C1/SS2 participants.

Figure 11

Error bar depicting the distribution of values for parameters (A) a(B)b(C)c(D)d(E)e and (F) f for each MASS-C1/SS2 participant. Here, the whiskers represent 2σ.

Figure 12

Variation of QPS spindles with varying values of (A) a(B)b(c)c(D)d(E)e and (F) f. Here, the parameter values are changed incrementally and the resulting effect on the QPS model is observed.

Figure 13

Variation in values of parameters (A) a, (B) b, (C) c, (D) d, (E) e, and (F) f of MASS-C1/SS2 participant 3 during an overnight recording.