Figure 1
Periodogram analysis of a time series. (A) Simulated time series with 16 hours of period 5.3 h alternating with 8 hours of period 1.62 h, plus white noise. (B) Fourier, MESA, and Lomb-Scargle periodograms all have similar large spikes at harmonics 24/4=6 h and 24/5=4.8 h, plus small spikes at harmonics 24/7=3.43 h, 24/8=3 h, and 24/9=2.67 h. Note the absence in all 3 periodograms of significant power at the ultradian periods 5.3 h and 1.62 h, marked by small arrows along the horizontal axis. (C) Inverse DFT of the first 18 harmonics. While the periodograms do not provide a direct means of detecting the ultradian periods of interest, if the underlying ultradian pattern is sufficiently regular (as is the case in this example), then taking the inverse DFT of the circadian harmonics can reveal what that pattern is.
Figure 2
AWT of the simulated activity time series. The heat map in (B) indicates the absolute value of the AWT coefficients for the simulated time series, shown again in (A) for ease of comparison. The short black curves in (B) are the wavelet ridges that provide an estimate of the instantaneous period, averaging 5.2 h and 1.6 h during the alternating intervals (close to the true values 5.3 h and 1.62 h). The amplitude is indicated by the color of the heat map. The amplitude of the time series alternates between 0.5 and 0.2, so the AWT also correctly estimates the amplitude during each time interval.
Figure 3
DWT of the simulated activity time series shown in Figure 1A, using the symlet6 filter. The wavelet details D1 through D7 are shown at the same scale as the time series itself so that the magnitudes can be directly compared.
Figure 4
Bout interval analysis using the DWT. (A) The first 2 days of the original simulated time series from Figure 1A is shown above the cleaned-up version, which equals the sum of the wavelet details D3-D7 from Figure 3. Peaks of the summed wavelet details (marked by diamonds) correspond to midpoints of the simulated activity bouts. (B) Length of time intervals between midpoints of activity bouts, measured as the distance between peaks of the summed wavelet details. The three large bouts of activity are separated by roughly 5.3 h (perturbed by the added noise), the four shorter bouts of activity are separated by around 1.6 h, and the transition intervals are around 3.5h. These values are correctly identified by the DWT interval analysis.
Figure 5
Example of the AWT applied to detect changes in period and amplitude over time. (A) Time series of wheel running (counts per 6 minute bin) for a female hamster in constant darkness. (B) Heat map of the magnitude of the AWT coefficients. The black curve is the wavelet ridge that indicates the instantaneous period, while the color of the heat map indicates amplitude. (C) Curves showing the amplitude (in blue) and period (in black), extracted from the wavelet ridge in (B), revealing that the oscillation of the amplitude is nearly antiphase to the oscillation of the period in this example. Hamster record courtesy of Eric Bittman and Emily Manoogian.
Figure 6
Example of the AWT applied to detect an ultradian period of activity. (A) Actogram displaying wheel running (6 minute bins) for a male hamster entrained to 14L:10D. (B) Heat map of the magnitude of the AWT coefficients. The wavelet ridge (the black curve) indicates the instantaneous period, while the color of the heat map indicates amplitude. While in the light, the hamster is not active so the AWT heat map shows dark blue, while in darkness the hamster typically displays an ultradian period around 4.5-5 h (indicated by the wavelet ridges in black on the hot spots). On day 3, two large bouts of activity are closely spaced, resulting in a lower ultradian period and a strong 12 h harmonic. The wavelet ridge at period 24 h indicates that the animal is successfully entrained to the 24 h LD cycle. The yellow spots around period 2 h are for the most part harmonics of the hotspots above them. Hamster record courtesy of Eric Bittman and Emily Manoogian.
Figure 7
Actograms displaying motion (6 minute bins) from 3 male hamsters entrained to 15L:9D, with averaged AWT heat maps and graphs of DWT-derived intervals between midpoints of activity. The AWT heat maps show the mean of the absolute value of the AWT coefficients, taken across 11 days. (A) The activity of this hamster exhibits a very short period near lights on and off and a longer ultradian period during the middle of the night. This pattern is reflected in both the AWT and the DWT-derived interval analysis. During the day, the bouts appear more randomly spaced, with no clear frequency emerging in either the AWT heat map or the interval graph, a pattern which also appears in the other two records. (B) This hamster appears to display two ultradian frequencies at night, a very short period of less than 0.5 h and a longer period that starts around 2 h shortly after lights-off and decreases through the night, again reflected in both the AWT heat map and the DWT-derived intervals analysis. (C) This hamster shows almost no activity in the first part of the night, after which it exhibits ultradian periods around 1 h and less than 0.5 h. Hamster records courtesy of Brian Prendergast.