Figure 1.
Hierarchically embedded cortisol pulses.
An example adapted from experimental data (black line) with six major cortisol pulses. For a Single Pulse Rise (U1) starting at time ∼3.5 hours, the single pulse peak is identified with and the starting nadir/valley is identified with a V1. The pulse rise is defined as the rise in cortisol concentration from the nadir (V1) to peak (P1). The pulse rise time (Uw1) is the time required for the cortisol concentration levels to rise from a sequence of pulses starting with the local nadir (V1) to the peak (P1). The Hierarchical (multiple) Pulse Rise (U2) occurs as the change in concentration from the first nadir (V2) to last nadir (P2), which is a local maximum nadir, in the rise portion of this hierarchically organized segment. The hierarchical pulse rise time (Uw2) is the elapsed time between V2 and P2 of the rising portion. The effect of the hierarchical pulses between V2 and P2 is an accumulation or increase in cortisol. Similarly, sequences of pulses associated with the dissipation or decrease in cortisol begin at time ∼4.8 hours.
Figure 2.
Hierarchically AdaPtive (HAP) Analysis Schematic.
The figure is divided into 3 sections: Time-series (left panel), Theory – Symbolic Representation (right panel) and Computational Steps (bottom panel). Solid arrows indicate theoretical steps; dotted lines with numbers indicate computational steps listed in the bottom panel.
Figure 3.
Qualitative time-series representations for a single participant (ID6).
The dotted vertical lines with lower case Roman numerals represent the same time points in the Time-Series, the Pulsicon, and the Production Graph for participant ID6. Shaded rectangles labeled with uppercase roman numerals represent the same temporal interval in the Time-Series, the Pulsicon, and the Production Graph. Time-Series: The 24-hour cortisol time-series. Nadirs identified in each of the four HAP iterations are connected with lines. The terminal HAP nadirs are identified with black circles. Pulsicon: The pulsicon for the time-series. Production Graph: Each rectangle represents productions: the application of rules defined in the CFL The five rectangular boxes represent the key features of the time-series. The first two rectangles (Labeled 1 and 2) show that there is a decreasing portion prior to the main portion of the cortisol time-series with the “>” in rectangle 2 (e.g., before left black circle). The main portion of the cortisol time-series is represented by the center rectangle (Labeled 3) with the production symbol (S). The production symbol (S) is a variable that can be replaced with any of the rules defined in the context free language. This section contains a hierarchy of embedded pulses. The last two rectangles (Labeled 4 and 5) from the production graph show that the cortisol time-series is rising after the main portion (rectangle labeled 3) of the cortisol time-series (e.g., after right black circle).
Figure 4.
Accumulation and dissipation rates for a single participant (ID6).
(A) Amplitude as a function of the rise or fall duration extracted from the first iteration of the HAP algorithm for participant ID6. Concentration rises (changes between local nadirs and subsequent local peaks) are shown with blue circles. Concentration falls (changes between local peaks and subsequent local nadirs) are shown with green triangles. Regression lines are shown separately for the rises (blue circles) and falls (green triangles). (B) A histogram of the instantaneous accumulation rates ( = rise amplitude/rise duration) from Panel A. (C) A histogram of the instantaneous dissipation rates ( = fall amplitude/fall duration) from Panel A.
Figure 5.
Multi-scale analysis of features identified with HAP from a single participant (ID6).
Each analysis iteration for participant ID6 is shown with a different symbol: 1-circle, 2-square, 3-up triangle, 4-down triangle. The symbol color indicates the timing of the event with the color code in right-most vertical panel: the sleep episode (blue), the first 8 hours of wake (green), or second 8 hours of wake (white). The dashed lines connect median value between iterations. (A) Cortisol accumulation rates for nadirs identified at each iteration. Accumulation rates for the first iteration are analogous to secretory rates. (B) Cortisol dissipation rates for nadirs identified at each iteration. Dissipation rates for the first iteration are analogous to clearance rates. (C) The nadir intervals identified at each iteration are shown. The event intervals at the first iteration are analogous to the inter-pulse interval. (D) The relationship between an accumulation rate and the immediately following dissipation rate.
Figure 6.
Simulated 24-hour cortisol concentration time-series profiles with gamma values from Table 1.
Within each panel, the left figure plots the time and amplitudes of each pulse and the right figure plots the resulting simulated dataset. The gamma value affects the clearance rate. (A) Gamma = 0.25 (B) Gamma = 0.75 (C) Gamma = 1.0 (D) Gamma = 1.5 (E) Gamma = 2.5, (F) Gamma = 5.0.
Table 1.
Amplitude (A) and secretion times (T) for simulated dataset*.
Figure 7.
HAP analysis of simulated cortisol test set.
(A–F) Simulated data generated from the data sets shown in Fig. 4 panels A–F. Simulated data are identified with blue markers. Peaks identified during the nadir selection algorithm are in red. Nadirs identified at a given recursive step are joined with a line, where line color represents a different recursive step.
Table 2.
Pulsicons for All Participants.
Table 3.
Accumulation and dissipation rates computed from the Level 1 iteration for all participants.
Table 4.
Hierarchical Features Summary by Iteration Level for a single participant (ID6).