Model of Luteinizing Hormone secretion.

The pituitary gland releases LH into the blood as successive spikes characterized by a quasi-instantaneous increase followed by slower (yet quite fast) decrease (see [4]). Hence, in our model, the LH release along a spike is approximated by a discontinuous function of time: the jump accounting for the instantaneous increase in the LH release is followed by a fast exponential decrease. The interspike interval is controlled by a function of time accounting for the varying release frequency. The spike amplitude is also subject to an inter-spike variability as well as to long-term changes partly due to the time variations in the stock of LH available for release. In our model, the amplitude is controlled by another function of time . Hence, the instantaneous release of LH (expressed in ng/ml/min) in the blood by the pituitary gland is given by:(1)where ⌊x⌋refers to the greatest integer smaller than x (integer part). The exponential decay rate is directly linked to the spike half-life, , (i.e. the time taken for the instantaneous released LH quantity to drop from the maximal spike value to half this value) through:

We based our choice of parameter values on the few experiments that have investigated the LH release by the pituitary gland in the ewe from synchronous sampling in the jugular blood and cavernous sinus [4]. Accordingly, we chose a spike half-life

We represent the continuously measured LH blood level (expressed in ng/ml) as the solution of:(2)where the LH release rate LH(t) is given by equation (1). Parameter a represents the instantaneous LH clearance rate from the blood. To be consistent with the one hour half-life of LH pulses (i.e. the time taken for the blood LH level to drop from the maximal pulse value to half this value) in the jugular blood, we have fixed .

Sampling protocol and assay variance.

To mimic the experimental protocol for LH data acquisition, we extract time series from the fine step numerical integration of equation (2). This process is intended to obtain a time series of N consecutive samples similar to experimental results, i.e. a finite sequence of N couples , with i = 1,2,…,N, where is the measured LH level at time .

In most experiments, the LH data are retrieved at a fixed frequency. We describe below the corresponding synthetic process: the samples are obtained each Ts minutes from the starting time t = r (expressed in minutes). Hence, the sampling times are(3)Note that the first sample is retrieved at time . Parameter r, chosen between 0 and Ts, allows one to shift the beginning of the sampling process while using the same set of data simulated from equation (2).

To take into account the inherent variability of the experimental sampling times, we compute times near the theoretical sampling time :(4)In experiments, one naturally expects the error on the sampling times to be bounded, otherwise it would mean that successive samples could be retrieved in inverse time order. We also assume that between these bounds, the errors are equally distributed. Hence, the random numbers , are generated from an uniform distribution (using the Mersenne Twister algorithm [9]). Then, we can compute the value of the solution of equation (2) at each of these times. This amounts to choosing the i-th sampling time with equal probability in an interval centered on . It is worth noticing that a truncated normal distribution (such that inversion between two consecutive sampling times cannot happen) for the sampling time errors does not significantly change the impact of the sampling process in comparison with a uniform distribution.

We also reproduce the LH assay variance by applying a multiplicative noise on each sample. We compute:(5)where the random numbers , are also generated from an uniform distribution (Mersenne Twister algorithm). Compared to a normal distribution, this choice allows us to lower the extremal errors while enhancing the frequency of medium ranged errors in the synthetic LH measured levels. Except for large deviation-induced phenomena (that remain very rare), the choice of a normal distribution for the assay error does not impact much the pattern of synthetic LH time series, as shown in the section “Algorithm robustness to assay error” of the Appendix S1 (see Figure S4 and Table S1).

The output of the sampling process is the time series defined by the N couples , of times and corresponding measured LH levels. Figure 2 illustrates the construction of the i-th sample in a time series.

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Figure 2. Computation of a synthetic sample.

Solution LHp(t) of equation (2) (green curve), retained sampling time (blue dotted line), real sampling time (red dotted line) randomly chosen in (blue interval), exact value of LH level at time (green dotted line), retrieved LH level (red dotted line) randomly chosen in (green interval). The output of the i-th step of the sampling process is the couple (magenta disc) of time and corresponding LH level.

https://doi.org/10.1371/journal.pone.0039001.g002

In the context of an experiment, there may be some uncertainty on the exact sampling times (i.e. the precise times at which the samples are retrieved). On the contrary, in our model of the sampling process, we can retrieve the sequence of effective sampling times for a given synthetic experiment. Figure 3 allows us to visualize the sequence (red dotted lines) compared to the registered sampling time sequence (blue dotted lines). Here, we set the sampling period Ts to 10 min and the shift constant r to 1 min, hence:

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Figure 3. Effect of the variability of the sampling times upon the synthetic time series.

The , are the effective sampling times leading to the red-colored LH time series. The , are the expected sampling times leading to the blue-colored LH time series. The original, non-sampled time series corresponds to the green line. One can observe an instance of great discrepancy between the LH level measured at time , which corresponds to the very beginning of the ascending part of a pulse, and the LH level measured at time , which corresponds to the maximum of the same pulse.

https://doi.org/10.1371/journal.pone.0039001.g003

The maximal error is fixed to 15% of the sampling period, so that f = 1.5 min and, for each i from 1 to N, .

Moreover, Figure 3 compares the results obtained without any variability on the sampling times (blue time series) with those obtained with an error of (red time series). It illustrates that this variability can occasionally imply a great difference near a pulse maximum. The 6th blue sample, obtained at , corresponds to the theoretical pulse maximum around 2.3 ng/ml. Yet the 6th red sample, obtained one minute earlier due to the variability of the sampling time, corresponds to the preceding minimal LH level. Consequently, the local maximum of the blue time series, obtained with the 6th sample, is noticeably greater than the local maximum of the red time series, which is obtained with the 7th sample.

Definitions.

For sake of clarity, we specify a few notions and terms that will be used in what follows. The definitions are illustrated by Figure 4. For a theoretical pulse (i.e. a peak in the signal LHp(t) triggered by a spike in LH(t)), we define:

• the theoretical pulse amplitude as the maximal value hit during the event.
• the theoretical pulse time as the time at which the level hits the theoretical pulse amplitude.

1. For a pulse in a time series (corresponding to a theoretical pulse), we define:

• the pulse amplitude as the maximal sample obtained during the pulse event,
• the pulse occurrence as the sample time at which the time series hits the pulse amplitude.

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Figure 4. Definition of pulse properties in the theoretical case versus experimental case.

For a theoretical pulse (i.e. a local maximum in the LHp(t) signal triggered by a spike in LH(t), we call “pulse time” the time at which LHp(t) admits a local maximum and “theoretical pulse amplitude” the value of LHp at this time. In a time series (either obtained from simulation and synthetic sampling protocol or experimental data), we call “pulse occurrence”, the time at which the time series admits a local maximum and “pulse amplitude” the corresponding value. Both the time values and the amplitude values are different in the theoretical and the experimental cases.

https://doi.org/10.1371/journal.pone.0039001.g004

Synthetic LH time series obtained from constant spike amplitude and frequency.

We first examine the effects of parameters r, f and b in case of a constant spike amplitude and constant interspike interval for a same sampling period Ts = 10 min:

• Case A: r = 1 min, f = 0 min, b = 0;
• Case B: r = 4 min, f = 0 min, b = 0;
• Case C: r = 4 min, f = 1.5 min, b = 0;
• Case D: r = 4 min, f = 1.5 min, b = 10%.

The top panel of Figure 5 displays the solution LHp(t) (green curve) of equation (2), i.e. the theoretical continuously measured LH blood level. Each panel from A to D shows the time series (blue stars) obtained through the sampling protocol for the values of parameters r, f, b specified above.

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Figure 5. Effect of the sampling process upon a LH level signal with constant amplitude and pulse frequency.

In all panels, . Top panel: theoretical continuously measured LH blood level (green curve). Panels A, B, C, D: sampling points (blue stars) of the time series obtained from the top panel signal through the sampling protocol. Panel A: first sampling time at r = 1 min, without any variability in the sampling process. Panel B: first sampling time at r = 4 min, without any variability in the sampling process. Panel C: first sampling time at r = 4 min, with variability in the sampling times . Panel D: first sampling time at r = 4 min, with variability both in the sampling times and the assays . The histograms correspond to the distribution of the levels at the basal line and the distribution of the amplitudes of the LH pulses, measured from the four cases A to D. The A and B time series, that only differ in the first sampling time, display constant (yet different) pulse amplitude. Red bars stand for case A (r = 1 min) value of the pulse amplitude (2.425 ng/ml) and level at the basal line (0.107 ng/ml). Green bars stand for case B (r = 4 min) value of the pulse amplitude (2.188 ng/ml) and level at the basal line (0.096 ng/ml). Blue bars stand for distributions of levels at the basal line and pulse amplitudes in case C (r = 4 min; f = 1.5 min) and case D (r = 4 min; f = 1.5 min; b = 10%, i.e. a variability of in the LH assays). In case D, the distributions of basal line levels (between 0.082 and 0.108 ng/ml) and pulse amplitudes (between 1.940 and 2.486 ng/ml) are wider than in case C (levels at the basal line between 0.092 and 0.101 ng/ml; pulse amplitude between 2.092 and 2.266 ng/ml), due to combined variabilities in the sampling times and assays.

https://doi.org/10.1371/journal.pone.0039001.g005

The histograms display, for each time series obtained in case C or D, the distribution of LH pulse amplitude (blue bars in left panels), i.e. local maxima, and LH levels at the basal line (blue bars in right panels), i.e. local minima. For sake of comparison, the constant amplitudes obtained in cases A and B have been marked with red and green bars respectively.

In cases A and B (f =  b = 0), one obtains a strictly periodic pattern of sampled LH levels since is a multiple of Ts. However, depending on the beginning of the sampling process r (1 min in time series A and 4 min in time series B), the maximum sample value varies from 2.425 ng/ml in case A (red bars in the histograms of Figure 5) to 2.188 ng/ml in case B (green bars). This shows the importance of the phase between the pulsatile LH signal and the periodic sampling process in the resulting observed pulse amplitude. It is worth noticing that this phase cannot be controlled at all in experimental conditions since the delay elapsed from the last pulse time is not known at the beginning of the experimental sampling process. The dependence of the basal line on this phase is weaker but still exists: it varies from 0.107 ng/ml in time series A to 0.096 ng/ml in time series B.

By comparing time series B and C of Figure 5, we can observe the impact of the variability in the sampling times (f = 0 min in case B and f = 1.5 min in case C). With variable sampling times, the pulse amplitude along time series is also variable, although the original continuous signal LHp(t) is perfectly periodic. This variability is shown in the histogram corresponding to case C in Figure 5: the various LH pulse (resp. basal line) amplitudes obtained in case C (blue bars) are scattered around the constant case B pulse (resp. basal line) amplitudes (green bar).

The impact of the variability in the assays is illustrated by the enhanced dispersal of the LH pulse and basal line amplitudes obtained in case D (blue bars in the histograms corresponding to case D in Figure 5) for which b = 10% compared to the case C amplitudes (b = 0).

In any case of either shifted (case B) or noised (cases C and D) time series, the pulse amplitudes are undervalued with respect to the genuine amplitude (correctly assessed only in case A). It may nevertheless happen that the effective sampling time coincides with a (genuine) pulse time. In that case, the pulse amplitude can be overvalued if the sign of the assay error is positive (instance of the blue bar on the right of the red bar in the left panel of case D).

Synthetic LH time series obtained from time-varying spike amplitude and frequency.

We now further examine the effects of the sampling process upon theoretical continuously measured LH level with time-varying pulse amplitude and frequency:

• Case E: constant spike amplitude and decreasing interspike interval, respectively

• Case F: decreasing spike amplitude and decreasing interspike interval, respectively

The sampling period Ts is the same in both cases: 10 min.

Figure 6 gives some examples of time series obtained in cases E and F. In each case, the green curve (rows 1 and 3) represents the solution LHp(t) (theoretical continuously measured plasma LH level) and the blue stars (rows 3 and 4) represent the time series obtained through the synthetic sampling protocol.

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Figure 6. Effect of the sampling process upon a LH level signal with regular increasing sampling frequency.

Case E (left panels): the pulse amplitude remains almost constant and the basal line increases regularly. Case F (right panels): the pulse amplitude decreases regularly and the basal line decreases regularly. Panels on row 1 represent the fine step simulation of LH blood level. Histograms on row 2 display the distribution of the LH pulse amplitudes and the distribution of the levels at the basal line, measured from the two theoretical LH level signals shown in row 1. A zoom on the distribution of the pulse amplitudes is shown as an insert in case E. Panels on row 3 represent the time series (blue stars) along the theoretical continuously measured LH level (green curve). Panels on row 4 represent the resulting LH measured time series (measured LH levels versus sampling times linked with segments). In both cases E and F, the sampling period is Ts = 10 min. In case E, the initial sampling time occurs at the first minute of the simulation (r = 1 min), without any variability in the sampling times (f = 0 min) or the assays (b = 0%). In case F, the initial sampling time occurs at the fourth minute of the simulation (r = 4 min), with variability both in the sampling times (f = 2 min) and the assays (b = 5%). Histograms on row 5 display the distribution of the LH pulse amplitudes and the distribution of the LH levels at the basal line, measured from the time series shown in row 4. While the distributions are regular in the theoretical time series, they become completely irregular in the sampled time series. As a result, the range of amplitudes is shortened. Regarding the distribution of the levels at the basal line, it is worth noticing that the measured values (E: between 0.125 and 0.519 ng/ml; F: between 0.094 and 0.258 ng/ml) are greater than the theoretical values (E: between 0.098 and 0.434 ng/ml; F: between 0.075 and 0.171 ng/ml). On the contrary, in case E, it is worth noticing that the theoretical pulse amplitudes vary from 2.379 to 2.425 ng/ml whereas measured pulse amplitudes vary from 1.447 to 2.395 ng/ml. In case F, all measured pulse amplitudes (between 0.302 and 1.959 ng/ml) are lower than the corresponding theoretical values (between 0.353 and 2.315 ng/ml).

https://doi.org/10.1371/journal.pone.0039001.g006

The histograms of Figure 6 detail, for cases E and F, the distribution of LH pulse amplitudes and successive LH levels at the basal line both for the theoretical continuously measured LH blood level (row 2, green bars) and the time series obtained through the sampling protocol (row 5, blue bars).

In case E, the spike frequency increases from 1 spike per 100 min to 1 spike per 50 min. Hence, the spike release arises more and more often along time, so that the LH blood pulses are successively triggered from a higher and higher basal level. Consequently, the basal line and, in a lesser extent, the pulse amplitude of the theoretical LH level undergo a small and smooth increase. Moreover, the number of samples per pulse decreases drastically as the pulse frequency increases, so that the LH time series looks noisier at the end than at the beginning of the time series (see case E, row 4 of Figure 6). This effect implies that the pulse amplitudes are spread out by the sampling protocol much stronger in case E (from 2.395 to 1.447 ng/ml) than in cases C and D (see case E, row 5 of Figure 6). Hence the time variations in the pulse frequency enhances the variability brought about by the sampling process.

Case F represents a situation that is naturally encountered in the physiological dynamics of LH secretion: the same increase in the spike frequency as in case E, combined with a decrease in the amplitude. As in the preceding cases, each of the LH pulse amplitude in the time series is smaller than the corresponding one in the theoretical LH level. Additionally in case F, the amplitude at the basal line is sensibly raised up by the sampling protocol. Hence, the difference between the pulse amplitude and the basal level is strongly deprecated, which adds to the noisy character of the ending of the time series.

Notations and definitions.

We consider a time series of N points obtained with a Ts-periodic sampling process and we note the sampling times:Hence, the k-th sampling time is (in particular, the first sampling time is ). We note either or the value corresponding to the k-th sample and call k a “time index”. For sake of simplicity in the following explanations, we will refer either to the time index k or the corresponding time .

The pulse detection algorithm is based on a sequence of different processes. Some of them consist in researching high amplitude peaks that can potentially be classified as pulses, others aim to remove, among the formerly selected peaks, those that do not fit other properties met by genuine pulses. In the following, we note the vector storing dynamically the time indexes at which the algorithm detects the summit of a potential pulse. s(P) is the size (number of elements) of P. The algorithm modifies vector P in such a way that indexes are always sorted in increasing order. Hence, at any time along the algorithmic process, s(P) pulses are detected, is the occurrence of the i-th detected pulse and is the amplitude of the i-th detected pulse.

In order both to moderate the importance of the amplitude and to account for several characteristics of the pulse shape, we need to introduce the notions of “height” and “magnitude” of a peak as well as that of “relative magnitude” between two peaks. Let us suppose that the i-th sample of the time series (occurring at time ) corresponds to a peak (i.e. and ). The height of this peak is defined as the difference between its amplitude and the lowest value of within the sampled time series, denoted by , i.e.:Let us assume additionally that a set of potential pulses P is identified from the time-series and the closest pulses registered in P before and after occur at and respectively (see Figure S1). We define the two minimal values of the time series between and on one hand, and between and on the other hand:

Then, we define the magnitude of the peak corresponding to the i-th sample as the geometric mean of and , i.e. .

Finally, let us consider two peaks occurring at and with . Let us call the minimal value of the time series between and :

We call “relative magnitude” between the two peaks the geometric mean between and .

The notion of peak height does not depend on the vector of potential pulses. It represents a normalization of the amplitude among the time-series with respect to the lowest value of the time serie. The peak magnitude can change as the vector P of identified pulses evolves and, consequently, will be used to compare a peak with its direct neighbors. The interest of the magnitude is to take into account the local baseline, without being sensitive to differences in the baseline from one side or the other of a peak. The relative magnitude between two peaks gives a semi-local reference for the pulse magnitude. In particular, the magnitude of a pulse can be usefully compared to the relative magnitude between its direct neighbors (i.e. the pulses just before and after it).

Pulse selection process.

• Initialization: We first fill vector P by selecting time indexes corresponding to great height samples. Even if, at this stage, we intend to recover a large enough set of potential pulse indexes, the time intervals between pulses should be consistent with the maximal frequency. Accordingly, we introduce a parameter Tp, called the nominal period, defined as the smallest time duration in which, from one pulse occurrence, one expects the following one.

Starting from the time index of the maximal sample in [0, 2Tp], we locate the time index of the minimal sample in and then we retrieve the time index of the maximal sample in (see Figure S3). We iterate the process along the whole time series to obtain the initial guess of potential pulse indexes .

This method is a trade-off between selecting all the peak indexes in the time-series (with the drawback that there will be too many of them if the time series is noisy) and selecting only local maxima corresponding to great amplitudes (with the drawback that there will be too few of them if the time series is smooth and the pulse frequency is low).

• Remove too small peaks: Once vector P is initialized, some of the registered indexes may correspond to small sample values. As illustrated in the first section, the pulse amplitude is strongly altered by the sampling process and, consequently, it cannot be used as an infallible criterium to select the pulses. Hence, we use multi-scale criteria based on the notions of height and magnitude to determine which indexes corresponds to too small peaks and to remove them from vector P.

1. –. Global relative criterium: We aim to remove peaks whose height is small in comparison with the height of all detected pulses. We define the “median height” as the median of the detected pulse heights. For each pulse, if the ratio between its own height and the median height is less than a threshold parameter , it is removed from P.

We have chosen the median instead of the (arithmetic or geometric) mean since it is less sensitive to the presence of great height peaks.

1. –. Semi-local relative criterium: We compare the magnitude of each peak with the relative magnitude between the immediately preceding and following pulses. If this comparison is not conclusive with respect to the threshold introduced above, we remove the peak index from vector P.

1. It is worth noticing that the geometric mean used to define the magnitudes provides robustness (compared to arithmetic mean, for instance) to this criterium with respect to possible local variations of the base line.

1. –. Global absolute criterium: We compare the magnitude of each pulse to an absolute threshold . If the comparison is not conclusive, the corresponding time index is removed from vector P.

This criterium precludes non significant elevations in the baseline to be considered as potential pulses. Hence, parameter corresponds to the assay detection threshold.

At this level of the selection process, vector P only contains the time indexes corresponding to peaks with sufficiently great height and magnitude (with respect to threshold and ) to be classified as pulses. However, as the initial guess for P may have skipped some potential pulses, the next step of the selection process consist in retrieving the missed pulses.

• Retrieve missed pulses: Between each pair of successive pulses registered in P, we examine each peak. If this peak fulfills the semi-local relative criterium, the corresponding time index is added to vector P.

By construction, such a retrieved peak almost automatically fulfills the other two global criteria.

• Shape-based criterium: The pulse duration has to be consistent with available knowledge on pulse half-life. For a fixed sampling rate, a detected pulse should extend over a minimum number of consecutive experimental data. Consequently, we intend to remove what we call “3-point peaks” for which the immediately preceding and following samples are local minima of the time series (see top panel of Figure S2). However, due to possible noise in the time-series, a pulse may appear as a 3-point peak. But, in this case, the pulse is expected to be not “too sharp” (see bottom panel of Figure S2). Thus we remove from P the time indexes corresponding to peaks with a “sharpness coefficient” greater than a chosen threshold .

The precise definition of “sharpness coefficient” will be detailed in Step 5 of the algorithm (see next subsection “Algorithmic pulse detection”). We only enlighten here that this criterium is based on the asymmetric shape of a pulse and allows one to get rid of the genuine peaks produced by occasional experimental errors.

Vector of InterPulse Interval (IPI) and IPI tunnel.

At a given step, we define the vector of InterPulse Intervals (IPI) from the current P vector by:As we aim to apply the algorithm mainly to hormonal time series, we designed a process to take into account some degree of regularity in the rate of change in the pulse frequency. Hence, given a vector of identified potential pulses, we introduce a cubic function fitted to in a least squares sense. Function gives an averaged, yet time evolving representation of the IPI built from the global sequence .

Then, we build a tunnel in delimited by the graphs of two piecewise linear functions of time and built from function . Precisely, for each pulse time , we define and to draw the lower and upper boundaries of the tunnel. Thus, parameters and tune the width of the so-called “IPI tunnel”; they represent a quantification of the pulse frequency regularity. The tunnel allows one to assess the regularity of the IPI time variations and can help the user to (i) classify a specific pulse as a potential outlier with respect to the pulse frequency properties of the series, (ii) identify and localize a possible rupture in the secretion rhythm.

To illustrate the use of the IPI tunnel, let us consider a time series for which the sequence of pulse indexes is easy to find by sight. We assume that the time series displays regular pulsatility, i.e. the pulse frequency undergoes smooth variations along time. Under this condition, function is a good approximation of the IPI sequence.

Let us first consider the case of a lack of detection: let P be the vector formed by the pulse indexes in Q except one (the ). In the course of the automatic pulse detection, this case may happen if the maximum amplitude corresponding to this pulse is low. Then, the corresponding IPI sequence is given by:(6)Under the regularity assumption, each IPI should be close to (or even in) the range delimited by the values of its neighbors and . On the contrary, in the case of vector P, the IPI, i.e. , is noticeably greater than the maximum of its neighbors. More precisely, its value is twice the mean of its neighbors and, consequently, is approximately twice the value.

Now, let us consider the case of an over-detection: let P be the vector formed by the pulse indexes in Q plus an extra pulse occurring at time lying between the and the pulse times stored in vector Q. Hence, . Then:(7)and moreover:

Under the regularity assumption, either or is too small compared to the expected IPI range delimited by and . In the less discriminating case, the extra pulse lies close to the middle of its neighbors . Then, and are almost equal to the half of the expected IPI . Hence, even if several combinations of and values exist, depending on the position of time k compared to and , one of the IPIs is always less than the half of the value, which indicates a potential over-detection.

An appropriate choice of the values of and allows one to discriminate the pulses that break the frequency regularity and indicate a rupture in the secretion rhythm.

The different steps of the algorithm are described as pseudo-code in the “Algorithm Description” of the Appendix S1. We have implemented the algorithm in the Scilab environment (http://www.scilab.org/) dedicated to numerical computing.

Nominal period Tp.

The value of the nominal period is chosen so as to favor the maximal number of correct detections, especially at the beginning of time series. According to the existence of high frequency series with short IPIs, there is a risk to detect two consecutive pulses in the same window if it is too large. The value of parameter Tp has been set to 40 min, which is a value close to the minimal observed period. The number of missed pulses is equal to 0 for Tp ranging from 40 to 70 min; it increases up to 2 for Tp = 80 min. Even in that latest case, the number of missed pulses is small enough to guarantee the algorithm robustness with respect to this parameter.

Relative magnitude threshold.

Compared to the performances of the algorithm with the default value (0.2), a decrease down to 0.1 or an increase up to 0.3 increase the total number of outliers (either false detections or missed detections). Moreover, this parameter can be adapted in order to correct outliers lying outside the tunnel, as previously seen.

3-point peak threshold.

The 3-point peak threshold is introduced to prevent non-asymmetric pulses from being detected. Figure S2 illustrates the identification of a 3-point peak pattern. is the ratio between the arithmetic mean of the amplitude of the neighboring points of rank 2 (green lines) and the geometric mean of the amplitude of the immediate neighbors (red lines). For instance, based on that criterion, the peak selected on panel A (dashed vertical line), is identified as a genuine 3-point peak. On the contrary, the peak selected on Figure S2, panel B (solid vertical line) is not identified as a 3-point peak, since it belongs to a genuine, asymmetric LH pulse with an exponential decrease, albeit locally noised (the LH level corresponding to the latter neighbor of rank 2 is a little higher than expected for a smooth exponential decrease). Over 15 analyzed potential 3-point peaks, the minimal value corresponding to a genuine 3-point peak was equal to 0.15, while the maximal value corresponding to a locally noised, genuine asymmetric LH pulse was equal to 0.07, so that there was no overlapping. Consistently, we chose a default set value (0.1) lying within the [0.07,0.15] range. This parameter is embedded within the algorithm, since there is no reason for the user to modify it. Indeed, the 3-point peaks rely on endocrinological considerations and take into account, either directly or indirectly, the typical duration of a LH pulse (around 30 min) and its asymmetric shape, that can be reconstructed from time series sampled at least every 10 min.

Lower and upper bounds of the IPI tunnel.

For the lower bound , the 0.6 default value does not lead to any IPI outlier, whereas a value of 0.5 leads to classify two genuine pulses as over-detected pulses, and a value of 0.4 leads to classify twelve genuine pulses as over-detected pulses (among more than 300 pulses). For the upper bound , the 0.6 default value allows one to identify every genuine outliers, without generating false outliers, but changing the value to 0.5 led to false additional under-detected pulses. However, the use of the tunnel bounds is mainly to draw the user’s attention to possible events of interest, and there is no direct sensitivity of the algorithm to their precise values.