Fig 1.
Schematic overview of neuro-behavioral recording applications.
(A) Use case: Systems for continuous neuro-behavioral recordings. Recording, energy, storage, connectivity, and usability demands placed on ambulatory brain recording systems can result in data loss. (B) Application: Cyclical neuro-behavioral signals–Inter-ictal epileptiform activity. (i) Representative example of inter-ictal epileptiform discharges (IEDs) recorded from intracranial electroencephalography (iEEG). (ii) Example of IED rates (IED per hour) recorded from a hippocampal iEEG electrode showing circadian changes in IED rate with increased IED rates during sleep. The period from 10 PM to 8 AM is noted in gray. (iii) Example of multiday IED rate recording showing multiday cycles. (iv) Random subsampling of the signal in iii to show how sparse sampling and data drops make it challenging to discern the underlying cycles of IED rate.
Fig 2.
Description of method objectives and signal assumptions.
(A) The core assumption of the model is that the underlying signal (ii) is the linear sum of (i) oscillations, a polynomial trend, and noise. Part (B) describes the overall workflow including (i) data input to the model, (ii) outputs, and (iii) estimated signal reconstructions. (Ci) Equation representing the core signal assumption that the observations come from a combination of oscillations, a polynomial trend, and noise or error. Notation includes y (m x 1 vector of observed data), Ψ (n x n discrete cosine transform (DCT) basis), Φ (m x n binary row subsampling matrix), x (n x 1 DCT coefficients), Τ (n x p Vandermonde matrix), z (p x 1 polynomial coefficients), (m x 1 error terms). (Cii) Expression for basis pursuit denoising containing x and z as unknowns, yielding a 2D minimization problem that is reduced to a 1D minimization problem (Ciii) by variable projection. (D) Schematic representation of the equations and sampling approach in (C).
Fig 3.
Basis Pursuit with Polynomial Detrending (BPWP) outputs for simulated inter-ictal epileptiform discharge (IED) timeseries.
Raw data show hourly rate of IEDs, updated every 20 minutes, from a simulated signal containing oscillations at 1, 7, 15, 21, 30, 50, and 100 days. Timeseries consists of over 20,000 samples. (B) Complex wavelet transform (CWT) spectrogram of the timeseries in A showing power in different cycles. Color bar indicates spectral power. (C) Results of ten-fold 75/25 cross-validation for δ parameter selection. Average mean square error with 95% confidence intervals for each δ value and sampling rate tested are shown. (D) Random samples from the raw data in A averaging at five samples per day. Timeseries consists of 1,800 total samples. (E) Underlying raw data are shown in gray. The estimated reconstruction of the underlying data based on the sparse samples in (D) is shown in orange. (F) The average CWT spectrum (average over time from the spectrogram in (B) is shown in gray). The method’s spectral output is shown in orange. Black stars denote significant peaks; peaks whose amplitude was above the 99th percentile of the distribution created by shuffling the input data and re-calculating the method 100 times. The insert shows the spectral outputs from the reshuffling in purple. The narrowband peaks from the method align with the central tendencies of the broadband CWT peaks.
Fig 4.
Basis Pursuit with Polynomial Detrending (BPWP) of real-world inter-ictal epileptiform discharges (IED) timeseries, participant 1.
(A) Raw data showing hourly rate of IEDs detected from the left hippocampus, updated every 20 minutes. Timeseries consists of over 20,000 samples. (B) Complex wavelet transform (CWT) spectrogram of the timeseries in A showing power in different cycles. Strong cycles are evident at one day and around on month. (C) Average of CWT spectrum (averaged over time from the spectrogram in) (B) shows cycles of IED rate including periods around one day, two-three weeks, one month, fifty days, and 100 days. (D) Random samples from the raw data in A averaging at 5 samples per day. Timeseries consists of 1,800 total samples. (E) Underlying raw data are shown in gray. The method’s estimated reconstruction of the underlying data based on the sparse samples in (D) is shown in orange. (F) The CWT spectrum for the raw data is shown in gray. The method’s spectral output is shown in orange. Black stars denote significant peaks; peaks whose amplitude was above the 99th percentile of the distribution created by shuffling the input data and re-calculating the method 100 times. The insert shows the spectral outputs from the reshuffling in purple. The amplitude of this noise floor is an order of magnitude smaller than the spectral output from the correctly ordered input data. The narrowband peaks from the method align with the central tendencies of the broadband CWT peaks. (G) The method’s estimated reconstruction of the underlying signal using only significant peaks from F with a period longer than two days is in orange. Overlayed black circles denote when seizures occurred. Seizures appear to prefer the peaks of the combined slow cycles derived from the method. (H) The method-based reconstruction was filtered in cycle ranges around one day, one week, two to three weeks, and one month then the signal Hilbert transform was used to identify the phase at which seizures occurred for each of these cycles. Polar histograms denoting the phase at which seizures occurred for each of these cycles indicate a cycle-specific phase preference for seizures. Stars denote p < 0.001 on the Omnibus test for uniformity, indicating that seizure phase is not uniformly distributed.
Fig 5.
Impact of varying sample density on Basis Pursuit with Polynomial Detrending (BPWP) outputs, participant 1.
(A) Raw data showing hourly rate of inter-ictal epileptiform discharges (IED) detected from the left hippocampus, updated every 20 minutes. Timeseries consists of over 20,000 samples. (Bi) Raw data are in gray and one random sample per day is in orange. (Bii) Raw data are in gray and the reconstructed signal using BPWP outputs based on input data of one random sample per day is in orange. (C) Average complex wavelet transform (CWT) spectrum from the raw data in (A) is in gray. The BPWP spectrum based on one sample per day input is shown in orange. Black stars denote significant peaks; peaks whose amplitude was above the 99th percentile of the distribution created by shuffling the input data and re-calculating the method 100 times. Random samples, signal reconstructions, and BPWP spectra are shown again for sampling rates of three and five per day in (D) and (E) and in (F) and (G) respectively. Agreement between BPWP output and raw data and CWT spectra improves as the signal is sampled more densely. Part (H) shows agreement between the BPWP and CWT spectra as a function of frequency of random sampling. For each sampling frequency, the raw data were resampled and BPWP was re-calculated 10 times. For each peak in the CWT spectrum, the offset between the period of the CWT peak and the nearest BPWP peak was calculated in terms of days and divided by the period of the CWT peak. This offset-to-cycle-length ratio was averaged across the 10 iterations and plotted as a log value on the y axis. The associated frequency of random sampling was plotted on the x axis. Shaded areas denote 95% confidence intervals. The offset ratio decreases and stabilizes as sampling density increases.
Fig 6.
Impact of data drops on Basis Pursuit with Polynomial Detrending (BPWP) outputs, participant 1.
(A) Raw data showing hourly rate of inter-ictal epileptiform discharges (IED) detected from the left hippocampus, updated every 20 minutes. Timeseries consists of over 20,000 samples. (Bi) Raw data are in gray and random sampling excluding 12-day data drops are in orange. (Bii) Raw data are in gray and the reconstructed signal using model output based on input data with 12-day data drops is in orange. (C) Average complex wavelet transform (CWT) spectrum from the raw data in (A) is in gray. The BPWP spectral output based on the sampling in Bi input is shown in orange. Black stars denote significant peaks; peaks whose amplitude was above the 99th percentile of the distribution created by shuffling the input data and re-calculating the method 100 times. Data drops of thirty- and sixty- days duration, signal reconstructions, and method spectra are shown in (D) and (E) and in (F) and (G) respectively. The total number of samples for BPWP is fixed across the conditions at n = 1307 which is approximately 4 samples per day assuming no drops.
Fig 7.
Basis Pursuit with Polynomial Detrending (BPWP) method performance, participant 1.
(A) Diagram depicting the approach to comparing the original inter-ictal epileptiform discharge (IED) rate timeseries (black) with the BPWP model output (orange). Each timeseries was bandpass filtered and the two filtered timeseries were used to calculate the Pearson correlation coefficient. (B) The Y axis shows the Pearson correlation coefficient for each bandpass filter with central periods listed on the X axis. P value for each point depicted was < 0.001. (C) Examples of filter outputs for the approximate daily (top), 19-day (middle), and 100 day (bottom) cycles. The original signal is in black, the BPWP output is in orange. The insert describes the daily cycle filter outputs from the 5 to 30-day time points.