Fig 1.
Introducing average burst duration profiles.
Average burst duration profiles are obtained by computing beta envelope average burst duration for a range of thresholds. An example is provided for three thresholds, where thick lines highlight the duration of individual bursts for the three thresholds in panel A, and the corresponding averages are identified with the same colour in panel B. Considering the time discretization of simple envelope models of the form dx = μ(x)dt + ζdW, where μ is the drift function, W is a Wiener process, and ζ is a constant noise parameter, we illustrate with two example drift functions the link between envelope dynamics (panels C1 and C2, one-dimensional vector field also sketched with blue arrows) and average burst duration profiles (panels E1 and E2). The envelope models produce the black envelopes in panels D1 and D2, and beta oscillations (shown in grey in panels D1 and D2) can be obtained by adding a constant frequency phase equation. In C1, when x moves away from the fixed point, it will be strongly attracted back. By contrast, in the case of C2, if x is around 1, there is weak attraction towards the fixed point, allowing x to stay at an elevated level for longer.
Fig 2.
Average burst duration profiles can have complex shapes.
Thick lines highlight the duration of individual bursts for two thresholds in the left panel. The corresponding averages are identified with the same colour in the average burst duration profile in the right panel. Longer bursts are still present at the 80th percentile, and shorter bursts are significantly more frequent at the 70th percentile than at the 80th percentile. In addition, high amplitude, longer bursts are sharp (quick rise and fall), and therefore their duration is only shortened slightly from the 70th to the 80th percentile. As a result, the average burst duration profile is non-monotonic.
Fig 3.
Power spectra and bursting features ON and OFF Levodopa (right hemispheres).
Each column corresponds to the right hemisphere of one of the eight patients. Each row corresponds to a feature, the ON state is in blue, and the OFF state in red. The first row shows power spectra, the second row average burst duration profiles, the third row average burst amplitude profiles, and last row envelope amplitude PDFs. Statistically significant differences under FDR control are indicated by black stars (three bursting features only). Error bars represent the SEM.
Fig 4.
Average burst duration profiles ON and OFF medication for data and GWR surrogates at ρ = 0 (right hemispheres).
In all panels, data profiles are solid lines, while linear surrogate profiles are dashed lines. The OFF medication state is indicated in red, and the ON state in blue.
Fig 5.
Sketch of burst duration metrics.
A: illustration of burst duration distance to linear surrogates in the OFF state (BDDLS OFF). BDDLS OFF is defined as the sum of squared differences between data and linear surrogate average burst duration profiles for the OFF condition divided by the square of a scale. The scale is taken as the mean value of the OFF linear surrogate average burst duration profile. BDDLS ON is defined in a similar way, and BDDLSdiff is BDDLS OFF medication minus BDDLS ON medication. B: DURdiff is defined as the sum of the differences across thresholds between burst duration profiles OFF and ON medication. In this figure, summation is indicated by the symbol +, division by the symbol /, and squaring by the symbol 2.
Table 1.
Statistical significance of medication state effect on BDDLS.
Showing p-values for the test that BDDLS is greater OFF than ON medication (sign rank test, all patients, both hemispheres, n = 16 per condition) as a function of the GWR surrogate parameter ρ. P-values in bold are smaller than 5%, while green indicates significance under FDR control.
Table 2.
Spearman’s correlations between BDDLSdiff and UPDRS score OFF medication.
Values are presented as a function of the GWR surrogate level ρ. P-values in bold are smaller than 5%, while green indicates significance under FDR control. Predictors and hemibody UPDRS OFF are averaged across sides (n = 8, d.f. = 6).
Fig 6.
Mapping of the Wilson-Cowan model onto the STN-GPe loop.
The excitatory population E and the inhibitory population I model the basal ganglia STN and GPe, respectively. Arrows denote excitatory connections or inputs, whereas circles denote inhibitory connections or inputs. The DBS electrode is implanted in the STN (indicated by a dashed light blue line) and records the STN LFP. The STN also receives an excitatory input from the cortex, while the GPe receives an inhibitory input from the striatum, and also has a self-inhibitory loop.
Fig 7.
Showing best fits to datasets 6R (ON: first row, OFF: second row), and 4L (ON: third row, OFF: fourth row). The first column shows twenty seconds of filtered LFP recording (A panels), while the same duration of model oscillatory activity output is plotted in the second column (B panels). Data and model PSDs are compared in the third column (C panels), and data and model average burst duration profiles are shown in the last column (D panels, SEM error bars). In the first and third rows, all model outputs correspond to fits of the linear WC model. In the second and fourth rows, dark red solid lines correspond to fits of the delayed non-linear WC model, and dark red dashed lines to fits of the linear WC model (shown for comparison, D panels only).
Fig 8.
Envelope model fits to all datasets.
The top two rows correspond to left hemispheres, the bottom two rows to right hemispheres, and each column corresponds to one patient. Patient’s average burst duration profiles and corresponding model fits are shown in the first and third rows (SEM error bars). The OFF state is represented in red for data, and in dark red for model fits. The ON state is represented in blue for data, and in dark blue for model fits. The drift functions (μ) of the fitted envelope models are shown in the second and fourth rows. The range of x values shown for each drift function corresponds to the range spanned by the corresponding model, and the light grey line represents μ = 0.
Fig 9.
First exit time and overshoot distribution.
Showing a discrete realisation of an OU process overshooting the threshold L by δ, with a sketch of the overshoot probability density at L in purple. The first exit time from x0 to L is the time taken to get below L for the first time when starting from x0.
Fig 10.
Simulations of average burst duration profiles for OU processes and a third degree polynomial model.
Average burst duration profiles from simulations of OU processes are compared to Eq (11) for a range of decay parameters and ζ = 1 in panel A. Similarly, average burst duration profiles from simulations of a third degree polynomial envelope model are compared to Eq (48) in panel B. Simulations are indicated by dashed lines (SEM error bars), and analytical results by dotted lines. Simulations consist of five repeats of 105 s, with a time step of 1 ms (OU process simulations use the exact updating Eq (17)).
Fig 11.
Inferring envelope dynamics with the passage method.
The method is applied to synthetic data (OU model in the first row, fifth degree polynomial model in the second row), and patient data (patient 6R, ON medication in the third row, and patient 6R, OFF medication in the last row). The recovered drift functions (μ) are shown in the first column, and the average burst duration profiles and the inverse CDFs simulated from the recovered dynamics are shown in the second and third columns, respectively. Ground truths are provided when available (black dashed lines). Synthetic data results are presented for 250 s and 1000 s of synthetic data. In the last column, 5 s of simulated data from the recovered dynamics are compared to training data.
Fig 12.
Comparison of the passage method and the direct method on synthetic data.
Showing the sum of squared errors between the inferred and ground truth μ functions for both methods as a function of the duration of synthetic data used for inference. The synthetic data were generated from an OU process (left) and a fifth degree polynomial (right). In both cases, the mean value of the sum of squared errors and the SEM error bars are obtained for a large number of repeats (150 repeats from 100 s to 250 s, 100 repeats from 350 s to 450 s, and 50 repeats from 500 to 1000 s). Significant differences are highlighted by black stars (t-tests under FDR control, same sample sizes as for error bars).
Fig 13.
Filtered LFP for patient 6L OFF (black), and corresponding GWR surrogates for a range of ρ levels (blue).
Most of the data temporal variability is already accounted for at ρ = 0.1. The plots share the same time axis.