Generative models implemented in osl-dynamics.

A) Hidden Markov Model (HMM) [24, 25]. Here, data is generated using a hidden state (θt) and observation model, which in our case is a multivariate normal distribution parameterised by a state mean (µk) and covariance (Dk). Only one state can be active at a given time point. Dynamics are modelled via state switching using a transition probability matrix (Aij), which forecasts the probability of the current state based on the previous state. B) Dynamic Network Modes (DyNeMo) [26]. Here, the data is generated using a linear combination of modes (µj and Dj) and dynamics are modelled using a recurrent neural network (RNN: f and g), which forecasts the probability of a particular mixing ratio (αt) based on a long history of previous values via the underlying logits (θt).

Preprocessing and source reconstruction.

First, the sensor-level recordings are cleaned using standard signal processing techniques. This includes filtering, downsampling and artefact removal. Following this, the sensor-level recordings are used to estimate source activity using a beamformer. Finally, we parcellate the data and perform corrections (orthogonalisation and dipole sign flipping). Acronyms: electrocardiogram (ECG), electrooculogram (EOG), independent component analysis (ICA), Linearly Constrained Minimum Variance (LCMV). These steps can be performed with the OHBA Software Library: https://github.com/OHBA-analysis/osl.

Methods for preparing training data.

A.I) Original (simulated) time series data. Only a short segment (0.2 s) is shown. Channel 1 (2) is a modulated sine wave at 15 Hz (30 Hz) with N (0, 0.1) noise added. A.II) Covariance of the original data. B) Amplitude Envelope (AE) data (solid red line) and original data (dashed blue line). C.I) Time-Delay Embedded (TDE) time series. An embedding window of ±7 lags was used. C.II) Covariance of TDE data. C.III) Spectral properties of the original data estimated using the covariance matrix of TDE data. Acronyms: Autocorrelation Function (ACF), Power Spectral Density (PSD).

TDE-HMM burst detection pipeline.

This is run on a single region’s parcel time course. Separate HMMs are trained for each region. A) Source reconstructed data is prepared by performing time-delay embedding and standardisation (z-transform). Following this an HMM is trained on the data and statistics that summarise the bursts are calculated from the inferred state time course. B) Subject-specific metrics summarising the bursts at a particular region are used in group-level analysis.

Dynamic functional network analysis pipeline.

A) First-level modelling. This includes data preparation (shown in the blue boxes), model training and post-hoc analysis (shown in the red boxes). The first-level modelling is used to derive subject-specific quantities. B) Group-level modelling. This involves using the subject-specific description from the first-level modelling to model a group.

Static functional network analysis pipeline.

A) The source reconstructed data is used to calculate metrics that describe networks. B) The subject-specific metrics are used to model a group. Acronyms: amplitude envelope correlation (AEC), power spectral density (PSD).

Burst detection: single region source reconstructed MEG data (left motor cortex) shows short-lived bursts of oscillatory activity.

A.I) Dynamic spectral properties of the first 20 s of the time series from the first subject. A.II) Amplitude envelope calculated after bandpass filtering the time series over the,β-band (top), α-band (middle) and 5/θ-band (bottom). B.I) The inferred state probability time course for the first 20 s of the first subject. B.II) The PSD of each state. B.III) Pearson correlation of each state probability time course with the amplitude envelopes for different frequency bands. B.IV) Distribution over subjects for summary statistics characterising the bursts. Note, no additional bandpass filtering was done to the source data when calculating the mean amplitude. The script used to generate the results in this figure is here: https://github.com/OHBA-analysis/osl-dynamics/blob/main/examples/toolbox_paper/ctf_rest/tde_hmm_bursts.py.

Dynamic network detection: a multi-region AE-HMM trained on the Elekta task MEG dataset reveals functional networks with fast dynamics that are related to the task.

A.I) For each state, group-averaged amplitude maps relative to the mean across states (top) and absolute amplitude envelope correlation networks (bottom). A.II) State probability time course for the first 8 seconds of the first subject. A.III) Distribution over subjects for the summary statistics for each state. B.I) State time courses (Viterbi path) epoched around the presentation of visual stimuli. The horizontal bars indicate time points with p-value < 0.05. The maximum statistic pooling over states and time points was used in permutation testing to control for the family-wise error rate. The script used to generate the results in this figure is here: https://github.com/OHBA-analysis/osl-dynamics/blob/main/examples/toolbox_paper/elekta_task/tde_hmm.py.

Dynamic network detection: a multi-region TDE-HMM trained on the Elekta task MEG dataset reveals spectrally distinct functional networks with fast dynamics.

A.I) For each state, group-averaged power maps relative to the mean across states (top), coherence networks (middle) and PSD averaged over regions (bottom), both the statespecific (coloured solid line) and static PSD (i.e. the average across states, dashed black line) are shown. A.II) State probability time course for the first 8 seconds of the first subject. A.III) Distribution over subjects for the summary statistics for each state. B.I) State time courses (Viterbi path) epoched around the presentation of visual stimuli. The horizontal bars indicate time points with p-value < 0.05. The maximum statistic pooling over states and time points was used in permutation testing to control for the family-wise error rate. The script used to generate the results in this figure is here: https://github.com/OHBA-analysis/osl-dynamics/blob/main/examples/toolbox_paper/elekta_task/tde_hmm.py.

Dynamic network detection: a multi-region TDE-HMM trained on the CTF rest MEG dataset identifies the same functional networks to those found with the Elekta task MEG dataset and reveals differences in the dynamics for young vs old groups.

A.I) For each state, group-averaged power maps relative to mean across states (top), absolute coherence networks (middle) and PSD averaged over regions (bottom), both the state-specific (coloured solid line) and static PSD (i.e. the average across states, dashed black line) are shown. A.II) State probability time course for the first 8 seconds of the first subject and run. A.III) Distribution over subjects for the summary statistics of each state. B.I) Comparison of the summary statistics for a young (18-34 years old) and old (34-60 years old) group. The star indicates a p-value < 0.05. The maximum statistic pooling over states and metrics was used in permutation testing to control for the family-wise error rate. The script used to generate the results in this figure is here: https://github.com/OHBA-analysis/osl-dynamics/blob/main/examples/toolbox_paper/ctf_rest/tde_hmm_networks.py.

Dynamic network detection: a multi-region TDE-DyNeMo trained on the CTF rest MEG dataset reveals spectrally distinct modes that are more localised than HMM states and overlap in time.

A.I) For each mode, group-averaged power maps relative to the mean across modes (top), absolute coherence networks (middle) and PSD averaged over regions (bottom), both the mode-specific (coloured solid line) and static PSD (i.e. the average across modes, dashed black line) are shown. A.II) Mode time course (mixing co-efficients) renormalised using the trace of the mode covariances. A.III) Pearson correlation between renormalised mode time courses calculated by concatenating the time series from eachsubject. A.IV) Distribution over subjects for summary statistics (mean and standard deviation) of the renormalised mode time courses. B.I) Comparison of the summary statistics for a young (18-34 years old) and old (34-60 years old) group. The maximum statistic pooling over modes and metrics was used in permutation testing to control for the family-wise error rate. The script used to generate the results in this figure is here: https://github.com/OHBA-analysis/osl-dynamics/blob/main/examples/toolbox_paper/ctf_rest/tde_dynemo_networks.py.

Static network detection: osl-dynamics can also be used to perform static network analyses (including functional connectivity).

In the CTF rest MEG dataset, this reveals frequency-specific differences in the static functional networks of young (18-34 years old) and old (34-60 years old) participants. Group-average PSD (averaged over subjects and parcels, A.I) power maps (A.II) coherence networks (A.III) and AEC networks (A.IV) for the canonical frequency bands (δ, θ, α, β. B.I) Power difference for old minus young (top) and p-values (bottom). Only frequency bands with at least one parcel with a p-value < 0.05 are shown, the rest are marked with n.s. (none significant). B.II) AEC difference for old minus young only showing edges with a p-value < 0.05. The maximum statistic pooling over frequency bands and parcels/edges was used in permutation testing to control for the family-wise error rate. The script used to generate the results in this figure is here: https://github.com/OHBA-analysis/osl-dynamics/blob/main/examples/toolbox_paper/ctf_rest/static_networks.py.

Reproducibility of the CTF rest MEG dataset single-region TDE-HMM analysis.

A) Variational free energy for 3 sets of 10 runs. B) Summary statistics for the best run from each set (indicated by the red dot in A).

Reproducibility of the Elekta task MEG dataset multi-region AE-HMM analysis. A) Variational free energy for 3 sets of 10 runs.

B) Summary statistics for the best run from each set (indicated by the red dot in A). C) Amplitude maps relative to the mean across states for the best run from sets 2 and 3. D) Network response to the visual task for sets 2 and 3. The horizontal bars indicate time points with p-values < 0.05. The maximum statistic pooling over states and time was used in permutation testing to control for the family-wise error rate.

Reproducibility of the Elekta task MEG dataset multi-region TDE-HMM analysis.

A) Variational free energy for 3 sets of 10 runs. B) Summary statistics for the best run from each set (indicated by the red dot in A). C) Power maps relative to the mean across states for the best run from sets 2 and 3. D) Network response to the visual task for sets 2 and 3. The horizontal bars indicate time points with p-values < 0.05. The maximum statistic pooling over states and time was used in permutation testing to control for the family-wise error rate.

Reproducibility of the CTF rest MEG dataset multi-region TDE-HMM analysis.

A) Variational free energy for 3 sets of 10 runs. B) Summary statistics for the best run from each set (indicated by the red dot in A). C) Power maps relative to the mean across states for the best run from sets 2 and 3. D) Comparison of the summary statistics for a young (18-34 years old) and old (34-60 years old) group. The star indicates a p-values < 0.05. The maximum statistic pooling over states and metrics was used to control for the family-wise error rate.

Reproducibility of the CTF rest MEG dataset multi-region TDE-DyNeMo analysis.

A) Variational free energy for 3 sets of 10 runs. B) Summary statistics for the best run from each set (indicated by the red dot in A). C) Power maps relative to the mean across modes for the best run from sets 2 and 3. D) Comparison of the summary statistics for a young (18-34 years old) and old (34-60 years old) group. The double star indicates a p-values < 0.01. The maximum statistic pooling over modes and metrics was used to control for the family-wise error rate.