Modeling time-varying brain networks with a self-tuning optimized Kalman filter

Abstract

Brain networks are complex dynamical systems in which directed interactions between different areas evolve at the sub-second scale of sensory, cognitive and motor processes. Due to the highly non-stationary nature of neural signals and their unknown noise components, however, modeling dynamic brain networks has remained one of the major challenges in contemporary neuroscience. Here, we present a new algorithm based on an innovative formulation of the Kalman filter that is optimized for tracking rapidly evolving patterns of directed functional connectivity under unknown noise conditions. The Self-Tuning Optimized Kalman filter (STOK) is a novel adaptive filter that embeds a self-tuning memory decay and a recursive regularization to guarantee high network tracking accuracy, temporal precision and robustness to noise. To validate the proposed algorithm, we performed an extensive comparison against the classical Kalman filter, in both realistic surrogate networks and real electroencephalography (EEG) data. In both simulations and real data, we show that the STOK filter estimates time-frequency patterns of directed connectivity with significantly superior performance. The advantages of the STOK filter were even clearer in real EEG data, where the algorithm recovered latent structures of dynamic connectivity from epicranial EEG recordings in rats and human visual evoked potentials, in excellent agreement with known physiology. These results establish the STOK filter as a powerful tool for modeling dynamic network structures in biological systems, with the potential to yield new insights into the rapid evolution of network states from which brain functions emerge.

Fig 1. Performance of KF and STOK on a simple simulated bivariate AR(1) process.

(A) Performance of the KF filter at recovering the imposed structure of AR coefficients (top panel; black dashed lines) under two extreme values of the adaptation constant (c = 0.0001, c = 1), highlighting the drastic variability of the estimates as a function of c: poor tracking performance is observed at the lowest (green line) and spurious noisy fluctuations at the highest c (blue line). The optimal c that minimizes the root-mean squared error (RMSE, blue line), lies at a point where KF and STOK performance are highly correlated (bottom panel; correlation shown by the grey dashed line). (B) Performance of the STOK filter, showing the high tracking ability and robustness to noise due to the self-tuning memory decay (top panel; orange line) which automatically increases tracking speed at relevant transition points between AR coefficient states (bottom panel; grey rectangles).

Fig 2. Comparison of the KF and STOK filters under the realistic simulation framework.

(A) Method for evaluating the performance of KF and STOK against simulated data (ground truth). Ground truth PDC was binarized setting to 1 all connections larger than 0. Estimated Partial Directed Coherence (PDC) [41] was binarized using different criteria, based on the quantile discretization of the estimates (criterion q; top panel). Signal detection indexes were calculated for each criterion and the area under the curve (AUC) was used as performance measure. The color code of the dots in the ROC plot (bottom panel) reflects the different criteria and correspond to the colorbar for estimated PDC strength (top panel). (B) Comparison of KF, STOK without regularization and STOK as a function of different SNR, showing the overall larger AUC using STOK. Error bars reflect 95% confidence intervals. (C) AUC curves for KF and STOK as a function of linear mixing. (D-E) Performance of the two filters as a function of the duration (D) and size (E) of AR coefficients. (F) Performance of the two filters with increasing sample size: regularization favors strongest connections and sparse networks as the network size increases (right panel), reducing overall weakest connections (left panel). (G) Correlation matrices at varying model orders for KF and STOK (top two panels) and their difference (bottom panel, STOK minus KF). (H) Correlations extracted at specific orders (p ∈ [2,15], with ground-truth model order = 6) showing the higher consistency of models estimated as p changes with STOK compared with KF.

Fig 3. Results in benchmark rat EEG.

(A) Layout of the multi-electrode grid used for recordings with the electrode and label codes used for all the plots. (B) Grand-average somatosensory evoked potentials at electrodes contralateral to stimulation (n = 10) showing the sequence of maximum voltage peaks, starting at e4 and propagating to e2-6. The gray line shows the evolution of the self-tuning memory parameter of the STOK filter. (C) Summed outflow in the gamma range (40–90 Hz) from all electrodes at the sampling rate of 500 Hz, revealing higher temporal precision with STOK filtering. (D-E) Criterion I and II: STOK and KF similarly identified e4 as the main driver at expected latencies (top panel), however, STOK recovered more temporally localized dynamics and evoked patterns in the total inflow of gamma activity from e4 to the two main targets e2-e6 (bottom panel). Colored squares at the bottom of each plot indicate time points of significance after bootstrap statistics (n = 10000, p < 0.05; see Results). (F-H) Same set of results using a sampling rate of 1000 Hz, revealing the compromised estimates of KF and the consistent and almost invariant results obtained with STOK.

Fig 4. Results in real human evoked potentials during visual motion discrimination.

(A) The visual motion discrimination paradigm presented during EEG recordings. Participants (n = 19) reported the presence of coherent motion in a briefly presented dot kinematogram (300 ms). (B) Shows grand-average event-related responses recorded at the scalp, with typical early (~100 ms) and late (~200 ms) components of visual processing. The orange line indicates the temporal dynamics detected by the self-tuning memory parameter c, that increases in anticipation of evident changes in the scalp signals. (C-D) Comparison of the non-parametric (wavelet) and parametric power spectrum densities (PSD) obtained with KF and STOK for one representative regions (MT+), with the violin plot showing the overall higher (and less variable) correlation between wavelet and STOK PSDs. (E) Global connectivity results from KF and STOK. Time-frequency plots show the results obtained with the two filters and their difference (STOK minus KF), graphically showing more evident dynamics obtained using the STOK filter. Line plots collapsing frequency and time highlight the statistical difference between STOK and KF results: STOK recovered multiple dynamic changes in overall connectivity patterns at physiologically plausible latencies (bottom plot) and characterized network desynchronization in the alpha range with higher precision (right-side plot). At frequencies (70–90 Hz) and latencies (180–200 ms) of interest for motion discrimination, the STOK revealed increased contribution to network activity (e.g., increased outflow) from visual regions, including MT+, and the frontal eye field (FEF; right-bottom plot).