Lag-invariant detection of interactions in spatially-extended systems using linear inverse modeling

Abstract

Measurements on physical systems result from the systems' activity being converted into sensor measurements by a forward model. In a number of cases, inversion of the forward model is extremely sensitive to perturbations such as sensor noise or numerical errors in the forward model. Regularization is then required, which introduces bias in the reconstruction of the systems' activity. One domain in which this is particularly problematic is the reconstruction of interactions in spatially-extended complex systems such as the human brain. Brain interactions can be reconstructed from non-invasive measurements such as electroencephalography (EEG) or magnetoencephalography (MEG), whose forward models are linear and instantaneous, but have large null-spaces and high condition numbers. This leads to incomplete unmixing of the forward models and hence to spurious interactions. This motivated the development of interaction measures that are exclusively sensitive to lagged, i.e. delayed interactions. The drawback of such measures is that they only detect interactions that have sufficiently large lags and this introduces bias in reconstructed brain networks. We introduce three estimators for linear interactions in spatially-extended systems that are uniformly sensitive to all lags. We derive some basic properties of and relationships between the estimators and evaluate their performance using numerical simulations from a simple benchmark model.

Fig 1. Set-up of the test model.

A. Source-space segment of length 4 mm (horizontal axis at height 0 mm), recording electrodes (black dots at height 0.5 mm), and the locations of two active sources of neuronal activity (red dots). Also shown are the sensitivities of the electrodes to activity of the two sources (two black curves). B. Observed spectral matrix of the recorded electric potentials that are induced by non-interacting sources of unit strength and in the presence of measurement noise (σ = 0.1).

Fig 2. Real, imaginary, and complex test-statistics.

A. Real-and imaginary part of the imaginary test-statistic as a function of lag. The strength of interaction was set to γ_k,l = 1 so that the curves range between zero and one. B. Sensitivity of the real, imaginary, and complex test-statistics at the true interaction-pair (k₀, l₀), as a function of lag C. Sensitivity of the imaginary test-statistic for all interaction-pairs and for a lag of 0 degrees. D. Sensitivity of the real test-statistic for all interaction-pairs and for a lag of 0 degrees. E. Sensitivity of the imaginary test-statistic for all interaction-pairs and for a lag of 90 degrees. F. Sensitivity of the real test-statistic for all interaction-pairs and for a lag of 90 degrees.

Fig 3. Subspace projections for suppression of spurious interactions.

A. Source-based estimator with correction in source-space. First the time-frequency coefficients of the sensor data are projected to source-space using an arbitrary linear inverse operator. Next the spectral matrix of the reconstructed time-frequency coefficients are estimated and subsequently corrected by applying a subspace projection. B. Sensor-based estimator with correction in source-space. First the spectral matrix of the sensor-space time-frequency coefficients is estimated. Next, the estimated sensor-space spectral matrix is projected to source-space using the tensor product of an arbitrary linear inverse operator with itself and subsequently corrected by applying a subspace projection. C. Sensor-based estimator with correction in sensor-space. First the spectral matrix of the sensor-space time-frequency coefficients is estimated. Next, the estimated sensor-space spectral matrix is corrected by applying a subspace projection and subsequently projected to source-space using the tensor product of an arbitrary linear inverse operator with itself.

Fig 4. Suppression levels.

A. Suppression levels of the source-based estimator as a function of regularization-level and for three interaction lags (0, 20, 40, 50, and 70 degrees). B. Same as in A. but for the sensor-based estimator with suppression in source-space. C. Same as in A. but for the sensor-based estimator with suppression in sensor-space. D. Suppression of projected sensor noise for all three estimators and as a function of regularization-level. In panels A, B, and C, the interaction lags increase in the direction of the arrows. In all panels, the parameters were set as in Table 1.

Fig 5. Bias-reduction through leakage correction.

A. Performance on the test-model of the uncorrected estimators as a function of noise-level. Performance is defined as one minus the bias and ranges between zero and one. B. Expected value of the source-based estimator for σ = 0.05 and λ = −2.2. C. Same as B but for the sensor-based estimator. D. Performance on the test-model of the corrected estimators as a function of noise-level. E. Expected value of the corrected source-based estimator for σ = 0.05 and λ = −3. F. Same as E but for the sensor-based estimator with correction in source-space. G. Same as E but for the sensor-based estimator with correction in sensor-space. In panels B, C, E, F, and G, the true interaction-pair is designated by the white-circles.

Fig 6. Detection power.

A. Detection power of the real uncorrected (red) and corrected (green) source-based test-statistic. D. Detection power of the imaginary uncorrected (red) and corrected (green) source-based test-statistic. G. Detection power of the complex uncorrected (red) and corrected (green) source-based test-statistic. Panels B, E, and H: Same format as panels A, D, and G, respectively, but for the sensor-based test-statistic with correction in source-space. Panels C, F, and I: Same format as panels A, D, and G, respectively, but for the sensor-based test-statistic with correction in sensor-space.

Fig 7. Reconstruction of functional networks.

A. Real part of the true spectral matrix. B. Imaginary part of the true spectral matrix. C. Absolute value of the true spectral matrix. D. Sensitivity matrices of the real (top row), imaginary (middle row), and complex (bottom row) uncorrected sensor-based test-statistic as a function of interaction strength γ ranging from 0 to 1 in steps in 0.1. E. Same as D. but for the corrected test-statistics.