Backward renormalization-group inference of cortical dipole sources and neural connectivity efficacy

Proper neural connectivity inference has become essential for understanding cognitive processes associated with human brain function. Its efficacy is often hampered by the curse of dimensionality. In the electroencephalogram case, which is a noninvasive electrophysiological monitoring technique to record electrical activity of the brain, a possible way around this is to replace multichannel electrode information with dipole reconstructed data. We use a method based on maximum entropy and the renormalization group to infer the position of the sources, whose success hinges on transmitting information from low- to high-resolution representations of the cortex. The performance of this method compares favorably to other available source inference algorithms, which are ranked here in terms of their performance with respect to directed connectivity inference by using artificially generated dynamic data. We examine some representative scenarios comprising different numbers of dynamically connected dipoles over distinct cortical surface positions and under different sensor noise impairment levels. The overall conclusion is that inverse problem solutions do not affect the correct inference of the direction of the flow of information as long as the equivalent dipole sources are correctly found.

Figure 1

(a)–(e) The renormalized lattices that represent a brain hemisphere used to solve the inverse problem at the different scales. (f) Each original lattice representing one hemisphere of the brain was inflated into a sphere [27]. Initially, for the coarsest lattice, an icosahedron was inscribed in the sphere. Regions of the sphere are identified with the icosahedron's face below. (g) The renormalization, in the inflated state is obtained by dividing each triangular face into four triangles. Deflation takes back to the lattice representations of the cortex. The dipoles possible positions at each scale are at the average centroid of all triangles in the original surface lying below the triangular faces derived through the renormalization procedure [(b)–(e)] or from the icosahedron faces (a).

Figure 2

The simulated scenarios. All use identical time series structures. In the different scenarios the number of electrically active time series that represent dipole currents are: (a) with five ($q_{1-5}=1$), (b) with two ($q_{1,5}=1,q_{2,3,4}=0$), and (c) and (d) with three ($q_{1,4,5}=1,q_{2,3}=0$). Crossed out nodes have $q_a=0$, uncrossed nodes $q_a=1$. Scenarios (c) and (d) differ only by the position of the sources. The actual values used to generate the time series of system (13) were: $a_{11}=.95,a_{12}=0.9025,a_{21}=.5,a_{31}-.4,a_{41}=-.5,a_{42}=\sqrt{2}/4,a_{43}=\sqrt{2}/4,a_{51}=-\sqrt{2}/4,a_{52}=\sqrt{2}/4$.

Figure 3

Localization error for different noise levels ($R_{\mathrm{NSR}}=0.01$, 0.1, and 0.25) for two-dipole scenario [Fig. 2]. Error measured in mm. Note that source 5 is deeper than source 1 and its localization is harder. Only BRG-VB finds both sources with small error. sLoreta finds source 1 with a small error but it increases for source 5. VB and MNE are equivalent on source 1 but VB is better for source 5 while MNE is more affected by the depth of source 5. Error bars are standard deviations obtained from 100 simulations.

Figure 4

For two-dipole scenario [Fig. 2] the simulated dipole cortical surface positions and the estimated currents using MNE, sLoreta, VB, and BRG-VB are contrasted. $R_{\mathrm{NSR}}=0.1$ independent Gaussian noise was added to scalp electrodes. The arrows represent the estimated (PDC above 0.1) and simulated connections between sources (points in yellow).

Figure 5

Two dipole scenario with sources 1 and 5 defined in Fig. 2. PDC connectivity reconstructed by the BRG-VB, VB, and sLoreta methods for noise level of $R_{\mathrm{NSR}}=0.1$. Left: forward connection from 1 to 5, right: feedback connection from 5 to 1. Dark red solid lines indicate significant PDC (red) and turn into green nonsignificant lines (green) as they fall below the dashed dark lines that represent the null connectivity hypothesis threshold level of $\alpha =0.05$ significance level. Gray shadows represent the confidence intervals [43].

Figure 6

Top (a) and (b): Original positions for two three-dipole scenarios [Figs. 2 and 2, respectively] with arrows representing the simulated connections. Bottom (c) and (d): The corresponding BRG-VB result overlaid onto the cortical surface at a representative time point with arrows representing the estimated connections (PDC $>0.1$) across sources. $R_{\mathrm{NSR}}=0.1$ independent Gaussian noise was added to scalp electrodes' potentials.

Figure 7

Connectivity plots between sources reconstructed using BRG-VB for the two three-dipole scenarios in Figs. 2 and 2. Top: Connectivity plots of Fig. 6 using both estimated signal and position for source 1 (reconstructed correctly) and (a) using the original source 5 position and (b) its estimated position (reconstructed at a wrong position). Bottom (c): Connectivity corresponding to Fig. 6 with its correctly reconstructed source positions for $R_{\mathrm{NSR}}=0.1$ for plot conventions as in Fig. 5.

Figure 8

Connectivity plots on sources estimated by the BRG-VB for artificial data with five interconnected dipoles corrupted with (a) practically noiseless ($R_{\mathrm{NSR}}=0.001$); (b) $R_{\mathrm{NSR}}=0.01$; (c) $R_{\mathrm{NSR}}=0.1$; and (d) $R_{\mathrm{NSR}}=0.25$. Lines are as described in Fig. 5.

Figure 9

Top: Original positions for the five-dipole scenario [Fig. 2]. Arrows represent the simulated connections across sources. Bottom: BRG-VB reconstruction results under increasing noise levels (a) practically noiseless ($R_{\mathrm{NSR}}=0.001$), (b) $R_{\mathrm{NSR}}=0.01$, (c) $R_{\mathrm{NSR}}=0.1$, and (d) $R_{\mathrm{NSR}}=0.25$. Reconstructed sources (in red) were overlaid onto the cortical surface at a representative sample point, where the estimated current is maximum at that point. Arrows represent the estimated connections across sources with significant PDC above 0.2.

Figure 10

Significant PDC maximum peak as a function of added electrode noise level (NSR) for sources estimated by the BRG-VB for the five-dipole scenario. The $x$ axis represents (a) $R_{\mathrm{NSR}}=0.001$; (b)$R_{\mathrm{NSR}}=0.01$; (c)$R_{\mathrm{NSR}}=0.1$; and (d)$R_{\mathrm{NSR}}=0.25$. The plotted boxes indicate the minimum and maximum significant PDC for 100 simulated trials. The percentage on each box indicates the portion of significant PDC values for all trials for each NSR.