Frequency-Resolved Functional Connectivity: Role of Delay and the Strength of Connections

Abstract

The brain functional network extracted from the BOLD signals reveals the correlated activity of the different brain regions, which is hypothesized to underlie the integration of the information across functionally specialized areas. Functional networks are not static and change over time and in different brain states, enabling the nervous system to engage and disengage different local areas in specific tasks on demand. Due to the low temporal resolution, however, BOLD signals do not allow the exploration of spectral properties of the brain dynamics over different frequency bands which are known to be important in cognitive processes. Recent studies using imaging tools with a high temporal resolution has made it possible to explore the correlation between the regions at multiple frequency bands. These studies introduce the frequency as a new dimension over which the functional networks change, enabling brain networks to transmit multiplex of information at any time. In this computational study, we explore the functional connectivity at different frequency ranges and highlight the role of the distance between the nodes in their correlation. We run the generalized Kuramoto model with delayed interactions on top of the brain's connectome and show that how the transmission delay and the strength of the connections, affect the correlation between the pair of nodes over different frequency bands.

Keywords: brain oscillation; connectome; correlation matrix; functional connectivity; functional network; hierarchical clustering; transmission delay.

Figure 1

Structural properties of the human connectome. (A) The normalized coupling weights (0 ≤ W ≤ 1) and (B) Euclidean distances (in mm) in the human connectome with 66 nodes (Hagmann et al., 2008). The squares show the modules and the nodes ordered for the structural module (community) they comprise. The color bar of the weight matrix has a log₁₀ scale. The background dark blue regions in A (< 10⁻⁵) and B (= 0) indicate the absence of edges between the areas. (C) Semi-log presentation of the distribution of the weights of structural connections that span five orders of magnitude. (D) Scatter plot shows the distribution of the normalized weights of the structural connections vs. the distance between the nodes. Here we have used a linear scale for both exes.

Figure 2

The correlation distributions. (A) The functional networks at five sample frequency ranges. The elements of the functional networks are the correlation indices σ_ij defined in the Methods. In each panel, the results of a simulation of the model with a given mean frequency are presented. From left to right the mean frequency is 3, 11, 23, 35, and 51 Hz which lie within the bands θ, α, β, γ, and high-γ, respectively. The frequencies are chosen from a normal distribution with the given mean value and standard deviation 0.1. The axes show the index of nodes which represent ROIs of the structural network. The gray dash lines indicate the boundary of the brain hemispheres. The coupling scale factor is K/N = 0.25 and the noise amplitude is 0.05. The initial phases are chosen from a uniform distribution in the range [−π, π]. The results are averaged over 200 realizations. (B) The distribution of correlations vs. weight (W) and distance (D) of the connections at each frequency. (C,D) The distribution of correlations vs. average natural frequencies of the nodes (ν₀). The colors in panels (C) and (D) show the corresponding connection's weights and distances, respectively.

Figure 3

(A) The scatter plots of the average correlation between the nodes C, vs. distance at five average frequencies, 3, 11, 23, 35, and 51 Hz corresponding to theta, alpha, beta, gamma, and high gamma, respectively The colors indicate the corresponding weights of the structural connections. For comparison, the structural connections W, are also shown in black dots. (B) The scatter plots of the average correlations between the nodes vs. weights of structural connections. The colors indicate the corresponding distances at different frequencies. (C) The distance between scatter plots of the correlations and weights of connections in (A). The inverse of this parameter is a measure of the similarity between the two matrices. (D) The slope of fitted lines in (B) vs. frequency.

Figure 4

The average correlation vs. frequency for connections whose strength lies in the range (A) 〈W〉 = 0.15 and (B) 〈W〉 = 0.25 and various distances indicated in the legends. The average correlation vs. frequency for connections whose distance lies in the range (C) 〈d〉 = 25 mm and (D) 〈d〉 = 35 mm and various strengths indicated in the legends. Correlation is calculated using the correlation index σ_ij defined in Methods. The colored areas show the results for p-value = 0.05. The width of the bins was set to 0.05 for the connection strengths and 16 mm for distances.