Negative Functional Connectivity and Its Dependence on the Shortest Path Length of Positive Network in the Resting-State Human Brain

Abstract

It is suggested that structurally segregated and functionally specialized brain regions are mediated by synchrony over large-scale networks in order to provide the formation of dynamic links and integration functions. The existence of negative synchrony, or negative functional connectivity (NFC), however, has been a subject of debate in terms of its origin, interpretation, relationship with structural connectivity, and possible neurophysiological function. The present study, which incorporated 20 cognitively healthy elderly human subjects, focused on testing the hypothesis that NFC significantly correlates with the shortest path length (SPL) in the human brain network. Our theoretical calculation, simulated data, and human study results support this hypothesis. In the human study, we find that (1) the percentage of NFC connections among all connections between brain regions significantly correlates with spatial Euclidian distance; (2) the strength of the NFC between the right amygdala and the left dorsolateral prefrontal cortex is significantly correlated with the SPL across the 20 human subjects; (3) such a significant relationship between the NFC and SPL exists in all the NFC connections in the whole brain; and (4) the correlations between the NFC and SPL also are frequency bandwidth dependent. These results suggest that an accumulated phased delay gives rise to the NFC, along the shortest path in the large-scale brain functional network. It is suggested that our study can be extended to examine a variety of neurological diseases and psychiatric disorders by measuring the changes of SPL and functional reorganization in the brain.

Introduction

R ecently, spontaneous blood oxygenation level-dependent (BOLD) signals measured by the resting-state functional magnetic resonance imaging method (R-fMRI) have been used to characterize regional functional connectivity and investigate changes in a variety of neurological and psychiatric disorders (Chen et al., 2011; Fox and Greicius, 2010; Fox and Raichle, 2007). Since the very inception of R-fMRI, negative functional connectivity (NFC) has been reported (Biswal et al., 1995). NFC refers to spontaneous BOLD signals in two brain regions that have a negative Pearson cross-correlation coefficient; sometimes it is also called an anticorrelation. NFC mechanisms in the context of network physiology are less understood and have been a subject of debate. Several studies demonstrated that the NFC could be an artifact introduced by a global signal regression procedure (Giove et al., 2009; Murphy et al., 2009; Weissenbacher et al., 2009), thereby suggesting careful interpretation. Others demonstrated that the data preprocessing method with the global signal regression may introduce negative correlations. Still, NFC exists even when no global signal regression is performed (Chang and Glover, 2009). Further investigation reported that several characteristics of anticorrelated networks are not attributable to global signal removal (Fox et al., 2009). It was found that NFC has predominantly long-range connections; this suggests a biological basis (Fox et al., 2009; Scholvinck et al., 2010; Schwarz and McGonigle, 2011). Nevertheless, studies have excluded NFC-related results to avoid uncertainty (Buckner et al., 2009; Meunier et al., 2009).

To understand the origin of the functional connectivity, an attempt has been made to investigate the relationship between R-fMRI measured connectivity and anatomical connectivity measured by diffusion tensor imaging (DTI) tractorgraphy. This infers that the spontaneous BOLD signals measured with R-fMRI are constrained by anatomical connectivity. It is reported that the R-fMRI functional connectivity reflects structural connectivity in the default-mode network (Buckner et al., 2009; Greicius et al., 2009). Direct comparison between R-fMRI functional connectivity and anatomical DTI connectivity shows their strong agreement in the whole brain, whereas the regions show that weak anatomical correlations have significant negative functional correlation (Honey et al., 2009; Skudlarski et al., 2008). These studies used the fully connected network model instead of a small-world network (Fox et al., 2006; Strogatz, 2001). As a result, functional connections are very pervasive and strong functional connections commonly exist between regions with no direct structural connection (Honey et al., 2009). Using a fully connected network model is not only considered impractical in inferring structural connectivity from functional connectivity, it is also deemed theoretically against the small-world hypothesis of the brain (Bullmore and Sporns, 2009).

In facing this challenge, an assumption could be made that functional connectivity is not limited or constrained only to direct anatomical connections. Indeed, it is also derived from indirect connections mediated via more distant grey matter regions (Koch et al., 2002). Early theoretical studies suggested that neurons activated by the same stimulus could be bound together by synchronization of their action potential (Abeles, 1982; von der Malsburg and Schneider, 1986). Ample evidence indicates that the human brain has been viewed as a complex small-world network in which information is continuously processed and transported between spatially distributed, but functionally linked regions with coherent temporal dynamics. Therefore, beyond the direct anatomical connection between two regions, indirect links via multiple nodes (also called vertices) could be ubiquitous, as manifested by functional connectivity patterns.

In the present study, we focus on identifying system-level features of NFC in the resting-state human brain and providing insight into its network physiology. We hypothesize that the strength of NFC is a result of the accumulated phase lags along the multinodal shortest pathway. Because the shortest pathway is defined along with positive connectivity networks, only positive functional connectivity (PFC) values are employed to calculate the shortest path length (SPL). Under this hypothesis, a PFC may have a shorter path length than the NFC. The longer the SPL is, the stronger the NFC can be. To test this hypothesis, we first will conduct a simulation study. Second, we will apply the simulated results to the whole human brain network. Third, we will examine whether two brain regions with no known direct anatomical connections between the amygdala (Amy) and dorsal lateral prefrontal cortex (dlPFC) have NFC and, if so, determine whether their NFC significantly correlated with their SPL in individual human subjects.

Theory

Structurally segregated and functionally specialized regions of the human brain form a complex “small-world” network (Bullmore and Sporns, 2009), which combines densely connected local neuron networks and a small fraction of long-range connections for supporting bottom-up, top-down, and cross-modal interactions through oscillations and synchrony (Engel et al., 2001). A comprehensive understanding of how the “small-world” structural networks can generate complex dynamics is still largely unknown. It is suggested that the formation of dynamic links and integration is mediated by synchrony over large-scale networks (Varela et al., 2001). Through inhibitory interneuron networks, oscillatory timing can transform unconnected principal cell groups into a temporal coalition (Konig et al., 1996) and the relationships between anatomical architecture and oscillatory patterns allow brain operations to be carried out simultaneously at multiple temporal and spatial scales (Buzsaki et al., 2004). Clearly, the synchronous activity of oscillating networks must be an intrinsic characteristic of the complex brain networks (Buzsaki et al., 2004). It is conceivable that the existence of negative synchrony may play an important role in oscillating networks. Mathematically, the NFC does not require direct structural connections; rather, it represents a large phase difference of the synchronized signals between brain regions. This relationship can be better expressed by vector analysis of the cross-correlation coefficient: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{upgreek}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}CC_ {XY} = \frac {X \cdot Y} {\mid X \mid \mid Y \mid} = cos \theta \tag {1}\end{align*}\end{document}

where vectors X and Y represent two time series from regions X and Y, and θ=arccos (CC_XY ) is the phase difference between X and Y vectors. This geometric representation shows that either the positive or negative cross-correlation coefficient is resulting from the phase difference between two vectors. Clearly, the generation of the NFC results from a negative value of cosθ. In the present study, we will test a hypothesis that the negative values of cosθ between two nodes are dependent on the SPL accumulated along the multinodal shortest path in a network of the resting-state human brain. We provide the simulation below and then apply it to the human brain analysis in order to demonstrate this relationship.

Materials and Methods

Subjects, imaging acquisition, and analyses

Human subjects

The study was conducted with Institutional Review Board approval and was in compliance with Health Insurance Portability and Accountability Act (HIPAA) regulations. Written informed consent was obtained from each participant. R-fMRI was employed to acquire the voxelwise time series from the 20 healthy elderly subjects (age: 74.6±6.6 years; 11 men and 9 women) recruited through the Memory Disorders Clinic at the Medical College of Wisconsin.

Data acquisition

Imaging was performed using a whole-body 3T Signa GE scanner with a standard quadrature transmit receive head coil. During the resting-state acquisitions, no specific cognitive tasks were performed, and the study participants were instructed to close their eyes and relax inside the scanner. Sagittal resting-state functional MRI (fMRI) datasets of the whole brain were obtained in 6 min with a single-shot gradient echo-planar imaging (EPI) pulse sequence. The fMRI imaging parameters were TE of 25 ms, TR of 2 sec, and flip angle of 90°; 36 slices were obtained without gap; slice thickness was 4 mm with a matrix size of 64×64 and field of view of 24×24 cm. High-resolution SPGR 3D axial images were acquired for anatomical reference. The parameters were TE/TR/TI of 4/10/450 ms, flip angle of 12°, number of slices of 144, slice thickness of 1 mm, and matrix size of 256×192.

To make sure that cardiac and respiratory frequencies did not account for any significant artifacts in the low-frequency spectrum, a pulse oximeter and respiratory belt were employed to measure these physiological noise sources. Further processing ensured minimizing of the potential aliasing effects.

Data preprocessing

We used Analysis of Functional NeuroImages (AFNI) software in this study (http://afni.nimh.nih.gov/afni/). The first five volumes of each raw resting-state functional imaging dataset were discarded to allow for T1 equilibration. Interleaved slice acquisition-dependent time shifts were corrected (AFNI command, 3dTshift). Spikes in time series data were removed (AFNI command, 3dDespike). Data were then motion corrected and detrended (AFNI commands: 3dvolreg and 3dDetrend). Cardiac aliasing was minimized (AFNI command: 3dretroicor-card). Respiratory volume variation was minimized on the basis of the respiratory belt signal (AFNI command: 3dretroicor-resp). The reference template in Talairach space, which contained 116 anatomically defined regions of interest (ROIs) (Tzourio-Mazoyer et al., 2002), was transformed and aligned to the SPGR images and echo-planar imaging resting-state functional images for each subject (AFNI command: 3dfractionize). This resulted in 116 mapped ROIs. The average time course within each ROI was extracted from the resting-state functional imaging datasets. The white matter mask in the Talairach space (http://afni.nimh.nih.gov/pub/dist/data/TT_wm+tlrc) that was not overlapped with the 116 regional masks was defined as the eroded white matter mask. This choice was made to extract pure white matter signal and avoid signals from the 116 ROIs. The eroded white matter mask and the cerebrospinal fluid mask (http://afni.nimh.nih.gov/pub/dist/data/TT_csf+tlrc) were transformed and aligned to the SPGR images and echo-planar images for each subject (AFNI command: 3dfractionize). The average time course within the cerebrospinal fluid or the eroded white matter mask, together with global signals, was removed as a nuisance regressor from the 116 regional time courses with linear regression. Finally, a band-pass filter was used to keep low-frequency fluctuations between 0.015 and 0.1 Hz.

Global signal regression

The concept of global signal regression is still in debate (Fox et al., 2009; Murphy et al., 2009), because it induces false-negative functional correlations. Therefore, the networks were analyzed with and without global regression procedures. We present the results obtained with and without the global signal regression procedure. In the present study, there were no significant differences related to global regression procedures.

Constructing functional network

If a network size is N (N vertices or nodes), there are N×(N−1)/2 possible edges in a fully connected network expressed in a matrix. The weighted distance of each edge between nodes i and j was defined as d_ij =1−CC_ij (Nakamura et al., 2009). The CC_ij is the Pearson cross-correlation coefficient between time series from nodes i and j. The adjacent matrix of CC_ij represents graph G, such that G={V,E,D}, consisting of a set of vertices \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$V = \{ V_1 , V_2 , \ldots , V_N \}$$\end{document} , a set of edges \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$E = \{ E_{ij} \mid i , j = 1 , 2 , \ldots , N \}$$\end{document} , and a set of associating weighted edge distances \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$D = \{ d_{ij} = 1 - CC_{ij} \mid i , j = 1 , 2 , \ldots , N \}$$\end{document} between brain regions i and j. As the CC_ij value could be positive or negative, and a negative CC_ij could indicate a result from the multinodal pathways, we only use positive CC_ij to calculate the SPL (there is no thresholding on the positive CC values).

Different frequency bands

To test our hypothesis that the strength of NFC is a result of the accumulated phase lags along the multinodal shortest pathway, it is necessary to determine the effect of the frequency bandwidth on the relationship between NFC and the phase lags. It is conceivable that the narrower a frequency bandwidth is, the less the variance of the relationship will be. In general, a frequency band between 0.015 and 0.1 Hz is often employed in resting-state functional connectivity studies. In this study, the frequency band of 0.015–0.1 Hz is separated into several narrowed bands with a bandwidth of 0.01 Hz to further investigate our hypothesis.

Simulations

Information flow from source node S to destination node D

Assuming that oscillatory timing can transform the unconnected principal cell groups into a temporal coalition (Konig et al., 1996) and that the information flow from source node S to destination node D passes through several brain nodes, the cross-correlation coefficient between the time series in regions S and D would be negative, if the accumulative phase difference along the path between S and D is larger than 90°. To simulate this scenario, we specify an information router consisting of six nodes and five connections (also called edges). Node 1 is a source and node 6 is a destination. A pathway that carries the information flow from node 1 to node 6 is predefined and the corresponding BOLD signals of the six nodes are generated, as shown in Figure 1A. The signals of six nodes are generated with the signal-to-noise ratio (SNR) of 2.5. The noise was randomly generated with a normal distribution. Each of these signals is consistent with the characteristics of 18 sinusoid waves, whose frequencies are from 0.015 to 0.1 Hz with 0.005 Hz steps. This frequency band is simulated according to the spontaneous BOLD signal (Biswal et al., 1995). Node 1 is the source that sends the signals to node 2 with a 30° delay for each frequency component. Nodes 3, 4, and 5 sequentially transfer the signal from one node to another to node 6 with the 30° delays in each step in the assumed pathway, as shown in Figure 1B. A 30° delay for a 6-node transferring network is large enough to cause a negative correlation. Yet, it is adequate to ensure that the largest delay between any pair of nodes is less than 180°. As the cross-correlation is a monotonic function only between 0° and 180°, the other scenario (e.g., the largest delay between nodes is large than 180° or less than 90°) is excluded in this simulation and will be further investigated in another study.

FIG. 1.

Simulations of signals and networks. (A) Simulated signals for the six different nodes. (B) Simulated signal-transferring network: node 1 is source and node 6 is destination with CC_ij value on each edge. (C) A matrix of the cross-correlation coefficient of six signals and the color bar represents the strength of CC_ij. (D) The network constructed from the matrix in C with CC_ij values for each edge. (E) A network in which each edge has a positive CC_ij; the number represents the path length of an edge (1−CC_ij); the bold route indicates the shortest path between source node 1 and destination node 6. (F) The same network shown in E, but each edge length is expressed in phase angle. (G) The linear correlation in frequency range of 0.015–0.1 Hz between the CC_ij of six negative connections in D and their corresponding shortest path lengths (SPL_CC). (H) Similar to G, except in the frequency band of 0.025–0.035 Hz. (I, J) Similar to G and H, but the relationship between negative functional connectivity (NFC) and SPL is expressed in θ and SPL_θ. (K, L) The noise-only time series data, without the simulated signals; they do not demonstrate a linear correlation between NFC and SPL in the network.

Effects of the Euclidean distance on NFC

To test whether the NFC strength is correlated with the spatial Euclidean distance between brain regions, and whether the percentage of NFC connections among all possible connections within and between brain regions is correlated with the Euclidean distance, the human brain was divided into 116 regions in the Talairach template. The coordinates of each region are located at the center of mass. As shown in Figure 2, the 116 regions were further divided into four parts according to the origin (the center (0, 0) of the anterior commissure) of the Talairach coordinates: the left anterior (LA: part 1), right anterior (RA: part 2), left posterior (LP: part 3), and right posterior (RP: part 4). Based on the permutation and combination, there were 10 different connection types: LA-LA, LA-RA, LA-LP, LA-RP, RA-RA, RA-LP, RA-RP, LP-LP, LP-RP, RP-RP. The spatial Euclidean distances for each connection type were obtained by averaging all Euclidean distances of all possible connections (excluding self loops) between two regions within a connection type. The Euclidean distance of two regions is the spatial distance between the centers of these two brain regions. The percentage of the NFC connections for each individual subject was obtained by dividing the number of NFC connections by the number of all possible connections within a connection type. The relationship between the averaged percentage of NFC and the averaged Euclidean distance for each connection type across 20 subjects was analyzed, using the linear regression method.

FIG. 2.

The relationships between NFC and the spatial Euclidean distance. In A–D, the dashed red lines divide the brain into four parts. The yellow lines indicate the negative connections between or within the brain parts. A yellow line is drawn when the mean of CC_ij value of a paired connection across the 20 subjects is negative. (A) Negative connections (yellow lines) between the left anterior and the right posterior parts and between the right anterior and the left posterior parts. (B, C, D) Other negative connections between or within the four brain parts. (E) The negative CC_ij values did not significantly correlate with the spatial Euclidean distance from a representative subject. (F) The linear relationship between the percentages of negative connections in 10 individual connection types and the corresponding averaged spatial Euclidean distances. There is an outlier, which is the connection type between the left and right anterior regions.

Determine the relationship between the NFC and the SPL calculated in CC values (SPL_cc)

According to the Graph theory, a path is a sequence of vertices, such that from each of its vertices there is an edge to the next vertex in the sequence. The shortest individual path length SPL_ij between two vertices i and j is the smallest sum of the weights for all possible paths from vertex i to vertex j in a weighted graph. The SPL_ij indicates how close a node is connected to another node in the network. The SPL_ij also provides a measure of the network's capacity for serial information transfer between nodes (Achard and Bullmore, 2007) and describes how efficiently a node is connected to the rest of the network (van den Heuvel et al., 2009). With the SPL_ij , a matrix can be formed, wherein an entry corresponds to the distance (SPL_ij ) between vertices i and j. If no path exists, SPL_ij equals infinity. The calculation of SPL_ij is as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{upgreek}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}{\rm SPL}_{ij} = \sum \nolimits_{{W_{E_{uv}}} {{\in}_{sp_{i \leftrightarrow j}}}{W_{E_{uv}}}} \ ( Howard \ et al., 2004) \tag {2}\end{align*}\end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$W_{E_{uv}} = 1 - CC_{uv}$$\end{document} and sp_i _↔j is the shortest path between i and j; the SPL_ij =∞ for any disconneted pairs i, j; \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$W_{E_{uv}}$$\end{document} is the weight of an edge between two directly connected nodes (u, v); u and v are possible nodes along the shortest path between i and j. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$W_{E_{uv}} = 1 - CC_{uv}$$\end{document} when link (u, v) exists (CC_uv >0). CC_uv is the cross-correlation coefficient value between u and v. The relationship between the SPL and the NFC is determined with linear regression analysis.

Determine the relationship between the NFC and the phase accumulation along the SPL calculated in phase (SPL_θ)

According to Eq. [1] in the Theory section, the phase between two vectors can be expressed as the arccosine function of the cross-correlation coefficient [4]. To determine the relationship between the NFC and the SPL_θ between nodes along the shortest path in phase, Eq. [2] was rewritten as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{upgreek}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}{\rm SPL}_{\theta} (i,j) = \sum \nolimits_{{\theta_{E_{uv}}}{_\in}_{sp_{I \leftrightarrow j}} \theta_{E_{uv}}} \tag {3}\end{align*}\end{document}

sp_i _↔j is the shortest path between i and j in the phase θ matrix; the SPL_θ (i, j)=∞ for any disconnected pairs i and j; \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$\theta_{E_{uv}}$$\end{document} is the phase difference of an edge between two directly connected nodes (u, v).

Testing models

In real data, a pair of negative-connected regions has two values: one is the functional SPL between the two regions, which is calculated from the positive functional network, and the second is the negative value from the cross-correlation coefficient between the two regions' time series. So, the first value is called the corresponding SPL of the NFC of this pair of regions. Then, we compare these two values to see whether they have a relationship. If there is a relationship, this indicates that the NFC can be predicted by positive correlation. Thus, we prove our hypothesis.

To test our hypothesis that the NFC resulted from SPL accumulated along the multinodal shortest path of positive functional connectivity, the following linear regression equations are employed: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{upgreek}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}CC_{ij} = a1 + b1 * SPL_{CC_{ij}} + \varepsilon \hbox{;} \quad\;\, CC_{ij} < 0 \tag {5}\end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{upgreek}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*}CC_{ij} = a2 + b2 * SPL_{ \theta_{ij}} + \varepsilon \hbox{;} \qquad CC_{ij} < 0 \tag {6}\end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{upgreek}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \theta_{ij} = a3 + b3 * SPL_{CC_{ij}} + \varepsilon \hbox{;} \quad \; \, \theta_{ij} > 90^{ \circ} \tag {7}\end{align*}\end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{upgreek}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}\begin{align*} \theta_{ij} = a4 + b4 * SPL_{ \theta_{ij}} + \varepsilon \hbox{;} \qquad \theta_{ij} > 90^{ \circ} \tag {8} \end{align*}\end{document}

Results

Simulated results

As shown in Figure 1, the signals from the six nodes (Fig. 1A) simulated a pathway from node 1 (source) to node 6 (destination; Fig. 1B) with cross-correlation coefficient values of each edge. A fully connected network of the six nodes was expressed in a matrix (Fig. 1C). The color bar represented the CC values between each pair of nodes, and the CC values for each connection were further provided in Figure 1D. In comparison to the defined network pathway in Figure 1B, the fully connected network in Figure 1D generated pervasive connections, including six negative connections drawn in blue lines.

Figure 1E shows the networks without the negative connections and the number along each edge represents the path length between two adjacent nodes; the path is defined as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$W_{E_{ij}} = 1 - CC_{ij}$$\end{document} (Nakamura et al., 2009). Figure 1E represents the relationship between CC_ij of the six negative connections and their SPL, SPL_ij . For example, there are many possible paths from node 1 to node 6; the shortest path is {1–3–4–6} with the SPL of 1.79, which corresponds to CC_ij of −0.33. Similarly, Figure 1F shows the networks without the negative connections and the number along each edge represents the phase θ_ij =arccos(CC_ij) between two adjacent nodes. Figure 1F represents the relationship between θ_ij of the six negative connections and their SPL, SPL_θ. For example, there are many possible paths from node 1 to node 6; the shortest path is {1–3–4–6} with the shortest path phase of 196.7°, corresponding to CC_ij of −0.33.

Intriguingly, all the SPLs of the six negative connections were significantly (p<0.0001) correlated with their corresponding negative correlation values, as shown in Figure 1G. Such a relationship was frequency bandwidth dependent, as shown in Figure 1G and H. By limiting simulated frequencies from 0.015 to 0.1 Hz to a narrower frequency band (0.015–0.025 Hz), both the NFC and corresponding SPL_CC have been changed. A similar effect of different frequency band on the relationship between the θ_ij and corresponding SPL_θ can be seen in Figure 1I and J. Further simulation demonstrated that using noise only time series data cannot produce the relationships between the negative CC value, θ, and the corresponding SPL_CC, SPL_θ, as shown in Figure 1K and L.

Relationship between NFC and spatial distance

It is known that functional connectivity strength is correlated with the spatial Euclidian distance (Honey et al., 2009; Salvador et al., 2005; Skudlarski et al., 2008). However, this relationship is established with all positive and negative CC values. With only NFC connections, the correlation is not significant, as shown in Figure 2E.

Although NFC strength does not correlate with spatial distance, the number of NFC connections still depends on it. When the spatial distance is longer, there are more NFC connections evident. Figure 2A–D illustrates the negative connections (yellow lines) within and between the four parts of a brain. A yellow line is drawn when a mean of the cross-correlation coefficients between two regions across the 20 subjects is negative. There is no such group mean negative connection within part 1 or part 2, although each individual subject could have negative cross-correlation coefficients in these parts. The mean spatial Euclidian distances of the 10 connection types are significantly correlated with the corresponding mean percentages of NFC connections across 20 subjects, as shown in Figure 2F (R ²=0.73, p=0.003). The longest spatial mean distances (about 91 cm) between part 1 and part 4, and between part 2 and 3, have the highest percentage of negative connections. The shorter spatial mean distances within part 1 or part 2 (about 50 cm), and within part 3 or part 4 (about 56 cm), have the lowest percentage of negative connections. The mean spatial mean distance (about 72.5 cm) between part 1 and part 3, or between part 2 and part 4, has negative connections of about 10.5%. The mean spatial mean distance (about 84 cm) between part 3 and part 4 has negative connections of about 10.8%. The outlier point occurred between part 1 (LA) and part 2 (RA) connection where the negative percentage is much higher than the corresponding Euclidian distance. It is suggested that the white matter fiber connections between the LA and RA represent a U-shape, which is longer than the calculated Euclidean distance of about 68.5 cm.

Negative connectivity between the right Amy and the left dlPFC correlated with SPL

It was found that the functional connectivity between the right Amy (R-Amy) and the left dlPFC (L-dlPFC) was negative in all 20 subjects, as shown in Figures 3A and 3B. In all 20 subjects, the NFC strengths (CC values) between the R-Amy and the L-dlPFC are significantly correlated with their SPL_CCs (R ²=0.76, p<0.000001, frequency band: 0.015–0.1 Hz; Fig. 3B). Interestingly, when we used a narrower frequency band of 0.025–0.035 Hz (Fig. 3C), the original negative cross-correlation coefficients between the two regions increased and became positive in several subjects with shorter SPLs. By replacing SPL_CC with SPL_θ, similar relationships were still evident, as shown in Figure 3D and F. After the CC value between two nodes was converted to θ value, the significant linear relationships between θ and SPL_CC, SPL_θ were still evident, as shown in Figure 3F–I. The fitting variations can be reduced when using the narrower frequency band, as shown in Figure 3I. Figure 4 shows the same results, but with global signal regression.

FIG. 3.

The NFC value is significantly correlated with the SPL between the right amygdala (R-Amy) and the left dorsal lateral prefrontal cortex (L-dlPFC) connection in the brains across the 20 human subjects. (A) The connection between the regions of the R-Amy and the L-dlPFC. The CC values of the connection are significantly correlated with SPL not only calculated in SPL_CC (B, C), but also in SPL_θ (D, E). This relationship also is frequency bandwidth dependent. As shown in C and E, some of the CC values became positive when the frequency band is limited to 0.025–0.035 Hz. This relationship between the NFC connection and SPL also holds when the CC values were converted to θ values, as shown in F–I. The larger the phases along the shortest paths, the longer the path length in either SPL_CC or SPL_θ will be. The regional time course signals were employed without the global signal regression procedure.

FIG. 4.

Same results as in Figure 3, but the global signal regression procedure was used.

Negative cross-correlation values are significantly correlated with their corresponding SPLs in the human brain

The methods used in simulation and regional analysis between the R-Amy and the L-dlPFC to determine the relationships between the negative CC values and corresponding SPL were applied to whole human brain networks with a network size of 116 nodes, as shown in the CC matrix from a representative subject in Figure 5A. A significant linear correlation exists between the negative cross-correlation value and the corresponding SPL_CC shown in Figure 5D (Eq. [5]) and SPL_θ shown in Figure 5G (Eq. [6]). This significant relationship and its variance of frequency bandwidth dependence were true for all human subjects, as summarized in Table 1. Because the variances of the frequency bandwidth dependence are very similar without statistical difference among different bands, only representative results are provided in Table 1. To verify that this relationship is an intrinsic characteristic in the human brain, individual's cross-correlation matrices were randomly permuted, as shown in Figure 5B. The scatter plot, shown in Figure 5E and H, indicates that the negative cross-correlation value was no longer correlated with the SPL of the same pairs of brain regions in the randomized CC matrix. The results with global signal regression are shown in Figure 6.

FIG. 5.

The NFC values in the whole brain are significantly correlated with their SPLs from a representative subject. (A) The cross-correlation coefficient (CC) matrix between 116 brain regions with a frequency band between 0.015–0.1 Hz. (B) A randomly permuted CC matrix from A. (C) The CC matrix from A by limiting frequency to a band of 0.025–0.035 Hz. (D, E, F) Correlations between the CC values of NFC and the SPL_CC, corresponding to the conditions of A, B, and C, respectively. (G–I) Correlations between CC values with SPL_θ. There are no significant correlations between the CC values and SPL, calculated in either SPL_CC (E) or SPL_θ (H) for the randomly permuted matrix. The correlations between CC values and SPLs under a different frequency bandwidth are different, as shown between D and F and between G and I. When the CC values were converted to θ values with arccosine (CC_ij) function for each pair of 116 brain regions (the phase matrices in J–L), the correlations in D–I are shown again in M–R. Results from statistical analysis across the 20 subjects are provided in Table 1. The regional time course signals were employed without the global signal regression procedure.

FIG. 6.

Same results as in Figure 5, but the global signal regression procedure was used. There was no randomized matrix result.

Table 1.

Summary of All Correlations Between CC and SPL_CC , θ and SPL_cc , CC and SPL_θ, and θ and SPL_θ

		CC and SPL_CC			θ and SPL_CC			CC and SPL _θ			θ and SPL _θ
		R²	Slope	p	R²	Slope	p	R²	Slope	p	R²	Slope	p
Without global signal regression
0.015–0.1 Hz	Mean	0.368	−0.536	0	0.366	0.355	0	0.393	−0.725	0	0.392	0.480	0
	SD	0.052	0.064	0	0.052	0.044	0	0.064	0.083	0	0.063	0.058	0
0.025–0.035 Hz	Mean	0.365	−1.002	0	0.359	0.745	0	0.910	−1.228	0	0.920	0.925	0
	SD	0.065	0.157	0	0.065	0.116	0	0.078	0.039	0	0.019	0.024	0
0.035–0.045 Hz	Mean	0.399	−1.004	0	0.392	0.741	0	0.900	−1.228	0	0.908	0.917	0
	SD	0.073	0.162	0	0.074	0.124	0	0.015	0.033	0	0.016	0.025	0
0.045–0.055 Hz	Mean	0.400	−1.027	0	0.394	0.760	0	0.897	−1.235	0	0.906	0.926	0
	SD	0.070	0.135	0	0.071	0.102	0	0.014	0.028	0	0.015	0.024	0
With global signal regression
0.015–0.1 Hz	Mean	0.384	−0.560	0	0.382	0.373	0	0.397	−0.765	0	0.396	0.510	0
	SD	0.042	0.038	0	0.042	0.027	0	0.051	0.051	0	0.051	0.037	0
0.025–0.035 Hz	Mean	0.441	−1.195	0	0.431	0.899	0	0.910	−1.239	0	0.921	0.948	0
	SD	0.070	0.096	0	0.071	0.072	0	0.010	0.028	0	0.011	0.016	0
0.035–0.045 Hz	Mean	0.477	−1.238	0	0.468	0.926	0	0.905	−1.244	0	0.915	0.945	0
	SD	0.055	0.078	0	0.056	0.06	0	0.011	0.017	0	0.011	0.012	0
0.045–0.055 Hz	Mean	0.454	−1.203	0	0.446	0.900	0	0.904	−1.247	0	0.914	0.947	0
	SD	0.067	0.094	0	0.068	0.07	0	0.011	0.028	0	0.012	0.014	0

SPL, shortest path length.

The relationship between the negative cross-correlation values and their SPLs in the human brain is frequency bandwidth dependent

There is a relatively large variance, as shown in Figure 5D, G. To study the sources of the variance, frequency band pass filter of 0.025–0.035 Hz was applied. Figure 5F (Eq. [5]) and Figure 5I (Eq. [6]) show that the negative cross-correlation values are significantly correlated with their SPLs at the narrower band. The linear regression fitting improved from the R ²<0.34 (Fig. 5G) to R ²>0.90 (Fig. 5I). The results with global signal regression are shown in Figure 6.

The negative cross-correlation values in phase, θ, significantly correlated with their corresponding SPLs in the human brain

Similar to the relationships between CC values and the SPLs, the phases of the NFC are significantly correlated with the accumulated phase (SPL_θ) and the SPL_CC along with the corresponding shortest path between two brain regions (Fig. 5J, M: Eq. [7] and Fig. 5P: Eq. [8]). This relationship is an intrinsic characteristic in the human brain, because after the phases are randomly permuted (Fig. 5K, N, Q), the phase of the NFC is no longer correlated with the SPL_θ and SPL_CC (Fig. 5N, Q). To study the sources of the variance in Figure 5M and Q, the frequency band pass filter of 0.025–0.035 Hz was applied. Figure 5L, O (Eq. [7]) and Figure 5R (Eq. [8]) show that the phase, θ, of NFC is significantly correlated with their SPL_θ and SPL_CC at the narrower band. The linear regression fitting between θ and SPL_θ improved from the R ²=0.335 to R ²=0.91, and the slope of 0.47 increased to 0.92 (Fig. 5P vs. in Fig. 5R, the Y axes labels are reversed in an upside-down manner). The results with global signal regression are shown in Figure 6.

Global signal regression

We conducted analyses with and without global signal regression in this study. Figures 4 and 6 represent analyses with global signal regression corresponding to Figures 3 and 5, respectively. They have similar results.

Discussion

NFC has been a subject of debate in terms of its origin, interpretation, artifacts, relationship with structural connectivity, and possible neurophysiological functions. In the present study, our main finding is that the NFC significantly correlates with the SPL (either in SPL_CC or SPL _θ) in the human brain network at resting-state conditions, suggesting that the NFC reflects the result of phase accumulation along the shortest path in brain functional networks. With theoretical calculation and simulated results, we demonstrate that the NFC is derived from the phase difference when simulated signals transfer along the multinodal pathway. We also demonstrate that significant linear correlations between NFC and the SPL are frequency bandwidth dependent. These results support the mechanistic hypothesis that the origin of NFC results from phase accumulation along the SPL in a network. One of our study limitations is that our simplest simulation only demonstrated the single input for each node. In a real brain, a node should receive more than one input.

The detailed relationships between anatomical architecture and synchronous patterns that allow brain operation in spatially segregated and functional integrated regions remain unclear. The present study provides mechanistic insight as to why the inference of structural connectivity from functional connectivity would be impractical (Honey et al., 2009). This is because a fully connected network model with a network size N could generate N×(N−1)/2 connections. These pervasive connections do not necessarily have one-to-one matched anatomical connections. Nevertheless, neural synchronization requires a minimal connectedness and structural basis (Nowak et al., 1995). In investigating the minimal structural connectedness, the present study, as related to NFC, suggests that the minimum anatomical connection between two nodes could be inferred along the SPL with diffusion spectroscopic imaging (DSI) experiments. Further study is needed to test this hypothesis by combining R-fMRI and DSI experiments from each individual subject (Honey et al., 2009). Alternatively, it is hypothesized that the observed NFC could be the result of inhibitory interneuron networks (Buzsaki et al., 2004) through which the oscillatory timing transforms unconnected principal cell groups into temporal coalitions (Konig et al., 1996). However, it is not known how the inhibitory interneuron networks could regulate the synchronized network and produce the NFC. Recent advancements in optogenetics may provide a method to address this question (Kravitz et al., 2010; Zhang et al., 2007).

In the study, simulation of a simple network has been applied to show the relationship between NFC and phase difference accumulation. The 30° phase difference between the two time series of two adjacent regions is a phase index, which combines a lot of phase differences of different frequencies. We did not simulate the scenario where the accumulated phase difference is larger than 180°, because the arccosine functional is not a monotonic function of an angle and the cross-correlation coefficient value can only be a one-to-one map, which represents a 0° to 180° space, rather than a 0° to 360°.

By applying the methods and concepts from the simulation, we demonstrate that the relationship between the NFC and the SPL is valid in the whole human brain network in the resting-state condition. When limiting the frequency band to 0.025–0.035 Hz, such a relationship becomes even stronger. Noticeably, this relationship was true for all 20 research subjects. Further, when we randomly permuted the individual's cross-correlation matrix, such relationships were lost. It is suggested that this relationship between NFC and the SPL is of an intrinsic nature of neural networks in these human subjects.

We have examined a specific neural circuit between the R-Amy and the L-dlPFC. These two regions are very important, as they are involved in processing cognitive and emotional activities, as well as their interactions (Pessoa, 2008). It is known that there is no direct anatomical connection between the R-Amy and the L-dlPFC (Amaral and Insausti, 1992; Ghashghaei and Barbas, 2002; Perez-Jaranay and Vives, 1991; Ray and Price, 1993; Siegle et al., 2007). It has been reported that the connection between R-Amy and L-dlPFC is negative (Roy et al., 2009). Moreover, we found that the cross-correlation coefficient values between these two brain regions are all nonpositive among all subjects with and without global signal regression. It would be interesting to see whether their negative cross-correlation is the result of their SPL. Our experimental results demonstrated that all 20 participants showed negative connectivity, and their negative connectivity was strongly correlated with their individual SPL, calculated either in SPL_CC or in SPL_θ.

In examining the possible neurophysiological significance of the NFC between the R-Amy and the L-dlPFC further, several studies have reported decreased negative connectivity strengths between the Amy and cortical regions in schizophrenics (Satterthwaite et al., 2010), persons with bipolar disorder (Chepenik et al., 2010), and heroin addicts (Ma et al., 2010). Reduced negative connectivity strength in the default-mode network was found in subjects with attention-deficit hyperactivity disorder (Cao et al., 2009) and increased sleep pressure (Samann et al., 2010). Based on the present study, it is suggested that the change in NFC in these diseases may be related to the changes in the SPL, thereby representing network reorganization (Wang et al., 2010). Although there is a lack of reports examining the NFC and the corresponding SPL in diseased patients, several studies showed that the averaged SPL among the positive connections in reward and memory circuits was decreased in heroin-addicted subjects (Yuan et al., 2010). Moreover, the normalized path length is significantly related to the intelligence quotient (van den Heuvel et al., 2009), and higher intelligence scores correspond to a shorter characteristic structural path length (Li et al., 2009). Our study extends the very real possibility that a variety of neurological diseases and psychiatric disorders may be studied by measuring the changes of SPLs and functional reorganization in the brain.

We did not find a common SPL between the R-Amy and the L-dlPFC among the 20 subjects; different subjects had different SPL. One possible reason is the dynamics of resting-state functional connectivity. The second reason is that different subjects may have a different path organization, because dendrites, axons, and synapses are dynamic during circuit development (Li et al., 2011). Also, it has been reported that differences in the negative correlation strengths of each individual in the default-mode network are significantly reflected in their individual behavior variance (Kelly et al., 2008). Another possible reason is that the SPL is not a measure of actual structural connectivity; rather, it reflects the phase difference of functional synchronization constrained by the large-scale anatomical structure. Although the mechanisms involved in large-scale integration are still largely unknown, the NFC may play an important role in the complex small-world network by providing long-range connectivity without requiring excessively long axons and long conduction delays (Buzsaki et al., 2004). It is suggested that the existence of the negative synchrony in a network might, in particular, contribute to modulating the so-called “binding problem,” that is, the problem of integrating structurally distributed information into coherent functionally representational patterns, where phase synchronization is crucial as a mechanism for large-scale integration (Varela et al., 2001).

Conclusion

In conclusion, the present study supported our hypothesis and demonstrated that NFC from either theoretically simulated datasets or the normal human brain significantly correlates with the SPL. The NFC value between the R-Amy and the L-dlPFC is significantly correlated with the SPL across the 20 human subjects. Such a significant relationship between the NFC and SPL exists in all NFC connections in the whole brain. The percentage of NFC connections among all connections significantly correlated with the corresponding Euclidean distance. The longer the distance is, the higher the NFC percentage will be. Further, the correlations between the NFC and SPL are frequency bandwidth dependent in that the narrower the frequency band, the less the variance of the correlation will be. These results suggest that the NFC may give rise to the phase delay of the synchronous signals along the shortest path in the large-scale brain functional networks.

Footnotes

Acknowledgments

The authors thank Carrie M. O'Connor, M.A., for editorial assistance, and Ms. Judi Zaferos-Pylant, B.S.M., and Yu Liu, M.S., for MRI technical support. This work was supported by National Institutes of Health (grant R01 AG20279 and NIH-NCRR CTSA program grant 1UL1RR031973).

Author Disclosure Statement

None of the authors of this article has reported any possible conflict of interests.

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Negative Functional Connectivity and Its Dependence on the Shortest Path Length of Positive Network in the Resting-State Human Brain

Abstract

Introduction

Theory

Materials and Methods

Subjects, imaging acquisition, and analyses

Human subjects

Data acquisition

Data preprocessing

Global signal regression

Constructing functional network

Different frequency bands

Simulations

Information flow from source node S to destination node D

Effects of the Euclidean distance on NFC

Determine the relationship between the NFC and the SPL calculated in CC values (SPLcc)

Determine the relationship between the NFC and the phase accumulation along the SPL calculated in phase (SPLθ)

Testing models

Results

Simulated results

Relationship between NFC and spatial distance

Negative connectivity between the right Amy and the left dlPFC correlated with SPL

Negative cross-correlation values are significantly correlated with their corresponding SPLs in the human brain

The relationship between the negative cross-correlation values and their SPLs in the human brain is frequency bandwidth dependent

The negative cross-correlation values in phase, θ, significantly correlated with their corresponding SPLs in the human brain

Global signal regression

Discussion

Conclusion

Footnotes

Acknowledgments

Author Disclosure Statement

References

Determine the relationship between the NFC and the SPL calculated in CC values (SPL_cc)

Determine the relationship between the NFC and the phase accumulation along the SPL calculated in phase (SPL_θ)