Stable functional networks exhibit consistent timing in the human brain

Network analysis is increasingly used to study the human brain. Chapeton et al. use intracranial EEG to identify effective connections in the brain that exhibit consistent timing across multiple temporal scales. Functional networks constructed from these connections are stable and exhibit a preferred direction for information propagation.

Abstract

Despite many advances in the study of large-scale human functional networks, the question of timing, stability, and direction of communication between cortical regions has not been fully addressed. At the cellular level, neuronal communication occurs through axons and dendrites, and the time required for such communication is well defined and preserved. At larger spatial scales, however, the relationship between timing, direction, and communication between brain regions is less clear. Here, we use a measure of effective connectivity to identify connections between brain regions that exhibit communication with consistent timing. We hypothesized that if two brain regions are communicating, then knowledge of the activity in one region should allow an external observer to better predict activity in the other region, and that such communication involves a consistent time delay. We examine this question using intracranial electroencephalography captured from nine human participants with medically refractory epilepsy. We use a coupling measure based on time-lagged mutual information to identify effective connections between brain regions that exhibit a statistically significant increase in average mutual information at a consistent time delay. These identified connections result in sparse, directed functional networks that are stable over minutes, hours, and days. Notably, the time delays associated with these connections are also highly preserved over multiple time scales. We characterize the anatomic locations of these connections, and find that the propagation of activity exhibits a preferred posterior to anterior temporal lobe direction, consistent across participants. Moreover, networks constructed from connections that reliably exhibit consistent timing between anatomic regions demonstrate features of a small-world architecture, with many reliable connections between anatomically neighbouring regions and few long range connections. Together, our results demonstrate that cortical regions exhibit functional relationships with well-defined and consistent timing, and the stability of these relationships over multiple time scales suggests that these stable pathways may be reliably and repeatedly used for large-scale cortical communication.

Introduction

Communication between brain regions plays an important role in cortical function, both at rest and during a variety of normal and pathological conditions (Fox et al., 2005; Kramer et al., 2008, 2011; Bullmore and Sporns, 2009; Rubinov and Sporns, 2010; van Straaten and Stam, 2013). As a result, the study of both structural and functional connections between brain regions has become an object of intensive investigation. At the cellular level, neuronal communication occurs through axons and dendrites and can be measured directly. Despite a growing body of research concerning large-scale functional networks (Bullmore and Sporns, 2009), communication between brain regions remains less well understood.

Unlike the study of single neurons, studying communication between brain regions requires indirect methods for assessing neuronal activity, such as changes in blood oxygenation levels using functional MRI or local field potentials using electrophysiology. These methods generally identify statistical relationships between the activity of different brain regions during either task-based paradigms or at rest (Bullmore and Sporns, 2009; Amini et al., 2010; Rubinov and Sporns, 2010; Friston, 2011; van Straaten and Stam, 2013). Interpreting these methods can be difficult, however, as an observed statistical relationship may reflect either functional communication or simply independent but similar patterns of activity (Friston, 2011; Buckner et al., 2013). Yet despite these challenges, there is growing evidence that there are ongoing functional relationships between brain regions that are likely shaped by underlying structural connectivity (Damoiseaux and Greicius, 2009, Gonzalez-Castillo et al., 2015). Indeed, the study of structural connections in the brain, most often carried out using imaging methods such as diffusion tensor imaging (DTI), can complement these functional studies. While highly informative, these structural maps do not by themselves identify whether two brain regions communicate, nor do they identify the direction of communication between them (Keller et al., 2014).

Measures of effective connectivity attempt to address this question and are premised on the fact that there should be a direction of communication as well as a predictive relationship between the activities of two neural populations (Reeke et al., 2005; Friston, 2011; Nigam et al., 2016). In individual neurons, measures of effective connectivity demonstrate that every time information is conveyed over the same structural pathway, the time required for communication is well defined and preserved (Finnerty et al., 2015; Nigam et al., 2016). Hence, one possible approach for identifying effective connections at larger spatial scales is to make consistent timing a requirement. If information is conveyed from one brain region to another via a stable pathway, as occurs at the scale of single neurons, then we should observe the same time delay for communication between them. Subsequently, any time delay identified through such measures of effective connectivity can then provide valuable information regarding the direction of activity propagation between brain regions.

Here, we take advantage of the millisecond precision afforded by intracranial electroencephalography (iEEG) to identify effective connections in the human brain that exhibit consistent timing. We explore these questions by examining iEEG captured during wakefulness from nine participants with medically refractory epilepsy who had subdural strip and grid electrodes implanted for seizure monitoring. We first ask whether functional relationships demonstrating precise timing exist in the human brain at this spatial scale. To avoid making any assumptions regarding the neural mechanisms of communication, we use a coupling measure based on time-lagged mutual information, which captures both linear and non-linear statistical relationships between two brain regions (Varma et al., 1997; Chen et al., 2000; Jeong et al., 2001; Na et al., 2002; Cho et al., 2012; Lenne et al., 2013; Nigam et al., 2016). We use a conservative thresholding criterion to define effective connections as only those with a statistically significant increase in average mutual information at a consistent time delay. This is an inherently directional measure, as it implies that knowledge of the activity in one region would allow an external observer to better predict subsequent activity in the other region.

After establishing the presence of connections exhibiting temporal consistency over several seconds, we then ask if and how these connections change over longer time scales. Recent studies have suggested that identifying reproducible networks reflecting stable pathways for communication is challenging even within the same subject, although recurring patterns have been identified over longer time scales (Bassett et al., 2011; Kramer et al., 2011; Gonzalez-Castillo et al., 2015). An important caveat, however, is that connections identified at each time point in these studies are not subject to the requirement of temporal consistency. Hence, it is not known whether connections that exhibit consistent time delays over several seconds maintain that consistency minutes, hours, or even days later. If such stability exists, this would suggest that these connections are reliably and repeatedly used by the brain for communication, and would provide valuable insights for both researchers and clinicians seeking to examine how information is conveyed between brain regions in both normal function and disease.

Materials and methods

Participants

Nine participants with medication-resistant epilepsy underwent a surgical procedure in which platinum recording contacts were implanted subdurally on the cortical surface. In all cases, the clinical team determined the placement of the contacts to best localize epileptogenic regions over the temporal lobes. Data were collected at the Clinical Center at the National Institutes of Health (NIH; Bethesda, MD). The research protocol was approved by the Institutional Review Board, and informed consent was obtained from the participants.

Intracranial recordings

IEEG data were recorded from subdural contacts (PMT Corporation, AdTech) using a Nihon Kohden EEG data acquisition system and sampled at 1000 Hz. Subdural contacts were arranged in both grid and strip configurations with an inter-contact spacing of 10 mm. Contact localization was accomplished by co-registering the postoperative CTs with the postoperative MRIs using both FSL Brain Extraction Tool (BET) and FLIRT software packages and mapped to both MNI and Talairach space. The resulting contact locations were subsequently projected to the cortical surface of a Montreal Neurological Institute N27 standard brain to identify anatomic locations (Dykstra et al., 2011). Preoperative MRIs were used when postoperative magnetic resonance images were not available.

We captured 30-s continuous iEEG recordings from each electrode, and define each 30-s recording as a block. We collected 10 blocks on two separate days for analysis, for a total of 20 blocks per participant. The 10 blocks of iEEG data from a given day were captured during a single 2-h recording, which we defined as a session. We reviewed video captured during seizure monitoring to confirm that participants were awake during every session, did not exhibit any seizure activity, and were not engaging in any formal behavioural testing. For each session, we collected blocks separated by 10 min on average. We rejected electrodes exhibiting obvious artefacts, epileptiform discharges, abnormal signal amplitude, and/or large line noise, retaining 69 ± 4 (mean ± SEM) electrodes for analysis in each participant (Mitra and Bokil, 2009; Bigdely-Shamlo et al., 2015) (for detailed information on data preprocessing see Supplementary material).

Time-lagged mutual information

Mutual information relies on generating an estimate of the joint probability distribution of voltage values from the two signals. We first normalized the continuous voltage traces separately for each electrode in each 1-s epoch to have zero mean and unit variance, and then discretized the resulting traces into voltage bins. We estimated the size of one unit of normalized voltage to be 42 ± 5 µV (Supplementary material). In almost all epochs considered, the normalized voltages for all electrodes were within −5 to 5, corresponding to a range of approximately ±210 µV. Therefore, we used 30 equally sized voltage bins for discretization based on Scott’s rule (bin size = 0.33 in units of normalized voltage; see Supplementary material) (Scott 1979). We created 2D histograms from the discretized voltage traces to estimate the joint and marginal probability distributions of the two voltage traces (Fig. 1B). Finally, we used these distributions to calculate the mutual information (Shannon, 1948; Cover and Thomas, 2005) between the two electrodes as:

$$I\left(X,Y\right)\hspace{0.5em}=\hspace{0.5em}{\displaystyle \sum }_{y\in Y\hspace{0.5em}}{\displaystyle \sum }_{x\in X\hspace{0.5em}}p\left(x,y\right)lo{g}_2\left(\frac{p\left(x,y\right)}{p\left(x\right)p\left(y\right)}\right)$$ (1)

where X and Y represent the distributions of discretized voltages for the two electrodes. p(x, y) represents the joint probability of observing a voltage of bin x on one electrode and a voltage of bin y on the other, whereas p(x) and p(y) represent the marginal probabilities of occurrence for each electrode separately. We averaged the mutual information values from all 1-s epochs within a block to determine the average mutual information, $Ī$?>, for that electrode pair during that block.

Figure 1.

Time-lagged mutual information as a measure of effective connectivity. (A) Local field potential signals, x(t) and y(t), are captured from two electrode contacts placed for seizure monitoring. (B) Joint and marginal probability distributions are estimated from discretized voltages from each time series. (C) The average mutual information between x(t) and y(t) is computed for every temporal offset, τ. (D) The relation between z-scored coupling W and temporal offset τ for an electrode pair that is not effectively connected. (E) The maximum coupling, W (τ_max), for this electrode pair occurs at a time delay of 9 ms; it exceeds a global threshold, and so this electrode pair is considered to be effectively connected.

We calculated $Ī$?> for the voltage traces from each electrode in a pair without any temporal offset between them, in which case the time delay is defined as τ = 0 and the average mutual information for that time delay is $\stackrel{}{Ī}\left(0\right)$?>. We also performed the same calculation while imposing time delays between the voltage traces of electrodes in a pair from −250 ms to 250 ms (Fig. 1C). We examined every time delay, τ, using 1-ms steps. Hence, τ = 0 captures time delays from −0.5 ms to +0.5 ms. In this manner, we calculated an average mutual information for each time delay, $\stackrel{}{Ī}\left(\tau \right)$?>, for each electrode pair (Fig. 1D and E).

Functional networks

To assess the effective connectivity between two electrodes, we used the maximum of $Ī\left(\tau \right)$?> across all time delays, τ. We defined the time delay at which this maximum occurs as the latency between those electrodes, τ_max, and its sign determines the direction of coupling. As all electrode pairs will exhibit a maximum for some time delay τ, we were interested in whether, for each electrode pair, $Ī({\tau }_{max\hspace{0.5em}})\hspace{0.5em}$?>is significantly larger than for any other time delay, $I(\tau \ne {\tau }_{max\hspace{0.5em}})$?>. To this end, we first defined the coupling between two electrodes, W(τ), as the z-scored value of $Ī\left(\tau \right)$?> over all τ:

$$W\left(\tau \right)\hspace{0.5em}=\hspace{0.5em}\frac{Ī\left(\tau \right)-{\mu }_{Ī}}{{\sigma }_{Ī}}$$ (2)

where ${\mu }_{Ī}$?> and ${\sigma }_{Ī}$?> represent the mean and standard deviation of $Ī$?> over all $\tau$?>. In the case where there is no preferred time delay between a pair of electrodes, the distribution of W(τ) values across all τ should be approximately normal (Fig. 1D). Under this assumption, we considered a connection to be significant if $W\left({\tau }_{max}\right)>{\Phi }^{-1}\left(0.05/N_{pairs}\right)$?> where Φ is the standard normal cumulative distribution function and N_pairs is the total number of electrode pairs in an individual participant. Because we tested each electrode pair separately, we defined a significance threshold as P < 0.05/N_pairs to account for multiple comparisons (Bonferroni correction). If W(τ_max) exceeds this threshold, we defined that electrode pair to be effectively connected with latency τ_max. To eliminate spurious coupling due to volume conduction, we eliminated all connections where the maximal coupling occurred with zero latency, τ_max = 0 (Nunez and Srinivasan, 2009; Chu et al., 2012) (Supplementary material).

We constructed functional networks using the identified connections by considering each electrode as a node. We represent all electrode pairs that exhibited an effective connection as defined above with edges connecting the corresponding nodes. We summarized the topological features of these functional networks using the average degree, characteristic clustering coefficient, characteristic path length, and global efficiency. A node’s degree is defined as the number of edges connected to that node. Its clustering coefficient is the fraction of the node’s partners that are also connected to each other, and its average path length is the average distance (in terms of edges) between that node and every other node in the network. Finally, efficiency is the average inverse distance between that node and every other node, where we defined the inverse distance between nodes with no paths between them to be zero. We calculated these metrics using modified scripts from the Brain Connectivity Toolbox (Rubinov and Sporns, 2010) and generated network visualizations using Gephi (Bastian et al., 2009). Except where otherwise noted, computational analyses were performed using custom written MatLab (MathWorks) scripts.

Statistical analysis

To assess the extent to which two functional networks are similar, we calculated the phi similarity coefficient, ϕ, between unweighted undirected versions of the adjacency matrices describing those networks (Cramer, 1945; Guilford, 1954). Each unweighted adjacency matrix describes whether an edge exists in each of the N_pairs possible connections between all nodes. We calculated ϕ between two networks as:

$$\varphi =\frac{\text{ad}-\text{bc}}{\sqrt{\left(\text{a}+\text{b}\right)\left(\text{a}+\text{c}\right)\left(\text{b}+\text{d}\right)\left(\text{c}+\text{d}\right)}}$$ (3)

where a is the fraction of all possible connections, N_pairs, that contain an edge in both networks. b is the fraction of all possible connections that contain edges in the first network, but not the second, while c is the fraction containing edges in the second network but not the first. Finally, d is the fraction of all possible connections that did not contain an edge in either network. For binary data such as unweighted undirected adjacency matrices, ϕ is equivalent to Pearson’s correlation coefficient. We can similarly quantify how similar an individual node’s connections are across different blocks by calculating the similarity ϕ between a node’s corresponding column in the adjacency matrix from one block, and its columns from different blocks (Fig. 3A, inset). We defined a node’s homogeneity as the average of ϕ for all combinations of blocks.

Figure 3.

Contributions to stability. (A) The spatial distribution of homogeneity for all nodes in a single participant is not uniform. The connections for a single node during all data blocks can be represented by a connectivity matrix (inset). Every row represents a possible connection with another node, and every column represents a data block. Every black bar represents a functional connection. (B) The correlation between node homogeneity and its degree for all nodes in a single participant. (C) The correlation between node homogeneity and its clustering coefficient for all nodes in the same participant. (D) Correlations of latency (τ_max, top) and coupling strength [W(τ_max), bottom] to distance between electrodes for all participants. Error bars represent 95% confidence intervals across all data blocks. (E) Scatter plots of scaled coupling pooled across participants versus distance. (F) The black bars represent the mean and standard error in scaled coupling within 15 mm bins. The red curve is the fit of a power law model to the data (adjusted R² = 0.97).

We statistically tested the similarities between functional networks by comparing them to the similarity we would expect by chance. To determine chance similarity, we computed how similar a network was to permuted versions of itself, where each permutation comprises a shuffling of the edge labels while preserving the degree distribution. To shuffle the edge labels, we randomly selected a pair of edges, and rewired their connections. Hence, given two connections A–B and C–D, we rewired their edges to yield A–D and C–B. If these new connections were already present, we chose a different pair. In this manner, every node, A, B, C, and D, retains the same degree but has different connections (Maslov and Sneppen, 2002). Because using the original identified functional network as a seed would reduce the surrogate similarity values, for each identified functional network, we first created one shuffled network with the specified degree distribution, and then used this as a seed to create 1000 surrogate networks, retaining the same degree distribution but with rewired edges. We computed all moments and statistics related to similarities and correlations on Fisher transformed versions of the distributions (Corey et al., 1998).

To examine whether connections exhibit conserved latencies across data blocks, we required a minimum sample size of four data blocks based on effect size and statistical power considerations. We considered a hypothetical case when a distribution of latencies is centred at 1 ms with a standard deviation of 0.5 ms. In this scenario, and given a temporal resolution of 1 ms, demonstrating a significant (P < 0.05) difference from zero with a statistical power of 0.8 using a one-tailed t-test would require a minimum sample size of four.

To investigate the direction of activity propagation, we represented each edge as a vector with a 3D direction. The spatial direction of each vector is determined by the spatial coordinates of the electrodes, extending from one electrode to the other depending on the latency of that pair. Because we were only interested in directionality, we assigned all vectors the same magnitude of one. To assess for the consistency of vector directions, we first projected these vectors on to the three standard orthogonal anatomic planes—sagittal, axial and coronal. To statistically test the distribution of spatial directions in each plane, we applied a Rayleigh z-test of non-uniformity to the distribution of angles for the projected vectors (Fisher, 1953; Berens, 2009).

Results

Functional connections exhibit consistent latencies

We collected iEEG from nine participants [three male; age 39.6 ± 3.6 years (mean ± SEM)] with medication-resistant epilepsy who underwent a surgical procedure for placement of subdural electrodes for seizure monitoring. We began by estimating the mutual information between local field potential (LFP) signals captured at every electrode (Fig. 1A; see ‘Materials and methods’ section). We divided voltage values from each LFP trace during 1-s epochs into discrete voltage bins and, for every electrode pair, estimated the joint and marginal probability distributions of voltages to calculate mutual information (Fig. 1B). We repeated this for all 1-s epochs in a 30-s block of data (98% overlap, 20 ms step size, n ≈ 1450 epochs), and calculated the average mutual information between that electrode pair over all 1-s epochs during that block.

We were interested in understanding how the average mutual information between electrode pairs depends on the time delay between them. Within every block, we calculated the average mutual information for every electrode pair using the LFP signal from one electrode and the LFP signal from the other electrode captured at a different time delay (τ; Fig. 1C). For every electrode pair, we calculated the average mutual information for all time delays between −250 ms and 250 ms in 1 ms steps (Fig. 1D and E). The time delay, which exhibits the maximum average mutual information, τ_max, reflects the latency and coupling direction for that electrode pair during that block. The magnitude reflects the extent to which mutual information consistently increases at this time delay averaged across all ∼1450 epochs during that block. All electrode pairs will exhibit some preferred time delay using this approach. However, we were interested in identifying only pairs with a significant increase in mutual information that occurs at a consistent time delay.

To this end, for each electrode pair we defined a coupling measure, W(τ), that characterizes the relative change in mutual information at every time delay, τ (see ‘Materials and methods’ section). Most electrode pairs did not exhibit a time delay, τ, which demonstrated a clear increase in coupling, W(τ), suggesting that there was no consistent temporal relation between them. We defined those pairs as unconnected (Fig. 1D). However, some electrode pairs demonstrated a single time delay, τ, during which coupling, W(τ), was clearly greater than during any other time delay (Fig. 1E). To identify these electrode pairs, we defined a conservative global threshold for each participant that corrected for multiple comparisons across all electrode pairs (Bonferroni adjusted α = 0.05/N_pairs; N_pairs = 2403 ± 267). We only considered electrode pairs with maximal coupling, W(τ_max), which exceed this threshold to be effectively connected (for analysis of coupling using linear correlation, see Supplementary material). Note that because we averaged mutual information over all epochs in a given block, an electrode pair can only exhibit coupling that exceeds this threshold if mutual information is consistently and repeatedly higher for one time delay than for all others. In this manner, we found that the average network density (the fraction of significant connections out of all possible connections) was 2.3% ± 0.5% across all participants.

Stability of connectivity and network topology

Having established the presence of electrode pairs that demonstrate significant increases in coupling at consistent time delays, we then built functional networks for every 30-s block of data using only these connections. We defined every electrode as a node, and every identified effective connection as a directed edge between the corresponding nodes (Fig. 2). For every participant, we constructed functional networks from 20 blocks of data, collected on two separate days of intracranial monitoring (10 blocks per session separated by 10.95 ± 0.16 min each; two sessions collected on separate days separated by 30.05 ± 3.64 h). Functional networks constructed from two separate blocks in the same participant exhibited a similar profile (Fig. 2A).

Figure 2.

Stability of network connectivity and topology. (A) Functional networks calculated for a single participant on different days exhibit similar profiles (inset, CT scan showing electrode locations). Nodes are placed according to 3D coordinates of the electrodes they represent, and each edge represents an effective connection. The colour of each node represents its degree. (B) The average similarity, $\stackrel{-}{\varphi }\hspace{0.5em}$?>, between networks from 10 blocks recorded in the same day, as well as between networks from different days, is significantly larger than zero for all participants (mean ± SD). (C) The probability densities of surrogate similarities and of Day 1 versus Day 2 similarities are shown in blue and red, respectively for Participant 1. The average similarity between networks from different days is significantly larger than the similarity between degree-matched surrogate networks. ϕ* represents P = 0.05. (D) The distributions of degree, characteristic clustering coefficient, characteristic path length, and global efficiency are not significantly different for each participant from day to day, but show differences between participants. Each pair of Tukey boxplots represents the distributions of network metrics for Day 1 (black) and Day 2 (red) for one participant. The whiskers cover points within 1.5 IQR of the first and third quartile (where IQR is the inter-quartile range); values outside of this range are considered outliers and are represented by individual points. The notches (triangles) are placed at the median ± 1.57 IQR/√n, where n is the number of networks from each day (n = 10).

To quantify the extent to which connections are preserved across different blocks in the same participant, we computed the phi similarity coefficient, ϕ, between the unweighted undirected versions of the adjacency matrices describing the functional networks constructed for each block (see ‘Materials and methods’ section). We found high levels of similarity ($\stackrel{-}{\varphi }\hspace{0.5em}$?> ≥ 0.35 for all participants) when comparing functional networks between data blocks within the same day, and when comparing networks between data blocks from different days (Fig. 2B). For both within and across day comparisons, the distribution of similarities in all participants was significantly greater than zero. If an electrode pair is connected in one block of data, it is very likely that the same pair will be connected minutes, hours, and even days later.

To ensure that the high levels of similarity we observe are not simply a consequence of comparing finite sized networks with similar degree distributions, we constructed 1000 degree-matched surrogate networks for each functional network constructed from each data block (see ‘Materials and methods’ section). We calculate the similarity between each true functional network and all of its surrogates. We then pooled the distribution of all surrogate similarities from all data blocks in each participant together to ensure the largest possible variance in the surrogate similarities (Fig. 2C). While all comparisons between true and surrogate networks have matching degree distributions, there is no such constraint when comparing the true functional networks to one another. Nevertheless, for all within and across day comparisons in all participants, the mean similarity between functional networks was significantly greater than the surrogate comparisons (P < 0.05, permutation test).

Although we observed similarity between adjacency matrices across blocks and days, it is possible that the addition or subtraction of a few edges from key nodes could significantly alter the overall network topology. To examine this possibility, we calculated basic network metrics of centrality, segregation, and integration (degree, clustering coefficient, and characteristic path length, respectively). For each functional network constructed during each block, we calculated the average degree and clustering coefficient across all nodes, the characteristic path length, and because networks are not fully connected, the global efficiency. To assess if the shape or location of the distributions of these metrics has significantly changed from one day to another, we used a two-sample Kolmogorov-Smirnov test. The distributions of these metrics from all 10 networks identified in one day did not significantly differ from those calculated for the 10 networks identified on the second day in any participant (Fig. 2D; D_max < 0.5, P_min > 0.05, where D_max and P_min are the maximum Kolmogorov-Smirnov test statistic and minimum P-value across all comparisons).

Contributions to network stability and coupling

After constructing functional networks using time-lagged mutual information to identify effective connections in all participants (Supplementary Fig. 3), we quantified the homogeneity of each node’s connections across blocks to assess contributions to overall stability (Fig. 3A). In individual participants, the spatial distribution of homogeneity was not uniform. Across all nodes, however, node homogeneity strongly correlated with node degree, averaged across all blocks in each participant [Spearman’s ρ = 0.8, P < 10⁻¹⁴ for this participant; Fig. 3B; across participants $\stackrel{-}{\rho }\hspace{0.5em}$?> = 0.83 ± 0.04, t(8) = 22.6, P < 10⁻⁸]. We found a similar relation between the homogeneity of each node and its average clustering coefficient across all blocks [Spearman’s ρ = 0.73, P < 10⁻¹¹ for this participant; Fig. 3C; $\stackrel{-}{\rho }\hspace{0.5em}$?> = 0.65 ± 0.03, t(8) = 19.27, P < 10⁻⁷].

As each identified effective connection is characterized by the Euclidean distance between the electrodes, we next examined the relation of both the magnitude of maximum coupling, W(τ_max), and the latency, τ_max, to distance. W(τ_max) was negatively correlated with distance in all participants [across participants $\stackrel{-}{\rho }\hspace{0.5em}$?> = −0.36 ± 0.04, t(8) = −8.35, P < 0.0001; Fig. 3D], suggesting that temporal consistency is more reliable at shorter distances. We fit the relation between maximum coupling, normalized within each participant ($\stackrel{\sim }{W}$?>), and distance, d (in mm), and found that it is best fit with a power law,

$$\stackrel{\sim }{W}\left(d\right)\hspace{0.5em}=\hspace{0.5em}0.88d^{-0.55},\hspace{0.5em}\text{adjusted}{R}^2\hspace{0.5em}=\hspace{0.5em}0.97$$ (4)

and can be well approximated as $\sim \frac{1}{\sqrt{d}}$?> (Fig. 3F; for fits with exponential and linear functions, see Supplementary material and Supplementary Fig. 4). Conversely, the latency, τ_max, was positively correlated with the Euclidean distance between electrodes across individual participants [across participants $\stackrel{-}{\text{ρ}}$?> = 0.25 ± 0.04, t(8) = 6.43, P < 0.001; Fig. 3D]. However, the relation between distance and latency was weak (Supplementary material), which is not surprising given that other structural factors such as axon diameter or myelination can strongly influence axonal conduction velocities, and hence, the time delay with which two brain regions communicate (Fields, 2008; Zatorre et al., 2012).

Stability of latencies

The stability of the connections we identified raises the possibility that the latencies, and hence the direction of these connections, are also preserved across different blocks. To investigate this, we examined the latency, τ_max, of every electrode pair identified as having an effective connection. Our aim was to statistically assess the consistency of these latencies, and therefore we only included electrode pairs that exhibited a connection during at least four blocks (n = 83 ± 10 across participants). Notably, if a connection was present in at least four blocks, then that connection was present in an average of 9.6 ± 0.3 blocks for each participant. If the direction of a specific connection was not preserved, then the latencies of that connection in different blocks will just as likely be positive as negative and will not be significantly different from zero. We examined this in all participants, and found 59% ± 3% of these effective connections have stable latencies that demonstrate a consistent direction and whose values are significantly different than zero (|t_min| ≥ 1.813, P_max < 0.05, unpaired t-test, where |t_min| and P_max are the minimum t-statistic and maximum P-value across all comparisons). After correcting for multiple comparisons, 30% ± 2% of the effective connections still have latencies whose values are significantly different than zero (|t_min| ≥ 3.903, P_max < 0.001, Bonferroni adjusted α = 0.05/n; Fig. 4A; see Supplementary Fig. 5 for all participants).

Figure 4.

Stability of connection latencies. (A) Latencies for individual electrode pairs during different data blocks for Participant 1. Only electrode pairs for which the distribution of τ’s was significantly different from zero are shown (Bonferroni corrected). Each point represents the latency during a data block exhibiting significant coupling. Each row of points represents the latencies for a single electrode pair for blocks with significant coupling. Black bars represent the mean and standard error of the distribution of latencies for each electrode pair. The counts on the right represent the number of data blocks, out of 20 possible, in which that electrode pair exhibited an effective connection. (B) For one electrode pair (star symbol in A), the maximum coupling, W (τ_max) and latency, τ_max, are shown for all data blocks. Data blocks with significant (non-significant) coupling are shown in red (black). Only one data block, with a low maximum coupling strength, exhibited a noticeably different latency. (C) Tukey boxplots of the difference in mean (left) and standard deviation (right) of latencies between data blocks without and with significant coupling, pooled across all participants. The whiskers cover points within 1.5 IQR of the first and third quartile, and outliers are designated as individual points. (D) Distribution of conduction velocity estimates obtained by dividing interelectrode distance by the latency, τ_max, for connections present in at least four blocks.

Our data suggest that when an electrode pair demonstrates an effective connection, assessed using our conservative coupling threshold, the latency of that connection, τ_max, is likely to be preserved. In individual electrode pairs, the latencies were similar even during data blocks with coupling that does not exceed that threshold (for a representative example, see Fig. 4B). We calculated the difference in mean latencies between all data blocks without and with significant coupling for every connected electrode pair. The distribution of these differences, pooled across all participants, does not differ from zero [Fig. 4C, left; t(212) = 0.498, P = 0.62]. However, when we calculated the difference in standard deviations of the latency, we found the standard deviations to be significantly larger during data blocks without significant coupling [Fig. 4C, right; t(212) = 3.565, P < 0.001]. Outliers contributing to this difference were all positive, suggesting that the latencies during blocks with low maximum coupling are more variable.

We used the identified latencies to estimate the velocity of signal propagation between electrode pairs that exhibits an effective connection in at least four blocks. We derive the velocity by dividing the inter-electrode distance by the latency, τ_max. The distribution of velocities ranges between 0.33 and 63.1 m/s (mean 9.8 ± 2.8 m/s; Fig. 4D). This distribution of velocities is consistent with the velocities observed in previous studies of non-human primate cortex, with values ranging from 0.3 m/s for connections within V1 (Girard, 2001) to 60 m/s for connections between the frontal eye fields and the lateral intraparietal area (Ferraina, 2002).

Anatomic networks

As each electrode has a location defined by its spatial coordinates, the effective connections we identified in our data therefore also provide information regarding the preferred direction of propagation. We represent each edge in each functional network as a 3D vector with a direction defined by the electrode locations and the latency between them (Fig. 5A; projections on to the sagittal plane). Given the temporal lobe coverage of the electrodes, these vectors mostly lie in the sagittal plane, and their directions within this plane are not constrained a priori. Nevertheless, 54 ± 11% of functional networks across participants exhibit a non-uniform distribution of directions in the sagittal plane (P < 0.05, Rayleigh test; Fig. 5A, inset).

Figure 5.

Network directionality. (A) Effective connections in an individual network in one participant can be represented as unit vectors pointing from one electrode to the other, with the direction being determined by the sign of that connection’s latency (inset, histogram of directions and mean vector for a single network projected on to the sagittal plane). (B) Projection of the mean vector for all networks from one participant on to the sagittal plane (radial scale in millimetres). The red vector indicates the overall mean vector across all networks for this participant. (C) The overall mean vectors for all participants show a preferred posterior-superior to anterior-inferior direction. (D) Histograms of directions for mean network vectors for all participants projected onto the sagittal, axial, and coronal planes. Each participant contributes 20 vectors, corresponding to the functional networks computed for each data block. The left and right columns correspond to participants with electrodes placed over the left and right temporal lobes, respectively.

We then used the vectors from all identified effective connections to calculate a mean vector for each functional network. In each participant, the directions of the mean vectors collected from each functional network also had a non-uniform distribution (P < 0.05, Rayleigh test; Fig. 5B). Based on the mean vectors calculated from all data blocks, we constructed an overall mean vector for each participant (red arrow in Fig. 5B). Notably, for all participants, the direction of the overall mean vector computed from all functional networks exhibits a preferred posterior-superior to anterior-inferior direction in the sagittal plane (Fig. 5C). The distributions of directions for mean vectors from all networks from all participants demonstrated non-uniformity in all three orthogonal anatomic planes, suggesting a preferred direction for activity propagation (Fig. 5D).

To anatomically characterize cortical connectivity, we examined the effective connections within and between every anatomical region containing electrodes from at least three participants. We defined the reliability of connections between any pair of anatomic regions as the fraction of total possible electrode pairs across all blocks that were identified as exhibiting an effective connection, averaged across all participants. Networks constructed using reliably connected anatomic regions exhibits small-world features (Humphries and Gurney, 2008), with many reliable connections existing between neighbouring regions, and few reliable connections between distant regions (Fig. 6; small-world coefficient = 2.6; Supplementary material).

Figure 6.

Anatomical network. Each node represents a distinct anatomical region containing electrode contacts from at least three participants, and each edge represents a connection between two regions. The reliability of a connection between two regions is represented by the colour of the edge. We display edges only if the reliability of that connection was in the top 15% of all connections. Because most regions contain multiple electrodes, the reliability of connections within each region was calculated and is represented by the colour of each node. Nodes that are coloured grey did not have enough electrode coverage to compute the within region reliability. AG = angular gyrus; C = cuneus; FG = fusiform gyrus; IFG = inferior frontal gyrus; IPL = inferior parietal lobule; ITG = inferior temporal gyrus; MeFG = medial frontal gyrus; MGF = middle frontal gyrus; MOG = middle occipital gyrus; MTG = middle temporal gyrus; PG = parahippocampal gyrus; PoG = postcentral gyrus; PrG = precentral gyrus; P = precuneus; SFG = superior frontal gyrus; SOG = superior occipital gyrus; SPL = superior parietal lobule; STG = superior temporal gyrus; SG = supramarginal gyrus; U = uncus.

Discussion

Our results demonstrate that there exist effective connections in the human brain that exhibit consistent increases in mutual information at specific time delays. Our data therefore provide evidence that the precise timing observed in individual neurons and their axonal projections (Finnerty et al., 2015; Nigam et al., 2016) is a feature of large-scale brain regions as well. Importantly, we found that identified effective connections and their latencies, as well as overall network topology, were preserved over minutes, hours and days. This stability is noteworthy, as the participants’ behaviour was relatively unconstrained. This suggests that the effective connections we identify may reflect stable pathways that are reliably used by the human brain for communication.

Identifying effective connections exhibiting consistent timing results in sparse, directed functional networks. The sparse networks we identify are only composed of 2–3% of electrode pairs, based on a conservative threshold that both requires consistent timing and reduces the effect of threshold on network density (Supplementary Fig. 1B). The relation between threshold and network density, however, is an active area of research, and will require further investigation. Nevertheless, within these sparse networks, we found individual electrodes, or nodes, that exhibit stronger homogeneity in their connections across data blocks. These nodes also have a high degree and clustering coefficient, suggesting that the overall network stability that we observe appears to be in large part mediated by highly connected nodes and their partners, and providing evidence that these nodes may represent critical areas that are important for robust communication across different brain regions. Hence, these stable connections and hubs may provide a basis for future studies of the dynamic flow of information during normal behaviour and cognition, and in disease.

Remarkably, we found that the latencies that define these effective connections are themselves highly preserved across multiple time periods. As many different paths may be available for communication between brain regions, it is noteworthy that the latencies we observe in our data are as consistent as they are. This consistency, not only during a single block but also over multiple time scales, suggests that these observed time delays may indeed reflect constraints governing the observed communication between regions. However, while consistent timing may be a necessary feature of a structural neural connection, it is not sufficient to implicate the existence of one. Moreover, while latencies provide a direction in which activity is propagated, directionality does not prove causality. One brain region may appear to drive the activity of another only because both are driven by a third region with different latencies (Bullmore and Sporns, 2009; Friston, 2011), and our approach does not rule out the possibility that regions from which we are not recording may be contributing to the observed latencies. Hence, directly relating these effective connections to underlying structural pathways via methods such as diffusion tensor imaging or through cortico-cortical evoked potentials (Keller et al., 2014) remains an important step for demonstrating the anatomic substrate through which such communication occurs.

We investigated the relation between distance and both the preferred latencies and maximum coupling in these connections. Although significant, the correlation between latency and distance was not very strong due to large variability in the observed latencies. This is not surprising given that other structural factors such as axon diameter or myelination also influence the time delay with which two brain regions communicate (Fields, 2008; Zatorre et al., 2012). More importantly, however, our estimates of conduction velocities between different brain regions, based on these latencies, are consistent with previous studies in non-human primates (Girard, 2001; Ferraina, 2002). Conversely, we found a stronger relationship, best fit by a power law but also well fit with an exponential decay, between coupling and distance, suggesting that it may be more difficult to communicate with a consistent timing over longer distances. The parameters estimated from these fits may help constrain distance-dependent models describing connectivity and the propagation of activity.

In all participants, we found a preferred direction for activity propagation from the posterior-superior to anterior-inferior temporal lobe that was consistent across all data blocks. Previous studies have identified both a ventral stream pathway for visual and auditory processing and a progression of cortical activity during episodic memory tasks that involves this same direction of activity (Ungerleider and Haxby, 1994; Hickok and Poeppel, 2000; Chalupa and Werner, 2004; Reddy and Kanwisher, 2006; Saur et al., 2008; Milner and Goodale, 2012; Pulvermuller, 2013; Burke et al., 2014; Greenberg et al., 2015). This agreement opens up the possibility that the connectivity can be pooled across participants to reveal general patterns of connectivity in the human brain. Indeed, we found many local connections and few long range connections that were reliable across participants, suggesting a small-world architecture in the human brain that is consistent with previous studies of large-scale functional and structural networks (Bullmore and Sporns, 2009). Extending these data across multiple brain regions, as is done with functional MRI studies of human brain connectivity, could provide a novel way of constructing such templates based on electrophysiology.

For the moment, our data can only speak to the presence of stable functional connections with consistent timing and not necessarily the mechanisms underlying such communication. Our choice for using mutual information offers an approach that is agnostic to the specific neural mechanisms of communication. Measures of connectivity such as spectral coherence and phase synchrony are premised on assumptions regarding frequency specific coupling (Fries, 2005; Bastos et al., 2015). Mutual information avoids any such assumptions, although this generality comes at the expense of being unable to determine the exact functional form of the relationship between two regions. Although many qualitative features of our results can be reproduced using a linear metric such as correlation (Supplementary Figs 1 and 2), there are differences hinting at non-linear interactions between brain regions that may be overlooked with correlation alone. Identifying the specific physiological processes that contribute to the coupling between brain regions, how such coupling may be mediated through previously identified mechanisms such as coherence and phase synchrony (Fries, 2005; Bastos et al., 2015), and how dynamic changes in such coupling may underlie the flow of information between brain regions will be important for understanding how these brain regions communicate.

Glossary

Abbreviation

iEEG

intracranial electroencephalography