Abstract
Communication and dynamic routing play important roles in the human brain to facilitate flexibility in task solving and thought processes. Here, we present a new network perturbation methodology and a corresponding analysis method to investigate and demonstrate the dynamic switching between different excitable pathways in the network. The methodology probes for dynamic changes in network communication pathways based on the relative phase offsets between two weak external oscillatory drivers. To investigate the feasibility and the properties of this method we use a computational modeling approach with delay-coupled neural mass models. In a model of the human connectome we show that network pathways have characteristic timescales and thus specific preferences for the phase lag between the regions they connect. For the analysis of dynamic switches of communication pathways we define the pathway-synchronization-facilitation index (PSF), which measures for a given pair of network nodes how their interaction is modulated by specific phase offsets. Our simulation results indicate that the PSF decreases with increasing shortest path length between the node-pair and increases with the number of different pathways by which the two nodes are connected. To further analyze the contribution of different interaction pathways to the communication between two network nodes, we define the pathway-activation index (PA). Our results show that most pairs of nodes in the connectome have interaction pathways that can be dynamically activated and that 60.1% of node pairs can switch their communication from one pathway to another depending on the phase offsets between the two nodes.
Significance A big challenge in elucidating information processing in the brain is to understand the neural mechanisms that dynamically organize the communication between different brain regions in a flexible and task-dependent manner. In this theoretical study, we present an approach to investigate the routing and gating of information flow along different pathways from one region to another. We show that stimulation of the brain at two sites with different frequencies and oscillatory phases can reveal the underlying effective connectivity. This yields new insights into the underlying processes that govern dynamic switches in the communication pathways between remote sites of the brain.
Introduction
Over the past decades it has been shown that the brain, facing a specific task or not, reveals a well-structured functional organization [4,6,29]. This has been specifically investigated for resting-state networks [2,5,9,24], but also for other networks when the brain is performing different tasks [5,17]. These findings lead to the idea that resting-state networks describe an inherent functional organization of the brain which is optimized to perform a wide range of tasks it encounters frequently [11,15]. If faced with a task that requires synchronization between brain areas not typically coupled at rest, this organization has to be altered temporarily in order to perform that task efficiently [18,26].
Within a complex network like the human brain, multiple structural pathways exist between most pairs of nodes given a sufficiently high spatial resolution. Since synchronization along such pathways seems to play an important role in the formation of functional clusters, we set out to identify general principles of how these pathways interact with each other during synchronization processes. Several studies have emphasized the importance of information transmission delay for synchronization processes as well as its role in the formation of functional clusters in the brain [3,4,7,8,12,14,16,28]. Thus, we hypothesize the time lag inherent to a communication path to be a key factor in the interaction between multiple pathways. These time windows are determined by axonal signal transmission delays as well as rise and decay properties of the post-synaptic response. Here, we focus on the former, expecting region-specific differences in the latter to be negligible for the long-range connections considered in this study. In particular, we predict that two brain regions trying to communicate at a certain frequency with a given phase offset will use only a fraction of their available communication paths. Further, we predict that the selection of communication paths will be influenced by their interaction time windows.
To test these hypotheses, we introduce an extrinsic stimulation set-up that allows to detect network interactions between pairs of nodes. This stimulation approach relies on the entrainment of a given pair of nodes to oscillate at the same frequency, but with a certain phase lag relative to each other. Comparing the coherence along different pathways over different stimulation phase offsets then reveals the phase preferences for different routes. While Figure 1A illustrates the extrinsic stimulation setup, Figure 1B motivates the use of different stimulation phase lags. It is important to note that even in the absence of any interaction through the network, there might be some induced trivial coherence between two stimulated nodes due to the external signal (Figure 1B). Thus, the coherence is measured for many different stimulation phase offsets and the measurement with the lowest coherence is chosen as the baseline. Any deviation in the coherence from this baseline can be attributed to induced changes in the coupling between the two stimulated nodes through the network, which may happen due to a switching in the pathways (compare Figure 1C and 1D). We propose that these differences in phase preferences at different pathways act as a switching and gating mechanism used by the brain to establish communication between remote brain areas when needed. Our method allows to investigate these mechanisms by probing the network for these dynamic switches in communication pathways.
In the next section, we define the computational model used for the evaluation of this methodology. We continue by demonstrating the phase offset preferences of different pathways in a simplified network of only 2-3 nodes. Subsequently, we move on to a human connectome model and show that coherence between stimulated nodes changes significantly over phase lags and how this effect relates to the connectedness and distance of the nodes. In a final step, we identify the pathways responsible for the interaction between the stimulated nodes, analyze their phase lag preferences and identify cases of phase-related switching between pathways. To this end, we evaluate how the activation of predefined structural pathways between stimulated nodes changes over different stimulation phase offsets.
Computational Model
To investigate switching and gating properties of networks based on phase relationships, we employ the computational model described in this section. Our computational model is based on the widely used Jansen-Rit neural mass model [21] which employs a mean-field approach to model the interaction between cell populations in the infragranular (green), granular (blue), and supragranular (red) layer, as illustrated with the relevant equations in Figure 2. The standard parametrization originally proposed by Jansen and Rit reflects cortical oscillatory activity in the alpha frequency band. Since the purpose of this article is the investigation of the effect of pathway time-scales on neural synchronization processes and not the effect of node time-scales, we decided to use this standard parametrization for each node in our network [21]. These parameters were chosen based on experimental findings in the neuroscience literature and are reported in Table S1 in the SI Appendix together with the definition of the transformation function σ(V) from average membrane potentials to firing rates (parameterized sigmoid).
To obtain the results reported below, the following two extensions were added to the standard Jansen-Rit model: First, we coupled multiple Jansen-Rit nodes via delayed, weighted connections between their infragranular pyramidal cell populations (yellow in Figure 2). Secondly, weak external drivers were applied at two stimulation sites influencing the average membrane potential of the infragranular layer with phase offset Δψ between the two drivers (purple in Figure 2).
Results
We first present results of simple simulations with only 2 or 3 nodes. Specifically, we show how the coherence depends on the transmission delay of the connections and on the relative phase offsets between the two external signals. Subsequently, we move on to the connectome simulation, where we first evaluate the overall similarity of the simulated functional connectivity with the functional connectivity obtained from electroencephalography (EEG) recordings. Finally, we present results of the connectome simulation showing a dependence of network communication on the stimulation protocol and the characteristic time-scales of communication paths.
Simple Models With 2 or 3 Nodes
The idea behind the extrinsic stimulation approach can be well explained using a simple toy-model of 2 directly coupled Jansen-Rit nodes, where each node is stimulated with a fext = 11 Hz sinusoidal signal with strength cext =0.25 mV. Figure 3 shows the coherence between the driven nodes for systematic changes in the phase offset between the stimuli and the distance between the coupled nodes. While uni-directionally coupled nodes can have preferences for any stimulation phase offset, as shown in Figure 3A, bi-directionally coupled nodes are more susceptible for stimulation at in- or anti-phase (see Figure 3B). This shows that the communication between coupled pairs of nodes can be modulated by stimulation and that communication channels can have characteristic stimulation phase offset preferences, depending on their length [22].
To quantify the modulation of communication, we define the pathway-synchronization-facilitation (PSF), measuring for a given pair of weakly stimulated network nodes ki and kj how their interaction is dependent on specific phase offsets: where coh(kj, kj, Δφ) is the coherence between network nodes ki and kj for stimulation phase offset Δφ. The PSF is high for node pairs if their coherence is high for one stimulation phase offset and low for another, i.e., the relative phase of the stimulation at the two sites matters strongly. The PSF curves in Figure 3A and 3B show that in both cases there is a PSF effect (PSF ¿ 0) and in the case of bi-directionally coupled nodes the strength of this effect depends on the distance between the nodes.
To extend this idea to communication via indirect pathways, we investigated synchronization between 2 nodes connected only indirectly via a third intermediate node. We used bi-directional couplings for both connections and both end nodes were stimulated as described previously. As can be seen in Figure 3C and 3D, the interaction between the two weakly stimulated nodes not only depended on the length of the connection, but potentially also on the relative position of the third node on the indirect path.
Connectome Model Without Stimulation
As shown in the previous section, the coherence in a network of only three nodes can already exhibit very complex dependencies on the stimulation phase offset. Next, we wanted to analyze network communication patterns in the case of a complex network with multiple competing pathways. To this end, we used a model of 33 delay-connected nodes, representing one hemisphere of the human connectome [14]. The structural connectivity matrix was obtained from diffusion tensor imaging (DTI) data as described in more detail in the SI Appendix. Figure 4A shows the sparse connectivity matrix Cmj used to connect the 33 regions and Figure 4B the corresponding distances Lmj.
For the same 33 regions, EEG resting-state recordings from the same subjects were used to calculate pairwise coherences in the 10 Hz range as shown in Figure 4C (details in SI Appendix). Similarly, we simulated the 33 connected neural-mass-models and processed the time-series of the pyramidal cells in the same way as the EEG data. This yielded a 33 x 33 functional connectivity matrix which we compared to the empirical functional connectivity by calculating the Pearson correlation coefficient.
The selection of parameters was based on the rationale to match the functional connectivity observed in the network model as good as possible to empirical EEG-based functional connectivity from human subjects. We performed a grid search over global structural connectivity scaling cnet and transmission velocity v to obtain the best match between modeled and empirical data. By fitting the velocity, we ensured that our pathway delays reflect realistic, empirically observed timescales of cortico-cortical interactions. We found the highest correlation (r = 0.57) for cnet = 20 and v = 3 m/s, so that we used these parameters for subsequent analyses (Figure 4D). Notice that this correlation is comparable with values of other bottom-up models reported in the literature [14,25], which is remarkable considering that we set a substantial amount of structural connections to 0.
Connectome Model With Stimulation
Based on this model of cortical activity we used the stimulation approach to investigate how pathways facilitate synchronization between network nodes at certain phase lags between the nodes. Specifically, we weakly stimulated different pairs of cortical regions with varying phase offsets between the two stimulation signals while measuring the coherence between the stimulated nodes at each phase offset. As argued above, finding differences in the coherence over stimulation phase offsets would indicate phase-specific communication modulation between the stimulated nodes. Before analyzing PSF effects in the connectome model, it was necessary to determine the optimal stimulation frequency and strength for this model. This was performed in two steps. First, we stimulated a single region in our network with a stimulus of varying frequency (4-22 Hz) and strength (0.01-2 mV) while evaluating the coherence between region and stimulus. The mean coherence (mean over 5 different stimulated nodes) for each parameter combination can be observed in Figure 5A. Since our main analysis will focus on coupling effects through different network paths between two stimulated nodes, we also calculated the coherence between stimulus and all network nodes (Figure 5B). This average coherence to the full network was strongest at 9-11 Hz, which is also the intrinsic frequency of unperturbed network nodes [27]. Interestingly, at this frequency the coherence to the directly stimulated node was weakest (compare 5A). Based on this, we set the frequency of our stimulus to 11 Hz, at which the network (and not only the directly stimulated node) was most susceptible for entrainment by an external stimulation.
In a second step, we stimulated pairs of nodes with 11 Hz stimuli. We varied the stimulus strength (0.25 – 1 mV) and the relative phase offset between the stimuli (0 — 2π), while evaluating the coherence between the stimulated nodes. All other parameters were chosen to be the same as for the previous simulation. As can be seen in Figure 5C and Figure 5D, the variance of the coherence over phase offsets depended on the stimulation strength. Based on visual inspection of the coherence patterns of 20 different region pairs, we chose our stimulus scaling to be cext = 0.5 mV, leaving the variance of the post-synaptic potential of the neural masses in a biologically plausible range and such that the external driver is relatively weak in comparison to the internal network dynamics. This gave us the final set of global model parameters which were used throughout all subsequent simulations.
The variability in the coherence between stimulated region pairs that we observed not only over stimulation phase offsets but also over different pairs (as depicted for 2 exemplary region pairs in Figure 5C and 5D) shows that the stimulated region pairs interacted with each other and that the interactions showed a characteristic profile of phase offset preferences. To statistically confirm the variance in the coherence between stimulated region pairs over phase offsets, we ran simulations with stimulations of each possible node pair. Again, we varied the phase offset between the two stimuli (16 equally spaced phase offsets between 0 and 2π) and evaluated the coherence between the stimulated nodes for each phase offset. Subsequently, those coherences were used to calculate the PSF for each region pair as defined in equation 1. Using a one-sample t-test, we found the PSF effect to be significantly larger than zero (mean = 0.1567, CI = [0.1445,0.1689], t = 25.2595, p ¡ 0.0001). Hence, we were able to show with our extrinsic stimulation approach that pathways facilitated synchronization between cortical nodes and that the facilitatory strength depended on the phase lag between the region’s average PSPs.
With the PSF effect established, we continued by investigating its dependence on certain features of the underlying structural connectivity graph. For this purpose, we searched for all possible pathways between each pair of stimulated nodes based on the structural connectivity matrix reported in Figure 4A. Since every stimulated pair of nodes was connected by at least one path via at most 6 edges, we restricted the search to pathways including 6 edges at maximum. With these pathways at hand, we started out by evaluating how the PSF effect changed with increasing network distance. An analysis of variance showed that the effect of shortest path length (minimum number of edges seperating a pair) on log(PSF) was significant, F(5,521) = 97.6141, p ¡ .0001. As can be seen in Figure 6A, we observed the trend that the PSF effect decreases with the number of nodes separating the stimulated nodes. Furthermore, as depicted in Figure 6B, this trend was supported by a significant correlation between the PSF effect and the length of the shortest pathway between the stimulated nodes (r = −0.56, p ¡ .0001), a measurement that is strongly related to both interregional distance and minimal number of separating edges. Thus, we conclude that there is a tendency for a decrease in the interaction of stimulated node pairs with increasing network distance between the nodes, where network distance can be measured either as the number of edges or as the summed up length of the edges of the shortest pathway connecting the nodes.
Next, we investigated the dependence of the PSF effect on the connectedness between the stimulated nodes. An analysis of variance showed that the effect of the number of connecting paths (only counting paths with 5 edges or less, all nodes with more than 5 connecting paths were pooled into one level) on log(PSF) was significant, F(5,501) = 10.0827, p ¡ .0001. The latter result can be observed in detail in Figure 6C.
Evaluation of Pathway Activation
Having described the influence of the external driver on the coherence between stimulated nodes, we next identified which particular pathways were involved in this interaction. For this analysis, we define the pathway activation (PA) for a pathway through n nodes ki with i = 1..n at a phase offset Δφ as the minimum of the pairwise coherences between neighboring pathway nodes:
In other words, if communication fails at any point along a pathway, leading to a reduced coherence between the involved nodes, this is considered to be a bottleneck for the information flowing through that pathway. We evaluated the pathway activation (PA) 2 for all pathways of up to n = 5 nodes connecting a given pair of stimulated nodes for all stimulation phase offsets. Doing this for each stimulated node pair, we found different classes of pathway interactions: Some pairs show only a very small selectivity for the stimulation phase offset (Figure 7A), while other node pairs were connected by paths with PA values with a strong dependence on the phase offset (Figure 7B, 7C). Moreover, some of these node pairs switched their interaction between different pathways depending on the stimulation phase offset, as shown in Figure 7C and the two switching pathways in Figure 7D and 7E.
To further analyze how the communication via specific pathways depends on the stimulation phase offset, we define the pathway phase selectivity (PPS) of a pathway P1 similar to the PSF as
Pathways with relatively constant PA values for all stimulation phase offsets have a low PPS (example in Figure 7A), while pathways with a high variation in the PA values have a high PPS (example in Figure 7B and 7C). The evaluation of PPS values for all node pairs results in a bimodal distribution (Figure 7F). The activation of pathways in the first mode at PPS = 0.1 is very hard to influence with phase offsets. But we also found many node pairs with pathways in the second mode at PPS = 0.35. The communication of these later node pairs can be modulated using different phase offsets.
In a next step, we analyzed the relationship of pathway-specific phase preferences (as shown in Figure 7A-C) to the phase preferences of the stimulated nodes (as shown in Figure 5C-D). We chose the most active pathway per node pair (averaged over all stimulation phase offsets) and calculated the phase difference between the stimulation phase offset with the highest coherence and the stimulation phase offset with the highest PA. The histogram of these phase differences is significantly different from uniform, χ2(15, N = 514) = 273.05,p < .001, and has a peak at 0 (Figure 7G). In contrast, a similar analysis for the second strongest pathway (excluding all pathways with overlapping sections with the strongest path), results in a histogram that does not differ from a uniform distribution, χ2(15, N = 514) = 14.06, p = 0.52 (Figure 7H). Therefore, we conclude that the pathway with the strongest PA shows a similar phase preference as the coherence between the two stimulated nodes.
Finally, we quantified the switching between the strongest and second strongest pathway per node pair. To this end, we define the pathway switching index (PSI) between pathways P1 and P2 as
The PSI is positive if the two pathways switch their activation depending on the stimulation phase offset, meaning that at one phase offset the first path is more active and at another phase offset the second path is more active. We found that 60.1% (309 of 514) of node pairs have a positive PSI between their non-overlapping strongest and second strongest pathways (Figure 7I). These results suggest that in this network of 33 nodes of the human connectome many node pairs have the capacity to switch their communication between at least two different pathways with a PA characteristic similar to the example shown in Figure 7C.
Discussion
We have carried a computational study of cortico-cortical synchronization processes that strongly emphasizes the role of phase relationships for dynamic switches in communication pathways. We introduced a novel method to detect network interactions between pairs of cortical regions via an extrinsic stimulation scheme. Using our method, we were able to quantify the influence of different pathways on cortico-cortical synchronization processes between all pairs of 33 brain regions and could further identify the pathways those region pairs use to interact with each other. These pathways represent communication channels with distinct interaction time windows. We found the ability of regions to communicate via these channels to depend on the phase lag at which they try to synchronize [19]. This finding is in line with the communication-through-coherence theory that predicts neural communication to critically depend on oscillatory phase differences [16]. Furthermore, it provides a mechanistic explanation for the dependency of the effect of extrinsic brain stimulation on the stimulation phase [20] and could guide future brain stimulation studies that are investigating phase-lagged neural synchronization, e.g., through multi-site transcranial stimulation, or optogenetics in combination with multielectrode recordings [32]. Our method is also applicable in future theoretical studies characterizing the dynamic properties of network graphs.
Since we were further able to demonstrate that for different stimulation phase offsets between the communicating regions, different communication channels may be employed, we believe that the switching between different synchronization phase lags could be a potential mechanism through which brain regions can dynamically change their effective communication channels. As suggested in [11,13], such a mechanism would provide the necessary flexibility to allow for the dynamic binding of remote neural representations into different concepts. Taken together, our results suggest a potential mechanism the human brain might have developed to use the physiological constraints imposed by coupling delays to its computational advantage.
Acknowledgments
This research has been funded by the DFG (SFB 936, projects A2/A3/C1/Z1).
Footnotes
↵2 holger.finger{at}uni-osnabrueck.de