Dissecting whole-brain conduction delays through MRI microstructural measures

Network models offer a powerful framework to study structure–function relationships in the human brain. Previous studies have highlighted similarities between structural and functional connectivity patterns (Honey et al., 2010; van den Heuvel et al., 2009), and simulation-based work has suggested the idea that structure can support and shape multiple functional configurations (Cabral et al., 2017). From the point of view of combining local dynamics with global interactions, these simulations can be logically separated into two components (Petkoski & Jirsa, 2019; Sanz-Leon et al., 2015): first, an oscillator used to model the behavior of neuronal populations in the gray matter; second, a set of rules describing how these oscillators interact with each other. The latter is where the structural connectivity estimated from white matter pathways comes into play. A fundamental and simple approach is given by the Kuramoto model (Kuramoto, 1984), that assumes the activity of a local system can be represented by their phase alone (Breakspear et al., 2010). In applications to brain dynamics, the Kuramoto model has been used both for slow oscillations as in metabolic activity (as estimated by functional magnetic resonance imaging), and fast ones as in electrical activity (as measured for example through magnetoencephalography) (Breakspear, 2017).

As the neuronal populations are distributed in space (i.e., through the cortical surface) and are interconnected by bundles of different length and with different features (e.g., myelination), assumptions about conduction velocity and therefore delays along those connections are needed to study global brain dynamics. As several studies focused on slow oscillations, a common assumption is that such conduction delays are negligible (de Lange et al., 2019; Gollo et al., 2017; Schmidt et al., 2015). In studies focused on faster oscillations, the conduction velocity is usually assumed to be constant, and therefore the delays scale linearly with the connection lengths (Cabral et al., 2011,2014). An alternative approach assumes instead that the delays are constant (Roberts et al., 2019), with higher conduction velocity compensating for longer connections. The validation of these assumptions is nontrivial: directly measuring delays not only involves invasive recordings with direct electrical stimulation (Silverstein et al., 2020), but also an extensive sampling effort over the whole cortex. An additional factor to consider is where the recordings and the stimulation are performed, as interfacing with the cortical surface leads to longer delays compared to directly accessing the white matter pathways (Shimono & Hatano, 2018).

Microstructural models fitted to magnetic resonance imaging (MRI) data provide a non-invasive alternative to directly measuring conduction delays. Such models exploit the sensitivity of MRI to biophysical quantities to estimate microscopic features of the neural tissue, and some of these features are determinants of conduction velocity. The first and fundamental determinant is the axonal diameter, as in myelinated axons conduction velocity is proportional to the inner (i.e., without accounting for the myelin sheath) diameter. Using diffusion MRI and ultra-high gradients (Huang et al., 2020; Jones et al., 2018), it is possible to estimate the axonal diameter in the human brain (at least for the largest ones), and recent studies have shown measures in line with the underlying histology (Huang et al., 2020; Veraart et al., 2020) and with good scan-rescan reliability (Veraart et al., 2021). Another important determinant of conduction velocity is the myelin content. The presence of the myelin sheath itself not only changes conduction dramatically, but also its amount increases velocity non-linearly (Drakesmith et al., 2019; Rushton, 1951). The specific contribution of myelin depends on the so-called g-ratio, which is the ratio between the inner and the outer diameter of the myelinated axon. While myelin indirectly influences diffusion phenomena, diffusion MRI is not specific to myelin content (Beaulieu, 2002), and therefore other techniques are required for quantitative measures. Fortunately, myelin influences several MRI-based measures (Piredda et al., 2021), including longitudinal and transverse relaxation rates, as well as magnetization transfer. These measures have generally been shown to be sensitive to myelin content according to the histological ground truth (Mancini et al., 2020). Using axonal diameter and myelin content measures, it is then possible to estimate the conduction velocity through single-neuron models (Arancibia-Cárcamo et al., 2017; Richardson et al., 2000) or through more simplified ones, such as the Rushton model (Rushton, 1951) or the Waxman model (Waxman & Bennett, 1972).

Leveraging these advanced MRI measures, in this study we aim to define whole-brain structural connectivity in terms of conduction delays, as estimated from microstructural measures. Our goal is first to characterize the conduction velocity and delay distributions as a function of the related connection lengths, and then to assess conduction delay approximations currently used in the literature.

We estimated a rich set of structural connectivity patterns in a group of healthy volunteers combining tractography and several microstructural measures: the axonal diameter, estimated from ultra-high gradient diffusion data using a spherical mean model (Fan et al., 2020); the myelin volume fraction, estimated as the macromolecular tissue volume (MTV) (S. Berman et al., 2018; Mezer et al., 2013); the g-ratio, computed using the approach proposed by Stikov and colleagues (Stikov et al., 2015); and finally, the conduction velocity, computed from the axonal diameter and the g-ratio (a relative measure of myelination) using the Rushton model (Rushton, 1951). All these measures were mapped onto the corresponding tractograms and subsequently used to weight the respective connectivity matrices, based on regions of interest (ROIs) from the Desikan-Killiany (Desikan et al., 2006) and the Lausanne (Cammoun et al., 2012) atlases, which were averaged to obtain group-level connectivity. Conduction delays were computed at the subject level as the ratio between the average connection length and the correspondent average conduction velocity for each connection.

As the first fundamental step, we characterized the relationships between each microstructural measure along a connection and its respective length (Fig. 1). Looking at the axonal diameter distribution as a function of connection length, we observed that most connections (83.95%) have an average diameter between 3 μm and 4 μm, with only very short connections (shorter than 0.05 m) presenting a more pronounced variability. Regarding the relationship between axonal diameter and myelin content (Fig. 2), we observed a linear trend between the g-ratio and the diameter (linear regression R²: 0.4263). Interestingly, when looking at the myelin volume fraction, which is an absolute measure of myelin, the trend disappears.

We then characterized the conduction velocity estimated using the Rushton formula in terms of connection length (Fig. 1—middle). The trend is very similar to the one between axonal diameter and length (Pearson’s correlation coefficient: 0.8317; the relationships between conduction velocity and, respectively, axonal diameter and g-ratio are showed in Figure S1), where for most connections the velocity distribution follows a constant trend. As a result, the conduction delay shows a highly linear relationship with the connection length (linear regression R²: 0.9635), with an angular coefficient equal to 0.07447. These trends were conserved when using a denser parcellation scheme such as the Lausanne atlas (Figure S2), and also at the single-subject level (Figure S3-S7).

The reciprocal of the angular coefficient from the length-delay relationship provides an estimated conduction velocity of 13.42 m/s. Using this value, we computed a delay distribution on the basis of the observed connection lengths under the assumption of fixed conduction velocity. In this way, we were able to compare the differences with the actual delay distribution computed from the axonal diameter values. For this comparison, we started analyzing path measures borrowed from the field of graph theory (Fig. 3). We first computed the shortest paths of the network between each pair of nodes for the two delay distributions. Despite being qualitatively similar, the delay distribution based on constant velocity shows shorter paths (25–50%) for subcortical connections. Taking into account the betweenness centrality for all the nodes of the network, we see both examples of underestimation and overestimation when comparing fixed conduction velocity with the estimated velocity distribution.

The final question we wanted to answer is about a more general dynamical behavior using the delay distributions: do these edge- and node-specific differences lead to a substantial difference when we simulate oscillation phenomena? To answer this question, we used the Kuramoto model (Kuramoto, 1984). In this model, each brain region is represented as an oscillator and characterized mainly by its phase (Breakspear et al., 2010). Each oscillator adjusts its own phase according to the phases of the other oscillators through the existing connections between them. Although the original formulation of the Kuramoto model does not assume delays in the interactions between oscillators, subsequent iterations explored the effect of such delays (Rodrigues et al., 2016; Yeung & Strogatz, 1999). Leveraging this delay formulation, we computed both global synchrony and metastability measures for a range of coupling factors (Fig. 4). We observed negligible differences between models based on either delay estimation, with both distributions leading to very similar synchronization patterns: the global synchrony measures show the critical regime (i.e., where the oscillators become rapidly synchronized) for the same coupling factor values, and likewise the metastability measures reach their peak in the same range.

In this work, we leveraged microstructural measures estimated from MRI to study the conduction delay distribution in the whole-brain structural network. Our approach focused on conduction velocity and its main two determinants: the axonal diameter and the g-ratio (Drakesmith et al., 2019). The related MRI-based measures have received a lot of attention over the years, and several studies focused on their validation using histology as ground truth (Mancini et al., 2020; Veraart et al., 2020), and on their potential use as biomarkers in pathology (Yu et al., 2019). Previous studies have started studying the relationship between conduction properties and MRI-based measures in two specific pathways, the corpus callosum (S. Berman et al., 2019; Horowitz et al., 2015) and the optic radiation (Shai Berman et al., 2020; Takemura et al., 2020; You et al., 2019). To the best of our knowledge, this is the first attempt to combine them to estimate conduction velocity and delay distributions across the whole brain.

To understand the conduction delay distribution in terms of its determinants, we first described how each microstructural measure and the resulting conduction velocity change as a function of connection length. We observed that both the axonal diameter and the conduction velocity follow an overall constant trend, and therefore, the computed conduction delays showed a remarkably linear relationship with the connection length. Not only does this result support the approximation of constant conduction velocity used in several computational studies (Cabral et al., 2011, 2014), but it is also in line with the few studies that quantified delays through invasive recordings, in macaques (Shimono & Hatano, 2018) and in humans (Silverstein et al., 2020). Specifically, Shimono and Hatano relied on existing electrocorticography (ECoG) datasets to estimate cortico-cortical delays (Shimono & Hatano, 2018), while Silverstein and colleagues used direct electrical stimulation in concurrency with recording intracranial electroencephalography (iEEG) (Silverstein et al., 2020). Given this qualitative agreement, it becomes clear that for quantitative predictions the next step for computational studies is defining a rigorous method to set either the proportionality constant used to estimate conduction velocity or, equivalently, the approximate constant velocity.

It is important to place the microstructural MRI measures in the appropriate context. Despite the use of an ultra-high gradient MRI scanner, we still have to deal with the diffusion resolution limit dictated by the current hardware (Huang et al., 2015; Jones et al., 2018). Specifically, with gradients of 300 mT/m, we are able to measure axons with a diameter of 2–3 μm (Drobnjak et al., 2016; Nilsson et al., 2017). In line with this limit, further MRI simulations for two specific microstructural approaches, AxCaliber (Drakesmith et al., 2019) and the spherical mean model (Fan et al., 2020), showed larger estimation errors for smaller axons. These results prevent current approaches to fully characterize the axonal diameter distribution in the central nervous system, but ultra-high gradient MRI still allows us to study the upper tail of such distribution. Histological validation work from Veraart and colleagues (Veraart et al., 2020) showed that we are sensitive to the fourth order moment of the axonal diameter distribution, and therefore we are only taking into account the largest axons. It follows that in these distributions most axons have a much smaller diameter and very few axons have the large ones we are sensitive at ultra-high gradients. Nevertheless, such large axons result in high conduction velocity, and therefore play an important role the propagation of faster oscillations (Ivanov et al., 2019). For a complete characterization of conduction phenomena in the brain, we would instead need a conduction velocity distribution for each connection. A similar limitation holds for the relative myelin content, the g-ratio: the so-called “aggregate” (i.e. voxel-level) g-ratio is approximately axon-area weighted (Campbell et al., 2018; West et al., 2016), and therefore larger axons have a greater weight.

There is also another interesting consideration for the role of myelin to make: the g-ratio, as estimated from MRI, does not seem to visibly alter the overall conduction velocity, whose distribution instead closely resembles that of the axonal diameter—at least for healthy subjects. This is expected: Drakesmith and colleagues (Drakesmith et al., 2019) already showed through simulations that the g-ratio, and therefore the myelin content, seems to have a limited role in modulating conduction velocity. While combining the axonal diameter and the g-ratio accounts for the 85.1% of the variance in conduction velocity, the g-ratio variance contribution is much smaller (around 12%). When using both the Waxman and the Rushton models, the largest myelin-driven effects on conduction velocity are predicted for very large axons (around 7–10 μm) and for very low g-ratio values (around 0.4–0.6). However, it is important to stress that this limited role may be the result of how we currently model conduction phenomena—in most cases through single-neuron models (Arancibia-Cárcamo et al., 2017). As some studies have started to show (H. Schmidt et al., 2021; Sheheitli & Jirsa, 2020), the presence of multiple axons running in parallel and the potential interactions among them may be an important factor to take into account. Another limitation is that the Rushton and Waxman models (Rushton, 1951; Waxman & Bennett, 1972) have been derived from peripheral nerve experiments and may have reached their limits when it comes to myelinated axons in the brain.

To further explore the validity of the conduction velocity approximation, we compared the conduction delay distribution computed from the axonal diameters with the one obtained with a constant conduction velocity estimated through linear regression. From a network modeling perspective, there are some differences for the overall delays between regions—as assessed with shortest path lengths—and for each region’s role in fast connections—as assessed with betweenness centrality. Despite these differences, simulating the actual dynamics with the Kuramoto model shows remarkably similar results in both cases. We conclude that, while the constant velocity approximation can be considered valid for whole-brain simulations, the observed differences may play a role when focusing the attention on specific regions or pathways.

As we relied on the concept of path for graph-based measures, it is necessary to discuss the methodological choice of how to define paths in brain networks. Given two nodes in a network, there are several approaches to characterize how the network topology constrains any communication process. Shortest paths have been widely used to characterize the brain structure and function (Fornito et al., 2016), however this approach assumes that each node has a global knowledge of the network topology—and so far there is little evidence that supports this assumption. Alternative approaches are diffusion processes and navigation, relying on the assumptions that communication processes are based, respectively on dispersive information flows (Mišić et al., 2015) and on greedy routing strategies (Seguin et al., 2018). In this study, we decided to rely on shortest paths as our goal was to approximate the shortest delay between two given regions for both the cases of estimated delay distribution and fixed conduction velocity.

Despite the overall agreement with primate electrophysiological studies, there is a discrepancy with rodent studies reporting constant delays in subcortico-cortical connections (Pelletier & Paré, 2002; Salami et al., 2003). In a study on thalamo-cortical connections (Salami et al., 2003), Salami and colleagues showed that stimulating the ventrobasal nucleus of the thalamus in ex vivo mice brain tissue evoked cortical responses with the same delays as stimulating the white matter between the thalamus and the cortex. Intriguingly, this feature was not present in mice with a lower postnatal age. These results deserve further investigations, as it is not clear if this feature may be limited to specific bundles of myelinated axons (Kimura & Itami, 2009). As explained by the authors, as a result of development the ventrobasal-white matter connection becomes ten times faster and therefore becomes negligible compared the ventrobasal-cortical connection. However, this does not tell us if distinct thalamo-cortical connections, with different lengths, would present the same overall delay, as later suggested (Pajevic et al., 2014). Extensive work on macaque brains from Innocenti and colleagues (Innocenti et al., 2013; Tomasi et al., 2012) did not report any correlation between diameter and length. Other studies reported isochronic properties in rodents for the amygdala (Pelletier & Paré, 2002) and the prefrontal cortex (McDougall et al., 2018), but again focusing on specific bundles. In any case, at the whole-brain level our results point out that constant delays seem unlikely to generally occur. As we have already discussed in detail, we are measuring the largest axons in each bundle and the resulting delays scale linearly with the connection length. This result implies that the rest of the axons in each bundle—smaller and, therefore, slower—would not be able to compensate for different connection lengths.

One final limitation we need to disclose is the limited number of subjects. As these measures required a dedicated ultra-high gradient MRI scanner and a substantial amount of scanning time, it is not possible to use any of the current publicly available large datasets to estimate the same measures. Nevertheless, to further support our conclusions, we took into account also the trends observed at the single-subject level, which replicate the results obtained at the group level.

To summarize, using microstructural measures estimated from MRI we were able to estimate the conduction velocity and delay distributions of white matter connections across the whole brain. These results support in general the current approximation of constant conduction velocity for simulation purposes, but also suggests that velocity distribution may contribute to provide a more detailed picture for specific areas and pathways. Overall, leveraging microstructural measures has the potential to further refine network models of the brain.