Edited by: Olaf Sporns, Indiana University, USA
Reviewed by: Joana R. B. Cabral, Universitat Pompeu Fabra, Spain; Juergen Kurths, Humboldt Universität, Germany
*Correspondence: Andreas Daffertshofer, Research Institute MOVE, VU University Amsterdam, Van der Boechorststraat 9, 1081 BT Amsterdam, Netherlands. e-mail:
This is an open-access article subject to a non-exclusive license between the authors and Frontiers Media SA, which permits use, distribution and reproduction in other forums, provided the original authors and source are credited and other Frontiers conditions are complied with.
In recent studies, functional connectivities have been reported to display characteristics of complex networks that have been suggested to concur with those of the underlying structural, i.e., anatomical, networks. Do functional networks always agree with structural ones? In all generality, this question can be answered with “no”: for instance, a fully synchronized state would imply isotropic homogeneous functional connections irrespective of the “real” underlying structure. A proper inference of structure from function and
The interplay between structural and functional brain networks has become a popular topic of research in recent years. It is currently believed that the topologies of structural and functional networks in various empirical systems may disagree (Sporns and Kötter,
Neurons synchronize their firing pattern in accordance with different behavioral states. On a larger scale, synchronous activities are considered to stem from meso-scale neural populations that oscillate at certain frequencies with certain amplitudes. That is, oscillatory activity may yield synchronization characteristics within a neural population or between populations (Salenius and Hari,
In a nutshell, we start off with a network of
The discussed structural connectivities differ qualitatively in their topology. In detail, we consider the fully connected isotropic network, a network with small-world topology generated by the Watts–Strogatz model (Watts and Strogatz,
To understand the qualitative relationship between macroscopically defined functional networks and the (underlying) structural connectivity, modeling local populations of neurons in terms of averaged properties like their mean voltage and/or firing rates appears very efficient. This mean-field-like approach has a long tradition and is typically referred to as neural mass modeling (Wilson and Cowan,
Here we chose for Wilson–Cowan as seminal neural mass model because it can readily be derived from microscopic descriptions like integrate-and-fire neurons, but also from more general models like Haken (
As said, we are going to put individual Wilson–Cowan models at every node
Within that population, every neuron receives input from all other neurons of the population. Furthermore, the excitatory units individually receive constant external inputs
The characteristics of this dynamical system range from a mere fixed-point relaxation to limit cycle oscillations (self-sustained oscillations) depending on parameter settings (Wilson and Cowan,
To combine Wilson–Cowan models in a network, different populations are now connected via their excitatory units by virtue of the sum of all
In words, all Wilson–Cowan oscillators, located at nodes
As the different Wilson–Cowan models display self-sustained oscillations, it seems obvious to describe them using their amplitude and phase dynamics. The required transforms and approximations are summarized in the Appendix and the outcomes reveal a phase dynamics similar to the seminal Kuramoto network of phase oscillators. The Kuramoto model and its link to the here-discussed network of Wilson–Cowan models will be briefly sketched in the following two sub-sections.
The collective behavior of a network of oscillators, whose states are captured by a single scalar phase
That is, the
Strictly speaking the system (2) does not represent the Kuramoto model in its original form as there the coupling between nodes
For the sake of legibility, however, we here refer to (2) also as the Kuramoto model.
As mentioned above, the effect of increasing
When deriving the Kuramoto network from the Wilson–Cowan oscillator network, the major ingredient is to average every oscillator over one cycle when assuming that its amplitude and phase change slowly as compared to the oscillator's frequency. That is, time-dependent amplitude and phase are fixed, the system is integrated over one period to remove all harmonic oscillations, and, subsequently, amplitude and phase are again considered to be time-dependent (Guckenheimer and Holmes,
with
with
which does resemble a Kuramoto network. In fact, by comparing this form with the dynamics (2) we find
In sum, the phase dynamics can, in good approximation, be cast into the form of a Kuramoto network provided the connectivity matrix is corrected by means of (3). This correction yields a non-trivial amplitude dependence of the connectivity at the level of the phase dynamics. Since
Given our interest in amplitude dependency, we finally add a note about “large” amplitudes. In line with the Appendix Eq.
Interestingly, the presence of large amplitudes yields, apart from slightly different coupling coefficients
More recently, several research groups started investigating the relationship between structural and functional connectivity, suggesting that functional connectivity may indeed resemble aspects of structural connectivity, at least to some extent (Lebeau and Whittington,
Synchronization was quantified via the phase locking index or the phase uniformity
This index agrees with the so-called Kuramoto order parameter and reflects the degree of divergence of the different phases in the network (not the relative phases). By varying the overall coupling strength
To study potentially “erroneous” simulations of the phase dynamics – and thus possible “misinterpretations” of structural connectivity when solely looking at functional networks defined via phase synchrony – we ignored for the Kuramoto network the amplitude dependency (3) of the connectivity matrix and simply identified
and
Recall that the connectivity in (5) differs from (2) by means of
Throughout simulations we fixed parameter settings as:
For the Kuramoto model the natural frequencies
By default, the constant input values
that by virtue of
The connectivity matrices
The original Kuramoto network comprises a fully connected homogeneous network – see above.
Although the Kuramoto network is usually studied for large size networks, we chose a network of 66 nodes in order to make a better comparison with the Hagmann dataset.
The model for generating small-world networks employed here was introduced by Watts and Strogatz (
Empirical networks are unlikely to have an organization that can be exactly described by one of the theoretical network models. To study a network that more realistically represents anatomical connections in the human brain we repeated our simulations on a network that was based on axonal pathways obtained by diffusion spectrum imaging. This dataset has been used to identify the so-called “structural core” of anatomical connections in the human cerebral cortex as described by Hagmann et al. (
The changes in synchronization
The different choices of
If the structural connectivity is isotropic, then amplitude distribution largely (if not fully) prescribes the functional connectivity pattern that thus clearly disagrees with the structural connectivity. In consequence, the current example revealed two strongly synchronized local clusters but the large difference between the input intervals prevented them from synchronizing with one another. It is important to note that, if amplitude effects were not taken into account, a full synchronization of the network would have been found.
With the current parameter settings no full global synchronization could be achieved in both the small-world and the Hagmann network. However, partial synchronization patterns could be observed that did not correspond with the structural connectivity but also not with the distribution of amplitudes (Figures
The introduction of network analysis to neuroscience has paved new ways for the study of neural network organizations. Particular focus has been on the search for complex networks since many of these networks – especially in the neuroinformatics context – are known for their efficiency when transferring and integrating information from local, specialized brain areas, even when they are distant (Sporns and Zwi,
Do these functional networks precisely match their underlying structural counterparts? In general, networks do not agree, especially when the functional networks are solely defined via (phase) synchronization patterns, which is common practice when studying electrophysiological signals, for instance, M/EEG. We have shown that, even if the local dynamics at every node of a network can be described as phase dynamics in the form of a Kuramoto network, the connectivity matrix at this level of phases does not necessarily agree with the connectivity at the level of neural mass models describing firing rates of local neural populations. The connectivity at the phase dynamics level has to be corrected by its amplitude dependency. This phase level is indeed closely related to the empirically assessed functional connectivity matrix as this, as said, is commonly defined through locking patterns of phases. If relying on Kuramoto-like approximations, the connectivity matrix has to be corrected via the relation (3) that may include non-trivial amplitude dependency. Especially, when the amplitudes differ from node to node, the connectivity at the level of phases can qualitatively differ from the structural connectivity at the level of neural mass or mean firing rates. That is, structural and functional connectivity may differ simply because of the latter's amplitude dependency.
In consequence, phase dynamics and, hence, synchrony patterns should always be analyzed in conjunction with the corresponding amplitude changes. Patterns of global synchrony (phase) may depend on local synchrony (amplitude). This may have profound impacts when linking, for instance, M/EEG studies to neural modeling. Amplitude there translates to (spectral) power, which typically differs between distinct behavioral states or due to pathology. Incorporating these amplitude changes will certainly help to understand how structural and functional network organizations in the cortex, in particular, and in the central nervous system, in general, may relate to one another.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
To show the link between the network of Wilson–Cowan models (1) and the Kuramoto network of phase oscillators (2) we adopt Schuster and Wagner's derivation (Schuster and Wagner, 1990). In contrast to their description of two coupled oscillators, however, we explicitly account for a network structure containing
When deriving the Kuramoto network, the strategy is to consider the Wilson–Cowan model in the oscillatory regime, i.e., in the presence of a stable limit cycle (Figure
More explicitly, let
i.e., Eq.
holds. In principle this can be any solution but here we identify
where we abbreviated
As said, this “mean”-centering allows for expanding the sigmoid function to the
Here the zero-th order
which represents a network of weakly non-linear, self-sustained oscillators. Conventionally its characteristics are studied after transforming the system into polar coordinates
where
with
As a last approximation, we consider the case in which all amplitudes
and
which is equivalent to (2).
For the sake of completeness we also list the natural frequencies Ω
with which Ω in (A.9) can be defined via averaging over nodes, i.e.,
When ignoring all coupling terms (i.e., setting
Which has the stationary solutions
provided the square-root exists; cf. Figure
We thank the Netherlands Organisation for Scientific Research for financial support (NWO grant # 021-002-047).
1At the individual neuron level, the dynamics reads:
where u, v, w, and z are positive constants representing coupling matrices within the local neural population – see, e.g., Schuster and Wagner (