RECHERCHE
Asymmetric high-order anatomical brain
connectivity sculpts effective connectivity
Arseny A. Sokolov
1,2,3,4, Peter Zeidman1, Adeel Razi1,5,6, Michael Erb7, Philippe Ryvlin3,
Marina A. Pavlova8, and Karl J. Friston1
1Wellcome Centre for Human Neuroimaging, Institute of Neurology, University College London, Londres, United Kingdom
2Department of Neurology, University Neurorehabilitation, University Hospital Inselspital,
University of Bern, Bern, Suisse
3Service de Neurologie and Neuroscape@NeuroTech Platform, Département des Neurosciences Cliniques,
Centre Hospitalier Universitaire Vaudois, Lausanne, Suisse
4Neuroscape Center, Weill Institute for Neurosciences, Department of Neurology, University of California San Francisco,
San Francisco, Californie, Etats-Unis
5Monash Institute of Cognitive and Clinical Neurosciences & Monash Biomedical Imaging,
Monash University, Clayton, Australia
6Department of Electronic Engineering, NED University of Engineering and Technology, Karachi, Pakistan
7Department of Biomedical Magnetic Resonance, University of Tübingen Medical School, Tübingen, Allemagne
8Department of Psychiatry and Psychotherapy, University of Tübingen Medical School, Tübingen, Allemagne
Mots clés: Effective connectivity, Structural connectivity, Network diffusion, Graph Laplacian
ABSTRAIT
Bridging the gap between symmetric, direct white matter brain connectivity and neural
dynamics that are often asymmetric and polysynaptic may offer insights into brain
architecture, but this remains an unresolved challenge in neuroscience. Ici, we used
the graph Laplacian matrix to simulate symmetric and asymmetric high-order diffusion
processes akin to particles spreading through white matter pathways. The simulated indirect
structural connectivity outperformed direct as well as absent anatomical information in
sculpting effective connectivity, a measure of causal and directed brain dynamics. Surtout,
an asymmetric diffusion process determined by the sensitivity of the network nodes to
their afferents best predicted effective connectivity. The outcome is consistent with brain
regions adapting to maintain their sensitivity to inputs within a dynamic range. Asymmetric
network communication models offer a promising perspective for understanding the
relationship between structural and functional brain connectomes, both in normalcy and
neuropsychiatric conditions.
un accès ouvert
journal
Citation: Sokolov, UN. UN., Zeidman, P.,
Razi, UN., Erb, M., Ryvlin, P., Pavlova,
M.. UN., & Friston, K. J.. (2020).
Asymmetric high-order anatomical
brain connectivity sculpts effective
connectivité. Neurosciences en réseau,
4(3), 871–890. https://est ce que je.org/10.1162/
netn_a_00150
EST CE QUE JE:
https://doi.org/10.1162/netn_a_00150
Informations complémentaires:
https://doi.org/10.1162/netn_a_00150
Reçu: 7 Janvier 2020
Accepté: 18 May 2020
Intérêts concurrents: Les auteurs ont
a déclaré qu'aucun intérêt concurrent
exister.
RÉSUMÉ DE L'AUTEUR
Auteur correspondant:
Arseny A. Sokolov
arseny.sokolov@chuv.ch
Éditeur de manipulation:
Randy McIntosh
droits d'auteur: © 2020
Massachusetts Institute of Technology
Publié sous Creative Commons
Attribution 4.0 International
(CC PAR 4.0) Licence
La presse du MIT
Measures of white matter connectivity can usefully inform models of causal and directed
brain communication (c'est à dire., effective connectivity). Cependant, due to the inherent differences
in biophysical correlates, recording techniques and analytic approaches, the relationship
between anatomical and effective brain connectivity is complex and not fully understood. Dans
this study, we use simulation of heat diffusion constrained by the anatomical connectivity of
the network to model polysynaptic (high-order) anatomical connectivity. The outcomes afford
more useful constraints on effective connectivity than conventional, typically monosynaptic
white matter connectivity. En outre, asymmetric network diffusion best predicts effective
connectivité. In conclusion, the data provide insights into how anatomical connectomes give
rise to asymmetric neuronal message passing and brain communication.
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Anatomical network diffusion and effective connectivity
INTRODUCTION
Multimodal neuroimaging analyses are expected to improve our understanding of structure-
function relationships in the brain (Toga et al., 2006; Honey et al., 2010; Sporns, 2014);
drawing on measures of structural, functional, and effective brain connectivity (Sporns et al.,
2000; Parc & Friston, 2013). Cependant, relating symmetric and static structural connectivity de-
rived from diffusion magnetic resonance imaging (dMRI) to time-varying and context-sensitive
functional dynamics (recorded by functional magnetic resonance imaging, IRMf, electroen-
cephalography, EEG, or magnetoencephalography, MEG) remains an unresolved technical and
conceptual challenge (Honey et al., 2009; Stephan et al., 2009; Pineda-Pardo et al., 2014;
Uludag & Roebroeck, 2014). White matter (WM) pathways are sufficient for communication
between brain regions, but functional brain dynamics can also be mediated through polysy-
naptic connections (Chiffre 1). En effet, previous studies suggested the direct structural pathways
inferred using dMRI account for only about 55% of measured resting-state functional connec-
tivity patterns (Koch et al., 2002; Honey et al., 2009; Deligianni et al., 2011; Becker et al.,
2016).
Current measures of anatomical and of resting-state functional connectivity are symmetric in
the sense that they do not enable an assessment of whether one orientation of a pathway may be
more prominent than the inverse (Friston, 2011). In contrast, models of effective connectivity
such as dynamic causal models (DCMs) indicate the weights of specific directions of interaction
(Friston et al., 2003), and recent data across species suggest that information about directed,
asymmetric connectivity may more appropriately reflect brain architecture (Kale et al., 2018;
Avena-Koenigsberger et al., 2019; Seguin et al., 2019).
Previous work has analysed the relationships between indirect anatomical connectivity and
resting-state functional connectivity (Honey et al., 2007; Deligianni et al., 2011; Abdelnour
et coll., 2014; Becker et al., 2016; Meier et al., 2016; Bettinardi et al., 2017; Liang & Wang,
2017; Abdelnour et al., 2018). Recent graph-theoretic research has demonstrated that conven-
tional, symmetric measures of brain WM architecture contain information on the differential
efficiency of afferent and efferent network communication (Avena-Koenigsberger et al., 2019;
Seguin et al., 2019). En outre, asymmetries in predicted communication efficiency were
found to reflect neurobiological concepts of functional hierarchy and were correlated with
directionality in resting-state effective connectivity analysed using spectral dynamic causal
modelling (Seguin et al., 2019). Thus far, formal integration of effective with anatomical con-
nectivity has only been implemented for direct and symmetric measures of structural connec-
tivité (Stephan et al., 2009; Sokolov et al., 2018; Sokolov et al., 2019). The primary motivation
for this study was to develop an integrative approach simulating symmetric and asymmetric
high-order (polysynaptic) structural connectivity and using the outcomes to constrain models
of task-related effective connectivity.
This central aim inspired the use of the graph Laplacian (GL: see Materials and Methods)
to compute polysynaptic symmetric and asymmetric structural connectivity. The GL is a con-
struct from spectral graph theory and represents the difference between the adjacency (indicat-
ing which network nodes are interconnected) and degree matrices (indicating the number of
nodes connected with each node). The GL can be used to simulate the diffusion of a conserved
quantity of particles over the network (Chiffre 2 and Supporting Information Video S1; Biggs,
1993). Surtout, the GL approach allows introducing asymmetry (see Materials and Meth-
ods and Supporting Information Figure S1): weighting (normalising) the structural adjacency
matrix to the in-degree implies that each target node has a fixed capacity to be influenced by
other nodes, and its relative sensitivity is determined by the probability of receiving inputs.
872
Effective connectivity:
A measure of the directed (causal)
influence of one neural system over
another using a model of neuronal
interactions.
Dynamic causal modelling:
A Bayesian framework which is used
to infer causal interactions between
coupled or distributed neuronal
systèmes (effective connectivity).
High-order structural connectivity:
Structurally unconnected regions
communicate polysynaptically to
engender indirect connectivity over
multiple hops.
Graph Laplacian:
A matrix representation of a graph
that combines node adjacency and
node degree in mathematical
formulation and belongs to spectral
graph theory.
Neurosciences en réseau
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Anatomical network diffusion and effective connectivity
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Spectral graph theory:
A study of the relationship between
a graph and the eigenvalues and
eigenvectors of its Laplacian matrix.
Adjacency matrix:
Square matrix representation of a
graph which is either binary
(presence or absence of connections)
or weighted (showing strength of
relations).
Eigenmode:
A stable state (c'est à dire., mode) of a
dynamic system in which all parts of
the system oscillate at the same
frequency.
Illustration of the relationship between anatomical and effective brain connectivity.
Chiffre 1.
(UN) Two network nodes n1 and n2 can have a structural pathway connecting them (blue double line)
that may underlie causal functional influence of network node n1 over n2 (effective connectivity;
orange arrow). (B) Effective connectivity from n1 to n2 may also be present in the absence of di-
rect structural connectivity, mediated by polysynaptic structural (grey double lines) and effective
connections through hidden nodes n3 and n4 (not specified in the network model). Modelling of
indirect structural connectivity may therefore provide better constraints on effective connectivity
than measures of direct structural connectivity alone.
Inversement, if we normalise to the out-degree, we assume each node has a fixed capacity to
influence other nodes, and the relative influence is proportional to efferent particle diffusion.
In our implementation, the GL matrix L is exponentiated and raised to the order τ, exp(L)τ.
At the start of the diffusion process, each node is equipped with a large number of “spikes.”
Each order τ represents a time step of the diffusion process, at which the spikes are distributed
to other nodes at a rate that is proportional to the connection strengths. Each increment of τ
thus indicates an extra path between any two given regions (nodes), via τ − 1 intermédiaire
nodes (par exemple., a third-order connection between two nodes means they are connected via two
other nodes; Figure 1B). In a continuous time interpretation of this process, we can equate
connectivity with the number of spikes accumulated as time progresses. This interpretation is
effectively a constrained diffusion process, where the diffusion coefficients are determined by
the GL and, finalement, every node is connected to every other node through multiple paths.
This equilibrium distribution is the principle eigenmode of the matrix exponential of the GL
(Figure 2H), revealing which nodes are more strongly involved in the diffusion or propagation
processus.
Diffusion simulation approaches capture the propensity of information or particle distribu-
tion along all possible paths in a network, and thus approximate network communicability
Neurosciences en réseau
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Anatomical network diffusion and effective connectivity
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Chiffre 2. Network diffusion simulated by exponentiation of the matrix exponential of the graph Laplacian. (UN) Based on the direct structural
adjacency matrix derived from probabilistic tractography, using Equations 1–4, we created (B) the matrix exponential of the graph Laplacian
(ici, normalised to the in-degree). Exponentiation of the matrix exponential of the graph Laplacian simulated diffusion of particles constrained
by the anatomical network and therefore yielded indirect structural connectivity. The 0th order is simply the identity matrix, meaning that each
node (square on the diagonal) is equipped with an equal number of particles (spikes). Each increase in order corresponds to an additional
propagation down another path or edge, and a subsequent distribution of spikes. Le (first-order) matrix exponential thus represents the first
small time step of diffusion. (C–G) As time (order) progresses, some nodes receive more particles (input) than others, until this closed system
reaches a state of saturation (equilibrium), corresponding to the principle eigenmode of the graph Laplacian. (H) This eigenmode is visible at
the 64th order (time step of diffusion). The bands reflect how many spikes every node has received (in-degree of each node) and approximate
the principle eigenvectors (Figure 4B). Please see Table 1 for abbreviated region labels.
Network communicability:
A graph-theoretic measure of the
ease of information propagation in a
network of interconnected regions.
Routing efficiency:
A measure of communication
efficiency, representing the average
inverse shortest path length between
all pairs of nodes in a complex
réseau.
Network diffusion:
A process simulating the propagation
of heat (or information) dans un
réseau.
Probabilistic tractography:
Estimation of white matter pathway
trajectories based on diffusion
MRI data.
Neurosciences en réseau
(Estrada & Hatano, 2008; Crofts & Higham, 2009). Network communicability has already
been used to characterise brain networks in normalcy and pathology, and in different species
(Crofts & Higham, 2009; Andreotti et al., 2014; Grayson et al., 2016; Shine et al., 2018).
In addition to routing efficiency representing shortest paths and thus easy and speedy com-
munication between network nodes, taking into account recurrent neuronal message passing
over multiple paths may afford more optimal approximations of brain dynamics (Bullmore &
Sporns, 2012; Avena-Koenigsberger et al., 2017). Using the matrix exponential of the GL al-
lows a path length-based correction of network communicability (Estrada & Hatano, 2008;
Crofts & Higham, 2009). Previous applications of the GL in neuroscience have suggested it
as a promising tool for modelling neurotransmitter diffusion in the synaptic junction (Barreda
& Zhou, 2011), simulating the spread of neurodegeneration (Raj et al., 2012) and comparing
resting-state functional with structural connectivity (Abdelnour et al., 2014, 2018).
The present study asks whether measures of simulated symmetric and asymmetric anatom-
ical network diffusion may usefully inform the effective connectivity that underwrites causal
and asymmetric interactions among distributed neuronal populations. We demonstrate the ap-
proach in the context of fMRI responses of a brain network to emotional body language, en utilisant
probabilistic tractography on high angular resolution diffusion imaging (HARDI) data from the
same cohort of normal individuals.
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Anatomical network diffusion and effective connectivity
MATERIALS AND METHODS
Participants
We used fMRI and HARDI data from 17 right-handed, male normal subjects (mean age 27.9
années) from a study on emotional body language processing. The cohort overlapped with that
analysed in previous research (Sokolov et al., 2012, 2014, 2018, 2019). The study was ap-
proved by the Ethics Committee of the University of Tübingen Medical School, Allemagne. Par-
ticipants provided informed written consent and were financially compensated.
fMRI and HARDI Data Recording and Preprocessing
A 3T scanner (TimTrio, Siemens Medical Solutions, Erlangen, Allemagne; 12 channel head
coil) was used for recording of three-dimensional T1-weighted structural MRI (magnetisation-
prepared rapid gradient echo, MPRAGE; 176 sagittal slices, TR = 2,300 ms, LE = 2.92 ms, TI =
1,100 ms, voxel size = 1 × 1 × 1 mm3), a field map for inhomogeneity correction, HARDI
data (two sessions with 64 diffusion gradient directions per subject; b-value = 2,600 s/mm2,
54 tranches axiales, TR = 7,800 ms, LE = 108 ms, slice thickness = 2.5 mm, matrix size = 88 ×
88, field of view = 216 mm) and functional echo-planar imaging (EPI; 171 volumes, 56 axial
slices, TR = 4,000 ms, LE = 35 ms, in-plane resolution 2 × 2 mm2, slice thickness = 2 mm,
1 mm gap).
Participants viewed animations of an arm represented by bright dots placed on the head and
main upper limb joints, facing to the right and knocking on an invisible door with different emo-
tional content (happy, angry, neutral; Pollick et al., 2001; Sokolov et al., 2011). In an event-
related design, the participants had to indicate which emotion was expressed by button press
(button assignment counterbalanced between participants). Stimulus duration was 1,000 ms,
and each stimulus category (emotion) was presented 30 times throughout the experiment.
To optimise event-related response function estimation, we applied jittering of stimulus on-
set intervals (entre 4,000 et 8,000 ms in steps of 500 ms) and stimulus order pseudo-
randomisation.
Preprocessing of fMRI data was performed using Statistical Parametric Mapping software
(SPM12, Wellcome Centre for Human Neuroimaging, Institute of Neurology, UCL, http://www.
fil.ion.ucl.ac.uk/spm) and included slice timing correction, realignment, unwarping, image co-
registration, segmentation-based normalisation, and smoothing. HARDI data preprocessing
with the FMRIB’s Diffusion Toolbox within the FMRIB Software Library (FSL5, Oxford Centre
for Functional MRI of the Brain, ROYAUME-UNI, http://www.fmrib.ox.ac.uk/fsl) consisted of brain extrac-
tion (Forgeron, 2002), motion and eddy current correction, followed by co-registration with the
anatomical reference image and normalisation to Montreal Neurological Institute (MNI) espace
using the FMRIB Linear Image Registration Tool (FLIRT; Jenkinson et al., 2002). Gradient di-
rections were adjusted according to the FLIRT parameters.
fMRI Analysis and DCM Specification
Analysis of fMRI data was conducted by first specifying a general linear model (GLM). Trials with
correctly classified emotional expression of point-light knocking (happy, angry, neutral) étaient
assigned distinct regressors, and regressors of no interest were modelled for trials with incorrect
classification (par exemple., neutral stimulus classified as happy), trials with missing responses, six head
motion parameters, and time series from WM and cerebrospinal fluid. The regressors were
convolved with the haemodynamic response function. High-pass filtering was performed (cut-
off 1/256 Hz), and serial autocorrelations were accounted for by a first-order autoregressive
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Anatomical network diffusion and effective connectivity
processus (coefficient of 0.2) plus white noise model. The GLM was applied to individual pre-
processed EPI data, and the contrasts happy versus neutral, angry versus neutral, and neu-
tral versus emotional knocking were specified. Individual contrast images were submitted to
second-level random effects analyses, and regional activations (at a p < 0.05 family-wise error
corrected voxel-wise threshold for multiple comparisons) were identified using the automated
anatomical labelling in SPM (Tzourio-Mazoyer et al., 2002) and the NeuroSynth.org database
(http://neurosynth.org; Yarkoni et al., 2011).
A one-state, bilinear, and deterministic DCM with mean-centred inputs and reciprocal ex-
trinsic connections between all nodes (full model) was created for each subject. This DCM
included seven regions showing differential activation with respect to correctly classified emo-
tional expressions of point-light knocking and three regions for activation versus baseline
(Table 1). For each region, time series were extracted as the first eigenvariate of all activated
voxels within a sphere with a radius of 8 mm, centred on each individual maximum (p < 0.05,
uncorrected). The individual maxima were found within 7 mm of the group activation coordi-
nate in every subject. The time series extraction was adjusted to remove effects that were not
related to the task such as motion. According to previous data on the architecture of the brain
network for body motion processing (Sokolov et al., 2018), driving input was specified on the
left middle temporal cortex, right fusiform gyrus, and right superior temporal sulcus. Modulat-
ing input of different emotional content was expressed in the DCM B-matrix (see Supporting
Information Methods) by modelling the influence of the corresponding regressors for happy,
neutral, and angry stimuli on all extrinsic connections in the network, as well as on intrin-
sic coupling within the seven nodes showing differential activation depending on emotional
content (Table 1).
Table 1. The regions forming the analysed network.
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MNI Coordinates
Y
X
Z
z-value
Cluster size
Anatomical label
Happy vs. neutral
R superior temporal sulcus (STS)
R caudate nucleus (CAU)
Angry vs. neutral
L midcingulate cortex (MCC)
L anterior cingulate cortex (ACC)
L insula (INS)
Neutral vs. emotional
Cerebellar vermis, lobule IX (CRB)
R amygdala (AMY)
50
10
− 6
− 8
−28
0
26
−38
18
−6
50
14
−46
−4
Active (stimulation vs. baseline)
L middle temporal cortex (MTC)
R fusiform gyrus (FFG)
R retrosplenial cortex (RSP)
−404
18
− 6
−784
−36
−54
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−48
18
−16
−48
−26
−48
12
4
5.82
5.46
5.21
5.08
4.87
6.02
5.93
5.90
5.78
5.72
186
120
192
168
134
206
182
362
238
267
Note. Seven regions with differential activation to emotional expressions of point-light knocking and three
regions showing activation not modulated by emotional content (at a p < 0.05 family-wise error corrected
voxel-wise threshold for multiple comparisons) were included in the analysis. Regional labels are provided
along with coordinates in MNI space, corresponding z-values, and cluster sizes.
Network Neuroscience
876
Anatomical network diffusion and effective connectivity
Direct Structural Adjacency Matrix
Individual preprocessed and normalised HARDI data were submitted to Bayesian Estimation of
Diffusion Parameters Obtained using Sampling Techniques with modelling of Crossing Fibres
(BEDPOSTX; Behrens et al., 2007) in FSL to obtain diffusion parameters for each voxel. Prob-
abilistic tractography with crossing fibres (PROBTRACKX; step length = 0.5 mm, number
of steps = 2,000, number of pathways = 5,000, curvature threshold = 0.2, modified Euler
integration; Behrens et al., 2007) was performed for each DCM node as a seed, and the other
DCM nodes as classification targets. The fibre pathway outputs were visually controlled for
plausibility. Structural connection strength between a seed region i and a classification target
region j was obtained by averaging the number of streamlines connecting every voxel in i
to one voxel of j, across both directions of tractography. Further averaging across all subjects
provided a symmetric group structural adjacency matrix (Figure 2A). We eschewed thresh-
olding and considered weighted adjacency matrices. Each element of the group adjacency
matrix Z was normalised to represent direct structural connection strength or probability ϕ,
relative to the greatest connection strength within the matrix. The between-region elements of
matrix Z were used to inform models of effective connectivity by direct structural connectiv-
ity, and for GL-based simulation of network diffusion to obtain measures of indirect structural
connectivity.
Graph Laplacian
We used the GL matrix L to construct a connectivity operator simulating diffusion of a con-
served quantity (heat, spikes) along direct and indirect pathways between the network nodes of
the structural adjacency matrix Z. As per definition, each column of the GL matrix L express-
ing probabilities ϕ of extrinsic structural connections has to sum to zero, which consequently
applies to any linear mixture of the columns of L.
To achieve this, we set the leading diagonal elements of L to the negative sum of the corre-
sponding column of Z (Supporting Information Figure S1):
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where D is the degree matrix Di,i = ∑n
i=1 Zi,j.
L = Z − D
We obtained our connectivity operator by calculating the matrix exponential Γ of L:
Γ = exp(L) = ∑∞
k=1
1
k!
Lk
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(1)
(2)
Subsequent exponentiation of Γ simulated distribution of particles (spikes) and thus yielded
indirect, high-order structural connectivity ψ after every time step τ:
Ψ(τ) = Γτ
(3)
Γ0 corresponds to the identity matrix, and the first order of the matrix exponential of the
GL Γ1 represents the distribution of particles ψ(1) after a first time point of diffusion τ = 1
(Figure 2B). With every increase of τ (order of Γτ), another time step is calculated and indirect
connections ψ(τ)i,j between nodes i and j become apparent or reinforced. Inherently, this sim-
ulated distribution of a conserved quantity constrained by the structural connectivity saturates
after a certain, unknown number of diffusion steps. We tested our hypothesis that this state of
equilibrium or saturation would provide the most informative priors on effective connectivity
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by comparing DCMs with indirect structural connectivity priors at different time steps (orders)
τ of the GL diffusion process.
Furthermore, we hypothesised that simulated network diffusion on asymmetric structural
adjacency matrices may introduce more plausible constraints on asymmetric effective brain
connectivity. Accordingly, we introduced three variants of the adjacency matrix Z as the ba-
sis for the diffusion process (Supporting Information Figure S1): (1) the symmetric adjacency
normalised along the rows, and (3)
matrix Z normalised to its maximum, (2) asymmetric Z
asymmetric Z
normalised along the columns in the following way:
′
′
′
Z
=
Z
W
(4)
with normalisation of Z to its out-degree (setting the sum of weights in each column to unity)
when the diagonal degree matrix is Wi,i = ∑n
i=1 Zi,j and normalisation of Z to its in-degree
(sum of weights in each row set to unity) for Wi,i = ∑n
j=1 Zi,j. When normalising to the out-
degree, we assume that each node has a fixed capacity to influence other nodes, and that the
relative influence is proportional to the efferent diffusion process along structural pathways.
Conversely, normalisation to the in-degree means that each target node has a fixed capacity
to be influenced by other nodes and its relative sensitivity is determined by the probability
of receiving input during the diffusion process. In what follows, we describe the evaluation
of which indirect structural connectivity ψ(τ) within each of the three plausible normalisa-
tion schemes underlying the diffusion operator Γτ afforded the best constraints on effective
connectivity.
Integration of Structural Connectivity with Dynamic Causal Modelling
After defining prior beliefs about the effective connectivity parameters, the estimation of DCMs
affords posterior estimates of the parameters as well as the evidence for the respective model
(Friston et al., 2003, Supporting Information Methods). The priors for extrinsic (off-diagonal;
between-region) connections in dynamic causal modelling form a multivariate normal dis-
tribution, defined by a vector of expectations and a prior covariance matrix Σy. By default,
the prior expectation is zero and the variance is equal across all extrinsic connections. The
greater the prior variance, the further the connectivity parameters can deviate from their prior
expectation of zero.
The priors also contribute to the calculation of the model evidence—the quantity used
to compare DCMs—which is the trade-off between model accuracy and complexity. In this
context, complexity is defined as the discrepancy between prior assumptions and posterior es-
timates, where greater complexity decreases model evidence. Optimising the priors according
to measures of structural connectivity may therefore increase model evidence, through a re-
duction of model complexity (Stephan et al., 2009; Sokolov et al., 2019).
However, the precise relationship between structure and function is unknown and likely
varies for different networks. Previous research provided support for the intuition that the
strength of direct structural connections relates to the prior effective connectivity in a posi-
tive monotonic fashion (Stephan et al., 2009; Sokolov et al., 2018, 2019). This means that for
lower structural connection strengths, the prior variance shrinks to a small value, precluding
strong effective connectivity. Conversely, for greater structural connection strengths, our prior
belief that the effective connectivity is close to zero can be relaxed by increasing the prior
variance.
Based on this rationale, to assess the utility of direct structural connectivity as priors for
effective connectivity, we used our previously developed structurally informed parametric
Model evidence:
The model evidence, or marginal
likelihood, represents the probability
of observing the measured data
under a certain model and is used
for Bayesian model comparisons.
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empirical Bayes (si-PEB) approach (Sokolov et al., 2019) to obtain the reduced prior covari-
ance Σy red from the probability ϕ for direct structural connectivity encoded in the symmetric
structural adjacency matrix Z normalised to its maximum:
Σy red =
Σy max
1 + exp(α − δ ∗ φ)
(5)
where maximum prior covariance is determined by the hyperparameter Σy max (range from
0.0625 to 0.25 in four equal steps), the sigmoid slope by δ (range from 0 to 16 in eight equal
steps) and sigmoid shift by α (range −2 to 2 in eight equal steps). This hyperparameter space
yielded 405 different mappings or models per network.
For indirect structural connectivity, across the three normalisation schemes (see Equation 4),
by assuming a simple linear positive relationship, we mapped the logarithm of indirect struc-
tural connection probability ψ(τ) afforded by the τ-th order of the diffusion operator Γτ to the
prior covariance as follows:
Σy red = Σy min + δ ∗ (log(ψ(τ) − e
−b) + b)
(6)
Here, similar to Equation 5, the three-dimensional hyperparameter model space is spanned by
the hyperparameters τ (range 1–64 in seven equal steps for both networks), Σy min (represent-
ing the default prior covariance, range 0.0156–0.0625 in seven equal steps for both networks),
and δ (sigmoid slope; range 0–0.25 in seven equal steps). The space thus contains 512 differ-
ent models per normalisation scheme, or 1,536 models in total. The use of log-transformed
structural probabilities ψ(τ) is motivated easily by noting that most structural connections
have a log normal distribution, and indeed have an exponential dependency upon distance
(Markov et al., 2013). Here, b is a small number that ensures the prior variance Σy min over
effective connectivity is lower bounded; in the absence of structural connectivity: Σy red =
Σy min.
Crucially, both model spaces (for direct and indirect structural connectivity) include flat
mappings (i.e., δ = 0), where structural constraints do not matter and Σy red is the same across
all extrinsic connections, thus representing an intrinsic control (null hypothesis) for the as-
sumption that structural constraints usefully shape effective connectivity.
Model Estimation and Evaluation
As we analysed second-level measures of structural connectivity, we used parametric empiri-
cal Bayes (PEB) (Friston et al., 2015; Supporting Information Methods) to make inferences on
effective connectivity at the group level. PEB is a hierarchical model, in which the average
group connectivity acts as an empirical prior on individual connectivity (Friston et al., 2016).
PEB estimation thus represents an iterative process between individual and group effective
connectivity. Therefore, PEB properly partitions within- and between-subject random effects
(Zeidman et al., 2019) and is robust to local minima problems (Friston et al., 2016). The indi-
vidual DCMs were estimated using the PEB scheme with a prior variance of 0.5 for all extrinsic
connections. Subsequently, the prior PEB variance of each extrinsic connection was adapted
(reduced) using Equations 5 and 6 for measures of direct and indirect structural connection
strengths, respectively.
The search for the models with the greatest evidence was afforded by Bayesian model re-
duction (BMR) (Friston et al., 2016; Supporting Information Methods). This recently introduced
statistical device enables analytical evaluation of large model spaces in a matter of seconds,
based on the estimation of a single, so-called “full” model. In contrast, the use of conven-
tional dynamic causal modelling would have required separate estimation of each of the 1,941
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alternative models (estimated processing duration: 1,300 days). By comparing the log evi-
dences of the different models, we assessed whether effective connectivity was better explained
by (1) indirect as opposed to direct measures of structural connectivity, (2) a particular order
of the connectivity operator Γτ, and (3) a particular normalisation scheme of the structural ad-
jacency matrix underlying Γ. Very strong evidence that one model provides a better account
for the observed data than another is concluded from a relative log-model evidence of three
(Penny, 2012), corresponding to a posterior probability of 95% or above.
RESULTS
Anatomical Network Diffusion Outperformed Direct Pathways in Sculpting Effective Connectivity
We assessed the value of simulated indirect (high-order) anatomical connectivity afforded by
network diffusion under the GL for sculpting effective connectivity, relative to models informed
by direct structural connectivity and to DCMs without anatomical information.
For indirect structural connectivity, we used a sigmoid function (Equation 6) with the hyper-
parameters δ (slope; range 0–0.25 in seven equal steps) and Σy min (lower boundary on prior
variance; range 0.0156–0.0625 in seven equal steps) to map the log-transformed group con-
nectivity values provided at eight different, equally distributed orders τ of the diffusion process
(from 1 to 64) onto prior variance of second-level effective connectivity.
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Figure 3. Mapping indirect anatomical to effective connectivity priors. (A) The mapping from indirect structural connectivity to prior
second-level variance for the optimal combination of hyperparameters τ = 64, δ = 0.15, and Σy min = 0.05. (B) BMR affords the poste-
rior probability for each model informed by indirect structural connectivity relative to a full, uninformed model with a uniform prior variance
of 0.5 for all extrinsic (between-region) connections. Higher luminosity represents greater posterior probability. At every time step (order τ) of
the graph Laplacian, Equation 6 is applied to map the resulting structural connectivity to prior variance on second-level effective connectivity,
defined by the sensitivity hyperparameter δ. The posterior probabilities computed from the log model evidences are shown for the optimal
prior second-level variance (Σy min = 0.05). The important aspect of this distribution is that the highest posterior probabilities (brightest grids)
are observed for a sensitivity hyperparameter δ substantially greater than zero (that would mean structural connectivity does not provide useful
constraints) and for higher orders τ of indirect structural connectivity.
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A grid search over the 1,536 candidate models resulting from the diffusion process un-
der three normalisation schemes (symmetric, weighted to the out-degree, weighted to the in-
degree) using BMR (overall computation time 5.98 seconds) indicated the best constraints on
effective connectivity were provided by the largest order (τ = 64) of a GL normalised to the
in-degree. The structure-function mapping at this order was governed by the hyperparameters
δ = 0.15 and Σy min = 0.05 (Figure 3).
This model clearly outperformed models informed by direct anatomical connectivity (log-
evidence difference 33.13 in favour of indirect structural connectivity) and those without
anatomical information (log-evidence difference 35.3 in favour of indirect structural connec-
tivity). Very strong evidence that one model provides a better account for the observed data
than another is concluded for a log-evidence difference of three or above (Penny, 2012).
The Graph Laplacian Principle Eigenmode Aligned with Effective Connectivity
As shown in Figure 3B, the evidence for DCMs of effective connectivity informed by indi-
rect structural connectivity priors increased with progression of the particle diffusion process
simulated by the GL, saturating at orders above τ = 50, corresponding to the principle eigen-
mode of the GL. This suggests the structural connectivity that matters for dynamical coupling
and effective connectivity is best conceived in terms of reciprocal message passing over long
(polysynaptic) paths, or periods of time.
Normalisation Schemes: Afference Matters
Bayesian model comparison (Penny, 2012) across the three normalisation schemes provided
consistently very strong evidence in favour of normalisation to the in-degree (log-evidence dif-
ference between this and the next probable normalisation scheme: 3.06; Figure 4). This result
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Figure 4. The role of asymmetry. (A) The bars represent the posterior probabilities of the most probable model in each of the three different
normalisation schemes (to the maximum, to the out-degree and to the in-degree). This analysis indicated very strong evidence in favour of
normalisation to the in-degree (posterior probability 96%). (B) The bars illustrate the first GL eigenvectors for each node (equivalent to the
horizontal bands in the equilibrium state, Figure 2H). They showed that the midcingular cortex (MCC), superior temporal sulcus (STS), middle
temporal cortex (MTC), and insula (INS) received most particles during the diffusion process (in descending order). (C) Crucially, these four
network nodes were also those with the highest functional input (in-degree) based on effective connectivity. Here, the functional input (bars)
for each node was defined as equalling the sum of squared weights over the respective rows of the effective connectivity matrix. Taken together,
this indicates that effective connectivity is best constrained by the input sensitivity of the network nodes in the diffusion process simulated by
the GL. Please see Table 1 for all abbreviated region labels.
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implied that effective connectivity is best predicted by the relative sensitivity of nodes to in-
coming information or afference and confirmed our hypothesis that introduction of asymmetry
in the diffusion process may offer a more plausible characterisation of asymmetric brain
dynamics.
When examining how the principle eigenmode of the GL normalised to the in-degree re-
lated to the functional afference of each node, one can see that the four nodes receiving the
greatest input during the GL diffusion process (midcingular cortex (MCC), superior temporal
sulcus (STS), middle temporal cortex (MTC) and insula (INS)) were also those with the highest
functional in-degree based on effective connectivity (Figure 4). This further speaks to the utility
and construct validity of the GL approach to inform effective by indirect structural connectivity.
Permutation Testing
Random permutations (n = 256) of the network nodes in the adjacency matrix (thereby preserv-
ing the distribution of edge weights) were used to assess how often a random structural adja-
cency matrix would afford a greater log-model evidence than reported above (after optimising
the normalisation, order, and sigmoid hyperparameters). This permutation testing suggested
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Figure 5. Permutation testing with random structural adjacency matrices. In order to assess how
often the graph Laplacian diffusion process over a random structural adjacency matrix would pro-
duce a log evidence greater than that afforded by the true structural adjacency matrix, 256 matrices
were formed by randomly permuting the network nodes, thereby preserving the symmetry and dis-
tribution of edge weights. The histogram shows the distribution of maximum log evidences over
these different models (under in-degree normalisation). The dashed line corresponds to a threshold
of p = 0.05 (a significant result should be located to the right of this line), while the solid line is the
observed log evidence for the best empirical model. The result indicates the improvement in model
evidence afforded by the true tractography matrix is significant (p = 0.01) in a classical sense, with
respect to a null distribution based on permutation testing.
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the improvement in evidence afforded by applying the GL to the actual structural adjacency
matrix was significant in a classical sense (p = 0.01), with respect to a null distribution of
largest log-model evidences (Figure 5).
DISCUSSION
This study makes several contributions to the understanding of structure-function relationships
in the brain. Based on previous research (Kale et al., 2018; Avena-Koenigsberger et al., 2019;
Seguin et al., 2019), we hypothesised that asymmetric polysynaptic anatomical connectiv-
ity would better predict the directed causal dynamics between neuronal populations (effec-
tive connectivity) than conventional (i.e., symmetric and monosynaptic) information on WM
pathways. The introduction of the GL allowed us to parameterise a diffusion process on the
structural graph, providing symmetrically and asymmetrically weighted adjacency matrices
of increasing order. Of note, other methods for modelling network communication based on
structural connectomes can inherently inform on asymmetry. Such approaches include navi-
gation (Seguin et al., 2018), search information (Goni et al., 2014), linear transmission models
(Misic et al., 2015), and diffusion efficiency (Goni et al., 2013). The novel approach presented
here enables hypotheses to be tested about the mapping from indirect anatomical connec-
tivity to effective connectivity via a (variational) Bayesian framework. Using a dataset with
fMRI and HARDI measures, we found that high-order structural connectivity simulated using
the GL greatly improved the evidence for DCMs of effective connectivity, compared to DCMs
informed by direct structural connectivity and anatomically uninformed models. Most impor-
tantly, input sensitivity during the diffusion process best predicted effective connectivity.
to
We introduced a computationally efficient approach to map indirect anatomical
effective connectivity, using hierarchical PEB models and BMR for DCMs (Friston et al., 2016;
Zeidman et al., 2019). This procedure is designed to account inherently for possible variations
in network architecture, normalisation scheme, value of anatomical information and mapping
between structural and effective connectivity, for any given study and context. By definition,
effective connectivity is determined by the context, such as the specific experimental task or
cognitive set (Friston et al., 2003). For this reason, we would not expect a universally optimal
set of priors on effective connectivity that could explain cognition per se. We sought to provide
an efficient method for finding the best effective connectivity priors for any specific context, in-
formed by indirect anatomical connectivity. An interesting future question will be whether the
utility of the GL approach to assess indirect structural connectivity generalises to models of ef-
fective connectivity for resting-state data. The value of understanding asymmetric coupling be-
tween functionally related regions at rest, through combination of neuronal and observational
models such as those used by dynamic causal modelling, has become increasingly recognised
(Friston et al., 2014; Razi et al., 2015, 2017). Recent work has already linked asymmetries in
indirect anatomical connectivity to resting-state effective connectivity (Seguin et al., 2019).
Using high-quality dMRI along with task-related and resting-state fMRI datasets such as
from the Human Connectome Project (Van Essen et al., 2013) may contribute to further test
and refine the outlined approach and conclusions, complementing previous research on these
datasets (Seguin et al., 2019). Furthermore, it will be of interest to perform large-scale analyses
integrating the structural connectome with whole-brain effective connectivity using the re-
cently introduced regressive dynamic causal modelling (Frassle et al., 2018). Such approaches
may also help to inform generative models of how WM pathways give rise to brain dynam-
ics (Robinson, 2012; Deco et al., 2013; Sanz Leon et al., 2013; Melozzi et al., 2017; Messe
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et al., 2018), and to better understand how network dynamics may shape cognitive function
and behaviour (Aertsen et al., 1994; Gerraty et al., 2018; Sokolov et al., 2018). Relating the
diffusion properties of brain networks to their functions in extension of previous approaches
based on direct anatomical connectivity (Deco et al., 2012; Senden et al., 2012; Hermundstad
et al., 2014) may further improve our conceptualisation of distributed information processing
in the brain.
Endowing the GL diffusion process with asymmetry clearly outperformed a symmetric dif-
fusion process in sculpting effective connectivity, suggesting that ensemble dynamics in the
brain may be shaped by the sensitivity of regions to their distributed input. These findings agree
with and extend recent work on resting-state functional connectivity in macaques and humans
showing that synchrony between nodes does not only depend on their direct or second-order
connectivity, but also the similarity of afferents they receive from the entire network and other
adjacent network characteristics (Adachi et al., 2012; Goni et al., 2014; Bettinardi et al., 2017).
Consequently, densely connected regions may not necessarily be in a best position to influ-
ence or to be influenced by other regions (Avena-Koenigsberger et al., 2017). This follows from
the fact that being locked into dense subgraphs may preclude a more widespread sensitivity to
distributed dynamics (Pillai & Jirsa, 2017). Clarifying whether and why some networks may be
better characterised in terms of their nodes’ sensitivity to inputs as opposed to their capacity
to influence other nodes is a promising avenue for future research on normal and altered brain
network function that can be pursued formally by the procedure described here.
The results presented here agree with and extend previous research employing network
communication models (Avena-Koenigsberger et al., 2019; Seguin et al., 2019). These studies
inferred the ease of sending and receiving information from undirected structural connec-
tomes. The differences between send and receive efficiencies were mapped onto functional
brain topography and the outcomes of separate resting-state effective connectivity analyses. For
instance, anatomical connectivity-based classification suggested that unimodal areas such as
the primary visual cortex or sensorimotor cortices were predominantly sending information,
whereas multimodal regions were mainly receivers (Seguin et al., 2019). The diffusion effi-
ciency approach, described as the (inverse) mean first passage time of a Markov chain process
(Goni et al., 2013; Seguin et al., 2019), is similar to in-degree normalisation. In contrast to the
present work, diffusion efficiency is derived using a matrix of transition probabilities. Nonethe-
less, measuring diffusion efficiency in a structural adjacency matrix yielded similar results,
revealing regional variability in input sensitivity and a rather uniform capacity to influence
other regions (Seguin et al., 2019). Future investigations are needed to fully explore the impli-
cations of the various measures to model diffusion processes.
The other important finding was that higher orders and the principle eigenmode of the GL
afforded better priors on effective connectivity than lower GL orders. This indicated that WM
connections and distributed neural dynamics give rise to brain communication through recur-
rent neuronal message passing over multiple paths. GL eigenmodes and eigenvalues are closely
related to network communicability, representing the ease of information transmission along
all possible paths in a network (Estrada & Hatano, 2008; Crofts & Higham, 2009; Andreotti
et al., 2014; Grayson et al., 2016; Shine et al., 2018). GL eigenmodes of structural adjacency
matrices exhibit a high degree of similarity between healthy subjects, as well as consistent and
meaningful alterations in developmental and virtual agenesis of the corpus callosum (Wang
et al., 2017). Laplacian eigenvalue spectra have been used for cross-species comparison of
anatomical networks and revealed specific characteristics of neural networks as opposed to
other network classes (de Lange et al., 2014). Furthermore, the anatomical graph energy
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(connectedness measure representing the sum of all absolute GL eigenvalues) has been shown
to be significantly lower in patients with Alzheimer’s disease than in controls (Daianu et al.,
2015). A greater number of apolipoprotein E4 gene copies predicted this reduction in graph
energy. The use and interpretation of metrics such as eigenmodes and eigenvalues afforded by
diffusion processes simulated by the exponentiation of a GL matrix could lead towards con-
sideration of more global network characteristics beyond the conventionally assessed single
hub or subgraph properties.
In clinical research, truly integrative computational, graph theoretic, or even simple correl-
ative analyses of multimodal connectivity remain rather sparse. However, the assessment and
comparison of network communicability using the GL may be of potential relevance to clinical
neuroscience. Implementation of the GL in patients to assess how local and global changes in
anatomical connectivity affect functional dynamics may shed new light on pathophysiology.
Other comparative network measures afforded by the GL are topological similarity, persis-
tent homology and graph diffusion distance (Hammond et al., 2013a; Bettinardi et al., 2017;
Liang & Wang, 2017). Furthermore, simulation of network diffusion by means of the GL has
been used to predict neuronal spreading and resulting brain atrophy patterns in Alzheimer’s
and frontotemporal dementia (Raj et al., 2012) and to infer sources of disease propagation in
mild cognitive impairment and Alzheimer’s dementia (Hu et al., 2016). Ultimately, the global
measures afforded by the GL may be used towards assessment of the relationships between
connectivity and behaviour at the network level (Sokolov et al., 2018). As efficiency and ease-
of-use are of primary significance in everyday clinical practice, this relatively straightforward
and rapid approach could potentially afford useful network biomarkers in neurological and
psychiatric disorders.
Dynamic causal modelling for fMRI has contributed to establishing or refining various
neuroanatomical and neurobiological concepts and hypotheses in functional realms such as
reading, mental imagery, memory retrieval, or body language reading (Chow et al., 2008;
Sokolov et al., 2012; Dijkstra et al., 2017; Ren et al., 2018; Sokolov et al., 2018). Inclusion of
electrophysiological data from EEG, MEG, or intracranial recording (David et al., 2013;
Almashaikhi et al., 2014; Proix et al., 2017), which enable more detailed biophysical modelling
due to their high temporal resolution, may offer additional insight. Indeed, structure-function
relationships appear to depend on different timescales of functional dynamics. Changes over
periods of several minutes largely reflect underlying anatomical connectivity, whereas dynam-
ics lasting a few seconds do so to a lesser degree (Honey et al., 2007; Shen et al., 2015; Cabral
et al., 2017). Interestingly, resting-state functional connectivity data derived from simultaneous
EEG and fMRI outperforms fMRI data alone in modelling structural connectivity, when compar-
ing these predictions to actual dMRI measures (Wirsich et al., 2017). Network models based
on anatomical connectivity and driven by EEG source activity accurately predict individual
fMRI resting-state patterns and reproduce neurophysiological phenomena and mechanisms
observed with different imaging modalities (Schirner et al., 2018). GL approaches to struc-
tural connectivity have also been used to improve EEG source localisation (Hammond et al.,
2013b). Another potentially exciting development could be the computational modelling of
sparse impulse stimulations as input in the GL diffusion process, similar to that implemented
for modelling of disease propagation sources (Hu et al., 2016). This could further approximate
the neuronal model within dynamic causal modelling (Friston et al., 2003) and information
flow in the brain.
In summary, we have studied symmetric and asymmetric diffusion processes based on
anatomical connectivity to optimise prior constraints on models of directed effective connec-
tivity. Bayesian model comparison indicated the best effective connectivity priors are provided
Network Neuroscience
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by indirect, high-order structural connectivity determined by the regional sensitivity to inputs
that would be seen under equilibrium states of particle diffusion within an anatomical net-
work. This may speak to a reappraisal of how we characterise the anatomical connectome,
when trying to understand asymmetric functional dynamics arising from structure of the sort
measured in neuroimaging.
ACKNOWLEDGMENTS
The authors wish to thank Richard Frackowiak, Patric Hagmann, Alexander Sokolov, and Klaas-
Enno Stephan for valuable discussion. Technical support was provided by Ric Davis, Jürgen
Dax, Chris Freemantle, Bernd Kardatzki, Rachael Maddock and Liam Reilly, and administra-
tive assistance by Marcia Bennett, David Blundred, Kamlyn Ramkissoon, Tracy Skinner, and
Daniela Warr.
SUPPORTING INFORMATION
Supporting Information for this article is available at https://doi.org/10.1162/netn_a_00150.
AUTHOR CONTRIBUTIONS
Arseny A. Sokolov: Conceptualization; Formal analysis; Funding acquisition; Investigation;
Methodology; Writing - Original Draft. Peter Zeidman: Methodology; Supervision; Writing -
Review & Editing. Adeel Razi: Conceptualization; Methodology; Writing - Review & Editing.
Michael Erb: Methodology; Supervision. Philippe Ryvlin: Supervision; Writing - Review &
Editing. Marina A. Pavlova: Conceptualization; Funding acquisition; Methodology; Resources;
Supervision; Writing - Review & Editing. Karl J. Friston: Conceptualization; Formal analysis;
Funding acquisition; Methodology; Resources; Supervision; Writing - Review & Editing.
FUNDING INFORMATION
Arseny A. Sokolov, Baasch-Medicus Foundation. Arseny A. Sokolov, Fondation Leenaards
(http://dx.doi.org/10.13039/501100006387). Arseny A. Sokolov, Schweizerische Neurologis-
che Gesellschaft (http://dx.doi.org/10.13039/100010766). Arseny A. Sokolov, Helmut Horten
Foundation. Arseny A. Sokolov, Synapsis Foundation Alzheimer Research Switzerland, Award
ID: 2019-CDA03. Marina A. Pavlova, Reinhold Beitlich Stiftung (http://dx.doi.org/10.13039/
501100003541). Marina A. Pavlova, BBBank Foundation. Marina A. Pavlova, Deutsche
Forschungsgemeinschaft (http://dx.doi.org/10.13039/501100001659), Award ID: DFG PA 847/
22-1. Karl J. Friston, Wellcome Trust (http://dx.doi.org/10.13039/100004440), Award ID:
088130/Z/09/Z.
REFERENCES
Abdelnour, F., Dayan, M., Devinsky, O., Thesen, T., & Raj, A. (2018).
Functional brain connectivity is predictable from anatomic
networks Laplacian eigen-structure. Neuroimage, 172, 728–739.
https://doi.org/10.1016/j.neuroimage.2018.02.016
Abdelnour, F., Voss, H. U., & Raj, A.
(2014). Network diffusion
accurately models the relationship between structural and func-
tional brain connectivity networks. Neuroimage, 90, 335–347.
https://doi.org/10.1016/j.neuroimage.2013.12.039
Adachi, Y., Osada, T., Sporns, O., Watanabe, T., Matsui, T.,
Miyamoto, K., & Miyashita, Y.
(2012). Functional connectivity
between anatomically unconnected areas is shaped by collective
network-level effects in the macaque cortex. Cerebral Cortex, 22(7),
1586–1592. https://doi.org/10.1093/cercor/bhr234
Aertsen, A., Erb, M., & Palm, G.
(1994). Dynamics of functional
coupling in the cerebral cortex: An attempt at a model-based
interpretation. Physica D: Nonlinear Phenomena, 75(1–3), 103–128.
Network Neuroscience
886
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
t
/
e
d
u
n
e
n
a
r
t
i
c
e
-
p
d
l
f
/
/
/
/
/
4
3
8
7
1
1
8
6
7
3
5
5
n
e
n
_
a
_
0
0
1
5
0
p
d
t
.
f
b
y
g
u
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Anatomical network diffusion and effective connectivity
Jung,
Almashaikhi, T., Rheims, S.,
J., Ostrowsky-Coste, K.,
Montavont, A., De Bellescize, J., Arzimanoglou, A., Keo Kosal, P.,
Guenot, M., Bertrand, O., & Ryvlin, P. (2014). Functional con-
nectivity of insular efferences. Human Brain Mapping, 35(10),
5279–5294. https://doi.org/10.1002/hbm.22549
Andreotti, J., Jann, K., Melie-Garcia, L., Giezendanner, S., Abela, E.,
Wiest, R., Dierks, T., & Federspiel, A. (2014). Validation of net-
work communicability metrics for the analysis of brain structural
networks. PLoS One, 9(12), e115503. https://doi.org/10.1371/
journal.pone.0115503
Avena-Koenigsberger, A., Misic, B., & Sporns, O. (2017). Commu-
nication dynamics in complex brain networks. Nature Reviews Neu-
roscience, 19(1), 17–33. https://doi.org/10.1038/nrn.2017.149
Avena-Koenigsberger, A., Yan, X., Kolchinsky, A., van den Heuvel,
M. P., Hagmann, P., & Sporns, O. (2019). A spectrum of routing
strategies for brain networks. PLoS Computational Biology, 15(3),
e1006833. https://doi.org/10.1371/journal.pcbi.1006833
Barreda, J. L., & Zhou, H. X. (2011). A solvable model for the dif-
fusion and reaction of neurotransmitters in a synaptic junction.
BMC Biophysics, 4, 5. https://doi.org/10.1186/2046-1682-4-5
Becker, C. O., Pequito, S., Pappas, G. J., Miller, M. B., Grafton,
(2016). Accurately pre-
S. T., Bassett, D. S., & Preciado, V. M.
dicting functional connectivity from diffusion imaging. In arXiv.
p1512.02602v02603.
Behrens, T. E., Berg, H. J., Jbabdi, S., Rushworth, M. F., & Woolrich,
M. W. (2007). Probabilistic diffusion tractography with multiple
fibre orientations: What can we gain? Neuroimage, 34(1), 144–155.
https://doi.org/10.1016/j.neuroimage.2006.09.018
Bettinardi, R. G., Deco, G., Karlaftis, V. M., Van Hartevelt, T. J.,
Fernandes, H. M., Kourtzi, Z., Kringelbach, M. L, & Zamora-
Lopez, G. (2017). How structure sculpts function: Unveiling the
contribution of anatomical connectivity to the brain’s sponta-
neous correlation structure. Chaos, 27(4), 047409. https://doi.org/
10.1063/1.4980099
Biggs, N. L. (1993). Algebraic graph theory. Cambridge: Cambridge
University Press.
Bullmore, E., & Sporns, O. (2012). The economy of brain network
organization. Nature Reviews Neuroscience, 13(5), 336–349.
https://doi.org/10.1038/nrn3214
Cabral, J., Kringelbach, M. L., & Deco, G.
(2017). Functional
connectivity dynamically evolves on multiple time-scales over
a static structural connectome: Models and mechanisms. Neuro-
image, 160, 84–96. https://doi.org/10.1016/j.neuroimage.2017.
03.045
Chow, H. M., Kaup, B., Raabe, M., & Greenlee, M. W.
(2008).
Evidence of fronto-temporal interactions for strategic inference
processes during language comprehension. Neuroimage, 40(2),
940–954. https://doi.org/S1053-8119(07)01089-0
Crofts, J. J., & Higham, D. J. (2009). A weighted communicability
measure applied to complex brain networks. Journal of the Royal
Society Interface, 6(33), 411–414. https://doi.org/10.1098/rsif.
2008.0484
Daianu, M., Mezher, A., Jahanshad, N., Hibar, D. P., Nir, T. M.,
Jack, C. R., Jr., Weiner, M. W., Bernstein, M. A., & Thompson,
P. M.
(2015). Spectral graph theory and graph energy met-
rics show evidence for the Alzheimer’s disease disconnection
syndrome in Apoe-4 risk gene carriers. Proceedings IEEE In-
ternational Symposium on Biomedical Imaging, 2015, 458–461.
https://doi.org/10.1109/ISBI.2015.7163910
David, O., Job, A. S., De Palma, L., Hoffmann, D., Minotti, L., &
Kahane, P.
(2013). Probabilistic functional tractography of the
human cortex. Neuroimage, 80, 307–317. https://doi.org/10.
1016/j.neuroimage.2013.05.075
Deco, G., Ponce-Alvarez, A., Mantini, D., Romani, G. L., Hagmann,
P., & Corbetta, M.
(2013). Resting-state functional connectiv-
ity emerges from structurally and dynamically shaped slow lin-
ear fluctuations. Journal of Neuroscience, 33(27), 11239–11252.
https://doi.org/10.1523/JNEUROSCI.1091-13.2013
Deco, G., Senden, M., & Jirsa, V.
(2012). How anatomy shapes
dynamics: A semi-analytical study of the brain at rest by a sim-
ple spin model. Frontiers in Computational Neuroscience, 6, 68.
https://doi.org/10.3389/fncom.2012.00068
de Lange, S. C., de Reus, M. A., & van den Heuvel, M. P. (2014).
The Laplacian spectrum of neural networks. Frontiers in Compu-
tational Neuroscience, 7, 189. https://doi.org/10.3389/fncom.2013.
00189
Deligianni, F., Robinson, E., Beckmann, C. F., Sharp, D., Edwards,
A. D., & Rueckert, D.
Inference of functional connec-
tivity from direct and indirect structural brain connections. In
2011 IEEE International Symposium on Biomedical
Imaging:
From Nano to Macro. Chicago, IL, pp. 849–852. https://doi.org/
10.1109/ISBI.2011.5872537
(2011).
Dijkstra, N., Zeidman, P., Ondobaka, S., van Gerven, M. A. J., &
Friston, K. (2017). Distinct Top-down and bottom-up brain con-
nectivity during visual perception and imagery. Scientific Reports,
7(1), 5677. https://doi.org/10.1038/s41598-017-05888-8
Estrada, E., & Hatano, N. (2008). Communicability in complex net-
works. Physical Revue E, 77(3 Pt 2), 036111. https://doi.org/10.
1103/PhysRevE.77.036111
Frassle, S., Lomakina, E. I., Kasper, L., Manjaly, Z. M., Leff, A.,
Pruessmann, K. P., Buhmann, J. M., & Stephan, K. E.
(2018).
A generative model of whole-brain effective connectivity. Neu-
roimage, 179, 505–529. https://doi.org/10.1016/j.neuroimage.
2018.05.058
Friston, K., Zeidman, P., & Litvak, V.
(2015). Empirical Bayes for
DCM: A group inversion scheme. Frontiers in Systems Neuro-
science, 9, 164. https://doi.org/10.3389/fnsys.2015.00164
Friston, K. J.
(2011). Functional and effective connectivity: A re-
view. Brain Connectivity, 1(1), 13–36. https://doi.org/10.1089/
brain.2011.0008
Friston, K. J., Harrison, L., & Penny, W.
(2003). Dynamic causal
modelling. Neuroimage, 19(4), 1273–1302. https://doi.org/
S1053811903002027
Friston, K. J., Kahan, J., Biswal, B., & Razi, A. (2014). DCM for rest-
ing state fMRI. Neuroimage, 94, 396–407. https://doi.org/10.1016/
j.neuroimage.2013.12.009
Friston, K. J., Litvak, V., Oswal, A., Razi, A., Stephan, K. E., van Wijk,
B. C., Ziegler, G., & Zeidman, P.
(2016). Bayesian model re-
duction and empirical Bayes for group (DCM) studies. Neuroim-
age, 128, 413–431. https://doi.org/10.1016/j.neuroimage.2015.
11.015
Network Neuroscience
887
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
t
/
e
d
u
n
e
n
a
r
t
i
c
e
-
p
d
l
f
/
/
/
/
/
4
3
8
7
1
1
8
6
7
3
5
5
n
e
n
_
a
_
0
0
1
5
0
p
d
t
.
f
b
y
g
u
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Anatomical network diffusion and effective connectivity
Gerraty, R. T., Davidow, J. Y., Foerde, K., Galvan, A., Bassett, D. S.,
& Shohamy, D. (2018). Dynamic flexibility in striatal-cortical cir-
cuits supports reinforcement learning. Journal of Neuroscience.
https://doi.org/10.1523/jneurosci.2084-17.2018
Goni,
J., Avena-Koenigsberger, A., Velez de Mendizabal, N.,
van den Heuvel, M. P., Betzel, R. F., & Sporns, O. (2013). Ex-
ploring the morphospace of communication efficiency in com-
plex networks. PLoS One, 8(3), e58070. https://doi.org/10.1371/
journal.pone.0058070
Goni, J., van den Heuvel, M. P., Avena-Koenigsberger, A., Velez
de Mendizabal, N., Betzel, R. F., Griffa, A., Hagmann, P.,
Corominas-Murtra, B., Thiran, J. P., & Sporns, O. (2014). Resting-
brain functional connectivity predicted by analytic measures of
network communication. Proceedings of the National Academy
of Sciences of the United States of America, 111(2), 833–838.
https://doi.org/10.1073/pnas.1315529111
Grayson, D. S., Bliss-Moreau, E., Machado, C. J., Bennett, J., Shen,
K., Grant, K. A., Fair, D. A., & Amaral, D. G. (2016). The rhesus
monkey connectome predicts disrupted functional networks re-
sulting from pharmacogenetic inactivation of
the amygdala.
Neuron, 91(2), 453–466. https://doi.org/10.1016/j.neuron.2016.
06.005
Hammond, D. K., Gur, Y., & Johnson, C. R. (2013a). Graph diffu-
sion distance: A difference measure for weighted graphs based
on the graph Laplacian exponential kernel. In: 2013 IEEE Global
Conference on Signal and Information Processing. Austin, TX.
p. 419–422.
Hammond, D. K., Scherrer, B., & Warfield, S. K. (2013b). Cortical
graph smoothing: A novel method for exploiting DWI-derived
anatomical brain connectivity to improve EEG source estima-
tion. IEEE Transactions on Medical Imaging, 32(10), 1952–1963.
https://doi.org/10.1109/TMI.2013.2271486
Hermundstad, A. M., Brown, K. S., Bassett, D. S., Aminoff, E. M.,
Frithsen, A., Johnson, A., Tipper, C. M., Miller, M. B., Grafton,
S. T., & Carlson, J. M. (2014). Structurally-constrained relation-
ships between cognitive states in the human brain. PLoS Com-
putational Biology, 10(5), e1003591. https://doi.org/10.1371/
journal.pcbi.1003591
Honey, C. J., Kotter, R., Breakspear, M., & Sporns, O. (2007). Net-
work structure of cerebral cortex shapes functional connectivity
on multiple time scales. Proceedings of the National Academy of
Sciences of the United States of America, 104(24), 10240–10245.
https://doi.org/10.1073/pnas.0701519104
Honey, C. J., Sporns, O., Cammoun, L., Gigandet, X., Thiran, J. P.,
Meuli, R., & Hagmann, P. (2009). Predicting human resting-state
functional connectivity from structural connectivity. Proceed-
ings of the National Academy of Sciences of the United States
of America, 106(6), 2035–2040. https://doi.org/10.1073/pnas.
0811168106
Honey, C. J., Thivierge, J. P., & Sporns, O. (2010). Can structure pre-
dict function in the human brain? Neuroimage, 52(3), 766–776.
https://doi.org/10.1016/j.neuroimage.2010.01.071
Hu, C., Hua, X., Ying, J., Thompson, P. M., Fakhri, G. E., & Li, Q.
(2016). Localizing sources of brain disease progression with net-
IEEE Journal of Selected Topics in Signal
work diffusion model.
Processing, 10(7), 1214–1225. https://doi.org/10.1109/JSTSP.
2016.2601695
Jenkinson, M., Bannister, P., Brady, M., & Smith, S. (2002). Improved
optimization for the robust and accurate linear registration and
motion correction of brain images. Neuroimage, 17(2), 825–841.
https://doi.org/10.1006/nimg.2002.1132
Kale, P., Zalesky, A., & Gollo, L. L. (2018). Estimating the impact
of structural directionality: How reliable are undirected connec-
tomes? Network Neuroscience, 2(2), 259–284.
Koch, M. A., Norris, D. G., & Hund-Georgiadis, M. (2002). An in-
vestigation of functional and anatomical connectivity using mag-
netic resonance imaging. Neuroimage, 16(1), 241–250. https://
doi.org/10.1006/nimg.2001.1052
Liang, H., & Wang, H. (2017). Structure-function network mapping
and its assessment via persistent homology. PLoS Computational
Biology, 13(1), e1005325. https://doi.org/10.1371/journal.pcbi.
1005325
Markov, N. T., Ercsey-Ravasz, M., Van Essen, D. C., Knoblauch, K.,
Toroczkai, Z., & Kennedy, H. (2013). Cortical high-density coun-
terstream architectures. Science, 342(6158), 1238406. https://
doi.org/10.1126/science.1238406
Meier, J., Tewarie, P., Hillebrand, A., Douw, L., van Dijk, B. W.,
Stufflebeam, S. M., & Van Mieghem, P. (2016). A mapping be-
tween structural and functional brain networks. Brain Connec-
tivity, 6(4), 298–311. https://doi.org/10.1089/brain.2015.0408
Melozzi, F., Woodman, M. M., Jirsa, V. K., & Bernard, C.
(2017).
The virtual mouse brain: A computational neuroinformatics plat-
form to study whole mouse brain dynamics. eNeuro, 4(3). https://
doi.org/10.1523/ENEURO.0111-17.2017
Messe, A., Hutt, M. T., & Hilgetag, C. C. (2018). Toward a theory of
coactivation patterns in excitable neural networks. PLoS Com-
putational Biology, 14(4), e1006084. https://doi.org/10.1371/
journal.pcbi.1006084
Misic, B., Betzel, R. F., Nematzadeh, A., Goni,
J., Griffa, A.,
Hagmann, P., Flammini, A., Ahn, Y. Y., & Sporns, O.
(2015).
Cooperative and competitive spreading dynamics on the human
connectome. Neuron, 86(6), 1518–1529. https://doi.org/10.
1016/j.neuron.2015.05.035
Park, H. J., & Friston, K.
(2013). Structural and functional brain
networks: From connections to cognition. Science, 342(6158),
1238411. https://doi.org/10.1126/science.1238411
Penny, W. D.
(2012). Comparing dynamic causal models using
AIC, BIC and free energy. Neuroimage, 59(1), 319–330. https://
doi.org/10.1016/j.neuroimage.2011.07.039
Pillai, A. S., & Jirsa, V. K. (2017). Symmetry breaking in space-time
hierarchies shapes brain dynamics and behavior. Neuron, 94(5),
1010–1026. https://doi.org/10.1016/j.neuron.2017.05.013
Pineda-Pardo, J. A., Bruna, R., Woolrich, M., Marcos, A., Nobre,
A. C., Maestu, F., & Vidaurre, D.
(2014). Guiding functional
connectivity estimation by structural connectivity in MEG: An
application to discrimination of conditions of mild cognitive im-
pairment. Neuroimage, 101, 765–777. https://doi.org/10.1016/
j.neuroimage.2014.08.002
Pollick, F. E., Paterson, H. M., Bruderlin, A., & Sanford, A. J. (2001).
Perceiving affect from arm movement. Cognition, 82(2), B51–61.
Network Neuroscience
888
l
D
o
w
n
o
a
d
e
d
f
r
o
m
h
t
t
p
:
/
/
d
i
r
e
c
t
.
m
i
t
.
/
t
/
e
d
u
n
e
n
a
r
t
i
c
e
-
p
d
l
f
/
/
/
/
/
4
3
8
7
1
1
8
6
7
3
5
5
n
e
n
_
a
_
0
0
1
5
0
p
d
t
.
f
b
y
g
u
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Anatomical network diffusion and effective connectivity
Proix, T., Bartolomei, F., Guye, M., & Jirsa, V. K. (2017). Individual
brain structure and modelling predict seizure propagation. Brain,
140(3), 641–654. https://doi.org/10.1093/brain/awx004
Raj, A., Kuceyeski, A., & Weiner, M. (2012). A network diffusion model
of disease progression in dementia. Neuron, 73(6), 1204–1215.
https://doi.org/10.1016/j.neuron.2011.12.040
Razi, A., Kahan, J., Rees, G., & Friston, K. J. (2015). Construct vali-
dation of a DCM for resting state fMRI. Neuroimage, 106, 1–14.
https://doi.org/10.1016/j.neuroimage.2014.11.027
Razi, A., Seghier, M. L., Zhou, Y., McColgan, P., Zeidman, P., Park,
H. J., Sporns, O., Rees, G., & Friston, K. J. (2017). Large-scale DCMs
for resting-state fMRI. Network Neuroscience, 1(3), 222–241.
https://doi.org/10.1162/netn_a_00015
Ren, Y., Nguyen, V. T., Sonkusare, S., Lv, J., Pang, T., Guo, L.,
Eickhoff, S. B., Breakspear, M., & Guo, C. C.
(2018). Effective
connectivity of the anterior hippocampus predicts recollection
confidence during natural memory retrieval. Nature Com-
munications, 9(1), 4875. https://doi.org/10.1038/s41467-018-
07325-4
Robinson, P. A.
(2012).
Interrelating anatomical, effective, and
functional brain connectivity using propagators and neural field
theory. Physical Review E, 85(1 Pt 1), 011912. https://doi.org/10.
1103/physreve.85.011912
Sanz Leon, P., Knock, S. A., Woodman, M. M., Domide, L.,
Mersmann, J., McIntosh, A. R., & Jirsa, V.
(2013). The virtual
brain: A simulator of primate brain network dynamics. Frontiers
in Neuroinformatics, 7, 10. https://doi.org/10.3389/fninf.2013.
00010
Schirner, M., McIntosh, A. R., Jirsa, V., Deco, G., & Ritter, P.
(2018). Inferring multi-scale neural mechanisms with brain net-
work modelling. eLife, 7. https://doi.org/10.7554/eLife.28927
Seguin, C., Razi, A., & Zalesky, A. (2019). Zalesky, A Inferring neural
signalling directionality from undirected structural connectomes.
Nature Communications, 10(1), 4289. https://doi.org/10.1038/
s41467-019-12201-w
Seguin, C., van den Heuvel, M. P., & Zalesky, A. (2018). Naviga-
tion of brain networks. Proceedings of the National Academy of
Sciences of the United States of America, 115(24), 6297–6302.
https://doi.org/10.1073/pnas.1801351115
Senden, M., Goebel, R., & Deco, G. (2012). Structural connectiv-
ity allows for multi-threading during rest: the structure of the cor-
tex leads to efficient alternation between resting state exploratory
behavior and default mode processing. Neuroimage, 60(4),
2274–2284. https://doi.org/10.1016/j.neuroimage.2012.02.061
Shen, K., Hutchison, R. M., Bezgin, G., Everling, S., & McIntosh,
(2015). Network structure shapes spontaneous func-
Journal of Neuroscience, 35(14),
A. R.
tional connectivity dynamics.
5579–5588. https://doi.org/10.1523/jneurosci.4903-14.2015
Shine, J. M., Aburn, M. J., Breakspear, M., & Poldrack, R. A. (2018).
The modulation of neural gain facilitates a transition between
functional segregation and integration in the brain. eLife, 7.
https://doi.org/10.7554/eLife.31130
Smith, S. M. (2002). Fast robust automated brain extraction. Hu-
man Brain Mapping, 17(3), 143–155. https://doi.org/10.1002/
hbm.10062
Sokolov, A. A., Erb, M., Gharabaghi, A., Grodd, W., Tatagiba, M. S.,
& Pavlova, M. A. (2012). Biological motion processing: The left
cerebellum communicates with the right superior temporal sul-
cus. Neuroimage, 59(3), 2824–2830. https://doi.org/10.1016/j.
neuroimage.2011.08.039
Sokolov, A. A., Erb, M., Grodd, W., & Pavlova, M. A.
(2014). A
Structural loop between the cerebellum and the superior tem-
poral sulcus: Evidence from diffusion tensor imaging. Cerebral
Cortex, 24(3), 626–632. https://doi.org/10.1093/cercor/bhs346
Sokolov, A. A., Kruger, S., Enck, P., Krageloh-Mann, I., & Pavlova,
M. A. (2011). A Gender affects body language reading. Frontiers
in Psychology, 2, 16. https://doi.org/10.3389/fpsyg.2011.00016
Sokolov, A. A., Zeidman, P., Erb, M., Ryvlin, P., Friston, K. J.,
(2018). A Structural and effective brain
& Pavlova, M. A.
connectivity underlying biological motion detection. Proceed-
ings of the National Academy of Sciences of the United States
of America, 115(51), E12034–E12042. https://doi.org/10.1073/
pnas.1812859115
Sokolov, A. A., Zeidman, P., Erb, M., Ryvlin, P., Pavlova, M. A., &
Friston, K. J. (2019). Linking structural and effective brain connec-
tivity: Structurally informed Parametric Empirical Bayes (si-PEB).
Brain Structure & Function, 224(1), 205–217. https://doi.org/10.
1007/s00429-018-1760-8
Sporns, O. (2014). Contributionsand challenges for network models
in cognitive neuroscience. Nature Neuroscience, 17(5), 652–660.
https://doi.org/10.1038/nn.3690
Sporns, O., Tononi, G., & Edelman, G. M. (2000). Theoretical neu-
roanatomy: Relating anatomical and functional connectivity in
graphs and cortical connection matrices. Cerebral Cortex, 10(2),
127–141.
Stephan, K. E., Tittgemeyer, M., Knosche, T. R., Moran, R. J., &
Friston, K. J.
(2009). Tractography-based priors for dynamic
causal models. Neuroimage, 47(4), 1628–1638. https://doi.org/
10.1016/j.neuroimage.2009.05.096
Toga, A. W., Thompson, P. M., Mori, S., Amunts, K., & Zilles, K.
(2006). Towards multimodal atlases of the human brain. Nature
Reviews Neuroscience, 7(12), 952–966. https://doi.org/10.1038/
nrn2012
Tzourio-Mazoyer, N., Landeau, B., Papathanassiou, D., Crivello, F.,
Etard, O., Delcroix, N., Mazoyer, B., & Joliot, M. (2002). Auto-
mated anatomical labeling of activations in SPM using a macro-
scopic anatomical parcellation of the MNI MRI single-subject
brain. Neuroimage, 15(1), 273–289. https://doi.org/10.1006/nimg.
2001.0978
Uludag, K., & Roebroeck, A. (2014). General overview on the merits
of multimodal neuroimaging data fusion. Neuroimage, 102 Pt 1,
3–10. https://doi.org/10.1016/j.neuroimage.2014.05.018
Van Essen, D. C., Smith, S. M., Barch, D. M., Behrens, T. E., Yacoub, E.,
Ugurbil, K., & WU-Minn HCP Consortium. (2013). The WU-minn
Human Connectome Project: An overview. Neuroimage, 80,
62–79. https://doi.org/10.1016/j.neuroimage.2013.05.041
Wang, M. B., Owen, J. P., Mukherjee, P., & Raj, A. (2017). Brain net-
work eigenmodes provide a robust and compact representation
of the structural connectome in health and disease. PLoS Com-
putational Biology, 13(6), e1005550. https://doi.org/10.1371/
journal.pcbi.1005550
Network Neuroscience
889
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2
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Anatomical network diffusion and effective connectivity
Wirsich, J., Ridley, B., Besson, P., Jirsa, V., Benar, C., Ranjeva, J. P.,
& Guye, M. (2017). Complementary contributions of concurrent
EEG and fMRI connectivity for predicting structural connectivity.
Neuroimage, 161, 251–260. https://doi.org/10.1016/j.neuroimage.
2017.08.055
Yarkoni, T., Poldrack, R. A., Nichols, T. E., Van Essen, D. C., &
Wager, T. D. (2011). Large-scale automated synthesis of human
functional neuroimaging data. Nature Methods, 8(8), 665–670.
https://doi.org/10.1038/nmeth.1635
Zeidman, P., Jafarian, A., Seghier, M. L., Litvak, V., Cagnan, H.,
Price, C. J., & Friston, K. J. (2019). A guide to group effective con-
nectivity analysis, part 2: Second level analysis with PEB. Neuro-
image, 200, 12–25. https://doi.org/10.1016/j.neuroimage.2019.
06.032
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