RECHERCHE
The role of node dynamics in shaping emergent
functional connectivity patterns in the brain
Michael Forrester
1, Jonathan J. Crofts
2, Stamatios N. Sotiropoulos
3,4,5,
Stephen Coombes
1, and Reuben D. O’Dea
1
1Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, ROYAUME-UNI
2Department of Physics and Mathematics, School of Science and Technology, Nottingham Trent University, Nottingham, ROYAUME-UNI
3Sir Peter Mansfield Imaging Centre, Queen’s Medical Centre, University of Nottingham, Nottingham, ROYAUME-UNI
4Wellcome Centre for Integrative Neuroimaging (WIN-FMRIB), University of Oxford, Oxford, ROYAUME-UNI
5National Institute for Health Research (NIHR) Nottingham Biomedical Research Centre,
Queen’s Medical Centre, Nottingham, ROYAUME-UNI
un accès ouvert
journal
Mots clés: Structural connectivity, Functional connectivity, Neural mass model, Coupled oscillator
théorie, Hopf bifurcation, False bifurcation
ABSTRAIT
The contribution of structural connectivity to functional brain states remains poorly
understood. We present a mathematical and computational study suited to assess the
structure–function issue, treating a system of Jansen–Rit neural mass nodes with
heterogeneous structural connections estimated from diffusion MRI data provided by the
Human Connectome Project. Via direct simulations we determine the similarity of functional
(inferred from correlated activity between nodes) and structural connectivity matrices under
variation of the parameters controlling single-node dynamics, highlighting a nontrivial
structure–function relationship in regimes that support limit cycle oscillations. To determine
their relationship, we firstly calculate network instabilities giving rise to oscillations, et le
so-called ‘false bifurcations’ (for which a significant qualitative change in the orbit is
observed, without a change of stability) occurring beyond this onset. We highlight that
functional connectivity (FC) is inherited robustly from structure when node dynamics are
poised near a Hopf bifurcation, whilst near false bifurcations, and structure only weakly
influences FC. Secondly, we develop a weakly coupled oscillator description to analyse
oscillatory phase-locked states and, furthermore, show how the modular structure of FC
matrices can be predicted via linear stability analysis. This study thereby emphasises the
substantial role that local dynamics can have in shaping large-scale functional brain states.
RÉSUMÉ DE L'AUTEUR
Patterns of oscillation across the brain arise because of structural connections between brain
régions. Cependant, the type of oscillation at a site may also play a contributory role. We focus
on an idealised model of a neural mass network, coupled using estimates of structural
connections obtained via tractography on Human Connectome Project MRI data. Using a
mixture of computational and mathematical techniques, we show that functional
connectivity is inherited most strongly from structural connectivity when the network nodes
are poised at a Hopf bifurcation. Cependant, beyond the onset of this oscillatory instability a
phase-locked network state can undergo a false bifurcation, and structural connectivity only
weakly influences functional connectivity. This highlights the important effect that local
dynamics can have on large-scale brain states.
Citation: Forrester, M., Crofts, J.. J.,
Sotiropoulos, S. N., Coombes, S., &
O’Dea, R.. D. (2020). The role of node
dynamics in shaping emergent
functional connectivity patterns in the
brain. Neurosciences en réseau, 4(2),
467–483. https://est ce que je.org/10.1162/
netn_a_00130
EST CE QUE JE:
https://doi.org/10.1162/netn_a_00130
Informations complémentaires:
https://doi.org/10.1162/netn_a_00130
Reçu: 2 Juillet 2019
Accepté: 31 Janvier 2020
Intérêts concurrents: Les auteurs ont
a déclaré qu'aucun intérêt concurrent
exister.
Auteur correspondant:
Michael Forrester
michael.forrester@nottingham.ac.uk
Éditeur de manipulation:
Petra Ritter
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
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
/
t
/
e
d
toi
n
e
n
un
r
t
je
c
e
–
p
d
je
F
/
/
/
/
/
4
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
un
_
0
0
1
3
0
p
d
t
.
F
b
oui
g
toi
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Node dynamics and functional connectivity
Diffusion tensor imaging:
A magnetic resonance imaging
technique that measures the diffusion
of water in tissue in order to produce
axonal fibre tract images.
Structural connectivity:
A pattern of anatomical links
between between distinct brain
régions. This is often measured using
diffusion tensor imaging.
Functional connectivity:
A pattern of statistical dependencies
between between distinct brain
régions. This is often measured using
coherence or correlation measures
between time-series.
Neural mass model:
A phenomenological model for the
activity of a neuronal population cast
as a system of ordinary differential
equations.
INTRODUCTION
Driven in part by advances in non invasive neuroimaging methods that allow characterisation
of the brain’s structure and function, and developments in network science, it is increasingly
accepted that the understanding of brain function may be obtained from a network perspective,
rather than by exclusive study of its individual subunits. Anatomical studies using diffusion MRI
allow estimation of structural connectivity (SC) of human brains, forming the so-called human
connectome (Sporns, 2011; Van Essen et al., 2013) which reflects white matter tracts connect-
ing large-scale brain regions. The graph-theoretical properties of such large-scale networks
have been well studied, highlighting key features including small-world architecture (Bassett &
Bullmore, 2006; Liao, Vasilakos, & Il, 2017), hub regions and cores (Oldham & Fornito, 2019;
van den Heuvel & Sporns, 2013), rich club organisation (Betzel, Gu, Medaglia, Pasqualetti,
& Bassett, 2016; Van Den Heuvel & Sporns, 2011), a hierarchical-like modular structure
(Meunier, Lambiotte, & Bullmore, 2010; Sporns & Betzel, 2016), and economical wiring
(Betzel et al., 2017; Bullmore & Sporns, 2012). The emergent brain activity that this struc-
ture supports can be evaluated by functional connectivity (FC) network analyses that describe
patterns of temporal coherence in neural activity between brain regions. These highly dynamic
patterns are widely believed to be significant in integrative processes underlying higher brain
fonction (Van Den Heuvel & Pol, 2010; van Straaten & Stam, 2013), and disruptions in SC and
FC networks are associated with many psychiatric and neurological diseases (Brun, Muldon,
& Bassett, 2015; Menon, 2011).
Cependant, the relationship between the brain’s anatomical structure and the neural activ-
ity that it supports remains largely unknown (C. J.. Honey, Thivierge, & Sporns, 2010; Parc &
Friston, 2013). En particulier, the divergence between dynamic functional activity and the rel-
atively static structural connections between populations is critical to the brain’s dynamical
repertoire and may hold the key to understanding brain activity in health and disease (Parc
& Friston, 2013), though current models have not yet been able to accurately simulate the
transitive states underpinning cognition (Petersen & Sporns, 2015). Empirical studies suggest
that while a structural connection between two brain areas is typically associated with a
stronger functional interaction, strong interactions can nevertheless exist in their absence
(Hermundstad et al., 2014; C. J.. Honey et al., 2010); moreover, these functional networks
are transient (Fox et al., 2005; Hutchison et al., 2013; Liegeois, Laumann, Snyder, Zhou, &
Yeo, 2017; Preti, Bolton, & Van De Ville, 2017), motivating more recent consideration of dy-
namic (rather than time-averaged) FC networks, which have been proposed to more accurately
represent brain function. An important example of SC-FC divergence is provided by resting-
state networks, such as the ‘default mode network’ and the ‘core network’ (Thomas Yeo et al.,
2011; Van Den Heuvel & Pol, 2010). These networks comprise brain areas that can be strongly
functionally connected at rest (Van Den Heuvel & Pol, 2010), but can also temporally vary. Dans-
deed, a neural ’switch’ has been proposed that facilitates transitions between resting-state net-
travaux (Goulden et al., 2014), and a theoretical study by Messé, Rudrauf, Benali, and Marrelec
(2014) estimated that nonstationarity of FC contributes to over half of observed FC variance.
Theoretical studies deploying anatomically realistic structural networks obtained through
tractography alongside neural mass models describing mean-field regional neural activity have
been used to further investigate the emergence of large-scale FC patterns (Breakspear, 2017;
Deco et al., 2013; C. J.. Honey, Kötter, Breakspear, & Sporns, 2007; Messé, Hütt, Konig, &
Hilgetag, 2015; Ponce-Alvarez et al., 2015; Rubinov, Sporns, van Leeuwen, & Breakspear,
2009). These findings suggest that through indirect network-level interactions, a relatively
static structural network can support a wide range of FC configurations, Par exemple, showing
Neurosciences en réseau
468
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
/
/
t
e
d
toi
n
e
n
un
r
t
je
c
e
–
p
d
je
F
/
/
/
/
/
4
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
un
_
0
0
1
3
0
p
d
.
t
F
b
oui
g
toi
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Node dynamics and functional connectivity
Hopf bifurcation:
The appearance or disappearance of
a periodic orbit through a local
change in the stability properties of
an equilibrium point under
parameter variation.
Human Connectome Project:
A consortium research project to
build a human connectome for
structural and functional
connectivité, and facilitate research
into brain disorders.
Weakly coupled oscillator theory:
A reduction of a network of weakly
interacting limit cycle oscillators to a
system of fewer dynamical equations
that describes the evolution of
relative phases between nodes.
that FC reflects underlying SC on slow time scales, but significantly less so on faster time scales
(C. Honey et al., 2009; C. J.. Honey et al., 2007; Rubinov et al., 2009).
In the context of mean-field models, simulated (typically time-averaged) FC has been found
most strongly to resemble SC when the dynamical system describing regional activity is close to
a phase transition (Stam et al., 2016), and strong structure–function agreement is reported near
Hopf bifurcations in Hlinka and Coombes (2012). De la même manière, analysis of the dynamical systems
underpinning neural simulations have shown to be a good fit to fMRI data when the system is
near to bifurcation (Deco et al., 2019; Tewarie et al., 2018). These results provide a possible
manifestation of the so-called critical brain dynamics hypothesis (Cocchi, Gollo, Zalesky, &
Breakspear, 2017; Shew & Plenz, 2013). In Crofts, Forrester, and O’Dea (2016), both SC and
FC are analysed together in a multiplex network, proposing a novel measure of multiplex
structure–function clustering in order to investigate the emergence of functional connections
that are distinct from the underlying structure. Deco, Kringelbach, Jirsa, and Ritter (2017)
consider dynamic FC, with transient FC states described as metastable states, and in Deco et al.
(2019), metastability of a computational model of large-scale brain network activity was used
to predict which structures of the brain could be influenced to force a transition between states
of wakefulness and sleep. Hansen, Battaglia, Spiegler, Deco, & Jirsa (2015) were also able to
observe dynamic transitions between states resembling resting-state networks in a noise-driven,
nonlinear mean-field model of neural activity.
In this paper, we adopt the mean-field neural mass approach and present a combined com-
putational and mathematical study, which significantly extends the related works of Hlinka
and Coombes (2012) and Crofts et al. (2016) to investigate how the detailed and rich dy-
namics of the intrinsic behaviour of neural populations, together with structural connectivity,
combine to shape FC networks. Thereby, we provide a complementary investigation to many
of the aforementioned studies which focus on the analysis of brain networks themselves, ou
those that employ statistical models, by instead investigating the relationship between network
structure and the emergent dynamics of these networks. Specifically, we consider synchrony
between neural subunits whose dynamics are described by the neural mass model of Jansen
and Rit (1995), and whose connectivity is defined by a tractography-derived structural net-
work obtained from data in the Human Connectome Project (HCP) (Van Essen et al., 2013).
Structure–function relations are interrogated by graph-theoretical comparison of FC and SC
topology under systematic variation of model parameters associated with excitatory/inhibitory
neural responses, and analysed by making use of techniques from bifurcation and weakly
coupled oscillator theory.
MÉTHODES
Neural mass model
We consider a network of interacting neural populations, representing a parcellation of the
cerebral cortex, such that each area (node) corresponds to a functional unit that can be rep-
resented by a neural mass model, and with edges informed by structural connectivity. Neural
mass activity is represented by the Jansen–Rit model (Jansen & Rit, 1995) of dimension m = 6,
that describes the evolution of the average postsynaptic potential (PSP) in three interacting neu-
ral populations: pyramidal cells (y0), and excitatory (y1) and inhibitory (y2) interneurons. These
populations are connected with strengths Ci (i = 1…4), representing the average number of
synaptic connections between each population. The Jansen–Rit model is mathematically de-
scribed by three second-order ordinary differential equations which are commonly rewritten
as six first-order equations by adopting the notation (y0, . . . , y5) for the dependent variables.
Neurosciences en réseau
469
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
t
/
/
e
d
toi
n
e
n
un
r
t
je
c
e
–
p
d
je
F
/
/
/
/
/
4
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
un
_
0
0
1
3
0
p
d
t
.
F
b
oui
g
toi
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Node dynamics and functional connectivity
The pairs (y0, y3), (y1, y4), et (y2, y5) are therefore associated with the dynamics of the pop-
ulation average of PSPs and their temporal derivatives. The quantity of primary interest herein
is y = y1 − y2, which is physiologically interpreted as the average potential of pyramidal
populations and the main contributor to signals generated in EEG recordings (Teplan, 2002).
Introducing an index i = 1, . . . , N to denote each node in a network of N interacting neural
populations, we write the evolution of state variables as:
˙y0i = y3i
,
˙y3i = Aa f
˙y1i = y4i
,
,
˙y2i = y5i
,
− 2ay3i − a2y0i
y1i − y2i
N
∑
j=1
(cid:0)
Pi + ε
(cid:1)
wij f
˙y4i = Aa
(
˙y5i = BbC4 f
y1j − y2j
+ C2 f
C1y0i
(cid:16)
(cid:17)
− 2by5i − b2y2i
.
(cid:0)
C3y0i
(1)
− 2ay4i − a2y1i
,
)
(cid:1)
Here f is a sigmoidal nonlinearity, representing the transduction of activity into a firing rate,
and with the specific form
(cid:0)
(cid:1)
F (v) =
νmax
1 + exp(r(v0 − v))
.
(2)
The model is identical to that presented in Jansen and Rit (1995) for a single cortical column,
but is completed by specifying the network interactions as a function of average membrane
potential of afferently connected pyramidal populations, encoded in a connectivity matrix
with elements wij (described in Structural and Functional Connectivity), with an overall scale
of interaction set by ε. The remaining model parameters, together with their physiological
interpretations and values (taken from Grimbert & Faugeras, 2006, and Touboul, Wendling,
Chauvel, & Faugeras, 2011), are given in Table 1. A schematic ‘wiring diagram’ for the model
indicating the interactions between different neural populations is shown in Figure 1.
The Jansen–Rit model, defined by Equation 1, can support oscillations that relate to impor-
tant neural rhythms, such as the well-known, alpha, beta, and gamma brain rhythms, and also
irregular, epileptic-like activity (Ahmadizadeh et al., 2018). De plus, the model is able to
replicate visually evoked potentials seen in EEG recordings (Jansen & Rit, 1995), from which
FC may be empirically measured (Srinivasan, Hiver, Ding, & Nunez, 2007).
Tableau 1. Parameters in the Jansen–Rit model, given by Equations 1 et 2 along with physiological
interpretations and values/ranges used in simulations, which were taken from Grimbert and Faugeras
(2006) and Touboul et al. (2011). En particulier, the values A and B, which modulate the strength
of excitatory and inhibitory responses respectively, were chosen as the key control parameters for
varying network activity.
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
t
/
/
e
d
toi
n
e
n
un
r
t
je
c
e
–
p
d
je
F
/
/
/
/
/
4
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
un
_
0
0
1
3
0
p
d
.
t
F
b
oui
g
toi
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Parameter
C1, C2, C3, C4 Average number of synapses between populations
Pi
Meaning
Basal extracortical input to main pyramidal excitatory
populations
Amplitude of excitatory, inhibitory PSPs respectively
Lumped time constants of excitatory, inhibitory PSPs
Global coupling strength
Coupling from node j to i
Maximum population firing rate
Potential at which half-maximum firing rate is achieved
Gradient of sigmoid at v0
UN, B
un, b
ε
wij
νmax
v0
r
Neurosciences en réseau
Value
135, 108, 33.75, 33.75
120 Hz
[2, 14] mV, [10, 30] mV
100 s−1, 50 s−1
0.1
[0, 1]
5 Hz
6 mV
0.56 mV−1
470
Node dynamics and functional connectivity
Chiffre 1. Wiring diagram for a Jansen–Rit network node, described by Equations (1 et 2). Ex-
citatory/inhibitory populations and synaptic connections are highlighted in red/blue, respectivement.
Interneurons (E, je) and pyramidal cells (PC) are interconnected with strengths Ci for i = 1…4. Aussi
shown is the expression for the external input to a PC population, consisting of a extracortical input
Pi, as well as contributions from afferently connected nodes.
In what follows, we consider the patterns of dynamic neural activity that arise under sys-
tematic variation of the model parameters A and B, these being chosen as the parameters of
interest because they govern the interplay between inhibitory and excitatory activity, lequel
would typically vary due to neuromodulators in the brain (Rich, Zochowski, & Booth, 2018).
It is known that a single Jansen–Rit node can support multistable behaviour, which includes
oscillations of different amplitude and frequency, mais, moreover, a network of these nodes can
also exhibit various stable phase-locked states. A small amount of white noise is added to the
extracortical input Pi on each node, in order to allow the system to explore a variety of these
dynamical states: Pi + dWi(t), where dWi(t) is chosen at random from a Gaussian distribution
with standard deviation 10−1 Hz and mean 0 Hz. For direct simulations of the network we
use an Euler–Murayama scheme, implemented in Matlab, with a fixed numerical time step of
10−4, which we have confirmed ensures adequate convergence of the method.
Structural and functional connectivity
The structural connectivity was estimated using diffusion MRI data recorded with informed
consent from 10 sujets, obtained from the HCP (Van Essen et al., 2013). Briefly, we explain
how this data is postprocessed to derive connectomic data, though we direct the reader to
Tewarie et al. (2019) and the references therein for a more detailed overview. Sixty thousand
vertices on the white/grey matter boundary surface for each subject (Glasser et al., 2013) étaient
used as seeds for 10,000 tractography streamlines. Streamlines were propagated through vox-
els with up to three fibre orientations, estimated from distortion-corrected data with a deconvo-
lution model (Jbabdi, Sotiropoulos, Savio, Graña, & Behrens, 2012; Sotiropoulos et al., 2016)
using the FSL package. The number of streamlines intersecting each vertex on the boundary
layer was measured and normalised by the total number of valid streamlines. This resulted in a
60,000 node structural matrix, which was further parcellated using the 78-node AAL atlas. Ce
was used to describe connections between brain regions, providing an undirected (symmet-
ric), weighted matrix whose elements wij define the strengths of the excitatory connections in
Equations 1. To enable a meaningful comparison between the network measures of SC and FC,
the former reflecting the density of tractography streamlines and the latter that of correlated
neural activity, we place them on a similar footing by thesholding and binarising, such that
only the top 23% of the weights (ordered by strength) are retained; voir la figure 2. Thresholding
is a widespread technique for removing spurious connections that may not in fact be a realistic
Neurosciences en réseau
471
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
/
/
t
e
d
toi
n
e
n
un
r
t
je
c
e
–
p
d
je
F
/
/
/
/
/
4
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
un
_
0
0
1
3
0
p
d
.
t
F
b
oui
g
toi
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Node dynamics and functional connectivity
Chiffre 2. The original structural matrix (UN) is derived from DTI data taken from the Human Con-
nectome Project database and parcellated on to a 78-region brain atlas. This is thresholded and
binarised to keep the top 23% strongest connections (B) and normalised by row so that ∑N
j=1 wij = 1
for all regions i) dans (C).
representation of brain connectivity. We note that our thresholding choice (that reduces the
number of connections, while ensuring that the overall modular structure is unchanged) est
commensurate with a recent study (Tsai, 2018), which employed DTI data averaged on the
same brain atlas as used herein to consider thresholding approaches suitable to remove weak
connections with high variability between (n = 30) different subjects. To generate nodal in-
puts with commensurate magnitudes, the structural connectivity matrix was normalised by row
so that afferent connection strengths for each node sum to unity. This normalisation process
permits some of the analysis that we undertake to help explain SC-FC relations (see Weakly
Coupled Oscillator Theory); cependant, we highlight that the results that we present herein are
not crucially dependent on such a choice and so our conclusions generalise (see Supporting
Information, section Mathematical Methods).
In view of the nonlinear oscillations supported by the network model given by (1), FC net-
works are obtained by computing the commonly-used metric of mean phase coherence (MPC;
Mormann, Lehnertz, David, and Elger (2000)), which determines correlation strength in terms
of the proclivity of two oscillators to phase-lock, giving a range from 0 (completely desynchro-
nised) à 1 (phase-locking). We choose yj = y1j − y2j as the variable of interest because of
its relation to the EEG signal, making it a good candidate to produce timeseries more readily
comparable with empirical data. Pairwise MPC measures the average temporal variance of the
phase difference ∆φjk(t) = φj(t) − φk(t), between two time-series indexed by j and k, où
here the instantaneous phase φj(t) is obtained as the angle of the complex output resulting
from application of a Hilbert transform to the time-series, yj(t). The MPC of the time-series
comprising M time-points tl (l = 1, . . . , M.) is defined as:
Rjk =
1
M.
M.
∑
l=1
(cid:12)
(cid:12)
(cid:12)
(cid:12)
ei∆φjk(tl)
.
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(3)
Structure–function relations are assessed by computing the Jaccard similarity coefficient
(Jaccard, 1912) of the nondiagonal entries of the binarised SC and FC matrices. This describes
472
Jaccard similarity:
An index for comparing members for
two sets of binary data to see which
are shared and which are distinct.
The higher the index, plus
similar the two sets.
Neurosciences en réseau
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
/
/
t
e
d
toi
n
e
n
un
r
t
je
c
e
–
p
d
je
F
/
/
/
/
/
4
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
un
_
0
0
1
3
0
p
d
.
t
F
b
oui
g
toi
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Node dynamics and functional connectivity
False bifurcation:
A qualitative change in the shape of a
periodic orbit without a change in
stability, say from a small to a large
amplitude oscillation, that occurs
under a very small parameter
variation.
Saddle-node bifurcation:
A local bifurcation in which two
equilibria of a dynamical system
collide and annihilate.
the relative number of shared pairwise links between the two networks, providing a natural
measure of structure–function similarity, ranging from zero for matrices with no common links
to unity for identical matrices.
Since the SC-FC correlation patterns of interest here arise naturally from global synchrony
or patterns of phase locking of oscillatory node activity, the local stability of oscillatory node
dynamics and of network (global or phase locking) synchrony is a natural candidate to ex-
In the following subsections we consider bifurcation, false
plain the structures we observe.
bifurcation, and weakly coupled oscillator theory approaches to address this.
Bifurcation analysis
Bifurcations for a single node are readily computed using
Single node and network bifurcations
the software package XPPAUT (Ermentrout, 2002), using A and B as the parameters of interest.
The result is a Hopf and saddle-node set in parameter space, which bounds a region of oscil-
latory solutions. We also observe a region of bistability bounded by fold bifurcations of limit
cycles, in which the types of activity described in Figure 4A and 4C can both exist. This is
shown in Figure 3. We refer the reader to Grimbert and Faugeras (2006), Touboul et al. (2011),
and Spiegler, Kiebel, Atay, and Knösche (2010) for a comprehensive analysis of the bifurcation
structure of the Jansen–Rit model.
The corresponding diagram for the full network requires numerical analysis of a much
higher dimensional system, described by N × m = 78 × 6 = 468 ODEs; this is compu-
tationally demanding, and so in the Supporting Information, section Mathematical Methods
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
/
t
/
e
d
toi
n
e
n
un
r
t
je
c
e
–
p
d
je
F
/
/
/
/
/
4
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
un
_
0
0
1
3
0
p
d
t
.
F
b
oui
g
toi
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Chiffre 3. Two-parameter bifurcation diagram in the (UN, B) plane in the single-node case of the
Jansen–Rit system of Equations 1. Other parameter values are as stated in Table 1. Red dashes are
Hopf bifurcations, black dots are false bifurcations, and blue lines represent saddle points. Là
is also a region of bistability, highlighted in yellow, which is bounded by saddle nodes and a set of
fold bifurcations of limit cycles. The pink and yellow shaded regions indicates parameter values for
which there exist stable oscillatory solutions. The three colored dots at B = 22, A = 7.0, 7.7, 9.0
indicate parameter values at which we observe distinctly different dynamics as shown in Figure 4.
Neurosciences en réseau
473
Node dynamics and functional connectivity
we develop a quasi-analytic approach by linearising the full network equations around a fixed
indiquer. The resulting equations can be diagonalised in the basis of eigenvectors of the structural
connectivité, leading to a set of N equations, each of which prescribes the spectral problem for
an m-dimensional system. Ainsi, each of these low-dimensional systems can be easily treated
without recourse to high-performance computing. De plus, this approach exposes the role
that the eigenmodes of the structural connectivity matrix has in determining the stability of
equilibria. We report the locus of Hopf and saddle-node sets for the network in Figure 5.
Comparison of Figures 3 et 5 shows that the bifurcation structure of steady states for the
full network is practically identical to that of the single node (even for moderate coupling
strength—here, ε = 0.1), highlighting the potential importance of single-node dynamics in
driving SC-FC correlations.
In Figure 4 we consider in more detail the types of activity that the network
False bifurcations
model (1) supports. En particulier, we observe that under changes to parameter values within
the oscillatory region (see highlighted parameter values in Figure 3), the time course of activity
shifts from single- to double-peaked waves, which could have consequences for synchronisa-
tion of oscillations and, moreover, FC. The points of transition are known as false bifurcations
since there is a significant dynamical change that occurs smoothly rather than critically. False
bifurcations in a neural context have previously been seen as canards in single-neuron models
(Desroches, Krupa, & Rodrigues, 2013) as well as in EEG models of absence seizures (Marten,
Rodrigues, Benjamin, Richardson, & Terry, 2009). In the latter case the false bifurcation cor-
responds to the formation of spikes associated with epileptic seizures (Moeller et al., 2008).
As illustrated in Figure 4 the false-bifurcation transition is characterised by the change from
a double-peaked profile (UN) to a sinusoidal-like waveform (C) via the development of a point of
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
/
/
t
e
d
toi
n
e
n
un
r
t
je
c
e
–
p
d
je
F
/
/
/
/
/
4
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
un
_
0
0
1
3
0
p
d
.
t
F
b
oui
g
toi
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Chiffre 4. Activity profiles of y = y1 − y2, the potential of the main population of pyramidal
neurons for a node in the Jansen–Rit network (1) in the absence of noise, with B fixed at 22 et
(UN) A = 9.0; (B) A = 7.7; (C) A = 7.0 and other parameter values as in Table 1. Subfigures in the
upper row are plots of the time series solution, whereas the bottom row shows the trajectories of sta-
ble orbits in the (oui, y′) plane. The chosen parameters lie at either side of the region where a smooth
transition between activity types occurs, corresponding to a false bifurcation (see highlighted param-
eter values in Figure 3). In B, an inflection point occurs and is highlighted as a red star on the orbit.
Neurosciences en réseau
474
Node dynamics and functional connectivity
inflection in the solution trajectory (B). Since this transition is not associated with a change in
stability of the periodic orbit, these false bifurcations are determined by tracking parameter sets
for which points of inflection occur. We refer the reader to Rodrigues et al. (2010) for details
on methods for detecting and continuing false bifurcations in dynamical systems. The result of
this computation is shown in Figure 3, where we observe the set of false bifurcations arising
from the breakdown of two branches of fold bifurcations of limit cycles. In the full network
(not shown), this computation is more laborious (and there is some delicacy in defining the
bifurcation since the network coupling leads nodes to inflect at marginally different parameter
valeurs); cependant, we obtain very similar results to those obtained in Figure 3 for a single node
(not shown).
Weakly coupled oscillator theory
Further insight into the phase relationship between nodes in a network can be obtained from
the theory of weakly coupled oscillators (voir, par exemple., Hoppensteadt and Izhikevich, 2012). Ce
technique reduces a network of limit cycle oscillators to a set of relative phases in a systematic
chemin. The resulting set of network ODEs is (N − 1)-dimensional, as opposed to the (Nm)-
dimensionality of the original system, and provides an accurate model as long as the overall
coupling strength is weak (|ε| ≪ 1). This is because when all oscillators lie on the same limit
cycle of a system, the interactions from pairwise-connected nodes can be considered as small
perturbations to the oscillator dynamics. De plus, the resulting set of network ODEs only
depends on phase differences, and it is straightforward to construct relative equilibria (oscilla-
tory network states) and determine their stability in terms of both local dynamics and structural
connectivité. A method to construct the phase interaction function, H, for the network is pro-
vided in the Supporting Information, section Mathematical Methods. Once this is known, le
dynamics for the phases of each node in the network, θi ∈ [0, 2π), takes the simple form:
˙θi = Ω + ε
N
∑
j=1
wijH(θj − θi),
i = 1, . . . , N − 1,
(4)
where Ω = 2π/T represents the natural frequency of an uncoupled oscillatory node with
period T, and the second term determines phase changes arising from pairwise interactions
between nodes. We emphasise that the T-periodic phase interaction function H(Ωt) =
H(Ω(t + T)) is derived from the full system given by Equation 1. For a given phase-locked
state θi(t) = Ωt + φi (where φi is the constant phase of each node), local stability is de-
H(Φ) avec
termined in terms of the eigenvalues of the Jacobian of Equation 4, denoted by
Φ = (φ1, . . . , φN)⊺, with components:
[
H(Φ)]ij = ε[H′(φj − φi)wij − δij
N
∑
k=1
H′(φk − φi)wik].
(5)
b
b
The globally synchronous steady state, φi = φ for all i, exists in a network with a phase
interaction function that vanishes at the origin (c'est à dire., H(0) = 0, which is not the case here), ou
for one with a row sum constraint, ∑j wij = Γ = constant for all i, which is true for our specific
structural matrix (for which Γ = 1). Note that the emergent frequency of the synchronous
network state is given explicitly by Ω + εΓH(0). Using the Jacobian in Equation 5, synchrony
is found to be stable if εH′(0) > 0 and all the eigenvalues of the graph Laplacian of the
structural network,
[L]ij = −wij + δij ∑
k
wik,
(6)
475
Neurosciences en réseau
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
/
/
t
e
d
toi
n
e
n
un
r
t
je
c
e
–
p
d
je
F
/
/
/
/
/
4
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
un
_
0
0
1
3
0
p
d
.
t
F
b
oui
g
toi
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Node dynamics and functional connectivity
lie in the right-hand complex plane. Since the eigenvalues of a graph Laplacian all have the
same sign (apart from, in this case, a single zero value), then local stability is entirely deter-
mined by the sign of εH′(0). Par exemple, for a globally coupled network with wij = 1/N,
then the graph Laplacian has one zero eigenvalue, et (N − 1) other degenerate eigenvalues
at −1, and so synchrony is stable if εH′(0) > 0.
It is therefore useful to consider the condition εH′(0) > 0 as a natural prerequisite for a
structured network to support high levels of synchrony (without recourse to exploring the full
Jacobian structure). A plot of εH′(0) is shown in Figure 5B. For completeness, cependant, the full
Jacobian was also computed in order to account for the potential influence of detailed structure
on the correspondence with the observed SC-FC agreement measured in simulations. To do
ce, the system given by Equation 1 was integrated with ε = 0.001 to a (stable) phase-locked
state, and relative phases computed. The eigenvalues of the Jacobian (Équation 5) were then
computed, providing an indication of solution attractivity. The largest nonzero eigenvalue for
each parameter choice is shown in Figure 5C.
It has been shown in Tewarie et al. (2018) that the eigenmodes of the structural connectiv-
ity matrix are predictive of emergent FC networks arising from an instability of a steady state.
The largest nonzero eigenvalue, which is related the most unstable eigenmode (or closest to
instability), was found to be a good predictor of resultant FC by computing the tensor product
of its corresponding eigenvector, v ⊗ v. Here we take this further by considering instabilities
In this case the Jacobian Equation 5 reduces to −εH′(0)Lij and
of the synchronous state.
the phase-locked state that emerges beyond instability of the synchronous state has a pattern
je
D
o
w
n
o
un
d
e
d
F
r
o
m
h
t
t
p
:
/
/
d
je
r
e
c
t
.
m
je
t
.
/
t
/
e
d
toi
n
e
n
un
r
t
je
c
e
–
p
d
je
F
/
/
/
/
/
4
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
un
_
0
0
1
3
0
p
d
.
t
F
b
oui
g
toi
e
s
t
t
o
n
0
9
S
e
p
e
m
b
e
r
2
0
2
3
Chiffre 5.
(UN) Jaccard similarity coefficient between SC and FC (measured by MPC in Equation 3)
when the Jansen–Rit network (Équation 1) supports an oscillatory solution, averaged over 30 concernant-
alisations of initial conditions chosen at random. Parameter values are given in Table 1. Warmer
colours indicate greater SC/FC correlation. Here we have superimposed the bifurcation diagram for
the network steady state, which shows the oscillatory region being bounded by Hopf/saddle-node
sets in solid/dashed white lines respectively; boxes are Bogdanov–Takens points. False bifurcations
in the single-node case are indicated by a black line but, because of its relative size, the bistable
region is not shown (though can be seen for the single node case in Figure 3).
(B) The value of
H′(0) (see Equations 4 et 5) in the A, B plane. When this value is positive/negative, the glob-
ally synchronised solution is stable/unstable (if it exists); (C) The largest nonzero eigenvalue of the
Jacobian for the full weakly coupled oscillator network (Équation 5), calculated at a stable phase-
locked state. More negative values indicate a stronger stability.
Neurosciences en réseau
476
Node dynamics and functional connectivity
determined by the a linear combination of eigenmodes of the graph Laplacian, since all eigen-
modes destabilise simultaneously. It is known that the graph Laplacian can be used to predict
phase-locked patterns (Chen, Lu, Zhan, & Chen, 2012) and has indeed been used to predict
empirical FC from SC (Abdelnour, Dayan, Devinsky, Thesen, & Raj, 2018). Following from
ce, the eigenmodes of the Jacobian in Equation 5 can be used as simple, easily computable
proxy for the FC matrix when the system is poised at a local instability. In Figure 7 we compare
the FC pattern from the (fully nonlinear) weakly coupled network with a linear prediction to
highlight its usefulness. Dans ce cas, MPC (Équation 3) is not ideally suited for our study be-
cause it struggles to discern between phase locking and complete synchrony, yet we consider
situations where stable phase locking naturally arises. Donc, FC in the weakly coupled
network is computed via the new metric of mean phase agreement (MPA), whereby patterns
of coherence are determined by a temporal average of relative phase differences:
ˆRjk =
1
M.
M.
∑
l=1
1
2
(cid:0)
1 + cos(∆φjk(tl))
.
(cid:1)
For comparison, we use the tensor product sum,
ˆR =
N∗
∑
je = 1
λivi ⊗ vi
(7)
(8)
, . . . , vN
of vk = (v1
k ), which denotes the kth eigenvector of the Jacobian for the synchronous
k
state. These are weighted by their corresponding eigenvalues, λk, and we include the N∗
unstable eigenmodes.
RÉSULTATS
Chiffre 5 shows plots in the (UN, B) parameter space highlighting our studies on the combined
influence of SC and node dynamics on FC. The region bounded by the bifurcation curves, ob-
tained via a linear instability analysis of the network steady state, is where the network model
supports oscillations as well as phase-locked states. In Figure 5A the Jaccard similarity between
SC and FC is computed from direct numerical simulations of the Jansen–Rit network model
(Équation 1). Beyond the onset of oscillatory instability (supercritical Hopf bifurcation), le
emergent phase-locked network states show a nontrivial correlation with the SC. This varies
in a rich way as one traverses the (UN, B) parameter space, showing that precise form of the
node dynamics can have a substantial influence on the network state. The highest correla-
tion between SC and FC coincides with a Hopf bifurcation of a network equilibrium (shown
as a solid white line), whilst a band of much lower correlation coincides with the fold bi-
furcations of limit cycles and false bifurcations of a single node (in black), reproduced from
Chiffre 3. En effet, it would appear that these mathematical constructs are natural for organising
the behaviour seen in our in silico experiments. We reiterate that we have confirmed that the
organising SC-FC features that we here identify are not crucially dependent on the binarisation,
thresholding and normalisation procedure, described in Structural and functional connectivity
and are qualitatively similar under variation of coupling strength (see Supporting Information,
section Mathematical Methods); moreover, results obtained via MPC and of MPA are indis-
In Figure 5B we show a plot of H′(0). Recall from Weakly
tinguishable (data not shown).
coupled oscillator theory that a globally synchronous state (which is guaranteed to exist from
the row sum constraint) is stable if εH′(0) > 0. Comparison with Figure 5A, highlights that
when synchrony is unstable (εH′(0) < 0) SC only weakly drives FC. Moreover, this instability
Network Neuroscience
477
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
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
a
_
0
0
1
3
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
Node dynamics and functional connectivity
region coincides with the region of bistability and the false bifurcation, stressing the important
role of these bifurcations for understanding SC-FC correlation.
Of course, there is a much finer structure in Figure 5A that is not predicted by considering
either the bifurcation from steady state, or the weakly coupled analysis of synchronous states,
and so it is illuminating to pursue the full weakly coupled oscillator analysis for structured net-
works. The eigenvalues of the Jacobian, corresponding to more general stable phase-locked
states, can be used to give a measure of solution attractivity. The largest eigenvalue is plotted in
Figure 5C. The most stable (nonsynchronous) phase-locked states occur in the neighbourhood
of the false bifurcations, as well as in the region of bistability and along the existence border
for oscillations, defined by a saddle node bifurcation. Furthermore, apart from near false bifur-
cations, stronger stability of the general phase-locked states corresponds with stronger stability
of global synchrony (Figure 5B).
To test the predictive power of the weakly coupled theory, in Figure 6 we compare the emer-
gent FC structure obtained from direct simulations of the Jansen–Rit network model (Equation 1)
against direct simulations of the weakly-coupled oscillator network (Equation 4). For the former,
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
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
a
_
0
0
1
3
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
Figure 6. Comparison of FC patterns from averages of realisations of the weakly coupled oscillator
model (Equation 4) with corresponding Jansen–Rit (Equation 1) simulations, with no noise present,
at A = 5, B = 19, computing averages over 600 realisations with initial conditions chosen at random
(other parameter values are given in Table 1). (A) ε = 0.01; (B) ε = 0.1; (C) ε = 1. These results show
how the weakly coupled theory becomes less predictive for stronger coupling strengths, resulting
in matrices with Jaccard similarity of 0.98, 0.76, and 0.65 (to 2 s.f.), respectively.
Network Neuroscience
478
Node dynamics and functional connectivity
the phases required to compute the mean phase agreement (Equation 7) are determined from
each time series by a Hilbert transform; in the latter case, the phase variables from Equation 4
are employed directly. Since the weakly coupled reduction of the Jansen–Rit model is deter-
ministic, these computations were ran in the absence of noise (dWi = 0 for all nodes). As
expected, we find excellent agreement between the modular FC structure in the case for very
weak coupling, with this agreement reducing with increasing ε, as quantified by a reduction
in Jaccard similarity (from 0.98 in panel A to 0.65 in C). This is a manifestation of the network
moving from a dynamical regime that can be well described by the weakly coupled reduction
(Equation 4) to one where stronger network interactions dominate. Since an analogous theory
does not exist for stronger coupling, we do not consider here how SC-FC relations arise from
network dynamics within a strongly coupled framework. Moreover, through the instability
theory of the synchronous state we can construct a proxy for the FC as described in Weakly
In Figure 7 we compare simulated FC with that predicted by ˆR
coupled oscillator theory.
(Equation 8; i.e., using the unstable eigenmodes of the Jacobian at synchrony), for parameter
values that lie just beyond the onset of instability of the globally synchronous state and near
the false bifurcation set (see Figure 5A and 5B). We observe that the key features of the FC
are captured by the eigenmode prediction; indeed the (weighted) Jaccard similarity coefficient
between predicted and simulated FC (both scaled to [0, 1]) is calculated to be 0.82. This is a
much more efficient way of simulating an emergent FC pattern, since it does not require brute
force forward integrations of the model, which may take a long time to converge.
All of these results highlight the strong impact that nodal dynamics can have on the corre-
lation between SC and FC, and the utility of bifurcation theory and phase oscillator reduction
techniques (that are naturally positioned to explain the generation of patterns of synchronous
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
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
a
_
0
0
1
3
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
Figure 7.
(A) FC prediction given by the linear combination of eigenmodes of the weakly coupled
oscillator system, given by tensor products of eigenvectors of the SC graph Laplacian (Equation 8),
with N∗ = N. (B) Direct simulation of the Jansen–Rit network model (Equation 1) with no noise
present. Parameter values are chosen as A = 6, B = 18, which lies near the existence border
for stable synchronous solutions (see Figure 5B); other parameter values are given in Table 1. The
(weighted) Jaccard similarity between the two FC networks (scaled to [0, 1] for comparability) is
calculated to be 0.82, indicating the predictive power of Equation 8.
Network Neuroscience
479
Node dynamics and functional connectivity
node and network activity) to provide insight into how SC-FC correlations are organised across
parameter space.
DISCUSSION
In this paper, we investigate the degree to which the dynamical state of neural populations, as
well as their structural connectivity, facilitates the emergence of functional connections in a
neural mass network model of the human brain. We have addressed this by using a mixture of
computational and mathematical techniques to assess the correlation between structural and
functional connectivity as one traverses the parameter space controlling the inhibitory and ex-
citatory dynamics and bifurcations of an isolated Jansen–Rit neural mass model. Importantly,
SC has been estimated from HCP diffusion MRI datasets. We find that SC strongly drives FC
when the system is close to a Hopf bifurcation, whereas in the neighbourhood of a false bifur-
cation, this drive is diminished. These results emphasise the vital role that local dynamics has
to play in determining FC in a network with a static SC. In addition, we show that a weakly
coupled analysis provides insight into the organisation of SC-FC correlation features across
parameter space, and can be exploited to predict emergent FC structure. Messé et al. (2014)
considered statistical models to predict FC from SC (in particular, a spatial simultaneous au-
toregressive model (sSAR), whose parameters can be estimated in a Bayesian framework) and
found, interestingly, that simpler linear models were able to fare at least as well. More re-
cently, Saggio, Ritter, and Jirsa (2016) were also able to make predictions of FC from empirical
SC data (and vice versa) using a simple linear model. Since the only free parameter of their
model for SC is the global coupling strength, results from this method are efficient and compu-
tationally inexpensive. We have not attempted to reproduce empirical data here, but we have
shown that similar predictions can be made using bifurcation theory and network reduction
techniques; such an approach allows us to consider in more detail, and explain, the influ-
ence of the rich neural dynamics supported by the Jansen–Rit model on SC-FC relationships.
Nevertheless, it is important to note that the FC structures we are concerned with are aver-
aged over long time scales and therefore represent a static FC state, as opposed to dynamic
FC (as discussed in Introduction). Use of such static FC networks as a clinical biomarker is
widespread; however, subject variability in FC means that their predictive power is restricted
to group analyses (Mueller et al., 2013). To capture the rich dynamic FC repertoire exhibited
in empirical resting-state data, for example, the distinct hierarchical organisation in switching
between FC states (Vidaurre, Smith, & Woolrich, 2017), will require alternative approaches.
One such approach is dynamic causal modelling, as employed in Goulden et al. (2014) and
Van de Steen, Almgren, Razi, Friston, and Marinazzo (2019) for empirical data.
The modelling work presented here is relevant in a wider neuroimaging context—for example,
epilepsy is often considered to be caused by irregularities in synchronisation (Lehnertz et al.,
2009; Mormann et al., 2003; Netoff & Schiff, 2002).
It is noteworthy that the changes in
synchrony patterns that we observe arise from local dynamical considerations as opposed to
large-scale structural ones. In the Jansen–Rit model, the bifurcations organising emergent FC
take the form of Hopf, saddle, fold of limit cycle, and false bifurcations. False bifurcations have
received relatively little attention in the dynamical systems community (a notable exception
being the work of Marten et al., 2009), although our results indicate that they may be signif-
icant for understanding how ‘synchronisability’ of brain networks is reduced during seizures.
This phenomena was reported in Schindler, Bialonski, Horstmann, Elger, and Lehnertz (2008),
which also found that synchronisability increases as the patient recovers from seizure state.
Network Neuroscience
480
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
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
a
_
0
0
1
3
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
Node dynamics and functional connectivity
A natural extension to the work presented here would be the inclusion of conduction delays,
characterised by Euclidean or path-length distances between brain regions, which are certainly
important in modulating the spatiotemperal coherence in the brain (Deco, Jirsa, McIntosh,
Sporns, & Kötter, 2009). These would manifest as constant phase shifts in the weakly coupled
reduction of the model (Ton, Deco, & Daffertshofer, 2014). For strongly coupled systems the
mathematical treatment of networks with delayed interactions remains an open challenge. Re-
cent work in this vein by Tewarie et al. (2019) focusses on the role of delays in destabilising
network steady states, and techniques extending the Master Stability Function to delayed sys-
tems (Otto, Radons, Bachrathy, & Orosz, 2018) may be appropriate for treating phase-locked
network states.
In summary, the findings reported here suggest that there are multiple factors which give
rise to emergent FC. While structure clearly facilitates FC, the degree to which it influences
emergent FC states is determined by the dynamics of its neural subunits.
Importantly, we
have shown that local dynamics has a clear influence on SC-FC correlation, as does network
topology and coupling strength. Our combined mathematical and computational study has
demonstrated that a full description of the mechanisms that dictate the formation of FC from
anatomy requires knowledge of how both neuronal activity and connectivity are modulated
and, moreover, exposes the utility of bifurcation theory and network reduction techniques. This
work can be extended to more complex neural mass models such as that derived in Coombes
and Byrne (2019), to further explore the relationship between dynamics and structure–function
relations in systems with more sophisticated models for node dynamics.
AUTHOR CONTRIBUTIONS
Investigation; Writing - Original Draft.
Michael Forrester:
Stephen Coombes: Supervi-
sion; Writing - Review & Editing. Jonathan Crofts: Supervision; Writing - Review & Editing.
Stamatios Sotiropoulos: Data curation; Writing - Review & Editing. Reuben O’Dea: Supervi-
sion; Writing - Review & Editing.
FUNDING INFORMATION
Michael Forrester, Engineering and Physical Sciences Research Council (http://dx.doi.org/10.
13039/501100000266), Award ID: EP/N50970X/1.
REFERENCES
Abdelnour, F., Dayan, M., Devinsky, O., Thesen, T., & Raj, A.
(2018). Functional brain connectivity is predictable from anatomic
network’s Laplacian eigen-structure. NeuroImage, 172, 728–739.
Ahmadizadeh, S., Karoly, P. J., Neši´c, D., Grayden, D. B., Cook,
M. J., Soudry, D., & Freestone, D. R. (2018). Bifurcation analysis
of two coupled Jansen-Rit neural mass models. PLoS One, 13(3),
e0192842.
Bassett, D. S., & Bullmore, E. (2006). Small-world brain networks.
The Neuroscientist, 12(6), 512–523.
Betzel, R. F., Gu, S., Medaglia, J. D., Pasqualetti, F., & Bassett, D. S.
(2016). Optimally controlling the human connectome: The role
of network topology. Scientific Reports, 6, 30770.
Betzel, R. F., Medaglia, J. D., Papadopoulos, L., Baum, G. L., Gur,
R., Gur, R., . . . Bassett, D. S. (2017). The modular organization
of human anatomical brain networks: Accounting for the cost of
wiring. Network Neuroscience, 1(1), 42–68.
Braun, U., Muldoon, S. F., & Bassett, D. S.
(2015). On human
brain networks in health and disease. In ELS (pp. 1–9). American
Cancer Society.
Breakspear, M. (2017). Dynamic models of large-scale brain activity.
Nature Neuroscience, 20(3), 340.
Bullmore, E., & Sporns, O. (2012). The economy of brain network
organization. Nature Reviews Neuroscience, 13(5), 336.
Chen, J., Lu, J.-a., Zhan, C., & Chen, G. (2012). Laplacian spectra
and synchronization processes on complex networks. In Hand-
book of optimization in complex networks (pp. 81–113). Springer.
Cocchi, L., Gollo, L. L., Zalesky, A., & Breakspear, M. (2017). Criti-
cality in the brain: A synthesis of neurobiology, models and cog-
nition. Progress in Neurobiology, 158, 132–152.
Coombes, S., & Byrne, A. (2019). Next generation neural mass mod-
els. In A. Torcini & S. F. Corinto (Eds.), Nonlinear dynamics in
computational neuroscience. Springer.
Network Neuroscience
481
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
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
a
_
0
0
1
3
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
Node dynamics and functional connectivity
Crofts, J. J., Forrester, M., & O’Dea, R. D. (2016). Structure-function
clustering in multiplex brain networks. EPL (Europhysics Letters),
116(1), 18003.
Deco, G., Cruzat, J., Cabral, J., Tagliazucchi, E., Laufs, H., Logothetis,
N. K., & Kringelbach, M. L. (2019). Awakening: Predicting ex-
ternal stimulation to force transitions between different brain
states. Proceedings of the National Academy of Sciences, 116(36),
18088–18097.
Deco, G., Jirsa, V., McIntosh, A. R., Sporns, O., & Kötter, R. (2009).
Key role of coupling, delay, and noise in resting brain fluctua-
tions. Proceedings of the National Academy of Sciences, 106(25),
10302–10307.
Deco, G., Kringelbach, M. L., Jirsa, V. K., & Ritter, P. (2017). The
dynamics of resting fluctuations in the brain: Metastability and
its dynamical cortical core. Scientific Reports, 7(1), 3095.
Deco, G., Ponce-Alvarez, A., Mantini, D., Romani, G. L., Hagmann,
P., & Corbetta, M. (2013). Resting-state functional connectivity
emerges from structurally and dynamically shaped slow linear
fluctuations. Journal of Neuroscience, 33(27), 11239–11252.
Desroches, M., Krupa, M., & Rodrigues, S. (2013). Inflection, ca-
nards and excitability threshold in neuronal models. Journal of
Mathematical Biology, 67(4), 989–1017.
Ermentrout, B. (2002). Simulating, analyzing, and animating dynam-
ical systems: A guide to XPPAUT for researchers and students
(Vol. 14). SIAM.
Fox, M. D., Snyder, A. Z., Vincent, J. L., Corbetta, M., Van Essen,
D. C., & Raichle, M. E. (2005). The human brain is intrinsically
organized into dynamic, anticorrelated functional networks. Pro-
ceedings of the National Academy of Sciences of the United
States of America, 102(27), 9673–9678.
Glasser, M. F., Sotiropoulos, S. N., Wilson, J. A., Coalson, T. S.,
Fischl, B., Andersson, J. L., . . . Jenkinson, M. (2013). The mini-
mal preprocessing pipelines for the human connectome project.
NeuroImage, 80, 105–124.
Goulden, N., Khusnulina, A., Davis, N. J., Bracewell, R. M., Bokde,
A. L., McNulty, J. P., & Mullins, P. G. (2014). The salience net-
work is responsible for switching between the default mode net-
work and the central executive network: Replication from DCM.
NeuroImage, 99, 180–190.
Grimbert, F., & Faugeras, O. (2006). Bifurcation analysis of Jansen’s
neural mass model. Neural Computation, 18(12), 3052–3068.
Hansen, E. C., Battaglia, D., Spiegler, A., Deco, G., & Jirsa, V. K.
(2015). Functional connectivity dynamics: Modeling the switch-
ing behavior of the resting state. NeuroImage, 105, 525–535.
Hermundstad, A. M., Brown, K. S., Bassett, D. S., Aminoff, E. M.,
Frithsen, A., Johnson, A., . . . Carlson, J. M. (2014). Structurally-
constrained relationships between cognitive states in the human
brain. PLoS Computational Biology, 10(5), e1003591.
Hlinka, J., & Coombes, S.
(2012). Using computational models
to relate structural and functional brain connectivity. European
Journal of Neuroscience, 36(2), 2137–2145.
Honey, C., Sporns, O., Cammoun, L., Gigandet, X., Thiran, J.-P.,
Meuli, R., & Hagmann, P. (2009). Predicting human resting-state
functional connectivity from structural connectivity. Proceedings
of the National Academy of Sciences, 106(6), 2035–2040.
on multiple time scales. Proceedings of the National Academy
of Sciences, 104(24), 10240–10245.
Honey, C. J., Thivierge, J.-P., & Sporns, O. (2010). Can structure pre-
dict function in the human brain? NeuroImage, 52(3), 766–776.
Hoppensteadt, F. C., & Izhikevich, E. M. (2012). Weakly connected
neural networks (Vol. 126). Springer Science & Business Media.
Hutchison, R. M., Womelsdorf, T., Allen, E. A., Bandettini, P. A.,
Calhoun, V. D., Corbetta, M., . . . Chang, C. (2013). Dynamic
functional connectivity: Promise,
issues, and interpretations.
NeuroImage, 80, 360–378.
Jaccard, P. (1912). The distribution of the flora in the alpine zone.
1. New Phytologist, 11(2), 37–50.
Jansen, B. H., & Rit, V. G. (1995). Electroencephalogram and visual
evoked potential generation in a mathematical model of coupled
cortical columns. Biological Cybernetics, 73(4), 357–366.
Jbabdi, S., Sotiropoulos, S. N., Savio, A. M., Graña, M., & Behrens,
T. E. (2012). Model-based analysis of multishell diffusion mr data
for tractography: How to get over fitting problems. Magnetic Res-
onance in Medicine, 68(6), 1846–1855.
Lehnertz, K., Bialonski, S., Horstmann, M.-T., Krug, D., Rothkegel,
A., Staniek, M., & Wagner, T. (2009). Synchronization phenom-
ena in human epileptic brain networks. Journal of Neuroscience
Methods, 183(1), 42–48.
Liao, X., Vasilakos, A. V., & He, Y. (2017). Small-world human brain
networks: Perspectives and challenges. Neuroscience & Biobe-
havioral Reviews, 77, 286–300.
Liegeois, R., Laumann, T. O., Snyder, A. Z., Zhou, J., & Yeo, B. T.
Interpreting temporal fluctuations in resting-state func-
(2017).
tional connectivity MRI. NeuroImage, 163, 437–455.
Marten, F., Rodrigues, S., Benjamin, O., Richardson, M. P., & Terry,
J. R. (2009). Onset of polyspike complexes in a mean-field model
of human electroencephalography and its application to ab-
sence epilepsy. Philosophical Transactions of the Royal Society
of London A, 367, 1145–1161.
Menon, V. (2011). Large-scale brain networks and psychopathology:
A unifying triple network model. Trends in Cognitive Sciences,
15(10), 483–506.
Messé, A., Hütt, M.-T., König, P., & Hilgetag, C. C. (2015). A closer
look at the apparent correlation of structural and functional con-
nectivity in excitable neural networks. Scientific Reports, 5, 7870.
Messé, A., Rudrauf, D., Benali, H., & Marrelec, G. (2014). Relating
structure and function in the human brain: Relative contributions
of anatomy, stationary dynamics, and non-stationarities. PLoS
Computational Biology, 10(3), e1003530.
Meunier, D., Lambiotte, R., & Bullmore, E. T. (2010). Modular and
hierarchically modular organization of brain networks. Frontiers
in Neuroscience, 4, 200.
Moeller, F., Siebner, H. R., Wolff, S., Muhle, H., Granert, O., Jansen,
O., . . . Siniatchkin, M. (2008). Simultaneous EEG-fMRI in drug-
naive children with newly diagnosed absence epilepsy. Epilepsia,
49(9), 1510–1519.
Mormann, F., Kreuz, T., Andrzejak, R. G., David, P., Lehnertz, K., &
Elger, C. E. (2003). Epileptic seizures are preceded by a decrease
in synchronization. Epilepsy Research, 53(3), 173–185.
Honey, C. J., Kötter, R., Breakspear, M., & Sporns, O. (2007). Net-
work structure of cerebral cortex shapes functional connectivity
Mormann, F., Lehnertz, K., David, P., & Elger, C. E. (2000). Mean
phase coherence as a measure for phase synchronization and its
Network Neuroscience
482
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
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
a
_
0
0
1
3
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
Node dynamics and functional connectivity
application to the EEG of epilepsy patients. Physica D, 144(3–4),
358–369.
Mueller, S., Wang, D., Fox, M. D., Yeo, B. T., Sepulcre, J., Sabuncu,
Individual variability in functional
M. R., . . . Liu, H.
connectivity architecture of the human brain. Neuron, 77(3),
586–595.
(2013).
Netoff, T. I., & Schiff, S. J.
(2002). Decreased neuronal synchro-
nization during experimental seizures. Journal of Neuroscience,
22(16), 7297–7307.
Oldham, S., & Fornito, A. (2019). The development of brain network
hubs. Developmental Cognitive Neuroscience, 36, 100607.
Otto, A., Radons, G., Bachrathy, D., & Orosz, G. (2018). Synchro-
nization in networks with heterogeneous coupling delays. Phys-
ical Review E, 97(1), 012311.
Park, H.-J., & Friston, K. (2013). Structural and functional brain
networks: From connections to cognition. Science, 342(6158),
1238411.
Petersen, S. E., & Sporns, O. (2015). Brain networks and cognitive
architectures. Neuron, 88(1), 207–219.
Ponce-Alvarez, A., Deco, G., Hagmann, P., Romani, G. L., Mantini,
D., & Corbetta, M. (2015). Resting-state temporal synchronization
networks emerge from connectivity topology and heterogeneity.
PLoS Computational Biology, 11(2), e1004100.
Preti, M. G., Bolton, T. A., & Van De Ville, D.
(2017). The dy-
namic functional connectome: State-of-the-art and perspectives.
NeuroImage, 160, 41–54.
Rich, S., Zochowski, M., & Booth, V. (2018). Effects of neuromodu-
lation on excitatory–inhibitory neural network dynamics depend
on network connectivity structure. Journal of Nonlinear Science,
1–24.
Rodrigues, S., Barton, D., Marten, F., Kibuuka, M., Alarcon, G.,
Richardson, M. P., & Terry, J. R. (2010). A method for detecting
false bifurcations in dynamical systems: Application to neural-
field models. Biological Cybernetics, 102(2), 145–154.
Rubinov, M., Sporns, O., van Leeuwen, C., & Breakspear, M.
(2009). Symbiotic relationship between brain structure and dy-
namics. BMC Neuroscience, 10(1), 55.
Saggio, M. L., Ritter, P., & Jirsa, V. K. (2016). Analytical operations
relate structural and functional connectivity in the brain. PLoS
One, 11(8), e0157292.
Schindler, K. A., Bialonski, S., Horstmann, M.-T., Elger, C. E., &
Lehnertz, K. (2008). Evolving functional network properties and
synchronizability during human epileptic seizures. Chaos: An
Interdisciplinary Journal of Nonlinear Science, 18(3), 033119.
Shew, W. L., & Plenz, D. (2013). The functional benefits of critical-
ity in the cortex. The Neuroscientist, 19(1), 88–100.
Sotiropoulos,S. N., Hernández-Fernández, M., Vu, A. T., Andersson,
J. L., Moeller, S., Yacoub, E., . . . Jbabdi, S. (2016). Fusion in dif-
fusion MRI for improved fibre orientation estimation: An appli-
cation to the 3T and 7T data of the human connectome project.
NeuroImage, 134, 396–409.
Spiegler, A., Kiebel, S. J., Atay, F. M., & Knösche, T. R. (2010).
Bifurcation analysis of neural mass models: Impact of extrinsic
inputs and dendritic time constants. NeuroImage, 52(3),1041–1058.
Sporns, O. (2011). The human connectome: A complex network.
Annals of the New York Academy of Sciences, 1224(1), 109–125.
Sporns, O., & Betzel, R. F. (2016). Modular brain networks. Annual
Review of Psychology, 67, 613–640.
Srinivasan, R., Winter, W. R., Ding, J., & Nunez, P. L. (2007). EEG
and MEG coherence: Measures of functional connectivity at dis-
Journal of Neuro-
tinct spatial scales of neocortical dynamics.
science Methods, 166(1), 41–52.
Stam, C., van Straaten, E., Van Dellen, E., Tewarie, P., Gong, G.,
Hillebrand, A., . . . Van Mieghem, P. (2016). The relation be-
tween structural and functional connectivity patterns in complex
brain networks. International Journal of Psychophysiology, 103,
149–160.
Teplan, M. (2002). Fundamentals of EEG measurement. Measure-
ment Science Review, 2(2), 1–11.
Tewarie, P., Abeysuriya, R., Byrne, Á., O’Neill, G. C., Sotiropoulos,
S. N., Brookes, M. J., & Coombes, S. (2019). How do spatially
distinct frequency specific MEG networks emerge from one un-
derlying structural connectome? the role of the structural eigen-
modes. NeuroImage, 186, 211–220.
Tewarie, P., Hunt, B., O’Neill, G. C., Byrne, A., Aquino, K., Bauer,
M.,
. . . Brookes, M. J. (2018). Relationships between neu-
ronal oscillatory amplitude and dynamic functional connectivity.
Cerebral Cortex, 1, 14.
Thomas Yeo, B., Krienen, F. M., Sepulcre, J., Sabuncu, M. R.,
Lashkari, D., Hollinshead, M., . . . Buckner, R. L. (2011). The
organization of the human cerebral cortex estimated by intrin-
sic functional connectivity. Journal of Neurophysiology, 106(3),
1125–1165.
Ton, R., Deco, G., & Daffertshofer, A. (2014). Structure-function
discrepancy: Inhomogeneity and delays in synchronized neural
networks. PLoS Computational Biology, 10(7), e1003736.
Touboul, J., Wendling, F., Chauvel, P., & Faugeras, O. (2011). Neu-
ral mass activity, bifurcations, and epilepsy. Neural Computation,
23, 3232–3286.
Tsai, S.-Y. (2018). Reproducibility of structural brain connectivity
and network metrics using probabilistic diffusion tractography.
Scientific Reports, 8(1), 11562.
Van Den Heuvel, M. P., & Pol, H. E. H. (2010). Exploring the brain
network: A review on resting-state fMRI functional connectivity.
European Neuropsychopharmacology, 20(8), 519–534.
Van Den Heuvel, M. P., & Sporns, O. (2011). Rich-club organiza-
tion of the human connectome. Journal of Neuroscience, 31(44),
15775–15786.
van den Heuvel, M. P., & Sporns, O. (2013). Network hubs in the
human brain. Trends in Cognitive Sciences, 17(12), 683–696.
Van de Steen, F., Almgren, H., Razi, A., Friston, K., & Marinazzo,
D. (2019). Dynamic causal modelling of fluctuating connectivity
in resting-state EEG. NeuroImage, 189, 476–484.
Van Essen, D. C., Smith, S. M., Barch, D. M., Behrens, T. E., Yacoub,
(2013). The WU-Minn human connectome
E., & Ugurbil, K.
project: An overview. NeuroImage, 80, 62–79.
van Straaten, E. C., & Stam, C. J. (2013). Structure out of chaos:
Functional brain network analysis with EEG, MEG, and func-
tional MRI. European Neuropsychopharmacology, 23(1), 7–18.
Vidaurre, D., Smith, S. M., & Woolrich, M. W. (2017). Brain net-
work dynamics are hierarchically organized in time. Proceedings
of the National Academy of Sciences, 114(48), 12827–12832.
Network Neuroscience
483
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
2
4
6
7
1
8
6
6
6
5
1
n
e
n
_
a
_
0
0
1
3
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
