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The entropy of resting-state neural dynamics is a
marker of general cognitive ability in childhood
Natalia Zdorovtsova1*, Edward J. Young2, Danyal Akarca1, Alexander Anwyl-Irvine1,
The RED Team1, The CALM Team1, Duncan E. Astle1,3
1 MRC Cognition and Brain Sciences Unit, University of Cambridge, Cambridge, UK
2 Department of Engineering, University of Cambridge, Cambridge, UK
3 Department of Psychiatry, University of Cambridge, Cambridge, UK
*Corresponding author: Natalia Zdorovtsova
Email: natalia.zdorovtsova@mrc-cbu.cam.ac.uk
Address: 15 Chaucer Road Cambridge CB2 7EF UK
Telephone: +44 1223 69896
Conflict of Interest Statement
The authors declare no competing financial interests.
Significance Statement
There is increasing evidence that the function of resting-state brain networks
contributes to individual differences in cognition and behaviour across development.
However, the relationship between dynamic, transient patterns of switching between
resting-state networks and neurodevelopmental diversity is largely unknown. Here,
we show that cognitive ability in childhood is related to the complexity of resting-state
brain dynamics. Additionally, we demonstrate that the probability of transitioning into
and remaining in certain ‘states’ of brain network activity predicts individual
differences in cognitive ability.
Acknowledgements
The authors were supported by the Medical Research Council program grant MC-
A0606-5PQ41, and D.E.A. and D.A. were supported by an Opportunity Award from
the James S. McDonnell Foundation. E.J.Y. is supported by the Engineering and
Physical Sciences Research Council. We would like to thank all members of the CALM
and RED Teams, particularly Alexander Anwyl-Irvine, Edwin Dalmaijer, and Sophie
Gibbons, for their help with recruitment, data collection, and data management.
Additionally, we would like to thank all of the children and parents for their
participation in the study, and the radiographers and MEG operators who support the
excellent paediatric scanning at the MRC CBSU.
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1 Abstract
Resting-state network activity has been associated with the emergence of individual
differences across childhood development. However, due to the limitations of time-
averaged representations of neural activity, little is known about how cognitive and
behavioural variability relates to the rapid spatiotemporal dynamics of these networks.
Magnetoencephalography (MEG), which records neural activity at a millisecond
timescale, can be combined with Hidden Markov Modelling (HMM) to track the
spatial and temporal characteristics of transient neural states. We applied HMMs to
resting-state MEG data from (n = 46) children aged 8-13, who were also assessed on
their cognitive ability and across multiple parent-report measures of behaviour. We
found that entropy-related properties of participants’ resting-state time-courses were
positively associated with cognitive ability. Additionally, cognitive ability was
positively correlated with the probability of transitioning into HMM states involving
fronto-parietal and somatomotor activation, and negatively associated with a state
distinguished by default-mode network suppression. We discuss how using dynamical
measures to characterise rapid, spontaneous patterns of brain activity can shed new
light on neurodevelopmental processes implicated in the emergence of cognitive
differences in childhood.
2 Introduction
2.1 The emergence of resting-state networks across childhood
development
Brain activity is characterised by intrinsic dynamics in the form of spatially-coherent
and spontaneous fluctuations, which form resting-state networks (RSNs). RSNs were
first discovered in the form of functionally-correlated features of fMRI timeseries
during resting-state scans (Biswal et al., 1995; Lowe et al., 1998; Xiong et al., 1999;
Cordes et al., 2000), and later taxonomised across a wide range of neuroimaging
modalities (Kiviniemi et al., 2003; van den Heuvel et al., 2009; Allen et al., 2011).
Further studies revealed a correspondence between the brain’s functional architecture
across tasks and at rest, indicating that these networks support distinct features of
sensory processing, cognition, and behaviour (Smith et al. 2009). As such, RSNs are
thought to have emerged across evolutionary timescales, and are known to develop
from infancy through adolescence, in order to serve specific functions (Yeo et al.,
2011). Understanding the nature of spontaneous patterns of brain activity, and how
they develop, is therefore an important challenge within developmental cognitive
neuroscience.
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2.2 Resting-state characteristics predict individual differences in
cognition and behaviour
Divergent RSN development has been observed in those with neurodevelopmental
conditions. Typical childhood development is characterised by a gradual segregation
between the default-mode network and task-positive networks across the brain.
Functional over-connectivity between these networks is associated with cognitive and
behavioural difficulties, particularly Attention-Deficit Hyperactivity Disorder (ADHD)
(Cortese et al., 2012; Sripada et al., 2014; Francx et al., 2015; Cai et al., 2018; de Lacy
& Calhoun, 2018; Jones et al., 2022). Children with conditions like ADHD have also
been found to demonstrate profiles of lower fronto-parietal activation at rest (Cortese
et al., 2012; Castellanos & Proal, 2012), as have those diagnosed with intellectual
disabilities in childhood (Ma et al., 2021). Given the fact that RSN function is
implicated in developmental divergence, there is value in outlining the precise
mechanisms by which RSN differences might correspond to population-level
variations in cognitive ability and behavioural difficulty.
2.3 Using dynamical measures to characterise the spatiotemporal
properties of spontaneous brain activity
As discussed previously, spontaneous functional network activity can be described
using static, time-averaged representations of neural activation. However,
computational models and empirical data suggest that the brain engages in rapid
switching between distinct states of functional connectivity (Rabinovich et al., 2008;
Nachstedt & Tetzlaff, 2017). In other words, states of neural activity are not merely
situated in space, but also in time, and transient, recurrent patterns of activity could
represent a meaningful mode by which information is processed by the brain. One
limitation of time-averaged representations is that they are unable to describe the
transient temporal dynamics of brain activity. An emerging field dedicated to
measuring these rapid dynamics strives to overcome some of the interpretability
constraints of time-averaged functional network identification techniques by
proposing a new set of data-driven, temporally-embedded methods of representing
neural activity.
One of these relatively novel methods is Hidden Markov Modelling (HMM), an
unsupervised machine learning technique that reconstructs a sequence of patterns as
a system of temporally-discrete states. Previously, HMMs have been used to extract
the underlying dynamical properties of neural data from MEG (Baker et al., 2014;
Vidaurre et al., 2018b; Quinn et al., 2018, Hawkins et al., 2019), EEG (Obermaier et
al., 2001; Williams et al., 2018; Dash & Kolekar, 2020; Marzetti, 2023), and fMRI
(Duan et al., 2005; Dang et al., 2017; Goucher-Lambert & McComb, 2019; Hussain et
al., 2023) at rest and in task settings. To our knowledge, there have not been any
studies of how resting-state neural dynamics vary in a developmental sample, or how
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these dynamics relate to the emergence of diverse profiles of behaviour and cognitive
ability.
2.4 The current study
In the current study, we used resting-state MEG data from children aged 8-13 to test
how the temporal properties of transient neural dynamics vary with age, gender,
behavioural difficulties, and general cognitive ability. We also explored the extent to
which the complexity of individual participants’ resting-state time-courses varied
across these neurodevelopmental features of interest.
3 Materials and Methods
3.1 Participants
Our MEG analysis sample, following exclusions (n = 46), included children aged 8-13
(M = 10.09, SD = 1.19). Children were all part of ongoing studies at the MRC Cognition
and Brain Sciences Unit, and all underwent an identical MEG protocol. Specifically,
all the children were originally part of two studies, either the Resilience in Education
and Development (RED) study (n = 40, PRE.2017.102) or the Centre for Attention,
Learning, and Memory (CALM) cohort (n = 6, 22/WM/0082), and agreed to take part
in this MEG session. Combined, this sample reflects a range of common behavioural
difficulties typically seen in mainstream education.
3.2 Cognitive Assessments and Behavioural Questionnaires
3.2.1 Wechsler Abbreviated Scales of Intelligence II – Matrix Reasoning Subtest
(WASI-II MR)
The Matrix Reasoning subtest of the Wechsler Abbreviated Scales of Intelligence II is
a general measure of cognitive ability and executive function. In this subtest, children
are presented with incomplete matrices of images and asked to select an image that
would suitably complete each matrix from a choice of four options. Children aged
9 years and older complete a possible total of 30 matrices, which become progressively
more difficult. The matrix reasoning test is finished when the child produces three
incorrect answers in a row. Trials correct were converted to age-standardised T-scores.
3.2.2 Strengths and Difficulties Questionnaire
The Strengths and Difficulties Questionnaire (SDQ) asked parents/carers to answer
25 questions measuring a variety of behavioural challenges (with possible responses
being ‘Not True,’ ‘Somewhat True,’ and ‘Certainly True’) based on their child’s
behaviour in the six months prior to assessment. A total SDQ score is calculated, in
addition to scores for five behavioural subscales: Hyperactivity, Conduct Problems,
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Emotional Regulation Problems, Peer Relationship Problems, and Prosocial
Behaviour (see Table 1). See Supplementary Figure 1 for plots representing how
cognitive, behavioural, and demographic traits were distributed across our combined
sample.
Table 1: Means, standard deviations, and range values were calculated for cognitive subtest
and behavioural subscale scores across our sample, which combined data from the RED study
and CALM cohorts. Here, we summarise these scores, in addition to some additional sample
characteristics.
Measure
Combined Sample
N
46
Age in Years
10.09 (± 1.19; range = 8-13)
Gender
47.8% male
WASI-II Matrix Reasoning
55.91 (± 9.70; range = 37-80)
SDQ (Total)
7.78 (± 5.83; range = 0-10)
SDQ (Hyperactivity)
4.11 (± 2.74; range = 0-10)
SDQ (Conduct Problems)
1.67 (± 1.99; range = 0-8)
SDQ (Peer Problems)
2.76 (± 2.52; range = 0-9)
SDQ (Emotion Regulation Problems)
2.83 (± 2.9; range = 0-10)
SDQ (Prosocial)
7.83 (± 2.58; range = 0-10)
Correlations were performed between each of the cognitive and behavioural measures
(see Figure 1). A significant negative relationship was found between WASI-II MR
scores and all subscales within the SDQ (rsubscales < -0.59, psubscales < 0.0002).
Additionally, scores on the WASI-II MR were positively correlated with the SDQ
Prosocial Behaviour subscale (r = 0.34, p = 0.0201).
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Figure 1: Here, we display a correlation matrix representing relationships between scores on
the WASI-II MR subtest, total SDQ scores, and SDQ subscale scores. Significant relationships
at p < 0.05 are marked by one asterisk, and significant relationships at p < 0.001 are marked
by two asterisks.
3.3 Resting-state MEG acquisition
MEG data were acquired using a high-density VectorView MEG system (Elekta-
Neuromag) with 102 magnetometers and 102 orthogonal pairs of planar gradiometers
(306 sensors in total). Five head position indicator (HPI) coils were attached to the
child’s head (one on each mastoid bone, two on the child’s forehead, and one on the
top of their head) in order to monitor changes in head position throughout the
recording. The positions of the HPI coils was recorded using a 3D digitizer
(FASTRACK, Polhemus) in addition to over 200 additional points distributed over the
scalp. Pulse was measured using an electrocardiogram electrode attached to each wrist
and eye movements were recorded using horizontal and vertical electrooculograms.
Data were sampled at 1Khz. Smaller children were seated on a booster seat to ensure
that the tops of their heads were in contact with the scanner and that they could remain
in a comfortable position for the duration of the scan.
Children were monitored by video camera throughout the scan. During the 10-minute
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resting-state scan, children were asked to sit as still as possible, close their eyes and let
their minds wander, without falling asleep.
3.4 Structural MRI acquisition
Out of the 46 participants in our MEG analysis sample (following outlier exclusions),
40 participants took part in a structural MRI scan, which yielded T1-weighted images
from a Siemens 3T Tim Trio system. For these images, a magnetization-prepared rapid
acquisition gradient echo sequence with 1mm isometric image resolution and 2.98ms
echo time was used. A natural (asymmetric) NIHPD Objective 1 scan template
intended for children in pre- to mid-puberty (aged 7.5 to 13.5) was used for the 6
participants who did not undergo a T1-weighted MRI scan (Fonov et al., 2011).
3.5 MEG Preprocessing and Source Reconstruction
3.5.1 Maxwell filtering and artefact removal
Maxwell filtering was performed using a script and repository of functions developed
by Alex Anwyl-Irvine called RED Tools, which implements MNE Python’s Maxfiltering
procedure (https://github.com/u01ai11/RED_Rest/tree/master/REDTools). Blinks,
saccades, and pulse-related artefacts were removed by running an automated
temporal independent components analysis (ICA), which applied MNE’s fastICA
function to the sensor-space time-courses. Following this, we performed an additional
ICA for which components were manually inspected, and any remaining ECG and EOG
components were removed. Additionally, components dominated by 50Hz noise were
removed to reduce electrical interference.
3.5.2 Co-registration and bandpass filtering
40 participants’ MEG data from our original sample (n = 52) were co-registered to
their T1-weigthed structural MRI image acquired using a 3T Siemens Tim Trio and an
MPRAGE sequence. A natural (asymmetric) NIHPD Objective 1 scan template
intended for children in pre- to mid-puberty (aged 7.5 to 13.5) was used for the
remaining 12 participants in our original sample (6 from the RED subsample and 6
from the CALM subsample) who did not undergo a T1-weighted MRI scan. Co-
registration was performed using the digitized scalp locations and fiducial markers
using an iterative closest point algorithm in SPM12 (Penny et al., 2011; Wellcome Trust
Centre for Neuroimaging, 2014). A forward model was fitted using a single sphere
homogeneous head shape model for each subject (Mosher et al., 1999). Then, data
were bandpass filtered to be between 1-30Hz in SPM12, as these slower frequencies
are better for considering functional connectivity with MEG (Luckhoo et al., 2012).
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3.5.3 Source-localisation and parcellation
The remaining preprocessing steps were implemented using the OHBA Software
Library (OSL v2.0.3; OHBA Analysis Group, 2017; https://github.com/OHBA-
analysis/osl-core) and OHBA’s Hidden Markov Model Library (HMM-MAR; Vidaurre
et al., 2016). First, a covariance matrix was computed across the whole time-course for
each participant and was regularized to 50 dimensions using PCA rank reduction.
Sensor normalisation was then performed across planar gradiometers and
magnetometers. Following this, we used a linearly-constrained minimum variance
beamformer to estimate whole-brain source-space activity for points in an 8mm grid
(Van Veen et al., 1997). The signal-space separation algorithm reduced the
dimensionality of the data, resulting in a set of estimated time-courses of brain activity
for each child for 3,559 source locations across the brain (Woolrich et al., 2011). At this
point in the preprocessing pipeline, we excluded 5 participants from our original
sample (n = 52) who had a very high predominance of bad segments across their time-
series (>60%), which was assessed using the OSL function ‘osl_detect_artefacts.m’.
We excluded an additional participant on the basis of their having a poor co-
registration solution. Upon visually inspecting the co-registration solution using SPM
12’s GUI, it became clear to us that scalp locations had been digitised improperly (with
points placed too far from the scalp). These exclusions reduced our MEG sample from
n=52 to n=46.
Following artefact-related exclusions, MEG data were further reduced down into a 38-
node parcellation following the method outlined in Quinn et al. (2018). A binarised
parcellation with 38 cortical regions was applied and a single time-course was
estimated per node from the first principal component across voxels. This further
reduced each time-course down to 38 parcels, as opposed to 3,559 voxels, and made it
possible to perform additional corrections for signal leakage.
3.5.4 Additional preprocessing steps
Following parcellation, further preprocessing was conducted according to OHBA’s
HMM-MAR library, which recommends an additional set of preprocessing steps prior
to the initialisation of the Hidden Markov Model: detrending, signal standardisation,
corrections for signal leakage, and downsampling. We first completed detrending,
which removes linear trends in the data for each channel separately, which was
followed by a standardisation of the signal across participants’ concatenated time-
courses. Next, symmetric multivariate orthogonalization was used to correct for signal
leakage introduced by source reconstruction with zero temporal lag according to the
methods specified in Colclough et al. (2015). Following this, the absolute signal
amplitude for each source at each timepoint was estimated using the Hilbert
transform. To reduce dimensionality in the data, we performed a PCA, which retained
the number of dimensions necessary to explain 95% of variance in the data. Finally,
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MEG time-courses were downsampled to 250Hz. For a full schematic of our
preprocessing procedure, please see Figure 2.
Figure 2: Here, we outline each step of our MEG data preprocessing pipeline, in addition to
the software packages and toolboxes used to complete each step of the pipeline.
3.6 Hidden Markov Modelling (HMM)
In the current study, we used the HMM-MAR toolbox, developed by OHBA to infer a
Hidden Markov Model from resting-state timeseries MEG data. The base code and set
of functions that we adapted to suit our analyses is publicly-available on GitHub
(https://github.com/OHBA-analysis/HMM-MAR). The analysis scripts for this study
are also publicly-available on GitHub
(https://github.com/nataliazdorovtsova/HMM_MEG).
3.6.1 Model description and specifications
Hidden Markov Models (HMMs) comprise a set of unsupervised machine learning
techniques that extract the spatial and temporal structure of timeseries of data by
inferring discrete number of mutually-exclusive states. The HMM assumes that
timeseries data, which are comprised of a set of observed features, can be described
using a sequence of a finite, hidden variables (HMM states). Within a single model,
HMM states are inferred on the basis that they belong to the same family of
distributions, but are each parameterised differently. HMM states correspond to
distinct patterns of brain activity that occur at different points across a timeseries.
More formally, if we take xt to represent the time-series data and st to represent a given
state at time point t, we assume that
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xt|st = k Multivariate Gaussian(μk, Σk),
where μk is a vector with (number of channels) elements containing the mean
activation and Σk is the (number of channels x number of channels) covariance matrix
that represents the activation relationships between channels when state k is active.
This is commonly referred to as the observational model. In this case, the observation
model characterises a multivariate Gaussian distribution of each state k by parameters
(μk, Σk). Although we use a multivariate Gaussian distribution to characterise states in
the current study, there are a number of different methods that can be applied
depending on the context-dependent theoretical goals of the researcher (e.g. Baker et
al., 2014; Vidaurre et al., 2016; Vidaurre et al., 2018a; Quinn et al., 2018; Gohil et al.,
2022). Here, we chose to use a multivariate Gaussian HMM with state-specific means
and covariances, which can be treated as a Multivariate Autoregressive (MAR) model
with an order equal to zero. This meant that the segmentation of states within our
model was based on instantaneous patterns of activation and connectivity—two
features of resting-state brain activity that we were interested in capturing for the
purposes of this study. Previously, Vidaurre et al. (2018b) also demonstrated that
multivariate Gaussian HMMs are suitable for exploring the temporal features of
spontaneous transitions between large-scale resting-state brain networks.
The sequence of HMM states across a time-course is characterised by modelling the
joint transition probabilities between state pairs. Put more simply, the probability Pr
of a given state at time point t depends on which state was active at time point t-1:
,
where Θl,k refers to the transition probabilities. Within matrix Θ, we can further
distinguish between the diagonal elements, Θkk, which control the persistence of each
state, and the off-diagonal elements, Θkl, which refer to transitions between mutually-
exclusive states. The observed data at each time point are modelled as a mixture of
Gaussian distributions, with weights given by wtk = Pr(st = k).
In the current study, we ran a Hidden Markov Model on concatenated time-series data
from all of the participants included in our dataset, which allowed us to obtain a group-
level estimate of the states. Whereas the states were calculated at the group level,
information pertaining to when a state is most likely to be active (the state time-
course) were calculated independently for each participant.
The HMM applies an inference algorithm to estimate the parameters of each state
(characterised by parameters μk and Σk), the probability of each state being active at
each time point (st), and the joint transition probabilities for each pair of states (Θl,k).
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3.6.2 Model outputs
The hmmmar.m function from the HMM-MAR toolbox produces a range of outputs
that can be used to estimate different features of HMM states. Below, we provide a
brief description of the state properties that we used within subsequent analyses.
Following HMM inference, the temporal characteristics of each state can be quantified
in terms of state fractional occupancies (the fraction of the total time spent in a state),
state lifetimes (the time spent in a given state before transitioning out of that state),
and interval lengths (the time it takes to re-enter a given state). Additionally, a
switching rate can be calculated for each participant. Switching rates provide a
measure of stability for each subject, since they represent the frequency of state
switching across an individual time-course.
Using the hmmmar.m output Ξ (which holds matrices containing joint posterior
probabilities of transitions between pairs of states), it is also possible to compute state
transition probability matrices for each individual participant, as well as for an entire
concatenated time-course. The relationships between transition probabilities and
other measures of interest can be investigated in their own right, as we shall explore
in the next section.
In the current study, we also used these probability matrices to derive an entropy rate
estimate for each participant’s MEG timeseries data. The entropy rate measures the
average uncertainty, or information, generated by a transition within a sequence. In
general, the entropy rate of a sequence of random variables (st) is defined as the limit
of H[s0,s1,…,sn]/n as n is taken to infinity. Although this is not feasible to evaluate the
entropy rate for general sequences, a sequence taken from a homogenous Markov
chain posesses a compact and computationally-efficient formula for the entropy rate.
Given a transition matrix Pr(st|st-1) defining an irreducible Markov chain over a finite
state space, there exists a unique invariant distribution for that chain, π, satisfying π(s)
=
. The entropy rate can then be computed as
. We can understand this formula as follows: the entropy
rate quantifies the average entropy of a transition within the sequence. Hence, another
equivalent formula (in the case of a strongly stationary process, such as a Markov
chain) is the limit of H[st+1|st]. At long timescales, the convergence theorem tells us
that the distribution of st is given by invariant distribution. Hence, the entropy rate is
the sum of the conditional entropies H[st+1|st = s’] weighted by the invariant
distribution .
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4 Statistical Analysis
4.1 HMM inference and calculation of state characteristics
HMM inference requires an a priori specification of the number of states used in the
model, k. Free energy metrics lend some objectivity to state number selection—in
theory, the ‘optimal’ number of states should be determined by the model that has the
smallest free energy (measured in arbitrary units). However, it is questionable whether
this practice lends itself to parsimony and theoretical coherence; the aim of the current
study was to establish whether a limited collection of neural states can track
differences in behaviour and cognition across our sample. Baker et al. (2014), for
instance, found that free energy often increases monotonically up to k = 15 states,
suggesting that an even higher number of states would be needed to yield a Bayes-
optimal solution. A similar limitation exists for more traditional dimensionality
reduction methods like Independent Components Analysis, which is driven by the
goals and constraints of the research question at hand. A smaller number of
prespecified components often yields canonical resting-state networks, whereas a
larger number can be used to extract finer-grained distinctions between patterns of
activity (Smith et al., 2011, Smitha et al., 2017).
In the current study, we trained 11 separate HMMs on our resting-state MEG dataset
with prespecified states ranging from k = 4 to k = 14. Free energy metrics and
information about cycles to model convergence can be found in Supplementary Table
1. After inspecting the topological features of states for each solution, we chose a HMM
with k = 7 in order to achieve a good representation of spatially-segregated states while
minimising redundancy. Varying the number of states between 4 and 14 did not appear
to change the topographical features of the most prominent HMM states, which appear
across solutions regardless of the addition of extra states (see Supplementary Figure 2
for plots of different model results).
4.2 Comparisons between measures of neural dynamics and measures of
cognition and behaviour
We first used an array of General Linear Models (GLMs) to explore how participants’
switching rates, entropy rates, state fractional occupancies, and maximum fractional
occupancies vary with age in order to explore developmental effects among our
participants (who were 8 to 13 years old). Gender was included as a regressor in these
models. Additionally, we ran a series of between-subjects t-tests to isolate any unique
relationships between gender and states’ temporal properties.
We then investigated how the state measures listed above vary with measures of
cognitive ability (WASI-II MR scores) and behaviour (SDQ Total and SDQ
Hyperactivity, Conduct Problems, Peer Problems, Emotion Regulation Problems, and
Prosocial Behaviour subscale scores). To control for potential confounds, we included
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gender and age as regressors in each of these models at the level of the cognitive and
behavioural outcome measures.
To control for multiple comparisons, we applied a False Discovery Rate (FDR)
correction with a 5% threshold in each of these analyses (see Supplementary Figures
3 and 4 for a schematic representation of the cognitive and behavioural GLMs).
5 Results
5.1 State characteristics for the seven-state HMM
A seven-state HMM revealed distinct spatial patterns of activity and variations in
oscillatory amplitude (see Figure 3). Each state-map represents the mean activation
profile of each parcel for the concatenated MEG dataset. State-specific increases and
decreases in oscillatory amplitude are plotted as yellow/orange and cyan/blue, and
represent neural activation and suppression, respectively.
State 1 is primarily characterised by DMN activation, and state 2 shows prominent
patterns of activation in visuo-temporal regions of the cortex. States 3 and 7 both
demonstrate patterns of default-mode suppression, although state 7 is also
characterised by concurrent left-parietal activation. Similarly, states 4 and 5 both show
fronto-temporal suppression, along with patterns of somatomotor and visual
activation, respectively. State 6, meanwhile, is characterised by fronto-parietal
activation and visual suppression.
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Figure 3: Here, we illustrate the results from our seven-state HMM. For each state, we plotted
the top 20% of positive activations and bottom 20% of negative activations on a cortical surface
using the HCP Workbench GUI. State labels correspond to our descriptions of the macroscopic
features of cortical activation and suppression patterns.
Using the state time-courses, it was possible to calculate some temporal properties of
each state. As illustrated in Figure 4, the temporal characteristics of the states vary
considerably. Notably, state 1 has the most variable distribution in both lifetimes and
intervals, which can be explained by the fact that it is the first state represented in
participants’ MEG time-courses—what seems to vary, in this case, is how long
participants remain in this first state that is characterised by DMN activation (see
Figure 5).
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Figure 4: Here, we plot the intervals (A) and lifetimes (B) of the states in our k = 7 HMM.
Note that state intervals and lifetimes were thresholded at 50ms, such that state appearances
that were sub-50ms were not included in the calculation of these temporal metrics.
A one-way ANOVA revealed significant differences between the states’ fractional
occupancies, F(6,315) = 81.42, p = 4.16 x e-61. Because intervals and lifetimes were
calculated for each individual state, and not between participants, it was not possible
to compare them in the same fashion as fractional occupancies. Nonetheless, their
means and standard deviations, along with those of states’ fractional occupancies, are
summarised in Table 2.
Table 2: Means and standard deviations (in brackets) for state intervals, lifetimes, and
fractional occupancies.
State
Intervals (ms)
Lifetimes (ms)
Fractional
occupancies
(proportion)
State 1 (DMN+)
1925.9 (± 3344)
129.49 (± 158.14)
0.0177 (± 0.0145)
State 2 (VT+)
293.9 (± 395.18)
69.09 (± 20.90)
0.1311 (± 0.0210)
State 3 (DMN-)
230.38 (± 192.23)
95.24 (± 94.50)
0.2330 (± 0.1085)
State 4 (SM+, FP-)
191.31 (± 143.90)
59.75 (± 10.24)
0.1578 (± 0.0314)
State 5 (V+, FT-)
223.52 (± 200.73)
63.05 (± 13.90)
0.1524 (± 0.0229)
State 6 (FP+, V-)
194.75 (± 149.74)
61.99 (± 14.08)
0.1542 (± 0.0320)
State 7 (LP+, DMN-)
196.42 (± 170.58)
61.20 (± 12.05)
0.1538 (± 0.0290)
In addition to calculating state lifetimes, intervals, and fractional occupancies, as well
as generating state-maps, we also calculated state transition probabilities between and
across participants. As anticipated, the most common timepoint-to-timepoint
transition was the ‘self-transition’, forming the diagonal of the state transition
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probability matrix. State-to-state transition probabilities, which are less probable than
self-transitions, are still readily-visualised when the diagonal of the matrix is zeroed-
out (see Figure 5). State transition matrices with all values retained were used in later
analyses.
Figure 5: In panel (A), we present the first 1000 timesteps (4 seconds sampled at 250Hz) of
the Viterbi path, which represents the maximum a posteriori sequence of states in a HMM.
Notably, state-switching is rapid, and can be detected at very short timescales. Using the joint
posterior probabilities of state transitions, we computed transition matrices for each of our
participants, as shown in (B). Panels (C) and (D), in which self-transitions have been
intentionally excluded for the purposes of plotting state-to-state transitions, display the
average transition probabilities across our entire sample (n = 46). No thresholds were applied
in the generation of these plots. The node sizes in panel (C) reflect the fractional occupancies
of each of the states.
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5.2 Entropy-related measures of neural dynamics are related to cognitive
ability, but not age, gender, or behaviour
First, we used a series of general Linear Models (GLMs) to investigate the effect of age
on states’ temporal properties. Gender was included as a control regressor in these
models. We did not find any significant effects of age on switching rates (t(43) = 0.7221,
padjusted = 0.4741), entropy rates (t(43) = 0.61, padjusted = 0.5451), state fractional
occupancies (t(43) < |-0.5611|, padjusted = 0.9333), or maximum fractional occupancies
(t(43) = 0.0763, padjusted = 0.9395).
Similarly, t-tests comparing these state measures between genders did not find any
significant effects for switching rates (t(44) = -0.3125, p = 0.7562), entropy rates (t(44)
= -0.3078, p = 0.7597), state fractional occupancies (t(44) < |-0.5020|, padjusted =
0.9851), or maximum fractional occupancies (t(44) = -0.1576, p = 0.8755).
Next, we investigated whether the temporal properties of states varied with six
measures of behaviour (SDQ Total and SDQ Hyperactivity, Conduct Problems,
Emotional Regulation Problems, Peer Problems, and Prosocial Behaviour subscales)
and one measure of cognitive ability (WASI-II Matrix Reasoning). To do this, we used
a series of GLMs in which age and gender were included as control regressors. No
significant relationships were found between any measures of behaviour or state
measures, t(43) < |-2.1504|, padjusted > 0.0932 (see Supplementary Table 2 for a full
summary of these non-significant results). We did, however, find significant
relationships between cognitive ability and entropy rates (t(43) = 2.2704, padjusted =
0.0284), and cognitive ability and switching rates (t(43) = 2.1688, padjusted = 0.0358).
Additionally, entropy rates and switching rates were found to be highly correlated (r =
0.9979, p < 0.00001), indicating that individual variations in entropy are almost fully
explained by state switching (see Figure 6).
Figure 6: Panels (A) and (B) display the significant linear relationships between switching
rates and entropy rates, respectively, and cognitive ability. Panel (C) shows the strong linear
relationship between switching rates and entropy rates.
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5.3 Switching is not random—state-specific fractional occupancies are
related to cognitive ability
Although we found positive correlations between state switching, entropy, and
cognitive ability, we wanted to investigate whether spending longer amounts of time
in specific states could further explain the relationships between neural dynamics and
cognition. Again, we used a series of GLMs in which age and gender were included as
control regressors. We found significant relationships between cognitive ability and
fractional occupancies in state 1 (t(43) = -2.7178, padjusted = 0.0162), state 3 (t(43) = -
2.7693, padjusted = 0.0162), state 4 (t(43) = 2.9467, padjusted = 0.0162), state 6 (t(43) =
3.2673, padjusted = 0.0152), and state 7 (t(43) = 2.6402, padjusted = 0.0162). An additional
GLM revealed a significant relationship between cognitive ability and maximum
fractional occupancies, t(43) = -2.6754, padjusted = 0.0106 (see Figure 7).
Figure 7: Panels (A1) to (A5) display the significant linear relationships between state
fractional occupancies and cognitive ability. Panel (B) displays the significant negative linear
relationship that was found between maximum fractional occupancies and cognitive ability.
5.4 Specific state transition probabilities are related to cognitive ability
Our next aim was to test whether specific state transitions were related to cognitive
ability. Using the transition matrices we previously extracted for each participant, we
ran a series of correlations between each cell of the 7-by-7 transition matrices and
WASI-II MR scores. Although a number of significant effects at p < 0.025 were initially
found (see Figure 8), indicating the presence of strong positive and negative
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correlations between cognitive ability and state transitions, these effects did not
survive corrections for multiple comparisons at the 95% confidence interval.
Figure 8: On the left, we illustrate significant relationships found between cognitive ability
and transitions into specific states at p < 0.025. Node sizes represent states’ fractional
occupancies, and green versus pink transition arrows represent positive and negative
correlations with cognitive ability, respectively. On the right, we have plotted a heatmap
representing correlations between cognitive ability and state transitions in a 7-by-7 matrix.
Notably, the columns for states 3, 4 and 6 show the highest correlation coefficients, suggesting
that transitions into these states are most strongly associated with cognitive ability.
Instead of performing correlations across 49 separate cells of the state transition
matrix, we decided to investigate how transitions into states 1-7, represented by the
columns of the matrix (which, unlike the rows, do not sum to 1), might relate to
individual differences in cognitive ability. We performed 7 correlations to this effect,
and found that three states were significantly correlated with cognitive ability
following corrections for multiple comparisons: state 3 (r = -0.3794, padjusted = 0.0217),
state 4 (r = 0.3889, padjusted = 0.0217), and state 6 (r = 0.4095, padjusted = 0.0217). We
did not find any significant relationships between transitions into states 1, 2, 5, or 7 (r
< |0.3154|, padjusted > 0.0573). While state 3 was negatively correlated with cognitive
ability, states 4 and 6 were positively correlated with cognitive ability (see Figure 9).
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Figure 9: Here, we provide a finer-grained illustration of the linear relationships between
transitions into specific HMM states and cognitive ability. Transitions into state 3 (A), state 4
(B), and state 6 (C) were found to be significantly correlated with cognitive ability following
corrections for multiple comparisons.
6 Discussion
By inferring an HMM using group-concatenated MEG data from our developmental
sample, we identified spatiotemporally-defined states that corresponded to well-
known RSNs, including the default mode, fronto-temporal, visual, and sensorimotor
networks. The spatial topographies of states in our HMM mirrored that of numerous
other studies that utilised this method to extract the underlying features of resting-
state MEG data, including Baker et al. (2014) and Becker et al (2020). State time-
courses were characterised by a predominance of self-transitions, and that states
exhibited transient (<100ms) average lifetimes. Across participants, there were
individual differences in how long participants spent in each state. Cognitive ability—
but not behaviour, gender, or age—was positively associated with participants’ state-
switching and entropy rates. Additionally, there were state-specific relationships
between cognitive ability and states’ fractional occupancies. The directionality of these
relationships was preserved in analyses exploring whether transitions into each of the
states is predictive of cognitive ability. Transitions into and time spent within DMN-
heavy states was associated with lower cognitive ability, whereas the opposite was true
of states with more fronto-parietal and sensory network activation profiles.
The broad alignment between our findings and those outlined in previous research
suggests that an HMM approach can be applied successfully in the study of transient
neural dynamics in the developing brain. Although data quality issues are known to
typically affect the reliability of MEG scans in children (e.g. Wehner et al., 2008; Pang,
2011), and particularly those with high levels of behavioural difficulties (Kaiser et al.,
2021), we believe our findings demonstrate that HMM inference is robust to these
potential issues, provided that data are preprocessed with enough attention to noise
removal and the exclusion of scans with high proportions of outliers.
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6.1 Time-course entropy and state-switching are positively associated with
cognitive ability
The switching and entropy rates of participants’ time-courses, which we found to be
highly intercorrelated in our sample, were positively associated with cognitive ability
(but not age, gender, or behaviour). Although state-switching and entropy were highly
correlated in the current study, they have different formal definitions, and therefore
have different possible interpretations. While the switching rate quantifies the extent
to which a participant engages in state-to-state transitions, as opposed to self-
transitions, the entropy rate can be regarded as a measure of the complexity of a
participant’s time-course. Since switching rates explained the majority of the variance
in participants’ entropy rates, it is reasonable to conclude that the information of MEG
time-series data increased in proportion to the number of rapid and largely stochastic
transitions between neural states. Had non-random and recurrent patterns of state
transition sequences dominated the time-series, entropy rates would have been lower
and more weakly associated with state-switching. Since the information of a time-
course in our sample relies so heavily upon the flexibility and speed with which state-
switching takes place, it may be useful to regard the neural entropy rate as a more
complex measure that is nonetheless heavily driven by a latent switching factor.
Cortical feedback and feedforward pathways are known to rapidly control local brain
network states, and this process is optimised to flexibly change the gain, precision, and
synchronisation of neural activity (Zagha & McCormick, 2014). Furthermore, dynamic
transitions between hidden neural states are believed to enable the flexible
reconfiguration of functional circuits across the brain, thereby enabling adaptive
cognitive and behavioural processes. A number of previous studies have highlighted
the positive associations between state switching and cognitive ability. For instance,
Taghia et al. (2018) found that task performance is strongly predicted by state-
switching at rest, and that considering task-related neural dynamics only minimally
improves the ability to predict task performance. Additionally, Cabral et al. (2017)
found that more flexible patterns of state-switching predict better cognitive
performance in older adults, and that this relationship is mediated by the tendency to
transition between particular states in the neural attractor landscape. In the context
of neurodevelopment, rapid switching between states is known to increase during
adolescence (Medaglia et al., 2018) and accompany motor-skill acquisition (Reddy et
al., 2018). Entropy, like state-switching, has also been found to increase throughout
childhood development, reflecting a gradual expansion in the diversity of neural
processes available to the child (Amalric & Cantlon, 2023). More broadly, brain
entropy has been positively associated with general intelligence in samples with large
age ranges, suggesting this brain-cognition relationship persists across the lifetime
(Saxe et al., 2018; Wang, 2021; Thiele et al., 2023). Although there are many different
operationalisations of brain signal complexity (e.g. Shannon entropy, multiscale
entropy, Fuzzy entropy, and microstate characteristics; see Keshmiri, 2020, for
review), the entropy rate of the neurophysiological HMM timeseries may also provide
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a useful means of exploring individual differences in the intrinsic complexity of neural
activity.
6.2 Tendency to stay within, and transition into, certain HMM states
predicts differences in cognitive ability
Although the relationship between cognitive ability and neural entropy rates was
largely defined by rapid, stochastic patterns of brain switching, we wanted to
investigate whether there were state-specific relationships between different
neurodevelopmental characteristics and neural dynamics. While we did not find any
relationships between the seven states’ fractional occupancies and age, gender, or
behaviour, cognition was positively associated with time spent in states 4, 6, and 7,
and negatively associated with time spent in states 1 and 3. Upon examining whether
transitions into certain states also characterised these relationships, we found a
similar profile of results. Transitions into states 4 and 6 had positive associations with
cognition, whereas the opposite was true for state 3.
The spatial topographies of these states may provide some insight into the reason for
this pattern of effects. One benefit of using MEG data to study these relationships is
its high temporal resolution, which allows one to record patterns of neural activation
and suppression (inhibition) that occur in very short time-windows. States 4 and 6, for
instance, were primarily characterised by fronto-parietal activation and suppression,
whereas state 3 was dominated by suppression across the default-mode network. As
mentioned in the Introduction, previous research indicates that the emergence and
spatiotemporal segregation of fronto-parietal networks supports executive function
across development (Keller et al., 2023), whereas the overactivation and hyper-
integrated spatiotemporal patterning of the DMN is associated with cognitive and
behavioural difficulties (Cortese et al., 2012). The fact that both fronto-parietal
activation and suppression predicts increases in cognitive ability may relate to large-
scale coordination of brain activity performed by fronto-parietal networks (Marek &
Dosenbach, 2018; Chen et al., 2022). Indeed, Gu et al. (2020) found that transitions
to, and between, different task-positive states was positively related to performance
on a cognitive task.
While DMN suppression has previously been viewed as a process that optimises for
goal-directed cognition (Anticevik et al., 2012; Leonards et al., 2023), it is possible that
an increased need to suppress DMN activity could also be viewed as a hindrance to
cognitive functioning. In the current study, we found that the time spent within states
corresponding to DMN activation and suppression was negatively associated with
cognitive ability. The same was true of transitions from other states into the DMN-
suppression state, which suggests that broad differences in DMN control may
contribute to cognitive differences in childhood. To build a coherent theoretical model
of the directionality of these effects, future studies of HMM states in childhood should
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aim to collect a wider breadth of data, and to assess how the activity of these states
changes throughout development.
6.3 Limitations
The primary limitations of the current study are its small sample size and relatively
constrained age range, which could account for why we did not find any effects of age
or gender on HMM state properties. Our sample size was reduced from n=52 to n=46
due to exclusions based on data quality, which is a common difficulty within research
in developmental cognitive neuroscience. While the size of our dataset was sufficient
for inferring robust HMM states, it is possible that we would have been able to explore
more granular neurodevelopmental effects had we had access to a larger sample. In
the same vein, a sample that included a wider age range would have enabled us to
investigate how neural dynamics change over time, rather than how they exist at one
phase of development. We encourage future research in this area to build upon our
findings with these core limitations in mind.
In this study, we also used a 38-node parcellation developed by Colclough et al. (2015)
and subsequently used in other studies that applied HMM to MEG data (e.g. Colclough
et al., 2017; Quinn et al., 2018). We chose this parcellation because the effective
dimensionality of MEG data in source-space is approximately 60-70 (Quinn et al.,
2018; Farahibozorg et al., 2018), and the number of parcels should be less than the
rank of the data in order for corrections for signal leakage to work. Although using this
parcellation allowed us to infer an HMM, it also reduced the spatial resolution at which
we were able to observe neural effects.
7 Conclusion
Using a multivariate Gaussian Hidden Markov Model, we inferred a seven-state model
of resting-state brain activity in a developmental sample. We found spatial and
temporal differences between each of the states identified in our model. Entropy-
related metrics of dynamic neural activity were positively associated with cognitive
ability. Particular states provided more clarity about the nature of these relationships;
DMN states were negatively associated, and fronto-parietal states were positively
associated, with cognitive ability in our sample.
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