A Predictor-Informed Multi-Subject Bayesian Approach for Dynamic Functional Connectivity Jaylen Lee1 Sana Hussain2 Ryan Warnick6 Marina Vannucci3 Isaac Menchaca2 Aaron R.

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A Predictor-Informed Multi-Subject Bayesian Approach for

Dynamic Functional Connectivity

Jaylen Lee1, Sana Hussain2, Ryan Warnick6, Marina Vannucci3, Isaac Menchaca2, Aaron R.

Seitz4, Xiaoping Hu2, Megan A. K. Peters2,5, and Michele Guindani7

1Department of Statistics, University of California, Irvine

2Department of Bioengineering, University of California Riverside

3Department of Statistics, Rice University

4Department of Psychology, University of California Riverside

5Department of Cognitive Sciences, University of California Irvine

6Microsoft Security Research

7Department of Biostatistics, University of California, Los Angeles

January 11, 2023

Abstract

Dynamic functional connectivity investigates how the interactions among brain regions vary over

the course of an fMRI experiment. Such transitions between diﬀerent individual connectivity

states can be modulated by changes in underlying physiological mechanisms that drive functional

network dynamics, e.g., changes in attention or cognitive eﬀort. In this paper, we develop a multi-

subject Bayesian framework where the estimation of dynamic functional networks is informed by

time-varying exogenous physiological covariates that are simultaneously recorded in each subject

during the fMRI experiment. More speciﬁcally, we consider a dynamic Gaussian graphical model

approach where a non-homogeneous hidden Markov model is employed to classify the fMRI time

series into latent neurological states. We assume the state-transition probabilities to vary over

time and across subjects as a function of the underlying covariates, allowing for the estimation of

recurrent connectivity patterns and the sharing of networks among the subjects. We further assume

sparsity in the network structures via shrinkage priors, and achieve edge selection in the estimated

graph structures by introducing a multi-comparison procedure for shrinkage-based inferences with

Bayesian false discovery rate control. We evaluate the performances of our method vs alternative

approaches on synthetic data. We apply our modeling framework on a resting-state experiment

where fMRI data have been collected concurrently with pupillometry measurements, as a proxy

of cognitive processing, and assess the heterogeneity of the eﬀects of changes in pupil dilation

on the subjects’ propensity to change connectivity states. The heterogeneity of state occupancy

across subjects provides an understanding of the relationship between increased pupil dilation and

transitions toward diﬀerent cognitive states.

1 Introduction

Functional connectivity (FC) has emerged as one of the most active research areas in functional mag-

netic resonance imaging (fMRI). The purpose of FC studies is to characterize the undirected statistical

arXiv:2210.01281v3 [stat.ME] 9 Jan 2023

dependencies between brain regions and thus gain a greater understanding of brain functioning (Fris-

ton et al., 1994; Hutchison et al., 2013). Simple approaches to studying FC rely on readily available

measures of temporal correlation, such as the partial correlations between two brain regions after

conditioning upon all other regions (Fornito et al., 2013; Friston, 2011). Such metrics assume that

interactions between brain regions are constant in space and time throughout the fMRI session (static

connectivity, Li et al., 2008). Rather, neuroscientists have become increasingly aware that functional

connectivity is dynamic and ﬂuctuating, i.e. non-stationary, and that studying the dynamics of FC

is crucial for improving our understanding of human brain function (Hutchison et al., 2013; Vidaurre

et al., 2017; Lurie et al., 2020). The term “chronnectome" has been introduced to describe the growing

focus on identifying time-varying, but reoccurring, patterns of coupling among brain regions (Calhoun

et al., 2014).

Recent studies have highlighted how subjects are more likely to experience particular connectivity

states when some underlying physiological conditions are present. For example, Chand et al. (2020)

have investigated the association between heart rate variations and FC. Similarly, in a sleep fMRI

study, El-Baba et al. (2019) have shown that transitions between connectivity states slow as subjects

fall into deeper sleep stages. As a further example, Kucyi et al. (2017) have described how connectivity

dynamics are associated with attentiveness in a pencil-tapping task. These studies, among others, have

motivated the need for models that provide a better understanding of how the transitions between

diﬀerent functional connectivity states are modulated by internal or external conditions measured

during the course of an experiment. In the experimental study we consider in this manuscript, we have

available fMRI data collected together with pupillometry measurements. Pupil dilation has become

increasingly popular in cognitive psychology to measure cognitive processing and resource allocation. It

is believed that the changes in pupil diameter reﬂect brain state ﬂuctuations driven by neuromodulatory

systems (Sobczak et al., 2021). For example, the pupil dilates more under conditions of higher attention

(Siegle et al., 2003). Thus, pupil dilation measurements can be seen as an index of eﬀort exertion,

task demand, or diﬃculty in an fMRI experiment (van der Wel and van Steenbergen, 2018). Thus, it

is of interest to understand how pupil dilation is associated with an increased probability of particular

connectivity states experienced by a subject during an experiment (Martin et al., 2021).

Many of the commonly used approaches for studying dynamic connectivity rely on multi-step infer-

ences. For example, in Calhoun et al. (2014) the fMRI time courses are ﬁrst segmented by a sequence

of sliding windows, and then precision matrices are estimated in each window. Finally, k-means clus-

tering methods are used to identify re-occurring patterns of FC states. Post-hoc analyses may be

employed to assess the association of the estimated dynamic connectivity states with other available

measurements, like pupil dilation measurements (Haimovici et al., 2017). However, the arbitrary choice

of the window length and the oﬀset may lead to spurious dynamic proﬁles and poor estimates of cor-

relations for each brain state (Lindquist et al., 2014; Shakil et al., 2016). Improvements were proposed

by Cribben et al. (2012, 2013) and Xu and Lindquist (2015), who developed change point detection

methods to recursively partition the fMRI time series into stable contiguous segments where networks

of partial correlations are estimated by employing the graphical lasso of (Friedman et al., 2008). These

methods do not require pre-specifying the segment lengths and can detect the temporal persistence

of the functional networks. However, they do not account for the possibility of states being revisited

and hence do not conform to the idea that the chronnectome exhibits recurrent patterns of dynamic

coupling between brain regions of interest (ROIs).

Other model-based approaches to dynamic connectivity consider the set of ROIs as the nodes (or

vertices) of an underlying graph and employ homogeneous hidden Markov models (HMMs) to detect

state transitions and infer a discrete set of latent connectivity states over time.Warnick et al. (2018)

develop a Bayesian HMM to model dynamic FC as the transition between state-speciﬁc graphs, each

graph being related to others via an underlying super-graph. Sourty et al. (2016) use product HMMs

to describe the evolution of the sliding-windows correlations and capture temporal dependencies across

resting-state networks. Chiang et al. (2015) used a Bayesian HMM to estimate the dynamic structure

of graph theoretical measures of whole-brain FC. Also, HMMs have been employed in time-varying

vector autoregressive (VAR) modeling frameworks for whole-brain resting state connectivity, where

the VAR coeﬃcients and the innovation covariance matrix are allowed to change with the latent states

(Vidaurre et al., 2017; Ting et al., 2018; Ombao et al., 2018). However, these implementations of

hidden Markov models typically assume that the probabilistic model underlying the state transitions

is constant throughout an experiment. Crucially, such a homogeneity assumption does not allow to

assess the modulatory eﬀect of time-varying physiological factors on the transitions, e.g. how changes

in vigilance measured via pupil dilation can impact state transitions (Lurie et al., 2020).

In this paper, we develop a multi-subject Bayesian framework where the estimation of dynamic

functional networks is informed by time-varying exogenous physiological covariates that are simul-

taneously recorded in each subject during the fMRI experiment. More speciﬁcally, we introduce a

multi-subject non-homogeneous HMM modeling framework where the transition probabilities between

states are shared between subjects and vary over time as a fucntion of the covariates. Our setting

allows for the estimation of unique connectivity state transitions for each subject. It also permits

group-based inferences, via recurring connectivity patterns and sharing of networks among the sub-

jects. With respect to the multi-step inference strategies described above, in our approach both the

dynamic connectivity states and their association with the physiological measurements are estimated

in a single modeling framework, accounting for all uncertainties. Kundu et al. (2018) have recently

proposed a two-step multi-subject fused-lasso approach for detecting change points in functional con-

nectivity. Diﬀerently from their proposal, our method does not assume that the experimental design

and the timing of the change points between connectivity states are shared among all subjects, and can

therefore be applied to more general experimental designs. Indeed, our approach allows for diﬀering

state transition behavior across multiple subjects by modeling the state transition parameters hierar-

chically. Our modeling approach further assumes sparsity in the network structures, by assuming a

shrinkage prior on the connectivity networks. Additionally, we propose a strategy for edge selection

that combines the posterior shrinkage-informed thresholding approach of Carvalho et al. (2010) with

the Bayesian False Discovery Rate controlling method of Müller et al. (2006).

We apply our modeling framework to a resting-state experiment where fMRI data have been col-

lected concurrently with pupillometry measurements, leading us to assess the heterogeneity of the

eﬀects of changes in pupil dilation on the subjects’ propensity to change connectivity states. Changes

in pupil diameter have been linked to attention and cognitive eﬀorts modulated by the activity of

norepinephrine-containing neurons in the locus coeruleus (LC). For example, Joshi et al. (2016) have

shown that LC activation predicts changes in pupil diameter that either occur naturally or are caused

by external events during near ﬁxation, as in many experimental tasks. Therefore, pupil dilation has

been used as a proxy for a metric of a person’s willingness to exert more eﬀort and involve a greater

mental eﬀort to complete a task. Recent methods for studying such association use a multi-step

approach, ﬁrst identifying switches in subjects’ state sequences and then calculating the diﬀerence

between the normalized pupil size before and after the estimated switch (see, e.g. Hussain et al., 2022).

In our application, we demonstrate how the model can recover expected change points in dynamic FC

states, as those states align quite well with the experimental events regulated by the behavioral task.

The rest of the paper is organized as follows. In section 2 we describe our proposed method and

edge selection procedure. We also give a brief synopsis of our Markov Chain Monte Carlo (MCMC)

approach to posterior inference. In section 3 we showcase our model performance on simulated data

and provide comparisons to the sliding window and homogenous HMM approaches. Lastly, in Section

4, we apply our model to the LC handgrip data with accompanying results and analysis. Section 5

concludes the paper with a discussion.

2 Methods

In this section, we describe our proposed predictor-informed multi-subject model for dynamic connec-

tivity. This is a single-step fully Bayesian approach that explicitly models the heterogeneity in the

dynamics of connectivity patterns across all subjects and – simultaneously – estimates the predictor

eﬀects on those dynamics. We achieve this by constructing a non-homogeneous Hidden Markov Model

(HMM) where the transition probabilities are functions of time-varying covariates.

2.1 An HMM model for dynamic functional connectivity

Let Yi

t= (Yi

t1, . . . , Y i

tR)Tdenote the vector of fMRI BOLD responses measured at time tin R regions of

interest (ROIs), t= 1, . . . , T on subject i= 1, . . . , N. We adopt a Gaussian graphical model framework,

and assume multivariate normality of the bold signals, that is Yi

t∼NR(µi

t,Ω−1,i

t), where µi

tis a mean

regression term and Ωi

tindicates a time-varying precision matrix, i.e. the inverse covariance matrix

at each time point. In graphical models, the zeros of the precision matrix correspond to conditional

independence; that is, if an oﬀ-diagonal element ωjkt = 0,j, k = 1, . . . , R, j 6=k, then the signals Yi

and Yi

tk(j6=k)are conditionally independent. The mean term µi

tcan be speciﬁed as a general linear

model (Friston, 1994) to capture activation patterns over time, as done for example in Warnick et al.

(2018). Here, however, since we are interested in capturing connectivity patterns through the modeling

of the time-varying precision matrix, we assume without loss of generality that the BOLD signal has

been mean-centered by removing any estimated trend and activation component. This “detrending” is

not uncommon for studying FC, especially for task-based fMRI data, where the data are ﬁrst mean-

centered, to remove any systematic task-induced variance, and the analysis is then conducted on the

time series of the residuals (see, e.g. Fair et al., 2007).

We propose to model the dynamics of FC using an HMM framework with Slatent states character-

izing FC and the brain transitions during the fMRI experiment. Our formulation captures the hetero-

geneity of individual-speciﬁc patterns of connectivity over time, since each subject’s fMRI data may be

characterized by speciﬁc change points and evolution of the brain’s functional organization. However,

we assume that the connectivity patterns may also be re-occurring and they can possibly be shared

among the subjects, thus allowing for group-based inferences. Let (s1, . . . , sT)be a T-dimensional

vector of categorical indicators st, such that st=sif state sis active at time t,s= 1, . . . , S. Then,

we assume the data follow a Gaussian graphical model at time tof the type

t|si

t=s, Ωs∼NR(0,Ω−1,i

s), s = 1, . . . , S, (1)

with subject-level precision matrices which, at each time, are characterized by the values of one among

Sprecision matrices, identifying which state is active at that time. Model (1) therefore implies con-

nectivity networks that vary by subjects and by state.

2.2 Modeling connectivity transitions as a function of observed physiological fac-

tors

Many neuroscience experiments involve the simultaneous collection of fMRI data together with physio-

logical, kinematics and behavioral data (Wilson et al., 2020). For example, our motivating application

considers a handgrip task where pupillometry dilation data (i.e., measurements of pupil dilation sizes)

are concurrently recorded. Pupillary dilation is regarded as a surrogate measure for activity in the

locus coeruleus circuit, which plays a central role in many cognitive processes involving attention and

eﬀort, and it is considered the main source of the neurotransmitter noradrenaline, a chemical released

in response to pain or stress. Neuronal loss in the locus coeruleus is known to occur in neurode-

generative disorders such as Alzheimer’s disease and related dementias as well as Parkinson’s disease

dementia. It is therefore important to understand how brain dynamics may be diﬀerentially modulated

as a function of pupil dilation in diﬀerent subjects.

Here, we propose to model the dynamics of FC by developing a non-homogeneous HMM framework

where estimation is informed by subject-level time-varying exogenous physiological covariates, e.g.

physiological factors like the pupillary data in our motivating application. More in detail, we assume

that switches between states are regulated by transition probabilities that vary over time and across

subjects as a function of Btime-varying subject-level covariates as

rst =P(st+1 =s|st=r) = exp(ξi

rs +xiT

tρi

l=1 exp(ξi

rl +xiT

tρi

l), r, s = 1, . . . , S, (2)

where xi

tdenotes a B×1vector of covariate values for subject iat time t, and ρi

s= (ρi

s1, . . . , ρi

sB)is

the corresponding B×1vector encoding the eﬀect of each covariate on the probability of transitioning

to state sfor subject i. The parameter ξi

rs deﬁnes a baseline transition probability from state rto state

sfor subject i, that is the transition probability without any covariate eﬀect. To ensure identiﬁability,

we deﬁne a state as reference. Without loss of generality, we set s= 1 as the reference state, and also

set the coeﬃcients ρi

1b,b= 1, . . . , B, and ξi

1·,i= 1,...N equal to zero. Thus, the state transition

coeﬃcients are interpreted with respect to the reference state, and we can re-express (2) in terms of

the log-relative odds of the transition from state rto state scompared to the transition from state r

to the reference state 1,

log(Qi

rst

r1t

) = ξi

rs +xiT

tρi

s, r, s = 1, . . . , S. (3)

In this formulation, the transition coeﬃcients exp(ρi

sb),b= 1,...B, are more naturally interpreted

as the relative change in odds of transitioning to state scompared to transitioning to state 1, after

a one unit change in xi

tb, holding all other covariates as constant. Similarly, the coeﬃcient exp(ξi

rs)

is interpreted as the expected odds of transitioning from state rto scompared to transitioning from

state rto 1, when the time-varying covariates, xi

t, are 0 or at a baseline/average value.

We assume independent Gaussian priors for the transition parameters ρand ξ. We further allow

for sharing of information in estimating the state transition structure across subjects, by employing

a hierarchical modeling formulation for the state transition parameters. More speciﬁcally, we assume

that the individual coeﬃcients ξi

rs and ρi

sb,b= 1, . . . , B, vary around population-level means, Zrs and

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APredictor-InformedMulti-SubjectBayesianApproachforDynamicFunctionalConnectivityJaylenLee1,SanaHussain2,RyanWarnick6,MarinaVannucci3,IsaacMenchaca2,AaronR.Seitz4,XiaopingHu2,MeganA.K.Peters2,5,andMicheleGuindani71DepartmentofStatistics,UniversityofCalifornia,Irvine2DepartmentofBioengineering,Univers...

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A Predictor-Informed Multi-Subject Bayesian Approach for Dynamic Functional Connectivity Jaylen Lee1 Sana Hussain2 Ryan Warnick6 Marina Vannucci3 Isaac Menchaca2 Aaron R.

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