Elsevier

NeuroImage

Volume 191, 1 May 2019, Pages 243-257
NeuroImage

Improved state change estimation in dynamic functional connectivity using hidden semi-Markov models

https://doi.org/10.1016/j.neuroimage.2019.02.013Get rights and content

Highlights

  • The HSMM and HMM perform similarly when estimating dynamic brain states.

  • The HSMM is more flexible than the HMM when estimating sojourn times.

  • The HMM and HSMM estimate different empirical sojourn distributions in rs-fMRI.

  • Multidimensional scaling reveals rs-fMRI states are reproducible.

  • Sojourn distributions are associated with sustained attention scores.

Abstract

The study of functional brain networks has grown rapidly over the past decade. While most functional connectivity (FC) analyses estimate one static network structure for the entire length of the functional magnetic resonance imaging (fMRI) time series, recently there has been increased interest in studying time-varying changes in FC. Hidden Markov models (HMMs) have proven to be a useful modeling approach for discovering repeating graphs of interacting brain regions (brain states). However, a limitation lies in HMMs assuming that the sojourn time, the number of consecutive time points in a state, is geometrically distributed. This may encourage inaccurate estimation of the time spent in a state before switching to another state. We propose a hidden semi-Markov model (HSMM) approach for inferring time-varying brain networks from fMRI data, which explicitly models the sojourn distribution. Specifically, we propose using HSMMs to find each subject's most probable series of network states and the graphs associated with each state, while properly estimating and modeling the sojourn distribution for each state. We perform a simulation study, as well as an analysis on both task-based fMRI data from an anxiety-inducing experiment and resting-state fMRI data from the Human Connectome Project. Our results demonstrate the importance of model choice when estimating sojourn times and reveal their potential for understanding healthy and diseased brain mechanisms.

Introduction

Interest in studying brain networks, where the brain is viewed as a system of interacting regions (nodes) that produce complex behaviors, has grown tremendously over the past decade. Networks have become a popular approach towards illustrating both the physiological connections (structural networks) and the coupling of dynamic brain activity (functional networks) linking different areas of the brain. It is within this paradigm shift that scientists have begun investigating how networks behave in healthy brains and how they are altered in neurological and psychiatric disorders (Stam, 2014).

The study of functional connectivity (FC), or the undirected association between two or more fMRI time series, has come to the forefront of research efforts in the field of neuroimaging. Studies have revealed information about the functional connections between different brain regions and local networks and have provided significant insights into the organization of functional communication in the brain. Brain networks can be created by performing a FC analysis, where the strength of the relationship between nodes is assessed using the functional magnetic resonance imaging (fMRI) time series associated with each node. The nodes can consist of individual voxels, (Handwerker et al., 2012; Hutchison et al., 2013; Leonardi and Van De Ville, 2015), pre-specified regions of interest (ROIs) (Chang and Glover, 2010), or sets of regions estimated using independent component analysis (ICA) (Allen et al., 2014). The relationship between nodes can be quantified using a variety of metrics, and we present the networks in this paper using both Pearson and partial correlation, as both are popular approaches.

To date, much of the brain connectivity analyses performed estimate one network state that does not change over time during a specific experimental scanning session. In other words, the network summarizes an aggregate state over time. However, recently there has been a growing interest in studying changes in connectivity over shorter time periods, i.e. on the second-to-minute scale (Hutchison et al., 2013; Calhoun et al., 2014). It is now generally believed that individuals transition in and out of different brain states during the course of the scanning session. Thus, there is a need to estimate the timing of the transitions between states, as well as the structure of the states themselves. Several approaches towards assessing time-varying connectivity have been suggested. These include using sliding window correlations (Allen et al., 2014), change point models (Cribben et al., 2012, 2013; Xu and Lindquist, 2015), wavelet transform coherence (Chang and Glover, 2010), and time series models (Lindquist et al., 2014; Choe et al., 2017).

There is a recent literature on Markov switching vector autoregressive (SVAR) models and dynamic connectivity. Samdin et al. (2017) developed a framework in which the dynamic connectivity systems are characterized by distinct VAR processes and allowed to switch between brain states, where the state evolution and directed dependencies are described by a Markov process and the SVAR parameters. They demonstrate that their approach is able to detect change points and provide reliable estimates of effective connectivity. In addition, Ting et al. (2018) propose a Markov-switching dynamic factor model, allowing for the dynamic network states to be driven by latent factors. A regime-switching SVAR factor process quantifies the dynamic effective connectivity. Lastly, Ombao et al. (2018) present approaches to modeling dynamic connectivity using EEG data recorded across replicated trials in an experiment.

In recent years, Hidden Markov models (HMMs) have also proven to be a useful modeling approach towards assessing FC. In this context, it is assumed that the time series data at each brain region can be described via a series of a fixed number of hidden/unknown brain states. Each state is characterized by a multivariate Gaussian distribution, which is parameterized by the mean and covariance. Of particular interest is the unique connectivity structure of each state, encapsulated within the covariance matrix.

HMMs (Baum et al., 1970) are a natural modeling approach to take, given that we have an observable sequence from a system in which it is not unreasonable to believe that the hidden states may be governed by a Markov process. They have been widely studied in statistics (Ephraim and Merhav, 2002) and have been applied extensively in many applications, such as in speech recognition and biological sequence analysis (Gales and Young, 2008; Yoon, 2009). In the context of dynamic FC, Eavani et al. (2013) introduced a HMM framework to uncover hidden states in resting-state fMRI data. They simultaneously modeled the underlying co-varying regions of interest in each state as a set of sparse rank-one basis matrices, such that non-negative combinations of these basis matrices act as priors for each of the HMM covariance matrices. Vidaurre et al. (2017) also utilized a HMM based analysis to discover a hierarchically organized temporal nature of resting-state networks. They showed that resting-state brain networks are organized into two metastates, where states within a metastate are highly likely to transition to other states within the same metastate. The authors provide evidence that the metastates are consistent across subjects, associated with behavior, and heritable. Moreover, Baker et al. (2014) applied HMMs to band-limited amplitude envelopes of source reconstructed MEG data and identified brain states that match up to previously established resting-state networks and which fluctuate at time scales two orders of magnitude faster than previously shown. Warnick et al. (2018) introduce a Bayesian approach to HMMs and dynamic functional connectivity, where they identify latent network states, but they do so in a manner where they assume the network/connectivity structures at each time point are related within one “super-graph”. They impose a sparsity inducing Markov random field prior on the presence of the edges in the super-graph.

A HMM approach to dynamic FC analysis is appealing for several reasons. First, the framework is able to handle large amounts of data to infer the network states, as well as the other model parameters such as the probabilities of transitioning between states. Moreover, these states and parameters can be estimated in a relatively computationally inexpensive manner. Second, while network states are estimated at the group level, individuality is respected in that information about when a state becomes active is subject-specific. Therefore, we are still able to obtain subject-specific estimates of an individual's state sequence and the amount of time he/she spends in each state. Third, the method is likelihood based, leaving room for model selection techniques to be employed when deciding upon the number of states to be estimated, as well as the number of brain regions to be included in the analysis. Fourth, the model is able to capture quick changes in brain connectivity, unlike the sliding window based approach.

Unfortunately, with all of the aforementioned strengths, comes a limitation and potentially unrealistic assumption of the HMM approach, which is that the sojourn time, the number of consecutive time points spent in a specific state, is geometrically distributed. A property of the geometric distribution is that the probability decreases as the sojourn time increases. Therefore, more weight is placed on shorter consecutive time points in a given state. This may not be appropriate in the context of functional connectivity analysis, as it suggests subjects are switching states very often, as opposed to potentially spending longer periods of time in the same state. For example, in some task-based fMRI experiments, participants are expected to spend several minutes in a particular brain state. It also suggests that the sojourn time for all states follow a very similar pattern, which may not be the case in practice.

A possible solution to this issue is to explicitly model and estimate the state sojourn distributions via Hidden semi-Markov models (HSMMs) (Yu, 2010). HSMMs differ in that the sojourn distributions are explicitly built into the likelihood function. One may specify the distributions to take a nonparametric form or a parametric form (such as a Poisson or Gamma distribution) and estimate the parameters directly. A more detailed comparison of HMMs vs. HSMMs can be found in Sections 2.4 Hidden Markov modeling, 2.5 Hidden semi-Markov modeling.

In this work, we propose using a HSMM approach to infer dynamic FC networks from fMRI data. We are particularly interested in modeling group level data in order to estimate several network states that are common across subjects in a particular study. Moreover, the parameter estimates we obtain from fitting the HSMM allow us to find each individual subject's most probable series of network states along with the amount of time he/she is in each state. The estimated model also lends additional information, such as the probabilities of transitioning from one network state to another, the sojourn distribution of each state, and the initial state probabilities.

We first conduct a simulation study, revealing that while both the HMM and HSMM are able to identify quick state switches with similar accuracy, the HSMM is superior to the HMM with regards to estimating sojourn times with fewer switches. This suggests that the HSMM is a more flexible model and the better model choice when estimating sojourn times and subject state sequences for data where subjects aren't expected to switch states at a rapid rate. We then demonstrate our approach on an anxiety-inducing experiment (Lindquist et al., 2007; Lindquist and McKeague, 2009), in which the algorithm was agnostic to alignment, and yet discovered the alignment near perfectly. These results suggest reliability of the method, which is quite encouraging for moving forward with analyzing resting-state data. Lastly, we present dynamic FC results using resting-state fMRI data from the Human Connectome Project (HCP), where we identify several states associated with different degrees of connectedness within and between sensory, motor, and higher order cognition regions. Moreover, we find that the sojourn distributions of several states are associated with scores on a sustained attention task, highlighting how informative they may be in understanding healthy and diseased brain mechanisms. Throughout the paper we make comparisons to the traditional HMM approach, illustrating that results can differ depending on which method is used.

Section snippets

Anxiety inducing experiment data and preprocessing

We fit our HMM and HSMM on 23 participants who performed an anxiety-inducing task while in the fMRI scanner (Denny et al., 2013). The experiment, approved by the University of Michigan institutional review board, was conducted in accordance with the Declaration of Helsinki.

The experimental paradigm was an off-on-off design, with an anxiety-provoking task occurring between two resting periods. In particular, participants were informed prior to the scan that they were to be given 2 min to prepare

Simulation study

A standard HMM, a HSMM with a nonparametric sojourn density, and a HSMM with a Poisson sojourn density were fit to the three simulated data sets described in Section 2.3, with the goal of assessing how accurate each model is in estimating the true transition matrix, sojourn densities, and states. Fig. 2 presents the four true states from which resting-state fMRI data were simulated in all three data sets, as well as the states estimated from each data set. In all three data sets, the results

Discussion

The study of functional connectivity (FC), or the undirected association between two or more fMRI time series, has come to the forefront of research efforts in the field of neuroimaging. While many of the brain connectivity analyses performed to date estimate a single static network structure for the length of time a subject is scanned, there is growing evidence that valuable information is obtained by estimating several networks over shorter time periods. In other words, it is desirable to

Acknowledgements

This work was supported by NIH grants P41 EB015909 and R01EB016061, as well as the Johns Hopkins University Provost's Postdoctoral Fellowship Program.

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