Modeling state-transition dynamics in resting-state brain signals by the hidden Markov and Gaussian mixture models

Abstract: Recent studies have proposed that one can summarize brain activity into dynamics among a relatively small number of hidden states and that such an approach is a promising tool for revealing brain function. Hidden Markov models (HMMs) are a prevalent approach to inferring such neural dynamics among discrete brain states. However, the impact of assuming Markovian structure in neural time series data has not been sufficiently examined. Here, to address this situation and examine the performance of the HMM, we compare the model with the Gaussian mixture model (GMM), which is with no temporal regularization and thus a statistically simpler model than the HMM, by applying both models to synthetic time series generated from empirical resting-state functional magnetic resonance imaging (fMRI) data. We compared the GMM and HMM for various sampling frequencies, lengths of recording per participant, numbers of participants, and numbers of independent component signals. We find that the HMM attains a better accuracy of estimating the hidden state than the GMM in a majority of cases. However, we also find that the accuracy of the GMM is comparable to that of the HMM under the condition that the sampling frequency is reasonably low (e.g., TR = 2.88 or 3.60 s) or the data is relatively short. These results suggest that the GMM can be a viable alternative to the HMM for investigating hidden-state dynamics under this condition.

• The paper demonstrates that HMMs uniquely capture longer state persistence compared to GMMs in high-frequency fMRI recordings.
• It employs large-scale empirical and synthetic data to quantify latent state estimation accuracy under various sampling conditions.
• The results advocate tailored model selection based on data resolution, balancing computational efficiency with decoding precision.

Comparative Modeling of Resting-State Brain Dynamics: HMM Versus GMM

Overview

The study "Modeling state-transition dynamics in resting-state brain signals by the hidden Markov and Gaussian mixture models" (2001.08369) presents a systematic comparison between two statistical frameworks—Hidden Markov Models (HMMs) and Gaussian Mixture Models (GMMs)—for estimating latent state dynamics underlying resting-state fMRI time series. The focus is on the conditions under which the Markov property assumed by HMMs provides a tangible estimation advantage over the temporally unregularized GMM. The analysis employs large-scale empirical fMRI data and controlled synthetic datasets matched on realistic acquisition parameters, offering quantifiable guidance on when each model is optimal for decoding hidden states in neural data.

Methodological Approach

The data comprise extensively preprocessed resting-state fMRI scans from the Human Connectome Project, with ICA-derived spatial components serving as the multi-dimensional observation space. Both HMM and GMM parameterization is restricted to two latent states per prior results on the functional relevance and heritability of such coarse-grain state parcellations.

Model fitting leverages standard EM and Baum–Welch inference for GMM and HMM, respectively, each initialized multiple times to mitigate local optima. Synthetic benchmarks are constructed by re-generating time series from fitted models, enabling the derivation of hard accuracy metrics defined as the correspondence of inferred state sequences to known ground truth. A suite of experiments explores the roles of sequence length, participant sample size, number of observed dimensions, and sampling frequency (TR), mirroring typical constraints in human neuroimaging studies.

Principal Results

The empirical comparison reveals that the latent state means and covariances inferred by GMM and HMM are closely matched, but HMM fitting leads to latent states with longer persistence (i.e., less frequent state flipping), consistent with Markovian memory incorporation. On synthetic data, the following regimes emerge:

• High-frequency, Longer Data: HMM consistently outperforms GMM in accurately recovering latent state sequences, particularly where TR is low (e.g., 0.72 s), producing accuracy saturating above 83% with moderate sample sizes ($n_{\text{pat}} \approx 20$).
• Low-frequency, Shorter Data: The estimation accuracy gap narrows, and GMM can match or surpass HMM, specifically at TR ≥ 2.88 s or small data settings. This effect is more pronounced as observation dimensionality ($N$) increases.
• Data Sufficiency and Scalability: For very short recordings or large $N$, GMM demonstrates greater robustness and computational expediency. HMM computational demands increase with data size and state cardinality.

Qualitative state sequence characteristics (e.g., state dwell time distributions) differ: GMM yields heavy-tailed, rapidly switching sequences; HMMs impose peaked dwell time distributions, leading to more temporally coherent states. The performance landscape is robust across both GMM- and HMM-generated synthetic data, although HMMs retain an edge when the true generative process is Markovian.

Implications

The results directly inform model selection for researchers analyzing multivariate neural time series, particularly in the context of fMRI. HMMs provide an informative inductive bias in high-temporal-resolution, high-sample scenarios, favoring their adoption as TRs decrease in modern acquisitions. Conversely, for lower-frequency data and studies with limited temporal or subject sampling, GMMs are not only computationally preferable but also yield comparable or improved classification of latent brain states due to reduced overfitting propensity.

The findings also re-emphasize a broader theoretical point: model selection for neural state decoding is nontrivial and context dependent. The absence of substantial Markovian dependency in low-resolution data undermines the principal advantage of HMMs. This highlights the need for tailored model application, potentially motivating automated or cross-validated model selection strategies specific to acquisition regimes and experimental design.

Future Directions

Potential extensions include examining model performance as the number of latent states increases, integrating Bayesian nonparametric or semi-Markovian generalizations to better accommodate unknown or non-exponential state dwell times, and leveraging task-based or behavioral annotation to constrain state inference. Critically, as acquisition technology advances and high-frequency large-N datasets become standard, focus should shift toward scalable HMM inference, improved initialization, and hybrid models incorporating both Markovian and non-Markovian patterns.

Conclusion

This study provides an authoritative quantitative framework for deciding between HMM and GMM for hidden-state inference in resting-state fMRI, centering the decision on data resolution, sample size, and computational tractability. The work sets benchmarks for future comparative evaluations, justifying routine consideration of GMM as a baseline and defining when HMMs deliver additional value in capturing the temporal organization of brain dynamics (2001.08369).