Overview

The Hierarchical Bayesian Parcellation framework is designed to derive a probabilistic brain parcellation from multiple fMRI datasets. The central quantity that the framework is the brain parcellation for each individual subject, \(\mathbf{U}^s\). The framework is partitioned into a spatial arrangement model, \(p(\mathbf{U}|\theta_A)\), the probability for a voxel to belong to a specific parcel (across the population), and a collection of dataset-specific emission models, \(p(\mathbf{Y}^{s,n}| \mathbf{U}^s;\theta_{En})\), the probability of each observed dataset given the individual brain parcellation. The individual brain parcellations can be then estimated as the posterior probability of \(\mathbf{U}^s\), given the arrangement and the emission models. The integration, message passing, and parameter estimation are implemented in the FullMultiModel class.

The framework can be used for two main purposes:

• To learn a new probabilistic brain parcellations across multiple fMRI datasets. For this purpose, you will build a FullMultiModel with one arrangement model and multiple emission models. The parameters of both arrangement model and emission models will be simultaneously estimated using the EM-algorithm. For more information, see the Atlas Training Example. In our papers, we show that the integration of multiple datasets can dramatically improve the quality of brain parcellations.

• To use an existing probabilistic atlas to obtain individualized brain parcellations for new subjects through the optimal integration of individual localizer data and the group atlas. For this purpose, you will build a FullMultiModel from one arrangement model and (typically) a single emission model, which models your specific localizer data. The parameters of the emission model will be estimated using the EM-algorithm, while the arrangement model will be frozen. For more information, see the Individual Parcellation Example. In Zhi et al. (2023), we show that the integration of as little as 10 min of individual data with the group atlas substantially improves the predictive power of the brain parcellation.

Arrangement Model

arrangements.py contains the implementation of different arrangement model classes used in the framework. The main types spatial arrangement models are:

• ArrangementModel: The base class for all arrangement models, which inherits from the Model class.

• ArrangeIndependent: The Independent arrangement model, which assumes that the brain locations are spatially independent. This is the arrangement model used in the Zhi et al. (2023).

• ArrangeIndependentSymmetric: The Independent symmetric arrangement model, which assumes that corresponding brain locations in the left and right hemisphere are assigned to corresponding parcels. While the boundaries are forced to be symmetric, the functional profiles for the left and right parcel are being estimated separately, which allows the study of functional lateralization (Nettekoven et al., 2024).

• ArrangeIndependentSeparateHem: An spatially independent arrangement model, which also constrains that there are matched pairs of parcels for regions in the left and right hemisphere. In contrast to the symmetric model, the boundaries are not constrained to be the same across the hemispheres. Used for the asymmetric version of our cerebellar atlas (Nettekoven et al., 2024).

• PottsModel: A potts model (Markov random field on multinomial variable) with K possible states. In our framework, we use this arrangement model only to simulate realistically looking brain parcellation maps. Due to computational requirement of Gibbs sampling, it is currently not used for inference.

• cmpRBM: A convolutional multinomial (categorial) restricted Boltzman machine for learning of brain parcellations for probabilistic input. It uses variational stochastic maximum likelihood for learning. Described in the 4th chapter of the Da Zhi’s dissertation (2023).

Emission Model

emissions.py contains the implementation of different emission model classes to calculate the data likelihood given the individual brain parcellation. The main active emission models are:

• EmissionModel: The base class for all emission models, which inherits from the Model class.

• MultiNomial: An emission model for discrete (label) data, such as a winner-take-all map of some characteristic of an individual subject (i.e. individualized resting-state networks).

• MixGaussian: The Gaussian mixture emission model with isotropic noise. We show in Zhi et al. (2023) that this emission model does not perform very well on task-based fMRI data.

• MixVMF: The von Mises-Fisher mixture emission model, which models the directionality of the functional profiles, but ignores the magnitude of the signal. The von Mises-Fisher distribution is use for data located on a hypersphere (vectors of unit length). This is the recommended emission model both for task-based and resting-state fMRI data.

Full Model

full_model.py contains the implementation of the FullMultiModel class that combines the arrangement and emission models. The class have the learning and inference details for different arrangment and emission models combination. The main methods are:

• Estep(): Calculates the posterior probability of ..math:mathbf{U}^s (the probabilistic brain parcellations) given the data and the current model parameters.

• fit_em(): Alternates between E-step and M-step (parameter update) to optimize the model parameters. When fitting the an emission model with a fixed arrangement model (for individual parcellation), a single optimization run is sufficient, as the algorithm typically converges to global maximum.

• fit_em_ninits(): When fitting both emissions and arrangement models simultaneously (when learning a new probabilistic atlas), local minima become a problem. This functions therefore runs the EM-algorithm starting with n_inits random initializations. It escapes local maxima by selecting the model with the highest likelihood after first few iterations. Check the paper for more algorithmic details.

Scope and related repositories

The HierarchBayesParcel repository implements the computational side of the hierarchical Bayesian parcellation framework. In the interest of making this toolbox as modular as possible, we do not provide the tools to extract the individual subject data in a group atlas space or to project the parcellations back into the volume of the surface.

For the illustrative examples, we are using the Functional_Fusion repository to import the preprocessed data as the input to the framework.

The analyzes and simulations reported in Zhi et al. (2023), can be replicated using the FusionModel repository.