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Constructing Connectome Atlas by Graph Laplacian Learning

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Abstract

The recent development of neuroimaging technology and network theory allows us to visualize and characterize the whole-brain functional connectivity in vivo. The importance of conventional structural image atlas widely used in population-based neuroimaging studies has been well verified. Similarly, a “common” brain connectivity map (also called connectome atlas) across individuals can open a new pathway to interpreting disorder-related brain cognition and behaviors. However, the main obstacle of applying the classic image atlas construction approaches to the connectome data is that a regular data structure (such as a grid) in such methods breaks down the intrinsic geometry of the network connectivity derived from the irregular data domain (in the setting of a graph). To tackle this hurdle, we first embed the brain network into a set of graph signals in the Euclidean space via the diffusion mapping technique. Furthermore, we cast the problem of connectome atlas construction into a novel learning-based graph inference model. It can be constructed by iterating the following processes: (1) align all individual brain networks to a common space spanned by the graph spectrum bases of the latent common network, and (2) learn graph Laplacian of the common network that is in consensus with all aligned brain networks. We have evaluated our novel method for connectome atlas construction in comparison with non-learning-based counterparts. Based on experiments using network connectivity data from populations with neurodegenerative and neuropediatric disorders, our approach has demonstrated statistically meaningful improvement over existing methods.

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Notes

  1. It is worth noting that the parcellation of the AAL template is defined based on anatomical structure. Although there are still quite a few fMRI studies using the AAL template, the interest is shifting towards using functional atlases which follow the insight of brain functions. However, we demonstrate the common functional brain networks using both structural and functional parcellations in order to show the proposed computational method can work with different atlases.

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Acknowledgments

This work was supported in part by NIH R21AG059065 and K01AG049089.

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Authors and Affiliations

  1. Department of Computer Science, University of North Carolina at Greensboro, Greensboro, NC, 27402, USA

    Minjeong Kim

  2. Intelligent Information Processing Laboratory and School of Automation, Hangzhou Dianzi University, Zhejiang, 310018, Hangzhou, China

    Chenggang Yan & Defu Yang

  3. Department of Psychiatry, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA

    Defu Yang & Guorong Wu

  4. Department of Psychology, Capital Normal University, Beijing, 100073, China

    Peipeng Liang

  5. Department of Neurology, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA

    Daniel I. Kaufer

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  1. Minjeong Kim
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Correspondence to Guorong Wu.

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Appendix

Appendix

To solve Eq. 12, we need to determine the proximity operator for h1 and h2, and the derivative of h3 as follows:

$$ \left\{\begin{array}{c} pro{x}_{h_1}=\max \left(0,\theta -\eta \right)\\ {} pro{x}_{h_2}=s\\ {}\nabla {h}_3=\gamma \left(4\theta +2{K}^T K\theta \right)\end{array}\right. $$

Besides, we introduce a step-size variable τ ∈ (0, 1 + ζ + ‖K‖2) to control the convergence, where \( {\left\Vert K\right\Vert}_2=2\sqrt{\frac{N\left(N-1\right)}{2}} \). The primal dual algorithm for Eq. 12 is summarized below:

figure a

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Kim, M., Yan, C., Yang, D. et al. Constructing Connectome Atlas by Graph Laplacian Learning. Neuroinform 19, 233–249 (2021). https://doi.org/10.1007/s12021-020-09482-8

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