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A Hierarchical Latent Space Network Model for Population Studies of Functional Connectivity

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Abstract

A major challenge in network neuroscience lies in understanding the organizational principles of the brain at different spatial scales. The brain is highly modular, in that brain regions naturally divide into densely connected subnetworks, which often themselves contain densely connected subnetworks. Modeling these complex hierarchies is a major technical challenge currently inhibiting progress in the field. We develop the hierarchical latent space model (HLSM) that can capture hierarchy at both the individual and population levels, account for multiple predictors of functional connectivity, and account for individual heterogeneity that manifests over a population. We apply several specifications of our model to healthy and paranoid schizophrenia patients collected from the Center for Biomedical Research Excellence project. We find that for both healthy and patient groups, the spatial location of two regions, in hemisphere and functional subnetwork, strongly influence their propensity to connect. We also find that alone, the spatial distance between two regions is significantly and inversely related to their connection probability, but that it is no longer significant once hemisphere and subnetwork locations have been controlled for. The HLSM also identifies increased heterogeneity in the connectivity of the healthy individuals over the patient group, suggesting a difference in overall connectivity patterns between the two populations.

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Funding

JDW was supported by NSF grant DMS-1830547. SJC was supported by NSF grants SES-1357622, SES-1461493, and SES-1514750, NIH R-34 DA043079, and the Alexander von Humboldt Foundation’s Fellowship for Experienced Researchers. ZLL was supported by NSF- SMA 1533500. The authors have intellectual property considerations for using cGERGM in a medical context. SJC, JDW, and ZLL are co-founders and equity holders in Cerenetics, Inc., a start-up company commercializing the technology described in this paper.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of San Francisco, San Francisco, CA, 94117-1080, USA

    James D. Wilson

  2. Department of Political Science, The Ohio State University, Columbus, OH, 43210, USA

    Skyler Cranmer

  3. Division of Arts and Sciences, NYU Shanghai, 1555 Century Avenue, Shanghai, 200122, China

    Zhong-Lin Lu

  4. Center for Neural Science and Department of Psychology, New York University, New York, NY, 10003, USA

    Zhong-Lin Lu

  5. NYU-ECNU Institute of Cognitive Neuroscience at NYU Shanghai, Shanghai, 200062, China

    Zhong-Lin Lu

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The authors have intellectual property considerations for using cGERGM in a medical context. SJC, JDW, and ZLL are co-founders and equity holders in Cerenetics, Inc., a start-up company commercializing the technology described in this paper.

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Appendix: The Effect of Thresholding on Coefficient Estimators

Appendix: The Effect of Thresholding on Coefficient Estimators

In the application study presented in “Results”, we tested the HLSM on unweighted networks that were thresholded so that binary edge values represented correlation values above 0.50. Here, we investigate the effect of the choice of threshold on the coefficient estimators that we obtain from the HLSM. To test this, we refit the Full Model unweighted HLSM on both groups, the 71 healthy individuals, and the 24 patients with paranoid schizophrenia across threshold values 0.3, 0.35,…, 0.70. For each threshold value, we fit the full model to each individual and report the coefficient estimators for the subnetwork effect, the hemisphere effect, and the spatial distance effect. These results are shown in Fig. 7.

Fig. 7
figure 7

Coefficient estimator distributions for the schizophrenia and healthy controls group across threshold values. For each threshold value τ, patient correlation networks were first binarized by setting all correlations above Tau to 1 and keeping the remaining values 0. The full model HLSM was subsequently fit to each network. The distribution of coefficient estimators across patients is shown for each threshold value

Full size image

Figure 7 suggests two important patterns. First, across threshold values, the coefficient estimators keep the same sign and statistical significance. This reveals that the interpretation discussed in “Results” is robust across values of the threshold. Secondly, there are noticeable trends in the coefficient estimators across threshold values. For example, the subnetwork effect tends to increase as the threshold increases, suggesting that as the network becomes more sparse (and contains fewer edges), the subnetwork community structure of the network has a stronger effect on connection probability within the population. It also appears that as the threshold increases, the variability in the estimators for the hemisphere and spatial distance effects increase. This trend intuitively suggests that estimator variability is inversely related to the number of edges in the network.

This investigation supports the analysis and discussion of the application in “Results” and provides some intuition as to the effect of thresholding on fitting the unweighted HLSM. In the future, a comparison between these unweighted models and the weighted HLSM for correlation networks should be studied. We plan to pursue this in future work.

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Wilson, J.D., Cranmer, S. & Lu, ZL. A Hierarchical Latent Space Network Model for Population Studies of Functional Connectivity. Comput Brain Behav 3, 384–399 (2020). https://doi.org/10.1007/s42113-020-00080-0

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