The missing link: Predicting connectomes from noisy and partially observed tract tracing data

Our understanding of the wiring map of the brain, known as the connectome, has increased greatly in the last decade, mostly due to technological advancements in neuroimaging techniques and improvements in computational tools to interpret the vast amount of available data. Despite this, with the exception of the C. elegans roundworm, no definitive connectome has been established for any species. In order to obtain this, tracer studies are particularly appealing, as these have proven highly reliable. The downside of tract tracing is that it is costly to perform, and can only be applied ex vivo. In this paper, we suggest that instead of probing all possible connections, hitherto unknown connections may be predicted from the data that is already available. Our approach uses a 'latent space model' that embeds the connectivity in an abstract physical space. Regions that are close in the latent space have a high chance of being connected, while regions far apart are most likely disconnected in the connectome. After learning the latent embedding from the connections that we did observe, the latent space allows us to predict connections that have not been probed previously. We apply the methodology to two connectivity data sets of the macaque, where we demonstrate that the latent space model is successful in predicting unobserved connectivity, outperforming two baselines and an alternative model in nearly all cases. Furthermore, we show how the latent spatial embedding may be used to integrate multimodal observations (i.e. anterograde and retrograde tracers) for the mouse neocortex. Finally, our probabilistic approach enables us to make explicit which connections are easy to predict and which prove difficult, allowing for informed follow-up studies

Magnitudes of source and target effects.

<p>Connections weights may be modulated properties of their end points, modeled here as source <b><i>δ</i></b> and target <b><i>ε</i></b> effects. A positive effect <i>δ</i>_<i>i</i> or <i>ε</i>_<i>i</i> means an increased likelihood of a connection originating from or terminating at node <i>i</i>, respectively. The effects have been scaled to percentages of the maximum latent distance () for easier interpretation.</p

Latent distances versus connection weights.

<p>For each connection, the latent distance is shown as well as the posterior expectation of the connection weight. The expectations of the <i>K</i> − 1 boundaries between the difference connection weight classes <i>b</i>_<i>k</i> are indicated with vertical lines.</p

For each of the different connectivity data sets, the table shows the number of source nodes, the number of target nodes, the numbers of observed and unobserved connections and finally the number of observed connection strength classes <i>K</i>.

<p>For each of the different connectivity data sets, the table shows the number of source nodes, the number of target nodes, the numbers of observed and unobserved connections and finally the number of observed connection strength classes <i>K</i>.</p

Macaque visual system connectivity.

<p><b>A</b>. The observed tracing data (left) [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005374#pcbi.1005374.ref044" target="_blank">44</a>], the corresponding predicted connectome (right), based on the 2D latent space model. <b>B</b>. The uncertainty associated with each of the predicted connections, indicated by the width of the 95% credible interval for the most uncertain class (see text). <b>C</b>. The predicted fraction of absent and present edges for either unobserved connections (left panel) or observed connections (right panel). <b>D</b>. The observed versus the predicted relative degree of the nodes in the network, for anterograde connections (left panel) and retrograde connections (right panel). The top five nodes with the largest differences in relative degree have their labels shown. A full listing is provided in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005374#pcbi.1005374.s001" target="_blank">S1 Appendix</a>.</p

Macaque cerebral cortex positioning.

<p>The left panel shows the node positions determined by the physical distances of the ROI. The connectivity in the left panel consists of the observed and present connections in the data. The right panel shows the posterior expectations of node distances as determined by the latent space model, with optimal dimensionality . Note positions are approximated using multidimensional scaling, based on the distances on the cortex and the posterior expectation of distances in the latent space.</p

Macaque visual system positioning.

<p>The left panel shows the node positions determined by the physical distances between the ROI. The connectivity in the left panel consists of the observed and present connections in the data. The right panel shows the posterior expectations of node distances as determined by the latent space model, with optimal dimensionality . Note positions are approximated using multidimensional scaling, based on the distances on the cortex and the posterior expectation of distances in the latent space.</p

Model performance.

<p>The prediction performance of the latent space model, the latent eigenmodel and the two baseline approaches, quantified using the negative log-likelihood (NLL), the mean absolute error (MAE), the false-positive rate (FPR) and the false-negative rate (FNR). All measures are obtained using ten-fold cross-validation. Error bars indicate one standard deviation over the ten folds. In the top row, the number of dimensions with the best generalization performance (i.e. the highest likelihood on hold-out data) is indicated with a vertical line. For the two macaque data sets, results are also shown for the fixed-positions model (see Section Link prediction). All scores have been normalized by the number of testing connections, for comparison between the different data sets.</p