Differential contributions of intra‐cellular and inter‐cellular mechanisms to the spatial and temporal architecture of the suprachiasmatic nucleus circadian circuitry in wild‐type, cryptochrome‐null and vasoactive intestinal peptide receptor 2‐null mutant mice

Abstract To serve as a robust internal circadian clock, the cell‐autonomous molecular and electrophysiological activities of the individual neurons of the mammalian suprachiasmatic nucleus (SCN) are coordinated in time and neuroanatomical space. Although the contributions of the chemical and electrical interconnections between neurons are essential to this circuit‐level orchestration, the features upon which they operate to confer robustness to the ensemble signal are not known. To address this, we applied several methods to deconstruct the interactions between the spatial and temporal organisation of circadian oscillations in organotypic slices from mice with circadian abnormalities. We studied the SCN of mice lacking Cryptochrome genes (Cry1 and Cry2), which are essential for cell‐autonomous oscillation, and the SCN of mice lacking the vasoactive intestinal peptide receptor 2 (VPAC2‐null), which is necessary for circuit‐level integration, in order to map biological mechanisms to the revealed oscillatory features. The SCN of wild‐type mice showed a strong link between the temporal rhythm of the bioluminescence profiles of PER2::LUC and regularly repeated spatially organised oscillation. The Cry‐null SCN had stable spatial organisation but lacked temporal organisation, whereas in VPAC2‐null SCN some specimens exhibited temporal organisation in the absence of spatial organisation. The results indicated that spatial and temporal organisation were separable, that they may have different mechanistic origins (cell‐autonomous vs. interneuronal signaling) and that both were necessary to maintain robust and organised circadian rhythms throughout the SCN. This study therefore provided evidence that the coherent emergent properties of the neuronal circuitry, revealed in the spatially organised clusters, were essential to the pacemaking function of the SCN.

The mean brightness time series for bioluminescence for each slice is shown to the right of each row. As noted in text, the analyses performed in this paper assess the amplitude of oscillation and are not sensitive to the baseline brightness of the image. While the y-axes have been uniformly scaled, due to the different lengths of the time series, the x-axes have different scales. The color scale is shown on the right side of the last panel. Figure S2: For each WT slice, we show (from left to right) the "map" showing the top six clusters produced by the k-medoids algorithm, and the mean brightness time series for each clusters, following previous work (Foley et al., 2011). In addition, we provide the Fourier power spectrum for each cluster. The color scale is shown on the right side of the first "map". Note that, due to the different lengths of the time series, the x-axes have different scales. Figure S3: The results of the k-medoid analysis for all WT samples are shown for the automatically determined ROIs. In the first three columns, from left to right, we show the "map" of the top six clusters in the tissue, the mean signals over the clusters, and the Fourier power spectra. The right hand panels show the same analyses for manually identified cells. Note that, due to the different lengths of the time series, the x-axes have different scales. Figure S4: Similarly to Figure S3, we show the results of the k-medoid analysis for all Cry-null samples with automatically determined ROIs. In the first three columns, from left to right, we show the top six clusters in the tissue, the mean signals over the clusters, and the Fourier power spectra. The right hand panels show the same analyses for manually identified cells. Note that, due to the different lengths of the time series, the x-axes have different scales. Figure S5: We show the same analyses as in Figure 6 for the right lobes of the SCN in that Figure   SM1 corresponds to the slice in Figure 6A and SM2 corresponds to Figure 6B. The spectral embedding analysis supports the results of cluster analysis and further demonstrates the occurrence of repeated local temporal features: in particular there is a recurrent 'start' impetus that exists in both WT and Cry-null slices.

The similarity graph of an SCN
The spectral cluster analyses in this paper are based on a graph theoretic model of the SCN. The similarity graph represents the SCN as a graph consisting of a collection of nodes and edges between those nodes. For our application, we choose nodes to be the superpixels (described in the main text) and we place an edge between two nodes weighted according to the similarity of their temporal brightness profiles computed according some chosen function, such as correlation or cosine distance. While we give the technical details below, the basic idea is that superpixels with similarly shaped time series will have connections with high weights between them while superpixels with significantly differently shaped time series will have low or zero weight connections.

Spectral Clustering Algorithm
This algorithm decomposes the tissue into regions with coherent behavior of luminescence time series subject to an important constraint, which is that spectral clustering produces a solution to L has a full complement of real eigenvalues. All of our graphs have one connected component and we order the eigenvalues from smallest to largest: The first nontrivial eigenvalue, λ 1 , is called the Fiedler value (Fiedler, 1968) and describes the algebraic connectivity of the graph. To finish the implementation of spectral clustering, we determine the number, l, of significant eigenvalues we wish to use. In the literature, this determination is often done using domain knowledge or estimation from a scree plot -we used the latter (see the main text for parameter estimation). Using the l associated eigenvectors, we construct the spectral embedding using the entries of eigenvectors associated to the first l nonzero eigenvalues as coordinates in ℝ . This embedding provides the geometric basis for spectral clustering -nodes are considered close to one another if they are close in this space, even if they are quite distant in other ways (e.g. spatially distant). Our last step in spectral clustering is to apply k-means to the spectral embedding to cluster the nodes together. To use spectral clustering we must specify three parameters, the scale parameter σ (described below), the number of eigenvectors l, and the number of clusters k.

Spectral embedding: Similarity model of the SCN
The spectral embedding is a mapping of the superpixels to the plane which allows us to see the extent to which the temporal and spatial structures of PER2 concentrations in the SCN are linked. Key to encoding the link between temporal and spatial structures in the spectral embedding is choosing a good adjacency matrix for our graph model of the functional connectivity SCN. To specify our adjacency matrix, we need to specify a notion of dissimilarity between the superpixels. Our notion is built from correlation between the time series of luminescence profiles for the superpixels. Precisely, we first normalize our time series by removing the mean time series via projection. If is the time series associated with superpixel i, we let where is the mean time series. We then construct the correlation matrix of the normalized time series -= ( � , � ). We convert the correlation to half the distance on the sphere by letting = sin � cos −1 2 �. In other words, if we further ensure that the normalized time series are of unit length, then is half the chordal distance on the unit sphere.
This distance matrix is a dissimilarity matrix, while an adjacency matrix for the graph needs to be a similarity matrix. We nonlinearly rescale the distance matrix to make a similarity matrix suitable for use as an adjacency matrix: The parameter σ is a scale parameter -it allows us to pick which distances (and hence correlations) we consider to be relevant and which are to be devalued (see below for parameter estimation).