Neuronal Graphs: A Graph Theory Primer for Microscopic, Functional Networks of Neurons Recorded by Calcium Imaging

Abstract

Connected networks are a fundamental structure of neurobiology. Understanding these networks will help us elucidate the neural mechanisms of computation. Mathematically speaking these networks are "graphs"-structures containing objects that are connected. In neuroscience, the objects could be regions of the brain, e.g., fMRI data, or be individual neurons, e.g., calcium imaging with fluorescence microscopy. The formal study of graphs, graph theory, can provide neuroscientists with a large bank of algorithms for exploring networks. Graph theory has already been applied in a variety of ways to fMRI data but, more recently, has begun to be applied at the scales of neurons, e.g., from functional calcium imaging. In this primer we explain the basics of graph theory and relate them to features of microscopic functional networks of neurons from calcium imaging-neuronal graphs. We explore recent examples of graph theory applied to calcium imaging and we highlight some areas where researchers new to the field could go awry.

Keywords: brain networks; calcium imaging; functional connectivity; graph theory; network analysis; neuronal networks.

Figure 1

Networks of neurons occur across scales and can be functional or structural. Microscopic networks (left), i.e., at neuron or synapse scale, are usually recorded with calcium imaging or electron microscopy techniques. Macroscopic networks (right), i.e., recordings of indistinguishable groups of neurons or brain regions, are often recorded using MRI techniques. Neuronal Graphs are microscopic (neuron-resolved), functional networks extracted from calcium imaging experiments.

Figure 2

Types of graphs and their representation. (A) An undirected and unweighted graph; this is occasionally used for graph analyses where edge weights cannot be consider, e.g., certain community detection algorithms. (B) A directed and unweighted graph. (C) An undirected and weighted graph where the thickness of the edge indicates the weight, this is the most common graph type in calcium imaging due to the limited temporal resolution that is usually captured. Directed, weighted graphs can be produced when the temporal resolution is high enough to infer some causality between neuron activations. (D) An adjacency matrix of a weighted graph, which is an N_V × N_V matrix where a value in index i, j is the weight between node i and j.

Figure 3

Extracting neuronal graphs from calcium imaging. (A) Well-developed algorithms now allow for automated neuron segmentation from calcium movies. After pairwise correlation with an appropriate metric a graph is extracted. (B) Often neuronal graphs are thresholded based on the pairwise correlation metric used, removing weak and potentially spurious edges; the remaining edges are either unweighted or weighted based on the original metrics. (C) Neuronal graphs can represent whole datasets, e.g., whole-brain calcium imaging, or sub-graphs of neural assemblies, e.g., only neurons activated by a particular stimuli, which may overlap such as in this example (blue and green subgraphs).

Figure 4

Graph degree, degree distribution, and density all reveal information about how connected the nodes of a graph are. More connected graphs may represent fast flow of information in a network. A not well connected graph (A) with low density and thus low mean degree and individual node degree as shown by the degree distribution (B). A “complete,” i.e., very well-connected, graph (C) with 100% density and thus high mean and node degree and a different degree distribution (D, c.f. B).

Figure 5

Paths, and especially shortest paths, in graphs give an idea of efficiency of flow or information transfer. Graphs with shorter average paths between nodes may represent networks with very efficient information transfer. (A) An example random path through a graph between the two green nodes. This is not the shortest path but just one of many potential paths. (B) An example shortest path between the same green nodes, there are multiple routes of the same shortest length, between the same two nodes in the same graph. This path represents one of the most efficient routes for information flow in a graph. (C) The distribution of shortest path lengths across all pairs of nodes in a graph can give an idea of flow efficiency in a network. A left-shifted distribution might be expected within connected brain network where neurons may be connected in a highly compact and efficient fashion for fast information processing.

Figure 6

Centrality measures indicate the relative importance of a node within a graph. There are a great number of centrality metrics that make different assumptions and provide different insights, here we provide two example metrics. (A) Closeness centrality gives a relative importance based on the average shortest path between a node and all other nodes. The smaller the average the more important the node (darker blue) and vice versa (whiter). (B) Betweenness centrality is similar but considers how many shortest paths must pass through a node and, in this example, clearly separates the central two nodes (dark blue) as much more important than the other nodes (which appear white). (C) Centrality can also be applied to edges instead of nodes; here more blue indicates a more central role in information flow for an edge.

Figure 7

Motifs represent repeating highly-localised topological patterns in the graph; remember, here “local” refers to directly connected nodes in the graph and not, necessarily, physically located neurons. (A) An example 3-motif in blue; there are a total of 55 3-motifs in this graph. (B) An equivalent 4-motif in blue; there are 132 4-motifs in this graph. (C) Histograms of graph motif counts can be used to create a signature or fingerprint for graphs that can then be compared between cases. Here, we see the histogram of all n-motifs in the graph used for (A,B).

Figure 8

Clustering provides another measure of connectivity and structure in a graph based on how nodes are locally connected; remember, here “local” refers to directly connected nodes in the graph and not, necessarily, physically located neurons. (A) Based the number of closed (blue) and open (green) triplets, the clustering coefficient can be calculated locally for every node. (B) Local clustering coefficients for nodes range from zero (white) to one (blue) and may vary a lot from the global (mean) clustering coefficient. Nodes with a high clustering coefficient may be involved in general aspects of information transfer and thus form an apex for clustering.

Figure 9

Community detection provides global clustering that can be either non-overlapping or overlapping depending on the algorithm used. (A) Non-overlapping communities assign each node to a community (blue or green) based on the choice of metric, often relating to number of connections. Such community detection algorithms could identify physical regions of the brain. (B) Over-lapping communities can assign a node to more than one community (black nodes) if they contribute to multiple communities. Such community detection algorithms could identify communication pathways through a network.

Figure 10

Random graphs, which can be used as in silico models or controls, can be generated in different ways giving the graphs different properties. Pseudocode showing the processes used to create Erdös-Rényi (A), Watts-Strogatz (B) and Barabási-Albert (C) model graphs. (D–F) Example graphs with N_V = 30 showing clearly different organisations for different generation models. (G–I) Probability distributions of node degree over graphs generated with N_V = 10, 000 showing lower average degree and increased tails in both the Watts-Strogatz (H) and Barabási-Albert (I) models.

Figure 11

A 4-Regular Ring Lattice on a 6 node graph. The blue edges connected to node 1 show why this graph is 4-regular graph—all nodes have exactly 4 edges connecting them to their 4 closet neighbours.