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.

Ensemble remodeling supports memory-updating

“…Based on theories suggesting that ensemble membership may change during learning to prevent overreliance on a single neuronal population 1,47,48 , we hypothesized that the decrease in ensemble activation strength may be a result of certain neurons decoupling from the rest of the ensemble. To investigate this possibility, we used graph theory 49 to infer and track the remodeled ensembles' neuron-pair relationships during Reversal. First, we split the Reversal session into 5-min bins.…”

A 3D in vitro model of the device-tissue interface: functional and structural symptoms of innate neuroinflammation are mitigated by antioxidant ceria nanoparticles

“…In order to assess the functional effects of wire implantations, we used AAV-mediated transgene delivery of a GCaMP6s calcium reporter (AA V1-hSyn1-mRuby2-GSG-P2A-GCaMP6s-WPRE-pA, AddGene) and recorded calcium activity in the tissue surrounding implants from DIV14-18. We used a mathematical method called graph theoretical analysis (GTA) to describe the complex functional interactions within a network [50,51]. Application of GTA to a graph made up of single cell 'nodes' is a characterization of network behavior of 'microcircuits' [52, 53], which are sensitive to subtle changes in network dynamics that precede gross histological and behavioral deficits in other neuroinflammatory and neurodegenerative disease states [54][55][56][57].…”

Nonparanormal graph quilting with applications to calcium imaging

“…Estimates of functional connectivity are then obtained by calculating measures of association between the fluorescence traces of pairs of neurons. Many different models have been proposed and used in order to analyse functional connectivity from calcium imaging data, ranging from measuring simple pairwise relationships with partial Pearson correlations (Ota et al, 2021; Sutera et al, 2015) and mutual information (O'Connor et al, 2022; Singh & Lesica, 2010; Stetter et al, 2012) to high‐dimensional multivariate methods such as component analysis (Brier et al, 2019; Cramer et al, 2023), directed acyclic graphs (Biswas & Shlizerman, 2020; Fallani et al, 2014; Nelson & Bonner, 2021; Pollonini et al, 2010; Schmidt et al, 2016) and the aforementioned probabilistic graphical models. The results from these data analyses have been used to better understand the underlying networks behind functional connectivity, including how neurons react to stimuli (Cramer et al, 2019; Stožer et al, 2013; West et al, 2022), the role of individual neurons in multi‐neuronal systems (Betzel, 2020; Mann et al, 2017) and how functional connectivity networks vary over time (Cohen, 2018).…”