From calcium imaging to graph topology

Abstract

Systems neuroscience is facing an ever-growing mountain of data. Recent advances in protein engineering and microscopy have together led to a paradigm shift in neuroscience; using fluorescence, we can now image the activity of every neuron through the whole brain of behaving animals. Even in larger organisms, the number of neurons that we can record simultaneously is increasing exponentially with time. This increase in the dimensionality of the data is being met with an explosion of computational and mathematical methods, each using disparate terminology, distinct approaches, and diverse mathematical concepts. Here we collect, organize, and explain multiple data analysis techniques that have been, or could be, applied to whole-brain imaging, using larval zebrafish as an example model. We begin with methods such as linear regression that are designed to detect relations between two variables. Next, we progress through network science and applied topological methods, which focus on the patterns of relations among many variables. Finally, we highlight the potential of generative models that could provide testable hypotheses on wiring rules and network progression through time, or disease progression. While we use examples of imaging from larval zebrafish, these approaches are suitable for any population-scale neural network modeling, and indeed, to applications beyond systems neuroscience. Computational approaches from network science and applied topology are not limited to larval zebrafish, or even to systems neuroscience, and we therefore conclude with a discussion of how such methods can be applied to diverse problems across the biological sciences.

Keywords: Calcium imaging; Graph theory; Systems neuroscience; Topology; Zebrafish.

Figure 1.

Rapid progress of systems neuroscience and the encoding of stimuli. (A) Across animal models of increasing complexity, we are now able to record the activity of many neurons. For example, we can record all of the neurons of C. elegans, about 2,000 neurons of D. melanogaster (Tainton-Heap et al., 2021), and about 80% of neurons in the larval zebrafish (D. rerio; X. Chen et al., 2018). High-density electrophysiological probes allow the recording of hundreds of units per session in mice, including deep structures within the brain (Steinmetz et al., 2019); in fact, researchers have recorded up to 10,000 neurons in a 0.3 mm³ portion of the brain using two-photon imaging (Stringer et al., 2019). (B). Stimulus encoding can occur at the single neuron level, where, for example, it could encode the direction of a stimulus. Stimulus encoding can also occur at the population level, where multiple units together can provide a better representation of the stimulus, including its direction and color. (C) Example of populations of neurons responding to the direction of water flow along the fish’s tail (Vanwalleghem et al., 2020). These neuronal populations show a tonic activity for the duration (10 s) of a water flow stimulus from the tail to the head (green arrow), or from the head to the tail (magenta arrow). The neurons were clustered in three categories: bidirectional response (blue), head to tail (pink), or tail to head (gold). Here we represent their spatial coordinates in the zebrafish brain (left). Telen = telencephalon, OT = optic tectum, TS = torus semicircularis (small dotted lines), ON = octavolateralis nucleus. (D) It is possible to predict neuronal activity from behavior, which in this example is the facial movement of a mouse. (Top) Motion energy was computed from consecutive frames (t and t + 1). (Middle) The principal components could then be used to predict the neuronal activity of 1,000 neurons. (Bottom) The predicted activity was remarkably similar (about 30%) to the real neuronal activity (sorted with an embedding for visualization). Adapted from Stringer et al. (2019).

Figure 2.

Calculating relationships between two units. (A) In systems neuroscience, a single unit may correspond to a neuron or a population of neurons. Two units may potentially interact in a variety of ways. For example, one can either activate or inhibit the other; the strength of these interactions can also vary, and feedback between units can also exist. As an example, units x (blue) and y (orange) have activities that change over time. At time i, unit x has activity x_i and unit y has activity y_i. (B) Multiple methods exist for quantifying relations between two variables. (Left) Pairwise similarity measures often see the leftmost portion of the activity traces as having high similarity and the middle portion as having low similarity; these measures may vary on their interpretation of the rightmost portion. (Middle) Linear regression calculates a line of best fit (darker gold) given the points (x_i, y_i) for each time i. This line of best fit minimizes the error between the predicted values of y_i, using x_i, and the true values y_i (lighter gold). (Right) Causal inference methods ask whether information about the activity of unit x between time i − k and the current time i improves the prediction accuracy of the future activity of unit y, above and beyond the prediction accuracy obtained by using the previous activity of unit y alone. (C) Applications of each method to larval zebrafish data. (Left) The pairwise similarity (correlation) between spinal neurons increases dramatically between 18 and 20 hours post fertilization (hpf), as can be seen by the blue and red activity traces moving in synchrony. Adapted from Warp et al. (2012). (Middle) Linear regression was used to quantify the responses to auditory stimuli; the coefficient is mapped to the color of each neuron in the brain, while the coefficient of determination R² is mapped to the size of the sphere. Adapted from Poulsen et al. (2021). (Right) The Granger causality approach was used to identify the flow of information between auditory brain regions in Vanwalleghem et al. (2017). Th = thalamus, TS = torus semicircularis, ON = octavolateralis nucleus, Hb = hindbrain.

Figure 3.

Systems neuroscience uses multiple methods to encode, represent, and analyze data. (A) Data consists of many units with recorded activity over time. (B) There are many options for encoding data into a mathematical representation. (C) Examples of representations. (Left) A network is composed of nodes and edges that connect exactly two nodes. (Middle) A simplicial complex is constructed from nodes and simplices that connect any number of nodes, and any subset of connected nodes. (Right) A hypergraph is built from nodes and hyperedges that connect any number of nodes. (D) Based on the type of representation used (network, simplicial complex, hypergraph), different analysis approaches become available. For example, representing data as a network allows one to detect many types of structure, and more recently very specific patterns such as core-periphery organization. Representing data as a simplicial complex permits the use of homology, which perceives the circular nature of the complex. Finally, a hypergraph representation enables a unique perspective on the community structure of the system. While one could use any of these representations to calculate, for example, community structure, we stress that each representation will provide a unique perspective on community structure, and furthermore that some representations are more amenable to particular analyses, for example, finding loops, than others (Torres et al., 2020). (E) Application of network theory to larval zebrafish data. WT and fmr1^-/- larval zebrafish were presented with increasing intensity of sound, from −21 dB to −9 dB, and their neuronal activity was recorded. Circle plots of the edges between brain region nodes in WT and fmr1^-/- zebrafish; an edge was placed if the correlation between their response to auditory stimuli is above 0.85. Node color indicates brain regions, in order: telencephalon, red; torus semicircularis, dark magenta; cerebellum, dark green; thalamus, orange; hindbrain without the Cb and ON, gray; octavolateralis nucleus, magenta; pretectum, light blue; optic tectum, blue; habenulae, yellow; tegmentum, light green. Adapted from Constantin et al. (2020). (F) Application of topological data analysis to the structural connectome. Example conserved cavity connecting visual processing regions shown in the brain (left) and abstracted for ease of interpretation (right). For visual simplicity, the cavity depiction in the brain does not include shading for 2-simplices, only edges. Adapted from A. E. Sizemore et al. (2018). (G) A hypergraph created from recordings of the on/off activity of brain regions during four tasks reveals that the green node A is only active in groups of three regions, whereas the blue node D is only active in groups of two regions. As such the green, magenta, and red nodes are all linked by a hyperedge. Adapted from Torres et al. (2020).