From single-neuron dynamics to higher-order circuit motifs in control and pathological brain networks

Abstract

The convergence of advanced single-cell in vivo functional imaging techniques, computational modelling tools and graph-based network analytics has heralded new opportunities to study single-cell dynamics across large-scale networks, providing novel insights into principles of brain communication and pointing towards potential new strategies for treating neurological disorders. A major recent finding has been the identification of unusually richly connected hub cells that have capacity to synchronize networks and may also be critical in network dysfunction. While hub neurons are traditionally defined by measures that consider solely the number and strength of connections, novel higher-order graph analytics now enables the mining of massive networks for repeating subgraph patterns called motifs. As an illustration of the power offered by higher-order analysis of neuronal networks, we highlight how recent methodological advances uncovered a new functional cell type, the superhub, that is predicted to play a major role in regulating network dynamics. Finally, we discuss open questions that will be critical for assessing the importance of higher-order cellular-scale network analytics in understanding brain function in health and disease.

Keywords: computational modelling; higher-order motifs; machine learning; network science; neuroimaging; neuropathology.

Figure 1:. A computational pipeline to record, model, and mine large-scale cellular-resolution networks

State of the art imaging technology and quantative analysis methods are enabling researchers to uncover network dyanamics across massive neuronal networks at single-cell resolution. In step 1 of this example pipeline, single-cell recordings from the larval zebrafish is performed using 2-photon Ca²⁺ imaging, capturing activity from thousands of neurons across the entire brain in both control and pathological (preseizure) conditions. In step 2, time-series neural activity from these brain states undergo computational modeling to infer the underlying cell-cell communication network using machine-learning based optimization (see Figure 2). The inferred directed graph can then be mined in step 3 using local clustering techniques that can extract important higher-order connectivity structures (see Figure 3). Reproduced in a modified form with permission from Hadjiabadi et al. (2021) (Hadjiabadi et al., 2021).

Figure 2.. Chaotic recurrent neural networks and FORCE learning

A: The network architecture of the chaotic recurrent neural network (RNN) is a graph consisting of a collection of nodes and tunable edges (Sompolinsky et al., 1988). Depending on the recording device used, the nodes can range from regions to individual neurons. The edges J_ij represent the influence between pairs of nodes and are signed (positive/negative) and weighted. Currently, networks with thousands of neurons can be modeled from whole-brain single cell recordings of larval zebrafish (Lovett-Barron et al., 2017; Andalman et al., 2019; Betzel, 2019; Hadjiabadi et al., 2021). The governing dynamics of the RNN allows for chaotic dynamics. Here, τ is the time constant, x_i is the inferred current within node i, g is the gain parameter and generates chaotic dynamics for g > 1, φ(.) is the tanh function, h_i is a noise term for node i, and z_i is the model generated unit activity at node i. The dynamics can be solved using numerical estimation methods. B: The learning rule for FORCE optimization ensures that the effective edges J_ij are updated at each training epoch. In the original formulation, the weight update is proportional to the error between the unit activity z_i and the target dervied activity f_i (Sussillo & Abbott, 2009). In recent work (Hadjiabadi et al., 2021), the error is scaled by the structural connectivity weight S_ij (see panel c). With single-cell imaging, that target activity is experimental GCaMP recordings. Note that z_i initially starts off as chaotic but eventually matches the target output, leading to an overall decrease in mean square error. C: Modifications to the original weight update step include incorporating structural data from the zebrafish neuroanatomical connectome as a biological constraint to guide training (Hadjiabadi et al., 2021). The aim is to modulate the distribution of effective weights such that pairs of neurons living in poorly connected brain regions exhibit low influence while pairs of neurons in highly connected brain regions have increased influence.The neuroanatomical connectome was built from quantitative analysis from thousands of neuron tracings and is represented by a weight matrix (Kunst et al., 2019).

Figure 3.. Higher-order motifs and local higher-order clustering

A: Motifs are subgraphs representing higher-order connectivity patterns of networks. These repeating subgraphs may be critical for understanding network function in massively complex systems (Milo et al., 2002). For example, feedfoward motifs and cycles have been implicated in sustaining cortical-like asynchronous spiking patterns in computational models (Bojanek et al., 2020), and the overrepresentation of feedforward motifs is furthermore predicted to emerge in the pathological brain, generating unrestrained excitation (Hadjiabadi et al., 2021). Moreover, bifan motifs, defined as two output neurons that individually target two input neurons, are critical for filtering and synchornization in signaling networks (Lipshtat et al., 2008), and have been linked to the circuit pathways underlying nictation in C. elegans (Benson et al., 2016). B-D: Local higher-order clustering can identify clusters dense in a specific motif of interest. The MAPPR algorithm (Yin et al., 2017) will identify clusters with the lowest conductance value, which can be thought of as islands isolated from the rest of the graph. As an illustration of how the motif conductance is quantified, we ue a toy undirected network structure and local cluster X. (B) An example seed node (asterisk) and its neighbors are shown. This seed node is a part of two triangle motifs (labels 1,2). (C) Zooming out, the seed node is part of a larger network, and resides inside a local cluster X containing 6 nodes. Each node in the cluster gets assigned a ‘motif degree’, which is the number of motifs each node is a part of (see B). (D) The motif conductance is calculated by considering how many motifs reside between the local cluster X and the rest of the network divided by the sum of the ‘motif degree’ for nodes inside the cluster X. In this case, there is 1 motif that cuts the cluster (gray), and the sum of ‘motif degree’ for nodes in X is 11. Altogether this gives a motif condutance of 1/11 = 0.091.

Figure 4.. Cellular superhubs identified through higher-order network analysis

A: Traditional hub neurons are identified by simple edge count and have unusually high degree of connectivity. Example outgoing hub neuron with numerous outgoing projections. B: An example superhub rich in feedforward motifs, identified using the MAPPR algorithm (Yin et al., 2017) for local higher-order clustering. In the zebrafish experiments illustrated in Figure 1 in the pathological preseizure brain state, feedforward motif conductance of outgoing hub neurons was significantly elevated compared to baseline state, indicating a propensity to project excitatory activity downstream (Hadjiabadi et al., 2021). C: Disconnecting superhubs from the epileptic brain stabilizes circuits. PCA analysis of simulated network dynamics after a single outgoing hub was perturbed in a modeled preseizure network before (top) and after (bottom) superhubs were disconnected. Reproduced, in modified form, with permission from Hadjiabadi et al. (2021) (Hadjiabadi et al., 2021).