Research Article

Neuroscience

Neural assemblies uncovered by generative modeling explain whole-brain activity statistics and reflect structural connectivity

Computational Neuroscience Lab, Department of Neurophysiology, Donders Center for Neuroscience, Radboud University, Netherlands
Sorbonne Université, CNRS, Institut de Biologie Paris-Seine (IBPS), Laboratoire Jean Perrin (LJP), France
Department of Physiology, Anatomy and Genetics, University of Oxford, United Kingdom
Blavatnik School of Computer Science, Tel Aviv University, Israel
Department Genes – Circuits – Behavior, Max Planck Institute for Biological Intelligence, Germany
Allen Institute for Brain Science, United States

Jan 17, 2023

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Abstract
Editor's evaluation
Introduction
Results
Discussion
Materials and methods
Data availability
References
Article and author information
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Abstract

Patterns of endogenous activity in the brain reflect a stochastic exploration of the neuronal state space that is constrained by the underlying assembly organization of neurons. Yet, it remains to be shown that this interplay between neurons and their assembly dynamics indeed suffices to generate whole-brain data statistics. Here, we recorded the activity from ∼40,000 neurons simultaneously in zebrafish larvae, and show that a data-driven generative model of neuron-assembly interactions can accurately reproduce the mean activity and pairwise correlation statistics of their spontaneous activity. This model, the compositional Restricted Boltzmann Machine (cRBM), unveils ∼200 neural assemblies, which compose neurophysiological circuits and whose various combinations form successive brain states. We then performed in silico perturbation experiments to determine the interregional functional connectivity, which is conserved across individual animals and correlates well with structural connectivity. Our results showcase how cRBMs can capture the coarse-grained organization of the zebrafish brain. Notably, this generative model can readily be deployed to parse neural data obtained by other large-scale recording techniques.

Editor's evaluation

Large scale recordings, sometimes involving 10s of thousands of neurons, are becoming increasingly common. Making sense of these recordings, however, is not easy. This paper introduces a new method, the compositional Restricted Boltzmann Machine, that overcomes this problem -- it can find structure in data, including both "cell assemblies" and structural connectivity, without inordinate computing resources (data from 40,000 neurons recorded from zebrafish can be analyzed in less than a day). This is a valuable contribution, both to those interested in data analysis, and to those interested in zebrafish.

https://doi.org/10.7554/eLife.83139.sa0

Introduction

The brain is a highly connected network, organized across multiple scales, from local circuits involving just a few neurons to extended networks spanning multiple brain regions (White et al., 1986; Song et al., 2005; Kunst et al., 2019). Concurrent with this spatial organization, brain activity exhibits correlated firing among large groups of neurons, often referred to as neural assemblies (Harris, 2005). This assembly organization of brain dynamics has been observed in, for example, auditory cortex (Bathellier et al., 2012), motor cortex (Narayanan et al., 2005), prefrontal cortex (Tavoni et al., 2017), hippocampus (Lin et al., 2005), retina (Shlens et al., 2009), and zebrafish optic tectum (Romano et al., 2015; Mölter et al., 2018; Diana et al., 2019; Triplett et al., 2020). These neural assemblies are thought to form elementary computational units and subserve essential cognitive functions such as short-term memory, sensorimotor computation or decision-making (Hebb, 1949; Gerstein et al., 1989; Harris, 2005; Buzsáki, 2010; Harris, 2012; Palm et al., 2014; Eichenbaum, 2018). Despite the prevalence of these assemblies across the nervous system and their role in neural computation, it remains an open challenge to extract the assembly organization of a full brain and to show that the assembly activity state, derived from that of the neurons, is sufficient to account for the collective neural dynamics.

The need to address this challenge is catalyzed by technological advances in light-sheet microscopy, enabling the simultaneous recording of the majority of neurons in the zebrafish brain at single-cell resolution in vivo (Panier et al., 2013; Ahrens et al., 2013; Wolf et al., 2015; Wolf et al., 2017; Migault et al., 2018; Vanwalleghem et al., 2018). This neural recording technique opens up new avenues for constructing near-complete models of neural activity, and in particular its assembly organization. Recent attempts have been made to identify assemblies using either clustering (Panier et al., 2013; Triplett et al., 2018; Chen et al., 2018; Mölter et al., 2018; Bartoszek et al., 2021), dimensionality reduction approaches (Lopes-dos-Santos et al., 2013; Romano et al., 2015; Mu et al., 2019) or latent variable models (Diana et al., 2019; Triplett et al., 2020), albeit often limited to single brain regions. However, these methods do not explicitly assess to what extent the inferred assemblies could give rise to the observed neural data statistics, which is a crucial property of physiologically meaningful assemblies (Harris, 2005). Here, we address this challenge by developing a generative model of neural activity that is explicitly constrained by the assembly organization, thereby quantifying if assemblies indeed suffice to produce the observed neural data statistics.

Specifically, we formalize neural assemblies using a bipartite network of two connected layers representing the neuronal and the assembly activity, respectively. Together with the maximum entropy principle (Jaynes, 1957; Bialek, 2012), this architecture defines the Restrictive Boltzmann Machine (RBM) model (Hinton and Salakhutdinov, 2006). Here, we use an extension to the classical RBM definition termed compositional RBM (cRBM) that we have recently introduced (Tubiana and Monasson, 2017; Tubiana et al., 2019a) and which brings multiple advances to assembly-based network modeling: (1) The maximum entropy principle ensures that neural assemblies are inferred solely from the data statistics. (2) The generative nature of the model, through alternate data sampling of the neuronal and assembly layers, can be leveraged to evaluate its capacity to replicate the empirical data statistics, such as the pairwise co-activation probabilities of all neuron pairs. (3) The cRBM steers the assembly organization to the so-called compositional phase where a small number of assemblies are active at any point in time, making the resulting model highly interpretable as we have shown previously for protein sequence analysis (Tubiana et al., 2019b).

Here, we have successfully trained cRBMs to brain-scale, neuron-level recordings of spontaneous activity in larval zebrafish containing 41,000 neurons on average (Panier et al., 2013; Wolf et al., 2017; Migault et al., 2018). This represents an increase of ∼2 orders of magnitude in number of neurons with respect to previously reported RBM implementations (Köster et al., 2014; Gardella et al et al., 2017; Volpi et al., 2020), attained through significant algorithmic and computational enhancements. We found that all cells could be grouped into 100–200 partially overlapping assemblies, which are anatomically localized and together span the entire brain, and accurately replicate the first and second order statistics of the neural activity. These assemblies were found to carry more predictive power than a fully connected model which has orders of magnitude more parameters, validating that assemblies underpin collective neural dynamics. Further, the probabilistic nature of our model allowed us to compute a functional connectivity matrix by quantifying the effect of activity perturbations in silico. This assembly-based functional connectivity is well-conserved across individual fish and consistent with anatomical connectivity at the mesoscale (Kunst et al., 2019).

In summary, we present an assembly decomposition spanning the zebrafish brain, which accurately accounts for its activity statistics. Our cRBM model provides a widely applicable tool to the community to construct low-dimensional data representations that are defined by the statistics of the data, in particular for very high-dimensional systems. Its generative capability further allows to produce new (synthetic) activity patterns that are amenable to direct in silico perturbation and ablation studies.

Results

Compositional RBMs construct Hidden Units by grouping neurons into assemblies

Spontaneous neural activity was recorded from eight zebrafish larvae aged 5–7 days post fertilization expressing the GCaMP6s or GCaMP6f calcium reporters using light-sheet microscopy (Panier et al., 2013; Wolf et al., 2017; Migault et al., 2018). Each data set contained the activity of a large fraction of the neurons in the brain ( $40709 \pm 13854$ ; mean ± standard deviation), which, after cell segmentation, were registered onto the ZBrain atlas (Randlett et al., 2015) and mapzebrain atlas (Kunst et al., 2019). Individual neuronal fluorescence traces were deconvolved to binarized spike trains using blind sparse deconvolution (Tubiana et al., 2020). This data acquisition process is depicted in Figure 1A.

Figure 1

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cRBMs construct Hidden Units by grouping neurons into assemblies.

(A) The neural activity of zebrafish larvae was imaged using light-sheet microscopy (left), which resulted in brain-scale, single-cell resolution data sets (middle, microscopy image of a single plane shown for fish #1). Calcium activity $Δ F / F$ was deconvolved to binarized spike traces for each segmented cell (right, example neuron). (B) cRBM sparsely connects neurons (left) to Hidden Units (HUs, right). The neurons that connect to a given HU (and thus belong to the associated assembly), are depicted by the corresponding color labeling (right panel). Data sets typically consist of $N \sim 10^{4}$ neurons and $M \sim 10^{2}$ HUs. The activity of 500 randomly chosen example neurons (raster plot, left) and HUs 99, 26, 115 (activity traces, right) of the same time excerpt is shown. HU activity is continuous and is determined by transforming the neural activity of its assembly. (C) The neural assemblies of an example data set (fish #3) are shown by coloring each neuron according to its strongest-connecting HU. 7 assemblies are highlighted (starting rostrally at the green forebrain assembly, going clockwise: HU 177, 187, 7, 156, 124, 64, 178), by showing their neurons with a connection $| w | \geq 0.1$ . See Figure 3 for more anatomical details of assemblies. R: Right, L: Left, Ro: Rostral, C: Caudal.

We trained compositional Restricted Boltzmann Machine (cRBM) models to capture the activity statistics of these neural recordings. cRBMs are maximum entropy models, that is, the maximally unconstrained solution that fits model-specific data statistics (Hinton and Salakhutdinov, 2006; Tubiana and Monasson, 2017; Gardella et al., 2019), and critically extend the classical RBM formulation. Its architecture consists of a bipartite graph where the high-dimensional layer of neurons $v$ (named ‘visible units’ in RBM terminology) is connected to the low-dimensional layer of latent components, termed Hidden Units (HUs) $h$ . Their interaction is characterized by a weight matrix $W$ that is regularized to be sparse. The collection of neurons that have non-zero interactions with a particular HU, noted $h_{μ}$ (i.e. with $| w_{i, μ} | > 0$ ), define its corresponding neural assembly μ (Figure 1B). This weight matrix, together with the neuron weight vector $g$ and HU potential $U$ , defines the transformation from the binarized neural activity $v (t)$ to the continuous HU activity $h (t)$ (Figure 1B). Figure 1C shows all recorded neurons of a zebrafish brain, color-labeled according to their strongest-connecting HU, illustrating that cRBM-inferred assemblies (hereafter, neural assemblies for conciseness) are typically densely localized in space and together span the entire brain.

Beyond its architecture (Figure 2A), the model is defined by the probability function $P (v, h)$ of any data configuration $(v, h)$ (see Materials and methods ‘Restricted Boltzmann Machines’ and ‘Compositional Restricted Boltzmann Machine’ for details):

P (v, h) = \frac{1}{Z} \exp (- E (v, h))

where $Z$ is the partition function that normalizes Equation 1 and $E$ is the following energy function:

E (v, h) = - \sum_{i} g_{i} v_{i} + \sum_{μ} U_{μ} (h_{μ}) - \sum_{i, μ} w_{i, μ} v_{i} h_{μ}

HU activity h is obtained by sampling from the conditional probability function $P (h | v)$ :

P (h | v) = \prod_{μ = 1}^{M} P (h_{μ} | v) \propto \prod_{μ = 1}^{M} \exp (- U_{μ} (h_{μ}) + h_{μ} \cdot \sum_{i} w_{i, μ} v_{i})

Conversely, neural activity is obtained from HU activity through:

P (v | h) = \prod_{i = 1}^{N} P (v_{i} | h) \propto \prod_{i = 1}^{N} \exp (g_{i} v_{i} + v_{i} \cdot \sum_{μ} w_{i, μ} h_{μ})

Equations 3 and 4 mathematically reflect the dual relationship between neural and assembly states: the Hidden Units $h$ drive ‘visible’ neural activity $v$ , expressed as $P (v | h)$ , while the stochastic assembly activity $h$ itself is defined as a function of the activity of the neurons: $P (h | v)$ . Importantly, the model does not include direct connections between neurons, hence neural correlations $⟨ v_{i} v_{j} ⟩$ can arise solely from shared assemblies. Moreover, this bipartite architecture ensures that the conditional distributions factorize, leading to a sampling procedure where all neurons or all HUs can be sampled in parallel. The cRBM leverages this property to efficiently generate new data by Monte Carlo sampling alternately from $P (h | v)$ and $P (v | h)$ (Figure 2B).

Figure 2 with 7 supplements see all

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cRBM is optimized to accurately replicate data statistics.

(A) Schematic of the cRBM architecture, with neurons $v_{i}$ on the left, HUs $h_{μ}$ on the right, connected by weights $w_{i, μ}$ . (B) Schematic depicting how cRBMs generate new data. The HU activity $h (t)$ is sampled from the visible unit (i.e. neuron) configuration $v (t)$ , after which the new visible unit configuration $v (t + 1)$ is sampled and so forth. (C) cRBM-predicted and experimental mean neural activity $⟨ v_{i} ⟩$ were highly correlated (Pearson correlation $r_{P} = 0.91$ , $P < 10^{- 307}$ ) and had low error ( ${nRMSE}_{⟨ v_{i} ⟩} = 0.11$ , normalized Root Mean Square Error, see Materials and methods - ‘Calculating the normalized Root Mean Square Error’ ). Data displayed as 2D probability density function (PDF), scaled logarithmically (base 10). (D) cRBM-predicted and experimental mean Hidden Unit (HU) activity $⟨ h_{μ} ⟩$ also correlated very strongly ( $r_{P} = 0.93$ , $P < 10^{- 86}$ ) and had low ${nRMSE}_{⟨ h_{μ} ⟩} = 0.15$ (other details as in C) (E) cRBM-predicted and experimental average pairwise neuron-HU interactions $⟨ v_{i} h_{μ} ⟩$ correlated strongly ( $r_{P} = 0.74$ , $P < 10^{- 307}$ ) and had a low error ( ${nRMSE}_{⟨ v_{i} h_{μ} ⟩} = 0.09$ ). (F) cRBM-predicted and experimental average pairwise neuron-neuron interactions $⟨ v_{i} v_{j} ⟩$ correlated well ( $r_{P} = 0.58$ , $P < 10^{- 307}$ ) and had a low error ( ${nRMSE}_{⟨ v_{i} v_{j} ⟩} = - 0.09$ , where the negative nRMSE value means that cRBM-predictions match the test data slightly better than the train data). Pairwise interactions were corrected for naive correlations due to their mean activity by subtracting $⟨ v_{i} ⟩ ⟨ v_{j} ⟩$ . (G) cRBM-predicted and experimental average pairwise HU-HU interactions $⟨ h_{μ} h_{ν} ⟩$ correlated strongly ( $r_{P} = 0.73$ , $P < 10^{- 307}$ ) and had a low error ( ${nRMSE}_{⟨ h_{μ} h_{ν} ⟩} = 0.17$ ). (H) The low-dimensional cRBM bottleneck reconstructs most neurons above chance level (purple), quantified by the normalized log-likelihood (nLLH) between neural test data v_i and the reconstruction after being transformed to HU activity (see Materials and methods - ‘Reconstruction quality’). Median normalized = ${nLLH}_{cRBM}$ 0.24. Reconstruction quality was also determined for a fully connected Generalized Linear Model (GLM) that attempted to reconstruct the activity of a neuron v_i using all other neurons $v_{- i}$ (see Materials and methods - ‘Generalized Linear Model’). The distribution of 5000 randomly chosen neurons is shown (blue), with median ${nLLH}_{GLM} = 0.20$ . The cRBM distribution is stochastically greater than the GLM distribution (one-sided Mann Whitney U test, $P < 10^{- 42}$ ). (I) cRBM (purple) had a sparse weight distribution, but exhibited a greater proportion of large weights $w_{i, μ}$ than PCA (yellow), both for positive and negative weights, displayed in log-probability. (J) Distribution of above-threshold absolute weights $| w_{i, μ} |$ per neuron v_i (dark purple), indicating that more neurons strongly connect to the cRBM hidden layer than expected by shuffling the weight matrix of the same cRBM (light purple). The threshold $Θ$ was set such that the expected number of above-threshold weights per neuron $E (# w_{i} > Θ) = 1$ . (K) Corresponding distribution as in (J) for PCA (dark yellow) and its shuffled weight matrix (light yellow), indicating a predominance of small weights in PCA for most neurons v_i. All panels of this figure show the data statistics of the cRBM with parameters $M = 200$ and $λ = 0.02$ (best choice after cross-validation, see Figure 2—figure supplement 1) of example fish #3, comparing the experimental test data test and model-generated data after cRBM training converged.

The cRBM differs from the classical RBM formulation (Hinton and Salakhutdinov, 2006) through the introduction of double Rectified Linear Unit (dReLU) potentials $U_{μ}$ , weight sparsity regularization and normalized HU activity (further detailed in Methods). We have previously demonstrated in theory and application (Tubiana and Monasson, 2017; Tubiana et al., 2019a; Tubiana et al., 2019b) that this new formulation steers the model into the so-called compositional phase, which makes the latent representation highly interpretable. This phase occurs when a limited number $m$ of HUs co-activate such that $1 ≪ m ≪ M$ where $M$ is the total number of HUs. Thus, each visible configuration is mapped to a specific combination of activated HUs. This contrasts with the ferromagnetic phase ( $m \sim 1$ ) where each HU encodes one specific activity pattern, thus severely limiting the possible number of encoded patterns, or the spin-glass phase ( $m \sim M$ ) where all HUs activate simultaneously, yielding a very complex assembly patchwork (Tubiana and Monasson, 2017). Therefore, the compositional phase can provide the right level of granularity for a meaningful interpretation of the cRBM neural assemblies by decomposing the overall activity as a time-dependent co-activation of different assemblies of interpretable size and extent.

Trained cRBMs accurately replicate data statistics

cRBM models are trained to maximize the $P (v, h)$ log-likelihood of the zebrafish data recordings, which is achieved by matching the model-generated statistics $⟨ v_{i} ⟩$ , $⟨ h_{μ} ⟩$ and $⟨ v_{i} h_{μ} ⟩$ (the mean neuronal activity, mean HU activity and their correlations, respectively) to the empirical data statistics (Equation 14). In order to optimize the two free parameters of the cRBM model – the sparsity regularization parameter $λ$ and the total number of HUs $M$ – we assessed the cRBM performance for a grid of $(λ, M)$ -values for one data set (fish #3). This analysis yielded an optimum for $λ = 0.02$ and $M = 200$ (Figure 2—figure supplement 1). These values were subsequently used for all recordings, where $M$ was scaled with the number of neurons $N$ .

We trained cRBMs on 70% of the recording length, and compared the statistics of model-generated data to the withheld test data set (the remaining 30% of recording, see Materials and methods ‘Train / test data split’ and ‘Assessment of data statistics’ for details). After convergence, the cRBM generated data that replicated the training statistics accurately, with normalized Root Mean Square Error (nRMSE) values of ${nRMSE}_{⟨ v_{i} ⟩} = 0.11$ , ${nRMSE}_{⟨ h_{μ} ⟩} = 0.15$ and ${nRMSE}_{⟨ v_{i} h_{μ} ⟩} = 0.09$ (Figure 2C-E). Here, nRMSE is normalized such that 1 corresponds to shuffled data statistics and 0 corresponds to the best possible RMSE, i.e., between train and test data.

We further evaluated cRBM performance to assess its ability to capture data statistics that the cRBM was not explicitly trained to replicate: the pairwise correlations between neurons $⟨ v_{i} v_{j} ⟩$ and the pairwise correlations between HUs $⟨ h_{μ} h_{ν} ⟩$ . We found that these statistics were also accurately replicated by model-generated data, with ${nRMSE}_{⟨ v_{i} v_{j} ⟩} = - 0.09$ (meaning that the model slightly outperformed the train-test data difference) and ${nRMSE}_{⟨ h_{μ} h_{ν} ⟩} = 0.17$ (Figure 2F, G). The fact that cRBM also accurately replicated neural correlations $⟨ v_{i} v_{j} ⟩$ (Figure 2F) is of particular relevance, since this indicates that (1) the assumption that neural correlations can be explained by their shared assemblies is justified and (2) cRBMs may provide an efficient mean to model neural interactions of such large systems ( $N \sim 10^{4}$ ) where directly modeling all $N^{2}$ interactions would be computationally infeasible or not sufficiently constrained by the available data.

Next, we assessed the reconstruction quality after neural data was compressed by the cRBM low-dimensional bottleneck. The reconstruction quality is defined as the log-likelihood of reconstructed neural data $v_{recon}$ (i.e. $v$ that is first transformed to the low-dimensional $h$ , and then back again to the high-dimensional $v_{recon}$ , see Materials and methods - ‘Reconstruction quality’). This is important to prevent trivial, undesired solutions like $w_{i, μ} = 0 \forall i, μ$ which would directly lead to ${⟨ h_{μ} ⟩}_{P (v, h)} = {⟨ h_{μ} ⟩}_{data} = 0$ (potentially because of strong sparsity regularization). Figure 2H shows the distribution of cRBM reconstruction quality of all neurons (in purple), quantified by the normalized log-likelihood (nLLH) such that 0 corresponds to an independent model $(P (v_{i} (t)) = ⟨ v_{i} ⟩)$ and 1 corresponds to perfect reconstruction (non-normalized $LLH = 0$ ). For comparison, we also reconstructed the neural activity using a fully connected Generalized Linear Model (GLM, see Materials and methods - ‘Generalized Linear Model’ and Figure 2, Figure 2—figure supplement 2H, blue). The cRBM nLLH distribution is significantly greater than the GLM nLLH distribution (one-sided Mann Whitney U test, $P < 10^{- 42}$ ), with medians ${LLH}_{cRBM} = 0.24$ and ${LLH}_{GLM} = 0.20$ . Hence, projecting the neural data onto the low-dimensional representation of the HUs does not compromise the ability to explain the neural activity. In fact, reconstruction quality of the cRBM slightly outperforms the GLM, possibly due to the suppression of noise in the cRBM estimate. The optimal $(λ = 0.02, M = 200)$ choice of free parameters was selected by cross-validating the median of the cRBM reconstruction quality, together with the normalized RMSE of the five previously described statistics (Figure 2—figure supplement 1).

Lastly, we confirmed that the cRBM indeed resides in the compositional phase, characterized by $1 ≪ m (t) ≪ M$ where $m (t)$ is the number of HUs active at time point $t$ (Figure 2—figure supplement 3A). This property is a consequence of the sparse weight matrix $W$ , indicated by its heavy-tail log-distribution (Figure 2I, purple). The compositional phase is the norm for the presently estimated cRBMs, evidenced by the distribution of median $m (t)$ values for all recordings (average $\frac{median (m)}{M}$ is 0.26, see Figure 2—figure supplement 3B). Importantly, the sparse weight matrix does not automatically imply that only a small subset of neurons is connected to the cRBM hidden layer. We validated this by observing that more neurons strongly connect to the hidden layer than expected by shuffling the weight matrix (Figure 2J). Further, we quantified the number of assemblies that each neuron was embedded in, which showed that increasing the embedding threshold did not notably affect the fraction of neurons embedded in at least 1 assembly (93–94%, see Figure 2—figure supplement 4). To assess the influence of $M$ and $λ$ on the inferred assemblies, we computed, for all cRBM models trained during the optimization of $M$ and $λ$ , the distribution of assembly sizes (Figure 2—figure supplement 5A-F). We found that $M$ and $λ$ controlled the distribution of assembly sizes in a consistent manner: assembly size was a gradually decreasing function of both $M$ and $λ$ (two-way ANOVA, both $P < 10^{- 3}$ ). Furthermore, for $M$ and $λ$ values close to the optimal parameter-setting ( $M = 200$ , $λ = 0.02$ ), the changes in assembly size were very small and gradual. This showcases the robustness of the cRBM to slight changes in parameter choice.

Sparsity facilitated that each assembly only connects to a handful of anatomical regions, as we quantified by calculating the overlap between cRBM assemblies and anatomical regions (Figure 2—figure supplement 6). We found that cRBM assemblies connect to a median of three regions (interquartile range: 2–6 regions). Importantly, the cRBM has no information about the locations of neurons during training, so the localization to a limited set of anatomical areas that we observe is extracted from the neural co-activation properties alone. For comparison, Principal Component Analysis (PCA), a commonly used non-sparse dimensionality reduction method that shares the cRBM architecture, naturally converged to a non-sparse weight matrix (Figure 2I, yellow), with fewer connected neurons than expected by shuffling its weight matrix (Figure 2K). This led to unspecific assemblies that are difficult to interpret by anatomy (Figure 2—figure supplement 6). As a result, sparsity, a cRBM property shared with some other dimensionality reduction techniques, is crucial to interpret the assemblies by anatomy as we demonstrate in the next section.

We next asked whether sparsity alone was sufficient for a generative model to accurately recapitulate the neural recording statistics. To address this question, we trained sparse linear Variational Autoencoders (VAEs) using the same parameter-optimization protocol (Figure 2—figure supplement 7A). Like cRBMs, linear VAEs are generative models that learn a latent representation of a dataset (Tubiana et al., 2019a). We observed that VAEs were not able to replicate the second-order statistics, and therefore were not able to reconstruct neural activity from latent representation (Figure 2—figure supplement 7B-D), even though they also obtained sparse representations (Figure 2—figure supplement 7E, F). Other clustering or dimensionality reduction methods, such as k-means, PCA and non-negative matrix factorization, have been used previously to cluster neurons in the zebrafish brain (Chen et al., 2018; Mu et al., 2019; Marques et al., 2020). However, because these methods cannot generate artificial neural data using their inferred assemblies, their quality cannot be quantitatively assessed as we have done for the cRBM (but see Tubiana et al., 2019a for other comparisons).

cRBM assemblies compose functional circuits and anatomical structures

Above, we have shown that cRBMs converge to sparse weight matrix solutions. This property enables us to visualize the cRBM-inferred neural assemblies as the collection of significantly connected neurons to an HU. Neurons from a given neural assembly display concerted dynamics, and so one may expect their spatial organization to reflect the neuroanatomy and functional organization of the brain. We here highlight a selection of salient examples of neural assemblies, illustrating that assemblies match well with anatomical structures and functional circuits, while the complete set of neural assemblies is presented in Video 1. In particular, we identified assemblies that together compose a neural circuit, are neurotransmitter-specific, encompass a long-range pathway, or can be identified by anatomy. The examples shown here are from a single fish (#3), but results from other fish were comparable.

Video 1

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posterframe for video — All neural assemblies of one example fish All 200 inferred assemblies of the example fish #2 of Figure 3 are shown in sequence.

Top: neural assembly. Bottom: HU activity of test data.

First, we identified six assemblies that together span the hindbrain circuit that drives eye and tail movements (Dunn et al., 2016; Wolf et al., 2017; Chen et al., 2018). We find two neural assemblies in rhombomere 2 which align with the anterior rhombencephalic turning region (ARTR, Ahrens et al., 2013; Dunn et al., 2016; Wolf et al., 2017, Figure 3A, B). Each assembly primarily comprises neurons of either the left or right side of the ARTR, but also includes a small subset of contralateral neurons with weights of opposite sign in line with the established mutual inhibition between both subpopulations. Two other symmetric assemblies (Figure 3C, D) together encompass the oculomotor nucleus (nIII) and the contralateral abducens nucleus (nVI, in rhombomere 6), two regions engaged in ocular saccades (Ma et al., 2014) and under the control of the ARTR (Wolf et al., 2017). Additionally, we observed two symmetric assemblies (Figure 3E, F) in the posterior hindbrain (in rhombomere 7), in a region known to drive unilateral tail movements (Chen et al., 2018; Marques et al., 2020) and whose antiphasic activation is also controlled by the ARTR activity (Dunn et al., 2016).

Figure 3

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cRBM assemblies compose functional circuits and anatomical structures.

(**A–I**) Individual example assemblies μ are shown by coloring each neuron $i$ with its connectivity weight value $w_{i, μ}$ (see color bar at the right hand side). The assembly index μ is stated at the bottom of each panel. The orientation and scale are given in panel A (Ro: rostral, C: caudal, R: right, L: left, D: dorsal, V: ventral). Anatomical regions of interest, defined by the ZBrain Atlas (Randlett et al., 2015), are shown in each panel (Rh: rhombomere, nMLF: nucleus of the medial longitudinal fascicle; nIII: oculomotor nucleus nIII, Cl: cluster; Str: stripe, P. nV TG: Posterior cluster of nV trigeminal motorneurons; Pa: pallium; Hb: habenula; IPN: interpeduncular nucleus). (**J–L**) Groups of example assemblies that lie in the same anatomical region are shown for cerebellum (Cb), torus semicircularis (TSC), and optic tectum (OT). Neurons i were defined to be in an assembly μ when $| w_{i, μ} | > 0.15$ , and colored accordingly. If neurons were in multiple assemblies shown, they were colored according to their strongest-connecting assembly.

Next, we observed assemblies that correspond to particular neurotransmitter expressions in the ZBrain atlas (Randlett et al., 2015), such as the excitatory Vglut2 (Figure 3G) and inhibitory Gad1b (Figure 3H) neurotransmitters. These assemblies consist of multiple dense loci that sparsely populate the entire brain, confirming that cRBMs are able to capture a large morphological diversity of neural assemblies. Figure 3I depicts another sparse, brain-wide assembly that encompasses the pallium, habenula (Hb) and interpeduncular nucleus (IPN), and thus captures the Hb-IPN pathway that connects to other regions such as the pallium (Beretta et al., 2012; Bartoszek et al., 2021).

Larger nuclei or circuits were often composed of a small number of distinct neural assemblies with some overlap. For example, the cerebellum was decomposed into multiple, bilateral assemblies (Figure 3J) whereas neurons in the torus semicircularis were grouped per brain hemisphere (Figure 3K). As a last example, the optic tectum was composed of a larger set of approximately 18 neural assemblies, which spatially tiled the volume of the optic tectum (Figure 3L). This particular organization is suggestive of spatially localized interactions within the optic tectum, and aligns with the morphology of previously inferred assemblies in this specific region (Romano et al., 2015; Diana et al., 2019; Triplett et al., 2020). However, Figure 3 altogether demonstrates that the typical assembly morphology of the optic tectum identified by our and these previous analyses does not readily generalize to other brain regions, where a large range of different assembly morphologies compose neural circuits.

Overall, the clear alignment of cRBM-based neural assemblies with anatomical regions and circuits suggests that cRBMs are able to identify anatomical structures from dynamical activity alone, which enables them to break down the overall activity into parts that are interpretable by physiologists in the context of previous, more local studies.

HU dynamics cluster into groups and display slower dynamics than neurons

HU activity, defined as the expected value of $P (h | v)$ (Equation 9), exhibits a rich variety of dynamical patterns (Figure 4A). HUs can activate very transiently, slowly modulate their activity, or display periods of active and inactive states of comparable duration. Figure 4B highlights a few HU activity traces that illustrate this diversity of HU dynamics. The top three panels of Figure 4B show the dynamics of the assemblies of Figure 3A-F which encompass the ARTR hindbrain circuit that controls saccadic eye movements and directional tail flips. HUs 99 and 161 drive the left and right ARTR and display antiphasic activity with long dwell times of ∼15s, in accordance with previous studies (Ahrens et al., 2013; Dunn et al., 2016; Wolf et al., 2017). HU 102 and 163 correspond to the oculomotor neurons in the nuclei nIII and nVI that together drive the horizontal saccades. Their temporal dynamics are locked to that of the ARTR units in line with the previously identified role of ARTR as a pacemaker for the eye saccades (Wolf et al., 2017). HUs 95 and 135, which drive directional tail flips, display transient activations that only occur when the ipsilateral ARTR-associated HU is active. This is consistent with the previous finding that the ARTR alternating activation pattern sets the orientation of successive tail flips accordingly (Dunn et al., 2016). The fourth panel shows the traces of the brain-wide assemblies of Figure 3G, I, displaying slow tonic modulation of their activity. Finally, the bottom panel, which corresponds to the collective dynamics of assembly 122 (Figure 3H), comprises short transient activity that likely corresponds to fictive swimming events.

Figure 4

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HU dynamics are bimodal and activate slower than neurons.

(A) HU dynamics are diverse and are partially shared across HUs. The bimodality transition point of each HU was determined and subtracted individually, such that positive values correspond to HU activation (see Materials and methods - ‘Time constant calculation’6.12). The test data consisted of three blocks, with a discontinuity in time between the first and second block (Materials and methods). (B) Highlighted example traces from panel A. HU indices are denoted on the right of each trace, colored according to their cluster from panel D. The corresponding cellular assemblies of these HU are shown in Figure 3A-I. (C) Top: Pearson correlation matrix of the dynamic activity of panel A. Bottom: Hierarchical clustering of the Pearson correlation matrix. Clusters (as defined by the colors) were annotated manually. This sorting of HUs is maintained throughout the manuscript. OT: Optic Tectum, Di: Diencephalon, ARTR: ARTR-related, Misc.: Miscellaneous, L: Left, R: Right. (D) A Principal Component Analysis (PCA) of the HU dynamics of panel A shows that much of the HU dynamics variance can be captured with a few PCs. The first 3 PCs captured 52%, the first 10 PCs captured 73% and the first 25 PCs captured 85% of the explained variance. (E) The distribution of all HU activity values of panel A shows that HU activity is bimodal and sparsely activated (because the positive peak is smaller than the negative peak). PDF: Probability Density Function. (F) Distribution of the time constants of HUs (black) and neurons (grey). Time constants are defined as the median oscillation period, for both HUs and neurons. An HU oscillation is defined as a consecutive negative and positive activity interval. A neuron oscillation is defined as a consecutive interspike-interval and spike-interval (which can last for multiple time steps, for example see Figure 1A). The time constant distribution of HUs is greater than the neuron distribution (Mann Whitney U test, $P < 10^{- 16}$ ).

Some HUs regularly co-activate, leading to strong correlations between different HUs. This is quantified by their Pearson correlation matrix shown in Figure 4C (top), which reveals clusters of correlated HUs. These were grouped using hierarchical clustering (Figure 4C, bottom), and we then manually identified their main anatomical location (top labels). These clusters of HUs with strongly correlated activity suggest that much of the HU variance could be captured using only a small number of variables. We quantified this by performing PCA on the HU dynamics, finding that indeed 52% of the variance was captured by the first three PCs, and 85% by the first 20 PCs (Figure 4D). We further observed that HU activity is bimodal, as evidenced by the distribution of all HU activity traces in Figure 4E. This bimodality can emerge because the dReLU potentials $U_{μ}$ (Equation 13) can learn to take different shapes, including a double-well potential that leads to bimodal dynamics (see Materials and methods - ‘Choice of HU potential’). This allows us to effectively describe HU activity as a two-state system, where $h_{μ} (t) > 0$ increases the probability to spike ( $P (v_{i} (t) = 1)$ ) for its positively connected neurons, and $h_{μ} (t) < 0$ decreases their probability to spike. The binarized neuron activity is also a two-state system (spiking or not spiking), which enabled us to compare the time constants of neuron and HU state changes, quantified by the median time between successive onsets of activity. We find that HUs, which represent the concerted dynamics of neuronal assemblies, operate on a slower time scale than individual neurons (Figure 4F, Figure 2—figure supplement 5G-L). This observation aligns with the expected difference between cellular and circuit-level time scales.

cRBM embodies functional connectivity that is strongly correlated across individuals

The probabilistic nature of cRBMs uniquely enables in silico perturbation experiments to estimate the functional connection $J_{i j}$ between pairs of neurons, where $J_{i j}$ is quantified by directly perturbing the activity of neuron $j$ and observing the change in probability to spike of neuron $i$ . We first defined the generic, symmetric functional connection $J_{i j}$ using $P (v_{i} | v_{j}, v_{k \neq i, j})$ (Equation 15) and then used $P (v)$ (Equation 12) to derive the cRBM-specific $J_{i j}$ (Equation 17, see Materials and methods - ‘Effective connectivity matrix’). Using this definition of $J_{i j}$ , we constructed a full neuron-to-neuron effective connectivity matrix for each zebrafish recording. We then asked whether this cRBM-inferred connectivity matrix was robust across individuals. For this purpose, we calculated the functional connections between anatomical regions, given by the assemblies that occupy each region, because neuronal identities can vary across individual specimen. We aggregated neurons using the L₁ norm for each pair of anatomical regions to determine the functional connection between regions (see Materials and methods - ‘From inter-neuron to inter-region connectivity’). For this purpose, we considered anatomical regions as defined by the mapzebrain atlas (Kunst et al., 2019) for which a regional-scale structural connectivity matrix exists to which we will compare our functional connectivity matrix.

This led to a symmetrical functional connectivity matrix for each animal, three of which are shown in Figure 5A-C (where non-imaged regions are left blank, and all eight animals are shown in Figure 5—figure supplement 1). The strength of functional connections is distributed approximately log-normal (Figure 5D), similar to the distribution of structural region-to-region connections (Kunst et al., 2019). To quantify the similarity between individual fish, we computed the Pearson correlation between each pair of fish. Functional connectivity matrices correlate strongly across individuals, with an average Pearson correlation of 0.69 (Figure 5E and F).

Figure 5 with 1 supplement see all

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cRBM gives rise to functional connectivity that is strongly correlated across individuals.

(A) The functional connectivity matrix between anatomical regions of the mapzebrain atlas (Kunst et al., 2019) of example fish #2 is shown. Functional connections between two anatomical regions were determined by the similarity of the HUs to which neurons from both regions connect to (Materials and methods). Mapzebrain atlas regions with less than five imaged neurons were excluded, yielding $N_{MAP} = 50$ regions in total. See Supplementary file 1 for region name abbreviations. The matrix is shown in *log*₁₀ scale, because functional connections are distributed approximately log-normal (see panel D). (B) Equivalent figure for example fish #3 (example fish of prior figures). (C) Equivalent figure for example fish #4. Panels A-C share the same *log*₁₀ color scale (right). (D) Functional connections are distributed approximately log-normal. (Mutual information with a log-normal fit (black dashed line) is 3.83, while the mutual information with a normal fit is 0.13). All connections of all eight fish are shown, in *log*₁₀ scale (purple). (E) Functional connections of different fish correlate well, exemplified by the three example fish of panels A-C. All non-zero functional connections (x-axis and y-axis) are shown, in *log*₁₀ scale. Pearson correlation $r_{P}$ between pairs: $r_{P} (# 2, # 3) = 0.73$ , $r_{P} (# 2, # 4) = 0.73$ , $r_{P} (# 3, # 4) = 0.78$ . All correlation p values $< 10^{- 20}$ (two-sided t-test). (F) Pearson correlations $r_{P}$ of region-to-region functional connections between all pairs of 8 fish. For each pair, regions with less than five neurons in either fish were excluded. All p values $< 10^{- 20}$ (two-sided t-test), and average correlation value is 0.69.

We conclude that similar functional circuits spontaneously activate across individuals, despite the limited duration of neural recordings (∼25 minutes), which can be identified across fish using independently estimated cRBMs. In the next section, we aggregate these individual matrices to a general functional connectivity matrix for comparison with the zebrafish structural connectivity matrix.

cRBM-inferred functional connectivity reflects structural connectivity

In the previous section we have determined the functional connections between anatomical regions using the cRBM assembly organization. Although functional connectivity stems from the structural (i.e. biophysical) connections between neurons, it can reflect correlations that arise through indirect network interactions (Bassett and Sporns, 2017; Das and Fiete, 2020). Using recently published structural connectivity data of the zebrafish brain (Kunst et al., 2019), we are now able to quantify the overlap between a structurally defined connectivity matrix and our functional connectivity matrix estimated through neural dynamics. Kunst et al., 2019 determined a zebrafish structural connectivity matrix between 72 anatomical regions using structural imaging data from thousands of individually Green Fluorescent Protein (GFP)-labeled neurons from multiple animals. We slightly extended this matrix by using the most recent data, filtering indirect connections and accounting for the resulting sampling bias (Figure 6A, regions that were not imaged in our light-sheet microscopy experiments were excluded). Next, we aggregated the functional connectivity matrices of all our calcium imaging recordings to one grand average functional connectivity matrix (Figure 6B).

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cRBM-inferred functional connectivity reflects structural connectivity.

(A) Structural connectivity matrix is shown in *log*₁₀ scale, updated from Figure 8C of Kunst et al., 2019. Regions that were not imaged in our experiments were excluded (such that $N_{MAP} = 50$ out of 72 regions remain). Regions (x-axis and y-axis) were sorted according to Kunst et al., 2019. Compared to Figure 8C of Kunst et al., 2019 additional structural data was added and the normalization procedure was updated to include within-region connectivity (see Materials and methods - ‘Extensions of the structural connectivity matrix’). See Supplementary file 1 for region name abbreviations. (B) Average functional connectivity matrix is shown in *log*₁₀ scale, as determined by averaging the cRBM functional connectivity matrices of all 8 fish (see Materials and methods - ‘Specimen averaging of connectivity matrices’). The same regions (x-axis and y-axis) are shown as in panel A. (C) The average functional and structural connectivity of panels A and B correlate well, with Spearman correlation $r_{S} = 0.39$ ( $P < 10^{- 20}$ , two-sided t-test). Each data point corresponds to one region-to-region pair. Data points for which the structural connection was exactly 0 were excluded (see panel D for their analysis). (D) The distribution of average functional connections of region pairs with non-zero structural connections is greater than functional connections corresponding to region pairs without structural connections ( $P < 10^{- 15}$ , two-sided Kolmogorov-Smirnov test). The bottom panel shows the evidence for inferring either non-zero or zero structural connections, defined as the fraction between the PDFs of the top panel (fitted Gaussian distributions were used for denoising).

For comparison, we also calculated the connectivity matrices defined by either covariance or Pearson correlation (Figure 6—figure supplement 1). The cRBM functional connectivity spans a larger range of values than either of these methods, leading to a more fine-grained connectivity matrix akin to the structural connectivity map (Figure 6B). This greater visual resemblance was statistically confirmed by calculating the Spearman correlation between structural and functional connectivity, which is greater for cRBM ( $r_{S} = 0.39$ , Figure 6C), than for covariance-based connectivity ( $r_{S} = 0.18$ , Figure 6—figure supplement 1 left) or correlation-based connectivity ( $r_{S} = 0.26$ , Figure 6—figure supplement 1 right). Hence, using recordings of ∼25 min on average, cRBMs were able to identify functional connections that resemble the anatomical connectivity between brain regions. Strong or weak functional connections are predictive of present or absent structural connections respectively (Figure 6D), and could thus potentially be used for inference in systems where the structural connectivity pattern is unknown.

Discussion

We have developed a cRBM model that accurately replicated the data statistics of brain-scale zebrafish recordings, thereby forming neural assemblies that spanned the entire brain. The objective of our study was threefold: first, to show that the cRBM model can be applied to high-dimensional data, such as whole-brain recordings, second, to prove that an assembly-based model is sufficient to generate whole-brain neural data statistics, and third, to describe the physiological properties of the assembly organization in the zebrafish brain and use it to create a functional connectivity map. We have shown that, after convergence, the cRBM-generated data not only replicated the data statistics that it was constrained to fit, but also extrapolated to fit the pairwise correlation statistics of neurons and HUs, leading to a better reconstruction of neural data than a fully connected GLM (Figure 2). These results thereby quantify how neural assemblies play a major role in determining the collective dynamics of the brain. To achieve this, cRBMs formed sparsely localized assemblies that spanned the entire brain, facilitating their biological interpretation (Figures 3 and 4, Figure 2—figure supplement 6). Further, the probabilistic nature of the cRBM model allowed us to create a mesoscale functional connectivity map that was largely conserved across individual fish and correlated well with structural connectivity (Figures 5 and 6).

The maximum entropy principle underlying the cRBM definition has been a popular method for inferring pairwise effective connections between neurons or assemblies of co-activating cells (Schneidman et al., 2006; Tavoni et al., 2017; Ferrari et al., 2017; Meshulam et al., 2017; Posani et al., 2018; Chen et al., 2019). However, its computational cost has limited this pairwise connectivity analysis to typically $\sim 10^{2}$ neurons. The two-layer cRBM model that we used here alleviates this burden, because the large number of neuron-to-neuron connections are no longer explicitly optimized, which enables a fast data sampling procedure (Figure 2B). However, we have shown that these connections are still estimated indirectly with high accuracy via the assemblies they connect to (Figure 2F). We have thus shown that the cRBM is able to infer the $\frac{1}{2} N^{2} \approx 10^{9}$ (symmetric) pairwise connections through its assembly structure, a feat that is computationally infeasible for many other methods. By implementing various algorithmic optimizations (Materials and methods - ‘Algorithmic Implementation’), cRBM models converged in approximately 8–12 hr on high-end desktop computers (also see Materials and methods - ‘Computational limitations’).

Previously, we have extensively compared cRBM performance to other dimensionality reduction techniques, including Principal Component Analysis (PCA), Independent Component Analysis (ICA), Variational Autoencoders (VAEs) and their sparse variants, using protein sequence data as a benchmark (Tubiana et al., 2019a). Briefly put, we showed that PCA and ICA could not accurately model the system due to their deterministic nature, putting too much emphasis on low-probability high-variance states, while VAEs were unable to capture all features of data due to the unrealistic assumption of independent, Gaussian-distributed latent variables. In this study, we repeated this comparison with sparse linear VAEs, and reached similar conclusions: VAEs trained using the same protocol as cRBMs failed to reproduce second-order data statistics and to reconstruct neural activity via the latent layer, while the learnt assemblies were of substantially lower quality (indicated by a large fraction of disconnected HUs, as well as a highly variable assembly size; Figure 2—figure supplement 7). Additionally, while PCA has previously been successful in describing zebrafish neural dynamics in terms of their main covariances modes (Ahrens et al., 2012; Marques et al., 2020), we show here that it is not appropriate for assembly extraction due to the absence of both a compositional and stochastic nature (Figure 2, Figure 2—figure supplement 6). Furthermore, we have shown that the generative component of cRBM models is essential for quantitatively assessing that the assembly organization is sufficient for reproducing neural statistics (Figure 2), moving beyond deterministic clustering analyses such as k-means (Panier et al., 2013; Chen et al., 2018), similarity graph clustering (Mölter et al., 2018) or non-negative matrix factorization (Mu et al., 2019) (see Supplementary file 2).

After having quantitatively validated the resultant assemblies, we moved to discussing the biological implications of our findings. Previous studies of the zebrafish optic tectum have identified neural assemblies that were spatially organized into single dense clusters of cells (Romano et al., 2015; Diana et al., 2019; Triplett et al., 2020). We have replicated these findings by observing the distinct organization of ball-shaped assemblies in the optic tectum (Figure 3L). However, our data extends to many other anatomical regions in the brain, where we found that assemblies can be much more dispersed, albeit still locally dense, consisting of multiple clusters of neurons (Figure 3). In sum, cRBM-inferred cell assemblies display many properties that one expects from physiological cell assemblies: they are anatomically localized, can overlap, encompass functionally identified neuronal circuits and underpin the collective neural dynamics (Harris, 2005; Harris, 2012; Eichenbaum, 2018). Yet, the cRBM bipartite architecture lacks many of the traits of neurophysiological circuits. In particular, cRBMs lack direct neuron-to-neuron connections, asymmetry in the connectivity weights and a hierarchical organization of functional dependencies beyond one hidden layer. Therefore, to what extent cRBM-inferred assemblies identify to neurophysiological cell assemblies, as postulated by Hebb, 1949 and others, remains an open question.

cRBM allowed us to compute the effective, functional connections between each pair of neurons, aggregated to functional connections between each pair of regions, by perturbing neural activity in silico. Importantly, we found that this region-scale connectivity is well-conserved across specimen. This observation is non-trivial because each recording only lasted ∼25 min, which represents a short trajectory across accessible brain states. It suggests that, although each individual brain may be unique at the neuronal scale, the functional organization could be highly stereotyped at a sufficiently coarse-grained level.

It would be naive to assume that these functional connections equate biophysical, structural connections (Das and Fiete, 2020). Both represent different, yet interdependent aspects of the brain organization. Indeed, we found that structural connectivity is well-correlated to functional connectivity, confirming that functional links are tied to the structural blueprint of brain connectivity (Figure 6). Furthermore, strong (weak) functional connections are predictive of present (absent) structural connections between brain regions, although intermediate values are ambiguous.

It will be crucial to synergistically merge structural and dynamic information of the brain to truly comprehend brain-wide functioning (Bargmann and Marder, 2013; Kopell et al., 2014). Small brain organisms are becoming an essential means to this end, providing access to a relatively large fraction of cells (Ahrens and Engert, 2015). To generate new scientific insights it is thus essential to develop analytical methods that can scale with the rapidly growing size of both structural and dynamic data (Helmstaedter, 2015; Ahrens, 2019). In this study, we have established that the cRBM can model high-dimensional data accurately, and that its application to zebrafish recordings was crucial to unveil their brain-scale assembly organization. In future studies, cRBMs could be used to generate artificial data whose statistics replicate those of the zebrafish brain. This could be used for further in silico ablation and perturbation studies with strong physiological footing, crucial for developing hypotheses for future experimental work (Jazayeri and Afraz, 2017; Das and Fiete, 2020). Lastly, the application of cRBMs is not specific to calcium imaging data, and can therefore be readily applied to high-dimensional neural data obtained by other recording techniques.

Materials and methods

Key resources table

Reagent type (species) or resource	Designation	Source or reference	Identifiers	Additional information
Software, algorithm	cRBM algorithm	This paper and Tubiana and Monasson, 2017	github.com/jertubiana/PGM	Materials and methods - ‘Restricted Boltzmann Machines’ , ‘Compositional Restricted Boltzmann Machine’ and Algorithmic Implementation
Software, algorithm	Fishualizer	Migault et al., 2018	bitbucket.org/benglitz/fishualizer_public
Software, algorithm	Blind Sparse Deconvolution	Tubiana et al., 2020	github.com/jertubiana/BSD
Software, algorithm	ZBrain Atlas	Randlett et al., 2015	engertlab.fas.harvard.edu/Z-Brain
Software, algorithm	mapzebrain atlas	Kunst et al., 2019	fishatlas.neuro.mpg.de
Software, algorithm	MATLAB (data preprocessing)	MathWorks	mathworks.com/products/matlab.html
Software, algorithm	Computational Morphometry Toolkit (CMTK)	NITRC	nitrc.org/projects/cmtk
Software, algorithm	Python	Python Software Foundation	python.org
Strain, strain background (Danio rerio, nacre mutant)	Tg(elavl3:H2B-GCaMP6f)	Quirin et al., 2016
Strain, strain background (Danio rerio, nacre mutant)	Tg(elavl3:H2B-GCaMP6s)	Vladimirov et al., 2014

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cRBMs construct Hidden Units by grouping neurons into assemblies.

cRBM is optimized to accurately replicate data statistics.

All neural assemblies of one example fish All 200 inferred assemblies of the example fish #2 of Figure 3 are shown in sequence.

cRBM assemblies compose functional circuits and anatomical structures.

HU dynamics are bimodal and activate slower than neurons.

cRBM gives rise to functional connectivity that is strongly correlated across individuals.

cRBM-inferred functional connectivity reflects structural connectivity.

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Thijs L van der Plas

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Jérôme Tubiana

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Guillaume Le Goc

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Geoffrey Migault

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Michael Kunst

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Herwig Baier

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Volker Bormuth

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Bernhard Englitz

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Georges Debrégeas

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