Learning dynamic representations of the functional connectome in neurobiological networks
Authors: Luciano Dyballa
Samuel Lang
Alexandra Haslund-Gourley
Eviatar Yemini
Steven W. Zucker
Venue: ICLR 2024 poster (2024)
Keywords: neural networks
non-negative tensor factorization
dynamic community detection
functional connectome
Primary area: applications to neuroscience & cognitive science
TL;DR: Tensor factorization and community detection are applied to brain-wide neuronal affinities to learn dynamic functional connectomes.

Abstract: The static synaptic connectivity of neuronal circuits stands in direct contrast to the dynamics of their function. As in changing community interactions, different neurons can participate actively in various combinations to effect behaviors at different times. We introduce an unsupervised approach to learn the dynamic affinities between neurons in live, behaving animals, and to reveal which communities form among neurons at different times. The inference occurs in two major steps. First, pairwise non-linear affinities between  neuronal traces from brain-wide calcium activity are organized by non-negative tensor factorization (NTF). Each factor specifies which groups of neurons are most likely interacting for an inferred interval in time, and for which animals. Finally, a generative model that allows for weighted community detection is applied to the functional motifs produced by NTF to reveal a dynamic functional connectome. Since time codes the different experimental variables (e.g., application of chemical stimuli), this provides an atlas of neural motifs active during separate stages of an experiment (e.g., stimulus application or spontaneous behaviors). Results from our analysis are experimentally validated, confirming that our method is able to robustly predict causal interactions between neurons to generate behavior.

Published as a conference paper at ICLR 2024
LEARNING DYNAMIC REPRESENTATIONS OF THE FUNCTIONAL CONNECTOME IN NEUROBIOLOGICAL NETWORKS
Luciano Dyballa', Samuel Lang", Alexandra Hashnd-GGurley',, Eviatar Yemini Steven W. Zucker'
DDppt. Computer Science, Yale University; 'Dept. Neurobiology, UMass Chan Medical School
ABSTRACT
The static synaptic connectivity of neuronal circuits stands in direct contrast the dynamics of their function. As in changing community interactions, differ- ent neurons participate actively in various combinations to effect behaviors at different times. We introduce an unsupervised approach to learn the dynamic affinities between neurons in live, behaving animals, and to reveal which commu- nities form among neurons at different times. The inference occurs in two major steps [ First, pairwise non-linear affinities between neuronal traces from brain- wide calcium activity are organized by non-negative tensor factorization (NTF). Each factor specifies which groups of neurons are most likely interacting for an inferred interval in time, and for which animals. Finally, a generative model that allows for weighted community detection is applied the functional motifs pro- duced by NTF to reveal dynamic functional connectome. Since time codes the different experimental variables (c.g.. application of chemical stimuli), this pro- vides an atlas of neural motifss active during separate stages of an experiment (e.g.. stimulus application or spontaneous behaviors). Results from our analysis are experimentally validated, confirming that method is able to robustly predict causal interactions between neurons to generate behavior.
INTRODUCTION
The connectome in ncurobiology might seem roughly analogous to the architecture of artificial neu- ral networks (ANNs) in artificial intelligence (AI), in the sense that it specifies the relevant structure of connections. In AI, engineering this architecture is an engaging part of problem formulation; in neurobiology, the structure of the connectome complicates the process of circuit dissection for several reasons. First, there are different types of connections between neurons: some are due to chemical synapses or electrical ones (gap junctions) (White al. 1986 Jarrell et al. 2012 Cook et al. 2019 Witvlict et al. 202T) 1), and others induced by, e.g., extrasynaptic signaling of monoamines or neuropeptides (Bentley al. 2016 Ripoll-Sánchez et al., 2022 Beets et al. 2023). Each one de- fines a distinct connectome (Fig, 1h). Secondly, unlike ANNs, in biology different connections can be engaged at any given moment (e.g., as result of neuromodulatory signals that alter system-wide states (Bargmann & Marder, 2013)). As in complex social networks, the effective-or functional connectome varies with task and animal state; these dynamics render the predictive power of the biological connectomes problematic. We agree with others working in different domains, e.g. [Bas- sett et al. (20TT): y; Skardingg al. (202T), that a functional connectome that is dynamic across time is required, given that number of phenomena have been observed wherein neurons flexibly change how they encode behavior a -eate-dependent manner (Atanas et al., (023). Our goal is to learn a representation of these time-varying communities of neurons organized by behavioral responses in a "simple" organism, Caenorhabditis elegans.
We present novel algorithm for learning the dynamic community organization within a weighted connectome, based on brain-wide activity measurements of each individual neuron. Our main con-
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tribution is to organize the connections estimated similarities between neural activity-ratiin than the neurons themselves. differs from previous attempts by considering the full system of similarities and animals across time, rather than taking a step-by-step statistical approach (Varshney et al., 2011 Bentley et al. 2016; Cook et al., 2019; Uzel et al. 2022). Our approach combines two previously unrelated components: (7) the use of tensor factorization reveal putative groups of related neurons/animals in time, which are then (ii) passed to a community detection algorithm. Al- though the emphasis here on the algorithm, our results are confirmed with experiments that silence specific neurons predicted by our method to directly measure their impact on behavior. Moreover, even though the dynamic functional connectomes we reveal are specific our approach is widely applicable to learning representations of dynamic communities of neurons in other organisms, as well as other domains, such fMRI and social networks.
elegans, we believe
behavioral neural activity
a
b
Syn
worms
MA
NP
static connectemes
dynamic functional connectome
Figure 1: From static dynamic connectomes. (a) Different modes of neural communication are represented by substantially different connectomes. Shown here are chemical synapses (Syn). gap junctions (GJ), monoamines (MA), and neuropeptide (NP) connectomes for the nematode worm C. elegans (image adapted from Bentley et al. (2016)). How these relate to behavior has been problematic, however. The immediate neighborhoods for neuron AWB are shown (b) from Syn and (c) from GJ; which governs AWB communication and when? (d, top) The dataset consists of activity traces from each neuron in C. elegans across time; dashed lines illustrate stimulus presentation of e.g. repulsive chemical that is directly sensed by the worm. Note how the traces differ across time and across worms. (d, bottom) Different modules, or communities of neurons, become active different time periods, revealing how neurons interact dynamically to encode behavior. Our goal is to infer these communities comprising the dynamic functional connectome.
2 METHODS
is now possible, using the "NeuroPAL." method (Yemini et al. 2021), to determine both the cell- type-identity of every neuron and also record its activity in C. elegans across time in awake, and even behaving animals (Dag et al. 2023). Previously, neural activity traces from multiple worms have been analyzed using traditional statistics such as correlation (e.g.. Kato al. (2015); Yemini et al. (202T); Susoy et al. (2021): Randi et al.
(2023) ), small local motifs (Uzel et al. (022), or
more complex predictive models (Cecere et al. 2021 Dag al.) 2023) which assume that most neurons will respond in roughly the same way when exposed to the same stimulus. However, both past and recent studies indicate different ways that neurons can change how they encode behavior in state-dependent manner (Bargmann & Marder, 2013] Gordus et al. 2015 Atanas et al., (023), despite identical repeated presentations of chemosensory stimuli intended to produce stereotypical behavior. Multiple worms therefore needed to determine whether "typical" functional networks might activate after the introduction of a particular stimulus.
The third dimension (individual worms) in the data suggests using tensor rather than matrix for representation of neural activity (Williams et al. 2018) Dyballa et al,, 2024). A naive tensor formu- lation would be build it using the temporal activity directly: WORMS x NEURONS TEMPORAT.
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TRACES (e.g., Williams al. (2018). However, this would restrict the similarity computation be- tween traces to a multi-linear form, which can be ambiguous for imaging data. We propose instead a novel tensor formulation with dimensions: TIME WORMS X PAIRWISE AFFINITIES, where instan- taneous affinities (that may vary over time) are precomputed from the temporal traces of each pair of neurons. In this way. non-linearities between neurons separately from the factorization (see below). Moreover, given that neural affinities are expected to be state-dependent and changing over time, the tensor components will be able to provide a summarized description of which communities or sub-networks are active and when. We demonstrate this in Results.
he introduced by performing the affinity computations
2.1 INTERPRETABLE, INSTANTANEOUS AFFINITIES; FIG.
Experimental recordings of C. elegans currently span several minutes to nearly half an hour in duration, during which it is likely that different groups of neurons will become active at different times. Affinities should thus be function of time, rather than the more traditional method of computing global-time measure such as correlation (e.g.. Le..Ymiiiii et al. (2021); Randi et al. (003)). We compute, instead, local non-linear similarities between pairs of neural trace curves, which we term differential affinity.
Building tmee-aryynng similarity measure between traces is a subtle problem. For example, two neurons that are mostly silent throughout experiment will have small Euclidean distance (and high cosine similarity) between them, even though there may be no evidence that the neurons participate in the same circuit. Or, if they happen to maintain high, constant levels of activity at some point in time, then their distances will also be locally small, but they reached those levels several seconds apart, is unlikely that the neurons
actually interacting. The instantaneous trace values not meaningful individually: the temporal history of how they arrived at that level is highly relevant. We rely on the information present in the rates of monotonic change in activity, and compute what we term local differential affinity.
We compare two neurons' derivatives during intervals in which both had constant sign, i.e., peri- ods of monotonic increase or decrease in activity. The intuition is that two neurons with coinciding changes in activity are likely to be influencing one another, or being influenced by a common phe- nomenon, and thereby participating in the same functional circuit. The details of this computation are given in the Appendix; examples of the instantaneous local differential affinity are shown in Fig. 2
Affinities show how similar two curves are, locally, terms of their absolute derivatives, which is motivated by the fact that two neurons with very similar but opposite sign derivatives are still likely to be interacting by means of some inhibitory mechanism. As a result, they can be interpreted as how likely it is for the two neurons to be interacting (either directly or indirectly, perhaps via common input). The instantaneous pairwise affinities d at a time t 1...., T may be bundled into an affinity matrix, A which is used in the tensor incorporating information across multiple worms.
2.2 DYNAMIC AFFINITY TENSOR DECOMPOSITION FIG. 3
We now have a set of affinity matrices {A()), }. each one encoding an affinity network at a time t, on which a community detection algorithm could be run. This would yield collection of "instan- tameous" network communities. However, which instants should be selected? How long should they be? Are they independent? Rather than addressing these questions individually, we exploit the summarization power of non-negative tensor factorization to automatically cluster affinity net- works across time (and animals!): we seek contiguous intervals during which given affinity pattern is approximately preserved across several worms. Recall that our tensor has dimensions: TIME x WORMS x PAIRWISE AFFINITIES. (We note as an aside that this tensor can also help with "com- pleting" affinity matrices containing missing data from few neurons.) See Fig. for graphical illustration of the following discussion.
If there are neurons, A(t) has dimension the connectivity structure in A(t) to be decomposable as the product of two matrices. Therefore, instead of stacking the 2-1) matrices in 4-way tensor, we vectorize each A(L) by reshaping it
n. Note that there is no expectation for parts of
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A001
45
02
ANHL
03
02
Figure 2: Computation of dynamic differential affinity. (a, top) Activity traces of two neurons, AWCL and ASER, across time. Note that they share marked periods similar trends in activity. (a, middle) Smoothed derivatives of the top traces. Note how the derivatives agree during periods in which both traces vary together. (a, bottom) Affinity trace across time computed between AWCL and ASER (see section 2.1 and Appendix A. for details). Note how the "bumps" coincide with regions of potential interest. (b) Similar plots for a different neuron pair. Again, our non-linear affinity measure has low value except for periods when both neurons increase decrease their activities, i.e., are more likely to be interacting.
vector of dimension n(n - 1)/2, the total number of pairs (for computational efficiency, since they are symmetric, only the upper-triangular part is used).
In Canonical Polyadic (CP) decomposition (Carroll Chang 1970 Harshman et al. 1970 Kolda & Bader 2009), the goal is to approximate a n-way tensor T R by sum of rank-1 tensors. Letting R be the number of components chosen, we may express the specific case
3-way tensor (Kolda 2006);
T
(3)
where o stands for the vector outer product and ( is called factor matrix containing the factors (k) its columns. Each rank-1 tensor thus formed by the outer product between each factor the same component (a component refers to each set of associated factors). Most tensor decomposition algorithms use squared reconstruction error objective function (Cichocki al. 2009):
er
1 2
min
T T/12.
XO, x0,xe, (
where ||
is the norm of the vectorized tensor.
Since the data consist of non-negative neural affinities, we adopt a non-negative tensor decomposi- tion, also known non-negative tensor factorization (NTF) (Bro & De Jong 1997). which adds non-negativity constraint (4) 2 0, vk to eq. 2 Non-negative factorization is popularly used for matrices (NMF), and achieves parts- based representation with more easily-interpretabb compo- nents (Lee & Scung] 1999. Cichocki et al., 2009); additionally, it has been shown produce more stable factors, is less prone to overfitting, and has high parameter efficiency (Williams et al. 2018). We use hierarchical alternating least squares (HALS) algorithm (Cichocki & Phan 2009). im- plemented in the tensortool Python library (Williams) (024). Its efficiency when applied to real-world datasets has been compared against several other algorithms in |haa & Cichocki (2008).
Each component resulting from our tensor will be constituted by three factors: one representing a pattern fa of (vectorized) pairwise affinities; vector fe representing periods of time where such pat-
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olegans worms (NeuroPAL)
neural activity trases
temporal peinwise alliniles
warms
wormd
meuuee polre
nourons
Imaging acquistion
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affinily computation
time
FRIDGY
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tompors tacker
time
component.
component
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graph representation
community detection
effnntiie
painwise
aftintty matrix
waighted natwork
functional connectome
Figure 3: The procedure for inferring dynamic functional connectomes in three stages. (a) Stage I: Building affinities. Neurons behaving worms are imaged to yield matrix of activity (in time) for each neuron for each worm. The differential affinity computation compares traces to yield tensor of affinities (in time) for each neuron pair for each worm. (b) Stage 2: Non-negative tensor factorization yields components that reveal affinity patterns found for certain worms (bar plot) different time intervals (temporal curve). For example, component 2 indicates that neuron pairs with high affinity in worms 4 and 6 are active early in the experiment, and the last factor applies mainly to worm 2. (c) Stage 3: From affinities to a functional connectome. The affinity factor from selected tensor component is rearranged an affinity matrix, from which equivalent weighted graph is implicitly built. A community detection algorithm then reveals groupings of neurons behaving similarly over the indicated time interval.
tern was observed; and a vector fu whose loadings, or coefficients, indicate how much each worm's affinities can be described by fa those times expressed in ft. These components, combined, can be seen as a soft multi-clustering of worms in terms of their activity patterns and the times at which they occur. See Fig. 3b and Appendix A.4
2.3 NEURAL AFFINITY COMMUNITIES; FIG.
Notice that, because the neurons sorted in arbitrary way, looking at the affinity factor matrices should not be particularly enlightening. However, because affinities are non-negative and bounded, they can be readily treated as adjacencies (weighted edges) in graph where the nodes correspond to individual neurons. This enables us to use a community-detectio algorithm to cluster similar neurons based on their activity over time.
Although several methods for dynamic community detection (DCD) exist, using both traditional ML. (Rossetti & Cazabel 2018) and deep learning (Skarding et al. 2021), the vast majority are focused on unweighted networks (Li et al., 2021). That is because in many applications dynamic network refers to a graph that changes its number of nodes or edges over time, not their weights (with few exceptions, e.g., Guo et al. (2014)). Of course, for purposes the edge weights (i.e., affinity factors from NTF) are essential, as they convey the likelihood that two neurons interact over some period of time (section 2.1
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temporal factor
wom factor
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20
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weighted graph
A10
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noutons
nourons
Figure 4: Tensor components for a real worm during an experiment which repellent and attractive stimuli were applied. The application of these stimuli is important, since it provides correspondence between the time axes. (a) One tensor component that selects out the interval during which repulsive NaCI (salt) was applied. All worms reacted, but worms 5, 6, and 7 showed higher responses for certain pairs neurons (from the affinity factor, shown as an unsorted matrix). The weighted graph visualized with traditional community structure revealed by the NWSBM algorithm (b) provides a clear view of the neuronal modules involved in salt perception. (c) A different tensor component that selects several time intervals when stimuli were applied, but is especially involved with the attractant 2,3 ..3-pentanedione. Most worms again implicated, but the community of neurons (d) qualitatively different.
embedding is difficult interpret, but the nested
Since our tensor decomposition organizes the temporal dimension ie.. which networks are active which times seek community detection method appropriate for handling edge weights. Additionally, we wish to prevent spurious aggregations of nodes due to random fluctuations in the affinity values (from possible noise in the activity traces). Therefore it is preferable to infer latent community structure that carries explanatory power, opposed adopting classical descriptive methods that attempt to find communitics according to some preconceived notion of good division of the network into groups.
classical model for community structure the stochastic block model (SBM) (Holland et al. 1983), which groups nodes according to their probabilities of connection the rest of the network. Statistical inference can be used to find the best- fittiniiig model to the data. A principled approach for this task is formulate generative models that allow this modular decomposition be found

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via statistical inference (Aicher al. 2015 Peixoto 2019). In particular, the nested weighted-SBM. algorithm (NWSBM) (Peixoto 2018 2014) extends the SBM to be applied to weighted networks by treating edge weights as covariates, and employs Bayesian approach capable of inferring the number communities from the data in an unsupervised fashion. Moreover, it requires no hyper- parameter tuning-
For example, let the partition of the network into C' communities be denoted as c = {G}, where G E [0, C 1] the community membership of node A model be defined that generates network G with probability
P PGI
where 0 is set of parameters controlling how the node partition affects the structure of G. Assum- ing there is only one choice of that is compatible with the generated network, we may write the Bayesian posterior probability
P P(c P(G P(G)
(4)
The selected model is the one found yield the smallest description length (Peixoto 2015), namely
L -= - P(G|O, In c),,c),
(5)
since eq. above can be rewritten as
P (ccG) exp(-L) (G))
(6)
Such problem is NP-hard in general, so the likelihood is optimized stochastically. Hence, instead of assuming that there is a single "best fit" to the model, an efficient multi-flip Markov-Chain Monte Carlo (MCMC) algorithm performs model averaging to increase the robustness of the results; addi- tional details can be found in PPexxooo| (2018). This algorithm was chosen after running benchmark tests against several other popular community detection methods (section 3.3 Table I).
3 RESULTS
Our approach was applied to the dataset from Yemini et al. (202I), in which calcium activity from 189 neurons in the head of C. elegans were recorded from total of individual worms. Each worm performed a series of behavioral tasks in response to stimuli: one gustatory repellent (NaCI) and two olfactory attractants (2-butanone and 2.3-pentanedione) were flushed into the worm surroundings over periods of 10 s. Different worms were exposed to the stimuli in 3 possible sequences. We restricted the data to sensory and inter-neurons, since those are more likely be directly responsive to the chemosensory stimuli used.
3.1 DYNAMIC FUNCTIONAL CONNECTOMES
Fig. fa.c shows informative factors resulting from running NTF on our tensor (section 2.3 Each affinity factor yields vectorized affinity matrix, which can be reorganized the upper-triangular part of a symmetric matrix (for ease visualization). This matrix in turn can be interpreted as a weighted adjacent matrix of network in which nodes represent individual neurons (section 2.3). The communities inferred by the NWSBM algorithm for each factor then apply to certain time win- dows and worms, weighted by their respective factor loadings, and represent transient functional connectome. Two examples are shown in Fig. 4p,d, with their respective temporal factors revealing that such neural interactions are mostly active during the presence of NaCl and pentanedione, re- spectively. A richly-connech individual community from the NaCI-sensing connectome is further analyzed in Fig.
3.2 VALIDATION EXPERIMENTS
Our algorithm predicted the involvement of multiple neurons in the salt-sensing (NaCI) circuit, as those neurons found clustered with the canonical NaCI-sensing neuron ASE. Among these predic- tions were several neurons not previously known play role in this salt-sensation circuit. For

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AWB
0.-
Noc3
ASE
115
IL.2
AWC
0.50
IL2D
0.00
AV.
60
52
123
25
150
250
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worn
Figure 5: A transient community for salt sensing. (a) We here highlight one community from the dendrogram shown in Fig. Fl (cf. inset below). By reverting back to the original activity traces, we can confirm that the individual neuronal responses agree with functional circuit organization, and that the affinity measure is meaningful.
(b) The temporal factor of the corresponding tensor compo- nent shows that this particular circuit is most prominently involved in a response to NaCI (salt), plus minor but negligiieee responses to the other stimuli (2-butanone and 2.3-pentanedione). (c) Individual activity traces for the most strongly connected neurons within the community, highlighted in proportion the instantaneous temporal loadings. Note that the traces are most similar precisely during the NaCI interval for several worms; traces are plotted from the three worms with highest loadings the corresponding worm factor (see Fig. 4p. (d) Note the complexity of the traces out- side this interval. In particular, even though in worm 7 all the selected neurons were correlated throughout the experiment, this was not true for other worms, which explains the NaCI specificity in the temporal factor. Notice that, because the affinities are computed from absolute derivatives, neurons will be strongly connected even when their traces have opposite signs.
example, our algorithm predicted that the worm's 's primary aversive olfactory neuron AWB Troemel et al. 1997) is substantially involved in salt sensation. Therefore, we decided to experimentally test this surprising prediction of a previously unknown role for AWB in sensing NaC1.
To do so, compared salt avoidance in worms where AWB was functional versus worms where AWB was silenced. Briefly, we used a standard "drop test" (Bazzicalupo Hilliard 2005) to mea- sure avoidance of salt, wherein drop of 160 mM NaCI bufter (as used in Yemini et al. (021)) was placed in front of worms, and their response was recorded as avoidance if they reversed, and non-avoidance if they continued moving forward. Each of our samples consisted of 25-50 worms that were drop tested, and collectively scored for their mean avoidance. We tested 12 such sam- ples of functional AWB controls versus 12 samples with AWB silenced. To silence AWB, we used published collection of worms (Wang et al. (017), and crossed an AWB neuron-specific driver (syIs666) to an inhibitory histamine-gatee chloride channel (HisCI1-sylss73) effector (Pokala et al.) 2014). These AWB>HisCII worms (strain YYY24, with genotype syls666:syls373) have a func- tional AWB that is silenced upon treatment with histamine.
We found that silencing AWB substantially increased salt avoidance (p-value 5.7 mean effect size = +25%); see Fig. A2 (Appendix A.3). Our results are particularly striking since AWB is an aversive neuron and thus silencing it would be expected to decrease avoidance rather than increase as we observed. This suggests the potential for oppositional interactions between salt sensation and olfactory aversion circuits. Our results not only validate the predictive power of our algorithm,
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Table 1: Mean NMI scores of algorithms applied the weighted-LFR benchmark
Algorithm
Net
Net 2
Net 3
Net 4
Net 5
Net 6
Net 7
Net 8
Net
NWSBM
0.28
0.33
0.39
0.65
0.39
0.54
0,25
0.51
0.64
Louvain
0.08
0.19
0.13
0.04
0.23
0.29
0.12
1.0
0.19
Combo
0.13
0.18
0.14
0.04
0.23
0.28
0.12
1.0
0.16
AS
0.18
0.16
0.16
0.47
0.20
0.12
0,28
0.20
0.20
NNSED
0.0
0.18
0.38
0.32
0.39
0.50
0.39
0.43
0,45
GNNS100
0.14
0.19
0.15
0.04
0.22
0.28
0.12
1.0
0.18
they further highlight its strength in predicting results that expert scientific researchers may find contradict assumptions they have based on published canonical neuronal roles.
3.3 COMPARISON WITH OTHER COMMUNITY DETECTION METHODS
The NWSBM method compared other popular algorithms for community detection, includ- ing classical methods such as Louvain (Blondel et al. 2008 Rossetti et al.) 2019) and Combo
(Sobolevsky et al.] 2014), based modularity maximization; asymptotic surprise (AS) (Traag et al. 2015): NNSED (Sun et al. 2017, RRozemeerzzki et al. 2020), a non-negative encoder-decoder
approach; and GNNS (Sobolevsky & Belyi] 2022 Belyi 2022), based recurrent graph neural networks. The quality of their results was evaluated using normalized mutual information (NMI) (Danon 2005). widely adopted measure of agreement between clustering assignments (For- tunato 2010). Because it requires ground truth not available for the vast majority of real network datasets with weighted edges-- the algorithms above were evaluated on benchmark of synthetic networks generated using the popular manrichinettiFFortunne Fortunato-Radicch (LFR) algorithm (Lanci- chinetti & Fortunato 2009), ) a standard tool for creating benchmarks with various types networks [Gopalan & Blei, 2013 Fortunato & Hric 2016 Yang et al. 2016). It provides several parameters that control the how the connections and their weights will be distributed within and across com- munities, allowing for rich variety of block structures to be generated. Our benchmark consisted of 9 different types of network (see Appendix [A.2] and Fig. A1 for details). Table shows the aver- age NMI scores obtained by each algorithm over ten instances of each type of network. Although most of the algorithms tested was able to score highest for at least one network type, NWSBM outperformed the others the majority of cases.
DISCUSSION AND CONCLUSION
We present here algorithm for dynamic connectome discovery in the nematode worm C. ele- gans. The key insight was to use the differential affinity between neurons in tensor factorization approach, rather than (the more common) use of individual neurons. The advantages were (i) that non-linear methods could be used in calculating affinities before they were tensorized; (ii) the tem- poral factor in each tensor component revealed experimental epochs during which functional circuits appeared; and (iii) the affinity factors could be remade into weighted graphs which community detection algorithms could be run. the end, the algorithm was able to make (surprising) pre- dictions about individual neurons that were involved in unexpected functional roles, which were experimentally confirmed.
While this last point illustrates role for machine learning in biological research, also showed that reverting the community structure back to the original affinities (Fig. (5) can explain how the original traces caused given pair of nodes to be ultimately grouped together.
Finally, we believe that our approach could be used more widely in understanding community be- havior not only in neuroscience, but also in social and ethological situations. Working with the higher-order affinities directly informed the biology in our case; this is most likely true for other dynamical interactions well.

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ACKNOWLEDGMENTS
Supported by NIH Grant IR01EY031059, NSF Grant 1822598.
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A APPENDIX
A.1 LOCAL DIFFERENTIAL AFFINITIES
We begin by computing each neuron i's derivative (t(t) over time, pre-smoothing the raw trace 7t) with a Gaussian filter of bandwidth 0.5 sec to avoid the amplification of any measurement noise (Fig. 2 top panels). Notice that periods where the derivative sign is constant must necessarily start and end at points where the derivative is either exactly zero changes sign, and are commonly termed "zero-crossings". Because sampling is discrete over time, under a sampling interval zero-crossing is computed as those time points / such that
g - T/2)g (1 - 0,
(7)
i.e. there a change of sign between two consecutive samples, and
lo:(t T/2)| + (gx + T/2)] O,
(8)
i.e., the derivatives are not both exactly zero. zero-crossing, and denote it as
When both conditions are met, we say g:(t) is
gt
We then partition (t) by those intervals where it has constant sign (i.e., those lying between cero-crossings); we call these humps. Thus the support of ge(t) (subset of the time domain dur- ing which the neuron's activity is not constant), may be represented as set of m time intervals
(2) b. ((m), where 6(") denotes the interval (fo. f1), such that
(to) ~ 0 (t2).
(10)
Thus, two bumps ( and 1/ from neurons i and j. respectively, will overlap when
6/ 0.
(11)
The affinity az between neurons and j at time is then defined as the fraction of overlap between the areas under their derivative bumps occurring at t: b, and by respectively. This is analogous the Jaccard index:
area (b.) n (by)
(12)
arca(b: b.) U (r;)
where area(b;) is the unsigned area under the bump b;:
ab)) = =/h t by dt.
(13)
with the integral taken over the time interval under each bump. Note that these affinity values lie in the range [0, 1], and because they are computed locally, will be sensitive to spurious fluctuations of activity. To prevent this, we take into account global information about the traces and weight this by the relative change in trace levels during those local bumps compared to the global range of values of (t).
Notice that the areas are unsigned, meaning we ignore the signs of the bumps. This motivated by the fact that neurons with very similar but opposite sign derivatives are still likely to be interacting by means of some inhibitory mechanism. This further contributes to making the affinities interpretable as indicating the likelihood of interactions between pairs of neurons, regardless of the specific physiological connectivity mechanisms involved.
A.2 WEIGITED COMMUNITIES BENCHMARK
A total of weighted networks with N=125 nodes (compatible with the size of our affinity net- works) were generated using the weighted LFR benchmark (Lancichinetti & Fortunato) 2009). The mean neighborhood size influences the resulting number of communities produced. How such neighborhoods are distributed between intra- and inter-community connections is determined by the
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topology ((4) and weight (Hw)) mixing parameters, respectively. The weights within node's neigh- borhood are sampled from power-law distribution with exponent 5. Finally, the degree sequence and community sizes drawn from power-law distributions with exponents -1 and -22- respec- tively. By choosing different sets of parameters for each network type, we aimed to vary several properties: the number of communities (determined by the k/N ratio the density of inter-block connections (Hwo/ Me ratio); as well as the overall weight distribution (3); we thus cover a variety of scenarios. Example weighted adjacency matrices from each style and their respective parameters are shown in Fig. A1.
f= 0.5
2
3
P=
5
Pp=0.50
ka 10
8
0.75 0.75
Figure A1: Family of synthetic networks generated using the weighted LFR benchmark. Each panel shows the weighted adjacency matrix of one instance of each type network, sorted by its ground- truth community membership. Default parameter values used: N 125. k 25, mazk 100, Mw 0.5, H4 = 0.5, 3 2, T = 2, T2 = Each network type varies a single parameter from this list (value shown above the matrix). (1-3) Varying while keeping other parameters fixed dramatically changes the weight distritution. (46) Smaller values of 14, increase the intra- block connections (irrespective of their weights). (7) A smaller k creates a larger number of small communities. (8-9) The uw/ de ratio controls the ratio between inter- and intra-block weights.
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A.3 EXPERIMENTAL VALIDATION RESULTS
Silencing AWB substantially increased salt avoidance 1--valu=-5.7e-11, mean effect size = +25%), validating the predictive power of our algorithm; see Fig. A2. Surprisingly, this suggests the poten- tial for oppositional interactions between salt sensation and olfactory aversion circuits.
0.8
NaCI Avoidance
0.5
0.00
160
00
0.1
+AWB
-AWB
Figure A2: Results for avoidance to 160 mM NaCI for AWB controls (+AWB) vs. silenced (-AWB). Two-tailed unpaired t-test P=5.7574e 11, mean effect size +25% avoidance when silencing AWB, N=12 12. red-95% CI, blue=1 SD.
A.4 SELECTION OF THE NUMBER OF TENSOR COMPONENTS
Our logic for selecting the number of tensor components, R, was to use as many as possible to minimize reconstruction error, provided the results across multiple random initializations remained stable (i.e., small variance). Small reconstruction error suggests a faithful representation, and small variability guarantees that the retained components are robust. Based Fig. А3μ, the error vari- ability (std. dev. across 15 runs) reaches a minimum when R 11411 (shaded area), then increases sharply for R 15. We therefore selected R=15. Fig. A3p shows that the distributions of emror for each choice of are separate until π = 15, after which they being to overlap; this further strengthens our confidence in our choice for the number of tensor components.
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3.00
R=t4
3.00
Re15
0.69
Rw17
011or (53
0.44
0.42
3.004
2.000
3.000
components
emfor (%)
Figure A3: Selecting the number of tensor components, R. (a) Reconstruction emror as a function of R. Although the mean error (blue curve) always decreases with increasing R, its variability (std. dev. across 15 runs, dashed gray curve) reaches minimum when R 14-15 (shaded area), then increases sharply for R > 15. (b) The empirical distributions of error for each choice of I remain they begin to overlap. These results motivate our choice
separate until R=15, starting at for the number of tensor components to use, namely 15.
16


Reviews:
[{"review_id": "2qnQXrY48L", "reviewer": "Reviewer_vGem", "summary": "The paper proposes a novel analysis method to infer time-varying functional connectomes from neuronal data in general and calcium imaging data in C. elegans in particular. The method is based on a three-step procedure. First, time-varying affinities between pairs of neurons are computed based on concurrent changes of the neuronal signal. Second, a non-negative tensor decomposition is employed to identify neuronal-temporal motifs of affinity across animals. Thirdly, community structure is inferred from the motifs using a stochastic block model. Taken together, these steps enable an interesting visualization of dynamical functional connectomes. The authors apply their method to experimental data recorded in C. elegans during a stimulus avoidance / attraction paradigm and identify a neuron previously not implicated in salt sensing. They then experimentally test its role by exposing worms with / without the neuron silenced to a salt stimulus and find that indeed the worms respond behaviorally as predicted.", "questions": "My most relevant question is regarding code availability -- why has the code not been made available, and how do the authors intend to remedy that situation?\n\nFurther questions are minor ones:\n\n* Since the affinity matrices are symmetric, vectorizing these (and using Euclidean norms) does not seem to be the right choice here? Have you looked into proper distance metrics for symmetric matrices [1]?\n* What does \"CP\" stand for on page 4? That abbreviation is not introduced?\n* Why was the set of neurons restricted to sensory and inter-neurons? Did the results change when using all neurons?\n* Would the method also work on other neuronal data modalities, e.g., spiking data?\n\n1. Vemulapalli, Raviteja, and David W. Jacobs. \"Riemannian metric learning for symmetric positive definite matrices.\" arXiv preprint arXiv:1501.02393 (2015).", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The authors identify and address a highly relevant problem, i.e., the issue that relationships between neurons are highly dynamic yet few algorithms are able to infer dynamical functional connectomes. Their new methods enables a new visualization of this complex, high-dimensional data and can be used to derive experimentally testable predictions on the time-dependent involvement of neurons in behaviorally relevant neuronal ensembles. The authors further test and validate one specific hypothesis experimentally in a behavioral experiment. Another strength is that the manuscript is very well written and easy to read.", "weaknesses": "Some of the algorithmic choices appear rather ad-hoc without a rigorous theoretical or neurophysiological justification. In particular, it is unclear to me why the problems in constructing a time-varying similarity measure, that the authors discuss in the second paragraph of Section 2.1, does not also apply to the derivatives of the calcium traces? Since the derivatives represent the influx/outflux of calcium, and are thus likely a better representation of the neurons' firing rates, I would think that similar problems persist? Also, it is not clear to me why the local differential affinities should be non-negative? One could argue that two neurons also form a network if one inhibits the other, which in my understanding would lead to a negative affinity?\n\nA further (and significant) weakness is that no link to code is provided in the manuscript. I believe that making all code publicly available is absolutely essential for reproducibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QWFupi6rtL", "reviewer": "Reviewer_6VUY", "summary": "This study presents an unsupervised method to identify the dynamic interactions between neurons. The approach has two main steps: first, based on calcium activity, the neural traces are organized to identify groups of neurons likely interacting over specific time intervals. Then, a generative model is applied to detect weighted communities in the functional patterns.", "questions": "Please see the Weaknesses.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is well-written and easy to follow. It is well-organized with illustrative figures. Also, the proposed method is simple, yet potentially effective for different tasks and domains. Moreover, compared to existing studies that use step-by-step statistical methods, this paper designs a method that allows considering the full system of similarities and animals across time. Also, presenting extensive experimental results supports the claim and the potential of the approach.", "weaknesses": "1. The main weakness of this paper is the lack of novelty in the model design. In fact, the proposed approach is a simple combination of existing methods and I cannot see a novel insight or a novel contribution from the machine learning side. \n\n2. It would be better to include more related baselines. In the literature, there are several learning methods that learn the structure of the brain networks. Based on the current set of baselines, the proposed method shows superior performance, but existing baselines are general graph learning methods and do not use special properties of the brain. Therefore, I suggest adding additional brain network-based graph learning approaches as baselines. \n\n3. There is a lack of discussion and experimental results about the scalability of the method and its efficiency. It would be great if the authors could provide more information about the efficiency of the method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UzXgshKWwo", "reviewer": "Reviewer_nAoq", "summary": "The goal of this work is to infer community structure in neural networks (specifically that of _C. elegans_) based on functional data. To do this, the authors 1) define a pairwise affinity score computable from single-neuron time series; 2) use tensor factorization to group these pairs into dynamical motifs; 3) feed tensor components from these motifs into a community detection algorithm. When applied to data recorded from individually identified _C. elegans_ neurons, this produces networks that are tested in validation experiments, which find that perturbation of key nodes does affect network structure and behavior related to, e.g., salt avoidance.", "questions": "- How sensitive are the results presented to the particular choice of affinity score? Would, e.g., a Spearman correlation between the two time series yield qualitatively similar results?\n- I might have missed this, but how did the authors decide how many tensor components to retain?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The problem of organizing functional data from neurons into more interpretable submodules that can be investigated in causal experiments is an important one.\n- The approach is reasonable and makes use of established techniques (tensor factorization, community detection).\n- Use of real neural data for both algorithm validation and suggesting perturbative experiments is a huge plus.", "weaknesses": "- I found the affinity score a bit _ad hoc_. I can understand the intuition, but it seems like there should be a more principled way to get at this information. Related, but along the opposite direction: why not include strong _anticorrelations_ in the affinity score? Shouldn't two neurons whose activity moves opposite to one another at most times be considered related/coupled?\n- The tensor factorization will tend to treat affinities independently of one another, though the $N(N-1)/2$ affinities result from only $N$ neuronal time series. That is, the tensor factorization does not respect the underlying geometry of the problem. It's unclear to me how big an issue this is in practice, but it might lead to issues with the method.\n- While the experimental data are a definite plus, it's always unclear how strongly they should be taken as validation of a particular data analysis method. In a strongly coupled network, ablating any one neuron is likely to have an effect, and it's not shown that that the method proposed here would necessarily outperform others for selecting which perturbations to apply.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)