Learning dynamic representations of the functional connectome in neurobiological networks

We appreciate your time and your valuable comments and questions. **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?** You are absolutely correct in your intuition, and in fact that is precisely the motivation behind our proposed affinity score computation. Firstly, we use first derivatives to eliminate the influence of the raw activity trace values (which are based on the calcium fluorescence signal and may not be a fair comparison across different neurons – see Sec. 2.1, 2nd paragraph). Secondly, we use absolute values specifically to allow for neurons with opposite changes in activity to have strong affinity regardless of the derivative sign. As stated in Appendix A.1: *"Notice that the areas are unsigned (…) This 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. 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."* This feature is further mentioned in the caption of Fig. 5 (last line): *"Notice that, because the affinities are computed from absolute derivatives, neurons will be strongly connected even when their traces have opposite signs."* In summary, every decision behind our affinity score has been carefully examined and extensively tested to arrive at a value that gives a reasonable measure of the likelihood that two neurons are interacting. Our experimental results further confirm the validity of our choices. Although the affinity derivation is developed in detail in Appendix A.1, based on your comments we now realize that this particular feature of our method should have been more clearly stated in the main text. So we have updated the manuscript to include in Sec. 2.1, 4th paragraph (new text in bold): *"Affinities show how similar two curves are, locally, in 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 two neurons to be interacting"* **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.** Although they originate from N time series, every affinity is computed independently for each pair of neurons. This is important for a number of reasons. For example a neuron $i$ might have an activity that is moderately similar to two other neurons, $j$ and $k$, while at the same time $j$ and $k$ have little affinity to each other. Here is an example to illustrate this subtlety, drawn directly from our experience with these neural data. Imagine a brief time window during which $i$'s activity decreases and soon after increases, whereas $j$ decreases in tandem with $i$ but does not increase afterwards. In contrast, $k$ does not decrease initially, but does increase together with $i$. Then both $j$ and $k$ are somewhat similar to $i$, but not to each other. This illustrates how, allowing the affinities between pairs of neurons to be treated independently, leads to a more appropriate measure of the likelihood of instantaneous interactions between them. **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?** In prior work we tried many of the standard approaches and found that none of them agreed with our natural, intuitive assessment of how two traces should relate to one another. This qualitative mismatch was, in fact, the main motivation for our developing a novel affinity score. We mentioned, in Sec. 2, second paragraph, that previous studies were limited to traditional statistics and were not able to reveal the same relationships we found between neurons. Moreover, from Sec. 2.1, first paragraph: Affinities should thus be a function of time, rather than the more traditional method of computing a global-time measure such as correlation. Spearman correlation, in particular, computes an average across all values, so it could not be used to produce a continuous similarity value over time. Our affinity method, on the other hand, proposes a principled way to define local time windows in which the rates of change can be compared (in absolute value, as explained above).

We thank the reviewer for their time and for their insightful comments and suggestions for the paper. **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.** Although our algorithm does employ two well-understood components, we stress the novel insight is in how they are used. While non-negative tensor factorization (NTF) has been used in analyzing neuronal responses (e.g., [1], [2]), no one has previously used it successfully to define neural communities. Our critical contribution is the tensor dimension: *we seek to organize the affinities between pairs of neurons directly*. It is these neural pairs that lead, through the pipeline, to dynamic neural communities. Attempts to build tensor dimensions on the neurons directly could not accomplish this level of community organization. To elaborate, instead of implicitly computing similarities between the traces within the NTF computation (which, under the hood, would cause their comparison to be based entirely on linear dot products), we explicitly pre-compute our nonlinear differential affinities. If we had worked with the activities directly it would have yielded tensor factorizations based on how similar the individual traces were; our approach goes one order higher and organizes the affinities directly. Working with individual neurons would produce factors based on how similar the raw traces were. There is another advantage to our approach to teasing apart the functional connectome in neural networks. We use a novel nonlinear differential affinity, which enables (i) a differential comparison between neural activity traces that is not dependent on the actual calcium fluorescence values, and (ii) accounts for both positive and negative relationships. Other recent studies have been limited to using traditional correlation-like measures of similarity between activity traces, and failed to find the neural interactions that we reveal in this paper (e.g., [3], [4], [5], [6]). Because most of the affinity computation is developed in Appendix A.1, we are adding a sentence to the main text of the manuscript to highlight this novelty. [1] Williams, A. H., et al. (2018). Unsupervised discovery of demixed, low-dimensional neural dynamics across multiple timescales through tensor component analysis. Neuron, 98(6), 1099-1115.\ [2] Dyballa, L., et al. (2023). Population encoding of stimulus features along the visual hierarchy. bioRxiv.\ [3] Susoy, V., et al. (2021). Natural sensory context drives diverse brain-wide activity during C. elegans mating. Cell, 184(20), 5122-5137.\ [4] Randi, F., et al. (2022). Neural Signal Propagation Atlas of C. Elegans. arXiv preprint arXiv:2208.04790.\ [5] Yemini, E., et al. (2021). NeuroPAL: a multicolor atlas for whole-brain neuronal identification in C. elegans. Cell, 184(1), 272-288.\ [6] Kato, S., et al. (2015). Global brain dynamics embed the motor command sequence of Caenorhabditis elegans. Cell, 163(3), 656-669.\ **It would be great if the authors could provide more information about the efficiency of the method.** Thank you for pointing this out. We realize this information is highly relevant in a context where experiments can collect increasing amounts of data. One of the reasons behind our choice of algorithm for NTF (HALS) was precisely its efficiency; it has been thoroughly benchmarked against other popular algorithms in [1], using a variety of real-world datasets. We have modified the manuscript to include this information (Sec. 2.2, 4th paragraph). For reference, our own experiments with running HALS on our dataset took about 35 s on a 3.1 GHz Dual-Core Intel Core i5 MaxBookPro laptop with 16GB of memory. [1] A.-H. Phan and A. Cichocki. Multi-way nonnegative tensor factorization using fast hierarchical alternating least squares algorithm (HALS). In Proc. of The 2008 International Symposium on Nonlinear Theory and its Applications, Budapest, Hungary, 2008.

I thank the authors for the new results and for answering my questions. I have raised my score to 6. While I appreciate the contributions of the paper from the neuroimaging and application side, I still believe that the novelty from the machine learning side is limited, which is the reason that I didn’t raise my score to 8.

We appreciate the attentive reading, your comments, questions, and suggestions. **why has the code not been made available, and how do the authors intend to remedy that situation?** We agree with the reviewer that the code should be fully available. Our plan is to make it available on github, so it may become a standard technique in computational neuroscience; we did not include a link due to the anonymity clause. To illustrate our intention, we now attach a zip file to the submission containing the source code for our method. **it is unclear to me why the problems in constructing a time-varying similarity measure (...) does not also apply to the derivatives of the calcium traces? Since the derivatives represent the influx/outflux of calcium (...) I would think that similar problems persist?** You are correct in that if the derivatives were to be compared using standard approaches such as Euclidean distance or cosine similarity, the same issues would persist. However, as discussed in detail in Appendix A.1, the comparison between derivatives is highly nonlinear: First, discrete time windows within which to compute local affinities are restricted to periods during which a neuron's derivative has constant sign, i.e., periods of monotonic influx or outflux of calcium. Then, the affinity is computed as the normalized intersection between the integrated calcium influx/outflux of each neuron (areas under the neurons' derivative curves). This approach acts to effectively eliminate the pitfalls presented in the second paragraph of Sec. 2.1, as has been empirically demonstrated. **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?** Yes: negative affinities are not only possible, they indeed frequently occur in our data. That is precisely the reason why we use absolute values: our goal is for the affinity to represent a (non-negative) “probability” that two neurons interact, regardless of the qualitative nature of that interaction (excitation or inhibition). Although the affinity derivation is developed in detail in Appendix A.1, based on your comments we now realize that this particular feature of our method should be more clearly stated in the main text. So we have updated the manuscript to include in Sec. 2.1, 4th paragraph (new text in bold): *"Affinities show how similar two curves are, locally, in 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 two neurons to be interacting"* **Since the affinity matrices are symmetric, vectorizing these (and using Euclidean norms) does not seem to be the right choice here?** Thanks for the suggestion as well as the interesting reference. Note, however, that in tensor factorization we do not compare affinity matrices as a whole. Rather, because tensor factorization is based on an alternating least-squares (ALS) approach, at each iteration the algorithm solves a problem of the form: $\mathbf{A} \leftarrow \textup{arg} \min_{\mathbf{A}} \left \|\mathbf{X}_{(1)} - (\mathbf{C}\odot \mathbf{B}) \mathbf{A}^{T} \right \|$ $\mathbf{B} \leftarrow \textup{arg} \min_{\mathbf{B}} \left \|\mathbf{X}_{(2)} - (\mathbf{C}\odot \mathbf{A}) \mathbf{B}^{T} \right \|$ $\mathbf{C} \leftarrow \textup{arg} \min_{\mathbf{C}} \left \|\mathbf{X}_{(3)} - (\mathbf{B}\odot \mathbf{A}) \mathbf{C}^{T} \right \|$ where $\mathbf{X}_{(i)}$ is the mode-$i$ matricization of the tensor $\mathbf{X}$ (following the notation from [1]). Therefore, the Euclidean distance here is never actually computed with respect to any of the symmetric affinity matrices (slices of $\mathbf{X}$), but rather to non-square, matricized versions of the entire tensor. It would therefore be non-trivial to adapt these techniques to our case. Of course, it is possible in the future that we will find an alternative metric which is better, but for now the standard approach suffices. [1] Rabanser, S., Shchur, O., & Günnemann, S. (2017). Introduction to tensor decompositions and their applications in machine learning. arXiv preprint arXiv:1711.10781. [1] Kolda, T. G., & Bader, B. W. (2009). Tensor decompositions and applications. SIAM review, 51(3), 455-500.