From Synapses to Dynamics: Obtaining Function from Structure in a Connectome Constrained Model of the Head Direction Circuit

Thank you for your thoughtful review, as well as for your helpful suggestions that will help us clarify and improve our paper! Below, we respond explicitly to each of the specific points and questions you raised. Given the constraints of the rebuttal guidelines, we are unable to include figures and data. However, we have described additional analysis that we have performed that we would gladly incorporate into the paper's final revision:

Could you please make it clear in lines 182-192, which of these loss functions are your unique contributions and which ones are used in the literature. I also think linear consistency loss needs a bit more clarification for someone who is not immediately expert (i.e., why does this incentivize linear changes wrt input)? For instance, most of the setup seems very similar to this seminal work: https://openreview.net/forum?id=8ox2vrQiTF Thank you for the great suggestion; we will include a more detailed discussion of our loss terms in the paper.

With regards to the linear consistency loss, the formula presented has a typo in it. The corrected loss term should read $\frac{1}{T} \frac{1}{2U} \sum_t^T \sum_{u=-U}^U \left( \frac{\delta \theta(t)}{|u|} - \mu\right)^2$ where $\mu = \frac{1}{T} \frac{1}{2U} \sum_t^T \sum_{u=-U}^U \frac{\delta \theta(t)}{|u|}$ which penalizes the variance of the ratio between changes in the internal state (decoded angle) and input velocity, ensuring that changes in internal state are proportional to changes in input drive. We will clarify this in the final manuscript. Likewise, thank you for bringing our attention to the connections between our work and the Schaeffer et al. (2023) paper, which also uses self-supervised learning to train a circuit to learn an internal representation of a spatial variable. We designed our loss terms independently based on biological constraints we believed should apply broadly across neural circuits. This has resulted in our linear consistency loss converging conceptually to the conformal isometry loss proposed by Schaeffer et al. (2023), both of which enforce that internal neural representations update proportionally with input magnitude. Specifically, we penalize the variance of the ratio between changes in the internal state (decoded angle) and input velocity, thus encouraging a consistent (linear) relationship of input velocity and velocity along the ring. However, there are clear distinctions between our approach and theirs: Schaeffer et al. (2023) impose explicit structural constraints (non-negativity, unit-norm) on their randomly initialized neural representations to learn Euclidean geometry directly. By contrast, we do not explicitly constrain neural activity; instead, our model starts from biologically measured connectomic structure, and implicitly learns biologically consistent representations through our self-supervised loss terms.

Specifically, in Schaeffer et al. (2023), the authors train a randomly initialized MLP where “the neural representations are enforced to be non-negative and unit-norm via the non-linearity”. In contrast, we initialize our network from the connectome (thus constraining the connectivity based on animal measurements), and do not explicitly constrain the neural representations, instead asking the network to implicitly learn good representations through our loss terms (stability, entropy, and minimum-speed) which are formulated based on broadly accepted biological principles of neural coding, i.e., efficient, stable representations. Additionally, our losses apply more generically across a range of neural circuits, in contrast to the separation and capacity losses from Schaeffer et al. (2023) which are more specific to spatial representations. Overall, while we view our loss terms as distinct from those in Schaeffer et al. (2023), we now see the clear relevance of their work to our paper, and will add references to their work in our paper.

Lines 159-160: I think you can also motivate this through gene expressions and more broadly biological motivations. For someone outside of neuroscience, it may not be clear why this is a smart choice.

Thank you for pointing this out. We will include more detailed discussion as to why parameterizing models at the cell-type level is motivated by biological evidence: neurons sharing cell types typically exhibit similar gene expression patterns, intrinsic electrophysiological properties, and connectivity motifs. These shared properties are thought to underpin similar functional roles within neural circuits. Cell-type-level parameterization thus aligns naturally with biological reality, reduces model complexity, and limits overfitting to individual neuron variability, while still capturing key functional diversity.

There could be a better explanation for why the linear consistency is not considered supervised learning. I think that might also be more appreciated by the NeurIPS audience.

Thank you for this feedback! We consider our linear consistency loss to be self-supervised because it enforces internal consistency of neural representations without providing explicit external labels or targets for neural activity. Instead of prescribing specific activity patterns or target states, it constrains neural updates solely based on internal representational properties (e.g., proportionality to input velocity). Thus, the training data itself—the circuit’s own dynamics under various inputs—generates the supervision signal, making the approach self-supervised rather than externally supervised. We will edit the text with a better explanation of this.

I do believe this study would be at a spotlight quality if there was an additional experiment with RELU or leaky-relu or selu or soft-plus nonlinearity. The choice of nonlinearity is arbitrary, as authors also point out. It would have been desirable to show the same results with a different nonlinearity, though I do not think this is absolutely necessary for acceptance.

Thank you for the great suggestion. We chose the sigmoid nonlinearity primarily because it closely resembles biological neural activation, which is characterized by saturating, bounded, and non-negative firing rates. We excluded non-linearities which are not lower-bounded and thus permit non-biological neural activity.

That said, we have also been interested in exploring how alternative activation functions such as ReLU would perform, particularly given its simplicity in theoretical analyses and success in the machine learning literature. However, changing the nonlinearity is a nontrivial modification to the model (e.g., firing rates will no longer be upper-bounded) and will thus require additional care beyond simply swapping out the activation function. We plan to examine such alternatives in future work, and agree that demonstrating that our results are robust to choice of nonlinearity will greatly strengthen the generality of our method.

As noted in my review, I think this is an interesting neuroscience work. I keep my stance that this work deserves acceptance.

As authors may be aware, there is an active reviewer discussion happening in parallel. As part of this discussion, an important point was raised and I would like to hear authors' response on the resulting question.

Could the authors provide a more principled and experimentally grounded roadmap for testing the predictions in figure 5? What can experimentalists do that will be a very clear causal test? Can you be more specific about underlying biology in general?

Thanks!

Sure, edited as requested. Although, I note clear disagreement. People do volunteer time, and thanking them and engaging with the argument is basic human decency. I dont get what is debatable here, but let's focus on the paper.

Dear Authors,

Thank you for the responses. I will admit, it is a bit underwhelming. A desirable one should have had new information, not restatement of claims we already read. For instance, what other cell types (maybe upstream?) are good targets. Regardless, I will keep my score since I do not see any clear argument for a score increase.

Thank you for your thorough review! Your questions and suggestions highlight important points that we will make sure to address to improve the clarity and interpretability of our manuscript. Below, we provide detailed responses addressing each of your points. Given the constraints of the rebuttal guidelines, we are unable to include figures and data. However, we have described additional analysis that we have performed that we would gladly incorporate into the paper's final revision.

Dependence on A Priori Functional Knowledge…

You raise an important point about the generalizability of our current loss formulation. Indeed, our linear consistency loss is motivated by the known integrative function of the Drosophila HD circuit. However, the critical underlying assumption—equivariant responses to input stimuli—is not circuit-specific and broadly applies to neural circuits that encode continuous variables (such as spatial position, head direction, or eye position). Thus, the overall framework we introduce does not inherently rely on detailed knowledge of neural representations or neuron-specific roles, and can broadly be applied to other integrator circuits in the brain. We are currently working towards extending our method beyond integrator circuits. For instance, we can remove assumptions about linearity and knowledge of the decoding function by redefining our linear consistency loss based on distances in the neuron activity space, rather than in the latent variable space, to enforce that larger changes to circuit inputs result in larger differences in population activity. This should work as long as the neural encoding of similar variables (i.e. nearby angles) are also similar, and we ensure that comparisons are within a reasonable range of inputs (resulting in internal representations where there is some overlap in neural activity). Nevertheless, we believe this present work, which focuses on integrator circuits, provides a valuable first step towards building more general models that leverage raw connectomics data to perform functional inference.

Clarity on Preprocessing and Hyperparameters…

The symmetrization process we used does not impose any type of rotational symmetry on the matrix but rather leverages known symmetry between brain hemispheres which means it is relatively light and does not rely on any parameterization. Nonetheless, we have also performed experiments in which this symmetrization is removed but cell types are divided based on their spatial location in the left vs. right hemispheres. In this setup, the model is able to learn to integrate. We can include this experiment in our appendix as well to show that the symmetrization process is not essential to our results. We have observed that a number of different hyperparameters do support integration and can include additional hyperparameter choices in the appendix. We included a number of hyperparameter sweeps in order to eliminate the effect of the hyperparameters on the interpretation of the ablation results. We also believe that our results in Figure 5 also demonstrates the robustness of our model at least with regard to initialization as a number of different seeds are able to ultimately integrate well.

On Connectome Preprocessing and Robustness… Details about the symmetrization process are included in the appendix. Are there other aspects which require additional clarification? We have additional results which show that similar results are achievable without any symmetrization at all if cell types are divided based on their membership in the left vs. right hemispheres. The most notable effect of eliminating the symmetrization is that the shape of the bump is altered slightly since there is more heterogeneity within the sections of the ring. However, the network is able to integrate smoothly and has a linear velocity response as in the network presented in the paper. On Generalizing the Self-Supervised Objective…

Thank you for highlighting this! Indeed, although our current linear consistency loss explicitly focuses on integration by enforcing input-output equivariance, the underlying principle of our method—using self-supervised learning to enforce internal consistency of neural representations—can naturally generalize to other neural computations. As mentioned earlier, relaxing the linearity assumption could readily extend our approach to circuits implementing memory or other persistent representations of continuous variables. In the case of divisive normalization, one could employ a proportional scaling consistency loss that ensures relative neural response patterns remain stable across varying input intensities, while still preserving similar response patterns for similar inputs. Overall, we believe that training connectome constrained networks using minimal, biologically-grounded self-supervised constraints provides a promising framework for data-driven discovery (or confirmation) of neural circuit function from connectomics data.

On the Role of P-EN Subpopulations…

Thank you for the great suggestion! To understand why P-ENb ablations were more detrimental than P-ENa ablations, we compared the learned synaptic strengths (recurrent, to/from EPG, and to/from GLNO; averages and variances were computed over 5 runs of each experiment) between the baseline and P-ENa/P-ENb ablation experiments.

At baseline, P-ENa neurons receive stronger input from GLNO (average weight 1.81) compared to P-ENb neurons (1.35), and also have stronger connections onto EPG neurons (0.75 vs. 0.42 for P-ENb). Thus, initially, P-ENa forms the primary pathway from GLNO (input) to EPG (internal compass) neurons.

However, when P-ENa neurons are removed, the network effectively compensates: GLNO-to-P-ENb connections strongly increase (from 1.35 to 2.28), P-ENb-to-EPG weights also increase (from 0.42 to 0.58), and EPG neurons boost their internal recurrence (from 0.06 to 0.16), restoring most of the circuit functionality.

In contrast, when P-ENb neurons are removed, the network shows minimal compensation and does not appear to reroute signals. GLNO-to-P-ENa connections remain unchanged (1.81 baseline vs. 1.83 after ablation), and P-ENa-to-EPG connections also remain similar (0.75 baseline vs. 0.74 after ablation). EPG recurrence stays low (0.03).

These findings suggest distinct and specialized roles for P-EN neuron subtypes: P-ENa neurons are the main pathway under normal conditions but are functionally redundant. In contrast, P-ENb neurons appear to fulfill a unique integrative role, perhaps due to specialized wiring or synaptic targeting, that make them uniquely essential for maintaining proper angular velocity integration.

Thank you again for suggesting this analysis, and we will include more details in the supplementary information.

On the Level of Abstraction (Cell-Type vs. Sub-Type)...

You raise an important point. We view this result as not as a contradiction but a refinement of our central claim: Our core method uses the broadest (cell-type-level) possible tuning to respect the granularity of current biological datasets and reduce overfitting. However, when key circuit elements are disrupted, introducing finer-grained tuning (e.g., ring neuron subtypes) can help restore function. This not only offers potential insights into the potentially redundant roles of different cell types and how they contribute to forming robust circuits, but may also illuminate important idiosyncrasies between cell (sub)types. Indeed, it is arguable that the ring neuron subtypes can each be considered their own cell type, as they each encode information about distinct aspects of the sensory environment, and synapse onto distinct dendritic locations on their postsynaptic neurons. This opens up the potential for a productive feedback loop, in which knowledge of cell types can be used for model training, and results from model training can refine cell type definitions in a hierarchical fashion.

Dear Authors, I still have some questions:

Turning the methods from Lappalainen et al. (2024) [1] into an unsupervised format is not novel; you should compare the advantages and disadvantages of the two methods with specific experimental results, rather than just demonstrating that a new method can run on a new dataset.

The self-supervised modeling of neural dynamics based on single neurons or cell types' internal representational consistency has also been seen in previous research [2,3], and you should clarify how you differ from them, can these methods achieve similar function?

[1] Lappalainen, J.K., Tschopp, F.D., Prakhya, S. et al. Connectome-constrained networks predict neural activity across the fly visual system. Nature 634, 1132–1140 (2024).

[2] Learning Time-Invariant Representations for Individual Neurons from Population Dynamics. NeurIPS 2023

[3] NetFormer: An interpretable model for recovering dynamical connectivity in neuronal population dynamics. ICLR 2025

Thank you for your review and for highlighting both the timeliness and relevance of our work. We appreciate your feedback and hope to improve our paper by incorporating your suggestions. Below, we provide detailed responses addressing each of the specific points you raised.

The fruit fly's head direction circuit is a well-studied system, and substantial existing knowledge about this circuit seem to have been incorporated into the design of the loss function here…

Thank you for bringing up this concern. We would like to clarify that while certain model components described in the paper are indeed tailored to the Drosophila HD circuit, the fundamental aspects of our approach are broadly applicable across other circuits with minimal modifications. We selected our loss functions based on general principles of neural computation that are broadly applicable across neural circuits, and believe our methods are readily generalizable to other circuits, particularly those that perform integration and representation of continuous variables.

Specifically, the minimum speed and entropy loss terms were chosen to reflect the general principles of efficient neural coding: the entropy loss encourages evenly distributed activity across the population such that the network representation does not heavily depend on the activity of single neurons, while the minimum speed loss encourages the network to react to stimuli. Thus, we use these loss terms to prevent the trivial solutions of constant or zero activity. Likewise, the stability loss relies on the assumption that the internal representation should not change in the absence of perturbations, which is widely applicable to networks involved in memory and integration (we acknowledge that in circuits with spontaneous dynamics, this specific loss term may be modified or removed). Finally, the linear consistency loss enforces equivariant representations, i.e., that changes in internal state are proportional to changes in input drive. This assumes equivariant representations—a widely held principle in systems that encode continuous variables (e.g., head direction, spatial location, eye position). We agree that these latter two terms are most naturally suited to circuits that perform integration, and are currently working on extensions to our model to generalize beyond integrator circuits in the brain as well. Nonetheless, we believe that this work provides a valuable step towards building models that leverage raw connectomics data to perform inference of circuit function.

We would also like to point out that while we do utilize some prior knowledge of network structure to compute our linear consistency loss, this knowledge is not required; our objectives can be easily generalized to systems where such prior knowledge is not available. Two concrete strategies to do this are:

1. decoder training: rather than assuming a known decoding function (e.g., population decoding of E-PGs), we can train a simple decoder (e.g., an MLP or linear readout) over a broader population to infer the encoded variable.
2. activity-similarity-based loss: even without a decoder, one can enforce that larger changes in the represented variable should produce more decorrelated activity across time, while small changes should produce similar activity patterns. This operates under the simple assumption the neural encoding of similar variables (i.e. nearby angles) are also similar. We are currently working to extend our model by testing these strategies in novel circuits with unknown or less-characterized functional roles.

Are all the terms of the loss function required to reproduce the results of this work? If so then, upon omission of terms, what results are not reproduced? If the results regarding the known properties of the neural circuit depend on the selection of the terms of the cost function, how can we be certain that the same wouldn't happen to the predictions regarding the unknown properties of the circuit?

This is indeed an important point, and we will include results from and discussion of ablation experiments of each of the individual loss terms in the supplementary information. However, we reiterate that all of the loss terms we use either assume nontrivial encoding in neural circuits and can generically apply across all neural circuits, or assume equivariance, which holds for a wide range of circuits performing continuous variable integration. While our complete model is currently limited to integrator circuits, we see this work as an important initial step towards building more general models that can infer dynamics from the connectivity measurements of networks more broadly. I will briefly describe some results from our ablation experiments with individual loss terms here:

• Linear consistency: The model is still able to integrate at a high velocity, but the integration velocity is not smooth nor linear.
• Stability: The model is able to integrate, but there are fewer stable fixed points along the ring, resulting in less robust maintenance of heading direction.
• Minimum speed: The model does not learn to integrate and collapses to a single fixed point.
• Entropy: The model is able to achieve similar performance without this loss term and thus it is not strictly required; however, including it improves training stability and avoids activity saturation or collapse.
• L1/L2 regularization: No major impact on integration ability; we include these primarily to promote biologically plausible (energy-efficient) neural activity

For the loss function terms, I couldn't find the definition for capital theta.

Thank you for pointing this out! $\Theta(u)$ represents the Heaviside step function which is defined to be 0 when $u \leq 0$ and 1 when u is positive. We will edit our paper to include this.

Thank you for your thorough and thoughtful review, and constructive feedback! We have responded below to your specific concerns and questions.

Can the authors elaborate on the "noisy connectome measurements" to help us better understand the problems? What are the potential sources of measurement errors? How likely are they to appear, and what will be the consequences?

In the paper, we provide a brief description of the noise in lines 30-31; we will expand our description to make the sources and consequences of connectome noise more clear. There are three primary sources for noise: connectomics data is primarily collected through electron microscopy, and errors can be introduced at multiple points throughout the process including image alignment of 2D slices, automatic detection of neuronal structures such as synapses, and manual proofreading. From our discussions with experimental labs, we have been made aware that human proofreading is a crucial but labor-intensive part of this process, and the amount of noise in connectome measurements varies depending on the level of proofreading. On the biological side, specific synapse numbers are also known to vary between animals and change across time, as animals learn and adapt in their environments. Thus, connectomic snapshots offer noisy measurements of biological networks, and require some level of parameter tuning to achieve a more accurate picture of network function.

What would the results be like if Z and, therefore, synaptic connectivity are not tuned at all?

We have actually run baseline experiments that vary the level of tuning to demonstrate the importance of cell-type parameterization. We briefly mention this in Section 5.3, but will expand upon it and also include more details in the SI. In short, tying parameters globally across cell types does not result in a solution that forms or integrates the bump, regardless of whether the synaptic connectivity gain is tuned or not (i.e., if Z=1).

Much prior domain knowledge is used in the design of the method, which limits its application to other circuits...

We have found that our method is robust to each of the specific assumptions you highlighted:

1. We used GLNO inputs to make our model more aligned with the biological circuit, but we have also previously tried injecting inputs through the PEN_a/b neurons as well as the E-PG neurons directly, and all work equally well. In fact, any source of initial net excitatory activity and time for the network to stabilize is sufficient to form a bump. It’s true that asymmetric inputs are needed for the bump to integrate, but we believe that local drive is a general assumption that applies broadly across the brain (e.g., retinotopy in vision).

2. Similar to 1, we chose local E-PG initializations to make our model more biologically aligned, but any source of symmetry breaking initialization will stabilize to a single bump through recurrent network interactions; we can generalize this by initializing neuron activities randomly instead.

3. Although we make use of the prior knowledge that E-PG neurons are known as the canonical ring neurons in the Drosophila brain, a more generic application of this approach could instead train a simple decoder (e.g., a MLP) to decode activity from a broader population of neurons involving more cell types. Alternatively, we can redefine our linear consistency loss by measuring change in the neuron activity space, rather than in the encoded variable. More broadly, we are also working to reformulate our model, including our loss terms, so that our method can be applied more generally to circuits beyond integrator networks in the brain.

Can you explain in more detail the "population coding" you used to decode the location?

Population coding is a common neural coding strategy where information, such as head direction, is represented by the combined activity of a group of neurons, rather than a single neuron. As defined in Bialek et. al. 1989, population coding assigns a preferred direction (the direction they maximally fire at) to each of the neurons in the population, such that the direction can be decoded as the sum of the neurons’ directions weighted by their activity. We will expand our definition in our paper to clarify this.

Can you elaborate on how the hyperparameters are optimized?

We describe the details of the hyperparameter search in the appendix, and will ensure this is referenced clearly in the main text. We performed a grid search over initial bias values, leak, and global synapse strength. We selected the best performing model based on the linearity of the model’s response to velocity stimuli.

Can you elaborate on why the linear consistency loss encourages a linear readout of the velocity input?

Thank you for pointing this out. The formula presented has a typo in it. The corrected loss term should read $\frac{1}{T} \frac{1}{2U} \sum_t^T \sum_{u=-U}^U \left( \frac{\delta \theta(t)}{|u|} - \mu\right)^2$ where $\mu = \frac{1}{T} \frac{1}{2U} \sum_t^T \sum_{u=-U}^U \frac{\delta \theta(t)}{|u|}$ which penalizes the variance of the ratio between changes in the internal state (decoded angle) and input velocity, ensuring that changes in internal state are proportional to changes in input drive. We will clarify this in the final manuscript.

The meaning of $\Theta ( \cdot )$ used in the loss function is unexplained.

Thank you for pointing out this oversight. $\Theta(u)$ is the heaviside step function which is defined to be 0 when $u \leq 0$ and 1 when u is positive; we will update the manuscript to include this.

Can you elaborate on why $C_{ij}$ is used in L1 and L2 regularization...

The $C_{ij}$ are used since the final synapse strength is the product of $Z_{ij}C_{ij}$, thus scaling the regularization to be proportional to the synapse strength itself. If we use the gains $Z_{ij}$ instead, then the regularization would be relatively stronger on the smaller weights. This is also a valid design choice, but we believe it intuitively makes more sense to penalize deviations from the connectome on the scale in which these deviations affect the dynamics, i.e. the scale of synaptic counts.

My primary concern lies in the generalizability of the proposed method...

You highlight a valid concern, and we will make sure to include a more detailed discussion about model generalization and limitations in our paper. We see the model described in the paper as an important step towards utilizing connectomes to characterize neural computations, and believe that it can be generalized naturally to a range of neural circuits for which the simple assumption of equivariant responses to inputs is applicable (as is known to be the case for many sensory circuits). As we mentioned earlier in our response, although we leveraged prior knowledge of population encoding of head direction in E-PGs in this paper, we can generalize this to other systems with minimal modifications by instead training an additional decoder on a broader set of neurons, and/or using a simpler approach based on measuring distances in the space of neural activity rather than a latent variable. In fact, we are now actively working to use these approaches to apply our method on novel circuits in the brain in which we do not know the precise roles of specific neuronal subtypes, or even circuit function.

We agree that finding an even more general, unsupervised approach that can relax even the assumption of equivariant representations would generalize our method even further to a larger set of networks. We see this as an important future direction that we are building towards, starting with our current model, which contributes to the field of connectome-constrained modeling by introducing a novel approach to understanding circuit function from structure.

The authors argue that replicating idealized continuous attractor dynamics requires finely tuned connectivity with perfect rotational symmetry, which is not observed in the connectome...

Your point about slow manifolds and finely tuned connectivity is apt and in fact, is in line with our model! Our model does not assume or enforce rotational symmetry, nor does it achieve (perfect) rotational symmetry through tuning. In many ways we are arguing the same point that you have raised: that despite some notable asymmetries (such as the variable number of neurons between each of the subsections of the ring) in the biological HD network, our minimal tuning is sufficient to produce a circuit that is still able to integrate. Our tuning process cannot correct for cell-level asymmetries, as we only adjust biophysical parameters on a cell-type level. This contrasts to previous works which build in assumptions about rotational symmetry in the network architecture, and cannot be used to make statements about the actual biological implementations of the network. Thus, we agree that it is very likely that biological circuits do employ slow manifolds and discrete fixed points without precise circuit symmetries in order to produce attractor-like behavior, and we believe that our method, applied to a circuit known for implementing attractor dynamics, is an important step in showing this is the case!

This raises an important question: how challenging is the problem if the system can function without the degree of tuning the authors assume?

We have performed controls that we can include in the supplementary information, where we have compared the results of untrained systems as well as systems that have been trained with fewer parameters. We find that the amount of noise in the data as well as our lack of complete knowledge of the biophysical parameters of the system necessitate additional tuning.