From Synapses to Dynamics: Obtaining Function from Structure in a Connectome Constrained Model of the Head Direction Circuit
Authors: Sunny Duan
Ling Liang Dong
Ila R Fiete
Venue: NeurIPS 2025 poster ()
Keywords: computational neuroscience
connectome
neural dynamics
attractor dynamics
self-supervised learning
structure-function mapping
Primary area: neuroscience_and_cognitive_science
TL;DR: We train a connectome-constrained model of the Drosophila HD circuit, and show that cell-type–level parameter tuning is both necessary and sufficient to recover accurate integration dynamics and reveal the functional roles of specific neuron types.

Abstract: How precisely does circuit wiring specify function? This fundamental question is particularly relevant for modern neuroscience, as large-scale electron microscopy now enables the reconstruction of neural circuits at single-synapse resolution across many organisms. To interpret circuit function from such datasets, we must understand the extent to which [measured] structure constrains dynamics. We investigate this question in the drosophila head direction (HD) circuit, which maintains an internal heading estimate through attractor dynamics that integrate self-motion velocity cues. This circuit serves as a sensitive assay for functional specification: continuous attractor networks are theoretically known to require finely tuned wiring, whereas connectomes reveal that biological wiring can be variable and omit key cellular parameters such as synaptic gains, neuronal thresholds, and time constants. 
We introduce a method that combines self-supervised and unsupervised learning objectives to estimate unknown parameters at the level of cell types, rather than individual neurons and synapses. Given the raw connectivity matrix, our approach recovers a network that robustly exhibits continuous attractor dynamics and accurately integrates a range of velocity inputs, despite  minimal parameter tuning on a connectome which notably departs from the symmetric regularity of an idealized ring attractor. We characterize how deviations from the original connectome shape the space of viable solutions. We also perform in-silico ablation experiments to probe the distinct functional roles of specific cell types in the circuit, demonstrating how connectome-derived structure, when augmented with minimal, biologically grounded tuning, can replicate known physiology and elucidate circuit function.

From Synapses to Dynamies: Obtaining Function from Structure in a Connectome Constrained Model of the Head Direction Circuit
Sunny Duan"
Ling Liang Dong"
Brain and Cognitive Sciences Massachusetts Institute of Technology sunnyd@mit. edu
Brain and Cognitive Sciences
Massachusetts Institute of Technology
loadingSmit. edu
Ila Fiete
Brain and Cognitive Sciences Massachusetts Institute of Technology f fiete@mit.edt edu
Abstract
How precisely does circuit wiring specify function? This fundamental question is particularly relevant for modern neuroscience, as large-scale electron mieroscopy now enables the reconstruction of neural circuits at single-synapse resolution across many organisms. To interpret circuit function from such datasets, we must understand the extent to which the measured structure constrains dynamics. We investigate this question in the Drosophila head direction (HD) circuit, which maintains internal heading estimate through attractor dynamics that integrate self-motion velocity cues. This circuit serves as a sensitive assay for functional specification: continuous attractor networks theoretically known require finely tuned wiring symmetries, whereas connectomes omit key cellular parameters such synaptic gains, neuronal thresholds, and time constants, and reveal that biological wiring can be heterogeneous. We introduce a method that combines self- supervised and unsupervised learning objectives estimate unknown parameters at the level of cell types, rather than individual neurons and synapses. Starting from the connectivity matrix, our approach recovers network that exhibits continuous attractor dynamics and accurately integrates range of velocity inputs, despite minimal parameter tuning on connectome that notably departs from the symmetric regularity of an idcalized ring attractor. We characterize how deviations from the original connectome shape the space of viable solutions. We also perform in-silico ablation experiments to probe the distinct functional roles of specific cell types in the circuit, demonstrating how monnectome-derivee structure, when augmented with minimal, biologically grounded tuning, can replicate known physiology and elucidate circuit function.
1 Introduction
Recent advances in large-scale electron microscopy have enabled the reconstruction of neural circuits synapse-level resolution, producing dense connectomic datasets across organisms including C. elegans (Cook et al., 2019), Drosophila (Scheffer et al., 20206) Dorkenwald et al., 2024), and mice MMCCONNS Consortium 2025). These datasets raise several fundamental questions: How precisely
"Equal contribution
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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can circuit function be determined from its wiring? Can we do so in the absence of key membrane and synaptic parameters such as cellular thresholds, gains, and time constants? Additionally, connectomes provide detailed synaptic connectivity maps but may contain potential errors from the data pipeline, including misalignment of electron microscopy sections and false positives or negatives in synapse detection SScheffer et al. (2020a). Thus, individual connectomes represent partial and sometimes noisy snapshots of the underlying biological networks that may fail to capture the necessary factors that play role in shaping circuit behavior.
We explore these questions in the Drosophila head direction (HD) circuit, a canonical example of a continuous ring attractor network in biology (Kim et al., 2017] Green et al. [0017] Jumer--vang al., 2017) that maintains an internal estimate of animal heading by integrating angular self self-motion
velocity inputs, Whereas theoretical studies of ring attractors (Skaggs et al. [994 Zhhng 1996]
Redish et al. 1996, Xie et al. 2002) show that they require finely tuned connectivity to support bump stability and smooth translation for velocity integration, we found that connectomic reconstructions of this circuit exhibit asymmetries and heterogeneities. Moreover, the Drosophila HD circuit involves
10 distinct cell types- many more than are theorized to be necessary in theoretical ring attractor models - raising open questions about the necessity and roles of specific circuit components.
Previous models of the HD circuit (Chang et al. 2023; Stentiford et al. 2024) have relied on hand- designed circuit architectures that impose connectivity motifs, in particular circular symmetry, known from earlier work to support ring attractor dynamics (Skaggs et al., 1994 Zhang 1996] Redish et al., 1996; Xie et al. 2002). While such models produce the expected network dynamics by design, they start with prior knowledge about circuit structure and the identities of the cell types involved in the core cireuit, thus limiting opportunities for data-driven discovery. Here, we introduce a framework that bridges this gap: starting from the original measured connectivity matrix, we optimize only small set of biologically grounded parameters at the cell-type level, such as cell-type-to-cell-yy synaptic gains and cell type-shared neuronal biases and time constants. The model is trained using a self-supervised linear consistency loss that simply enforces the internal bump movement to be proportional to the input velocity, in addition to several regularization terms that ensure the model learns a non- con-trivial solution that remains close to the original connectome. Notably, this learning objective requires neural activity recordings behavioral labels, enabling inference of functional dynamics from only structural measurements.
Our trained model recovers high-fidelityy continuous attractor dynamics and robustly performs angular velocity integration across wide range of inputs. also reproduces experimental findings including realistic bump width and bidirectional bump motion dependent asymmetric input from the left and right P-EN subpopulations (Seelig and Jayaraman| 2015] Turner- et al. 2017). We find that tuning parameters at the cell-type level achieves performance comparable to models with more parameters, while global parameter tying fails to recover attractor Sebavior-indicatti type-specific tuning offers a practical tradcoff between model flexibility and parsimony, performing comparably to fully parameterized models while preserving biological interpretability. We also explore the space of viable solutions under different levels of simulated noise and characterize the diversity of model solutions as a function of their initial starting points on the loss landscape. Finally, conduct in silico ablation experiments to probe the roles of different cell types.
that cell-
Key Contributions
We introduce self-supervised comnectome-constraineee framework that recovers functional dynamics of integrator circuits directly from noisy connectome measurements
We show that parameterization at the level of cell types is sufficient and necessary to recover integrating network dynamics
We characterize how initial conditions shape the diversity of learned solutions.
We perform in silico ablations on the trained networks, yielding novel biological insights on the functional roles of specific neuron classes in the Drosophila HD circuit.

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2 Background
2.1 Continuous Attractor Models
Continuous attractor networks encode analog variables, such as orientation or position, via localized bumps of persistent activity that shift smoothly across the neural population. These dynamics require precise tuning of recurrent connectivity, typically balancing local excitation and broad inhibition (Khona and Fietel 2022). Even minor deviations in synaptic strength or timing can cause bump instability, drift, or collapse. This sensitivity makes such networks a strong testbed for evaluating the extent to which connectomes alone can specify circuit dynamics.
2.2 Drosophila HD Circuit
The Drosophila HD circuit resides in the central complex and comprises several cell types (populations of cells that exhibit shared morphological, genetic, and connectivity profiles) thought to implement a ring attractor (Kim et al., 2017 Turner-Evans et al., 2020). E-PG neurons in the ellipsoid body (EB) are connected in an anatomical ring-like structure and comprise the core set of cells that track animal heading via localized bump population activity (Seelig and Jayaraman (013). P-EN neurons convey angular velocity signals: left and right P-EN neurons asymmetrically project to E-PG neurons shift the bump based on the directional velocity, and are further subdivided into P-ENa and P-END subpopulations whose differential roles in the circuit have yet to be ascertained (Turner-Evans ct al. 2017). Inhibitory inputs from Delta7 and ring neurons are believed to normalize and gate activity (Green al.) [0117] Hulse et al. 2023), while P-EG neurons are thought to contribute to bump stabilization (Pisokas et al. 2020). although this has not yet been confirmed experimentally.
These cell types outnumber those idealized ring attractor models (Skaggs al. 1994 Zhang 1996] Redish et al. 1996 Xie et al.) 2002). which typically define a core set of two to three cell types to implement local excitation, global inhibition, and bump movement, raising open questions about the role and necessity of each cell type.
For our experiments, we use connectomic data from the Drosophila FlyEM hemibrain dataset (Scheffer et al. 2020b) after simple preprocessing step that takes advantage of the natural symmetry between the left and right brain hemispheres correct synapse counts through symmetrization process (Section Ib. We focused our analysis on a subpopulation of neurons previously implicated in head direction (HD) processing, comprised of population of 439 neurons spanning six cell types: E-PG, Delta7, P-EG, P-EN, GLNO, and ring neurons. Notably, the ring neuron population, which is responsible for conveying sensory inputs to the circuit, are divided by lineage into ER and ExR subtypes, which can further be divided into a diverse set of 29 different subtypes of ring neurons.
The connectome provides signed synapse counts (where signs are inferred based on measured neurotransmitters (Davis et al.) 2020)) but lack key parameters such as synaptic gains or time constants, and include asymmetries that likely result from biological and measurement noise. This motivates the need for methods that can infer minimal functional parameters from structure while remaining faithful to the measured connectivity.
3 Related Work
Connectome- models have been used to study neural circuit function across a range of scales and assumptions. [Chang et al. (2023) tested whether the required global inhibition in the fly head-direction (HD) circuit is provided by Delta7 or ring neurons, using spiking models hand-constructed from comnectome-derived motifs and evaluating bump-based metrics across manual parameter sweeps. Stentiford et al. (2024) built a detailed spiking model of the central complex using prespecified phystological parameters, connectivity, and receptive fields, and showed that the models could learn visual -eeediing associations via Hebbian plasticity. |Pospisil et al. (2024) inferred whole- brain "effectome" from optogenetic perturbation data, using the connectome as a structural prior along with full-brain activity recordings to infer the effective weights of a linear model. Other works (Mi et al. 2023 Lu et al. 2025) have leveraged neural recordings to infer connectivity structure. Our work is most similar to Lappalainen et al. (2024), who train deep network constrained by the fly visual motion connectome. However, they assume an idealized, perfectly tiled columnar architecture, and optimize cell-type parameters via supervised learning on an optic flow task. In contrast, we employ
3

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unsupervised training directly on the raw, measured connectome. Finally, though not a connectome- constrained model, /Shaefferr et al. (2023) also used self-supervised objectives train a circuit to learn an internal representation of a spatial variable, and introduce conformal isometry loss similar in concept to our linear consistency loss to enforce proportional updates of neural representations with input magnitude. However, whereas they train randomly initialized network with activity-level constraints, we instead initialize a network with measured connectomic structure, and train it to learn biologically consistent representations through cell-type-level parameterization and self-supervised loss terms without explicit neural activity constraints.
Our approach differs in several key ways. We operate directly on the raw connectome without archi- tectural simplification. activity recordings, or external supervision. We introduce a sulf-super learning objective based on internal representational consistency, which enforces that changes in internal velocity reflect changes input velocity without requiring activity measurements or task labels, and optimize only a small set of biologically interpretable, cell-type-leell parameters. Our method recovers continuous attractor dynamics despite known physiological asymmetries and irregu- larities of the connectome. Because the objective encodes generic dynamical constraints rather than task-specific labels, our framework can be extended to other circuits hypothesized to implement a coarse computational function, such as integration, memory, or normalization, offering scalable approach to functional inference from connectomic structure without activity recordings.
Sillgiicll cata New www.ttanenttttee
Turee paromete's
Synopss counts M.
KPG
Synapee strengt
Figure 1:
The HD circuit and model setup. a. The connectivity pattern of the PEN and EPG neurons. The left/right PEN neurons asymmetrically project to the EPG neurons, causing the ring to rotate clockwise or counterelockwise depending on net input drive. b. Ring neurons provide uniform inhibition to EPG neurons. while Delta7 neurons target EPG neurons far away on the ring. Schematic of tuning procedure. Fixed synapse counts and signs inferred from neurotransmitters are combined with trained cell-type-specifie gain parameters to produce the final network weights. Nodes are colored by cell type.
4 Methods
4.1
Model description
Rate-based neuron dynamics model We modeled the dynamics of this neural circuit using rate-based dynamics model. The firing rate æj of each neuron in the circuit evolves according to the following differential equation:
dr, = -Ex to Wtnr + + 2t(
T
dt
A
where (1) is the time-varying firing rate of neuron j. тj represents global intrinsic time constant of each neuron and é is a global constant determining the scale of activity and, together with T, the rate of activity decay in the absence of input. a-)) denotes the sigmoid function, Wix represents the synaptic weight from neuron neuron j. by is the threshold parameter for neuron j. and (t) represents any (potentially time-varying) external inputs injected into neuron j.
Connectomics-basee model parameterization To model conectomically defined circuits, we define the weights in Eq. as Wiy wa + (i)z)) Dgg)sgn where the connectivity matrix (C) and the signs (sgn E {+1)) of the interaction are fully determined by the empirical synapse counts measured in the connectome and from recorded neurotransmitters, respectively (Scheffer et al. 2020b). The

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value Wy is a common gain factor that relates synapse count to connection strength; Z,y corresponds to an additional, unknown differential synapse-specific gain; changes in this parameter permit deviations structure. 'The connectome does not specify neural time constants, neural thresholds, and the neural non-linearity (transfer function). Thus, the parameters {Z, b, 7 are the subjects of optimization.
from connectomically-defiee
To examine the extent which the connectome alone provides sufficient information for detailed dynamics, we constrained the optimization problem with biological inductive biases by assuming that the gain and bias parameters Z, b are shared across all neurons of a given cell type. This parameterization at the level of cell types is biologically motivated, as neurons of the same type exhibit similar gene expression, intrinsic olectrophysiological properties, and connectivity patterns Davie et al. 2018) Burer--Evang et al.) (020). Thus, instead of optimizing parameters Zza. bj for each for each synapse and neuron, we optimized a much smaller set of cell-type-specifie parameters 7AB bb where A, B.... E C index cell type. In other words, ZAB is the shared gain factor for the all synapses from neurons of Type to those of Type A and b is the threshold of all neurons of type A. The dynamics are therefore given by: ΣΣ
dx (t) --1)(() dt
wo(1 + + sgn P Caaa b + +u((t)
(2)
BEC
This reduces the number of optimized parameters from 4 193, 161 to just 7 57 parameters, relative to full per-synapse parameterization.
4.2 Task-based optimization
Velocity integration task We optimize the free network parameters over a set of simulation trials using a set of hiologically-motivae self-supervised and unsupervised objectives under the assumption that the circuit performs integration of its input signals.
State initialization For each simulation trial, initialized the firing rates of single E-PG neuron to I, representing bump on the ring, and initialized the activities of all other neurons to O.
Dynamics were simulated according to Eq. for fixed duration of Teim = 2s using the Dormand- Prince method (Dopri5) (Dormand and Prince 1980) provided in the Diffrax (Kidger 2021) library in Jax. During each trial, constant velocity input u [-U, U'] representing clockwise (negative), counter-clockwise (positive), or zero velocity) was applied to the network. Following experimental evidence that GLNO neurons provide angular velocity signals to the HD system (Hulse et al. 2023 the velocity input was selectively injected into either the left or right GLNO neurons.
Training Objectives We define a set of biologically motivated objectives based on the assumption that the network is an integrator, i.e. that it should update its internal state as a linear function of the velocity input, and maintain its existing state in the absence of inputs. Notably, we do not specify particular activity profiles in the network, nor do we ask the state to change by a prespecified amount in response to specific velocity inputs.
In each trial, we evaluate the network state at discrete time points t= == (1,.... At separated by interval At. We use population coding (Bialek et al. 1989) to decode the network state (location along the ring, 0(t) each from the EPG neuron population by calculating the heading as the vector sum of neurons' preferred directions weighted by their firing rates. The change in the represented
angle over the interval At was computed as 50(t) ((t)-
Af
Entropy: Encourages diverse activation across the population by penalizing overly concen- trated or uniform activity patterns.
N
Centropy
ΣΣ
(t) (t)
T
Where (,(()
e0 Σ e
is the softmax of the neural activities a given time point.
Stability: Encourages stability, i.c., no changes in state given zero input (u 0).
1 T
Sstability =
Σ 50(t) 1(u = 0) T
5

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Minimum Speed: Prevents the model from learning a trivial zero solution.
T
speed =
Σ
Σ
80(1)
20
w=-U
where θ is the heavy-side step function.
Linear Consistency: Encourages the model to update its internal state as a linear function of the velocity input
1
T
U
80(t) [u]
Lincarity
) ([u))
2
10m-1
U
where 20 Σ
AB(C)
L1 and 1.2 Regularization: Imposes costs for changes in weights changes, with the penalty size corresponding to the number of synapses that are impacted by each weight.
LL
E
AB AABICs
LJ JEB
L1
Σ
(ABCC:)
JEB
Total loss:
Liaa B1G Cineurit 32Lspeed + BaLenropy BeLLI + BBLL2 Where B1, B2, 83, Br. 3r are hyperparameters.
(3)
Optimization
The initial values of ZAB are all O. Initial values of 64 are a constant b for all cell types. The quantities wo, b, l were chosen by hyperparameter optimization. All trainable parameters optimized minimize the total loss function Leoual using Adam (Kingma 2014.
5
Results
5.1
Robust integrator dynamics from minimal training of a comnectome-coostraineee model
We optimize model parameters according the procedure outlined above (Figure 2h). Allowing only a single gain factor (wy) does not yield activity bumps; allowing two global gain Tactors (one for all excitatory synapses and another for all inhibitory ones) yields an activity bump but no integration. By tuning the small set of cell-type parameters defined above, we recover a dynamical system that is able integrate bidirectional velocity inputs as well as maintain stable heading representation in the presence of noise, displaying near-zero drift under constantly injected multiplicative noise or below % of neuron activity (Figure 2p. c, d). Despite not specifying for output shape our training, we find that the model produces activity bump with a width (Figure (e) close to previous experimental reports of x/ (Kim al. 2017). The model also accurately integrates velocity input, maintaining internal state updates that scale proportionally with input magnitude, and minimal pinning region (no bump movement to non-zero velocity inputs), enabling accurate integration over small velocity inputs and velocity direction changes (Figure g)
5.2 Tuned network exhibits characteristies of a ring aftractor network
To evaluate whether the trained model exhibits dynamics characteristic of continuous ring attractor, we analyzed the geometry of its internal state space. Principal component analysis of EPG activity during continuous, -areingyyigg velocity inputs in both directions reveals that the network's 's popu- lation dynamics are constrained to a one-dimensional manifold (Figure 33,, b). This observation is confirmed by local intrinsic dimensionality analysis using local PCA (Kmbhatia and Leen 1997), which estimates the dimensionality of the dynamics to be approximately (Figure 3b). The correla- tion dimension Grassberger and Procaccia (1983), which quantifies the global scaling properties of the attractor, also close to I (Figure Bi), consistent with smooth, now-dimensional integrator.

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Weights
Weights
Perrett Change Magohade
Abecilute Change Magettuce
Figure 2: Tuned network weights produce robust integrator dynamics. a. Initial and trained network weights. b. The activity bump of the EPG neurons across no input and directional input conditions. The bump exhibits minimal drift for noise levels below 10% of neuron activity d. Slices of population activity for EPG and leftright PEN neurons, colored by time. Asymmetric activation of PEN populations drives bump movement in the corresponding direction. e. The width of the activity bump, when calculated over left, right, and all neurons, closely matches experimental measurements of nt/2 f. The internal heading velocity increases proportionally the magnitude of input velocity, indicating accurate integration. g. The input-output velocity relationship remains linear across biologically plausible input range.
To further assess the topological structure of the dynamics, we apply persistent homology to the E-PG state trajectories using the Ripser package (Bauer 2021 Tralic et al. 2018). This analysis yields persistence diagram consistent with ID topological ring (Figure ). matching theoretical expectations for a circular continuous attractor. Together, these analyses demonstrate that the network does not merely perform velocity integration in the output space, but that its internal dynamics form a structured, low-dimensional ring manifold, in line with theoretical models of the Drosophila HD circuit.
PCA, 24
12-
2312
4114
634
-14
2:90
400
Combitin Démandon
CH
12
03-
-34
0.2
Figure 3: State space geometry of the HD circuit a. 2D projections of EPG activity via PCA show that the dynamics lie on one-dimensional ring manifold. b. Ring manifold structure is preserved in a 3D projection
PCA space. c. Local PCA estimates the intrinsic dimensionality of the dynamics to be 1. d. The local correlation dimension of the network is also approximately 1. e. Persistent homology analysis yields a persistence diagram consistent with a ID topological ring.
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5.3 The connectome as structural prior for functional dynamics
Given that finetuning the HD network by training cell-type parameters is sufficient to obtain a network that produces ring attractor dynamics despite noise in the connectome, we hypothesized that the connectome provides a structural prior that supports robust integrator dynamics. To test this, we trained models initialized with increasingly perturbed versions of the connectome. We found that small amounts of Gaussian noisc, when preserving connection signs, still yielded integrating solutions. However, introducing sign flips degraded performance, and introducing more drastic perturbations, in- cluding shuffling weights within cell-type blocks as well as across the entire connectome, consistently failed to produce functional networks even after extensive training (Figure 4a).
Figure 4: Visualizing the space of solutions From top to bottom: initialization weights and activity bumps of networks initialized with connectomes perturbed with gaussian noise (a" 10), gaussian noise with random sign flips (p 0.05), permutation of weights within cell-type blocks, and permutation all weights. b. Training runs with multiple initial weight matrix conditions. Left: Initial and learned cell-type weights (Z (ZAB) centered at 1.0. Right: Heatmap depicting the neural activity over time when left and right inputs are given, c. Visualization of the surface fit by kernel PCA. The training trajectory points are used fit the surface and are depicted by the blue dots. d. Visualization of training trajectories along the loss landscape. Trajectories are labeled similarly to for comparison.
Cell-type parameterization as the correct level of abstraction Although the connectome provides essential structural information, learning sell-type-specific parameters critical for producing functional network. We trained networks using simplified parameterizations by either tying parameters globally across all cell types, or by learning only broad excitatory and inhibitory cell type parameters (see Section A.2 Figure A4). Neither approach yielded a network capable of velocity integration. Overall, our results showed that network capable of angular velocity integration does not emerge generically from any connectome, nor from arbitrary choices of biophysical parameters.
Diversity of solutions from initial positions on the loss landscape We next explored the de- generacy of the loss landscape by adding different amounts of noise to the network parameters at initialization, and tracking their trajectories across training (Figure 4p). We visualize these trajectories on the fitted loss landscape (Section AA.) of solutions using kernel PCA (Schölkopf al. 1998). We find that the final solutions of the network depend on the proximity of the initial parameters; networks close by tend converge solutions within the same basin, while networks initialized far apart tend to arrive at different solutions. Despite this divergence, networks that arrived at different solutions still produced functional integrator dynamics and reside in the overall loss basin, indicating some degeneracy in the solution space but a common structural motif (Figure 4 d).
5.4 Cell-type ablations
Given functional model of the Drosophila head direction circuit, we next asked: what are the unique roles of each of the cell types within the circuit? To address this question, performed ablation

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experiments by zeroing the output weights of specific cell-type populations in our model. Because the model weights were optimized on the full set of cell types, we retrained the ablated networks to give each model fair chance at recovering performance by altering the weights of the remaining cell types. Thus, we test which sets of neurons are strictly necessary the sense that their function cannot be substituted by simply tuning the weights of other cell types within the network. As with the optimization of the original model, we sweep over a range of hyperparameters and select a set of the best performing models based on linear integration consistency and maximum bump velocity for evaluation (Section A.4). Across all experiments, ablated models exhibit less stability of the bump in the presence of large amounts of noise compared to the original network even after training (Figure
Koto offic por velcity input istes livel 0.120
5C lead 0,500
P velecity Irput
Mean Vilic Resporas
12.5
atditonn
Bumro wabecity
para
0.0
120
156
input vebicty
noce 0.1000
204
PAMR DUTO
208
200
Figure 5: Cell-type ablation experiments. Rate of diffusion of the bump with zero inputs in the presence of multiplicative noise at 10% of neuron activity. Non-integrating solutions (Delta7 and P-EN ablations) not shown. b. Rate of diffusion in a noiseless condition. Mean bump velocity as function of input velocity, colored by ablation. d. Slope of mean velocity response, calculated over inputs in the range of [0, 2]. e. Mean bump width of different ablation conditions. Red line indicates experimentally observed width. f. Sample EPG activity profiles by ablation, given the same input profile as in Figure?) 2
Both Delta7 and ring neurons have been hypothesized to provide the crucial global inhibition to implement ring attractor in the Drosophila (Chang et al. 2023). Although we found that ablating the ring neuron population reduced the range of represented velocities along the ring and displayed more drift in the presence of noise, ablating the Delta7 neurons had far stronger effect: the network was no longer able to form a bump, let alone integrate velocity (Figure 5D. Notably, we found that able to recover bump formation and integration when relaxed the
the aablted network block matrix assumption and individually tune the weights of all the ring neuron subtypes (Section A. indicating that the Drosophila HD circuit relies on multiple types of inhibition, in contrast to theoretical ring attractor models.
The P-EG neurons have previously been hypothesized to play role in stabilizing the bump in the Drosophila HD circuit (Pisokas et al. 2020). although experimental evidence in support of this is lacking. We find support for this hypothesis, the P-EG-ablated model produces a network that exhibits more drift under even low amounts of noise even after training (Figure b).
Finally, although the role of the P-EN neurons in providing asymmetric drive to move the bump in the E-PG neuron population has been well characterized, the roles of the P-ENa and P-ENb subpopulations remain poorly understood. We confirm the role of the P-EN neurons by showing that a PbN-ablated network is unable to maintain or move a bump even after training (Figure We then separately ablated the P-ENa and P-ENb neuron subpopulations and found that while ablating

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either subpopulation increased drift under noise, the P-ENb ablation had a stronger effect, leading to greater drift and reduced range bump velocities.
Conclusion and Discussion
We present method that leverages biological priors and parameterization at the level of cell types to recover circuit dynamics from noisy connectome measurements using self-supervised learning. Without access to biophysical parameters, external task labels, activity data, model is able to recover robust continuous attractor dynamics capable of stable velocity integration in spite of observed asymmetries and small network size in the Drosophila head direction circuit. Interestingly. this result implies that connectomes, without biophysical gain parameters but with knowledge of connectivity signs, may be close being lottery tickets (Frankle and Carbin 2018). We characterize the solution space and show that multiple viable solutions emerge across different initializations, allowing estimation of parameter uncertainty through noise injection. We performed in silico experiments via targeted cell-type-specific ablations to elucidate the mechanistic roles of different cell types within the circuit.
Despite being able to recover circuit dynamics, the fidelity of the inferred parameters in this study to the underlying biological parameters may be limited in part by the fidelity of the connectome, because we performed additional (albeit minor) prepprocessing the connectivity matrix and we found multiplicity of possible solutions. In addition, made assumptions about neurotransmitter identity and synaptic signs based on cell types.
We believe our self-supervised learning approach will be powerful for future work on inferring function and activity from connectomes, relative to supervised approaches which require prior knowledge about circuit activity states. However, our stability and linear consistency losses terms assumed equivariance to imputs and involved population decoding, limiting their direct application to circuits that integrate continuous variables, such oculomotor integrators (Seung) 1996), grid cells (Burak and Ficte 2009), time cells (Kraus et al.) (013), or circuits involved evidence accumulation (Shadlen and Newsome 200T). However, by modifying the combination of losses used during training. An important future direction is to generalize approach by exploring alternative unsupervised objectives that relax these assumptions. Additional future directions include extending the model to study multi-modal sensory integration with the internal state, and validating model predictions through closed-loop perturbation experiments.
believe framework could be easily extended
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A Technical Appendices and Supplementary Material
A.1 Weight Symmetrization
To enforce bilateral symmetry in the neural network architecture while preserving functional con- nectivity patterns, we implemented weight matrix symmetrization procedure. Let W E RVV be the initial weight matrix, where is the number neurons. For cach neuron i, we define its lateralization 4 E {L, R) and its lateralized label (removing lateralization information).
The symmetrization process operates on connection groups defined by the tuple (s, t, l. 4t), where and are the lateralized labels of the source and target neurons, and and 4, are their lateralizations. For each unique connection group, we compute the symmetrized weight:
Waym (s, 4,)
Σ 2nat 1.
(s,,,, (4, 44,44
where Nst is the number of connections in the group and is the indicator function. The final symmetrized weight matrix Wyy is constructed as:
Waym (1,44 if Jusyym (7,, (1,44 Wis
W sym..
otherwise if
0
This procedure ensures that left-right symmetric connections have identical weights (W sysym. = W Wsymj's for lateralized pairs), while preserving the network's functional connectivity patterns and maintaining the overall network structure and strength. The result of this symmetrization is shown in Figure A1
Figure Al: Weight matrix symmetrization.
All experiments we're re with weight symmetrization but eliminating weight symmetrization is possible if left and right populations of each cell type are treated as separate cell populations which can be tuned independently. This doubles the number of cell types but comparable results achievable using this parameterization, even when weight symmetrization is not applied[AX A2
A.2 Single parameter and E/I controls
In addition to the cell-type parameterization of the model, we also performed baseline experiments using single global parameter parameterize the synapse gains across all cell-types (thus tuning differential gains between cell types), as well as using two parameters to describe the excitatory and inhibitory neuron gains separately. These reduced parameterizations failed produce solutions capable of forming or integrating a bump even after model training, demonstrating the importance of cell-type parameterization (Figure A4).
A.3 Kernel PCA and loss landscape visualization
In order to visualize the optimization landscape, we ran training runs with differently seeded initial parameters. We tracked the parameter values throughout training. We then used kernel PCA
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Rightlrput:
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Figure A2: Neural dynamics after weight symmetrization is removed but left and right hemispheres allowed have different biophysical parameterizations.
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Figure A3: Visualization of the dynamics of untrained network.
(Schölkopf et al. 1998) with radial basis function kernel to fit the training trajectories and used Kernel Ridge regression to learn the mapping from the basis to the original parameter space which we use to sample parameter values in order to construct two dimensional loss landscape.
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Activity
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Figure A4: Baseline experiments. Learned weights and ring activity after optimizing only a single global gain parameter Wy (top), or optimizing two parameters describing excitatory and inhibitory synaptic gains (bottom).
A.4 Hyperparameter sweeps and model selection
For each ablation experiment (including the no-ablation condition), trained models across set of 90 hyperparameters by using grid search over the initial bias values, leak, and global synapse strength. We additionally randomly initialized the block weights using normal distribution centered at with variance of 2 percent the parameter value and swept over five seeds for each hyperparameter setting. In Figure we chose the best models from each condition to evaluate. We define best as the models with the smallest minimum speed loss (i.e., the models with the greatest range of velocity integration), after filtering for R2 of > 0.7 and temporal consistency loss of 0.013.
A.5 Delta7 ablations with full ring parameterization
Figure A5: Activity bumps in Delta7 ablation, full ring parameterization.
angular velocity over time
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Figure A6: Internal vs. input velocity in Delta7 ablation, full ring parameterization.
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A.6 Sample weights from ablation experiments
In Weigets
Frod diss
Fillll Weigtte
Fina Not N0-9975
Paruar CI erags Magnitck
Aboofata Charge Magritadd
Pasarri Change Magritude
Chunge Magentads
(a) P-EG ablation
(b) Ring ablation
atante J1
I-ilal Weighte
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Percent Change Hagritude
Abookute Charge in Magn bude
F5R
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(c) Delta7, full ring parameterization
(d) P-EN ablation
Figure A7: Ablation effects on trained synaptic weights: P-EG, Ring, Delta7, and P-EN.
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atente pena
sobre pento
I-bial Weights
Change Hagnituce
Abnokute Charge Nagritute
Peccent Change Hagnitude
Abrokatte Change Magnhude
(a) P-ENa ablation
(b) P-ENb ablation
Figure A8: Ablation effects on trained synaptic weights: P-ENa and P-ENb.
A.7 Additional Supplementary Methods
Velocity input Velocity input is driven by differential activity between the left and right GLNO neurons. In our experiments, we drove the velocity using either the left or right GLNO neurons at given time. It is sufficient to drive differential activity between the left and right GLNO neurons to drive the velocity input which can shown in A10
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Ablata Encropy loos
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Abiate man speed loss
Abiaze regulantation
Abiaca compora consistency loos
Aeguuarr Welectty
Ablare stationary loss
Figure A9: Neural Dynamics after ablating various loss terms. We show example neural trajectory
on the left hand side rotating in both directions as well as the angular velocity over time under different input conditions
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Figure A10: Activity trace when reversing sign of velocity input
A.8 Supplementary Figures
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Figure A11: Synapse statisties. Distribution of incoming and outgoing synapse weights before and after training of full connectome-initializeeed network.
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Reviews:
[{"review_id": "k0v73Ds5hq", "reviewer": "Reviewer_4Q4m", "summary": "The authors address the challenge of deriving functional network dynamics from incomplete and noisy connectomic data of the Drosophila head direction (HD) circuit. They propose a method to infer missing neuronal parameters—such as thresholds and time constants—and to correct potential errors in synaptic connectivity. The resulting model accurately integrates angular velocity inputs and exhibits continuous attractor dynamics. Importantly, the authors demonstrate that tuning parameters at the cell-type level suffices to achieve the desired dynamics. They also perform in silico ablation experiments to elucidate the functional roles of specific neuron types.", "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?\n- What would the results be like if $Z$ and, therefore, synaptic connectivity are not tuned at all?\n- Much prior domain knowledge is used in the design of the method, which limits its application to other circuits. To explore its full potential, I wonder how the method would behave if all of this knowledge were removed. By domain knowledge, I mean 1. external inputs are only injected to left/ right GLNO neurons; 2. a single E-PG neuron's activity is initialized to 1 while others are set to 0; 3. location is decoded from the activity of E-PG neurons.\n- Can you explain in more detail the \"population coding\" you used to decode the location?\n- Can you elaborate on how the hyperparameters are optimized?\n- Can you elaborate on why the linear consistency loss encourages a *linear* readout of the velocity input?\n- The meaning of $\\Theta(\\cdot)$ used in the loss function is unexplained.\n- Can you elaborate on why $C_{ij}$ is used in L1 and L2 regularization? Are small synapse counts more subject to measurement errors?", "limitations": "- My primary concern lies in the generalizability of the proposed method. The approach—particularly the training objectives and input-output relationships—appears tailored specifically to the Drosophila head direction circuit. A broader discussion of the scope and limitations of this generalization would significantly strengthen the paper.\n\n- The authors argue that replicating idealized continuous attractor dynamics requires finely tuned connectivity with perfect rotational symmetry, which is not observed in the connectome. While this claim is common in theoretical models, I am not fully convinced that such symmetry and fine-tuning are strictly necessary in biological implementations. Dynamical systems with slow manifolds and discrete fixed points can exhibit ring-attractor-like behavior without strict rotational symmetry or extensive tuning. It is plausible that the Drosophila HD circuit leverages such robustness.\nThis raises an important question: how challenging is the problem if the system can function without the degree of tuning the authors assume? For example, what do the dynamics look like with the initial connectivity matrix and neuronal parameters? Including a comparison between this baseline model and the final optimized model would help clarify the extent to which the proposed optimization is necessary and what specific gains it affords.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Overall, I found the question interesting, the analysis solid, the results convincing, and the paper well-written. \nWhile I am generally positive about the contribution, I outline some high-level concerns in the \"Limitations\" section below.", "was_revised": true, "final_justification": "The authors address the challenge of inferring functional network dynamics from incomplete and noisy connectomic data of the Drosophila head direction (HD) circuit. The proposed self-supervised learning approach, guided by biologically motivated training objectives, is both interesting and promising. The method also appears readily applicable—or adaptable with minimal modifications—to other integrator circuits. \nAdditionally, the results suggest that the connectome already contains substantial information about circuit function, as parameter tuning at the cell-type level proves sufficient. \nOverall, I find the paper to be well-executed and complete with respect to the problem it seeks to address, and I recommend its acceptance to NeurIPS.", "rebuttal": "Thank you for your thorough and thoughtful review, and constructive feedback! We have responded below to your specific concerns and questions. \n\n> 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?\n\nIn 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. \n\n> What would the results be like if Z and, therefore, synaptic connectivity are not tuned at all?\n\nWe 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). \n\n> Much prior domain knowledge is used in the design of the method, which limits its application to other circuits...\n\nWe have found that our method is robust to each of the specific assumptions you highlighted:\n1. 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).\n2. 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.\n\n3. 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.\nMore 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.\n\n>Can you explain in more detail the \"population coding\" you used to decode the location?\n\nPopulation 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. \nAs 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.\n\n> Can you elaborate on how the hyperparameters are optimized?\n\nWe 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.\n\n> Can you elaborate on why the linear consistency loss encourages a linear readout of the velocity input?\n\nThank you for pointing this out. The formula presented has a typo in it. The corrected loss term should read \n$ \\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.\n\n> The meaning of $\\Theta ( \\cdot )$ used in the loss function is unexplained.\n\nThank 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.\n\n> Can you elaborate on why $C_{ij}$ is used in L1 and L2 regularization...\n\nThe $C_{ij}$ 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.\n\n>My primary concern lies in the generalizability of the proposed method...\n\nYou 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. \n\nWe 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.\n\n> The authors argue that replicating idealized continuous attractor dynamics requires finely tuned connectivity with perfect rotational symmetry, which is not observed in the connectome...\n\nYour 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!\n\n> This raises an important question: how challenging is the problem if the system can function without the degree of tuning the authors assume?\n\nWe 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."}, {"review_id": "Tw8qhqIX5f", "reviewer": "Reviewer_rEqo", "summary": "In this work, the Authors propose a model to specify a circuit's function based on its known connectivity. To this end, they use a standard model for the dynamics of neural activity and learn its parameters (gain, bias, time constant) individually for each cell type to optimize a regularized task-specific loss. The model is then applied to the fruit fly's head direction circuit data from the FlyEM dataset where the Atuhors recover some known properties if the circuit (integrator dynamics; ring attractor network) and confirm/refine the roles of individual cell types in this circuit.", "questions": "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?\n\nHow to extend this model to other circuits? How much prior knowledge of the circuit's function is required for the successful deployment of this model?", "limitations": "The loss function appears to require substantial information about the function of the neural circuit -- the property that the model has to predict.\n\nMy current scores on the paper reflect the uncertainty about how this cost function was formulated for the described task and how to formulate it for new tasks. I am happy to revisit my scores based on the interaction with the Authors.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nThe work deals with a highly important neuroscience problem of understanding the function/dynamics of a circuit bases on (incomplete, noisy) data describing its connectivity. Indeed, the connectivity data, typically obtained with the electron microscopy, while providing the information about the connectivity graph between the cells, lacks other important information including the strength of connectivity and other neuronal property. Additionally, the data is often noisy, featuring only the synaptic connections that were established with high confidence.\n\nThis work is also timely: as more high-volume high-resolution connectivity data becomes available for different species (worm, fruit fly, mouse), an opportunity present to study various circuits using these datasets, necessitating an algorithm that goes from the connectivity graph to a dynamical system. This paper presents such an algorithm.\n\nThe approximations/assumptions made in this paper are reasonable. For the neuronal dynamics, a standard equation is used. The parameters for this equation are unified within each cell type. These assumptions are then tested by recovering the known dynamics of the studied neural circuit and the known roles of its constituent cell types.\n\nThe results of the ablation studies are interesting and novel. Alongside with recovering the known properties of the studied neural circuit and its cells, the ablation studies here have helped to propose some novel properties of the circuit's cell subtypes.\n\nThe text is mostly well-written, well-structured and easy to follow.\n\nWeaknesses.\n\nThe 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. This includes the linear consistency, the stability, the minimum speed, and the entropy loss terms. I didn't see the ablation experiments for the components of the loss function but I assume that, since all of them are presented in the manuscript, all of them may be necessary to reproduce the known properties of the circuit in the proposed model. If that is true, then a successful model may require a substantial knowledge of the studied system -- something that it otherwise is supposed to provide. This property may also limit the application of the model to new, less studied systems.\n\nFor the loss function terms, I couldn't find the definition for capital theta.", "was_revised": true, "final_justification": "The Authors have clarified where they see the novelty/impact of the paper. Their responses have addressed some of my concerns.\nThe other concerns still remain, e.g., to further increase the impact of this work, one needs to *show* that the model can generalize to other circuits and to to just *suggest* that this might be the case.\nThus, based on the discussion with the Authors, I am raising the score by 1 point.", "rebuttal": "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.\n\n>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…\n\nThank 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. \n\nSpecifically, 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. \n\nWe 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:\n\n1. 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.\n2. 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. \nWe are currently working to extend our model by testing these strategies in novel circuits with unknown or less-characterized functional roles.\n\n> 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?\n\nThis 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. \nI will briefly describe some results from our ablation experiments with individual loss terms here:\n* Linear consistency: The model is still able to integrate at a high velocity, but the integration velocity is not smooth nor linear.\n* 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.\n* Minimum speed: The model does not learn to integrate and collapses to a single fixed point.\n* 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. \n* L1/L2 regularization: No major impact on integration ability; we include these primarily to promote biologically plausible (energy-efficient) neural activity\n\n\n> For the loss function terms, I couldn't find the definition for capital theta.\n\nThank 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."}, {"review_id": "TgaE1E2E7t", "reviewer": "Reviewer_VhDz", "summary": "This paper presents a computational framework to infer functional dynamics directly from a raw, synapse-level neural connectome, addressing the fundamental question of how anatomical structure specifies circuit function.  The authors use the Drosophila head direction (HD) circuit, a known continuous attractor network, as a testbed.  Their core contribution is a method that optimizes a small, biologically-interpretable set of parameters at the cell-type level (e.g., synaptic gains between cell types, neuronal biases) rather than for individual neurons or synapses.  The model is trained using a self-supervised objective that enforces linear velocity integration, without requiring any neural activity recordings or behavioral labels.  The resulting model successfully reproduces robust attractor dynamics, matching key experimental findings.  The authors then use this functional model to perform in silico ablation experiments, probing the specific roles of different cell types and generating new, testable hypotheses about the circuit's operation.", "questions": "On Connectome Preprocessing and Robustness: The paper states that the connectome data underwent a \"symmetrization process\" (§11.7).  Could the authors please elaborate on the specifics of this procedure in the main text or a supplement? How essential is this step to achieving the reported results? A demonstration that the model is not overly sensitive to this specific preprocessing choice would strengthen the paper. My evaluation would improve if the authors can show that the symmetrization is a minor, justified correction and that the model's success is not brittle with respect to this choice or the key hyperparameters.\n\nOn Generalizing the Self-Supervised Objective: The paper suggests the framework could be extended to other functions like \"memory, or normalization\".  This is an exciting prospect. Could the authors provide a concrete example of how the self-supervised loss function could be formulated for a different computation, for instance, divisive normalization, without relying on target activity patterns? Clarifying this would help the reader understand the true scope and generalizability of the proposed methodology beyond integrator circuits.\n\nOn the Role of P-EN Subpopulations: The finding that ablating the P-ENb subpopulation has a more detrimental effect than ablating P-ENa is a specific and novel prediction.  Can the authors leverage their trained model to provide a more mechanistic hypothesis for this difference? For instance, do the learned synaptic gain parameters (Z_AB) or the underlying connectivity counts (C_ij) reveal a systematic difference in how these two populations contribute to the circuit that explains the ablation result? Linking the emergent functional difference back to the structural or parameterized weights would provide a deeper insight.\n\nOn the Level of Abstraction (Cell-Type vs. Sub-Type): The finding that a Delta7-ablated network can be partially rescued by tuning the weights of individual ring neuron subtypes is fascinating and seems to run counter to the main thesis that cell-type level tuning is sufficient.  Does this suggest that the optimal level of parameterization is circuit-dependent? Could the authors comment on the principles that might guide a researcher in choosing the right level of abstraction (e.g., cell-type, subtype, or even single-neuron) when applying this framework to a new circuit?", "limitations": "Yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\nSignificance: The central idea of finding a \"middle ground\" of parameterization at the cell-type level is a powerful and important contribution.  It provides a principled approach to make connectomic data functional, moving beyond hand-tuned models without resorting to overly complex, unconstrained optimization. The self-supervised training paradigm, which obviates the need for large-scale neural recordings, is highly significant and enhances the scalability of this approach to other circuits where such data is unavailable. \n\n\n\nQuality: The manuscript is of high technical quality. The authors do not just show that their model can integrate velocity; they perform a rigorous analysis of the learned dynamics using PCA, intrinsic dimensionality analysis, and persistent homology to demonstrate that the model has indeed learned a low-dimensional ring attractor manifold.  The ablation studies are particularly strong, as they involve retraining the ablated networks, providing a much more compelling test for the necessity of a given cell type than simply removing it from a pre-trained model. \n\n\n\nClarity: The paper is exceptionally well-written, clear, and logically structured. The motivation is well-established, the methods are described precisely, and the figures are clean, informative, and directly support the claims made in the text. Figure 1, in particular, provides an excellent and intuitive schematic of the core methodology. \nOriginality: While connectome-constrained modeling is an active area of research, the specific combination of self-supervised learning with a cell-type level parameterization applied directly to a raw, asymmetric connectome is novel.  The approach successfully recovers function from a biologically realistic (i.e., noisy and imperfect) structure, which is a key departure from models that assume idealized connectivity. \n\n\nWeaknesses:\n\nDependence on A Priori Functional Knowledge: The self-supervised loss function is predicated on knowing that the circuit performs linear velocity integration.  This is a reasonable assumption for the well-studied HD system, but it limits the framework's applicability for true de novo discovery of function in less-understood circuits. The authors acknowledge this, but it remains a primary limitation on the generalizability of the current method.\n\nClarity on Preprocessing and Hyperparameters: The paper mentions a \"symmetrization process\" applied to the connectome synapse counts as a preprocessing step.  The details are deferred to an appendix, but this step could be critical. If the raw connectome requires significant \"correction\" to work, it slightly weakens the claim of obtaining function directly from the measured structure. Additionally, while the paper mentions that key global parameters were chosen via hyperparameter optimization, the sensitivity of the results to these choices, as well as the loss weights, is not explored. This leaves open the question of the model's robustness.", "was_revised": true, "final_justification": "I still believe that this work arrives at a common conclusion by applying a specific data analysis method to a particular dataset.\n\nThe method is not novel and there is no new insight from this work.\n\nI keep my score.", "rebuttal": "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.\n\n> Dependence on A Priori Functional Knowledge…\n\n\nYou 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.\n\n\n> Clarity on Preprocessing and Hyperparameters…\n\n\nThe 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.\nWe 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.\n\n> On Connectome Preprocessing and Robustness…\nDetails 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.\n> On Generalizing the Self-Supervised Objective…\n\n\nThank 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. \n\n\n> On the Role of P-EN Subpopulations…\n\nThank 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. \n\nAt 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.\n\nHowever, 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.\n\nIn 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). \n\nThese 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.\n\nThank you again for suggesting this analysis, and we will include more details in the supplementary information.\n\n> On the Level of Abstraction (Cell-Type vs. Sub-Type)...\n\nYou 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."}, {"review_id": "0Rud5Xe9KN", "reviewer": "Reviewer_hYrM", "summary": "Authors conduct a computational study on the head direction circuit in fruit fly by using the recently published connectome dataset. They train networks that are constrained on the connectome and show that a continuous attractor (which is often studied with hand-picked models in the literature) emerges from a self-supervised training. Overall, the authors provide several distinct insights into the neurobiology of the head circuit with a rigorous and well-executed study.", "questions": "I have the following suggestions:\n\n- 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\n\n- 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.\n\n- 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. \n\n- 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.", "limitations": "NA.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "The methodology is sound, the ablation studies are on point, the training process is exceptionally clever. I am having trouble finding things that are wrong with this work, see my final point below for suggestions for improvements.", "was_revised": true, "final_justification": "As noted in my review, I think this work is an interesting addition to Neurips.", "rebuttal": "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:\n\n\n> 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\nThank you for the great suggestion; we will include a more detailed discussion of our loss terms in the paper. \n\nWith regards to the linear consistency loss, the formula presented has a typo in it. The corrected loss term should read \n$\\frac{1}{T} \\frac{1}{2U} \\sum_t^T \\sum_{u=-U}^U \\left( \\frac{\\delta \\theta(t)}{|u|} - \\mu\\right)^2$ 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.\nLikewise, 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. \n\nSpecifically, 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.\n\n> 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.\n\nThank 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. \n\n> 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.\n\nThank 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.\n\n> 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.\n\nThank 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.\n\nThat 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."}]

Author rebuttal:

Meta-review: The goal of this work is to use recently collected data on the fly connectome to construct a working model of the fly head direction circuit based solely on connectivity data augmented with theoretically-motivated learning objectives. While a small number of previous papers have attempted the same structure-to-function task, these efforts have been in early sensory systems with primarily feedforward architecture, whereas the head direction system is multiple synapses removed and recurrent. The authors find that constructing a working attractor model requires inferring parameters at the level of cell types, with several consistency losses necessary to impose either known biological properties (activity sparsity) or necessary functional properties (correct integration behavior). The authors then demonstrate the robustness of their learned network solution, proposing hypotheses to be tested in future experiments.

Reviewers' responses were mixed. While many found the work well-written and neuroscientifically interesting, there was debate as to the novelty of the approach relative to other connectome-based models, and some reviewers felt that the modeling strategy used to achieve success here may not generalize well to other settings.

During rebuttal, the authors emphasized that this work differs from other connectome-constrained models in its focus on a system quite removed from the sensory periphery and recurrent circuits. Authors conceded that their approach relies heavily on the known function of the circuit, though they also argued that several of their loss objectives could be part of a more general strategy in other circuits, particularly when those circuits involved integrator or attractor behavior. Authors also maintained that while the head direction system is well-studied, their model does offer insights about the system achieves stability and can generate novel experimental predictions. They also noted that this work is distinct from several other noted works in neuroscience using self-supervised learning, which attempt to infer structure (connectivity or cell types) from functional data.

In the end, all reviewers agreed that the work is well-executed and clearly written, and while not all agree about its novelty, the innovative combination of inferring cell-type-specific parameters with self-supervised consistency losses to produce a _functional_ connectome-constrained model is a significant advance. Moreover, while the approach here may be somewhat tailored to the head direction circuit, it contains a number of insights and ingredients that are likely to inform and be adapted to other systems. All these factors argue for acceptance.
Decision: Accept (poster)