Connectome-to-Function: Conditional Generative Latent Representations for Reservoir Computing

# Connectome-to-Function: Conditional Generative Latent Representations for Reservoir Computing
Source: [https://arxiv.org/html/2609.06093](https://arxiv.org/html/2609.06093)
Xingyu LiuYuanhao JiaYunhang XiaoHairuo XueFeihan SunGuozhang Chen\\corresponding

###### Abstract

Connectomes, graph\-level maps of neurons and their synaptic connections, provide a structural basis for understanding how brain circuits support function and computation\. However, mapping connectome structure to computation remains difficult because these graphs are high\-dimensional, sparse, and sensitive to local structural variation\. Existing approaches often depend on hand\-crafted structural descriptors or task\-specific predictors, which limits their ability to represent connectomes in a form that is both generative and functionally meaningful\. We propose a conditional generative latent framework that encodes connectome graphs into a compact structural space while using available node\-level conditions to guide reconstruction and generation\. From this space, the model can reconstruct observed connectivity with a mean edge\-reconstruction AUC up to 0\.910 and generate new candidate connectomes, enabling a unified analysis of graph structure and computational behavior\. Using connectome\-derived graphs as recurrent computational substrates, we found that the learned latent space captures functional variation across reservoir\-computing experiments, with cross\-validatedR2R^\{2\}values up to approximately 0\.87\. Interpretability analysis further revealed task\-specific structural mechanisms: in our examples, memory performance is associated with reciprocal recurrent connectivity, whereas prediction and classification are more strongly associated with spectral properties of the recurrent network\. These findings suggest an AI\-for\-science approach to linking neural connectivity to computation and provide a generative and interpretable basis for studying how distinct structural mechanisms shape computational capacity\.

1National Key Laboratory for Multimedia Information Processing, School of Computer Science, Peking University, Beijing, China

2School of Electronics Engineering and Computer Science, Peking University, Beijing, China

3School of Computer Science, Beijing University of Posts and Telecommunications, Beijing, China

4College of Engineering, Peking University, Beijing, China

5Yuanpei College, Peking University, Beijing, China

guozhang\.chen@pku\.edu\.cn

## Introduction

A central scientific question in neuroscience is how the structure of a neural circuit gives rise to its computational function\. Connectomes provide increasingly detailed descriptions of neural wiring across scales\([Sporns, Tononi, and Kötter 2005](https://arxiv.org/html/2609.06093#bib.bib37)\), and network neuroscience has related topology to integration, segregation, and wiring cost\([Bullmore and Sporns 2009](https://arxiv.org/html/2609.06093#bib.bib9);[Rubinov and Sporns 2010](https://arxiv.org/html/2609.06093#bib.bib33)\)\. Recent single\-cell connectomes further resolve synaptic connectivity together with neuronal 3D position and cell identity\([Dorkenwald et al\. 2024](https://arxiv.org/html/2609.06093#bib.bib14);[The MICrONS Consortium 2025](https://arxiv.org/html/2609.06093#bib.bib40)\), making this structure–function question accessible at the level of local circuits\. Reservoir computing provides a controlled way to probe such relationships because recurrent connectivity is kept fixed while only a readout is trained\([Jaeger 2001](https://arxiv.org/html/2609.06093#bib.bib18);[Maass, Natschläger, and Markram 2002](https://arxiv.org/html/2609.06093#bib.bib27);[Suárez et al\. 2024](https://arxiv.org/html/2609.06093#bib.bib38)\)\. This leads to our central question:*can measured connectome structure be organized into a continuous representation that helps explain variation in circuit computation?*

![Refer to caption](https://arxiv.org/html/2609.06093v1/Teaser.png)Figure 1:Connectome\-to\-function latent space\. Observed connectomes are encoded into a generative latent space where coordinates predict reservoir performance across copy memory, Lorenz system prediction, and sequential MNIST\. Arrows indicate task\-specific functional directions, and bottom values report cross\-validatedR2R^\{2\}\.Existing approaches capture only parts of this problem\. Graph descriptors characterize empirical connectomes through predefined statistics\([Sporns, Tononi, and Kötter 2005](https://arxiv.org/html/2609.06093#bib.bib37);[Bullmore and Sporns 2009](https://arxiv.org/html/2609.06093#bib.bib9);[Rubinov and Sporns 2010](https://arxiv.org/html/2609.06093#bib.bib33)\), while generative network can synthesize graph structure\([Vértes et al\. 2012](https://arxiv.org/html/2609.06093#bib.bib44);[Betzel et al\. 2016](https://arxiv.org/html/2609.06093#bib.bib6);[Betzel and Bassett 2017](https://arxiv.org/html/2609.06093#bib.bib7);[Akarca et al\. 2021](https://arxiv.org/html/2609.06093#bib.bib3);[Barabási and Barabási 2020](https://arxiv.org/html/2609.06093#bib.bib4);[Kipf and Welling 2016](https://arxiv.org/html/2609.06093#bib.bib21);[Simonovsky and Komodakis 2018](https://arxiv.org/html/2609.06093#bib.bib35)\); however, these approaches are primarily designed to reproduce structural properties rather than to reveal how structural variation maps to computation\. Shuvaev et al\. showed that a compact generative encoding can preserve task\-relevant network structure, evaluating compression by the performance of the generated network rather than weight reconstruction\([Shuvaev et al\. 2024](https://arxiv.org/html/2609.06093#bib.bib34);[Gaier and Ha 2019](https://arxiv.org/html/2609.06093#bib.bib15)\)\. However, their representation is obtained for task\-trained artificial networks with a specific task performance entering the optimization criterion\. Conversely, connectome\-based reservoir studies directly evaluate biological wiring as a computational substrate\([Damicelli, Hilgetag, and Goulas 2022](https://arxiv.org/html/2609.06093#bib.bib13);[Morra and Daley 2023](https://arxiv.org/html/2609.06093#bib.bib29);[Costi et al\. 2025](https://arxiv.org/html/2609.06093#bib.bib11);[Sumi et al\. 2023](https://arxiv.org/html/2609.06093#bib.bib39)\), but largely study fixed connectomes or rewired variants\. Thus, a key gap remains: a*generative representation learned solely from measured biological connectivity*in which computational organization can be discovered without using function to train the representation\.

To address this gap, we propose a conditional generative latent framework for sparse connectome graphs\([Sohn, Lee, and Yan 2015](https://arxiv.org/html/2609.06093#bib.bib36)\)\. The model encodes each local circuit into a compact structural coordinate while using neuronal location and cell type as generation conditions, enabling both reconstruction and controlled sampling of candidate connectomes\. Experiments show that this latent space is structurally faithful and functionally informative\. The model reaches an edge\-reconstruction AUC of0\.9100\.910and better preserves higher\-order topology than a naive VAE baseline\([Liu, Li, and Chen 2026](https://arxiv.org/html/2609.06093#bib.bib24)\)\. When generated graphs are evaluated as reservoirs on memory, forecasting, and classification tasks\([Suárez et al\. 2024](https://arxiv.org/html/2609.06093#bib.bib38)\), their latent coordinates predict performance with cross\-validatedR2R^\{2\}values of approximately0\.460\.46–0\.870\.87\. Interpretability analysis further reveals task\-specific structural mechanisms: memory is associated with reciprocal recurrent connectivity, whereas prediction and classification are more strongly associated with spectral properties of the recurrent network\. Thus, the learned coordinates provide a generative graph representation aligned with measurable computational variation\.

![Refer to caption](https://arxiv.org/html/2609.06093v1/Overall_framework.png)Figure 2:Framework overview\. \(a\) The conditional VAE encodes directed local circuits sampled from connectomes with node coordinates and cell type into a 32\-dimensional structural latent coordinate, then decodes edge probabilities under node\-level conditions\. \(b\) Decoded graphs define fixed Dale\-signed reservoirs evaluated on copy memory, Lorenz system prediction, and sequential MNIST; a predictor maps latent coordinatesziz\_\{i\}to functional scoresFiF\_\{i\}\.In summary, our main contributions are as follows:

- •We propose a conditional generative framework for connectome graphs that learns latent representations by conditioning on neuronal cell type and spatial organization\.
- •We identify a structure\-function space predictive of reservoir\-computing performance in the learned latents, across various tasks including memory, dynamic prediction, and classification\.
- •We conduct an interpretability analysis of the learned connectome space, linking task\-relevant functional directions to distinct structural mechanisms\.

## Related Work

#### Graph Descriptor and Generative Representation Learning

Connectomes have traditionally been characterized using graph\-theoretic descriptors that summarize properties such as small\-worldness, modularity, efficiency, clustering, and wiring cost\([Sporns, Tononi, and Kötter 2005](https://arxiv.org/html/2609.06093#bib.bib37);[Bullmore and Sporns 2009](https://arxiv.org/html/2609.06093#bib.bib9);[Rubinov and Sporns 2010](https://arxiv.org/html/2609.06093#bib.bib33)\)\. While these measures provide interpretable summaries of network organization, they are predefined, post\-hoc descriptive statistics and therefore offer limited access to the high\-dimensional space of possible connectome structures\. Mechanistic generative network models address this limitation by synthesizing graphs under prescribed wiring rules and structural constraints\([Vértes et al\. 2012](https://arxiv.org/html/2609.06093#bib.bib44);[Betzel et al\. 2016](https://arxiv.org/html/2609.06093#bib.bib6);[Betzel and Bassett 2017](https://arxiv.org/html/2609.06093#bib.bib7);[Akarca et al\. 2021](https://arxiv.org/html/2609.06093#bib.bib3);[Barabási and Barabási 2020](https://arxiv.org/html/2609.06093#bib.bib4)\), yet they lack continuous, data\-driven latent spaces\. In parallel, graph representation learning provides continuous representations of discrete graph structures: variational graph autoencoders and related models encode graphs or their nodes into latent variables to reconstruct adjacency patterns\([Kipf and Welling 2016](https://arxiv.org/html/2609.06093#bib.bib21);[Simonovsky and Komodakis 2018](https://arxiv.org/html/2609.06093#bib.bib35)\), while conditional variational frameworks enable generation under external attributes\([Sohn, Lee, and Yan 2015](https://arxiv.org/html/2609.06093#bib.bib36)\)\. Although models such as GraphVAE can generate small graphs under graph\-size constraints\([Simonovsky and Komodakis 2018](https://arxiv.org/html/2609.06093#bib.bib35)\), general\-purpose graph generators remain difficult to apply directly to neural microcircuits\. Common decoding or graph\-matching schemes are not well suited to the extreme sparsity of cortical connectivity, and existing models rarely integrate 3D spatial coordinates, discrete cell identities, and permutation\-invariant graph\-level representation within a single conditional generative process\.

#### Connectome\-Based Computation and Reservoir Models

Recent network\-neuroscience studies have leveraged empirical connectomes as recurrent architectures, often using reservoir\-computing models to relate static topology to computation\([Suárez et al\. 2024](https://arxiv.org/html/2609.06093#bib.bib38)\)\. Prior work has shown that connectome\-derived reservoirs can support memory tasks\([Damicelli, Hilgetag, and Goulas 2022](https://arxiv.org/html/2609.06093#bib.bib13)\), chaotic time\-series prediction under fruit\-fly\-connectome constraints\([Morra and Daley 2023](https://arxiv.org/html/2609.06093#bib.bib29)\), and Drosophila\-based forecasting analyses involving empirical topology and weight distributions\([Costi et al\. 2025](https://arxiv.org/html/2609.06093#bib.bib11)\)\. Related biological reservoir studies also suggest that modular neuronal organization can improve generalization\-filter behavior\([Sumi et al\. 2023](https://arxiv.org/html/2609.06093#bib.bib39)\)\. Although these studies show that empirical connectomes can support reservoir\-computing tasks, they mainly evaluate fixed empirical networks or simple rewired variants\. They do not learn a continuous generative latent space that can both produce new connectome structures and predict functional variation across tasks\.

#### Structure\-to\-Function Mapping and Performance Prediction

Investigating how network topology dictates computational capacity forms a shared frontier across machine learning and systems neuroscience\. Neural architecture search \(NAS\) studies typically analyze artificial architectures and use architecture–performance pairs or initialization\-based proxies to estimate task performance\([You et al\. 2020](https://arxiv.org/html/2609.06093#bib.bib46);[Wen et al\. 2020](https://arxiv.org/html/2609.06093#bib.bib45);[Abdelfattah et al\. 2021](https://arxiv.org/html/2609.06093#bib.bib1)\)\. In systems neuroscience, connectome\-constrained models show that measured biological wiring can support mechanistic predictions for a specified computation\([Lappalainen et al\. 2024](https://arxiv.org/html/2609.06093#bib.bib22)\), while spatially embedded recurrent networks demonstrate how structural and functional organization can emerge under biological constraints in artificial systems\([Achterberg et al\. 2023](https://arxiv.org/html/2609.06093#bib.bib2)\)\. Across these lines of work, structure–function relationships are typically studied either by evaluating task\-trained artificial architectures, by using task performance to supervise architecture prediction, or by optimizing models constrained by a particular biological circuit\. Here, we take a complementary approach: we learn a continuous generative representation from measured biological connectivity without task\-performance supervision, and subsequently test whether the learned space predicts variation in reservoir performance across tasks\.

## Method

### Conditional VAE for Connectome

#### Problem Setup and Notation

Single\-cell\-level structure is high dimensional\. To better address the structure\-to\-function question, we first learn a low\-dimensional latent space for circuit structure\. Specifically, a local circuit structure is considered as an attributed directed graphG=\(V,E,X\)G=\(V,E,X\), whereVVandEEare the sets of vertices and edges andXXdenotes the node\-annotation matrix withN=\|V\|N=\|V\|\. Further, we used adjacency matrixA∈\{0,1\}N×NA\\in\\\{0,1\\\}^\{N\\times N\}, whereAi​j=1A\_\{ij\}=1indicates a connection from neuronjjto neuronii\. We decompose the node annotations as

X=Concat⁡\(Xloc,Xtype\)X=\\operatorname\{Concat\}\\left\(X\_\{\\mathrm\{loc\}\},X\_\{\\mathrm\{type\}\}\\right\)whereXloc∈ℝN×3X\_\{\\mathrm\{loc\}\}\\in\\mathbb\{R\}^\{N\\times 3\}contains the spatial coordinates of the neuronal somas andXtype∈\{0,1\}N×NtypeX\_\{\\mathrm\{type\}\}\\in\\\{0,1\\\}^\{N\\times N\_\{\\mathrm\{type\}\}\}encodes neuron types \(excitation or inhibition\) as one\-hot vectors\. We parameterize this latent space by assigning each graph aDD\-dimensional latent variablez∈ℝDz\\in\\mathbb\{R\}^\{D\}\. To this end, the encoder, parameterized byϕ\\phi, defines the approximate posterior distributionqϕ​\(z∣A,X\)q\_\{\\phi\}\(z\\mid A,X\), while the decoder, parameterized byθ\\theta, defines the conditional likelihoodpθ​\(A∣z,X\)p\_\{\\theta\}\(A\\mid z,X\)\. Thus, the latent variablezzcaptures the structural information of the connectome graph conditioned on the observed node features\.

#### Conditional Variational Objective

Conditioning on the node featuresXX, we formulate connectome structure modeling under a conditional variational framework\([Kingma and Welling 2014](https://arxiv.org/html/2609.06093#bib.bib20);[Rezende, Mohamed, and Wierstra 2014](https://arxiv.org/html/2609.06093#bib.bib32)\)\. Specifically, we assume a standard Gaussian prior overzzthat is independent of the node conditionsXX, i\.e\.,p⁡\(z∣X\)=p⁡\(z\)=𝒩⁡\(0,I\)p\(z\\mid X\)=p\(z\)=\\mathcal\{N\}\(0,I\)\. Under this generative assumption, the conditional log\-likelihoodlog⁡pθ​\(A∣X\)\\log p\_\{\\theta\}\(A\\mid X\)admits the following evidence lower bound \(ELBO\):

log⁡pθ​\(A∣X\)\\displaystyle\\log p\_\{\\theta\}\(A\\mid X\)≥ℒELBO​\(θ,ϕ,A,X\)\\displaystyle\\geq\\mathcal\{L\}\_\{\\mathrm\{ELBO\}\}\(\\theta,\\phi;A,X\)=𝔼qϕ​\(z∣A,X\)​\[log⁡pθ​\(A∣z,X\)\]\\displaystyle=\\mathbb\{E\}\_\{q\_\{\\phi\}\(z\\mid A,X\)\}\\left\[\\log p\_\{\\theta\}\(A\\mid z,X\)\\right\]−DKL\(qϕ\(z∣A,X\)∥p\(z\)\)\.\\displaystyle\-D\_\{\\mathrm\{KL\}\}\\left\(q\_\{\\phi\}\(z\\mid A,X\)\\,\\\|\\,p\(z\)\\right\)\.
For training, we introduce a coefficientβ\\betato control the strength of the latent regularization, following the beta\-VAE formulation\([Higgins et al\. 2017](https://arxiv.org/html/2609.06093#bib.bib17)\)\. The resulting objective is

ℒCVAE=ℒrecon\+β​ℒKL,\\mathcal\{L\}\_\{\\mathrm\{CVAE\}\}=\\mathcal\{L\}\_\{\\mathrm\{recon\}\}\+\\beta\\mathcal\{L\}\_\{\\mathrm\{KL\}\},
where

ℒrecon=−𝔼qϕ​\(z∣A,X\)​\[log⁡pθ​\(A∣z,X\)\],\\mathcal\{L\}\_\{\\mathrm\{recon\}\}=\-\\mathbb\{E\}\_\{q\_\{\\phi\}\(z\\mid A,X\)\}\\left\[\\log p\_\{\\theta\}\(A\\mid z,X\)\\right\],ℒKL=DKL\(qϕ\(z∣A,X\)∥p\(z\)\)\.\\mathcal\{L\}\_\{\\mathrm\{KL\}\}=D\_\{\\mathrm\{KL\}\}\\left\(q\_\{\\phi\}\(z\\mid A,X\)\\,\\\|\\,p\(z\)\\right\)\.

#### Architecture

We formulate the problem as conditional connectome graph generation, where the graph is generated conditioned on node\-level information\. The connectome data exhibit two salient properties: first, the 3D neuron coordinates provide explicit spatial structure; second, the graph representation and the learned latent embedding should be invariant to node permutations\. To account for both properties, we introduce PointNet\+\+ as an essential encoding layer on the encoder side, where it hierarchically fuses node attributes with spatial coordinates into local geometric embeddings\([Qi et al\. 2017](https://arxiv.org/html/2609.06093#bib.bib30)\)\. These embeddings are then processed by a GAT to integrate adjacency information\([Velickovic et al\. 2018](https://arxiv.org/html/2609.06093#bib.bib43)\), and by a Transformer\-based graph global encoder with a learned CLS token to obtain a permutation\-invariant graph\-level representation\([Vaswani et al\. 2017](https://arxiv.org/html/2609.06093#bib.bib42)\)\. Two projection heads finally parameterizeμ\\muandlog⁡σ2\\log\\sigma^\{2\}, from which the latent variable is sampled via the standard reparameterization trick\([Kingma and Welling 2014](https://arxiv.org/html/2609.06093#bib.bib20);[Rezende, Mohamed, and Wierstra 2014](https://arxiv.org/html/2609.06093#bib.bib32)\)\.

On the decoder side, we follow the same design principle\. The conditional template is first encoded by PointNet\+\+ to produce node\-wise condition embeddings, and a Transformer decoder combines these embeddings with the latent code to reconstruct node representations\. An edge predictor then maps the decoded node representations to a directed edge\-probability matrix, on which Bernoulli sampling is performed to obtain binary adjacency matrices for downstream tasks\. In this way, our decoder preserves the conditional generation setting of the reference model, while making the spatial structure of the input explicit throughout decoding\. Please see Supplementary Materials for details\.

### Linking Latent Structure to Function

#### Connectome\-based Reservoir Computing

To evaluate the functional properties of each connectome graph, we instantiate it as a recurrent reservoir and measure its computational performance across tasks\. To follow Dale’s rule\([Dale 1935](https://arxiv.org/html/2609.06093#bib.bib12)\), we assign a signdj∈\{−1,1\}d\_\{j\}\\in\\\{\-1,1\\\}to presynaptic neuronjjaccording to its inhibitory or excitatory type, respectively, and construct the recurrent matrix asWi​j=Ai​j​djW\_\{ij\}=A\_\{ij\}d\_\{j\}\. To make reservoir dynamics comparable across graphs, we normalize the recurrent matrix to a fixed spectral radius before task evaluation\.

We use the resulting matrixWWas the recurrent connectivity in the standard reservoir\-computing model\([Jaeger 2001](https://arxiv.org/html/2609.06093#bib.bib18);[Maass, Natschläger, and Markram 2002](https://arxiv.org/html/2609.06093#bib.bib27);[Lukoševičius and Jaeger 2009](https://arxiv.org/html/2609.06093#bib.bib26)\)\. Given a sequence input\{xt\}t=1T\\\{x\_\{t\}\\\}\_\{t=1\}^\{T\}, the computation consists of input injection, recurrent state updates, and linear readout:

at\\displaystyle a\_\{t\}=Win​xt\+W​ht−1,\\displaystyle=W\_\{\\mathrm\{in\}\}x\_\{t\}\+Wh\_\{t\-1\},ht\\displaystyle h\_\{t\}=\(1−η\)​ht−1\+η​tanh⁡\(at\),\\displaystyle=\(1\-\\eta\)h\_\{t\-1\}\+\\eta\\tanh\(a\_\{t\}\),y^t\\displaystyle\\hat\{y\}\_\{t\}=Wout​ht\.\\displaystyle=W\_\{\\mathrm\{out\}\}h\_\{t\}\.Here,hth\_\{t\}denotes the reservoir state,η\\etais the leaking rate,WinW\_\{\\mathrm\{in\}\}maps the input to the reservoir state space, andWoutW\_\{\\mathrm\{out\}\}maps the state to the task output\.

During training,WinW\_\{\\mathrm\{in\}\}is randomly initialized and then held fixed, as isWW\. OnlyWoutW\_\{\\mathrm\{out\}\}is fitted to the task targets, yieldingWout⋆W\_\{\\mathrm\{out\}\}^\{\\star\}\. Nothing changes in testing phase\. Thus, performance between local circuits measures how recurrent topology affects reservoir dynamics and the information available to a linear output\.

#### Latent\-to\-Function Regression

Given the connectome graphs\{Gi\}i=1M\\\{G\_\{i\}\\\}\_\{i=1\}^\{M\}, we first encode each graph with the CVAE encoder to obtain latent vectors\{zi\}i=1M\\\{z\_\{i\}\\\}\_\{i=1\}^\{M\}, where eachziz\_\{i\}summarizes the structural pattern of the corresponding local circuit\. We then examine whether differences across these structural latent representations predict variation in computational function by decoding each latent code into a connectome and evaluating the resulting graph as a reservoir\.

Because both the latent codeziz\_\{i\}and the node\-level conditionsXiX\_\{i\}can influence the decoded connectivity, we fix the node conditions across latent codes to isolate structural variation attributable tozz\. Specifically, we randomly select a local circuit as a shared condition template, denoted byX0X\_\{0\}\. We repeat the analysis using 10 different templates and observe no qualitative change in the results\. Each latent codeziz\_\{i\}is then decoded under the same conditionX0X\_\{0\}:

Gi′∼pθ​\(A∣zi,X0\)\.G\_\{i\}^\{\\prime\}\\sim p\_\{\\theta\}\(A\\mid z\_\{i\},X\_\{0\}\)\.
The function of each generated graph is evaluated using the reservoir\-computing pipeline\. Because both graph generation and reservoir evaluation involve stochasticity, we estimate a robust functional score for each latent code\. Specifically, for eachziz\_\{i\}, we generate 10 binary adjacency matrices under the shared conditionX0X\_\{0\}, evaluate each graph across 3 random seeds, and average the resulting task scores\. This yields one aggregated functional scoreFiF\_\{i\}for each latent codeziz\_\{i\}\.

We then fit a regression model from the latent representation to the predicted functional scoreF^i\\hat\{F\}\_\{i\},

fψ:zi→Fi,F^i=fψ​\(zi\)\.f\_\{\\psi\}:z\_\{i\}\\rightarrow F\_\{i\},\\qquad\\hat\{F\}\_\{i\}=f\_\{\\psi\}\(z\_\{i\}\)\.Reliable cross\-validated prediction indicates that the learned latent representation contains structural variation that is informative of computational function\.

![Refer to caption](https://arxiv.org/html/2609.06093v1/reconstruction_latent_topology.png)Figure 3:Connectome reconstruction fidelity and latent topological organization\. \(a\) Representative binary adjacency matrices for the original connectome, the naive VAE baseline, and the full CVAE reconstruction\. \(b\) PCA projection of the learned 32\-dimensional latent vectors from 498 connectome samples, colored by clustering coefficient and mean degree\. The smooth variation of these graph\-level descriptors over the PC1–PC2 plane indicates that the latent space preserves continuous macroscopic topological variation\.Table 1:Reconstruction comparison between the full CVAE and naive VAE baseline\. AUC measures edge\-level reconstruction\. Deg\., Eff\., Clust\., Assort\., Mod\., Trans\., and Louv\. denote mean degree, global efficiency, clustering coefficient, assortativity, modularity, transitivity, and directed Louvain modularity relative errors\([Blondel et al\. 2008](https://arxiv.org/html/2609.06093#bib.bib8)\)\.

## Experiments

![Refer to caption](https://arxiv.org/html/2609.06093v1/latent_to_function_predictions.png)Figure 4:Predicted versus measured reservoir performance from 32\-dimensional CVAE latent coordinates for copy memory, Lorenz system prediction, and sequential MNIST\.### Experimental Setup

#### Dataset and Preprocessing

We utilize the IARPA MICrONS connectome dataset\([The MICrONS Consortium 2025](https://arxiv.org/html/2609.06093#bib.bib40)\), which comprises spatial coordinates, cell\-type annotations, and single\-cell synaptic connectivity mapping from mouse cortical volumes\. Following prior work\([Liu, Li, and Chen 2026](https://arxiv.org/html/2609.06093#bib.bib24)\), we model local circuits as functional columns traversing the cortical lamina and extract fixed cylindrical subvolumes aligned orthogonally to the layer boundaries\. These columnar subvolumes are arranged across the tangentialxx\-zzplane following a hexagonal close\-packing layout\. Along thezzaxis, the plane is partitioned into non\-overlapping spatial regions, with the two outer regions jointly assigned to the training set and the two central regions assigned to the validation and test sets, respectively\. Each sampling cylinder has a radius of27\.18​μ​m27\.18~\\mu\\mathrm\{m\}\(For reference, the overall volume spans roughly1387​μ​m×519​μ​m1387~\\mu\\mathrm\{m\}\\times 519~\\mu\\mathrm\{m\}acrossxxandzz\)\. To maximize data utilization and spatial coverage, neighboring sampling cylinders within the same split are allowed to overlap by approximately30%30\\%in cross\-sectional area\. This sampling design provides broad and approximately uniform coverage of column locations while reducing bias arising from local variations in neuronal density\. For each extracted cylinder, we construct a directed subgraph containing solely the enclosed somatic nodes and their internal synaptic edges, thereby restricting the analysis to local circuit units with comparable spatial scale and orientation\.

To accommodate variable neuron counts across microcircuits \(100–330 nodes\), each graph is padded to a uniform size ofNmax=330N\_\{\\max\}=330nodes\. We record the number of valid nodes for each graph and mask the padded nodes during model training and evaluation\. This preprocessing procedure yields the graph dataset\{Gi\}i=1M\\\{G\_\{i\}\\\}\_\{i=1\}^\{M\}used in the subsequent experiments, whereM=498M=498\.

Table 2:Candidate partial mediators for task\-specific latent functional directions\.rα,Sr\_\{\\alpha,S\}andrS,Fr\_\{S,F\}are Pearson correlations for the two marginal associations\.βS\|α\\beta\_\{S\|\\alpha\}is the mediator coefficient in the controlled modelF∼α\+SF\\sim\\alpha\+S\.βα\\beta\_\{\\alpha\}is the coefficient ofα\\alphain the total\-effect modelF∼αF\\sim\\alpha, andβα\|S\\beta\_\{\\alpha\|S\}is the coefficient ofα\\alphaafter adding the mediator inF∼α\+SF\\sim\\alpha\+S\. Red\. is the fractional direct\-effect reduction\. CW5 denotes normalized length\-5 closed walks\.

### Connectome Reconstruction and Latent Geometry

#### Graph\-Level Fidelity

To evaluate whether the conditional latent model recovers connectome topology at the probabilistic level, we compare it with a naive VAE baseline\([Liu, Li, and Chen 2026](https://arxiv.org/html/2609.06093#bib.bib24)\)that excludes both the PointNet\+\+ spatial encoder and conditional feature inputs\. Edge\-level reconstruction results are reported in Table[1](https://arxiv.org/html/2609.06093#Sx3.T1)\. The full CVAE consistently outperforms the baseline in edge\-probability ranking, as quantified by the area under the ROC curve \(AUC\)\. This stronger AUC suggests that the model learns an informative relative ordering over candidate edges, assigning higher probabilities to connections that are more likely under the observed connectome distribution\. Such probabilistic ranking is a natural fidelity criterion for sparse cortical microcircuit graphs, where the generative task is to recover the topology of connectivity patterns rather than merely reproduce individual sampled edges\.

At the graph level, the two models differ more sharply in their ability to preserve connectome topology \(Figure[3](https://arxiv.org/html/2609.06093#Sx3.F3)a and Table[1](https://arxiv.org/html/2609.06093#Sx3.T1)\)\. The naive baseline approximates density\-sensitive statistics such as mean degree and global efficiency, but fails to preserve higher\-order topological features\. In contrast, the full CVAE substantially reduces relative errors in modularity, transitivity, and clustering coefficient, indicating that it retains higher\-order modular and transitive structure beyond marginal edge density\.

#### Latent Distributional Statistics

We examine the learned 32\-dimensional latent space with PCA while retaining the full latent vectors for downstream analyses\. Across 498 connectome samples, the first two principal components explain 56\.13% of the variance\. As shown in Figure[3](https://arxiv.org/html/2609.06093#Sx3.F3)b, graph\-level descriptors such as clustering coefficient and mean degree change smoothly over the PC1–PC2 plane\. This pattern suggests that the learned latent space preserves continuous variation in macroscopic connectome topology, even though the model is trained through node\- and edge\-level reconstruction objectives\.

### Connectome Latents Predict Reservoir Function

#### Downstream Tasks and Metrics

To assess the relation between connectome latent representations and macroscopic function, we evaluate the generated connectome graphs on three complementary reservoir\-computing tasks\. The copy task measures delayed memory for symbolic sequences, following standard memory benchmarks in reservoir computing\([Jaeger 2002](https://arxiv.org/html/2609.06093#bib.bib19)\); Lorenz system prediction evaluates closed\-loop prediction and multi\-step rollout stability in a chaotic dynamical system\([Lorenz 1963](https://arxiv.org/html/2609.06093#bib.bib25)\); and sequential MNIST tests the classification separability of input\-driven reservoir states using the MNIST benchmark\([LeCun et al\. 1998](https://arxiv.org/html/2609.06093#bib.bib23)\)\. The functional metrics are defined in the Supplementary\.

Table 3:Latent\-to\-function regression performance across reservoir\-computing tasks\. Entries report 5\-fold cross\-validatedR2R^\{2\}from the 32\-dimensional CVAE latent coordinates to the sample\-level task metric\. Higher values are better for all entries\. XGB and GPR denote XGBoost and Gaussian process regression, respectively\.
#### Cross\-Validated Latent\-to\-Function Predictability

Regression from the 32\-dimensional CVAE latent coordinates to task performance shows that the learned structural representation predicts functional variation across connectome microcircuits\. We compare both linear and nonlinear predictors, including Ridge regression, Gaussian process regression \(GPR\)\([Rasmussen and Williams 2006](https://arxiv.org/html/2609.06093#bib.bib31)\), and XGBoost\([Chen and Guestrin 2016](https://arxiv.org/html/2609.06093#bib.bib10)\), with the full comparison reported in Table[3](https://arxiv.org/html/2609.06093#Sx4.T3); the corresponding predicted\-versus\-measured relationships are shown in Figure[4](https://arxiv.org/html/2609.06093#Sx4.F4)\. The strongest structure–function association appears in sequential MNIST, with a best cross\-validatedR2R^\{2\}of approximately0\.870\.87, whereas Lorenz system prediction yields a weaker but still measurable predictive signal, with a best cross\-validatedR2R^\{2\}of approximately0\.460\.46\. Furthermore, when accounting for model complexity, simple models such as Ridge regression achieve cross\-validated performance close to the best nonlinear models across tasks\. This result suggests that the CVAE latent space organizes part of the graph\-topological variation along functional directions that can be captured by relatively simple predictive models\.

### Interpreting the Latent Space

The goodness of fit in structure\-to\-function regression reveals that the latent space contains task\-relevant structural variation, but it does not identify which graph properties account for this relationship\. We therefore perform a mediation\-style interpretability analysis to test whether task\-specific structural descriptors explain part of the association between a latent functional direction and reservoir performance\([Baron and Kenny 1986](https://arxiv.org/html/2609.06093#bib.bib5);[MacKinnon, Fairchild, and Fritz 2007](https://arxiv.org/html/2609.06093#bib.bib28)\)\.

#### Gradient of Predicted Function in Latent Space

We standardize the latent vectors across samples to obtainz~i\\tilde\{z\}\_\{i\}and fit a Ridge regression to predict the functional score:

F^i=b\+z~i⊤​w\.\\hat\{F\}\_\{i\}=b\+\\tilde\{z\}\_\{i\}^\{\\top\}w\.Thus,wwis the gradient of the functional predictor with respect to the standardized latent coordinates: it specifies the direction in latent space along which the predicted functional score increases most rapidly\. We normalize this gradient as

w¯=w∥w∥2\\bar\{w\}=\\frac\{w\}\{\\lVert w\\rVert\_\{2\}\}and define

αi=z~i⊤​w¯\.\\alpha\_\{i\}=\\tilde\{z\}\_\{i\}^\{\\top\}\\bar\{w\}\.Therefore,αi\\alpha\_\{i\}is the signed projection of sampleii’s latent representation onto the task\-specific functional direction\. It summarizes the latent variation most associated with task performance and can be directly compared with interpretable structural descriptors\.

#### Understanding Latent Gradient with Graph metrics

Next, we seek to identify specific structural descriptorSmS\_\{m\}that can explain part of the association between the task\-specific latent coordinateα\\alphaand the performance scoreFF\. Specifically, a descriptorSmS\_\{m\}is treated as a candidate partial mediator only if it follows the empirical pattern expected from a mediation\-style mechanism: variation alongα\\alphashould be associated with variation inSmS\_\{m\};SmS\_\{m\}should be associated with task performance; and, after controlling forSmS\_\{m\}, the direct association betweenα\\alphaandFFshould be reduced but not necessarily eliminated\.

To assess whether a candidate descriptor satisfies these requirements, we fit four ordinary least squares regression models:

Sm∼α,F∼Sm,F∼α\+Sm,F∼α\.S\_\{m\}\\sim\\alpha,\\qquad F\\sim S\_\{m\},\\qquad F\\sim\\alpha\+S\_\{m\},\\qquad F\\sim\\alpha\.
We identifySmS\_\{m\}as a candidate partial mediator if it satisfies four criteria: \(1\) the associationα→Sm\\alpha\\rightarrow S\_\{m\}is statistically significant and aligned with the hypothesized structural mechanism; \(2\)Sm→FS\_\{m\}\\rightarrow Fis statistically significant in the direction of superior task performance; \(3\)SmS\_\{m\}retains a significant independent effect in the joint modelF∼α\+SmF\\sim\\alpha\+S\_\{m\}\(with coefficientβSm\|α\\beta\_\{S\_\{m\}\\mid\\alpha\}\); and \(4\) controlling forSmS\_\{m\}attenuates the direct effect ofα\\alpha, i\.e\.,\|βα\|Sm\|<\|βα\|\|\\beta\_\{\\alpha\\mid S\_\{m\}\}\|<\|\\beta\_\{\\alpha\}\|\. We quantify this attenuation as the fractional direct\-effect reduction:

Red=1−\|βα\|Sm\|\|βα\|,\\mathrm\{Red\}=1\-\\frac\{\|\\beta\_\{\\alpha\\mid S\_\{m\}\}\|\}\{\|\\beta\_\{\\alpha\}\|\},whereβα\\beta\_\{\\alpha\}denotes the total effect ofα\\alphainF∼αF\\sim\\alpha, andβα\|Sm\\beta\_\{\\alpha\\mid S\_\{m\}\}is its direct effect after adjusting forSmS\_\{m\}inF∼α\+SmF\\sim\\alpha\+S\_\{m\}\.

### Task\-specific Structural Mechanisms

Equipped with the mediation criteria defined above, we systematically evaluated candidate graph properties spanning local wiring motifs to global spectral profiles\. In Table[2](https://arxiv.org/html/2609.06093#Sx4.T2), we report the most salient candidate mediators identified for each task along with their effect attenuation statistics\. Across the tasks, these primary descriptors point to distinct structural mechanisms:

- •Copy memory: Mixed Excitatory–Inhibitory Reciprocal MicrocircuitsThe copy task requires stable recurrent feedback to maintain delayed information\. The strongest candidate mediator is the proportion of reciprocal connections involving mixed excitatory and inhibitory neurons\. Such microcircuits provide recurrent feedback while limiting the excessive signal amplification that can arise in purely excitatory loops\. Closed walks of length five provide a secondary topological mechanism, suggesting that short recurrent motifs also contribute to memory maintenance\.
- •Lorenz system prediction: Non\-normality and Dominant Singular ModesMulti\-step forecasting of Lorenz dynamics places stronger demands on dynamical stability\. The latent functional direction is primarily associated with reduced matrix non\-normality and a smaller dominant singular value, both of which suppress transient amplification in the reservoir dynamics\. These spectral changes are consistent with improved rollout stability in chaotic prediction, where non\-normal transient amplification can cause small state perturbations to grow rapidly\([Trefethen and Embree 2005](https://arxiv.org/html/2609.06093#bib.bib41);[Hennequin, Vogels, and Gerstner 2012](https://arxiv.org/html/2609.06093#bib.bib16)\)\.
- •Sequential MNIST: Singular\-Spectrum Dispersion and Stable RankSequential MNIST performance depends on the separability of reservoir states induced by different input classes\. The main candidate mediators are the dispersion of the singular\-value spectrum and the stable rank of the recurrent matrix\. A more distributed singular spectrum and a higher stable rank indicate that the reservoir can support a richer set of dynamic representations, thereby improving linear separability for classification\.

Taken together, these results suggest that the same connectome latent space encodes task\-specific structural factors rather than a single generic notion of graph quality\. Delayed memory is associated with local reciprocal microcircuits, whereas dynamical forecasting and sequential classification are more closely linked to global spectral properties of the reservoir matrix\. Although the mediation analysis is observational, it provides a concrete account of how latent structural variation relates to distinct computational functions\.

## Conclusion

We proposed a conditional generative framework that learns compact latent representations of connectome graphs from neuronal cell type and spatial organization\. Using generated connectomes as recurrent reservoirs, we found that the latent space predicts functional variation across memory, dynamic prediction, and classification\. Further interpretability analysis linked task\-relevant latent directions to distinct structural descriptors: mixed excitatory–inhibitory reciprocal cores for memory, and Dale\-signed spectral properties for prediction and classification\. These results suggest that sparse connectome graphs can be organized into a latent space that connects circuit topology with measured computational behavior\.

#### Limitations

First, all experiments are conducted on local cortical microcircuits extracted from the MICrONS dataset, and the generalizability of the learned latent space to connectomes from other brain regions, species, or spatial scales remains to be established\. Second, computational function is evaluated through reservoir computing, which provides a controlled and interpretable proxy for circuit computation but does not capture the full range of biological neural dynamics or learning mechanisms\. Finally, although the proposed framework generates structurally plausible connectomes and preserves structure\-function relationships, the biological validity of the synthesized circuits has not been experimentally verified\. Future work will investigate larger and more diverse connectome datasets, incorporate more biologically realistic dynamical models, and validate generated connectomes against additional anatomical and physiological constraints\.

## Acknowledgement

This work was supported by the National Key R&D Program of China, Project Number 2025YFA1016700\. This work was also supported by the National Natural Science Foundation of China \(NSFC\) under Grant No\. 62576011\. This work was supported in part by the Beijing Major Science and Technology Project under Contract no\. Z251100008125055\. This work was also supported by Beijing Academy of Artificial Intelligence \(BAAI\)\.

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