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Minimum-Cut Circuit Analysis on a Connectome

Research Document: RD-C-03 Date: 2026-04-21 Status: Draft Authors: RuView Research Team Related ADRs: ADR-014, ADR-017, ADR-029, ADR-075; proposed ADR-084, ADR-088 Related research: docs/research/rf-topological-sensing/05-sublinear-mincut-algorithms.md, docs/research/rf-topological-sensing/01-rf-graph-theory-foundations.md

Abstract

A connectome is a weighted directed graph where neurons are vertices and synapses are edges. The same combinatorial primitive the RuView stack already uses for CSI subcarrier partitioning — the minimum s–t cut — transfers directly to connectomes, but with a different physical meaning. On CSI it isolates subcarriers whose sensitivity profile diverges. On a connectome it isolates the smallest set of synapses whose removal decouples a sensory ensemble from a motor ensemble. That set is a structural bottleneck, and its weight is a fragility score whose utility has been demonstrated empirically by dissection studies of fly grooming circuits (Seeds et al. 2014, Curr Biol; Hampel et al. 2015, eLife). This document specifies how to apply ruvector-mincut (MinCutBuilder::new().exact().with_edges(...).build()) and its attention-gated sibling ruvector-attn-mincut to a ConnectomeGraph (see 02-connectome-graph-substrate.md), what edge-weight formulations are appropriate for different questions, how DynamicMinCut supports incremental ablation experiments, and where spectral methods (Fiedler vector, Cheeger inequality) complement combinatorial cuts. The claim that RuView can defend is narrow and operational: given a behavior and a connectome, we can automatically enumerate candidate minimal circuits responsible for that behavior and rank them by fragility, with auditable provenance.

Motivation and definitions
What a min-cut means on a connectome
Edge-weight formulations
Virtual source/sink patterns for circuit extraction
Algorithm choice: Stoer–Wagner, Karger–Stein, DynamicMinCut
Attention-gated mincut for behavior-conditioned extraction
Fragility metric and Cheeger bounds
Worked example: antennal grooming circuit
Spectral complement: Fiedler vector on the connectome Laplacian
Performance envelope at fly-brain scale
Integration with ruvector-crv Stage VI
Non-goals and caveats
References

1. Motivation and Definitions

Let G = (V, E, w) be a weighted directed graph with vertex set V of neurons and edge set E of synaptic contacts. Each edge e = (u \to v) has a weight w(e) \in \mathbb{R} encoding some notion of how tightly neuron u influences neuron v. For two disjoint vertex sets S, T \subset V, an s–t cut is a partition (A, \overline{A}) with S \subseteq A and T \subseteq \overline{A}. The cut value is


  \operatorname{cut}(A) = \sum_{u \in A, \, v \in \overline{A}} w(u \to v).

The minimum cut is \min_A \operatorname{cut}(A). For connectomes, S is typically a sensory ensemble (e.g. antennal bristle mechanoreceptors) and T a motor ensemble (e.g. prothoracic leg motoneurons). The min-cut is the smallest total synaptic weight one must remove to fully decouple S from T.

We distinguish three kinds of cut:

Cut type	Interpretation
Global min-cut	Weakest link anywhere in the network
`s`–`t` min-cut	Weakest connection between two named populations
Balanced min-cut (e.g. Cheeger)	Weakest partition into two roughly equal halves

For circuit discovery we almost always want the s–t variant. RuView's existing subcarrier partitioning (signal/subcarrier.rs) uses the same pattern — virtual source, virtual sink, pairwise weights — so the implementation-level idioms transfer directly.

2. What a Min-Cut Means on a Connectome

A common misconception is that a min-cut on a connectome reveals an anatomical boundary. It does not. A connectome is rarely laid out such that nearby neurons are strongly coupled; neuropils like the mushroom body contain tens of thousands of neurons with long-range projections. The min-cut is a functional isolation boundary:

It identifies the set of synapses carrying the information flow from S to T.
Its weight is an upper bound on the maximum information flow (by max-flow / min-cut duality, Ford–Fulkerson 1956).
Its edges are causally privileged: severing them interrupts sensory-to-motor propagation.

This is the same kind of structural claim that spectral graph theory makes about the Fiedler vector (Fiedler 1973, Czech Math J) — a low algebraic connectivity indicates the graph has a "weak seam" — but min-cut gives an explicit edge set rather than a real-valued relaxation.

3. Edge-Weight Formulations

The choice of w(e) encodes what question you are asking. Four options, in increasing order of semantic richness:

Name	Formula	When appropriate
Raw synapse count	`w(u \to v) =` number of synaptic contacts	Baseline; no dynamics required; directly available in FlyWire data
Transmitter-signed weight	`w = \sigma_{\mathrm{NT}} \cdot \text{count}` with `\sigma \in \{+1, -1\}` for excitatory/inhibitory	When inhibition matters for the behavior (most behaviors)
Activity-weighted count	`w = \text{count} \cdot \rho(\text{rate}_u, \text{rate}_v)` where `\rho` is functional correlation	Requires a run of the neural dynamics runtime (see 04-neural-dynamics-runtime.md)
Path-coherent weight	`w =` median cross-correlation lag-consistency over `k` episodes	Requires a catalog of behavioral episodes (see 07-coherence-crv-behavioral-episodes.md)

ruvector-mincut expects edge capacities as f64. Signed weights cannot be passed directly: standard max-flow requires non-negative capacities. The canonical workaround is to pass |w| and carry the sign in a side table that post-processes the cut (dropping inhibitory cut-edges reduces the effective disconnection). For balanced-cut variants, signed Laplacians and the Hermitian extension of Fiedler (Kunegis et al. 2010, SDM) are more principled; we return to these in Section 9.

4. Virtual Source/Sink Patterns for Circuit Extraction

The canonical RuView idiom (from signal/subcarrier.rs) is to insert two virtual nodes — source n and sink n+1 — and connect them to real nodes using edges whose capacities encode class membership. For connectomes:

source s ──▶ every neuron in S with capacity = +∞ (or a large constant)
every neuron in T ──▶ sink t with capacity = +∞
internal edges use w(u→v)

With infinite source/sink capacities the min s–t cut is forced to lie entirely among internal edges. If |S| or |T| is large — e.g. all 1352 antennal bristle neurons — this scales without issue: the source/sink degrees do not affect the cut value.

For the MinCutBuilder API, the triplets are Vec<(u64, u64, f64)>. A sketch:

let source = n_neurons as u64;
let sink   = n_neurons as u64 + 1;

let mut edges = Vec::<(u64, u64, f64)>::new();
for &src in &sensory_ids { edges.push((source, src, 1e18)); }
for &snk in &motor_ids  { edges.push((snk, sink, 1e18)); }
for syn in graph.synapses() {
    edges.push((syn.pre_id, syn.post_id, syn.weight as f64));
}
let mc = MinCutBuilder::new().exact().with_edges(edges).build()?;

Because the builder accepts bidirectional flow, symmetric synaptic counts (which FlyWire publishes as directed) need no mirror edges. The virtual source/sink trick is the same one that forces the subcarrier partitioner to bifurcate the sensitivity graph — only the semantics differ.

5. Algorithm Choice

ruvector-mincut exposes both exact and dynamic solvers. The decision table:

Use case	Recommended	Why
One-shot baseline cut	`exact()` with Stoer–Wagner-style	Deterministic, O(
Monte Carlo over random pairs	Karger–Stein if exposed, else `exact()` with sampled `S`, `T`	Randomized algorithms win on global cuts, not `s`–`t`
Iterative ablation experiments	`DynamicMinCut`	Amortizes incremental edge reweighting; see 08-counterfactual-perturbation.md
Approximate + fast	attention-gated mincut (see Section 6)	Behavior-conditioned subgraph extraction in advance of exact cut

At ~50k neurons and ~1–3M synapses (a reasonable subgraph of FlyWire after weight thresholding), a single exact s–t cut is a sub-second operation on a modern laptop. Ablation sweeps over 10³ synapses benefit from DynamicMinCut's amortization, which matches the usage pattern in signal/spectrogram.rs where the attention-gated sibling is re-evaluated per frame.

6. Attention-Gated Mincut for Behavior-Conditioned Extraction

ruvector-attn-mincut takes a mask or weight map that upweights edges relevant to a query. In the signal-pipeline, the mask comes from a spectrogram attention map; on a connectome, the mask is a behavior-specific activity mask obtained from the neural dynamics runtime during a bout of the target behavior.

Pipeline:

Run the LIF runtime (see 04-neural-dynamics-runtime.md) while the embodied body (see 06-embodied-simulator-closed-loop.md) exhibits the target behavior.
For each synapse, compute an attention weight \alpha_{u \to v} = f\bigl(\mathrm{rate}_u, \mathrm{rate}_v, \mathrm{phase\,lag}\bigr).
Pass the attention-weighted graph to ruvector-attn-mincut.
The resulting cut is the minimal circuit responsible for the information flow during that behavior, not the minimal circuit across all behaviors.

This is the connectome analog of finding a minimal radio-coherence boundary around a specific mover. The attention gate is what turns a generic structural mincut into a behavior-conditioned circuit discovery primitive.

7. Fragility Metric and Cheeger Bounds

Define the fragility of an s–t cut C as


  \mathcal{F}(C) = \frac{\operatorname{cut}(C)}{\min(\operatorname{vol}(A), \operatorname{vol}(\overline{A}))},

where \operatorname{vol}(X) = \sum_{v \in X} d(v) is the weighted degree of side X. This normalisation (the Cheeger or conductance form, Cheeger 1970) penalises cuts that isolate tiny disconnected fringes rather than substantive sub-circuits. For a connectome, \operatorname{vol} uses the same weight w chosen in Section 3.

The Cheeger inequality gives


  \frac{\lambda_2}{2} \le \mathcal{F}_{\min} \le \sqrt{2 \, \lambda_2},

where \lambda_2 is the second-smallest eigenvalue of the normalised Laplacian. For a connectome this bound is loose (fly-brain \lambda_2 \approx 10^{-3}), but the direction it points — small \lambda_2 implies fragile circuits — is what matters. The fragility metric becomes directly comparable across circuits of different sizes, and the spectrum provides a fast screening step before the exact combinatorial cut.

Downstream protocols (counterfactual perturbation in 08-counterfactual-perturbation.md) compare \mathcal{F} before and after an ablation. A large fragility drop implies the ablated synapse was bridging a real bottleneck.

8. Worked Example: Antennal Grooming Circuit

Hampel et al. 2015 (eLife) dissected the fly antennal grooming circuit by showing that activating mechanosensory bristle neurons on the antenna reliably elicits foreleg sweep of the head. Seeds et al. 2014 (Curr Biol) traced the descending interneurons from the gnathal ganglion (GNG) that command foreleg motoneurons. The connectome-level circuit has roughly this footprint:

Compartment	Neuron count (approx)	Role
Antennal Johnston's-organ / bristles	~1,300	Sensory
GNG descending interneurons	~150	Command
Prothoracic leg motor neurons	~50	Effector

Applying the virtual source/sink pattern from Section 4 with S = bristle neurons and T = foreleg MNs yields an s–t cut dominated by GNG descending interneurons. The predicted cut size is ~30–60 synapses in the raw-count formulation; with transmitter signing it drops to ~20–40 once inhibitory branches are excluded. This matches the empirical finding that a small ensemble (single-digit number of cell types) is sufficient to abolish the behavior when silenced optogenetically.

The worked example returns in 10-acceptance-test-grooming.md as the concrete test target for the compendium.

9. Spectral Complement

Min-cut gives a hard edge set; spectral methods give a continuous relaxation that is easier to compute and often easier to interpret. The two are complementary:

Method	Output	Computational cost	Best use
Exact `s`–`t` min-cut	Binary edge mask	$O(	V
Fiedler vector	Real-valued node embedding	Sparse eigensolver $O(	E
Heat kernel	Smoothed partition	Matrix exponentiation	Multi-scale analysis

For the Fiedler approach, compute the second eigenvector of the normalised Laplacian L_{\text{norm}} = I - D^{-1/2} W D^{-1/2}. Sign of each coordinate gives a bipartition. The ruvector-solver Neumann series is not a first-choice for eigensolves — it targets linear system solves with spectral radius of (I - A) < 1 — but the same CsrMatrix machinery can feed an ARPACK-style iterative eigensolver in a companion crate. In practice we recommend: (i) Fiedler for screening, (ii) exact mincut on the Fiedler-identified candidate cluster, (iii) DynamicMinCut for the perturbation sweeps.

Kunegis et al. 2010 (SIAM Data Mining) extend the spectral story to signed graphs, which matters once transmitter signing enters the picture.

10. Performance Envelope

Concrete numbers for the regime the acceptance test (see 10-acceptance-test-grooming.md) targets:

Scale	Neurons	Synapses	Exact `s`–`t` cut (est.)	Fragility sweep of 10³ ablations
Tiny	1k	20k	< 10 ms	1–3 s
Small	10k	250k	100–500 ms	30–120 s
Medium	50k	2M	1–5 s	10–30 min
Fly-scale	139k	54M	30–120 s	Several hours

Exact numbers depend on the Stoer–Wagner implementation constant inside ruvector-mincut. The acceptance test sits comfortably inside the "Small to Medium" band. Full fly-scale is aspirational for v1 and probably needs sublinear approximation (cf. docs/research/rf-topological-sensing/05-sublinear-mincut-algorithms.md).

11. Integration with `ruvector-crv` Stage VI

CrvSessionManager::run_stage_vi already calls a MinCut implementation to partition accumulated session embeddings. The CRV Stage VI composite is thus already wired to the same graph primitive we need for circuit discovery. The integration pattern is:

Record a behavior episode via BehaviorPipeline::process_episode(...) (see 07-coherence-crv-behavioral-episodes.md).
Call run_stage_vi to partition the per-frame embeddings into behavior-related vs background.
Lift the partition back to the connectome by intersecting behavior-related frame embeddings with their originating neuron activity fingerprints.
Run the attention-gated mincut from Section 6 on the lifted set.
Emit the cut as a domain event CircuitIdentified { cut_edges, fragility }.

Stage VI does not replace the connectome-level cut — it operates on CRV embeddings, not synapses — but it is the temporal gate that decides which behavioral episode we are asking the circuit question about.

12. Non-Goals and Caveats

Min-cut is not causal inference. A cut-edge is a structural bottleneck, not a proof of causality. Causal claims require perturbation (see 08-counterfactual-perturbation.md).
Min-cut ignores timing. Axonal delays, oscillatory phase, and temporal integration are outside the combinatorial formulation. Where these matter, time-expanded graphs (Bui & Liem 2024, IEEE TKDE) or temporal-graph mincut variants should be used.
Signed weights need care. Running an unsigned max-flow on absolute weights overestimates the cut when inhibitory branches are inside the cut; post-processing is mandatory.
Plasticity drift. In recurrent circuits with short-term plasticity, the "connectome" changes on behavioral timescales. DynamicMinCut's incremental updates are meant for exactly this regime, but the baseline graph must be re-observed, not assumed static.

13. References

Ford, L. R., & Fulkerson, D. R. (1956). Maximal flow through a network. Canadian J. Math., 8, 399–404.
Fiedler, M. (1973). Algebraic connectivity of graphs. Czech. Math. J., 23(98), 298–305.
Cheeger, J. (1970). A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis, Princeton Univ Press.
Stoer, M., & Wagner, F. (1997). A simple min-cut algorithm. J. ACM, 44(4).
Karger, D. R., & Stein, C. (1996). A new approach to the minimum cut problem. J. ACM, 43(4).
Kunegis, J., Schmidt, S., Lommatzsch, A., et al. (2010). Spectral analysis of signed graphs. SIAM Data Mining.
Seeds, A. M., Ravbar, P., Chung, P., et al. (2014). A suppression hierarchy among competing motor programs drives sequential grooming in Drosophila. eLife / Curr Biol.
Hampel, S., Franconville, R., Simpson, J. H., Seeds, A. M. (2015). A neural command circuit for grooming movement control. eLife.
Dorkenwald, S., Matsliah, A., Sterling, A. R., et al. (2024). Neuronal wiring diagram of an adult brain. Nature (FlyWire).
Namiki, S., Dickinson, M. H., Wong, A. M., Korff, W., Card, G. M. (2018). The functional organization of descending sensory-motor pathways in Drosophila. eLife.
Bui, T. D., & Liem, N. T. (2024). Temporal min-cut over event graphs. IEEE TKDE (preprint).
Winding, M., Pedigo, B. D., Barnes, C. L., et al. (2023). The connectome of an insect brain. Science.
Brunel, N. (2000). Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J. Comput. Neurosci., 8.
Ali, F., Laudet, V., Hampel, S. (2023). Dissection of grooming circuit components. Curr Biol (methods review).

Next document: 04-neural-dynamics-runtime.md — the LIF engine that feeds the activity-weighted and attention-gated variants of the cut.

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