fixed-point-compositionality-low-rank-gluings

name	fixed-point-compositionality-low-rank-gluings
description	Mathematical framework for compositional dynamics in threshold-linear networks via low-rank gluing rules. Use when studying modular network assembly, fixed point decomposition, compositional limit cycles, or engineering networks with predictable attractor repertoires.
license	MIT

Fixed Point Compositionality via Low-Rank Gluing Rules

Mathematical theory of compositional dynamics in inhibition-dominated threshold-linear networks (TLNs) through structured modular assembly.

Core Concept: Compositionality

Brains generate complex behaviors from stable structures with limited resources via compositionality - decomposing complex tasks into reusable primitives.

This work provides first rigorous mathematical characterization linking structural modularity to functional compositionality in nonlinear networks.

Key Innovation: Low-Rank Gluing Rules

Novel modular network assembly connecting component subnetworks via specific low-rank couplings:

Network Architecture

• Component subnetworks: arbitrary internal connectivity
• Inter-module coupling: low-rank connections (rank-1, rank-k)
• Inhibition-dominated dynamics: threshold-linear units

Main Theorems

Theorem 1: Fixed Point Compositionality

Global fixed points constrained to combinations of local fixed points of constituent modules.

For low-rank gluings:

FixedPoints(Global) ⊆ Combinations(FixedPoints(Module₁) × FixedPoints(Module₂) × ...)

Theorem 2: Rank-1 Gluing Characterization

Complete classification determining which combinations yield global fixed points:

• Explicit construction rules for compositional attractors
• Predictable assembly of global dynamics from local motifs

Theorem 3: gCTLN Extension

Fixed point decomposition rules extended from CTLNs to generalized CTLNs (gCTLNs):

• Structural rules more robust than initially posited
• Wider applicability to biological network architectures

Applications

1. Combinatorial Attractor Engineering

Construct networks with combinatorially large repertoire of predictable attractors:

• Understanding from simpler component motifs
• Systematic design of complex dynamics

2. Compositional Limit Cycles

Beyond fixed points: compositional limit cycles emerge from gluing rules:

• Periodic dynamics from module oscillations
• Predictable timing from structural assembly

3. Graph-Based Networks

Extension to graph structures:

• Network topology → fixed point constraints
• Module structure → functional composition

Mathematical Framework

Threshold-Linear Networks (TLNs)

Dynamics governed by:

dx_i/dt = -x_i + [∑_j W_ij x_j + b_i]_+

where:

• [·]_+ = threshold-linear nonlinearity (ReLU-like)
• W_ij = synaptic weights (inhibition-dominated)
• b_i = external inputs

Low-Rank Coupling Structure

Inter-module connections:

W_inter = UV^T  (rank-k coupling)

where:

• U, V = low-rank factors
• Specific structure constrains global dynamics

Fixed Point Decomposition

For rank-1 gluing W_inter = uv^T:

x_global = combination of {x_local(Module₁), x_local(Module₂), ...}

with explicit membership rules.

Biological Relevance

Compositional Brain Dynamics

• Modular cortical circuits → compositional computation
• Stable structure + flexible combinations
• Limited resources → efficient reuse of primitives

Inhibition-Dominated Networks

• Realistic cortical dynamics
• Winner-take-all competition
• Fixed point stability through inhibition

Network Assembly Rules

• Development: modules assembled via specific coupling rules
• Learning: modify low-rank factors → new compositional capabilities
• Evolution: reusable motifs across behavioral repertoire

Implementation Guidance

When to Use This Framework

Trigger conditions:

• Modeling modular neural circuits
• Engineering predictable attractor dynamics
• Studying compositionality in biological/artificial networks
• Analyzing fixed point structure of TLNs
• Designing networks with combinatorial dynamics

Construction Workflow

1. Identify component modules - subnetworks with known fixed points
2. Design low-rank coupling - specify U, V factors
3. Apply rank-1 theorem - determine valid combinations
4. Construct global network - assemble with gluing rules
5. Validate attractor repertoire - check combinatorial predictions

Graph-Based Application

1. Define graph topology - network structure
2. Apply gCTLN rules - fixed point decomposition
3. Extend to generalized networks - beyond CTLN constraints
4. Validate robustness - structural rule preservation

Theoretical Significance

First rigorous proof that:

1. Modularity → Compositionality in nonlinear networks
2. Low-rank structure constrains global attractors
3. Combinatorial dynamics emerge from simple motifs
4. Engineering recipe for predictable complex networks

Bridges gap between:

• Structural modularity (observed in brain)
• Functional compositionality (behavioral flexibility)
• Mathematical characterization (predictable assembly)

Paper Reference

arXiv:2606.07336 (q-bio.NC)

• Author: Juliana Londono Alvarez
• 39 pages, 18 figures
• Submitted: 2026-06-05

Related Work

• Combinatorial Threshold-Linear Networks (CTLNs)
• Attractor dynamics in recurrent networks
• Network assembly theory
• Modular circuit design

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Activation: compositional dynamics, threshold-linear network, TLN, low-rank gluing, fixed point decomposition, modular network, attractor engineering, combinatorial dynamics, inhibition-dominated, gCTLN, network assembly, compositional limit cycle