Dynamic Curvature Adaptation: A Unified Geometric Theory of Cortical State and Pathological Collapse

Abstract

We propose the Curvature Adaptation Hypothesis (CAH): nervous systems optimize task-dependent information transport by dynamically unlocking the latent hyperbolic geometry inherent in their hierarchical structure. We identify a plausible biophysical actuator—the Martinotti-cell subtype of Somatostatin (SST) interneurons—which targets distal apical dendrites to regulate the apical-somatic conductance ratio (γ), to serve as a geometric switch. Using Optimal Transport (OT) simulation and finite-size scaling analysis, we demonstrate that modulating (γ) drives the network through a sharp, non-linear phase transition from a stable Euclidean regime (κ ≈ 0) to a deep hyperbolic regime (κ < 0). Crucially, this transition is scale-invariant across network depths (N=3,5,7), reflecting the fractal nature of the underlying tree topology. We show that this geometric transition is remarkably robust to degree-preserving topological scrambling. This indicates that the hyperbolic regime is driven by local synaptic availability rather than global architectural order. To validate this optimal transport mathematics in biologically realistic wetware, we implement a Spiking Neural Network (SNN) simulation of the VIP-SST-Pyramidal microcircuit, demonstrating that VIP-mediated disinhibition natively induces the highly synchronized 40Hz gamma-band pillars required to initiate the macroscopic hyperbolic plunge. Furthermore, we simulate two distinct pathological states: (1) Hyper-Integration, where the introduction of VIP-like hub nodes abolishes the flat regime, forcing the network into permanent hyperbolicity (analogous to the hyper-associative phenotypes observed in mania or psychosis); and (2) Geometric Collapse, where synaptic pruning (>30% spine loss) prevents the network from accessing deep negative curvature even under high coupling (analogous to the structural synaptic loss observed in neurodegenerative disorders like Alzheimer's disease). These results suggest that geometry is not a static anatomical feature but a dynamic functional state, actively tuned by the interplay of inhibitory interneurons. We provide a reproducible simulation protocol and a proposed optogenetic experiment in Macaque PFC to make the Curvature Adaptation Hypothesis (CAH) empirically falsifiable.

Summary

This project introduces the Curvature Adaptation Hypothesis (CAH), a novel theoretical framework proposing that the mammalian cortex optimizes information transport by dynamically "warping" its functional manifold into hyperbolic regimes. We identify the Martinotti-cell subtype of Somatostatin (SST) interneurons—which target distal apical dendrites— as the biological actuator for this geometric switch, regulating the apical-somatic conductance ratio (γ) to trigger a non-linear phase transition into negative curvature (κ<0).

Key Findings

• Scale-Invariant Phase Transition: Using Optimal Transport (OT) simulations on stochastic Galton-Watson trees, we demonstrate that the transition from Euclidean to Hyperbolic geometry is a scale-invariant property of hierarchical networks.
• Topological Robustness: Our results reveal that the capacity for hyperbolic depth is robust to degree-preserving topological scrambling, suggesting that local synaptic density, rather than global architectural perfection, is the primary driver of the geometric manifold.
• Spiking Neural Network Validation: A PyNEST simulation of the VIP-SST-Pyramidal microcircuit, establishing the dynamical biological framework for the geometric switch.
• The Geometric Trilogy of Disease: We computationally model three distinct cognitive states:
  1. Healthy Regime: Tunable phase transition between flat and hyperbolic manifolds.
  2. Manic Regime: "Geometric Inelasticity" caused by high-centrality VIP-like hub nodes, forcing the system into permanent hyperbolicity.
  3. Neurodegenerative Regime: "Geometric Collapse" caused by synaptic pruning (>30% loss), preventing the manifold from accessing the depth required for complex hierarchical integration.

Metabolic Implications

We provide a biophysical energy ROI analysis demonstrating that the local maintenance "tax" required for SST-gating is offset by a global "signaling tax haven," where hyperbolic geodesics minimize the metabolic cost of hierarchical inference.

Related Works

• Pender, M. A. (2026). The Manifold Chip: Silicon Architecture for Dynamic Curvature Adaptation via Dual-Gated Analog Shunting. 10.5281/zenodo.18717807 https://doi.org/10.5281/zenodo.18717807
• Pender, M. A. (2026). Geometry-Aware Plasticity: Thermodynamic Weight Updates in Non-Euclidean Hardware. 10.5281/zenodo.18761137. https://doi.org/10.5281/zenodo.18761137
• Pender, M. A. (2026). Logic as a Hyperbolic Actuator: Evidence for VIP-Mediated Phase Transitions in Transformer Attention Manifolds. 10.5281/zenodo.18627785 https://doi.org/10.5281/zenodo.18627785
• Pender, M. A. (2026). The Fermi Paradox, Dark Matter, and the Scale Invariance of the Curvature Adaptation Hypothesis. 10.5281/zenodo.18913772. https://doi.org/10.5281/zenodo.18913772
• Pender, M. A. (2026). The Metabolic Phase Transition: Qualia as a Topological Solution to the Landauer Limit in High-Dimensional Manifolds. 10.5281/zenodo.18655523.  https://doi.org/10.5281/zenodo.18655523

Repository Contents

The python scripts are included here, but you may also find them at:
https://github.com/MPender08/dendritic-curvature-adaptation

Manuscript: Full pre-print detailing the mathematical derivation and biophysical mechanism.
Simulation Suite: Python-based implementation (NetworkX, POT) including:

The script energy_ROI_tracker.py depends on the physics engine in run_CAH_scaling_analysis.py. Please ensure both files are downloaded to the same directory before running.

Note: NEST is required to run the PyNEST simulation.

pip install networkx numpy matplotlib pot tqdm joblib scipy

python run_CAH_scaling_analysis.py: Finite-size scaling and robustness tests.