Dynamic Curvature Adaptation: A Dendritic Gating Mechanism for Hyperbolic Inference and Metabolic Efficiency

Abstract We propose the Curvature Adaptation Hypothesis (CAH): nervous systems optimize task-dependent information transport by dynamically unlocking the latent hyperbolicgeometry inherent in their hierarchical structure. We identify a plausible biophysical actuator—Somatostatin (SST) interneuron-mediated dendritic shunting—that regulates 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. Conversely, we demonstrate that synaptic pruning (spine loss) catastrophically degrades this capacity, identifying ‘Geometric Collapse’ as a distinct marker of neurodegeneration. 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 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 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 “Scrambled Topology” 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 Somatostatin (SST) interneuron-mediated dendritic shunting 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 (κ30% 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 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 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. pip install networkx numpy matplotlib pot tqdm joblib scipy run_CAH_scaling_analysis.py: Finite-size scaling and robustness tests. run_CAH_with_Hubs.py: Simulation of hyper-integrative/manic states. run_CAH_Pruning.py: Simulation of neurodegenerative collapse. energy_ROI_tracker.py: Metabolic expenditure modeling.