Dynamic Curvature Adaptation: Geometric Regulation of Cortical State and Pathological Collapse

Curvature Adaptation Hypothesis entry-point: https: //doi. org/10. 5281/zenodo. 19634691 Abstract We propose the Curvature Adaptation Hypothesis (CAH): nervous systems regulate effective information geometry in order to manage the thermodynamic and transport burdens imposed by hierarchical inference. CAH identifies 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, thereby modulating access to distinct routing regimes. Using Optimal Transport (OT) modeling and finite-size scaling analysis, we show that varying γ can drive a sharp transition from a relatively flat baseline (κ ≈ 0) into a more transport-efficient negative-curvature regime (κ < 0). This transition remains robust across network depths (N=3, 5, 7) and under degree-preserving topological scrambling, indicating that curvature-sensitive routing is not a fragile artifact of one exact global architecture. To bridge these transport mechanics with biologically plausible dynamics, we implement a Spiking Neural Network (SNN) simulation of the VIP-SST-Pyramidal microcircuit and use the Earth Mover’s Distance W1 as a geometric proxy for signaling cost. The results suggest that local inhibitory control can, in principle, purchase a macroscopic reduction in distributed transport burden, yielding a real thermodynamic return on investment under the right structural conditions. We then use this framework to generate falsifiable predictions for pathological connectomics, modeling Mania as a failure of geometric regulation and neurodegeneration as a loss of access to efficient hierarchical routing. Effective information geometry is not merely a static anatomical backdrop, but a controllable physical variable with consequences for integration and pathology. Summary This project details the Curvature Adaptation Hypothesis (CAH), a theoretical framework proposing that the mammalian cortex may optimize information transport by dynamically regulating its effective information geometry in response to hierarchical demand. In its foundational form, CAH identifies the Martinotti-cell subtype of Somatostatin (SST) interneurons—which target distal apical dendrites—as a plausible biological actuator of this control process, regulating the apical-somatic conductance ratio (γ) and thereby modulating access to distinct routing regimes. Using Optimal Transport (OT) modeling, finite-size scaling analysis, and Spiking Neural Network (SNN) simulations, this work argues that curvature-sensitive routing is a real biophysical and thermodynamic candidate resource rather than a static geometric backdrop. Key Findings Scale-Robust Geometric Transition: Using Optimal Transport (OT) simulations on stochastic Galton-Watson trees, the project shows that increasing γ can drive a sharp transition from a relatively flat baseline into a more transport-efficient negative-curvature routing regime, with the transition profile remaining robust across hierarchical depths. Topological Robustness: Degree-preserving topological scrambling preserves access to the transition, suggesting that curvature-sensitive routing is not a fragile artifact of one exact global architecture. This supports the view that preserved local degree structure and synaptic availability make a substantial contribution to the system’s geometric capacity, even when fine global order is disrupted. Spiking Neural Network Validation: A PyNEST simulation of the VIP-SST-Pyramidal microcircuit provides a biologically plausible microcircuit mechanism for curvature-sensitive routing control, showing how local inhibitory dynamics can open access to a more strongly integrated large-scale routing regime. Pathological Failures of Geometric Regulation: The project computationally models three distinct cognitive conditions: Healthy Regime: Flexible, regulated access to more transport-efficient routing conditions under hierarchical demand. Manic Regime: Excessive VIP-like hub integration reduces the selectivity of geometric regulation, biasing the system toward less regulated, shortcut-dominated integration. Neurodegenerative Regime: Synaptic pruning raises geometric resistance and narrows the system’s access to the routing conditions required for efficient hierarchical separation and integration. Metabolic Implications We provide a biophysical energy ROI analysis showing that the local maintenance cost required for SST-mediated gating can be repaid by a larger global reduction in signaling burden. Rather than treating hyperbolicity as a simple end state, the framework identifies curvature-sensitive routing as a constrained thermodynamic strategy through which hierarchical networks can reduce costly intermediate relay operations. Related Works This work is expanded upon directly in: Pender, M. A. (2026). A Control-Law Extension of the Curvature Adaptation Hypothesis in Hierarchical Transport Networks. Zenodo. https: //doi. org/10. 5281/zenodo. 19270109 Pender, M. A. (2026). Beyond Mean Curvature: Lower-Tail Routing Structure in Controlled Hierarchical Networks. https: //doi. org/10. 5281/zenodo. 19324674 Pender, M. A. (2026). The Biological q10 Tail as a Layered System: Rich-Club Scaffold, SST Actuation, and Thalamocortical Thickening. Zenodo. https: //doi. org/10. 5281/zenodo. 19477954 CAH Framework: Pender, M. A. (2026). The Curvature Adaptation Hypothesis: Dynamic Information Geometry as a Regulated Resource in Neural Computation. Zenodo. https: //doi. org/10. 5281/zenodo. 19634691 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 energyROIₜracker. py depends on the physics engine in runCAHₛcalingₐnalysis. 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 runCAHₛcalingₐnalysis. py: Finite-size scaling and robustness tests. python runCAHwithHubs. py: Simulation of hyper-integrative/manic states. python runCAHPruning. py: Simulation of neurodegenerative collapse. python energyROIₜracker. py: Metabolic expenditure modeling. python biologicalₘanifold. py: PyNEST SNN simulation.