Entropic curvature on hyperbolic semantic manifolds indexes anxiety-like cognitive dynamics: an interpretable geometric biomarker

Background

Human semantic cognition unfolds as stochastic trajectories over high-dimensional symbolic networks whose large-scale geometry is hyperbolic and whose discrete Ollivier-Ricci curvature is predominantly negative [3]. Predictive-processing accounts of mood and anxiety disorders posit that altered precision-weighting and aberrant exploration of internal state spaces underlie ruminative and intrusive cognition [4,5]. We hypothesised that such alterations should manifest as quantifiable shifts in the local curvature–entropy coupling of semantic trajectories — yielding an interpretable, open-source geometric biomarker complementary to neuroimaging and ecological momentary assessment.

Aims

(i) To define an entropic-curvature index integrating geometric stability with informational uncertainty over a large empirical semantic graph; (ii) to test whether anxiety-biased cognitive regimes produce statistically robust deviations in this index; (iii) to identify regime-shift signatures consistent with predictive-processing accounts, and to outline a route towards clinical validation against multimodal patient data.

Methods

The Small World of Words English corpus (SWOW-EN; >12,000 cue words) [1] was represented as a weighted directed graph G=(V,E,w) and embedded in the Poincaré disk. For each node v we computed local Ollivier-Ricci curvature κ(v) under the Lin–Lu–Yau formulation [2] and local Shannon entropy H(v) = −Σ P(v,u) log P(v,u). We defined the Entropic-Curvature index C_ent(v) = κ(v)·(1 − H(v)/log deg(v)). Cognitive trajectories (n = 10,000 per regime) were simulated through Langevin dynamics and biased Markov chains. Normative regimes used low-temperature, valence-balanced exploration; anxiety-biased regimes increased stochastic temperature with negative-valence priming, modelling reduced precision weighting [4]. Effect sizes were computed via Cohen’s d with bootstrap confidence intervals (10⁴ resamples); regime-shift detection used percolation-style order parameters on the giant cluster; long-range dependence was estimated through detrended fluctuation analysis (Hurst exponent). Robustness was assessed across alternative graph sparsifications and entropy normalisations.

Results

Anxiety-biased regimes showed substantially higher variance in C_ent relative to normative regimes (Cohen’s d = 1.84; 95% CI 1.79–1.89), with approximately 47% longer residence times in high-entropy hubs and elevated entropy production rates. A sharp regime transition emerged near ⟨k⟩²/N ≈ 2.5, consistent with percolation-like trapping in maladaptive attractors. Anxiety-biased trajectories yielded lower estimated Hurst exponents (median 0.42 vs 0.58, p < 10⁻³), suggesting multifractal-like dynamics absent in normative exploration. Findings were stable across sparsification thresholds and across independent random seed batches.Conclusions. Entropic curvature on hyperbolic semantic manifolds provides an interpretable, computationally tractable index of the stability–uncertainty coupling underlying cognitive trajectories. The observed regime shift and altered fractal scaling are compatible with — though not yet diagnostic of — anxiety-related cognitive dynamics; clinical validation against fMRI/EEG and ecological momentary assessment data is required before any inferential claim regarding individual patients can be sustained. Ongoing parallel work extends the framework to clinical cohorts (unpublished).The pipeline is fully open-source and generates falsifiable predictions for mechanistic studies of mood and anxiety disorders and their pharmacological treatment pathways and better modelling.

References

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