Scale-Invariant Geometry of Neural Dynamics Across Biological Timescales

Neural systems operate across temporal scales spanning microseconds to seconds, yet whether geometric principles unify dynamics across these scales remains unknown. We measured neural trajectory curvature —quantified via condition number of population activity—across biophysical ion channel models, spiking neural networks, rate-based recurrent networks, and human EEG (n=42). Curvature follows a power law 𝜅 ∝ 𝜏𝛼 across four orders of magnitude (𝛼 = −1.51 ± 0.08, 95% CI [−1.65, −1.40], 𝑅2 = 0.97, permutation p=0.018). This scaling replicates in an independent task dataset (n=14, Go/No-Go; 𝛼 = −1.86 ± 0.15, no significant task difference: Δ𝛼 = −0.29 ± 0.17, p=0.094) and generalizes cross-species (mouse single-unit recordings: 𝛼 = −0.64 ± 0.08, 𝑅2 = 0.935) and cross-modality (MEG: 𝛼 = −0.48). Theorydata calibration across seven manipulations shows strong correlation (Spearman 𝜌 = 0.929, p=0.0025). Trial-level curvature predicts reaction time (r=0.132, 𝑝 < 0.001, 𝑅2 = 1.7%). Computational modeling links scaling to dimensionality reduction in recurrent dynamics. As a biological validation, schizophrenia patients (n=27) show reduced curvature versus controls (n=56; Cohen’s d=-0.77, p=0.0014), correlating with symptom severity (r=0.48, p=0.021). These data demonstrate a scale-invariant geometric principle governing neural dynamics from ion channels to cognition, with quantifiable disruption in psychiatric illness.