A Mathematical Model of Reliability-Dependent Sensory Reweighting in Quiet Standing

Human quiet standing relies on vestibular, proprioceptive, and visual information whose contributions change with sensory context. Posturography characterizes this reweighting through responses to visual and support-surface perturbations, but a compact account of how sensory reliability propagates from state estimation to postural action remains incomplete. We develop a minimal quiet-standing model in which sensory reliability is encoded by channel-specific precision parameters that weight sensory prediction errors. A one-link inverted pendulum with intrinsic ankle stiffness estimates posture by minimizing an active-inference free-energy objective and selects ankle torque by minimizing the same objective under a restorative sensory goal. A fixed-point analysis predicts in closed form the perturbation-induced belief bias, each channel's state-update contribution, the steady posture, and a linear stability condition from the precisions, goal dynamics, and intrinsic stiffness. Closed-loop simulations in 115 trials matched the predicted equilibria to within 0.001degrees wherever the loop settled, and the two conditions predicted unstable diverged or drifted. Reducing an unreliable channel's precision eightfold reduced perturbation-driven posture shifts by 82%, with a matching decrease in that channel's state-update contribution; intrinsic stiffness left this reduction essentially unchanged for visual and sensory-only perturbations but bounded it for physical support tilts; a global precision reduction did not reduce the shift; and a within-trial precision switch showed belief updating, not a separate action gain, carrying the change to ankle torque. The results give a compact mathematical account of postural sensory reweighting as relative, context-selective precision control in a closed multisensory loop.