Logarithmic coding leads to adaptive stabilization in the presence of sensorimotor delays.

Animals respond to sensory stimuli with motor actions, which in turn generate new sensory inputs. This sensorimotor loop is constrained by time delays that impose a trade-off between responsiveness and stability. Additionally, as the relationship between a motor command and the corresponding sensory feedback is context-dependent, the response must be adapted in real time. It is generally believed that this adaptation process relies on an internal model that is continuously updated through prediction error minimization. Here, we experimentally reveal an alternative strategy based on a simpler feedback mechanism that does not require any internal model. We developed a virtual reality system for the miniature transparent fish Danionella cerebrum that enables in vivo brain-wide imaging during fictive navigation. By systematically manipulating the feedback parameters, we dissected the motor control process that allows the animal to stabilize its position using optic flow. The sensorimotor loop can be fully described by a single delay differential equation, whose solutions quantitatively capture the observed behavior across all experimental conditions. Both behavioral and neural data indicate that the observed adaptive response arises from logarithmic nonlinearities at the sensory (Weber-Fechner law) and motor (Henneman's size principle) ends. These fundamental properties of the nervous system, conserved across species and sensory modalities, have traditionally been interpreted in terms of efficient coding. Our findings unveil a distinct functional role for such nonlinear transformations: ensuring stability in sensorimotor control despite inherent delays and sensory uncertainty.