A logarithmic theory of visuomotor stabilization

Although many animals rely on visual information to navigate, optic flow is inherently ambiguous as it confounds information about motion speed and object distance. As a result, the visual feedback produced by a given motor command is context-dependent and requires an appropriately adapted response. Recent experiments have investigated how the fish Danionella cerebrum use visual cues to stabilize their position against simulated external currents. Logarithmic sensorimotor transformations have been proposed to enable adaptive responses to perturbations while preventing delay-induced instabilities. Here, we develop the theoretical framework introduced for continuous locomotion to show how logarithmic coding naturally gives rise to this adaptive behavior. The system is modeled by a nonlinear delay differential equation, which is analyzed using dynamical systems theory. We further analyze experimental data to uncover the mechanisms underlying swimming initiation and positional drift correction. Finally, we extend our framework to intermittent locomotion, resulting in a nonlinear difference equation, and show that it still produces robust adaptive behavior. This is motivated by the literature on zebrafish, where visuomotor stabilization has been extensively studied, but burst-and-coast swimming obscures the underlying adaptation mechanism. We show that our theory can reproduce the experimental results reported for motor adaptation in zebrafish without invoking internal models. Overall, our results highlight logarithmic coding as a unifying principle for visuomotor stability across continuous and intermittent locomotor regimes.