A universal power law optimizes energy and representation fidelity in visual adaptation.

Sensory systems continuously adapt their responses based on the probability of encountering a given stimulus. In the mouse primary visual cortex (V1), the average population response is a power law of the prior probability of stimuli in the environment. For a given stimulus type (e.g., oriented gratings), the power law's exponent is invariant to changes in statistical environments, enabling predictions of average population responses to new environments. Here, we aim to provide a normative explanation for the power law behavior. We develop an efficient coding model where neurons adjust their firing rates through multi-objective optimization, hypothesizing that the neural population adapts to enhance stimulus detection and discrimination while reducing overall neural activity. We show that a power law that matches the one observed experimentally can emerge from our model. We interpret the exponent as reflecting a balance between energy efficiency and representational fidelity in adaptation. Furthermore, we account for the invariance of the power law's exponent across environmental changes by linking it to the dependence of tuning curve modulation on stimulus probability. Finally, we explain that variations in the exponent with different stimulus types (e.g., natural movies) result from changes in the minimal distances between neural representations, in agreement with experimental findings. We conclude that a universal power law of adaptation can be explained as a trade-off between representation fidelity and energy cost.