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 population response magnitude is a power law of the stimulus probability 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 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 optimization of a weighted objective, hypothesizing that the neural population adapts to enhance stimulus detection and discrimination while reducing overall neural activity. We show that a model balancing representational fidelity and energy efficiency matches the power law observed experimentally for a wide range of parameters, while models of adaptation with alternative coding objectives and resource constraints are unable to reproduce this empirical observation. 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 how variations in the exponent with different stimulus types (e.g., natural stimuli) 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.