Intrinsic geometry of trial-to-trial variability in primary visual cortex is optimal for robust representation of visual similarity

How neuronal populations construct robust representations of the sensory world despite neural variability remains unclear. Here, we show that trial-to-trial variability in mouse primary visual cortex follows a simple rule: for each stimulus, the mean and variance of spike counts across neurons show a highly stereotyped relationship, with a slope of 1 on a log-log scale. To test how this geometry of trial-to-trial variability affects sensory representations, we numerically manipulated the slope of the log-mean vs. log-variance relationship. We found that the intrinsic geometry of trial-to-trial variability, with slope 1, enables representations of distinct sensory inputs to have minimal overlap while being continuous. At this slope, representational similarity was maximally consistent across neuronal subsets, both within and across mice. By contrast, at slope 0, when variance is uniform across neurons, visual representations were more efficient but less robust. Simulations of recurrent networks, where excitatory and inhibitory neurons formed locally balanced clusters with stronger connections, showed that larger clusters produced steeper slopes and more robust representations. Together, these results suggest that the geometry of trial-to-trial variability mediates a tradeoff between efficiency and robustness, and that the intrinsic geometry is near optimal for robust visual representations.