Saliency Thresholds in Neural Code and its Relation to the Power-Law, Gaussian, and Lambert W Function

The cortical neurons’ response properties are peculiar in that despite the variability in the stimulus distribution the response has a stereotypical heavy-tail distribution. For example, a visual cortical model(orientation energy)’s response results in an invariant power-law-like response distribution, regardless of the stimulus image. An interesting observation is that when this response distribution is compared with a normal (Gaussian) distribution with a matched standard deviation, the intersection where the power law distribution exceeds the matched Gaussian distribution is linearly correlated with the saliency threshold. (The same orientation energy model, when fed with a white noise image, results in a normal-distribution-like response, justifying its use as a baseline.) Further analysis reveals that this intersection point can be analytically computed using the Lambert W function, and it is also linearly correlated with the standard deviation of the response. These results point to an interesting theoretical juncture where the power law, Gaussian, and Lambert W function meet, and relate to an important threshold in neural code. In additional computational experiments, we show how some of these results can be replicated using Convolutional Neural Networks with recurrent shared weights. These results reveal a fundamental mathematical relationship linking three ubiquitous functions in natural systems, indicating a potentially universal principle in neural computation. (AA, JHP: equal contribution.)