Relationship between Sigmoid and Gaussing Distribution
Yes sure, the core idea is that the sigmoid function approximate very nicely the cumulative density function of a Gaussian distribution. Therefore, if the input of a neuron is a continuous Gaussian distribution, the probability of the output being inferior to a certain value will follow (almost) the shape of the sigmoid function.

A detailed explanation can be found p16 of the following article:

https://www.researchgate.net/publication/220233479_Catastrophic_Forgetting_and_the_Pseudorehearsal_Solution_in_Hopfield-type_Networks