Developmental and evolutionary constraints on olfactory circuit selection

Across species, neural circuits show remarkable regularity, suggesting that their structure has been driven by underlying optimality principles. Here we ask whether we can predict the neural circuitry of diverse species by optimizing the neural architecture to make learning as efficient as possible. We focus on the olfactory system, primarily because it has a relatively simple evolutionarily conserved structure and because its input- and intermediate-layer sizes exhibit a tight allometric scaling. In mammals, it has been shown that the number of neurons in layer 2 of piriform cortex scales as the number of glomeruli (the input units) to the 3/2 power; in invertebrates, we show that the number of mushroom body Kenyon cells scales as the number of glomeruli to the 7/2 power. To understand these scaling laws, we model the olfactory system as a three-layer nonlinear neural network and analytically optimize the intermediate-layer size for efficient learning from limited samples. We find, as observed, a power-law scaling, with the exponent depending strongly on the number of samples and thus on longevity. The 3/2 scaling seen in mammals is consistent with observed longevity, but the 7/2 scaling in invertebrates is not. However, when a fraction of the olfactory circuit is genetically specified, not learned, scaling becomes steeper for species with a small number of glomeruli and recovers consistency with the invertebrate scaling. This study provides analytic insight into the principles underlying both allometric scaling across species and optimal architectures in artificial networks.