Summary Complicated neuronal circuits can be genetically encoded, but the underlying developmental algorithms remain largely unknown. Here, we describe a developmental algorithm for the specification of synaptic partner cells through axonal sorting in the Drosophila visual map. Our approach combines intravital imaging of growth cone dynamics in developing brains of intact pupae and data-driven computational modeling. These analyses suggest that three simple rules are sufficient to generate the seemingly complex neural superposition wiring of the fly visual map without an elaborate molecular matchmaking code. Our computational model explains robust and precise wiring in a crowded brain region despite extensive growth cone overlaps and provides a framework for matching molecular mechanisms with the rules they execute. Finally, ordered geometric axon terminal arrangements that are not required for neural superposition are a side product of the developmental algorithm, thus elucidating neural circuit connectivity that remained unexplained based on adult structure and function alone.

Localized roll patterns are structures that exhibit a spatially periodic profile in their center. When following such patterns in a system parameter in one space dimension, the length of the spatial interval over which these patterns resemble a periodic profile stays either bounded, in which case branches form closed bounded curves ("isolas"), or the length increases to infinity so that branches are unbounded in function space ("snaking"). In two space dimensions, numerical computations show that branches of localized rolls exhibit a more complicated structure in which both isolas and snaking occur. In this paper, we analyze the structure of branches of localized radial roll solutions in dimension 1+ε, with 0 < ε 1, through a perturbation analysis. Our analysis sheds light on some of the features visible in the planar case.

A recent approach to the Beck-Fiala conjecture, a fundamental problem in combinatorics, has been to understand when random integer matrices have constant discrepancy. We give a complete answer to this question for two natural models: matrices with Bernoulli or Poisson entries. For Poisson matrices, we further characterize the discrepancy for any rectangular aspect ratio. These results give sharp answers to questions of Hoberg and Rothvoß (SODA 2019) and Franks and Saks (Random Structures Algorithms 2020). Our main tool is a conditional second moment method combined with Stein's method of exchangeable pairs. While previous approaches are limited to dense matrices, our techniques allow us to work with matrices of all densities. This may be of independent interest for other sparse random constraint satisfaction problems.