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Animal behaviour arises from computations in neuronal circuits, but our understanding of these computations has been frustrated by the lack of detailed synaptic connection maps, or connectomes. For example, despite intensive investigations over half a century, the neuronal implementation of local motion detection in the insect visual system remains elusive. Here we develop a semi-automated pipeline using electron microscopy to reconstruct a connectome, containing 379 neurons and 8,637 chemical synaptic contacts, within the Drosophila optic medulla. By matching reconstructed neurons to examples from light microscopy, we assigned neurons to cell types and assembled a connectome of the repeating module of the medulla. Within this module, we identified cell types constituting a motion detection circuit, and showed that the connections onto individual motion-sensitive neurons in this circuit were consistent with their direction selectivity. Our results identify cellular targets for future functional investigations, and demonstrate that connectomes can provide key insights into neuronal computations. Reconstruction of a connectome within the fruitfly visual medulla, containing more than 300 neurons and over 8,000 chemical synapses, reveals a candidate motion detection circuit; such a circuit operates by combining displaced visual inputs, an operation consistent with correlation based motion detection. Three papers in this issue of Nature use the retina as a model for mapping neuronal circuits from the level of individual synaptic contacts to the long-range scale of dendritic interactions. Helmstaedter et al. used electron microscopy to map a mammalian retinal circuit of close to a thousand neurons. The work reveals a new type of retinal bipolar neuron and suggests functional mechanisms for known visual computations. The other two groups study the detection of visual motion in the Drosophila visual system — a classic neural computation model. Takemura et al. used semi-automated electron microscopy to reconstruct the basic connectome (8,637 chemical synapses among 379 neurons) of Drosophila's optic medulla. Their results reveal a candidate motion detection circuit with a wiring plan consistent with direction selectivity. Maisak et al. used calcium imaging to show that T4 and T5 neurons are divided into specific subpopulations responding to motion in four cardinal directions, and are specific to 'ON' versus 'OFF' edges, respectively.

With Q ⊂ R N a bounded open connected set, the initial value problem for the nonhomogeneous nonlinear heat equation ∂u/∂t - ΔM = f(x, t, u) with zero boundary conditions and zero initial condition is solved in a generalized sense in the region Ω × (0, T) under new one-sided conditions on f(x, t, s). In particular, this result greatly improves on the well-known one-sided result of Brezis and Nirenberg, [2, p. 302]. The techniques used here are completely different from those employed by the last mentioned authors and carry over even to the case when the space operator may be singular elliptic and quasilinear in the lower order terms. So the main result is presented in this singular quasilinear context.