Synaptic wiring of neurons in Caenorhabditis elegans is largely invariable between animals. It has been suggested that this feature stems from genetically encoded molecular markers that guide the neurons in the final stage of synaptic formation. Identifying these markers and unraveling the logic by which they direct synapse formation is a key challenge. Here, we address this task by constructing a probabilistic model that attempts to explain the neuronal connectivity diagram of C. elegans as a function of the expression patterns of its neurons. By only considering neuron pairs that are known to be connected by chemical or electrical synapses, we focus on the final stage of synapse formation, in which neurons identify their designated partners. Our results show that for many neurons the neuronal expression map of C. elegans can be used to accurately predict the subset of adjacent neurons that will be chosen as its postsynaptic partners. Notably, these predictions can be achieved using the expression patterns of only a small number of specific genes that interact in a combinatorial fashion.

The nervous system contains trillions of neurons, each forming thousands of synaptic connections. It has been suggested that this complex connectivity is determined by a synaptic ''adhesive code,'' where connections are dictated by a variable set of cell surface proteins, combinations of which form neuronal addresses. The estimated number of neuronal addresses is orders of magnitude smaller than the number of neurons. Here, we show that the limited number of addresses dictates constraints on the possible neuronal network topologies. We show that to encode arbitrary networks, in which each neuron can potentially connect to any other neuron, the number of neuronal addresses needed scales linearly with network size. In contrast, the number of addresses needed to encode the wiring of geometric networks grows only as the square root of network size. The more efficient encoding in geometric networks is achieved through the reutilization of the same addresses in physically independent portions of the network. We also find that ordered geometric networks, in which the same connectivity patterns are iterated throughout the network, further reduce the required number of addresses. We demonstrate our findings using simulated networks and the C. elegans neuronal network. Geometric neuronal connectivity with recurring connectivity patterns have been suggested to confer an evolutionary advantage by saving biochemical resources on the one hand and reutilizing functionally efficient neuronal circuits. Our study suggests an additional advantage of these prominent topological features-the facilitation of the ability to genetically encode neuronal networks given constraints on the number of addresses.complex networks ͉ design principles ͉ evolution M uch of the complexity of neuronal networks lies in the pattern of synaptic connections between neurons (1). The human brain contains Ϸ10 11 neurons, each forming Ϸ10 4 synaptic connections (2). How do neurons know how to make the right connections? Roger Sperry's ''chemoaffinity hypothesis'' postulates that neurons select their targets by recognizing distinct molecular markers displayed on the neuronal surfaces (3). The implication of this hypothesis is that neuronal connectivity is genetically encoded. Candidate genetic loci for the synaptic connectivity code have been found in several organisms including Caenorhabditis elegans (4, 5), Drosophila (6), and vertebrates (7). Recent work has suggested that in vertebrates, protocadherins-cell adhesion proteins of the cadherin superfamily may constitute this adhesive code (7-13). Protocadherins are localized to the synaptic junctions, where they form contacts with similar proteins on neighboring neurons (13,14).A naïve design for genetically encoding neuronal networks would assign each neuron a unique address and a list of addresses that it should connect to. This design potentially permits the genetic encoding of any arbitrary network but poses the problem of requiring diversity of the markers constituting the neuronal addresses. Although...

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