Geometric constraints on neuronal connectivity facilitate a concise synaptic adhesive code.

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.