Wiring economy and volume exclusion determine neuronal placement in the Drosophila brain

Abstract

Wiring economy has successfully explained the individual placement of neurons in simple nervous systems like that of Caenorhabditis elegans [1-3] and the locations of coarser structures like cortical areas in complex vertebrate brains [4]. However, it remains unclear whether wiring economy can explain the placement of individual neurons in brains larger than that of C. elegans. Indeed, given the greater number of neuronal interconnections in larger brains, simply minimizing the length of connections results in unrealistic configurations, with multiple neurons occupying the same position in space. Avoiding such configurations, or volume exclusion, repels neurons from each other, thus counteracting wiring economy. Here we test whether wiring economy together with volume exclusion can explain the placement of neurons in a module of the Drosophila melanogaster brain known as lamina cartridge [5-13]. We used newly developed techniques for semiautomated reconstruction from serial electron microscopy (EM) [14] to obtain the shapes of neurons, the location of synapses, and the resultant synaptic connectivity. We show that wiring length minimization and volume exclusion together can explain the structure of the lamina microcircuit. Therefore, even in brains larger than that of C. elegans, at least for some circuits, optimization can play an important role in individual neuron placement.

Figure 1. A lamina cartridge in Drosophila melanogaster reconstructed in three dimensions from serial EM images

(A) Single EM image in which the cell profiles have been segmented and digitally labeled with different colors. (B) Three-dimensional shapes of the neurons in the reconstructed cartridge; see Fig. S1 for shapes of individual neurons. (C) Schematic version of (B). Neurons connect with lateral branches, here illustrated for neuron C2. R1–R6: photoreceptors terminals 1 to 6; L1–L5: lamina monopolar cells 1 to 5; L4+x and L4-y: incoming L4 collaterals from the two neighboring anterior cartridges along the +x and -y axes, respectively; Am: amacrine cells; T1: T medulla neuron 1; C2–C3: centrifugal medulla neurons C2 and C3; Lawf: lamina wide-field cells; ep. glia: epithelial glia; s. glia: satellite glia; m. glia: marginal glia. To enable their visualization only four of the six photoreceptors, five of the six T1 branches and two of the eleven Lawf branches are shown.

Figure 2. Connectivity in the lamina cartridge

(A) Connectivity matrix; see Table S1 for an extended matrix including neuronal subtypes. Hotter colours indicate larger number of synapses. (B)-(F) Representation of neurons and connecting neighbours for (B) a photoreceptor, (C) an amacrine neuron, (D) centrifugal neurons C2 and C3, (E) a large wide-field neuron Lawf and (F) large monopolar cell L2.

Figure 3. Wiring economy in 2-D

(A) Pairwise cost between two neurons used to model soft volume exclusion for short distances and wiring length cost for distances larger than the sum of the radii of the two profiles of partner neurons. r_i is the radious of the i neuron. (B) Configuration of the lowest cost found by SA when the initial configuration is the actual one. White lines connect to actual positions marked by black squares. (C) The configuration computed in (B) is non-trivial because it includes both connections to close and distant neurons, illustrated for neuron C2. (D) The actual cost of the computed configuration in (B) (arrow) is significantly lower than random configurations obtained by permuting large neuron profiles with other large neuron profiles, or small profiles with small (dark gray histogram), and 240 other converged configurations found by SA using random initial configurations (white histogram).

Figure 4. Roles of connectivity and axon sizes

(A) Error of six wiring economy configurations with respect to the experimental configuration. Green line: includes two manipulations of neuronal radii, one forcing all neurons to have the same size ( homogeneous case ) and another one in which the larger neurons are modeled as the smaller ones and vice versa (the inverted case ). Blue line: includes two manipulations of the connectivity matrix, one making all connecting neurons use the same number of synapses ( homogeneous case ) and the other one in which neuron pairs with the largest number of synapses are modeled as the ones with the smaller numbers ( inverted case ). (B) Wiring economy in simplified models. Model I: Wiring economy configuration in a simplified model in which the dark-blue neurons (photoreceptor terminals) are connected only to one light-blue one (L1 or L2). Model II: Wiring economy configuration including now both L1 and L2 neurons. Model III: Wiring economy configuration with photoreceptor terminals and 30 smaller neurons having a random connectivity matrix. Model IV: Same as Model III but including neurons L1 and L2.

Figure 5. Wiring economy in 3D

From a wiring economy computation in 3D, configurations in (A) distal, (B) central and (C) proximal regions of the cartridge. The neurons are placed in their computed positions with the experimental areas. Lines connect the computed to the observed positions.