Upper bound on ganglion size

The separation of neuronal somata from the neuropil, as typical for unipolar neurons, is thought to save tissue volume, see Sec. 2.3.1. When neuronal somata are located inside the neuropil, neurites have to circle around the somata (Fig. 7.4A). This, besides others, elongates the neurites, for which the negative consequences, such as an increase in signal transmission delay, have been summarized by Rivera-Alba et al. [133]. Whether a ganglion-like structure with separated soma layer, or somata embedded within the neuropil minimizes tissue volume depends on the relative size of soma and connecting neurite (the stem neurite in the case of unipolar cells). Under the assumption that tissue minimization is indeed evolutionary favorable¹¹, one can derive an upper bound for the size of a ganglionic structure, because, with a sufficiently large number of soma sheets, the length of the stem neurites required to reach the neuropil implies a larger volume than is saved by an separation of the somata and the neuropil, see Fig. 7.4.

Hence, the advantage for volume minimization of a separated soma layer depends not only on neuronal morphology, but also on the number of neurons.

The constraint of minimal tissue volume allows for an educated guess on the maximal number of soma sheets. Tissue volume is reduced when the neuropil is void of somata, as then a dendritic or axonal neurite does not need to circumvent somata (Fig. 7.4). Yet, the exclusion of somata from the neuropil requires a stem neurite which connects the

11Minimization of volume (or wiring length) is in the literature common regarded as an evolutionary pressure, see Sec. 2.3.1. Yet, I am not aware of empirical evidence for this claim. While it makes sense that neuronal systems should not be larger than they need to be, to me it seems to be of only minor importance compared to other evolutionary pressures such as an increase in complexity.

7.6. Wiring optimization in ganglionic structures soma with the dendritic and axonal trees. Each additional sheet of somata increases the distance between somata and neuropil, and hence lengthens the stem neurite by at least one soma diameter. This increases the required volume with each additional soma sheet. The exclusion of somata from the neuropil hence saves volume as long as the volume decrease due to straight dendrites is larger than the volume increase due to longer stem neurites.

The minimization of volume can be bounded in the simple model depicted in Fig. 7.4.

For a soma intermingled within the neuropil, the volume is increased by the difference between the volume of the straight dendrite, and the encircling dendrite. In any neuropil, this would affect several dendrites (and axons). Yet, the largest volume saving occurs for the thickest dendrite because the change in length between straight and circling is the same for thick and thin dendrites, such that the volume saving is larger for thicker dendrites. For simplicity, all dendrites but a single, thick dendrite are hence ignored. The volume saved in this case is compared with the minimal volume of the stem neurite required to locate the soma on top of the some layer consisting ofnsheets of somata. This comparison allows to predict under which condition the exclusion of somata from the neuropil saves tissue volume.

For a soma with diameter swithin the neuropil that is encircled by a dendrite of diameterd, the dendrite volume required for wrapping around the soma isV_around = π/4d²(π/2s), while a straight passage without soma in the way amounts toV_direct = π/4d²s. An exclusion of the soma from the neuropil is useful as long as the difference of both is larger than the stem neurite volume required to exclude the soma. The diameter dof the thickest part of a dendrite is often related to the stem neurite diameter, and a value of 0.5dis supported by data, as shown in the first publication, Supplemental Information. For an estimate of the volume decrease, the stem neurite diameter is denoted as p∗^{d, where p} is the ratio between stem neurite diameter and neurite diameter. An exclusion of the soma will save tissue volume only if the minimal volume required by the stem neurite is smaller than the decrease in volume due to the straight dendrite,

Vstem < V_around−^Vdirect

π/4(pd)²(n−¹)s < π/4d²s(π/2−¹) p²(n−¹) < _π/2−¹≈^0.57

wherenis the number of stacked somata in the soma layer, see Fig. 7.4. Choosing p=0.5 as in the first publication, we findn<3.3 as condition for volume-minimizing soma exclusion. This simple estimate predicts that, from three layers of somata on, additional somata should rather be located inside the neuropil, instead of adding them as an additional sheet of somata on the surface of the ganglion. This estimate is conservative, because the stem neurite will be longer than estimated as it often passes also part of the neuropil, and as it may also need to encircle somata when passing underlying sheets. Note that this educated guess compares a dendrite which typically participates in signal transmission with the stem neurite which is typically only partly depolarized by the signal. Partial depolarization is probably energetically

less costly than full depolarization of a dendrite, such that exclusion of somata may be energetically favorable even when increasing tissue volume.

Figure 7.4.:A: Neurites have to circle around the soma if the soma is located inside the neuropil (right), which requires more volume than if the neurite can pass through the space taken up by the soma (left). B: A stem neurite connects soma and neuropil. An exclusion of the somata from the neuropil reduces the overall volume of the ganglion if the volume reduction illustrated in A is larger than the volume of the stem neurite.

The prediction of maximal three soma layers per ganglion demands for a quantitative analysis of the soma layer number for different animals. While this is left for future research, many species considered in the first publication show indeed only a small number of soma layers.

As a side note, it seems that a living example for an upper bound on the number of soma sheets is provided by the octopus. The octopus has mostly unipolar neurons.

While many neurons are arranged in ganglia as known from other invertebrates, its neurons are partly arrange in layers rather than spherical ganglia [61, 115]. In the light of the hypothesis of this section, the disintegration of spherical ganglia in the octopus could result from its large number of neurons, too large for a pure ganglionic arrangement. Moreover, in the highly developed optical lobe of octopus, multipolar neurons can also be found besides the typically unipolar shape [61, 115]. Maybe this could be explained by one of the advantages of a multipolar neuron discussed in Sec.

7.2, such as a facilitation of recurrent connections. While I consider it as unlikely that intelligence is constrained by neuronal morphology or arrangement, as there is general agreement that intelligence rather arises from the connection pattern of neurons, it is nevertheless interesting that brain activity in octopuses shows oscillations in the field potential, similar to vertebrates and in contrast to most other invertebrates [16], and that the large brain of the octopus allows it to compete on a cognitive level with vertebrates of the sea [82]. In summary, the octopus is one example that shows increased neuron number and a deviation of the ganglionic arrangement. The question whether this relation is indeed causal, as suggested by the hypothesis above, is left for future research.

7.7. Recapitulation

7.7. Recapitulation

The first project of this thesis shows that signal transmission can be optimized by a central or externalized soma¹². In addition, further advantages of unipolar and multipolar morphologies were discussed, with an emphasis on signal processing and the spatial environment of the soma. The discussion portrays (energy-)efficient signal transmission as a mayor driving force for the divergent evolution of neurons with central or externalized soma. As detailed above, I hypothesize that the divergent evolution arises from different solutions to the problem that a large central soma poses for signal transmission: While invertebrates externalized the large soma that evolved to support complex neurons, vertebrates retained the central soma location, but externalized part of the somatic machinery into proximal dendrites.

12Energetic or other advantages may also have driven the development of unipolar neurons in vertebrates (e.g., spinal dorsal root ganglion cells), and multipolar neurons in invertebrates (e.g., octopus gravity receptor system [25]).