How Axon and Dendrite Branching Are Governed by Time, Energy, and Spatial Constraints

Neurons are connected by complex branching processes - axons and dendrites - that collectively process information for organisms to respond to their environment. Classifying neurons according to differences in structure or function is a fundamental part of neuroscience. Here, by constructing new biophysical theory and testing against our empirical measures of branching structure, we establish a correspondence between neuron structure and function as mediated by principles such as time or power minimization for information processing as well as spatial constraints for forming connections. Specifically, based on these principles, we use undetermined Lagrange multipliers to predict scaling ratios for axon and dendrite sizes across branching levels. We test our predictions for radius and length scale factors against those extracted from neuronal images, measured for cell types and species that range from insects to whales. Notably, our findings reveal that the branching of axons and peripheral nervous system neurons is mainly determined by time minimization, while dendritic branching is mainly determined by power minimization. Further comparison of different dendritic cell types reveals that Purkinje cell dendrite branching is constrained by material costs while motoneuron dendrite branching is constrained by conduction time delay over a range of species. Our model also predicts a quarter-power scaling relationship between conduction time delay and species body size, which is supported by experimental data and may help explain the emergence of hemispheric specialization in larger animals as a means to offset longer time delays. Author summary Neurons are the basic building blocks of the nervous system, responsible for information processing and communication in animals. They consist of a centralized cell body and two types of processes - axons and dendrites - that connect to one another. Previous studies of the differences among neuron cell types have focused on comparisons of either structure or function separately, without considering combined effects. Based on theory for structure of and flow through biological resource distribution networks, we develop a new model that relates neuron structure to function. We find that differences in structure between axons and dendrites as well as between dendrites of different cell types can be related to differences in function and associated evolutionary pressures. Moreover, using our mathematical model, we find that the conduction time delay of electrical signals systematically varies with species body size - neurons in larger species have longer delays - providing a possible explanation for hemispheric specialization in larger animals.

Neurons are fundamental structural units of information processing and communication Single neuron cells have structural similarities to cardiovascular networks, with 48 centralized cell bodies analogous to the heart and branching processes analogous to 49 blood vessels. We propose that the branching structures of axons and dendrites result 50 from optimizing organismal function subject to biophysical constraints. We consider 51 biophysical properties of neurons that might play an important role in governing 52 structure, using data to guide our evaluation of the relative importance of different 53 functions. 54 An important evolutionary function of neuronal networks involves transferring large 55 amounts of information between brain regions in a short amount of time [Laughlin and 56 Sejnowski, 2003]. On the individual cell level, the various morphological forms observed 57 in neurons are various adaptations of basic principles such as limiting signal time delay 58 [Ramón y Cajal, 1995]. Thus, it is important to consider conduction time as an 59 important design principle that governs neuronal branching structures. 60 Indeed, foundational work by Cuntz et al. has used graph theory to quantify and 61 study how connections among axons and dendrites determine conduction time delay. 62 This approach focuses on the tradeoff between wiring costs and conduction time, 63 represented as path length [Cuntz et al., 2010]. The results formalize the laws set forth 64 by Ramon y Cajal, leading to a graph-theoretical algorithm that generates biologically 65 accurate synthetic axonal and dendritic trees [Cuntz et al., 2011]. 66 space-filling. Axon and dendrite radius relate to resistance and thus signaling speed and 69 conduction time. Space-filling constrains the possible connections, branching, and 70 network structure of neurons. Consequently, in this paper we take a similar approach to 71 Cuntz et al. [Cuntz et al., 2010, Chklovskii, 2004] but now incorporate the dependence 72 of conduction time on fiber radius and myelination -insulation that surrounds the fiber 73 and facilitates signal transduction [Squire et al., 2013] -along with constraints of 74 space-filling. 75 As the speed of information processing increases, energy loss due to dissipation also 76 increases [Laughlin and Sejnowski, 2003]. Signaling in the brain consumes a substantial 77 amount of energy [Attwell and Laughlin, 2001], which suggests that energy expenditure 78 is another important factor constraining the design. Moreover, the relationship between 79 metabolic rate and conduction time plays an important role in determining axon 80 function in species across scales of body size  The WBE framework 81 relies on the assumption that resource distribution networks are optimized such that the 82 energy used to transport resources is minimized [West et al., 1997], specifically, by 83 minimizing power lost to dissipation in small vessels [Savage et al., 2008]. 84 Building on biological and physical principles that constrain electrophysiological 85 signaling and information processing in neurons, we build models that predict a suite of 86 neuron morphologies based on which biological or physical principle is under the 87 strongest selection or pressure. Our model includes both conduction time and energy 88 efficiency while also incorporating additional factors set forth by Ramón y Cajal's laws 89 such as the material costs and space filling [Ramón y Cajal, 1995]. We make theoretical 90 predictions for how branch radius and length change across branching generation for 91 both axons and dendrites. We compare these predictions to our empirically measured 92 data to make conclusions about the functional basis for morphological differences 93 observed across cell types. We also use this model to predict how conduction time delay 94 in neurons changes with neuron size, another prediction that is supported by empirical 95 data.
Here, P T OT is the power lost due to dissipation, T T OT is the time delay for a signal 108 travelling across the network, r is the branch radius, l is the branch length, k is the 109 branching generation of the network (with 0 being the trunk and N being the tips), N is 110 the total number of levels of the network, n is the branching ratio, M is the mass of the 111 neuron process, and d is the dimension of space into which the neuron processes project. 112 The branching ratio, n, is equal to 2 for a bifurcating network. We use optimization 113 methods to calculate scaling relationships between the radius and length of successive 114 branches, r k+1 r k and l k+1 l k , as shown in Figure 1.

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In Equation 1, the first term is the power loss due to dissipation, given by l k r 2 k n k . For a neuronal network, we define the power loss by the equation, 117 P = I 2 0 R net , where I 0 is the ionic current and R net is the resistance to current flow in 118 the network. Axons and dendrites can be approximated as wires through which current 119 flows and encounters resistance from the neuron fiber. The resistance is given by where A k is the cross sectional area of the wire, and l k is the length of the is proportional to the square root of the diameter for an unmyelinated fiber [Hodgkin, 135 1954], and proportional to the diameter for a myelinated fiber [Rushton, 1951]. Thus, an 136 value of 0 corresponds to an unmyelinated fiber, and a value of 1 2 corresponds to a 137 myelinated fiber. 138 We can switch between models that optimize either conduction time or power usage 139 by varying α between 0 and 1, corresponding to the following two equations.
In these two equations, the governing optimization principle (first term) is 141 constrained by brain region volume (second term), neuron size (third term), and space 142 filling (fourth term). These quantities are held constant during the optimization. The Thus, the energetic cost of maintaining the resting membrane potential is captured in 158 the total network volume.

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In the section on the allometry calculation, we show that minimizing power (Eq. 3) 160 subject to a conduction time delay constraint leads to a 1 4 -power scaling between calculation can be found in Text S1.

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We will first minimize P by differentiating with respect to radius at an arbitrary 172 level k, setting the result equal to 0.
Solving for the Lagrange multiplier, we have: Using the fact that the Lagrange multiplier is a constant and thus the denominator 175 must be constant across levels, we can solve for the scaling ratio: To find the length scaling ratio, we minimize P with respect to length at an 177 arbitrary level k, and set the result equal to 0.
We solve for the Lagrange multiplier λ k by substituting λ, as calculated in (6). As 179 before, using the fact that the denominator must be constant across levels and 180 substituting in the scaling ratio in (7) for radius, we can solve for the scaling ratio for 181 length: This method is used to solve for the scaling ratios for radius and length for the other 183 cases and compared to empirical results. These findings are summarized in Table 1 in

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We begin by setting the derivative of P* with respect to radius and length equal to 190 zero to solve for the multipliers λ and λ k , respectively, at the stationary point.

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Substituting the expression for λ k back into the original expression for P * , we get an 192 expression that simplifies to the original power term that it minimized, N k=0 l k r 2 k n k . For 193 simplicity, we replace the power term with P and the time delay constraint term with T 194 and rearrange. This calculation is shown in detail in Text S2. We will now take the derivative of this term with respect to M, the mass, and set it 200 Previous results have shown that the energetic cost, which we have interpreted here 202 as power loss due to dissipation, decreases with increasing body weight of animals at a 203 linear rate . Thus, we can express ∂P ∂M generally as a negative 204 constant, −C. We can rewrite the above expression as follows: Solving and applying the initial condition that T=0 when M=0, we have: Thus, from this equation, we have extracted the scaling relationship, a mixed power 207 law relationship including a 1 4 -power law and a linear term with relative weights. Figure  intervals. This database provides 3D reconstruction data that are organized in text files 218 by pixels, in files that specify a pixel ID label for each point, the x,y,z spatial 219 coordinates, the radius of the fiber at each point, and a parent pixel ID, referring to the 220 adjacent pixel previously labelled. The scaling ratios for radius and length can be 221 obtained by organizing this data in terms of branches. This is accomplished by finding 222 the pixels at which the difference between the child pixel ID and the parent pixel ID is 223 greater than 2, which can be defined as branching points. Based on the branching 224 points, a branch ID and parent branch ID can be assigned to each of the pixels.

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The radius can be extracted from each of the branches by taking each of the radius 226 values in each branch and averaging them by the following formula, defining each 227 branch as branch k, where the pixels i range from 1 to N k , where N k is the last pixel of 228 each branch: The length of each branch can be extracted by summing up the Euclidean distances 230 between each of the points in the branch by the following formula: Once the radius and length of each of the branches is found, the scaling ratios are 232 computed by dividing the daughter radius or length, respectively, by the corresponding 233 value for the parent branch. Through this method and using the Python library 234 matplotlib, we generate histograms to visualize the distributions. For the radius 235 distributions, we find a large peak at r k+1 r k = 1.0, which is likely due to the resolution 236 limit of the images. After a certain level, the radius for each of the branches is 237 equivalent to the pixel size itself. Thus, in our distributions for radius, we focused on the data for scaling ratios that are less than 1.0. We use solid black lines to denote the 239 mean values in the data, and error bars represent twice the Standard Error of the Mean 240 (SEM), the standard deviation divided by the square root of the number of data points. 241 We look at neuron reconstructions from both axons and dendrites, and from a range 242 of cell types, brain regions, and species. More detailed information about the source of 243 each of the individual reconstructions can be found in Text S4. neurons were taken from a range of brain regions: the midbrain, the hippocampus, the 262 antennal lobe, the optic lobe, and the ventral nerve cord.

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To study peripheral nervous system neurons, we sampled from reconstruction data 264 that was labelled by region on NeuroMorpho.Org. This data, shown in Figure 4 was calculated by considering the animal leg length predicted from the average body 276 mass. We use a log-log plot, shown in Figure 5, to obtain a power law relationship 277 between body mass and conduction time, where the slope is equal to the power.

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We compared theoretical predictions for scaling ratios calculated from objective  theoretical predictions for length, as we have detailed in Table S1 and Table S2. Given 314 the lack of agreement between the theoretical predictions for length scaling ratios and 315 the data, we focused on radius scaling ratios in this analysis. 0.68 ± 0.004, with theoretical predictions from the four different calculations. We find 327 that the dendrite radius scaling ratio mean is closest to the theoretical predictions from 328 the objective functions minimizing power. The mean lies in between the optimal scaling 329 ratios for function P, so n −1/2 ≈ 0.71, which holds volume to be fixed, and function P*, 330 so n −2/3 ≈ 0.63, which holds time delay to be fixed. Later, in Figure 3, we looked at  The closest theoretical predictions for the dendrite scaling ratio mean are the optimal scaling ratios for function P, minimizing power with fixed volume, n −1/2 ≈ 0.71, and for function P*, minimizing power with fixed time delay, n −2/3 ≈ 0.63. The closest theoretical predictions for the axon scaling ratio mean are the optimal scaling ratios for function T, minimizing time delay, the myelinated case with = 1 2 , n −1/3 ≈ 0.79, and the unmyelinated case with = 0, n −2/5 ≈ 0.76. We restricted radius scaling ratio data to values that are less than 1.0. The representative reconstruction images show the characteristic differences in morphology between dendritic and axonal trees. The dendritic tree, shown on the left, is taken from an elephant cerebellar Golgi cell [Jacobs et al, 2014]. The axonal tree, with a representative long parent branch, is taken from a mouse touch receptor [Lesniak et al, 2014]. We have restricted radius scaling ratio data to values that are less than 1.0. The black solid lines denote the mean values in the distributions, shown with error bars, and the red, green, blue, and magenta dashed lines represent the theoretical predictions for various objective functions. The closest theoretical prediction for Purkinje cells is the optimal scaling ratio for function P, minimizing power with fixed volume, n −1/2 ≈ 0.71. The closest theoretical prediction for motoneurons is the optimal scaling ratio for function P*, minimizing power with fixed time delay, n −2/3 ≈ 0.63. The representative image for the Purkinje cell is from a mouse [Murru et al., 2019] and the representative image for the motoneuron is from a cat spinal motoneuron [Cullheim et al., 1987].

Peripheral Nervous System Neurons
In order to test this theoretical result, we analyzed experimental data to determine 378 that the 1 4 -power mass term is more significant than the linear term, as is shown in 382 more detail in Text S3. Furthermore, we used a log-log plot to determine the power of 383 the relationship, plotting the log of the conduction time delay data against the log of 384 the average body mass of each species. This plot is shown in Figure 5. Our results indicate that the radius scaling ratio mean for axons is closest to the 402 prediction that minimizes time for conduction through myelinated fibers, which 403 supports this notion that information processing speed is a key principle governing the 404 structure of axons.

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In contrast, dendritic trees are relatively short, have more extensive branching, and 406 generally do not conduct action potentials [Rall, 1977]. Previous theoretical work on 407 wiring optimization in cortical circuits similarly proposes that there are differing 408 evolutionary selection pressures governing axons and dendrites. Rather than conduction 409 time delay, the key principle behind dendritic structure is passive cable attenuation 410 [Chklovskii, 2000]. Our results suggest that dendrites are optimized to minimize power, 411 which is related to a voltage drop, with a volume constraint that we have interpreted as 412 a cost in materials. Thus, minimizing power in our theoretical framework is effectively 413 minimizing the attentuation of the passive signals in dendrites.

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There is a great deal of diversity in the branching structures of dendritic trees, and 415 the differences in scaling ratio distributions among the different types gives us important 416 insights into their distinct functional roles. We found that the structure of Purkinje cells 417 and motoneurons are both governed by power minimization, and Purkinje cell structure 418 is constrained by volume while motoneuron structure is constrained by time delay. The 419 predictions and results from the data for Purkinje cells and motoneurons are supported 420 by previous theoretical and experimental results [Hillman, 1979]. We conclude that time 421 plays a greater role in optimizing the structure for motoneuron dendrites.

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Efficiency in information processing is a key function of neurons in the sensorimotor 423 system, and our results emphasize that function as a key feature governing their 424 structural design. When organisms are exposed to environmental stimuli, it triggers a 425 response in the motor system that must be executed very rapidly. Some of these  where the first level begins at the tips, and higher levels are determined when two 493 branches of the same level combine. This has been applied to other networks in biology 494 [Turcotte et al., 1998]. Hermann Cuntz's group has also applied this ordering method to 495 analyze dendritic trees, finding differences in branching metrics across neuron cell types 496 [Vormberg et al., 2017]. We hypothesize that applying this labeling scheme to define 497 branching levels for length will give a distribution of scaling radios that looks more like 498 the normal distributions observed for radius scaling ratios, and means that agree more 499 closely with our theoretical predictions. This is a major goal of our future work, both and length scaling ratios in order to extract more information from the data.

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Throughout this model, we have assumed that branching is symmetric -the radius 507 and length of daughter branches are identical. Previous work has attempted to capture 508 asymmetry in cardiovascular networks and plants [Brummer et al., 2017]. Another 509 major goal of our future work is applying this theoretical framework to look at 510 branching of neuron processes, and using branching properties related to asymmetry to 511 compare different cell types.

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Beyond the scaling ratios for successive branches in the individual neuron processes, 513 it is interesting to consider allometric scaling relationships of species size and functional 514 properties. Previous work on cardiovascular networks has extracted an allometric 515 scaling relationship that relates species size (or mass) with volume [Savage et al., 2008]. 516 Moreover, previous work on scaling has shown an allometric scaling relationship 517 between single cell neurons and animal body mass [Savage et al, 2007], and when brains 518 grow in size, they require more extensive axonal trees to traverse greater distances 519 [Bekkers and Stevens, 1990]. Building on these ideas from our theoretical formulation of 520 the objective function that minimizes power subject to the constraint of fixed 521 conduction time delay, we were able to extract a functional scaling relationship between 522 species size and time delay for unmyelinated fibers. We derived that there is a mixed thus improving the efficiency of information processing. 534 We conclude that neuron function places profound constraints on neuron   This material is based upon work supported by the National Science Foundation