Geometrical factors determining dendritic domain intersection between neurons: a modeling study

Overlap between dendritic trees of neighboring neurons is a feature of nervous systems. Overlap allows neurons to share common afferences and defines the topography of the circuit they belong to. Proximity is also a requirement for dendritic communication, including dendro-dendritic synaptic contacts. We simulated overlap dynamics between pairs of ventral tegmental area dopamine neurons, a population characterized by diverse and extensive dendritic domains. Using each neuron’s 3D convex hull (CH) as a proxy for dendritic domain size and shape, we examined intersection versus cell-to-cell distance curves for 210 pairs of neurons, and found that decay dynamics were diverse and complex, indicating that intersection between real dendritic domains does not comply with spherical or isotropic shape assumptions. We re-examined intersection dynamics for the same 210 pairs, but this time with either: a) a normalized volume corresponding to the average volume of all CHs, b) normalized shapes, using an average CH shape, or CHs from neurons exhibiting average dendritic distribution, isotropy or morphology, or c) a normalized cell body position, which we artificially placed in each CHs’ centroid. All three interventions significantly increased pair intersection and simplified decay dynamics, yet shape normalization had the strongest influence. Critically, shape uniformity was also the most relevant factor for increased dendritic domain pair intersection using the neurons’ original brain position and distances. We applied this experimental approach to other populations using reconstructions from neuromorpho.org database and found that shape and intersection dynamics were cell-type dependent. We conclude that intersection between neurons is not only maximized by proximity, but that individual-specific dendritic domain geometries have a profound impact too. The results predict that one biological solution for circuits requiring selective connectivity is to exhibit greater heterogeneity in size, cell body location, and especially shape, among their constituent neuronal elements.

examined intersection versus cell-to-cell distance curves for 210 pairs of neurons, and found that 23 decay dynamics were diverse and complex, indicating that intersection between real dendritic 24 domains does not comply with spherical or isotropic shape assumptions. We re-examined 25 intersection dynamics for the same 210 pairs, but this time with either: a) a normalized volume 26 corresponding to the average volume of all CHs, b) normalized shapes, using an average CH 27 shape, or CHs from neurons exhibiting average dendritic distribution, isotropy or morphology, or 28 c) a normalized cell body position, which we artificially placed in each CHs' centroid. All three 29 interventions significantly increased pair intersection and simplified decay dynamics, yet shape 30 normalization had the strongest influence. Critically, shape uniformity was also the most relevant 31 factor for increased dendritic domain pair intersection using the neurons' original brain position 32 and distances. We applied this experimental approach to other populations using reconstructions 33 from neuromorpho.org database and found that shape and intersection dynamics were cell-type 34 dependent. We conclude that intersection between neurons is not only maximized by proximity, 35 but that individual-specific dendritic domain geometries have a profound impact too. The results 36 predict that one biological solution for circuits requiring selective connectivity is to exhibit 37 3 greater heterogeneity in size, cell body location, and especially shape, among their constituent 38 neuronal elements. 39 40

41
Overlap between the dendritic trees of adjacent neurons is a common feature of the nervous 42 system (Harris & Spacek, 2017;Nieuwenhuys et al., 1998;Ramon y Cajal, 1904). Neurons whose 43 respective dendrites share a given locus are likely to share common afferent inputs (Braitenberg 44 & Schüz, 1998;Packer et al., 2013;Peters & Feldman, 1976). Also, the topographical organization 45 of a projection system is partly defined by the degree to which spatially adjacent neurons share 46 or exclude sensory or afferent inputs (Nieuwenhuys et al., 1998). The role of dendritic fields' 47 overlap may be particularly important in brain structures such as the neocortex of large 48 mammals, which in large-brain exhibit a trend to decreased neuronal density (neuronal cell body 49 density) and a more extensive neuropil (Herculano-Houzel, 2011). In addition, proximity also 50 ensures dendro-dendritic paracrine transmission (Brombas et al., 2017;Rice & Patel, 2015) and 51 it is a requirement for establishing dendro-dendritic synapses (Groves & Linder, 1983;Hinds, 52 1970). The role of dendritic overlap between adjacent neurons in determining integration or 53 segregation of afferent signals may play is also expected to be critical in species in which neuronal 54 density (neuronal cell bodies) decreases and neuropil increases, as it happens in the neocortex 55 of large-brain mammals (Herculano-Houzel, 2011) 56 57 Dendritic domains overlap has been studied at a population level in somatosensory and visual 58 systems in the context of neurons' receptive field development (Jan & Jan, 2010;Lefebvre et al., 59 6 To address these questions, we studied adult mouse ventral tegmental area (VTA) dopamine (DA) 85 neurons, a population involved in reinforcing behavior, and whose neurons are morphologically 86 characterized by long radiating and overlapping dendrites (Montero et al., 2021) that sub-serve 87 integration of multiple and heterogeneous input sources (Geisler & Zahm, 2005). We used the 88 neurons' convex hulls polyhedrons (CHPs) generated from previously 3D reconstructed VTA DA 89 neurons (Montero et al., 2021). CHPs enclose the minimal convex volume in which the dendritic 90 domains locate and have been used as proxies for dendritic domain size and shape (Felix et  Consequently, in this study we set the following aims: 1) to test the assumption that CHPs are 95 adequate proxies for dendritic domain size and shape and, similarly, that convex hulls 96 intersections are adequate proxies for dendritic domain and proximity, 2) to model the influence 97 of cell body to cell body distance on intersected volume in the mediolateral (ML), dorsoventral 98 (DV) and anteroposterior (AP) axes, 3) to model the influence of CHPs size, shape and cell body 99 placement inside the dendritic domain, on intersection, 4) to test whether results are replicable 100 when examining natural pair intersection using the original placement of neurons inside the brain 101 10 form a facet and the volume of the CHP. The volumes of CHPs obtained using convhulln coincided 154 with the volumes reported for CH by Neurolucida Explorer (MBF Bioscience). 155

156
Obtaining the volume of the intersected polyhedron required a transformation of the way the 157 CHPs were encoded. For that, using MATLAB, we started from the original CHP and obtained the 158 vertices. By using the linspace function, vertices forming each CHP facet were joined using 1000 159 points per CHP edge. Thus, we ended with a transformed CHP defined by points lined up along 160 edges. To obtain the intersected polyhedron between pairs of CHPs, we proceeded as follows: 161 using the inhull function (D'Errico, 2021), the points from each transformed CHP edge that are 162 inside the other CHP of the pair were obtained, and then the convhulln function was used to 163 obtain the intersected CHP and the intersected volume. 164 2.4.1 Intersection ratio 165 It was reasoned that because the intersected volume depends strongly on the volume of 166 individual neurons CHPs (for instance, the intersected volume of two large but minimally 167 intersected neurons could be the same as the intersected volume of two small but largely 168 intersected neurons), it was necessary to find a ratio that normalized for the volume. As such, it 169 was decided to work with an intersection ratio (IR), that is defined as follows: 170 We decided to normalize by the non-intersected volume, and not by the total added volume of 173 both individual neurons, so that the ratio approaches 100 for maximally overlapping CHPs. In any 174 event, the results in the vast majority of analyses did not vary between ratios using either the 175 non-intersected volume or the total added volume as divisors (not shown), nor when total 176 intersected volume was used. 177

Dendritic length and nearest neighbor inside intersected volume
178 First, using the Segment points-dendrites data from Neurolucida Explorer in MATLAB, each 179 neuron dendritic segment was resampled, in order that the distance between dendritic segments 180 were equal to 0.001 µm (new dendritic segments). Then, after determining the intersected 181 volume between pairs of CHPs, the inhull function was used to search which new dendritic 182 segments from each neuron where inside the intersected volume. Following, the dendritic length 183 inside the intersection (in µm) corresponded to: number of new dendritic segments (inside the 184 intersected CHP) x 0.001. 185 After determining which dendritic points were inside the intersected volume, the nearest 186 neighbors between the new dendritic segments of the pair of neurons were determined using 187 the knnsearch function. Then, the nearest neighbor from the pair of neurons was the minimum 188 value from the output matrix from knnsearch. 189

190
The following methods were developed using custom MATLAB scripts. For each of these 191 methods, IR was calculated at increasing distances between cell bodies (i.e., cell bodies of each 192 neuron, except for normalized cell body, see below), from 0 to 400 µm, at 10 µm steps. One 193 neuron of the pair was moved at these distances across the ML axis, then the DV axis, and finally 194 the AP axis. Any single ray will intersect the facets of the two CHPs at distances d1 and d2. Then, a point at 207 mid-distance between d1 and d2 is defined, and the exercise is repeated for all rays emanating 208 from the apex. A new polyhedron will be created by all points located at mid-distance between 209 the facets of both CHPs at every solid angle. This procedure can be projected to a larger number 210 of neurons CHP, in which case the points of the new polyhedron will correspond to the average 211 of all n distances at which rays intersect the facets of all CHPs in a given ray direction. The isotropy index was used to estimate how different from a sphere each CHP was. Each neuron 241 CHP was centered in the origin (coordinate 0,0,0), using the cell body as the reference point. 242 Then, a sphere, that had the same volume as the CHP, was also centered in the origin (in relation 243 to the sphere centroid). Following, the intersection volume was obtained. The isotropy index 244 correspond to the ratio between the intersected volume and the CHP volume. To obtain the 245 average isotropy index neuron, the median of isotropy index from the population was calculated. 246 For the average morphology neuron, the following parameters obtained from Neurolucida 247 Explorer were used: dendritic length, dendritic tree number, maximum dendritic order, number 248 of segments and convex hull volume. For each parameter, the absolute deviation from the mean 249 was obtained (absolute value of the subtraction between each data and the mean of the 250 corresponding parameter). Then, for each neuron, the sum of the absolute deviations from the 251 mean was calculated. The neuron with lower value corresponded to the average morphology 252 neuron. 253 For average wedge neuron, the dendritic wedge data for the 4 anterior and 4 posterior quadrants 254 of each neuron was obtained. Then, the absolute deviation from the mean was calculated for 255 each quadrant. Following, for each neuron, the sum of the absolute deviations from the mean 256 was calculated. The neuron with lower value corresponded to the average wedge neuron. We found that for 84% of the pairs, nearest neighbor distances lower than 25 µm were observed. 306 Moreover, a significant correlation between the intersection volume and the dendritic length 307 inside the intersection was found ( Figure 2B3, r = 0.9685, p < 0.001, Spearman correlation). 308 Overall, these results show that the intersection of neurons CHPs is a representative proxy of the 309 intersection between dendritic fields. 310 311 3.2 CHP intersection dynamics: modeling the influence of CHP volume, 312 shape and cell body position 313 Following method validation, we evaluated how the intersection between pairs of neurons CHPs 314 changes when the distance between CHPs increases ( Figure 3A). We used an intersected ratio 315 (IR, more details in methods) as a normalized index of intersection between pairs of neurons 316 CHPs. Then, IR values were obtained from pairs of VTA DA neurons, first with their cell bodies 317 aligned (distance between cell bodies = 0 µm) and then after one of the neuron CHP was moved 318 across the ML, DV or AP axes (in 10 µm steps, until 400 µm). In Figure 3B we show some 319 representative examples of IR vs distance curves. We found a high variability in IR as a function 320 of distance. For instance, in Figure 3B (ML), the green curve shows a pair of CHPs whose IRs first 321 increased, then reached a peak, and then decreased, in contrast to the magenta and red curves, 322 whose IR values only decreased. These results indicate that IR between neuronal pairs depends 323 on the distance between their cell bodies, but also on other geometrical factors. 324 To understand how the differences in geometry between neuronal pairs affect intersection, we 325 separately modeled the influence of neurons CHP volume, shape, and cell body position ( Figure  326 4, see methods) by running the same previous simulations (as in Figure 3), but this time with all 327 pairs having either a normalized volume, a normalized shape or a normalized cell body position. 328 More precisely, we first transformed all CHPs to a normalized volume, which was equal to the 329 population mean for VTA DA neurons (0.047 mm 3 ). In Figure 4A,B we show a simpler description 330 of the procedure using only two neurons CHPs. Their respective volumes are 0.069 mm 3 and 331 0.023 mm 3 ( Figure 4A), therefore one of them decreases its volume to the average value ( Figure  332 19 4B, neuron CHP 14), and the other increases its volume to the average value ( Figure 4B, neuron 333 CHP 3). Note that the CHP shape remains unchanged with this procedure. 334 To examine the effect of differences in CHP shape on IR we reasoned that, similarly, this could be 335 assessed using a normalized shape from the population of CHPs. We set to find a CHP that we 336 defined as the CHP of the polyhedron formed by the points located at average distances from 337 the surfaces of all CHP, with their cell bodies aligned. After the normalized CHP is defined, its 338 volume can be modified to the volume values of the original neurons CHPs. In Figure 4C we show 339 a simpler description of the procedure using only two neurons CHPs. The CHPs of the previous 340 pair now were combined to generate a normalized shape following the averaging rules already 341 explained ( Figure 4C, top row, more details in Supplementary Figure 1 and methods). Then, the 342 new normalized CHP was given back the original CHP volumes ( Figure 4C, bottom row). 343 The normalized shape of all VTA DA neurons is shown in Figure 5A. As it can be appreciated, the 344 average shape exhibits the essential geometrical features of this dopaminergic population, 345 particularly the dendritic domain size and the extension in the ML, DV and AP axes. In frontal 346 view, we observe that the CHP was oriented dorsomedially to ventrolaterally, with the cell body 347 at mid position, in agreement with the distribution of VTA neurons dendritic field, as described 348 in (Montero et al., 2021). 349 Finally, we took into account the fact that, as it can be noted in Figure 1B, the cell bodies of real 350 VTA DA neurons are normally displaced from the center of the dendritic domain. We considered 351 this a relevant factor for intersection and therefore defined a third normalization procedure, in 352 which the cell body of each CHP was moved to the CHP' centroid. Again, in Figure 4D we show a 353 simpler description of the procedure using only two neurons CHPs (more details in methods). 354 20 We applied the modeling procedures described above to our population of VTA DA neurons 355 ( Figure 6). In each condition and in the ML, DV or AP axes, all the possible pairs of VTA DA neurons 356 were analyzed (210 pairs in total, using repeating pairs given that a neuron "a" in a pair is first 357 tested moving away from neuron "b", and then neuron "b" is tested moving away from neuron 358 "a"). Compared to the original data (no changes in CHP geometry), a significant increase in IR was 359 found in the normalized volume condition across the ML ( Figure 6A1), DV ( Figure 6A2) and AP 360 ( Figure 6A3) axes (p < 0.05 using Wilcoxon signed rank test with Sidak multiple comparisons 361 correction, more details of statistical differences between conditions in Figure 6A), but the 362 magnitude of change in IR was low. For the normalized cell body position condition, significantly 363 higher IR values were observed in the three axes, compared to the original data and also to the 364 normalized volume condition ( Figure 6A1-3). However, the modeling condition that showed the 365 largest effect on IR values across each axis, was the normalized shape condition ( Figure 6A1-3). 366 These results indicate that differences in IR values between pairs of VTA DA neurons CHPs are 367 influenced, in decreasing order, by the CHP shape, cell body position, and volume. 368 To give further sustain to the results describing the influence of shape normalization on 369 intersection (see discussion), we used CHPs from neurons that exhibited average values for 370 morphology, isotropy and dendritic distribution characteristics, as follow: 1. average isotropy 371 neuron: for each neuron CHP, an isotropy index was calculated to determine how much a CHP is 372 similar to a sphere of the same volume (more details in methods); then the median population 373 value was found (0.46), and the neuron with the closest value was selected (in this case neuron 374 7 ( Figure 1B)); 2. average morphology neuron: using the dendritic length, dendritic tree number, 375 maximum dendritic order, number of segments and convex hull volume, we found the neuron 376 21 with the lower absolute deviation of the mean from the sample (more details in methods), which 377 corresponded to neuron 14 ( Figure 1B); 3. average dendritic distribution neuron: using the 378 dendritic wedge data for the 4 anterior and 4 posterior quadrants ( Figure 2A3), we found the 379 neuron with the lower absolute deviation from the mean from the sample (more details in 380 methods), which corresponded to neuron 1 ( Figure 1B). Each average neuron was then analyzed 381 using the same approach as for the normalized shape analysis: starting with one of the three 382 average neurons, we created 15 new CHPs, all with the average neuron CHP shape, and then 383 gave to each new CHP the respective 15 volumes from the VTA DA neurons population. And then 384 determining IR values between 0 to 400 µm (10 µm steps) across the ML, DV and AP axes. 385 Figure 6B shows the IR curves for average isotropy neuron, average morphology neuron and 386 average dendritic domain neuron. Each condition was compared with the original data analysis 387 across each axis. All three average neurons showed significantly higher IR values than the original 388 condition (p < 0.05 using Wilcoxon signed rank test with Sidak multiple comparisons correction, 389 more details of statistical differences between groups in Figure 6B) and almost identical 390 magnitudes to that of CHP shape normalization in 6A. 391 We wanted to further examine the diversity of distance versus IR curves shown previously ( Figure  392 3), this time focusing on the types of decay curves observed. For the original data, we found that 393 51% of the curves for all pairs and axes tested, described an initial decay ("decay curves", as the 394 magenta or red curves in Figure 3B, ML), whereas 49% of curves exhibited an initial rise, a peak, 395 and then a decay ("peak curves", as the green curve in Figure 3B, ML). After volume 396 normalization, the proportion of decay and peak curves remained similar to the original data 397 (51% vs 49%). Interestingly, in the normalized cell body condition, an increase in decay curves 398 22 cases was observed (57%, versus 43% of peak curves). Yet the larger effect was found in the shape 399 normalization condition, where peak curves disappeared and thus 100% of the curves ended as 400 decay curves. Overall, these results are in line with the previous findings of this section, 401 highlighting that CHP shape is the main factor that contributes to the variability in intersection 402 dynamics. 403  Figure 5B shows the normalized shape of these neuronal populations. 411

CHP intersection dynamics: examples from other neuronal types
We first computed the IR dynamics of the neuronal CHPs (original data) of each region, and 412 qualitatively examined differences between cell types ( Figure 7A). MN neurons show higher IR 413 values in the three axes, followed by MD, SNc rat and mouse neurons, and PC cells, although each 414 population shows specific differences in each axis. 415 As with VTA neurons (Figure 6), we modeled CHP geometry for PC, SNc, MD and MN neurons 416 populations, using the normalized volume, normalized shape and normalized cell body position 417 conditions ( Figure 7B). We show the most representative axis of each neuronal population. As in 418 the original condition analysis (Figure 7A), mouse and rat DA neurons showed similar curve 419 profiles after the application of each modeling conditions in the ML axis ( Figure 7B1-2). 420 23 Specifically, in both SNc mouse ( Figure 7B1) and rat ( Figure 7B2) analysis, normalized volume 421 showed increased IR values than the original condition (p < 0.05, Wilcoxon signed rank test with 422 Sidak multiple comparisons correction, more details of statistical differences between conditions 423 in Figure 7B1-2), but the magnitude of change in IR is low. The normalized cell body location 424 modeling also showed significantly higher values than the original and also the normalized 425 volume conditions in both SNc mouse and rat neurons, but the magnitude of change was higher 426 in SNc rat. The largest effect for both mouse and rat SNc DA neurons was observed in the 427 normalized shape condition, which was significantly higher than the other three conditions. 428 These results were very similar to those found with VTA DA neurons ( Figure 6A), indicating a 429 common underlying dendritic organization. 430 Interestingly, modeling conditions applied on MD neurons only significantly modified IR values 431 on short distances on the ML axis (p < 0.05, Wilcoxon signed rank test with Sidak multiple 432 comparisons correction, more details of statistical differences between conditions in Figure 7B3). 433 These results indicate that neither volume, shape or position of the cell body is relevant to explain 434 the intersection dynamics in MD neurons. 435 For MN, no significant differences were observed between conditions, although normalized 436 shape showed a trend of higher values than the other conditions ( Figure 7B4). The lack of 437 significant differences could be related to the low sample of neurons in this region (3 neurons, 6 438 pairs analyzed at each distance). 439 Finally, for PC, we first show the ML axis as a positive control for our results, considering the 440 mainly 2D structure of the PC dendritic field, and minimal extension in the ML axis. In this sense, 441 the modeling conditions only showed an effect at a distance between cell bodies of 0 µm, with a 442 24 fast decrease of IR values at 10 µm ( Figure 7B5'). A different result was found at the AP axis 443 ( Figure 7B5''), a more biologically relevant condition, considering the default sagittal 444 arrangement of these neurons in the cerebellum. A significant difference between the original 445 and normalized volume condition was observed, but the magnitude of the difference was low (p 446 < 0.05, Wilcoxon signed rank test with Sidak multiple comparisons correction, more details of 447 statistical differences between conditions in Figure 7B5''). Higher IR values were found after 448 normalizing cell body location and shape, until 50 µm. In summary, these results indicate that 449 intersection dynamics appear to be cell-type specific. 450

451
As a final approach to characterize the uses of the CHP intersection method for neurons, we again 452 used our sample of VTA DA neurons, but now to analyze IR dynamics using the real positions of 453 the neurons in the region (as determined in (Montero et al., 2021)). As seen in Figure 8A Figure 8B4). For the original data, high variability in IR values across distance was found 459 ( Figure 8B1), as data show a low R 2 for an exponential decay non-linear regression (R 2 = 0.4231). 460 A similar IR profile was found in the normalized volume condition (Figure 8B2), confirming a low 461 effect of volume variability in intersection dynamics (as also found in Figure 6A), with an even 462 lowest R 2 for an exponential decay fit (R 2 = 0.3232). Interestingly, shape and cell body location 463 normalization drastically changed the data curve profile and reduced the variability of IR values 464 25 across distances ( Figure 8B3-4), as evidenced by a higher exponential decay fit (normalized 465 shape: R 2 = 0.8373; normalized cell body: R 2 = 0.7595). Finally, we analyzed the IR values of the 466 closer 50% (0 to 340 µm, n = 53 pairs, Figure 8C1) and farther 50% (340 to 807 µm, n = 52 pairs, 467 Figure 8C2) of distances between cell bodies. For the closer half of distances ( Figure 8C1), we 468 found significantly higher IR values in the normalized shape conditions compared to the original 469 data (p < 0.001, Friedman test followed by a Dunn's multiple comparisons test). On the other 470 hand, for larger distances (Figure 8C2), original and normalized volume conditions were 471 significantly higher than both normalized shape and cell body conditions (details in Figure 8C2). 472 Overall, the results using real neurons positions in the VTA confirm the principles described in 473 appropriate proxy, given that the vast majority of space within a neuronal CHP is not occupied 488 by dendritic segments. And, similarly, whether intersection between CHPs reflects dendritic 489 domain intersection in any biologically meaningful way. We examined the shape assumption by 490 testing whether CHP and dendritic length space distribution were comparable. We first 491 confirmed that larger CHP does indeed contain more dendritic length ( Figure 2A1). Then we 492 examined the spatial distribution of CHP volume and dendritic length and found that they were 493 indistinguishable from each other, as tested using wedge analysis in Figure 2A3. These results 494 support the contention that CHPs are adequate proxies for dendritic distribution. 495 We then tested the CHP intersection assumption by studying the proximity between the 496 dendrites from two neurons located inside the intersected volume. We found that the median 497 nearest neighbor distance between dendrites from different neurons inside the intersected 498 volume was 4.6 µm, and that in almost 85% of the cases, at least one pair of dendrites located at 499 less than 25 µm ( Figure 2B2). To put these results in perspective, a 25 µm distance is similar to 500 the longest axis of DA neurons cell bodies (see Figure 1 in (Meza et al., 2018;Montero et al., 501 2021)). Finally, we determined that the total length of shared dendrites inside the intersected 502 volume tightly correlates with the CHPs intersected volume ( Figure 2B3). These data support the 503 assumption that CHP intersection is an appropriate proxy for the intersection of neuronal 504 dendritic fields. 505 Having confirmed these assumptions, it is important to stress, however, that CHP does not 506 represent and are therefore not useful to describe other aspects of dendritic organization, such 507 as dendritic density or architectural complexity inside the CHP, as CHPs only considers the outer 508 limits of a dendritic domain. Thus, other systematic modeling approaches must be used if one 509 wants to study the influence of dendritic density or architecture on intersection. We consider 510 those possible approaches, which we imagine would require combining methods that model 511

515
In order to study the influence of shape on intersection, an additional challenge in this study was 516 to find a CHP whose shape best represented an otherwise diverse population. We took the direct, 517 though methodologically more challenging approach, to compute the average 3D CHP shape 518 from the population of 15 neurons CHPs. This approach allowed us to model intersection using a 519 28 normalized shape, while maintaining the original volume and cell body position of CHPs. Now, 520 because the resulting average CHP is smooth and ovoid-shaped, one reasonable concern is that 521 the observed increases in intersection are merely due to these characteristics, and not due to the 522 use of a normalized shape To address this issue, we chose three CHPs whose respective neurons 523 exhibited average morphology (Montero et al., 2021), isotropy (see methods), and spatial 524 dendritic distribution ( Figure 2A3 and methods). In this way, we could use CHPs with average 525 characteristics, but whose CHP maintained its pointed, original shape, thus discarding an artificial 526 influence from the smother, ovoid-shaped average CHP. 527

Role of distance, volume, cell body location and shape on
528 intersection 529 Our results show that in addition to cell-to-cell distance predictably decreasing intersection, 530 distance versus intersection relationship was not straightforward and had to depend on other 531 geometrical factors. Indeed, intersection was affected by the relative position of each neuron to 532 its pair (i.e. whether one locates besides, above/below, or in front/behind from the other). This 533 is an expected consequence of the dendritic domain of neurons not being spherical (i.e. in that 534 maximal extensions in ML, DV or AP axes differ from each other). The location of the cell body, 535 which we used to identify each neuron and showed displacement from the dendritic domain's 536 centroid, was also a relevant factor influencing intersection dynamics. Yet perhaps the most 537 noteworthy finding of this study, however, is that dendritic domain intersection is significantly 538 increased by shape homogeneity between neurons (Figure 6-8). Or, said conversely, that 539 29 dendritic domain exclusion between adjacent neurons is significantly increased by heterogeneity 540 in dendritic domain size and shape. 541 We found that individual pairs distance vs intersection decay dynamics were diverse. Among 542 them, some curves exhibited a peak, i.e., an initial rise followed by the expected decay. We found 543 this result interesting because it challenges the intuitive assumption that neurons whose cell 544 bodies locate next to each other should have a maximal chance of sharing afferents inputs. We 545 initially assumed that this specific dynamic (peak curves) mostly resulted from the displaced 546 location of the cell bodies within their respective CHP, resulting in dendritic domains sharing little 547 volume with their cell bodies aligned (for instance, in a hypothetical scenario in which two 548 Purkinje-like cells had their cell bodies aligned, but their respective dendritic domains pointing in 549 opposite directions). Interestingly, however, although cell body position normalization 550 diminished the proportion of peak curves, shape normalization completely eliminated them, 551 again emphasizing the critical role of CHP shape in determining intersection dynamics. 552

553
The principles that underlie intersection ultimately depend on the size and spatial organization 554 of VTA DA dendrites. As detailed in (Montero et al., 2021), the dendritic tree of individual mouse 555 VTA DA neurons conforms to early descriptions (Kline & Felten, 1985;Phillipson, 1979;Ramón-556 Moliner & Nauta, 1966) characterizing them by long radiating dendrites, low branching patterns, 557 and overall conforming to the "isodendritic" cell type typical of reticular neurons (Jones, 1995;558 Ramón-Moliner & Nauta, 1966). 559 The mechanisms involved in specification of VTA neurons dendritic trees have not been studied 560 in detail yet (Brignani & Pasterkamp, 2017). We speculate that dendritic morphogenesis of VTA 561 30 DA neurons is likely to rely on the catenin signaling pathway, which plays a critical role in VTA DA 562 neurons neurogenesis and migration (Tang et al., 2009) and, as studied in other brain regions, is 563 essential for dendritic growth and branching (Arikkath, 2009). On the other hand, given the highly 564 overlapping pattern of VTA DA neurons dendritic trees, there must also exist mechanisms that 565 allow the co-existence of dendrites from neighboring cells that belong to the same cell class 566

614
The broad spectrum of dendritic domain arrangements among neuronal types has been 615 conceptualized as representing different types of connectivity, either of the selective, sampling, 616 or space filling type (Harris & Spacek, 2017). The former type is characterized by small dendritic 617 fields occupying a restricted locus, and represents an efficient solution to relay nervous signals 618 from one neuronal stage to another as parallel, separate channels. The sampling strategy (and to 619 some extent the space filling's) connectivity strategy, on the other hand, is characterized by 620 broader dendritic fields encompassing different loci, and represents a solution to increase 621 integration and redundancy in signal transferring (Harris & Spacek, 2017). Our results allow us to 622 predict that, for a given number of neurons and a given dendritic length, systems that require 623 parallel and exclusive transferring of nervous signals may accomplish it by promoting 624 heterogeneity in size, cell body location and specially shape of dendritic domains. Conversely, 625 33 systems that require signal integration and redundancy may accomplish it by promoting 626 uniformity in dendritic domain size, cell body location and shape. 627 628 34 ACKNOWLEDGMENTS 629 We thank Luciana López-Juri for her insightful comments during the progress of this work. This