Spatial segregation between populations of ponto-cerebellar neuronsStatistical analysis of multivariate spatial interactions.код для вставкиСкачать
THE ANATOMICAL RECORD 231:510-523 (19911 Spatial Segregation Between Populations of Ponto-Cerebellar Neurons: Statistical Analysis of Multivariate Spatial Interactions JAN G. BJAALIE, PETER J. DIGGLE, ALFEO NIKUNDIWE, TEVFIK KARAGULLE, AND PER BRODAL Department of Anatomy, Institute of Basic Medical Sciences, Oslo, Norway (J.G.B., A.N., T.K., P.B.), and Mathematics Department, School of Engineering, Computing, and Mathematical Sciences, Lancaster University, Lancaster, United Kingdom (P.J.D.) ABSTRACT This study applies terms and methods for describing spatial interactions between multivariate spatial point patterns, which are, to our knowledge, new in neurobiology. We consider two categories of points, type 1 and 2, distributed within a certain reference volume (such as a nucleus of the brainstem or a cortical area). The points may, for example, represent different categories of labelled cells or axonal fields of termination. We say that there is spatial neutrality between points of type 1 and 2 if the types are signed by random labelling. If a mechanism drives the two point categories together, we say that the point patterns are positively associated. Conversely, if a mechanism drives type 1 and 2 points apart, we say t h a t they are segregated. By comparing two cumulative distribution functions of distances between points, we can distinguish neutrality, positive association, and segregation. One function, H,,(t), is the cumulative distribution function of the distance t between a pair of randomly selected points of type 1 and 2. The other, H,,(t), is the corresponding function for a pair of points randomly selected without reference to type. Plots of the estimated difference between these two functions give a n indication of positive association, neutrality, or segregation. A statistical test, based on simulations of random (neutral) distributions, can be used to see whether deviations from neutrality are significant. We apply the analysis described above to a major pathway of the brain, namely the ponto-cerebellar projection. Different types of cells in the pontine nuclei are retrogradely labelled with the fluorescent tracers Rhodamine-B-isothiocyanate, Fluoro-Gold, and Fast Blue. The tracers are injected in adjacent or more distant folia of the cerebellar paraflocculus. The location of the somata of labelled cells are recorded and the total distribution reconstructed in three dimensions and displayed on a dynamic graphics workstation. We ask whether different units (folia) in the paraflocculus receive information from the same population, from two different positively associated populations, or from segregated cell populations. We find a statistically significant tendency for cell populations projecting to adjacent folia to be positively associated, although there are few cells containing multiple labels. Populations of neurons projecting to folia wider apart are significantly segregated. From inspections of the reconstructions, using real-time rotations, we find that the swarms of labelled neurons tend to accumulate in shells or lamellae in the pons. Within the lamellae, the cells are aggregated in clusters and bands with empty holes (containing unlabelled ponto-cerebellar cell bodies, presumably projecting to other cerebellar targets) in between. By determining the average distance to a reference plane for each cell population, we find that cell populations shift in a ventro-medial direction as the injection sites move from the medial part of the dorsal paraflocculus toward the lateral part and into the ventral paraflocculus. We therefore conclude that there is a continuous shift in location of pontocerebellar cell populations, corresponding to specific shifts in cerebellar target regions. To understand the wiring patterns of the brain in functional terms depends on comprehensive and reliable of the topological arrangement Of neurOnS and terminal axonal fields. In traditional neuroanatomy, neuronal somata or fields of termination 0 1991 WILEY-LISS, INC. Received January 30, 1991; accepted March 22, 1991. Address reprint requests to Jan G. Bjaalie, Dept. of Anatomy, University of Oslo, P.O. BOX1105 Blindern, N-0317 Oslo 3, Norway. MULTIVARIATE SPATIAL POINT PATTERNS (the position of which we refer to as events) are presented as points in two-dimensional drawings of tissue sections. The shape and arrangement of individual categories of points and their spatial interactions and relationships are then described on the basis of such drawings, usually in a qualitative manner. With the use of computer technology and three-dimensional reconstructions, however, i t is possible to improve the visualization of shape and arrangement. Furthermore, on the basis of a file of data, one may even employ formal statistical tests, e.g., to distinguish random and clustered patterns (cf. Bjaalie and Diggle, 1990; Bjaalie et al., 1990). Very little has been done, however, on quantitative estimations of the spatial interactions between different categories of events. Currently, the terms overlap and segregation (or separation) are often used to describe spatial relations (see, e.g., Bjaalie and Brodal, 1989; Goldman-Racik and Selemon, 1990). When events of type 1 and 2 occupy the same space in some parts of the reference volume, we usually say t h a t overlap occurs. When they do not occupy the same space, we say t h a t the two types of events are segregated. This terminology, however, is not very precise. For example, if there is spatial neutrality between two point patterns, both overlap and segregation (in the sense used above) may nevertheless occur. Thus “overlap-by-chance’’ is different from overlap caused by a mechanism driving two types of events together in space. We introduce the term positive association to describe the tendency for different point patterns to be located together in space. Segregation between clustered point patterns may also occur simply because, by chance, clusters of different categories fail to overlap. In contrast, a mechanism during development may be responsible for segregation. Mathematical analyses are required both to distinguish “overlapby-chance’’ from positive association, and “segregationby-chance” from “real” segregation between two point patterns. Along the same lines of reasoning, we need measures of the degree of positive association or segregation. The present study aims at establishing more precise terms as well as procedures for quantitative analyses of spatial interactions in two- and three-dimensional space. We have initially applied such analyses on the cortico-ponto-cerebellar system, which provides the quantitatively dominating input to the cerebellum (for reviews on basic organization, see Brodal 1982, 1987; Brodal and Bjaalie, 1987). In this article, we study the amount of positive association or segregation between well-defined populations of ponto-cerebellar neurons. Thus we consider the two- and three-dimensional organization of the neurons of origin of the projection to a limited and macroscopically well-defined region of the cerebellum-the paraflocculus (see, e.g., Larsell, 1970). The paraflocculus has a regular folial pattern, and the dorsal and ventral paraflocculus are continuous. It is thus ideally suited for a comprehensive and quantitative study on how the distribution of pontocerebellar cell somata varies with the location of their cerebellar target region. We ask whether different units of the paraflocculus (single folia or smaller) receive information from the same populations, from positively associated populations of ponto-cerebellar neurons, or if not, to what de- 511 gree they are segregated. To this end, we inject two or three different retrograde fluorescent tracers into separate folia. A priori, we assume that a high frequency of double- or triple-labelled neurons together with positive association between differently labelled cells in the pontine nuclei indicate that the same information is conveyed to the various folia injected. The converse would be a situation with few or no double- or triple-labelled neurons, and with spatially segregated populations, indicating the existence of separate channels for the transfer of information from the pontine nuclei to the cerebellar cortex. Our hope is that such quantitative information and the use of more precise terminology to describe spatial interactions may provide new insight that in t u r n will improve our functional understanding of the connectivity of the brain. EXPERIMENTAL PROCEDURES This study employed 15 adults cats of both sexes, ranging from 1.9 to 5.1 kg in weight. Under deep pentobarbital anaesthesia (30 mglkg body weight, administered intraperitoneally), supplemented with 0.2-0.5 ml of Hypnorm (Janssen), the posterolateral quadrant of the cerebral hemisphere was removed to expose the cerebellar paraflocculus. Aqueous solutions of fluorescent tracers, 2% Rhodamine-B-isothiocyanate (Sigma), 2% or 6% Fluoro-Gold (Fluorochrome Inc.) and 2% Fast Blue (Sigma), were injected into different folia of the dorsal and ventral paraflocculus with a Hamilton syringe fitted with a glass micropipette. Small amounts of the tracers (0.03-0.10 pI) were deposited. A summary of the volumes and types of tracers used in the individual animals is shown in Table 1. Injection sites are presented in Figures 1 and 2. After survival periods of 6-7 days, the cats were anaesthetized and perfused transcardially with 1 litre of warm physiological saline, followed by 2 litres of warm 4% paraformaldehyde in 0.1 M phosphate buffer (pH 7.4f, and finally with 1 litre of buffered cold 10% sucrose. The brain was left in situ for 2-3 hours a t room temperature to allow further fixation. It was then dissected out and the cerebellum was photographed and inspected macroscopically for obvious signs of the injection site. RITC leaves a red spot at the injection site, and thus the folium injected with this tracer was sketched with respect to the other folia. The brains were soaked overnight in 30%buffered sucrose at 4°C. Some cases were prepared for three-dimensional computer reconstruction of the distribution of labelled cells, and to secure perfect alignment of the sections, fiducial marker holes were drilled in the tissue block comprising the pontine nuclei. Sections were cut on a freezing microtome at 40 pm. The pons was cut transversely and the cerebellum parasagittally. ;Irnp dP FB F1 F-G RITC VP Abbreviations average perpendicular distance to the ventromedial surface of the peduncle dorsal paraflocculus Fast Blue flocculus Fluoro-Gold Rhodamine-B-isothiocyanate ventral paraflocculus 512 J.G. BJAALIE ET AL. TABLE 1. Case summaries Cat CCoL 486 489 Volume injected (pl) RITC F-G FB 0.06 0.06 0.06 0.06 impl. 490 0.06 493 0.05 495 0.05 488 impl. 0.08 0.05 0.10 0.03 498 0.06 0.09 0.06 499 0.08 500 0.06 0.08 0.06 0.08 502 0.033 0.06 503 0.03 505 0.02 510 0.03 511 0.05 0.033 0.033 0.033 0.053 0.06 514 0.04 0.063 0.04 was collected in a separate well and mounted serially for the purpose of complete reconstruction of the distribution of retrogradely labelled cells. Location The unstained sections were studied with a Leitz incident light fluorescence microscope providing excitadP1 tion light of 530-560 nm for RITC labelled neurons vP2 and 340-380 nm for F-G and FB labelled neurons. The dP distribution of retrogradely labelled pontine cells as dP well a s the boundaries of the cerebellar injection sites dP were plotted at 30 x magnification with a n x-y recorder VP attached to the microscope stage. All cells labelled with dP a given tracer, including double labelled cells, were dP plotted. VP For three-dimensional reconstructions, drawings of dP complete series of sections (or every second section) VP were aligned with the aid of the drilled holes, vessels, dP and surface contours. The location of each cell and a VP few contours of the pontine nuclei for each case were VP digitized and saved on file. The data files thus contain dP x,y,z coordinates for each cell; the z coordinate was deVP fined by the section number and thickness of the secVP tions. Since a substantial number of labelled cells (up VP to 14.078 for the RITC-500 population, cf. Fig. 1)was dP plotted and subsequently digitized, a method for didP rectly defining coordinates €or each cell without digitidP zation was also developed. With this procedure, analog dP signals from the microscope potentiometres were transdP mitted directly to the computer via a n analog to digital dP converter (for further details, cf. Bjaalie 1991). dP The reconstructions were displayed on a Tektronix dP XD88-34 3D graphics superworkstation, with the TekdP tronix 4G graphics pipeline. Different populations of labelled cells were coded with pseudo colors. Each popdP dP ulation was inspected from different angles of view, using real-time rotations in hardware, either alone or dP together with the other populations reconstructed in VP the same animal. VP The statistical methods employed are described in VP Results. VP Crus I1 'Dorsal paraflocculus. 'Ventral paraflocculus. 36%concentration; otherwise, concentration for all tracers was 2%. In most cases, the cerebellum was encased in a mixture of gelatin and 30% buffered sucrose and hardened overnight in 4% paraformaldehyde prior to sectioning on a freezing microtome (Switzer, 1979; Blackstad et al., 1981). This embedding protocol prevents the cerebellar folia from falling apart during sectioning, thereby making possible exact reconstruction of the injection sites. In cases aimed only at providing information from single sections (not computer reconstructed), adjacent sections from the pons were collected in groups of five in individual wells containing phosphate buffer. One series was mounted on chrome-alum-coated slides, air dried overnight, and cleared rapidly in xylene and coverslipped with Eukitt. Another series was counterstained with thionin, dehydrated in graded alcohols, cleared also in xylene, and coverslipped with Eukitt. In cases used for computer reconstructions (CCoL 488, 490, 495, 499, 500, and 5021, each section of the pons RESULTS Cell Distribution and Spatial Relationships Judged From 3-0 Reconstructions Altogether, 14 populations of labelled cells in the pontine nuclei were available as three-dimensional computer reconstructions. Twelve of these-two in each of cats CCoL 490,499, and 502, and three in each of cats CCoL 495 and 500-comprise large parts (from one-third to two-thirds) of the pontine nuclei. They could thus be used to visualize the structural arrangement of pontine cells projecting to single folia in the paraflocculus. Reproducible trends in spatial arrangement (clustering, banding, etc.) of the labelled cells, as well as the numerical capacity of the projections, are to be dealt with in detail elsewhere (A. Nikundiwe, J.G. Bjaalie, and P. Brodal, in preparation). Quantitative data on spatial interactions, including statistical tests on positive association and segregation as well as attempts to measure a possible segregation process, are considered below. Here, we give some comments on the information directly available by inspection of the reconstructions. Each of the reconstructed populations of labelled neurons were studied on our graphics workstation. They could thus be viewed from all possible angles. MULTIVARIATE SPATIAL POINT PATTERNS 513 511 Fig. 1. Location of injection sites in the paraflocculus (tracer-case number). For orientation, see the flattened map of the cerebellum (upper right) with the paraflocculus drawn with thick lines. Figure 3 shows a typical population of labelled neurons viewed a t seven different angles. In Figure 3C, the pons is seen from the top (rostral end), the plane of sectioning is facing the viewer. The view demonstrated in Figure 3G shows the same reconstruction rotated 90” (compared to the situation in Fig. 3C) about the x axis, so that the ventral surface of the pons is facing the viewer. The labelled cells are seen forced into their respective section planes (z coordinates were defined as section number and thickness of the sections, see above). Intermediate views, Figure 3D-F, strengthen the impression that the cells are aggregated, with holes or empty zones in between. In some places, the aggregates form snakelike patterns; in other places they are more fractionated and occur as large clusters, often interconnected with tiny strands of cells. The shape of the pontine nuclei and the corresponding curving of the swarm of labelled cells make it difficult to distinguish (on the two-dimensional pictures presented here) between real aggregation and pseudo-aggregation occurring in certain angles of view. Essential for the understanding of the arrangement of the labelled cells, however, is the view presented in Figure 3A. Here, the swarm of labelled cells appears as a narrow zone, curving along the mediolateral extent of the pontine nuclei. We call this the angle of “maximum lamellarity.” The reconstruction is seen from the rostral end of the pons, a bit from the dorsal and lateral side. Thus in conclusion, each population of labelled cells tends to be confined to a lamella. Within a lamella, the cells are aggregated into clusters and bands with empty holes in between (containing the somata of ponto-cerebellar neurons presumably projecting to other cerebellar targets). The spatial relationships between two (or more) populations of cells labelled with different tracers can to advantage be studied with the graphics workstation. Choosing different pseudo-colours to represent the different types of labelled cells, one gets a fairly good visual impression of overlap and separation between different populations in single animals. We have cho- sen to present double-labelled cells as separate points, one for each tracer. It should be noted that the number of double labelled cells were generally less than 10%.’ Thus in zones of the reconstruction where overlap between different labels are seen, most of the overlap is due to the fact that cells containing different labels are located together. Whether or not the overlap observed is simply “overlap-by-chance” or whether there is a process driving driving cells of different categories together in space (positive association), cannot easily be decided by inspection only. Interpretation of observed segregation between differently labelled cells may also pose problems. In all animals, there are clusters of spatially separated cells. In some cases, the separation is so obvious and the degree of overlap so small that there is no doubt that a segregation mechanism is responsible for the distribution patterns. In other cases, the degree of separation is smaller, and there seems to be more overlap. Consequently, it is difficult to decide whether there is a segregation mechanism operating or whether there is actually spatial neutrality between the different cell populations. In the color illustrations (Figs. 4,5), two typical cases illustrating spatial relationships between multiple populations are shown. Figure 4 illustrates a case in which adjacent folia of the dorsal paraflocculus were injected with different tracers. The two categories of labelled cells are partly overlapping and partly separated. The angle of “maximum lamellarity” (defined above) demonstrates that the two lamellae of cells are similarly located. Figure 5 shows a case in which three tracers were injected in the dorsal paraflocculus. Two of the tracers were injected in adjacent folia, whereas the third was injected many folia away from the others (cf. Fig. 1: injection sites RITC-500, F-G-500, and FB-500). A complicated three-dimensional mosaic of patches or cell clusters is seen. At the angle of ’After control injections of two tracers in the same folium, more than 80% of the labelled cells were double labelled. 514 J.G. BJAALIE ET AL. C C o L 500 C C O L 490 RlTC 0 F-G FB Fig. 2. Drawings of sections through injection sites in two animals. Note that the total extent of each injection site is always smaller than the size of a single folium, although some injections cover more than one folium. “maximum lamellarity,” the different categories of cells, also contains a n internal loose zone of cells close cells appear as partly segregated. There seems to be a to the peduncle. The need for objective measures for shift in location of the lamellae from a position fairly spatial interactions should be clear from the two examclose to the peduncle (the corticospinal and corticobul- gles demonstrated in Figures 4 and 5. Our main impression, however, from inspection of bar fiber tract) for the F-G-500 population (red dots), to a site far more ventrally and medially for the the computer reconstructions and plots of single secRITC-500 population (blue dots). It seems t h a t the tions, is that there are certain reproducible trends in third population, the FB-500, occupies a n intermedi- distribution of labelled ponto-cerebellar neurons. Popate position (yellow dots), but it is difficult to discern ulations projecting to the medial part of the dorsal its precise relationship to the other two populations. paraf locculus are located fairly close to the peduncle, Furthermore, the RITC-SO0 population (projecting to whereas populations projecting to more lateral parts a folium in the “bend region” at the transition between are more externally situated (ventrally and medially). the dorsal and ventral paraflocculus, cf. Fig. 11, in ad- The cell populations projecting to the “bend region”dition to the main ventromedially located lamella of the transition between the dorsal and ventral parafloc- MULTIVARIATE SPATIAL POINT PATTERNS Fig. 3. Photographs of the screen of a dynamic graphics workstation demonstrating the location of the somata of 2.864 retrogradely labelled ponto-cerebellar neurons (population projecting to injection site F-G-490, cf. Fig. 1).The reconstruction comprises a consecutive series of 35 sections through the rostral 28%of the pontine nuclei. The thin lines represent the outline of the pontine nuclei, medial to the left. Seven different views (A-G) are shown. The plane of sectioning (transverse) is indicated as a dashed line in the inset (lower right). 515 In C , the reconstruction is seen with the plane of sectioning facing the viewer. In G , the plane of sectioning is seen “on edge” with the rostral end of the pontine nuclei upwards (reconstruction is rotated 90” about the x-axis compared to C). Intermediate views are shown in D-F. The angle of “maximum lamellarity” is demonstrated in A. From this angle of view, the pons is viewed from the rostral end, a bit from the dorsal and lateral side, compared to the view in C. The view presented in B is intermediate between A and C. Fig. 4. Photographs of the screen of a dynamic graphics workstation showing the location of the somata of altogether 9.948 retrogradely labelled neurons in cat CCoL 490. F-G labelled cells (2.864) are shown as yellow, and RITC labelled cells (7.084) as red. The reconstruction comprises a consecutive series of 35 sections through the rostral 28% of the pontine nuclei. Injection sites are shown in Figure 1 (F-G-490 and RITC-4901. The thin lines represent the outline of the pontine nuclei. Upper picture: The same view as demonstrated for the F-G population in Figure 3C (plane of sectioning facing the viewer, medial to the left, and ventral downward). Middle picture: The same view as for the F-G population in Figure 3E (rotated 45” about the x-axis compared to the plane of sectioning). Lower picture: The angle of view of “maximum lamellarity” (see also Fig. 3A). Fig. 5. Photographs of the screen of a dynamic graphics workstation showing the location of the soma of altogether 28.471 retrogradely labelled neurons in cat CCoL 500. RITC labelled cells (14.048) are shown as blue, F-G labelled cells (10.084) as red, and FB labelled cells (4.309) as yellow. The rostral 56% of the pontine nuclei was reconstructed (35 sections, every second). The thin lines represent the outline of the pontine nuclei. The two upper pictures show the same angle of view as demonstrated for another reconstruction in Figure 3E (see also inset, lower right, in Fig. 31, i.e., a rostro-ventral view of the pontine nuclei (angle of view of 45” to the plane of sectioning). Medial is to the left, and rostral upward. Upper picture: F-G population (red) and FB population (yellow). Middle picutre: Same as upper picture with the addition of the RITC population (blue).Lower picture: same as middle picture but rotated so as to be seen from the angle of “maximum lamellarity.” 517 MULTIVARIATE SPATIAL POINT PATTERNS culus-in addition contain an extra internal lamella of cells close to the peduncle. When injections are placed farther along the ventral paraflocculus, the general impression is that labelled cell populations are even more externally placed than those projecting to the dorsal paraflocculus. This pattern of organization is further tested and elucidated with statistical methods in the next section. ce.3 17 1 21 Statistical Analyses: Segregation, Random Labelling, or Positive Association ? The limitations imposed by pure inspection of threedimensional reconstructions make it worthwhile to search for objective criteria for positive association and segregation. Positive association and segregation are essentially deviations from spatial neutrality (“overlap-by-chance” and “segregation-by-chance”). Neutrality, however, is not an unambiguous term. We consider two fundamentally different definitions of neutrality in a multivariate spatial point pattern. Either, the component patterns are statistically independent of one another, or they are formed by random labelling (Diggle, 1986). For reasons considered below (see Discussion), we use random labelling as a benchmark hypothesis, and search for deviations from this. We say that random labelling occurs if cells projecting to two different target regions are a random sample from the totality of labelled cells, ignoring the target region. We do not know whether the fluorescent tracers label all neurons projecting to the target region or not, or what fraction of cells that are labelled. We assume, however, that the unlabelled cells are a random sample of the totality of cells projecting to the target region in question (i.e., that there is “random thinning,” see Discussion). Accordingly, we require a measure of segregation that is invariant under random thinning of the spatial distribution pattern. Also, we require a measure that is symmetric in the sense that the spatial segregation between cell population 1 vs. 2 should be the same as that between population 2 vs. 1. To achieve this, we proceed as follows. Let H,,(t) denote the cumulative distribution function of the distance t between a pair of randomly selected cells belonging to target regions 1 and 2, and H,,(t) the corresponding function for a pair of cells randomly selected without reference to their target regions. If there is random labelling, H12(t) = H,,(t) for all t, whereas if the patterns are segregated there will be a tendency for distances between cells projecting to different target regions to be larger than distances between cells chosen without reference to the target region, leading to H,,(t) < H,,(t), a t least for small values of t. Conversely, if the patterns are positively associated in the sense that cells from different target regions tend to be located together, we expect that H,,(t) > H,,(t), a t least for small values or t. In the present context, positive association could arise through a functional dependency between the patterns, as, for example, if one pattern is a perturbation of the other. One should also note that the occurrence of double-labelled cells increases the probability for finding positive association. The above considerations suggest that a useful diagnosticfor segregation is a plot of the estimated difference, D(t) = Al2(t) - A,,(t), against t. We calculate the 31 Fig. 6. System for numbering of folia in the paraflocculus, beginning with 1 in the dorsal paraflocculus, ending with 32 in the ventral paraflocculus. estimates H,,(t) and A,,(t) as the observed proportions of intercell distances less than or equal to t. Examples of such plots are given in Figures 7 and 8. For clarity, we assign numbers to each folium of the paraflocculus, beginning with 1 in the dorsal paraflocculus, ending with 32 in the ventral paraflocculus, as shown in Figure 6. We first consider plots of D(t) obtained for combinations where tracers are injected in widely separated folia (6-15 folia between injection sites, cf. 5 of the data sets in Fig. 7). Over a range of values oft, we find H,,(t) < H,,(t), suggesting that the two cell populations compared are spatially segregated (Fig. 7). There is considerable individual variation in the behavior of D(t). For example, compared to other cases with widely separated injection sites, D(t) is close to zero for cat 495 (folium 14 vs. 22, Fig. 7). In cases where adjacent folia are injected (Fig. 81, we find both H,,(t) < H,,(t) (suggesting segregation, in cat 499) and H,,(t) > H,,(t) (indicating positive association, in cats 490 and 500). Furthermore, for comparisons of adjacent folia, the overall data show that the deviation from H,,(t) = H,,(t) (random labelling) is generally smaller than for cases where tracers are injected in widely separated folia. The curves presented in Figures 7 and 8, however, only tell us whether there is a tendency for patterns to be segregated or positively associated. They do not give precise quantitative information on the strength of the possible segregation or positive association. Therefore, we canI;lot make a direct comparison between the estimates D(t) for different animals. The reason for this is that D(t) depends on the size and shape of the threedimensional region within which the complete cell distributions are observed (the “observation region”). In principle, we could eliminate these dependencies on the observation region by using the reduced second moment measure or K function (cf. Bjaalie and Diggle, 1990), which is essentially the H function with a correction for edge effects. However, the edge corrections in three dimensions are technically feasible only for simple shapes of the observation region such as cuboidal regions. 518 J.G. BJAALIE ET AL. Cat 495 ( f o l i u m 14vs.20 ) Cat 495 ( f o l i u m 14vs.22) I II - I I c c I 0 0 0 0 <I -0.010 <I - - I I c c I .N.. 2 <I <I -0 0301 I I I I 1000 2000 3000 4 000 - 0 020 -0030 t tI ' I I I I 1000 2 000 3000 L 000 t (rnm) (mrn) Cat 500 ( folium 1 vs. 16 ) Cat 495 ( folium 2Ovs.22 ) 0 I 1 0 --- 0 <I - 0 0 1 0 I I - I c N r'l <I - 0 0 2 0 <I -0.020 - 0 030 -0 030 1000 2000 3000 L 000 I ' I I I 1 1000 2000 3000 LOOO (mm) (mrn) Cat 500 ( f o l i u m lvs.15) t Cat 502 ( folium 2vs.12) I - c 0 0 <I I c - I N <I I - 0 030 1000 2000 3000 I I LOOO (rnrn) Fig. 7. Plots of the estimated difference, H,,(t) - Ho,,(t), against distance t, for different animals and combinations of injection sites (cf. Fig. 6 for numbering of folia). HJt) is the cumulative distribution function of the distance t between a pair of randomly selected points of type 1 and 2. H,,(t) is the corresponding function for a pair of points randomly selected without reference to type. The curves suggest segregation between populations of labelled cells projecting to widely separated folia, since HJt) < H,,(t) over a range of values o f t . MULTIVARIATE SPATIAL POINT PATTERNS The next logical step in our analysis is to construct a formal test of significance to see whether deviations from Hl,(t) = H,,(t) (random labelling or neutrality), presented in Figures 7 and 8, are significant. Our test of significance is based on 99 simulations of random distributions. Each simulation randomly gives a type 1 or type 2 label to each of the totality of cells, irrespective of the tracer-label. We compute the statistic Cat 490 ( f o l i u m 12vs.13 ) -- 519 0 c 0 <I -0.010 -I c N <I -0.020 -0.030 I I 1000 2000 I I t 3000 4.000 (mm) Cat 499 ( folium 20 vs. 21 ) 0 <I -0.01ot I I c 1 2 <r -0.020 -0.030 I I 1.000 2.000 I I 3.000 4.000 (mml t I -0010 c -0.030 I c I 0 <I I I I 1000 2.000 I I 3.000 4.000 (mm) Quantitative Description of a Possible Segregation Mechanism Having established the existence of spatial segregation between the locations of cells projecting to separated target regions, our second objective is to give a quantitative description of the phenomenon. Specifically, we want to develop a measure to test further the hypothesis that with a gradual change in target region (from dorsal to ventral paraflocculus, nos. 1-32 in Fig. 6), there is a corresponding drift of the shell-like spaces containing labelled cells away from the ventromedial surface of the peduncle and, further, that the extent of this drift differs at different rostrocaudal levels of the pons. To investigate this hypothesis, we take sections from up to eight different rostrocaudal levels of the pons in each cat and, within each section, convert the cell locations to perpendicular cell distances from the ventromedial border of the peduncle. In practice, we first digitize a reference line, corresponding to the ventro- Cat 500 ( f o l i u m 1 5 v s . 1 6 ) - recompute values of u2, us, . . . ,ulo0 after each of the 99 independent random distributions of the totality of cells, and rank the value of ul, amongst the ul, u2, . . ., uloo . If u1 ranks kth largest, the p-value of the test is p = k/100 (Bernard, 1963; Besag and Diggle, 1977). The results obtained in this way for each of the total of nine combinations shown in Figures 7 and 8 are p = 0.01 for all combinations. Accordingly, all deviations from H12(t) = H,,(t) shown in Figures 7 and 8 are statistically significant.2 We therefore conclude that there is a segregation process that moves cell populations projecting to widely separated folia apart from each other. For cell populations projecting to adjacent folia (Fig. B), we have evidence in favor of both positive association between cell populations a s well a s segregation. A possible interpretation is that a n inherent positive association between populations projecting to adjacent folia becomes attenuated and is eventually reversed with a progressive increase in spatial segregation between populations projecting to more widely .separated folia. This leads us to conclude that populations of ponto-cerebellar neurons are confined to shelllike tissue volumes, the spatial location of which gradually move in accordance with specific target region in the paraflocculus (from folium 1 to 32, cf. Fig. 6). t Fig. 8. Plots of the estimated difference, fi,,(t) - A,,(t), against distance t, for different animals and combinations of injection sites (cf. Fig. 6 for numbering of folia). The curves suggest positive association between populations of labelled cells projecting to adjacent folia, in two of the data sets, since H,,(t) > H,,(t) over a range of values oft. Presentation is otherwise as in Figure 7. 'In one remaining case, cat 488, only a small fraction of the pontine nuclei (the rostra1 7%) was reconstructed. Therefore, compared to other cases, only relatively few labelled neurons were recorded (511 F-G labelled cells; 1,179RITC labelled cells). Injections were placed in folium no. 13 (RITC) and 14 (F-G) (cf. Fig. 6). Deviations from neutrality were not statistically significant (P = 0.10). 520 J.G. BJAALIE ET AL. Cat 4 9 5 Cat 4 9 0 1500 - 1500 -5 a E 12 13 0 - -E, 1500 1500 1000 a E, IU - - --- 500 . . . - ' ,1 -5 1000 1000- a 16 500 - 1 I I I 2 4 6 8 01 I I I I 2 L 6 8 6 8 pons ( l e v e l 1 pons ( level 1 Cat 4 9 9 Cat 500 t 15ooc I I 1 tt I I 1 I 2 L 6 8 0 2 L pons ( l e v e l 1 pons ( l e v e l 1 Cat 502 Cat 511 15001 12//', / \ / - - ' / I 500 , ._-- 2 -f ' 1000 I I a Iu 5001 I 0 1 I I I 2 4 6 8 pons ( l e v e l 1 01 I 1 I I 2 L 6 8 pons ( level 1 Fig. 9. Plots of the average perpendicular distance to the ventromedial surface of the peduncle (d,,,), for cells in a population, against rostro-caudal level within the pontine nuclei. Level 1 is the most rostral, level 8 the most caudal. Numbers assigned to each curve correspond to the location of injection site (folium numbers, cf. Fig. 6). medial border of the peduncle; we then use an algorithm to calculate the perpendicular distance from a labelled cell to a line defined by the two closest (digitized) points on the reference line. In the computer reconstructed cases, we already have the x,y coordinates available for the labelled cells at each section level. For remaining animals, we digitize the x,y position of each category of labelled cells as a basis for computing the distances. Plots of the average perpendicular distance to the ventromedial surface of the peduncle (d,,,) against the rostrocaudal level within the pontine nuclei clearly demonstrate how the different populations of labelled cells in each animal are shifting in location (Big. 9). As an example, we take cat 511 (Fig. 9, lower right). This animal received injections in three widely separated folia (nos. 2,21, and 30 in the terminology presented in Fig. 6; see also Fig. 1). The average perpendicular distance to the ventromedial surface of the peduncle, at six different rostrocaudal levels within the pontine nu- 1500 MULTIVARIATE SPATIAL POINT PATTERNS t I f 0 1000 500 0 1 10 20 30 paraflocculus ( 1 - 3 0 ) Fig. 10. Trend curve for d,,, a t pontine level 2 (defined in legend to Fig. 9) based on data pooled for all animals presented in Figure 9, and plotted against location of injection site in the paraflocculus (cf. Fig. 6 ) . The trend curve demonstrates that with a change in target region (from dorsal to ventral paraflocculus, folium nos. 1-32 in Fig. 6 ) , there is a corresponding increase in the average distance to the peduncle for cells in a population projecting to a single folium. clei are shown (FB labelled cells were not found at level 1). At all levels, the mean distance to the peduncle increases with the progression in target region number. The increase was strongest in rostra1 parts of the pontine nuclei (level 2: from a mean distance of 613 pm for cells projecting to folium no. 2, to a mean distance of 959 pm and 1,470 pm for target 21 and 30 cells, respectively). This overall trend was reproduced in several other cases (see, e.g., cats 502,500, and 499 in Fig. 9). Even the mean distances for cells projecting to adjacent targets (such as 15 vs. 16 in cat 500, and 20 vs. 21 in cat 499) showed an increase with increasing target numbers, although this was not always the case (cf. virtually no shift in location of cells projecting to targets 12 and 13 in cat 490, Fig. 9). Also, in one case (cat 495, Fig. 9) a t some levels in the pontine nuclei, a partly reverse trend of the one described above was seen. Nevertheless, our material supports the hypothesis of a systematic shift in location of cells with progression in target region in the paraflocculus. There is considerable individual variability in the absolute values of average perpendicular distances to the ventromedial surface of the peduncle. The conclusions about a shift in location of cell groups are therefore based upon reproducible changes from animal to animal. Still, if results for dvmpare pooled for all animals presented in Figure 9 and plotted against location of injection site (folium nos. 1-32) in the paraflocculus, the overall trend is clear, despite individual variations (Fig. 10). DISCUSSION The present investigation uses computer threedimensional reconstructions for studying the spatial organization of large populations of points (such as the point coordinates corresponding to the center of individual neuronal somata). Moreover, we present quantitative data of spatial relationships between reconstructed multivariate point patterns, such as multiple populations of neurons within the same reference volume in the brain. Statistical methods are developed in 521 order t o objectively distinguish patterns created by spatial neutrality (random labelling) from patterns created by either a process driving different point types together in space (positive association) or apart from each other (segregation). Using such a quantitative approach, we have shown that following injections of two tracers in adjacent cerebellar folia, populations of labelled cells in the pontine nuclei are significantly positively associated. In contrast, populations projecting to target regions wider apart are significantly segregated. We also find that with gradual change in target region (from dorsal to ventral paraflocculus), there is a corresponding drift of the spatial distribution of pontine cell groups in a ventromedial direction. Methodological Considerations The validity of the patterns observed in the computer three-dimensional reconstructions depends on several methodological factors. One is the alignment of the sections. The fiducial marker holes together with blood vessels passing from section to section provide the basis for our manual alignment of complete series of sections. Using careful microtomy a t stable temperature, most sections were free from distortions. In a few sections, twisting or compression nevertheless occurred. This was adjusted for by manually aligning different parts of the section a t a time. Thus we feel certain that the alignment of sections was sufficiently accurate for the present analysis. The observations of well-defined bands and clusters of cells in the reconstructions (and not blurred distributions) also indicate successful alignment. The accuracy of the plotting of the labelled cells is another important methodological factor. The plotting was performed with an accuracy with regard to x and y coordinates of single cells of about * l O pm. The z coordinate was defined by section number and thickness. We regard these inaccuracies as insignificant in the present context, since we do not deal with intricate details a t a small scale, but rather with the overall distributions and spatial relationships. With regard to the possibility of errors by double counts of cells, our system for plotting enabled us to control that all cells were included, and virtually none had been plotted twice. It is likely, however, that fragments of cell bodies occurring in two adjacent sections could sometimes be plotted twice. Thus we did not check for double counts in the careful way done by Bjaalie and Diggle (1990). However, the double counts occur with equal probability for the different types of labelled cells. The properties of the fluorescent tracers, in terms of uptake and efficiency of labelling, are more likely to influence our findings. Fast Blue is generally regarded as a very sensitive fluorescent tracer. In our hands, efficiency of labelling (number of labelled cells per injected amount of tracer) varied a t least with a factor of five. The Fluoro-Gold tracer gave more stable results in terms of number of labelled cells from experiment to experiment. On the average, the efficiency of labelling was higher than that of Fast Blue. The most efficient tracer, however, was RITC. In terms of variability in number of labelled cells, it held an intermediate position between Fast Blue and Fluor-Gold. (Further considerations on numerical capacities of the labelled neuronal populations are to dealt with in a study in 522 J.G. BJAALIE E T AL. preparation by J.G. Bjaalie.) However, regardless of individual variability in efficiency of labelling, the labelled neurons were distributed throughout comparable regions of the pontine nuclei, and the arrangement of clusters and bands were apparently not influenced by the number of labelled cells. This observation strongly supports the view t h a t unlabelled neurons projecting to the injection site were randomly distributed, i.e., that the patterns observed are the result of a “random thinning” (for further considerations on random thinning, see Bjaalie and Diggle, 1990). Our statistical tests are designed so as not to be influenced by this methodological phenomenon. A few methodological remarks should also be made concerning the statistical analyses. In our search for deviations from spatial neutrality, we have chosen random labelling as a benchmark hypothesis. As mentioned above (see Results, section on Statistical Analyses), however, independence might also be considered as a reasonable definition of spatial neutrality. We must therefore consider the difference between these two kinds of spatial neutrality. This is important because positive association and segregation in relation to independence could be mechanistically different from positive association and segregation in relation to random labelling. Statistical independence is a situation in which the component patterns are generated by independent stochastic processes (see also Diggle, 1986). Such a situation is unlikely to occur in our material. The density distribution of the labelled cells over the reference volume (the pontine nuclei) is inhomogeneous, and because of this there is necessarily a spatial dependence between different categories of labelled cells, because all types are more likely to occur within the more dense regions. Therefore, the rejection of independence is not very informative in our context. Random labelling, in contrast, is a kind of neutrality between type 1 and 2 events quite different from independence. One should note t h a t in the case of random labelling, the component patterns are generally dependent, even if the differentiation into type 1 and type 2 events is completely random (Diggle, 1986). In a biological context, it seems reasonable to say t h a t a “label” or sign is attached to each neuron when its target region(s) are determined at a certain stage during development. Our statistical tests takes into consideration distances between events (the location of individual labelled cells), and therefore, deviations from random labelling tell us something about possible mechanisms operating on the single neuron level. It is not meant to represent a final solution for determining the complicated spatial interactions between different categories of neurons. Nevertheless, the approach we have used provides objective results understandable in relation to subjective evaluation of the data. Does Precise Terminology on Spatial interactions improve Our Understanding of the Connectivity of the Brain? This study deals with spatial relationships between neuronal populations in a new way. We generally assume that two neurons t h a t are located close together are more likely to be influenced by the same afferent fibers than neurons located far apart. Does it matter, however, whether they are located close together sim- ply by chance or due to a mechanism driving them together in space (positive association)? Perhaps it does not matter for the operations of the individual neurons, but it may nevertheless be of importance for the operations of large neuronal populations. The present approach therefore may serve to focus attention on the connectivity of populations of neurons, rather than of individual neurons. Thus a positive association between two neuronal populations increases the probability of similar information transfer, compared to a situation where there is only “overlap-by-chance.’’ Along the same lines of reasoning, i t is more likely that the different kinds of information can be transferred through neuronal populations t h a t are segregated due to a specific mechanism, compared to populations that are only incidently separated. Furthermore, as briefly considered below, the mathematical approach has made it possible for us to suggest a new model for organization of neuronal populations in the pontine nuclei. We therefore believe that the terminology used in the present work provides a n advantage over the conventional terminology. We replace subjective evaluations with formal tests, and we introduce methods for quantitative estimations of the basis for intricate wiring patterns. The Biological Significance of a Gradual Shift in Location of Neuronal Cell Populations Our data suggest t h a t there are gradual shifts in location of pontine neuronal populations projecting to gradually changing locations of cerebellar target regions. It seems reasonable to assume that this gradual shift in location also leads to a gradual change in the information flow transferred by the populations. Consequently, it appears that adjacent cerebellar spots of about the size of a folium (or smaller) receive only slightly different inputs. This would seem to be cornpatible with a model of a continuous map of mossy fiber input to the cerebellum, rather than a patchy of scattered map (for a review of parallel computer maps and brain maps, see Nelson and Bower, 1990) a s suggested by previous anatomical and physiological data (for references, see Brodal, 1982,1987; Welker 1987,1990).As pointed out above, the statistical approach is essential as a basis for our conclusions. Pure inspection of spatial interactions between cell clusters indeed suggests a mosaic segregated cell distribution, as a product of a patchy mapping process. One could therefore speculate that our hypothesis of continuous shifts of spatial distributions of cells would actually apply also to other systems where segregated cell groups are described (see, e.g., Goldman-Rakic and Selemon, 1990). This could be tested by applying our methodological approach. In developmental terms, one could think of gradients of neuroactive substances guiding the growth or signing of different neurons to distinct target regions, thus creating continuous maps. Gradients would be more parsimonious to code genetically than a tremendous number of specific abrupt changes. Our methodological approach may therefore also be a useful tool in developmental studies, by enabling the investigator to detect and describe changes not already revealed by qualitative methods only. Of specific interest to us is the relation between our MULTIVARIATE SPATIAL POINT PATTERNS hypothesis for mapping of cerebellar mossy fiber input and that of previous electrophysiological micromapping studies (for reviews, see Welker 1987, 1990). Welker and collaborators have found that, within a cerebellar folium, there is a “fractured somatopic map” (patchy map) in parts of the cerebellum investigated in both cat and rat. (The paraflocculus has not been studied in this way, however.) It is not obvious how these data can be reconciled with our hypothesis, unless we assume t h a t a certain set of functionally defined patches are repeated from folium to folium. In t h a t case, in principle, the small shift in location of pontocerebellar neurons projecting to adjacent folia could correspond to small changes in the exact organization of repeated combinations of patches. Some of our observations, however, are difficult to reconcile even with this assumption. Thus regardless of the size of cerebellar spots injected with a tracer, we always find a widespread distribution of ponto-cerebellar labelled cells. Admittedly, our injection sites are not anywhere near the size of the single patches of Welker. Previous investigators, however, have made extremely small injections of horseradish peroxidase-covering only a tiny fraction of a folium-in the cerebellar paramedian lobule (Hoddevik and Walberg, 1979). They find that the few labelled cells, nevertheless, are widely distributed. We are therefore inclined to believe that pontocerebellar cell bodies projecting to a minor part of a folium (a few functionally defined patches) are distributed within the same lamellae as the neurons projecting to a larger target region (an entire folium). It remains to be determined, however, whether injections of minute amounts of multiple tracers into different parts of a single folium would result in a positive association between the various groups of labelled cells or, if in fact, at such a smaller scale, there would be a n element of segregation corresponding to differential input to functionally distinct patches. Thus our findings raise further questions about the principles of organization of ponto-cerebellar pathways. ACKNOWLEDGMENTS We thank Kari Ruud and Gunnar F. Lothe for expert technical assistance. 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