{"title": "Bayesian morphometry of hippocampal cells suggests same-cell somatodendritic repulsion", "book": "Advances in Neural Information Processing Systems", "page_first": 133, "page_last": 139, "abstract": null, "full_text": " \n\n \n\n \n\n \n\n1 \n\nBayesian morphometry of hippocampal cells \nsuggests same-cell somatodendritic repulsion \n\n \nGiorgio A. Ascoli * \n\nascoli@gmu.edu \n\nAlexei Samsonovich \nKrasnow Institute for Advanced Study a t George Mason University \n\n \n\nFairfax, VA 22030-4444 \n \n\n asamsono@gmu.edu \n\n \n\nAbstract \n\nVisual inspection of neurons suggests that dendritic orientation may be \ndetermined both by internal constraints (e.g. membrane tension) and by \nexternal vector fields (e.g. neurotrophic gradients). For example, basal \ndendrites of pyramidal cells appear nicely fan-out. This regular \norientation is hard to justify completely with a general tendency to \ngrow straight, given the zigzags observed experimentally. Instead, \ndendrites could (A) favor a fixed (\u201cexternal\u201d) direction, or (B) repel \nfrom their own soma. To investigate these possibilities quantitatively, \nreconstructed hippocampal cells were subjected to Bayesian analysis. \nThe statistical model combined linearly factors A and B, as well as the \ntendency to grow straight. For all morphological classes, B was found \nto be significantly positive and consistently greater than A. In addition, \nwhen dendrites were artificially re-oriented according to this model, the \nresulting structures closely resembled real morphologies. These results \nsuggest that somatodendritic repulsion may play a role in determining \ndendritic orientation. Since hippocampal cells are very densely packed \nand their dendritic trees highly overlap, the repulsion must be cell-\nspecific. We discuss possible mechanisms underlying such specificity. \n\n1 Int r oduc t i on \n\nThe study of brain dynamics and development at the cellular level would greatly benefit \nfrom a standardized, accurate and yet succinct statistical model characterizing the \nmorphology of major neuronal classes. Such model could also provide a basis for \nsimulation of anatomically realistic virtual neurons [1]. The model should accurately \ndistinguish among different neuronal classes: a morphological difference between classes \nwould be captured by a difference in model parameters and reproduced in generated \nvirtual neurons. In addition, the model should be self-consistent: there should be no \nstatistical difference in model parameters measured from real neurons of a given class \nand from virtual neurons of the same class. The assumption that a simple statistical model \nof this sort exists relies on the similarity of average environmental and homeostatic \nconditions encountered by individual neurons during development and on the limited \namount of genetic information that underlies differentiation of neuronal classes. \n\nPrevious research in computational neuroanatomy has mainly focused on the topology \nand internal geometry of dendrites (i.e., the properties described in \u201cdendrograms\u201d) [2,3]. \nRecently, we attempted to include spatial orientation in the models, thus generating \n\n\f \n\n \n\n \n\n2 \n\nvirtual neurons in 3D [4]. Dendritic growth was assumed to deviate from the straight \ndirection both randomly and based on a constant bias in a given direction, or \u201ctropism\u201d. \nDifferent models of tropism (e.g. along a fixed axis, towards a plane, or away from the \nsoma) had dramatic effects on the shape of virtual neurons [5]. Our current strategy is to \nsplit the problem of finding a statistical model describing neuronal morphology in two \nparts. First, we maintain that the topology and the internal geometry of a particular \ndendritic tree can be described independently of its 3D embedding (i.e., the set of local \ndendritic orientations). At the same time, one and the same internal geometry (e.g., the \nexperimental dendrograms obtained from real neurons) may have many equally plausible \n3D embeddings that are statistically consistent with the anatomical characteristics of that \nneuronal class. The present work aims at finding a minimal statistical model describing \nlocal dendritic orientation in experimentally reconstructed hippocampal principal cells. \n\nHippocampal neurons have a polarized shape: their dendrites tend to grow from the soma \nas if enclosed in cones. In pyramidal cells, basal and apical dendrites invade opposite \nhemispaces (fig. 1A), while granule cell dendrites all invade the same hemispace. This \nbehavior could be caused by a tendency to grow towards the layers of incoming fibers to \nestablish synapses. Such tendency would correspond to a tropism in a direction roughly \nparallel to the cell main axis. Alternatively, dendrites could initially stem in the \nappropriate (possibly genetically determined) directions, and then continue to grow \napproximately in a radial direction from the soma. A close inspection of pyramidal \n(basal) trees suggests that dendrites may indeed be repelled from their soma (Fig. 1B). A \ntypical dendrite may reorient itself (arrow) to grow nearly straight along a radius from the \nsoma. Remarkably, this happens even after many turns, when the initial direction is lost. \nSuch behavior may be hard to explain without tropism. If the deviations from straight \ngrowth were random, one should be able to \u201cremodel\u201d th e trees by measuring and \nreproducing the statistics of local turn angles, assuming its independence of dendritic \norientation and location. Figure 1C shows the cell from 1A after such remodeling. In this \ncase basal and apical dendrites retain only their initial (stemming) orientations from the \noriginal data. The resulting \u201ccotton ball\u201d s uggests that dendritic turns are not in dependent \nof dendritic orientation. In this paper, we use Bayesian analysis to quantify the dendritic \ntropism. \n\n2 Me t hods \n\nDigital files of fully reconstructed rat hippocampal pyramidal cells (24 CA3 and 23 CA1 \nneurons) were kindly provided by Dr. D. Amaral. The overall morphology of these cells, \nas well as the experimental acquisition methods, were extensively described [6]. In these \nfiles, dendrites are represented as (branching) chains of cylindrical sections. Each section \nis connected to one other section in the path to the soma, and may be connected on the \nother extremity to two other sections (bifurcation), one other section (continuation point), \nor no other section (terminal tip). Each section is described in the file by its ending point \ncoordinates, its diameter and its \"parent\", i.e., the attached section in the path to the soma \n[5,7]. In CA3 cells, basal dendrites had an average of 687(\u2013 216) continuation points and \n72(\u2013 17) bifurcations per cell, while apical dendrites had 717(\u2013 156) continuation points \nand 80(\u2013 21) bifurcations per cell. CA1 cells had 462(\u2013 138) continuation points and \n52(\u2013 12) bifurcations (basal), 860(\u2013 188) continuation points and 120(\u2013 22) bifurcations \n(apical). In the present work, basal and apical trees of CA3 and CA1 pyramidal cells were \ntreated as 4 different classes. Digital data of rat dentate granule cells [8] are kindly made \navailable by Dr. B. Claiborne through the internet (http://cascade.utsa.edu/bjclab). Only \n36 of the 42 cells in this archive were used: in 6 cases numerical processing was not \naccomplished due to minor inconsistencies in the data files. The \u201crejected\u201d cells were \n1208875, 3319201, 411883, 411884A, 411884B, 803887B. Granule dendrites had \n\n\f \n\n \n\n \n\n3 \n\n549(\u2013 186) continuation points and 30(\u2013 6) bifurcations per cell. Cells in these or similar \nformats can be rendered, rotated, and zoomed with a java applet available through the \ninternet (www.cns.soton.ac.uk) [7]. \n\n \n\n \nFigure 1: A: A pyramidal cell (c53063) from Amaral\u2019s archive. B: A zoom-in from panel \nA (arrows point to the same basal tree location). Dotted dendrites are behind the plane. C: \nSame cell (c53063) with its dendritic orientation remodeled assuming zero tropism and \nsame statistics of all turn angles (see Results). \n\nIn agreement with the available format of morphological data (described above), the \nprocess of dendritic growth1 can be represented as a discrete stochastic process consisting \nof sequential attachment of new sections to each growing dendrite. Here we keep the \ngiven internal geometry of the experimental data while remodeling the 3D embedding \ngeometry (dendritic orientation). The task is to make a remodeled geometry statistically \nconsistent with the original structure. The basic assumption is that neuronal development1 \nis a Markov process governed by local rules [4]. Specifically, we assume that (i) each \nstep in dendritic outgrowth only depends on the preceding step and on current local \nconditions; and (ii) dendrites do not undergo geometrical or topological modification \nafter their formation (see, however, Discussion). In this Markov approximation, a \nplausible 3D embedding can be found by sequentially orienting individual sections, \nstarting from the soma and moving toward the terminals. We are implementing this \nprocedure in two-step iterations (1). First, at a given node i with coordinates ri we select a \nsection i+1, disregard its given orientation, and calculate its most likely expected \ndirection n'i+1 based on the model (here section i+1 connects nodes i and i+1, and n \nstands for a unit vector). For a continuation point, the most likely direction n'i+1 is \ncomputed as the direction of the vector sum ni + vi. The first term is the direction of the \nparent section ni, and reflects the tendency dendrites exhibit to grow relatively straight \ndue to membrane tension, mechanical properties of the cytoskeleton, etc. The second \nterm is a local value of a vector field: vi = v(ri), which comprises the influence of external \nlocal conditions on the direction of growth (as specified below). Finally, we generate a \nperturbation of the most likely direction n'i+1 to produce a particular plausible instance of \na new direction. In summary, the new direction ni+1 is generated as: \n \n1 Although we resort to a developmental metaphor, our goal is to describe accurately \nthe result of development rather than the process of development. \n\n\f \n\n \n\n \n\n4 \n\nn\nn\n\ni\n'\n\n+\n1\n\n+\n1\n\ni\n\n=\n||\n\ni\n\nnT\n+\nn\n\ni\n\n'\n+\n1\ni\nv\n\ni\n\n,\n.\n\n \n\n \n\n \n\n \n\n \n\n(1) \n\nHere Ti is an operator that deflects n'i+1 into a random direction. If we view each \ndeflection as a yaw of angle a i, then the corresponding rolling angle (describing rotation \naround the axis of the parent dendrite) is distributed uniformly between 0 and 2p . The \nprobability distribution function for deflections as a function of a i is taken in a form that, \nas we found, well fits experimental data: \n\n(\n)\n(cid:181)T\nP i\n\na\ns\n\ni\n\n,\n\ne\n\n \n\n \n\n \n\n \n\n \n\n(2) \nwhere s << 1 is a parameter of the model. At bifurcation points, the same rule (1), (2) is \napplied for each daughter independently. A more accurate and plausible description of \ndendritic orientation at bifurcations might require a more complex model. However, our \nsimple choice yields surprisingly good results (see below). The model (1), (2) can be used \nin the simulation of virtual neuronal morphology. In this case one would first need to \ngenerate the internal geometry of the dendrites [1-5]. Most importantly, model (1), (2) \ncan be used to quantitatively assess the significance of the somatocentric (radial) tropism \nof real dendrites. Assuming that there is a significant preferential directionality of growth \nin hippocampal dendrites, the two main alternatives are (see Introduction): \n\nHA: The dominating tropic factor is independent of the location of the soma. \nHB: The dominating tropic factor is radial with respect to the soma. \n\nThe simplest model for the vector field v that discriminates between these alternative \nhypotheses includes both factors, A and B, linearly: \n\n+=\n\na\n\nv\n\ni\n\nbn\n\n.r\ni\n\n \n\n \n\n \n\n \n\n \n\n(3) \n\nHere a = (ax, ay, az) is a constant vector representing global directionality of cell-\nindependent environmental factors (chemical gradients, density of neurites, etc.) \ninfluencing dendritic orientation. nr\ni is the unit vector in the direction connecting the \nsoma to node i, thus representing a somatocentric tropic factor. In summary, ax, ay, az, b \nand s are the parameters of the model. Finding that the absolute value a = |a| is \nsignificantly greater than b would support HA. On the contrary, finding that b is greater \nthan a would support HB. Based on a Bayesian approach, we compute the most likely \nvalues of a, b and s by maximization of the likelihood of all experimentally measured \norientations (taken at continuation points only) of a given dendritic tree: \n\n(\n)\na\n**,\n\nb\n\n=\n\narg\n\nmax\n{\n}\ns\na\n,\n\nb\n\n,\n\nP\n\ni\n\n(\nT\ni\n\n)\n\n=\n\narg\n\nmin\n{\n}\na\nb\n\n,\n\na\n\ni\n\n,\n\n \n\n \n\n \n\n(4) \n\nwhere a i is given by (1)-(3) with experimental section orientations substituted for ni, ni+1, \nasterisk denotes most likely values, and the average is over all continuation points. Given \na* and b*, the value of s * can be found from the average value of a i computed with a = \na* and b = b*. The relation results from differentiation of (4) by s . The same relation \nholds for the average value of a computed based on the probability distribution function \n(2) with s = s *. Therefore, computed from the neurometric data with a = a* and b \n= b* is equal to based on (2) with s = s *. The model is thus self-consistent: the \nmeasured value of s * in a remodeled neuron is guaranteed to coincide on average with \nthe input parameter s used for simulation. In addition, our numerical analysis indicates \nself-consistency of the model with respect to a and b, when their values are within a \npractically meaningful range. \n\n-\n(cid:213)\n\f \n\n \n\n \n\n5 \n\n3 Re sul t s \nResults of the Bayesian analysis are presented in Table 1. Parameters a and b were \noptimized for each cell individually, then the absolute value a = |a| was taken for each \ncell. The mean value and the standard deviation of a in Table 1 were computed based on \nthe set of the individual absolute values, while each individual value of b was taken with \nits sign (which was positive in all cases but one). The most likely direction of a varied \nsignificantly among cells, i.e., no particular fixed direction was generally preferred. \n\n \nTable 1: Results from Bayesian analysis (mean \u2013\ndeflection angle, a and b are parameters of the model (1)-(3) computed according to (4). \n \n\n standard deviation). a\n\n is the minimized \n\n \n\nOriginal data \n\na \n\nDataset \nCA3-bas 16.4 \u2013\nCA3-apic 15.2 \u2013\nCA1-bas 16.6 \u2013\nCA1-apic 19.1 \u2013\nGranule 19.1 \u2013\n\n 2.3 0.49 \u2013\n 1.9 0.36 \u2013\n 1.6 0.49 \u2013\n 2.0 0.30 \u2013\n 2.7 1.01 \u2013\n\nB \n 0.17 0.08 \u2013\n 0.16 0.12 \u2013\n 0.26 0.14 \u2013\n 0.20 0.16 \u2013\n 0.64 0.17 \u2013\n\na \n 0.05 12.0 \u2013\n 0.07 12.0 \u2013\n 0.10 14.2 \u2013\n 0.15 17.3 \u2013\n 0.11 11.0 \u2013\n\nA \n\na \n\nZ coordinate set to zero \n\n 2.4 0.42 \u2013\n 2.9 0.29 \u2013\n 1.9 0.48 \u2013\n 2.4 0.22 \u2013\n 1.9 0.36 \u2013\n\nb \n 0.15 0.06 \u2013\n 0.23 0.10 \u2013\n 0.31 0.16 \u2013\n 0.17 0.11 \u2013\n 0.16 0.07 \u2013\n\n 0.05 \n 0.14 \n 0.12 \n 0.10 \n 0.05 \n\n \n\nThe key finding is that a is not significantly different from zero, while b is significantly \npositive. The slightly higher coefficient of variation obtained for granule cells could be \ndue to a larger experimental error in the z coordinate (orthogonal to the slice). In several \ngranule cells (but in none of the pyramidal cells) the greater noise in z was apparent upon \nvisual inspection of the rendered structures. Therefore, we re-ran the analysis discarding \nthe z coordinate (right columns). Results changed only minimally for pyramidal cells, and \nthe granule cell parameters became more consistent with the pyramidal cells. \n\nThe measured average values of the model parameters were used for remodeling of \nexperimental neuronal shapes, as described above. In particular, b was set to 0.5, while a \nwas set to zero. We kept the internal geometry and the initial stemming direction of each \ntree from the experimental data, and simulated dendritic orientation at all nodes separated \nby more than 2 steps from the soma. A typical result is shown in Figure 2. Generally, the \nartificially re-oriented dendrites looked better than one could expect for a model as \nsimple as (1) \u2013 (3). This result may be compared with figure 1C, which shows an \nexample of remodeling based on the same model in the absence of tropism (a = b = 0). \nAlthough in this case the shape can be improved by reducing s , the result never gets as \nclose to a real shape as in Fig. 2 C, D, even when random, uncorrelated local distortions \n(\"shuffling\") are applied to the generated geometry. Thus, although the tendency to grow \nstraight represents the dominant component of the model (i.e., b<1), somatocentric \ntropism may exert a dramatic effect on dendritic shape. Surprisingly, even the asymmetry \nof the dendritic spread (compare front and side views) is preserved after remodeling. \nHowever, two details are difficult to reproduce with this model: the uniform distribution \nof dendrites in space and other subtle medium-distance correlations among dendritic \ndeflections. In order to account for these properties, we may need to consider spatially \ncorrelated inhomogeneities of the medium and possible short range dendrodendritic \ninteractions. \n\n\f \n\n4 Di sc ussi on \n\n \n\n \n\n6 \n\nThe key results of this work is that, according to Bayesian analysis, dendrites of \nhippocampal principal cells display a significant radial tropism. This means that the \nspatial orientation of these neuronal trees can be statistically described as if dendrites \nwere repelled from their own soma. This preferential direction is stronger than any \ntendency to grow along a fixed direction independent of the location of the soma. These \nresults apply to all dendritic classes, but in general pyramidal cell basal trees (and granule \ncell dendrites) display a bigger somatocentric tropism than apical trees. \n \n\n \nFigure 2: Dendritic remodeling with somatocentric tropism. A, B: front and side views \nof cell 10861 from Amaral' s archive. C, D: Same views after remodeling with parameters \na = 0, b = 0.5, s = 0.15 (corresponding to = 17\u0002\u0001\u0004\u0003\u0006\u0005\b\u0007\n\t\f\u000b\u000e\r\u0010\u000f\u0012\u0011\u0014\u0013\n\u0013\u0002\u0015\u0017\u0016\u0018\u0011\u0012\t\u001a\u0019\u001b\u0013\u000e\r\u001c\u0016\u001e\u001d\u001f\u0011 \u0016!\u000b\b\t#\"\u0006\u0019\u001a\u0007\u0018\u0013\u001c\u000f$\t#\t\nstem were taken in their original orientations; all subsequent experimental orientations \nwere disregarded and regenerated from scratch according to the model. \n\n \n\nAssuming that dendrites are indeed repelled from their soma during development, what \ncould be a plausible mechanism? Principal cells are very densely packed in the \nhippocampus, and their dendrites highly overlap. If repulsion were mediated by a \ndiffusible chemical factor, in order for dendrites to be repelled radially from their own \nsoma, each neuron should have its own specific chemical marker (a fairly unlikely \npossibility). If the same repulsive factor were released by all neurons, each dendrite \nwould be repelled by hundreds of somata, and not just by their own. The resulting \ntropism would be perpendicular to the principal cell layer, i.e. each dendrite would be \npushed approximately in the same direction, independent of the location of its soma. This \nscenario is in clear contrast with the result of our statistical analysis. Thus, how can a \ngrowing dendrite sense the location of its own soma? One possibility involves the \nspontaneous spiking activity of neurons during development. A cell that spikes becomes \nunique in its neighborhood for a short period of time. The philopodia of dendritic growth \n\n\f \n\n \n\n \n\n7 \n\ncones could possess voltage-gated receptors to sense transient chemical gradients (e.g., \npH) created by the spiking cell. Only dendrites that are depolarized during the transient \nchemical gradient (i.e., those belonging to the same spiking cell) would be repelled by it. \nAlternatively, depolarized philopodia could be sensitive to the small voltage difference \ncreated by the spike in the extracellular space (a voltage that can be recorded by tetrodes). \n\nThe main results obtained with the simple model presented in this work are independent \nof the z coordinate in the morphometric files, i.e. the most error-prone measurement in \nthe experimental reconstruction. However, it is important to note that any observed \ndeviation of dendritic path from a straight line, including that due to measurement errors, \ncauses an increase in the most likely values of parameters a and b. Another possibility is \nthat dendrites do grow almost precisely in straight lines, and the measured values of a and \nb reflect distortions of dendritic shapes after development. In order to assess the effect of \nthese factors on a and b, we pre-processed the experimental data by adding a gradually \nincreasing noise to all coordinates of dendritic sections. Then we were able to extrapolate \nthe dependence of a*, b* and * on the amplitude of noise in order to estimate the \nparameter values in the absence of the experimental error (which was conservatively \ntaken to be of 0.5 m m). For basal trees of CA3 pyramidal cells, this analysis yielded an \nestimated \u201ccorrected\u201d value of b between 0.14 and 0.25, with a remaining much smaller \nthan b. Interestingly, our analysis based on extrapolation shows that, regardless of the \nassumed amount of distortion present in the experimental data, given the numbers \nmeasured for CA3 basal trees, positive initial implies positive initial b. In other \nwords, not only measurement errors, but also possible biological distortions of the \ndendritic tree may not be capable of accounting for the observed positivity of the \nparameter b. Although these factors affect our results quantitatively, they do not change \nthe statistical significance nor the qualitative trends. However, a more rigorous analysis \nneeds to be carried out. Nevertheless, artificially reoriented dendrites according to our \nsimple model appear almost as realistic as the original structures, and we could not \nachieve the same result with any choice of parameters in models of distortion without a \nsomatocentric tropism. In conclusion, whether the present Bayesian analysis reveals a \nphenomenon of somatodendritic repulsion remains an (experimentally testable) open \nquestion. What is unquestionable is that the presented model is a significant step forward \nin the formulation of an accurate statistical description of dendritic morphology. \n \nAc k nowl e dg me nt s \nThis work was supported in part by Human Brain Project Grant R01 NS39600, funded jointly by \nNINDS and NIMH. \n\nRe f er e nce s \n[1] Ascoli G.A. (1999) Progress and perspectives in computational neuroanatomy. Anat. Rec. \n\n257(6):195-207. \n\n[2] van Pelt J. (1997) Effect of pruning on dendritic tree topology. J. Theor. Biol. 186(1):17-32. \n[3] Burke R.E., W. Marks, B. Ulfhake (1992) A parsimonious description of motoneurons dendritic \n\nmorphology using computer simulation. J. Neurosci. 12(6):2403-2416. \n\n[4] Ascoli G.A., J. Krichmar (2000) L-Neuron: a modeling tool for the efficient generation and \n\nparsimonious description of dendritic morphology. Neurocomputing 32-33:1003-1011. \n\n[5] Ascoli G.A., J. Krichmar, S. Nasuto, S. Senft (2001) Generation, description and storage of \n\ndendritic morphology data. Phil. Trans. R. Sci. B, In Press. \n\n[6] Ishizuka N., W. Cowan, D. Amaral (1995) A quantitative analysis of the dendritic organization \n\nof pyramidal cells in the rat hippocampus. J. Comp. Neurol. 362(1):17-45. \n\n[7] Cannon R.C., D. Turner, G. Pyapali, H. Wheal (1998) An on-line archive of reconstructed \n\nhippocampal neurons. J Neurosci. Meth. 84(1-2):49-54. \n\n[8] Rihn L.L., B. Claiborne (1990) Dendritic growth and regression in rat dentate granule cells \n\nduring late postnatal development. Dev. Brain Res. 54(1):115-124 \n\n\f", "award": [], "sourceid": 2088, "authors": [{"given_name": "Giorgio", "family_name": "Ascoli", "institution": null}, {"given_name": "Alexei", "family_name": "Samsonovich", "institution": null}]}