{"title": "Optimal Sizes of Dendritic and Axonal Arbors", "book": "Advances in Neural Information Processing Systems", "page_first": 108, "page_last": 114, "abstract": null, "full_text": "Optimal sizes of dendritic and axonal arbors \n\nDmitri B. Chklovskii \n\nSloan Center for Theoretical Neurobiology \n\nThe Salk Institute, La Jolla, CA 92037 \n\nmitya@salk.edu \n\nAbstract \n\nI consider a topographic projection between two neuronal layers with dif(cid:173)\nferent densities of neurons. Given the number of output neurons con(cid:173)\nnected to each input neuron (divergence or fan-out) and the number of \ninput neurons synapsing on each output neuron (convergence or fan-in) I \ndetermine the widths of axonal and dendritic arbors which minimize the \ntotal volume ofaxons and dendrites. My analytical results can be sum(cid:173)\nmarized qualitatively in the following rule: neurons of the sparser layer \nshould have arbors wider than those of the denser layer. This agrees with \nthe anatomical data from retinal and cerebellar neurons whose morphol(cid:173)\nogy and connectivity are known. The rule may be used to infer connec(cid:173)\ntivity of neurons from their morphology. \n\n1 Introduction \n\nUnderstanding brain function requires knowing connections between neurons. However, \nexperimental studies of inter-neuronal connectivity are difficult and the connectivity data \nis scarce. At the same time neuroanatomists possess much data on cellular morphology \nand have powerful techniques to image neuronal shapes. This suggests using morphologi(cid:173)\ncal data to infer inter-neuronal connections. Such inference must rely on rules which relate \nshapes of neurons to their connectivity. \n\nThe purpose of this paper is to derive such rule for a frequently encountered feature in the \nbrain organization: a topographic projection. Two layers of neurons are said to form a topo(cid:173)\ngraphic projection if adjacent neurons of the input layer connect to adjacent neurons of the \noutput layer, Figure 1. As a result, output neurons form an orderly map of the input layer. \n\nI characterize inter-neuronal connectivity for a topographic projection by divergence and \nconvergence factors defined as follows, Figure 1. Divergence, D, of the projection is the \nnumber of output neurons which receive connections from an input neuron. Convergence, \nC, of the projection is the number of input neurons which connect with an output neuron. I \nassume that these numbers are the same for each neuron in a given layer. Furthermore, each \nneuron makes the required connections with the nearest neurons of the other layer. In most \ncases, this completely specifies the wiring diagram. \n\nA typical topographic wiring diagram shown in Figure 1 misses an important biological de(cid:173)\ntail. In real brains, connections between cell bodies are implemented by neuronal processes: \naxons which carry nerve pulses away from the cell bodies and dendrites which carry signals \n\n\fOptimal Sizes of Dendritic and Axonal Arbors \n\n109 \n\nn J \n\nn 2 \n\nFigure 1: Wiring diagram of a topographic projection between input (circles) and output \n(squares) layers of neurons. Divergence, D, is the number of outgoing connections (here, \nD = 2) from an input neuron (wavey lines). Convergence, C, is the number of connections \nincoming (here, C = 4) to an output neuron (bold lines). Arrow shows the direction of \nsignal propagation. \n\na) ... ~~ ... \n\nwiring diagram \n\nb) ... QQQQQQQQQQQQ ... \n\n[5 \n\n[5 \n\nType I \n\nc) \"'9~?9~?'\" \n\nType II \n\nFigure 2: Two different arrangements implement the same wiring diagram. (a) Topographic \nwiring diagram with C = 6 and D = 1. (b) Arrangement with wide dendritic arbors and \nno axonal arbors (Type I) (c) Arrangement with wide axonal arbors and no dendritic arbors \n(Type II). Because convergence exceeds divergence type I has shorter wiring than type II. \n\ntowards cell bodies.[I] Therefore each connection is interrupted by a synapse which sepa(cid:173)\nrates an axon of one neuron from a dendrite of another. Both axons and dendrites branch \naway from cell bodies fonning arbors. \n\nIn general, a topographic projection with given divergence and convergence may be imple(cid:173)\nmented by axonal and dendritic arbors of different sizes, which depend on the locations of \nsynapses. For example, consider a wiring diagram with D = 1 and C = 6, Figure 2a. Nar(cid:173)\nrow axonal arbors may synapse onto wide dendritic arbors, Figure 2b or wide axonal arbors \nmay synapse onto narrow dendritic arbors, Figure 2c. I call these arrangements type I and \ntype II, correspondingly. The question is: which arbor sizes are preferred? \n\nI propose a rule which specifies the sizes of axonal arbors of input neurons and dendritic \narbors of output neurons in a topographic projection: High divergence/convergence ratio \nfavors wide axonal and narrow dendritic arbors while low divergence/convergence ratio \nfavors narrow axonal arbors and wide dendritic arbors. Alternatively, this rule may be for(cid:173)\nmulated in tenns of neuronal densities in the two layers: Sparser layer has wider arbors. \nIn the above example, divergence/convergence (and neuronal density) ratio is 116 and, ac(cid:173)\ncording to the rule, type I arrangement, Figure 2b, is preferred. \n\nIn this paper I derive a quantitative version of this rule from the principle of wiring economy \nwhich can be summarized as follows. [2, 3, 4, 5, 6] Space constraints require keeping the \nbrain volume to a minimum. Because wiring (axons and dendrites) takes up a significant \nfraction of the volume, evolution has probably designed axonal and dendritic arbors in a \nway that minimizes their total volume. Therefore we may understand the existing arbor \nsizes as a result of wiring optimization. \n\n\f110 \n\nD. B. Chklovsldi \n\nTo obtain the rule I formulate and solve a wiring optimization problem. The goal is to find \nthe sizes ofaxons and dendrites which minimize the total volume of wiring in a topographic \nwiring diagram for fixed locations of neurons. I specify the wiring diagram with divergence \nand convergence factors. Throughout most of the paper I assume that the cross-sectional \narea of dendrites and axons are constant and equal. Therefore, the problem reduces to the \nwire length minimization. Extension to unequal fiber diameters is given below. \n\n2 Topographic projection in two dimensions \n\nConsider two parallel layers of neurons with densities nl and n2. The topographic wiring \ndiagram has divergence and convergence factors, D and C, requiring each input neuron to \nconnect with D nearest output neurons and each output neuron with C nearest input neurons. \nAgain, the problem is to find the arrangement of arbors which minimizes the total length of \naxons and dendrites. For different arrangements I compare the wirelength per unit area, L. I \nassume that the two layers are close to each other and include only those parts of the wiring \nwhich are parallel to the layers. \n\nI start with a special case where each input neuron connects with only one output neuron \n(D = 1). Consider an example with C = 16 and neurons arranged on a square grid in \neach layer, Figure 3a. Two extreme arrangements satisfy the wiring diagram: type I has \nwide dendritic arbors and no axonal arbors, Figure 3b; type II has wide axonal arbors and \nno dendritic arbors, Figure 3c. I take the branching angles equal to 120\u00b0, an optimal value \nfor constant crossectional area. [ 4] Assuming \"point\" neurons the ratio of wire length for type \nI and type II arrangements: \n\nLr \n-L ~0.57. \nII \n\n(1) \n\nThus, the type I arrangement with wide dendritic arbors has shorter wire length. This con(cid:173)\nclusion holds for other convergence values much greater than one, provided D = 1. How(cid:173)\never, there are other arrangements with non-zero axonal arbors that give the same wire \nlength. One of them is shown in Figure 3d. Degenerate arrangements have axonal arbor \nwidth 0 < Sa < 1/ vnI where the upper bound is given by the approximate inter-neuronal \ndistance. This means that the optimal arbor size ratio for D = 1 \n\n(2) \n\nBy using the symmetry in respect to the direction of signal propagation I adapt this result \nfor the C = 1 case. For D > 1, arrangements with wide axonal arbors and narrow dendritic \narbors (0 < Sd < 1/ vnv have minimal wirelength. The arbor size ratio is \n\n(3) \n\nNext, I consider the case when both divergence and convergence are greater than one. \nDue to complexity of the problem I study the limit of large divergence and convergence \n(D, C \u00bb 1). I find analytically the optimal layout which minimizes the total length of ax(cid:173)\nons and dendrites. \n\nNotice that two neurons may form a synapse only if the axonal arbor of the input neuron \noverlaps with the dendritic arbor of the output neuron in a two-dimensional projection, Fig(cid:173)\nure 4. Thus the goal is to design optimal dendritic and axonal arbors so that each dendritic \narbor intersects C axonal arbors and each axonal arbor intersects D dendritic arbors. \nTo be specific, I consider a wiring diagram with convergence exceeding divergence, C > D \n(the argument can be readily adapted for the opposite case). I make an assumption, to be \n\n\fOptimal Sizes of Dendritic and Axonal Arbors \n\n111 \n\nb) \n\nwiring diagram \n\nType I \n\na) \n\nc) \n\nType II \n\nType I' \n\nFigure 3: Different arrangements implement the same wiring diagram in two dimensions. \n(a) Topographic wiring diagram with D = 1 and C = 16. (b) Arrangement with wide \ndendritic arbors and no axonal arbors, Type I. (c) Arrangement with wide axonal arbors and \nno dendritic arbors, Type II. Because convergence exceeds divergence type I has shorter \nwiring than type II. (d) Intermediate arrangement which has the same wire length as type I. \n\nFigure 4: Topographic projection between the layers of input (circles) and output (squares) \nneurons. For clarity, out of the many input and output neurons with overlapping arbors only \nfew are shown. The number of input neurons is greater than the number of output neurons \n(C / D > 1). Input neurons have narrow axonal arbors of width Sa connected to the wide but \nsparse dendritic arbors of width Sd. Sparseness of the dendritic arbor is given by Sa because \nall the input neurons spanned by the dendritic arbor have to be connected. \n\n\f112 \n\nD. B. Chklovskii \n\nverified later, that dendritic arbor diameter Sd is greater than axonal one, Sa. In this regime \neach output neuron's dendritic arbor forms a sparse mesh covering the area from which sig(cid:173)\nnals are collected, Figure 4. Each axonal arbor in that area must intersect the dendritic arbor \nmesh to satisfy the wiring diagram. This requires setting mesh size equal to the axonal arbor \ndiameter. \n\nBy using this requirement I express the total length of axonal and dendritic arbors as a func(cid:173)\ntion of only the axonal arbor size, Sa. Then I find the axonal arbor size which minimizes the \ntotal wirelength. Details of the calculation will be published elsewhere. Here, I give an in(cid:173)\ntuitive argument for why in the optimal layout both axonal and dendritic size are non-zero. \nConsider two extreme layouts. In the first one, dendritic arbors have zero width, type II. In \nthis arrangement axons have to reach out to every output neuron. For large convergence, \nC \u00bb 1, this is a redundant arrangement because of the many parallel axonal wires whose \nsignals are eventually merged. In the second layout, axonal arbors are absent and dendrites \nhave to reach out to every input neuron. Again, because each input neuron connects to many \noutput neurons (large divergence, D \u00bb 1) many dendrites run in parallel inefficiently car(cid:173)\nrying the same signal. A non-zero axonal arbor rectifies this inefficiency by carrying signals \nto several dendrites along one wire. \n\nI find that the optimal ratio of dendritic and axonal arbor diameters equals to the square \nroot of the convergenceldivergenceratio, or, alternatively, to the square root of the neuronal \ndensity ratio: \n\n(4) \n\nSince I considered the case with C > D this result also justifies the assumption about axonal \narbors being smaller than dendritic ones. \n\nFor arbitrary axonal and dendritic cross-sectional areas, ha and hd, expressions ofthis Sec(cid:173)\ntion are modified. The wiring economy principle requires minimizing the total volume oc(cid:173)\ncupied by axons and dendrites resulting in the following relation for the optimal arrange(cid:173)\nment: \n\n(5) \n\nNotice that in the optimal arrangement the total axonal volume of input neurons is equal to \nthe total dendritic volume of the output neurons. \n\n3 Discussion \n\n3.1 Comparison of the theory with anatomical data \n\nThis theory predicts a relationship between the con-/divergence ratio and the sizes of axonal \nand dendritic arbors. I test these predictions on several cases of topographic projection in \ntwo dimensions. The predictions depend on whether divergence and convergence are both \ngreater than one or not. Therefore, I consider the two regimes separately. \n\nFirst, I focus on topographic projections of retinal neurons whose divergence factor is equal \nor close to one. Because retinal neurons use mostly graded potentials the difference between \naxons and dendrites is small and I assume that their cross-sectional areas are equal. The \ntheory predicts that the ratio of dendritic and axonal arbor sizes must be greater than the \nsquare root of the input/output neuronal density ratio, Sd/ Sa > (ndn2)1/2 (Eq.2). \nI represent the data on the plot of the relative arbor diameter, Sd/ Sa, vs. the square root \nof the relative densities, (ndn2)1/2, (Figure 5). Because neurons located in the same layer \nmay belong to different classes, each having different arbor size and connectivity, I plot data \n\n\fOptimal Sizes of Dendritic and Axonal Arbors \n\n113 \n\ns~/s \u2022. .--_______ ...,--______ --.\" \n\n50 \n\nI. \n\n.. I \n\nU5 \n\nuz \n\nA C \n\no \n\nC=l \n\n5 \n\nI. \n\nZO \n\nFigure 5: Anatomical data for several pairs of retinal cell classes which form topographic \nprojections with D = 1. All the data points fall in the triangle above the Sd/ Sa = \n(ndn2)1/2 line in agreement with the theoretical prediction, Eq.2. The following data has \nbeen used: 0 - midget bipolar -+ midget ganglion,[7, 8, II]; U - diffuse bipolar -+ parasol \nganglion,[7, 9]; 'V - rods -+ rod bipolar,[lO]; b. - cones -+ HI horizontals.[12]; 0 - rods -+ \ntelodendritic arbors of HI horizontals,[ 13]. \n\nfrom different classes separately. All the data points lie above the Sd/ Sa = (ndn2)1/2line \nin agreement with the prediction. \n\nSecond, I apply the theory to cerebellar neurons whose divergence and convergence are both \ngreater than one. I consider a projection from granule cell axons (parallel fibers) onto Purk(cid:173)\ninje cells. Ratio of granule cells to Purkinje cells is 33()(),[14], indicating a high conver(cid:173)\ngence/divergence ratio. This predicts a ratio of dendritic and axonal arbor sizes of 58. This \nis qualitatively in agreement with wide dendritic arbors of Purkinje cells and no axonal ar(cid:173)\nbors on parallel fibers. \n\nQuantitative comparison is complicated because the projection is not strictly two(cid:173)\ndimensional: Purkinje dendrites stacked next to each other add up to a significant third di(cid:173)\nmension. Naively, given that the dendritic arbor size is about 400ILm Eq.4 predicts axonal \narbor of about 7 ILm. This is close to the distance between two adjacent Purkinje cell arbors \nof about 9 ILm. Because the length of parallel fibers is greater than 7 ILm absence of axonal \narbors comes as no surprise. \n\n3.2 Other factors affecting arbor sizes \n\nOne may argue that dendrites and axons have functions other than linking cell bodies to \nsynapses and, therefore, the size of the arbors may be dictated by other considerations. Al(cid:173)\nthough I can not rule out this possibility, the primary function ofaxons and dendrites is to \nconnect cell bodies to synapses in order to conduct nerve pulses between them. Indeed, if \nneurons were not connected more sophisticated effects such as non-linear interactions be(cid:173)\ntween different dendritic inputs could not take place. Hence the most basic parameters of \naxonal and dendritic arbors such as their size should follow from considerations of connec(cid:173)\ntivity. \n\nAnother possibility is that the size of dendritic arbors is dictated by the surface area needed \n\n\fJ14 \n\nD. B. Chklovskii \n\nto arrange all the synapses. This argument does not specify the arbor size, however: a com(cid:173)\npact dendrite of elaborate shape can have the same surface area as a wide dendritic arbor. \n\nFinally, agreement of the predictions with the existing anatomical data suggests that the rule \nis based on correct principles. Further extensive testing of the rule is desirable. Violation \nof the rule in some system would suggest the presence of other overriding considerations in \nthe design of that system, which is also interesting. \n\nAcknowledgements \n\nI benefited from helpful discussions with E.M. Callaway, EJ. Chichilnisky, H.J. Karten, \nC.P. Stevens and TJ. Sejnowski and especially with A.A. Koulakov. I thank G.D. Brown \nfor suggesting that the size of axonal and dendritic arbors may be related to con-/divergence. \n\nReferences \n\n[1] Cajal, S.R.y. (1995a). Histology of the nervous system p.95 (Oxford University Press, New(cid:173)\n\nYork). \n\n[2] Cajal, S.R.y. ibid. p.1l6. \n[3] Mitchison, G. (1991). Neuronal branching patterns and the economy of cortical wiring. Proc R \n\nSoc Lond B Bioi Sci 245, 151-8. \n\n[4] Chemiak, C. (1992). Local optimization of neuron arbors, Bioi Cybem 66,503-510. \n[5] Young, M.P. (1992). 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J Neurosci 16,2724-2739. \n\n\f", "award": [], "sourceid": 1732, "authors": [{"given_name": "Dmitri", "family_name": "Chklovskii", "institution": null}]}