{"title": "Wiring Optimization in the Brain", "book": "Advances in Neural Information Processing Systems", "page_first": 103, "page_last": 107, "abstract": null, "full_text": "Wiring optimization in the brain \n\nDmitri B. Chklovskii \n\nSloan Center for \n\nTheoretical Neurobiology \n\nThe Salk Institute \nLa Jolla, CA 92037 \n\nmitya@salk.edu \n\nCharles F. Stevens \n\nHoward Hughes Medical Institute \nand Molecular Neurobiology Lab \n\nThe Salk Institute \nLa Jolla, CA 92037 \nstevens@salk.edu \n\nAbstract \n\nThe complexity of cortical circuits may be characterized by the number \nof synapses per neuron. We study the dependence of complexity on the \nfraction of the cortical volume that is made up of \"wire\" (that is, ofaxons \nand dendrites), and find that complexity is maximized when wire takes \nup about 60% of the cortical volume. This prediction is in good agree(cid:173)\nment with experimental observations. A consequence of our arguments \nis that any rearrangement of neurons that takes more wire would sacrifice \ncomputational power. \n\nWiring a brain presents formidable problems because of the extremely large number of con(cid:173)\nnections: a microliter of cortex contains approximately 105 neurons, 109 synapses, and 4 \nkm ofaxons, with 60% of the cortical volume being taken up with \"wire\", half of this by \naxons and the other half by dendrites. [ 1] Each cortical neighborhood must have exactly the \nright balance of components; if too many cell bodies were present in a particular mm cube, \nfor example, insufficient space would remain for the axons, dendrites and synapses. Here \nwe ask \"What fraction of the cortical volume should be wires (axons + dendrites)?\" We ar(cid:173)\ngue that physiological properties ofaxons and dendrites dictate an optimal wire fraction of \n0.6, just what is actually observed. \n\nTo calculate the optimal wire fraction, we start with a real cortical region containing a fixed \nnumber of neurons, a mm cube, for example, and imagine perturbing it by adding or sub(cid:173)\ntracting synapses and the axons and dendrites needed to support them. The rules for per(cid:173)\nturbing the cortical cube require that the existing circuit connections and function remain \nintact (except for what may have been removed in the perturbation), that no holes are cre(cid:173)\nated, and that all added (or subtracted) synapses are typical of those present; as wire volume \nis added, the volume of the cube of course increases. The ratio of the number of synapses \nper neuron in the perturbed cortex to that in the real cortex is denoted by 8, a parameter \nwe call the relative complexity. We require that the volume of non-wire components (cell \nbodies, blood vessels, glia, etc) is unchanged by our perturbation and use 4> to denote the \nvolume fraction of the perturbed cortical region that is made up of wires (axons + dendrites; \n4> can vary between zero and one), with the fraction for the real brain being 4>0. The relation \nbetween relative complexity 8 and wire volume fraction 4> is given by the equation (derived \nin Methods) \n\n8 - -\n\n1 (1-4\u00bb2/34> \n-\n4>0\u00b7 \n\n-,\\5 1 - 4>0 \n\n(I) \n\n\f104 \n\nD. B. Chklovskii and C. F Stevens \n\n2 \n\n-(I) -?1 \n\n)( \n..!! 1 \nQ.. \nE \no \nu \n\n:\\ = .9 \n\n:\\=1 \n\nO~----r---~-----.----~--~ \n0.0 \n1.0 \n\n0.6 \n\n0.8 \n\n0.2 \n\n0.4 \n\nWire fraction (+) \n\nFigure I: Relative complexity (0) as a function of volume wire fraction (e/\u00bb. The graphs are \ncalculated from equation (1) for three values of the parameter A as indicated; this parameter \ndetermines the average length of wire associated with a synapse (relative to this length for \nthe real cortex, for which (A = 1). Note that as the average length of wire per synapse \nincreases, the maximum possible complexity decreases. \n\nFor the following discussion assume that A = 1; we return to the meaning of this parameter \nlater. To derive this equation two assumptions are made. First, we suppose that each added \nsynapse requires extra wire equal to the average wire length and volume per synapse in the \nunperturbed cortex. Second, because adding wire for new synapses increases the brain vol(cid:173)\nume and therefore increases the distance axons and dendrites must travel to maintain the \nconnections they make in the real cortex, all of the dendrite and unmyelinated axon diam(cid:173)\neters are increased in proportion to the square of their length changes in order to maintain \nthe intersynaptic conduction times[2] and dendrite cable lengths[3] as they are in the actual \ncortex. If the unmyelinated axon diameters were not increased as the axons become longer, \nfor example, the time for a nerve impulse to propagate from one synapse to the next would \nbe increased and we would violate our rule that the existing circuit and its function be un(cid:173)\nchanged. We note that the vast majority of cortical axons are unmyelinated.[l] The plot of \no as a function of e/> is parabolic-like (see Figure 1) with a maximum value at e/> = 0.6, a \npoint at which dO/de/> = O. This same maximum value is found for any possible value of \ne/>o, the real cortical wire fraction. \n\nWhy does complexity reach a maximum value at a particular wire fraction? When wire and \nsynapses are added, a series of consequences can lead to a runaway situation we call the \nwiring catastrophe. If we start with a wire fraction less than 0.6, adding wire increases the \ncortical volume, increased volume makes longer paths for axons to reach their targets which \nrequires larger diameter wires (to keep conduction delays or cable attenuation constant from \none point to another), the larger wire diameters increase cortex volume which means wires \nmust be longer, etc. While the wire fraction e/> is less than 0.6, increasing complexity is \naccompanied by finite increases in e/>. At e/> = 0.6 the rate at which wire fraction increases \nwith complexity becomes infinite de/>/dO --t 00); we have reached the wiring catastrophe. \nAt this point, adding wire becomes impossible without decreasing complexity or making \nother changes - like decreasing axon diameters - that alter cortical function. The physical \ncause of the catastrophe is a slow growth of conduction velocity and dendritic cable length \nwith diameter combined with the requirement that the conduction times between synapses \n(and dendrite cable lengths) be unchanged in the perturbed cortex. \n\nWe assumed above that each synapse requires a certain amount of wire, but what if we could \n\n\fWiring Optimization in the Brain \n\n105 \n\nadd new synapses using the wire already present? We do not know what factors determine \nthe wire volume needed to support a synapse, but if the average amount of wire per synapse \ncould be less (or more) than that in the actual cortex, the maximum wire fraction would still \nbe 0.6. Each curve in Figure 1 corresponds to a different assumed average wire length re(cid:173)\nquired for a synapse (determined by A), and the maximum always occurs at 0.6 independent \nof A. In the following we consider only situations in which A is fixed. \nFor a given A, what complexity should we expect for the actual cortex? Three arguments \nfavor the maximum possible complexity. The greatest complexity gives the largest num(cid:173)\nber of synapses per neuron and this permits more bits of information to be represented per \nneuron. Also, more synapses per neuron decreases the relative effect caused by the loss or \nmalfunction of a single synapse. Finally, errors in the local wire fraction would minimally \naffect the local complexity because d(} / dqJ = 0 at if> = 0.6. Thus one can understand why \nthe actual cortex has the wire fraction we identify as optimal. [ 1] \n\nThis conclusion that the wire fraction is a maximum in the real cortex has an interesting \nconsequence: components of an actual cortical circuit cannot be rearranged in a way that \nneeds more wire without eliminating synapses or reducing wire diameters. For example, \nif intermixing the cell bodies of left and right eye cells in primate primary visual cortex \n(rather than separating them in ocular dominance columns) increased the average length of \nthe wire[4] the existing circuit could not be maintained just by a finite increase in volume. \nThis happens because a greater wire length demanded by the rearrangement of the same \ncircuit would require longer wire per synapse, that is, an increased A. As can be seen from \nFigure I, brains with A > 1 can never achieve the complexity reached at the maximum of \nthe A = 1 curve that corresponds to the actual cortex. \nOur observations support the notion that brains are arranged to minimize wire length. This \nidea, dating back to Cajal[5], has recently been used to explain why retinotopic maps ex(cid:173)\nist[6],[7], why cortical regions are separated, why ocular dominance columns are present in \nprimary visual cortex[4],[8],[9] and why the cortical areas and flat worm ganglia are placed \nas they are. [ 10-13] We anticipate that maximal complexity/minimal wire length arguments \nwill find further application in relating functional and anatomical properties of brain. \n\nMethods \n\nThe volume of the cube of cortex we perturb is V, the volume of the non-wire portion is \nW (assumed to be constant), the fraction of V consisting of wires is if>, the total number of \nsynapses is N, the average length of axonal wire associated with each synapse is s, and the \naverage axonal wire volume per unit length is h; the corresponding values for dendrites are \nindicated by primes (s' and h'). The unperturbed value for each variable has a 0 subscript; \nthus the volume of the cortical cube before it is perturbed is \nVo = Wo + No(soho + s~h~). \n\n(2) \n\nWe now define a \"virtual\" perturbation that we use to explore the extent to which the actual \ncortical region contains an optimal fraction of wire. If we increase the number of synapses \nby a factor () and the length of wire associated with each synapse by a factor A, then the \nperturbed cortical cube's volume becomes \n\nVo = Wo + A(} (Nosoho ~ + Nos~h~ ~~) (V/VO)1/3 . \n\n(3) \n\nThis equation allows for the possibility that the average wire diameter has been perturbed \nand increases the length of all wire segments by the \"mean field\" quantity (V/VO)1/3 to \ntake account of the expansion of the cube by the added wire; we require our perturbation \ndisperses the added wire as uniformly as possible throughout the cortical cube. \n\n\f106 \n\nD. B. Chklovskii and C. F. Stevens \n\nTo simplify this relation we must eliminate h/ ho and h' / h&; we consider these terms in tum. \nWhen we perturb the brain we require that the average conduction time (s/u, where u is the \nconduction velocity) from one synapse to the next be unchanged so that s/u = so/uo, or \n\n~ = !.... = '\\so(V/VO)1/3 = '\\(V/VO)1/3. \nuo \n\nSo \n\nSo \n\n(4) \n\nBecause axon diameter is proportional to the square of conduction velocity u and the axon \nvolume per unit length h is proportional to diameter squared, h is proportional to u 4 and the \nratio h/ ho can be written as \n\n(5) \n\nFor dendrites, we require that their length from one synapse to the next in units of the cable \nlength constant be unchanged by the perturbation. The dendritic length constant is propor(cid:173)\ntional to the square root of the dendritic diameter d, so 8/.fd = 80/ ViIO or \n\n~ = (;~)2 = (,\\(V/VO)1/3f =,\\2(V/Vo)2/3. \n\n(6) \n\nBecause dendritic volumes per unit length (h and h') vary as the square of the diameters, \nwe have that \n\n~~ = (~) 2 =,\\4 (V/VO)4/3. \n\n(7) \n\nThe equation (2) can thus be rewritten as \n\nV = Wo + No(soho + s~h~)B,\\5 (V/Vo)5/3 . \n\n(8) \nDivide this equation by Vo, define v = VIVo, and recognize that Wo/Vo = (1 - 4>0) and \nthat 4>0 = No(soho + s&h&)/Vo ; the result is \n\n(9) \n\nBecause the non-wire volume is required not to change with the perturbation, we know that \nWo = (1 - 4>o)Vo = (1 - 4\u00bbV which means that v = (1 - 4>0)/(1 - 4\u00bb; substitute this in \nequation (9) and rearrange to give \n\n= ~ ( 1 - 4> ) 2/3 .!t \nB \n4>0 . \n\n,\\5 1 - 4>0 \n\n(1) \n\nthe equation used in the main text. \n\nWe have assumed that conduction velocity and the dendritic cable length constant vary ex(cid:173)\nactly with the square root of diameter[2],[ 14] but if the actual power were to deviate slightly \nfrom 112 the wire fraction that gives the maximum complexity would also differ slightly \nfrom 0.6. \n\nAcknowledgments \n\nThis work was supported by the Howard Hughes Medical Institute and a grant from NIH to \nC.F.S. D.C. was supported by a Sloan Fellowship in Theoretical Neurobiology. \n\n\fWiring Optimization in the Brain \n\n107 \n\nReferences \n\n[1] Braitenberg. V. & Schuz. A. Cortex: Statistics and Geometry of Neuronal Connectivity (Springer. \n1998). \n\n[2] Rushton. W.A.H. A Theory of the Effects of Fibre Size in Medullated Nerve. J. Physiol. 115. \n101-122 (1951). \n\n[3] Bekkers. J.M. & Stevens. C.F. Two different ways evolution makes neurons larger. Prog Brain Res \n83. 37-45 (1990). \n\n[4] Mitchison. G. Neuronal branching patterns and the economy of cortical wiring. Proc R Soc Lond \nB Bioi Sci 245.151-158 (1991). \n\n[5] Cajal. S.R.Y. Histology of the Nervous System 1-805 (Oxford University Press, 1995). \n[6] Cowey. A. Cortical maps and visual perception: the Grindley Memorial Lecture. Q J Exp Phychol \n31. 1-17 (1979). \n\n[7] Allman J.M. & Kaas J.H. The organization of the second visual area (V II) in the owl monkey: a \nsecond order transformation of the visual hemifield. 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MD. 1977). \n\n\f", "award": [], "sourceid": 1728, "authors": [{"given_name": "Dmitri", "family_name": "Chklovskii", "institution": null}, {"given_name": "Charles", "family_name": "Stevens", "institution": null}]}