{"title": "Branching Law for Axons", "book": "Advances in Neural Information Processing Systems", "page_first": 197, "page_last": 204, "abstract": null, "full_text": "Branching Law for Axons \n\nDmitri B. Chklovskii and Armen Stepanyants \n\nCold Spring Harbor Laboratory \n\n1 Bungtown Rd. \n\nCold Spring Harbor, NY 11724 \n\nmitya@cshl. edu \n\nstepanya@cshl.edu \n\nAbstract \n\nWhat determines the caliber of axonal branches? We pursue the \nhypothesis that the axonal caliber has evolved to minimize signal \npropagation delays, while keeping arbor volume to a minimum. We \nshow that for a general cost function the optimal diameters of \nmother (do) and daughter (d], d2 ) branches at a bifurcation obey \nh \na ranc mg aw: \ne envatIOn re les on t e \nfact that the conduction speed scales with the axon diameter to the \npower V (v = 1 for myelinated axons and V = 0.5 \nmyelinated axons). We test the branching law on the available \nexperimental data and find a reasonable agreement. \n\nd v+2 d v+2 d v+2 Th d \" \n0 = ] + 2 \n\nfor non(cid:173)\n\nl' \n\n. \n\nb \n\nh\u00b7 \n\n1 \n\n1 Introduction \n\nMulti-cellular organisms have solved the problem of efficient transport of nutrients \nand communication between their body parts by evolving spectacular networks: \ntrees, blood vessels, bronchs, and neuronal arbors. These networks consist of \nsegments bifurcating into thinner and thinner branches. Understanding of branching \nin transport networks has been advanced through the application of the optimization \ntheory ([1], [2] and references therein) . Here we apply the optimization theory to \nexplain the caliber of branching segments in communication networks, i.e. neuronal \naxons. \n\nAxons in different organisms vary in caliber from O. ll1m (terminal segments in \nneocortex) to lOOOl1m (squid giant axon) [3]. What factors could be responsible for \nsuch variation in axon caliber? According to the experimental data [4] and cable \ntheory [5], thicker axons conduct action potential faster, leading to shorter reaction \ntimes and, perhaps, quicker thinking. This increases evolutionary fitness or, \nequivalently, reduces costs associated with conduction delays. So, why not make all \nthe axons infinitely thick? It is likely that thick axons are evolutionary costly \nbecause they require large amount of cytoplasm and occupy valuable space [6], [7]. \nThen, is there an optimal axon caliber, which minimizes the combined cost of \nconduction delays and volume? \n\n\fIn this paper we derive an expression for the optimal axon diameter, which \nminimizes the combined cost of conduction delay and volume. Although the relative \ncost of del ay and volume is unknown, we use this expression to derive a law \ndescribing segment caliber of branching axons with no free parameters. We test this \nlaw on the published anatomical data and find a satisfactory agreement. \n\n2 Derivation of the branching law \n\nAlthough our theory holds for a rather general class of cost functions (see Methods), \nwe start, for the sake of simplicity, by deriving the branching law in a special case \nof a linear cost function. Detrimental contribution to fitness , It , of an axonal \nsegment of length , L , can be represented as the sum of two terms, one proportional \nto the conduction delay along the segment, T, and the other - to the segment \nvolume, V: \n\n(1) \nHere, a and f3 are unknown but constant coefficients which reflect the rel ative \ncontribution to the fitness cost of the signal propagation delay and the axonal \nvolume. \n\nIt =aT+ jJV. \n\n5rr--,----.---.----,---.----.---.--7TO \n\n4.5 \n\n4 \n\n3.5 \n\n2.5 \n\n2 \n\n1.5 \n\n0.5 \n\ndelay cost \n\n~l/d \n\n1 \n\n1.5 \n\n2 \n\n2.5 \n\n3 \n\n3.5 \n\n4 \n\ndiameter, d \n\nFigure 1: Fitness cost of a myelinated axonal segment as a function of its diameter. \nThe lines show the volume cost, the delay cost, and the total cost. Notice that the \ntotal cost has a minimum. Diameter and cost values are normalized to their \nrespective optimal values. \n\nWe look for the axon caliber d that minimizes the cost function It. To do this, we \nrewrite It as a function of d by noticing the following relations: i) Volume, \n\n\fV=!!...Ld 2 . \n' \n\n4 \n\nii) Time delay, T=.!::....; \ns \n\niii) Conduction velocity s=kd \n\nfor \n\nmyelinated axons (for non-myelinated axons, see Methods): \n\n(2) \n\nThis cost function contains two terms, which have opposite dependence on d, and \nhas a minimum, Fig. 1. \n\nNext, by setting -\n\n= 0 we find that the cost is minimized by the following axonal \n\na~ \nad \n\ncaliber: \n\n1/3 \nd=~ \n\n)\nlrkfJ \n\n( \n\n(3) \n\nThe utility of this result may seem rather limited because the relative cost of time \ndelays vs. volume, a/ fJ ' is unknown. \n\nFigure 2: A simple axonal arbor with a single branch point and three axonal \nsegments. Segment diameters are do, d \" and d 2 . Time delays along each segment are \nto, t\" and t2. The total time delay down the first branch is T,= to +f\" and the second -\nTz= to +f2\u00b7 \n\nHowever, we can apply this result to axonal branching and arrive at a testable \nprediction about the relationship among branch diameters without knowing the \nrelative cost. To do this we write the cost function for a bifurcation consisting of \nthree segments, Fig. 2: \n\n(4) \n\nwhere to \n\nsegment 1, \n\nis a conduction delay along segment 0, \n\n- conduction delay along \nt2 - conduction delay along segment 2. Coefficients a 1 and a 2 \n\nt1 \n\n\frepresent relative costs of conduction delays for synapses located on the two \ndaughter branches and may be different. We group the terms corresponding to the \nsame segment together: \n\n(5) \n\nWe look for segment diameters, which minimize this cost function. To do this we \nmake the dependence on the diameters explicit and differentiate in respect to them. \nBecause each term in Eq. (5) depends on the diameter of only one segment the \nvariables separate and we arrive at expressions analogous to Eq.(3): \n\n( JI/3 \n\nd = 2al \nkfJn \nI \n\n' \n\n( Jif3 \n\nd = 2a2 \nk {In \n\n2 \n\nIt is easy to see that these diameters satisfy the following branching law: \n\ndg = d? +d~ . \n\n(6) \n\n(7) \n\nSimilar expression can be derived for non-myelinated axons (see Methods) . In this \ncase, the conduction velocity scales with the square root of segment diameter, \nresulting in a branching exponent of 2.5 . \n\nWe note that expressions analogous to Eq. (7) have been derived for blood vessels, \ntree branching and bronchs by balancing metabolic cost of pumping viscous fluid \nand volume cost [8], [9]. Application of viscous flow to dendrites has been \ndiscussed in [10]. However, it is hard to see how dendrites could be conduits to \nviscous fluid if their ends are sealed. \n\nRail [11] has derived a similar law for branching dendrites by postulating \nimpedance matching: \n\n(8) \n\nHowever, the main purpose of Rail's law was to simplify calculations of dendritic \nconduction rather than to explain the actual branch caliber measurements. \n\n3 Comparison with experiment \n\nWe test our branching law, Eq.(7), by comparing it with the data obtained from \nmyelinated motor fibers of the cat [12] , Fig. 3. Data points represent 63 branch \npoints for which all three axonal calibers were available. Eq.(7) predicts that the \ndata should fall on the line described by: \n\n(9) \n\nwhere exponent TJ = 3 . Despite the large spread in the data it is consistent with our \npredictions. In fact, the best fit exponent, TJ = 2.57 , is closer to our prediction than \nto Rail ' s law, TJ = 1.5. \nWe also show the histogram of the exponents TJ obtained for each of 63 branch \npoints from the same data set, Fig. 4. The average exponent, TJ = 2.67 , is much \n\n\fcloser to our predicted value for myelinated axons, \n'7 = 1.5. \n\n'7 = 3, than to RaIl's law, \n\n'\"tj\"'\" \n'--.. \n'\"tj '-< \n\n0.9 \n\n0.8 \n\n0.7 \n\n0.6 \n\n0.5 \n\n0.4 \n\n0.3 \n\n0.2 \n\n0.1 \n\nRaZZ's law, \n1] = 1.5 \n\nO L-~~~--~~~---L---L--~--~~~~ \no \n\n0.6 \n\n0.8 \n\n0.5 \n\n0.9 \n\n0.1 \n\n0.2 \n\n0.3 \n\n0.4 \n\n0.7 \n\nFigure 3: Comparison of the experimental data (asterisks) [12] with theoretical \npredictions. Each axonal bifurcation (with d, =F- d2 ) is represented in the plot twice. \nThe lines correspond to Eq.(9) with various values of the exponent: the RaIl's law, \n'7 = 1.5 , the best-fit exponent, '7 = 2.57 , and our prediction for myelinated axons, \n'7 = 3. \n\nAnalysis of the experimental data reveals a large spread in the values of the \nexponent, '7. This spread may arise from the biological variability in the axon \ndiameters, other factors influencing axon diameters, or measurement errors due to \nthe finite resolution of light microscopy. Although we cannot distinguish between \nthese causes, we performed a simulation showing that a reasonable measurement \nerror is sufficient to account for the spread. \n\nFirst, based on the experimental data [12], we generate a set of diameters do, d, \nand d 2 at branch points, which satisfy Eq. (7). We do this by taking all diameter \npairs at branch point from the experimental data and calculating the value of the \nthird diameter according to Eq. (7). Next we simulate the experimental data by \nadding Gaussian noise to all branch diameters, and calculate the probability \ndistribution for the exponent '7 resulting from this procedure. The line in Fig. 4 \nshows that the spread in the histogram of branching exponent could be explained by \nGaussian measurement error with standard deviation of O.4.um. This value of \nstandard deviation is consistent with 0.5.um precision with which diameter \nmeasurements are reported in [12]. \n\n\f14 \n\n12 \n\n10 RaIl's \n\n8 \n\n6 \n\n2 \n\n0 \n0 \n\naverage \nexponent \n\npredicted \nexponent \n\n2 \n\n3 \n\n6 \n\nFigure 4: Experimentally observed spread in the branching exponent may arise from \nthe measurement errors. The histogram shows the distribution of the exponent '7, \nEq. (9) , calculated for each axonal bifurcation [12]. The average exponent is \n'7 = 2.67 . The line shows the simulated distribution of the exponent obtained in the \npresence of measurement errors. \n\n4 Conclusion \n\nStarting with the hypotheses that axonal arbors had been optimized in the course of \nevolution for fast signal conduction while keeping arbor volume to a minimum we \nderived a branching law that relates segment diameters at a branch point. The \nderivation was done for the cost function of a general form , and relies only on the \nknown scaling of signal propagation velocity with the axonal caliber. This law is \nconsistent with the available experimental data on myelinated axons. The observed \nspread in the branching exponent may be accounted for by the measurement error. \nMore experimental testing is clearly desirable. \n\nWe note that similar considerations could be applied to dendrites. There, similar to \nnon-myelinated axons, time delay or attenuation of passively propagating signals \nscales as one over the square root of diameter. This leads to a branching law with \nexponent of 5/2. However, the presence of reflections from branch points and \nactive conductances is likely to complicate the picture. \n\n5 Methods \n\nThe detrimental contribution of an axonal arbor to the evolutionary fitness can be \nquantified by the cost, Q:. We postulate that \nis a \nmonotonically increasing function of the total axonal volume per neuron, V , and all \nsignal propagation delays, Tj\n\n, from soma to j -th synapse, where j = 1,2,3, ... : \n\nthe cost function , Q:, \n\n(10) \n\n\fBelow we show that this rather general cost function (along with biophysical \nproperties ofaxons) is minimized when axonal caliber satisfies the following \nbranching law : \n\n( 11) \n\nwith branching exponent '7 = 3 for myelinated and '7 = 2.5 for non-myelinated \naxons. \n\nAlthough we derive Eq. (11) for a single branch point, our theory can be trivially \nextended to more complex arbor topologies. We rewrite the cost function, ([, in \nterms of volume contributions, ~, of i -th axonal segment to the total volume of \nthe axonal arbor, V , and signal propagation delay, ti , occurred along i -th axonal \nsegment. The cost function reduces to: \n\n(12) \n\nNext, we express volume and signal propagation delay of each segment as a \nfunction of segment diameter. The volume of each cylindrical segment is given by: \n\n1r \n4 \n\nV =-Ld, \n\nI \n\nI \n\nI \n\n2 \n\n(13) \n\nwhere Li and di are segment length and diameter, correspondingly. Signal \npropagation delay, t i , is given by the ratio of segment length, Li , and signal speed, \nSi' Signal speed along axonal segment, in turn, depends on its diameter as : \n\nwhere V = 1 for myelinated [4] and V = 0.5 for non-myelinated fibers [5]. As a \nresult propagation delay along segment i is: \n\n(14) \n\n(15) \n\nSubstituting Eqs. (13), (15) into the cost function, Eq. (12) , we find the dependence \nof the cost function on segment diameters, \n\nt1'(1r Lod 2 1r ~d2 \n~ -\n\n+-\n4 \n\n0 \n\n4 \n\nI \n\n1r ~d2 Lo \n+-\n4 \n2 ' kd v \no \n\n-+ - -+ -\nkd v \n2 \n\n~ Lo \nkd v ' kd v \n0 \n\nI \n\n~ J \n\n. \n\n(16) \n\nTo find the diameters of all segments, which minimize the cost function ([, we \ncalculate its partial derivatives with respect to all segment diameters and set them to \nzero: \n\n\f~=Q:'!!...' d -Q:' v~ =0 \nad \n2 \n\nv 2 '--2 2 \n\nT2 kd v+1 \n\n2 \n\nBy solving these equations we find the optimal segment diameters: \n\n(17) \n\n2v(Q:~ +Q:;. ) \n\nI \n\n2 \n\ndv+2 = \no \n\nk1rQ:~' \n\n2vQ:' \nd v+2 =----l.. \nk1rQ:~ , \n1 \n\n2vQ:' \nd v+2 =----!J.. \nk1rQ:~ . \n2 \n\n(18) \n\nThese equations imply that the cost function is minimized when the segment \ndiameters at a branch point satisfy the following expression (independent of the \nparticular form of the cost function, which enters Eq. (18) through the partial \nderivatives Q:~ , Q:~ , and Q:~ ): \n\nI \n\n2 \n\nd\"=d\"+d\" \n2 , \n\no \n\n1 \n\nl] = v+2. \n\n(19) \n\nReferences \n\n[I] Brown, J. H., West, G. B., and Santa Fe Institute (Santa Fe N.M.). (2000) Scaling in \nbiology. Oxford; New York: Oxford University Press. \n\n[2] Weibel, E. R. (2000) Symmorphosis : on form and function in shaping life. Cambridge, \nMass.; London: Harvard University Press. \n\n[3] Purves, D . (1997) Neuroscience. Sunderland, Mass.: Sinauer Associates. \n\n[4] Rushton, W. A. H. (1951) A theory of the effects of fibre size in medullated nerve. J \nPhysiol 115, 10 1-122. \n\n[5] Hodgkin, A. L. (1954) A note on conduction velocity. J Physioll25, 221-224. \n\n[6] Cajal, S. R. y. (1999) Texture of the Nervous System of Man and the Vertebrates, Volume \nl. New York: Springer. \n\n[7] Chklovskii, D. B., Schikorski, T., and Stevens, C. F. (2002) Wiring optimization in \ncortical circuits. Neuron 34,341-347. \n\n[8] Murray, C. D. (1926) The physiological principle of minimum work. 1. The vascular \nsystem and the cost of blood volume. PNAS 12, 207-214. \n\n[9] Murray, C. D. (1927) A relationship between circumference and weight in trees and its \nbearing on branching angles. J Cen Physioll0, 725-729. \n\n[10] Cherniak, C., Changizi, M., and Kang D.W. (1999) Large-scale optimization of neuron \narbors. Phys Rev E 59,6001-6009. \n\n[11] Rail , W. (1959) Branching dendritic trees and motoneuron membrane resistivity. Exp \nNeuroll,491-527. \n\n[12] Adal, M. N., and Barker, D . (1965) Intramuscular branching of fusimotor fibers. J \nPhysiol 177, 288-299. \n\n\f", "award": [], "sourceid": 2265, "authors": [{"given_name": "Dmitri", "family_name": "Chklovskii", "institution": null}, {"given_name": "Armen", "family_name": "Stepanyants", "institution": null}]}