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A mathematical model for the growth of dendritic trees
243
Neuroscience Letters, 54 (198 5) 243-249 El sevier Scien tific Publish er s Ireland Ltd .
NSL 03168
A MATHEMATICAL MODEL FOR THE GROWTH OF DENDRITIC TREES WILLIAM IRELAND l . ' , J ACK HEIDEL 2 . " and ETSURO UEMURA 1
'D epartment of A natomy, College of Veterinary Medicine, Io wa State University, and Irepartment of Mathematics, Io wa State University, A mes, IA 50011 (U.S.A .) i
(Rec eived September 7th, 1984; Revised version received December Sth, 1984; Accepted December 9th, 1984)
Key words: dendrite - term inal dendrite - non-terminal dend rite - cortex - length - branch order - rat
A theoretical model for calculating the total length of dendritic trees is presented . Predictions of velocity of d endritic gro wth, time when br an ch ing begi ns and mean max imal extent o f dendritic tree derived f ro m the model for ap ica l den drites in en torhina l cortex o f th e rat are presented .
Mathematical modeling of the dendritic tree has been pursued in two ways: modeling of the metric aspects of the tree concern ed with length of bra nches and branching probabilities [6, 7], and modeling of the topology of the trees, i.e. the branching patterns [1, 10]. Recently, modeling has been concerned with determining whether dendritic trees grow by term inal addition of branches or by branching of non-terminal segments. A topological analysis of dendritic branching in the developing cerebral cortex supports the terminal growth hypothesis in this structure [9]. The total length of all t he branches of a dendritic tree is of importance. If the mean maximal extent of the dendritic tree is less than th e membrane space constant A, then the tree may be considered equipotential. If the mean maximal extent of the tree is longer than A, howev er, then synaptic input to branches removed from the cell body may be attenuated significantly before it reaches the cell body [3]. It would be of value to determine a mathematical model for predicting the length of a dendritic tree. Such a model has been suggested by a data set on the length and branch orders of dendrites of cells in the cerebral cortex of rats of different ages. Determining the total length of a dendritic tree is one of the features of the model presented in this paper. Also , fun ction s are presented which wiII give the number of terminal branches and the mean maximal extent of the tree as a func tion of time. "Auth or fo r correspondence. " "Pr esen t address: Depart men t o f M athe ma tics , Uni versit y o f Ne brask a at Omaha, Omaha, NE 68182, U .S .A . 0304-3940 / 85/ $ 03.30 © 1985 Elsevier Scientific P ubl ishers I relan d Ltd.
244
Sprague-Dawley rats aged 5, 21, 34 and 95 days (2 males and 2 females per age group) were used in this study. The rats were euthanatized, and their fresh brains were fixed in Golgi-Cox solution for 7 days. A O.S-mm coronal slice of cerebral cortex at the level of the median eminence was processed by the Golgi -Cox method. The sampled tissue was embedded in low-viscosity nitrocellulose, serially sectioned at 130 urn and mounted on slides. A computer-assisted light microscope [8] was used for quantitative analysis of the apical dendrites of 60 randomly sampled Golgi-impregnated pyramidal neurons of the entorhinal cortex of each rat at each age. Dendrites were ordered centrifugally. Dendrites were classified as terminal or non-terminal and their lengths were measured. From this raw data, the mean dendrite lengths of terminal and nonterminal branches were computed on a per order basis. Also, the mean total length of the dendritic tree, the probability of branching of each order and the mean number of terminal branches were computed. The mean maximum extent of the dendritic trees [which is the mean distance measured along the branches from soma to dendritic tip, f(t)] at different ages can be calculated from the data as the mean maximum branch order times the nonterminal branch length plus the added length for terminal branches. Bifurcation probabilities were used to compute mean maximal branch orders. The proportion of a population of cells whose maximum branch order is n is expected to be PI' P 2 · ••• P(n-I)' (1- P n) where P; is the probability that a branch of order n will bifurcate. To compute the mean maximal branch order, an average of the branch orders weighted by the proportion of cells with that maximal branch order was computed. The model we propose to describe the change in length of the total dendritic tree as a function of time is: dL(t)/dt= V(t)· £(t)
(1)
where L(t) is the total length of the dendritic tree as a function of time t, V(t) is the velocity of growth of a terminal branch as a function of time t and e(t) is the number of terminal branches as a function of time t. This model assumes that length, velocity and the number of terminal branches are functions of time, and says that the change in total length of a dendritic tree per unit time is equal to the product of the velocity of growth of the growing branches times the number of these branches. Implicit in this is the assumption that growth in length occurs only in terminal branches. The model can be derived by considering the analogy between a growing dendritic tree and the growth of bacterial colonies (see Appendix). For the sake of simplicity, it is assumed that the rate of growth of a branch depends only on time and not on the particular branch. In other words, all of the terminal branches are growing at the same rate at the same time. This assumption, although simple, is a valid first approximation to modeling the growth of the den-
245 TABLE I BASIC QUANTITIES FROM THE DATA USED TO FIT THE MODEL Age in days 5 21 34 95
Number of observations
Mean total length (I'm)
Mean number of terminal branches
f(t) (urn)
Distance function
240 240 240 240
244.70 468.73 635.36 977.52
4.9167 6.3667 6.8751 7.9008
110.02 122.82 137.44 180.12
dritic tree. It can be weakened considerably by regarding the more general equation from which Eqn. 1 is derived as a special case. This is discussed in the Appendix. In an attempt to verify the model from the data available, equations were fit to the total length of the dendritic tree and to the number of terminal branches as a function of time using the SAS NUN procedure (non-linear least squares method) [5]. The derivative of the total length function was divided by the function e(t) to estimate V(t). This quotient was integrated numerically using the trapezoid rule to find values for a distance function f(t), which could be compared with the distance function calculated from the data. The basic quantities for the model are presented in Table 1. Fitting the length as a function of time yielded the equation: L(t) = 1120.19 - 964.27
e - 0.0191
(F= 1040.18, df=3,1, P
(2)
The first derivative of this equation is: dL(t)
~=:
18.73
e-0.OJ91
(3)
The most biologically reasonable equation to fit the mean number of terminal branches as a function of time was: f.(t),=;7.70-10.39 e- 0 .05(l + 15.85)
(F=536.18, df=3,2, P<0.005)
(4)
This yields an expression for velocity V(t), by dividing the expression for dL(t)/dt by the expression for e(t). Predicted velocities at different times are presented in Table II. TABLE II PREDICTIONS FROM THE MODEL Age in days
Number of observations
Predicted terminal growth velocity (urn/day)
Predicted values of the distance function (urn)
5 21 4
240 240 240 240
4.22 2.08 1.44 0.40
98.58 140.05 160.95 207.12
95
246
From the data, it was found that the probabilities of branching decreased from 0.98 for first order dendrites according to the equation: P n=0.98/vn (F= 17.38, df= 1,13, P<0.005)
where P; is the probability that a branch of order n will bifurcate and n is the branch order. In our data, the mean non-terminal branch length was 32.52 urn and showed no trends over time. Terminal branches had a length greater than this, which increased with age. The added length at 5 days was 2.7 urn, at 21 days it was 12.9 urn, at 34 days it was 24.6 urn and at 95 days it was 68.9 urn, The data on f'(r ) is presented in Table 1. The predictions for f(t) derived from the expressions for V(t) by integration are presented in Table II. The data available were detailed enough to support the model proposed. In part, this is because of the power of the computerized method used for collecting them. However, certain limitations are evident. First, there are not enough time periods sampled. This leads to uncertainty in the constants to the equations we found by non-linear least squares. Second, the data represent the means of many observations. Therefore, our predictions represent what could be expected of the group of neurons in aggregate. These limitations may explain the discrepancy between predicted and calculated f(t)'s. One consequence of the equation for L(t) is that the age when dendrites first start to grow is predicted to be about 8 days before birth. This finding is probably only applicable to the part of the cortex sampled since it is known that different parts of the brain mature at different times [2]. Our finding that non-terminal branches were of a constant length while terminal branches were longer bears out the observations of others [6]. Furthermore, it appears that growth occurs only at terminal branches because these became longer with age while non-terminal branches did not. Thus, one of the assumptions on which the model is based appears to be true. Eqn. 4 predicts that with increasing time the mean number of terminal branches becomes constant at a value of 7.699, which is not too different from our data. Furthermore, the equation predicts that the number of terminal branches becomes 0, i.e. the cell has no branches at 10 days before birth. This value is of the same order as the time when branching begins, predicted by Eqn. 2. The velocity of growth predicted by our equations begins at a high rate and then declines over time. The initial rapid growth predicted might be explained by the fact that the growing dendritic tips are physically nearer the soma at a young age and grow more rapidly because they would be nearer the source of membrane and cytoplasmic constituents. The decrease is reasonable because the dendritic tree must have an upper limit to its size.
247
APPENDIX
The model used in this paper, Eqn. 1, can be derived as a special case of a more general partial differential equation. This suggests that the restrictive conditions on V and £ made in our development above could be relaxed. The partial differential equation is derived by analogy with the McKendrick, Von Forester, Rubinow equation used in modeling the growth of cell colonies [4]. The equation is: an
at
+ a(Vn) =0
(AI)
au
where n(t,u) is the cell density function with t=time and u=maturity, and V(t,u) is the velocity of maturation. The number of cells as a function of time t, N(t) is given by: Ul
N(t) =
J net, u)du
Uo
and the boundary condition is:
where Uo is maturity level at birth and Ut is maturity level at mitosis. To derive a partial differential equation analogous to Eqn. AI, proceed as follows. Let e(t, r) be the length density of the terminal (growing) branches of the dendritic tree where t= time, and r= branch length, and V(t, r) is the velocity of growth. Then £ (t, r)LI r is the total dendritic length at time t of all terminal branches between rand r+Llr in length. Thus, e (t +,1 t, r)LI T = e(t, r)LI T + V(t, r)e (t, -r),1 t - V(t, r + LIr)e (t, r +,1 r)LI t
Divide by LltLlr and write: e(t+,1t,r)-£(t,r) LIt
+
V(t,r+Llr)£(t,-r+Llr) _ V(t,T)£(t,-r) =0
LIT
Llr
Taking the limit as LI t, LI r--' 0 yields: (A2)
the desired partial differential equation, analogous to Eqn. AI. The corresponding boundary condition is: e(t, 0) Vet, 0) = 2P(t)e (t, m¢) V(t, m
where P(t) is the probability of bifurcation as a function of time t and m¢ is the length of a non-terminal branch.
248
We now show that Eqn . I, the model for this paper, is a special case of Eqn. A2. Suppose that V(t, r) = V(t) , a function of time only. Let ek(t, r ) be the length density of the kth order terminal br anches only. Let LkU) be the total length of the terminal branches of order k . Then, 00
Lk(t)=J ek(t,r)d, o
_k=J ae k(t, r) d r
and so
dL
00
dt
0
dt
But using Eqn. A2 , we ha ve dL
_ k= _
dt
awe) ae J__ k dr=- V(t) J -dr= 0 ar 0 aT
co
ce
V(t)[ek(t, oo)- fk(t,O»)
since V = V(t) and fk(t, r) are continuous. Since ek(t, 00) = 0 (all branches have finite length), we obtain dL k = V(t)ek(t, 0) dt
(A3)
H ere , ek(t, O) is the number of terminal branches o f order k. This is equivalent to Eqn. 1 for terminal branches of order k. Letting Li= total length of all terminal branches, we obtain (by summing Eqn. A3 over all terminal orders): dL * = V(t)e(t) dt where £(t) is the number of terminal branches. Finally, let L(t) = the total length of all branches , terminal and non-terminal. Then no ting that dL /dt= dL * / d t (only terminal branches are changing in length), we obtain dL = VU) eu) dt which is exactly Eqn. 1. Thus Eqn. 1 is a special case of Eqn, A2, which holds for more general functions o f f and V.than does Eqn. I. For example V in Eqn . A2 need not be a function of time only. Hopefully Eqn. A2 can be used to obtain more refined and detailed models of dendritic growth . This work was supported by Grants NS17107 and RR00167 from the National Institutes of Health. Wisconsin Reg ional Primate Research Center publication No. 24-013 . I Hollingsworth, T . and Berry , M., Network a na lysis of dendrit ic fields of pyramidal cells in neo co rtex and Purk inje cells in th e cerebellum of the rat, Phil. Trans . Roy. So c. La nd . B, 270 (1975) 227-265. 2 Jones, E.G ., Development of conne ctivity in the cerebra l co rtex. In W.M . Cowan (Ed .), Stud ies in Developmental Neurobiology, Essays in Honor of Vikt or H amburger , Oxford University Press, Oxford, 1981, p. 375.
249 3 Koch, c., Poggio , T. and Torres, V., Retinal ganglion cells: a functional interpretation of dendritic morphology, Phil. Trans. Roy. Soc. Lond. Ser, B, 298 (1982) 227-264 . 4 Rubinow, 5.1., Mathematical problems in the biological sciences , Society for Industrial and Applied Mathematics , Philadelphia, J 973. 5 S.A .S . Users Gu ide, S.A.S. Institute, Cary, NC , U .S.A. , 1979, p. 317. 6 Ten Hoopen, M . and Reuver, H .A., Probabilistic analysis of dendritic branching patterns of cortical neurons, Kybernetik, 6 (1970) 176-188. 7 Ten Hoopen, M. and Reuver , H.A., Growth patterns of neuronal dendrites - an attempted probabilistic description, Kybernetik , 8 (1971) 234-239 . 8 Uernura, E. and Lopes, E. , A computer-assisted light micro scope for analys is of dendritic branches, Amer. Assoc . Vel. Anal. (1982) Abstract 29 . 9 Uylings , H .B.M ., Verwe r, R.W .H ., Van Pelt, J. and Parnavelas, J.G., Topological analysis of dendritic growth at var ious stages of cerebral development, Acta Stereo!. , 21 (1983) 55-62 . J 0 Van P elt, J. and Verwer, R.W .H ., The exact probabilities of branching patterns under terminal and segmental growth hypotheses, Bull. Math. BioI., 45 (1983) 269-285.
Neuroscience Letters, 54 (198 5) 243-249 El sevier Scien tific Publish er s Ireland Ltd .
NSL 03168
A MATHEMATICAL MODEL FOR THE GROWTH OF DENDRITIC TREES WILLIAM IRELAND l . ' , J ACK HEIDEL 2 . " and ETSURO UEMURA 1
'D epartment of A natomy, College of Veterinary Medicine, Io wa State University, and Irepartment of Mathematics, Io wa State University, A mes, IA 50011 (U.S.A .) i
(Rec eived September 7th, 1984; Revised version received December Sth, 1984; Accepted December 9th, 1984)
Key words: dendrite - term inal dendrite - non-terminal dend rite - cortex - length - branch order - rat
A theoretical model for calculating the total length of dendritic trees is presented . Predictions of velocity of d endritic gro wth, time when br an ch ing begi ns and mean max imal extent o f dendritic tree derived f ro m the model for ap ica l den drites in en torhina l cortex o f th e rat are presented .
Mathematical modeling of the dendritic tree has been pursued in two ways: modeling of the metric aspects of the tree concern ed with length of bra nches and branching probabilities [6, 7], and modeling of the topology of the trees, i.e. the branching patterns [1, 10]. Recently, modeling has been concerned with determining whether dendritic trees grow by term inal addition of branches or by branching of non-terminal segments. A topological analysis of dendritic branching in the developing cerebral cortex supports the terminal growth hypothesis in this structure [9]. The total length of all t he branches of a dendritic tree is of importance. If the mean maximal extent of the dendritic tree is less than th e membrane space constant A, then the tree may be considered equipotential. If the mean maximal extent of the tree is longer than A, howev er, then synaptic input to branches removed from the cell body may be attenuated significantly before it reaches the cell body [3]. It would be of value to determine a mathematical model for predicting the length of a dendritic tree. Such a model has been suggested by a data set on the length and branch orders of dendrites of cells in the cerebral cortex of rats of different ages. Determining the total length of a dendritic tree is one of the features of the model presented in this paper. Also , fun ction s are presented which wiII give the number of terminal branches and the mean maximal extent of the tree as a func tion of time. "Auth or fo r correspondence. " "Pr esen t address: Depart men t o f M athe ma tics , Uni versit y o f Ne brask a at Omaha, Omaha, NE 68182, U .S .A . 0304-3940 / 85/ $ 03.30 © 1985 Elsevier Scientific P ubl ishers I relan d Ltd.
244
Sprague-Dawley rats aged 5, 21, 34 and 95 days (2 males and 2 females per age group) were used in this study. The rats were euthanatized, and their fresh brains were fixed in Golgi-Cox solution for 7 days. A O.S-mm coronal slice of cerebral cortex at the level of the median eminence was processed by the Golgi -Cox method. The sampled tissue was embedded in low-viscosity nitrocellulose, serially sectioned at 130 urn and mounted on slides. A computer-assisted light microscope [8] was used for quantitative analysis of the apical dendrites of 60 randomly sampled Golgi-impregnated pyramidal neurons of the entorhinal cortex of each rat at each age. Dendrites were ordered centrifugally. Dendrites were classified as terminal or non-terminal and their lengths were measured. From this raw data, the mean dendrite lengths of terminal and nonterminal branches were computed on a per order basis. Also, the mean total length of the dendritic tree, the probability of branching of each order and the mean number of terminal branches were computed. The mean maximum extent of the dendritic trees [which is the mean distance measured along the branches from soma to dendritic tip, f(t)] at different ages can be calculated from the data as the mean maximum branch order times the nonterminal branch length plus the added length for terminal branches. Bifurcation probabilities were used to compute mean maximal branch orders. The proportion of a population of cells whose maximum branch order is n is expected to be PI' P 2 · ••• P(n-I)' (1- P n) where P; is the probability that a branch of order n will bifurcate. To compute the mean maximal branch order, an average of the branch orders weighted by the proportion of cells with that maximal branch order was computed. The model we propose to describe the change in length of the total dendritic tree as a function of time is: dL(t)/dt= V(t)· £(t)
(1)
where L(t) is the total length of the dendritic tree as a function of time t, V(t) is the velocity of growth of a terminal branch as a function of time t and e(t) is the number of terminal branches as a function of time t. This model assumes that length, velocity and the number of terminal branches are functions of time, and says that the change in total length of a dendritic tree per unit time is equal to the product of the velocity of growth of the growing branches times the number of these branches. Implicit in this is the assumption that growth in length occurs only in terminal branches. The model can be derived by considering the analogy between a growing dendritic tree and the growth of bacterial colonies (see Appendix). For the sake of simplicity, it is assumed that the rate of growth of a branch depends only on time and not on the particular branch. In other words, all of the terminal branches are growing at the same rate at the same time. This assumption, although simple, is a valid first approximation to modeling the growth of the den-
245 TABLE I BASIC QUANTITIES FROM THE DATA USED TO FIT THE MODEL Age in days 5 21 34 95
Number of observations
Mean total length (I'm)
Mean number of terminal branches
f(t) (urn)
Distance function
240 240 240 240
244.70 468.73 635.36 977.52
4.9167 6.3667 6.8751 7.9008
110.02 122.82 137.44 180.12
dritic tree. It can be weakened considerably by regarding the more general equation from which Eqn. 1 is derived as a special case. This is discussed in the Appendix. In an attempt to verify the model from the data available, equations were fit to the total length of the dendritic tree and to the number of terminal branches as a function of time using the SAS NUN procedure (non-linear least squares method) [5]. The derivative of the total length function was divided by the function e(t) to estimate V(t). This quotient was integrated numerically using the trapezoid rule to find values for a distance function f(t), which could be compared with the distance function calculated from the data. The basic quantities for the model are presented in Table 1. Fitting the length as a function of time yielded the equation: L(t) = 1120.19 - 964.27
e - 0.0191
(F= 1040.18, df=3,1, P
(2)
The first derivative of this equation is: dL(t)
~=:
18.73
e-0.OJ91
(3)
The most biologically reasonable equation to fit the mean number of terminal branches as a function of time was: f.(t),=;7.70-10.39 e- 0 .05(l + 15.85)
(F=536.18, df=3,2, P<0.005)
(4)
This yields an expression for velocity V(t), by dividing the expression for dL(t)/dt by the expression for e(t). Predicted velocities at different times are presented in Table II. TABLE II PREDICTIONS FROM THE MODEL Age in days
Number of observations
Predicted terminal growth velocity (urn/day)
Predicted values of the distance function (urn)
5 21 4
240 240 240 240
4.22 2.08 1.44 0.40
98.58 140.05 160.95 207.12
95
246
From the data, it was found that the probabilities of branching decreased from 0.98 for first order dendrites according to the equation: P n=0.98/vn (F= 17.38, df= 1,13, P<0.005)
where P; is the probability that a branch of order n will bifurcate and n is the branch order. In our data, the mean non-terminal branch length was 32.52 urn and showed no trends over time. Terminal branches had a length greater than this, which increased with age. The added length at 5 days was 2.7 urn, at 21 days it was 12.9 urn, at 34 days it was 24.6 urn and at 95 days it was 68.9 urn, The data on f'(r ) is presented in Table 1. The predictions for f(t) derived from the expressions for V(t) by integration are presented in Table II. The data available were detailed enough to support the model proposed. In part, this is because of the power of the computerized method used for collecting them. However, certain limitations are evident. First, there are not enough time periods sampled. This leads to uncertainty in the constants to the equations we found by non-linear least squares. Second, the data represent the means of many observations. Therefore, our predictions represent what could be expected of the group of neurons in aggregate. These limitations may explain the discrepancy between predicted and calculated f(t)'s. One consequence of the equation for L(t) is that the age when dendrites first start to grow is predicted to be about 8 days before birth. This finding is probably only applicable to the part of the cortex sampled since it is known that different parts of the brain mature at different times [2]. Our finding that non-terminal branches were of a constant length while terminal branches were longer bears out the observations of others [6]. Furthermore, it appears that growth occurs only at terminal branches because these became longer with age while non-terminal branches did not. Thus, one of the assumptions on which the model is based appears to be true. Eqn. 4 predicts that with increasing time the mean number of terminal branches becomes constant at a value of 7.699, which is not too different from our data. Furthermore, the equation predicts that the number of terminal branches becomes 0, i.e. the cell has no branches at 10 days before birth. This value is of the same order as the time when branching begins, predicted by Eqn. 2. The velocity of growth predicted by our equations begins at a high rate and then declines over time. The initial rapid growth predicted might be explained by the fact that the growing dendritic tips are physically nearer the soma at a young age and grow more rapidly because they would be nearer the source of membrane and cytoplasmic constituents. The decrease is reasonable because the dendritic tree must have an upper limit to its size.
247
APPENDIX
The model used in this paper, Eqn. 1, can be derived as a special case of a more general partial differential equation. This suggests that the restrictive conditions on V and £ made in our development above could be relaxed. The partial differential equation is derived by analogy with the McKendrick, Von Forester, Rubinow equation used in modeling the growth of cell colonies [4]. The equation is: an
at
+ a(Vn) =0
(AI)
au
where n(t,u) is the cell density function with t=time and u=maturity, and V(t,u) is the velocity of maturation. The number of cells as a function of time t, N(t) is given by: Ul
N(t) =
J net, u)du
Uo
and the boundary condition is:
where Uo is maturity level at birth and Ut is maturity level at mitosis. To derive a partial differential equation analogous to Eqn. AI, proceed as follows. Let e(t, r) be the length density of the terminal (growing) branches of the dendritic tree where t= time, and r= branch length, and V(t, r) is the velocity of growth. Then £ (t, r)LI r is the total dendritic length at time t of all terminal branches between rand r+Llr in length. Thus, e (t +,1 t, r)LI T = e(t, r)LI T + V(t, r)e (t, -r),1 t - V(t, r + LIr)e (t, r +,1 r)LI t
Divide by LltLlr and write: e(t+,1t,r)-£(t,r) LIt
+
V(t,r+Llr)£(t,-r+Llr) _ V(t,T)£(t,-r) =0
LIT
Llr
Taking the limit as LI t, LI r--' 0 yields: (A2)
the desired partial differential equation, analogous to Eqn. AI. The corresponding boundary condition is: e(t, 0) Vet, 0) = 2P(t)e (t, m¢) V(t, m
where P(t) is the probability of bifurcation as a function of time t and m¢ is the length of a non-terminal branch.
248
We now show that Eqn . I, the model for this paper, is a special case of Eqn. A2. Suppose that V(t, r) = V(t) , a function of time only. Let ek(t, r ) be the length density of the kth order terminal br anches only. Let LkU) be the total length of the terminal branches of order k . Then, 00
Lk(t)=J ek(t,r)d, o
_k=J ae k(t, r) d r
and so
dL
00
dt
0
dt
But using Eqn. A2 , we ha ve dL
_ k= _
dt
awe) ae J__ k dr=- V(t) J -dr= 0 ar 0 aT
co
ce
V(t)[ek(t, oo)- fk(t,O»)
since V = V(t) and fk(t, r) are continuous. Since ek(t, 00) = 0 (all branches have finite length), we obtain dL k = V(t)ek(t, 0) dt
(A3)
H ere , ek(t, O) is the number of terminal branches o f order k. This is equivalent to Eqn. 1 for terminal branches of order k. Letting Li= total length of all terminal branches, we obtain (by summing Eqn. A3 over all terminal orders): dL * = V(t)e(t) dt where £(t) is the number of terminal branches. Finally, let L(t) = the total length of all branches , terminal and non-terminal. Then no ting that dL /dt= dL * / d t (only terminal branches are changing in length), we obtain dL = VU) eu) dt which is exactly Eqn. 1. Thus Eqn. 1 is a special case of Eqn, A2, which holds for more general functions o f f and V.than does Eqn. I. For example V in Eqn . A2 need not be a function of time only. Hopefully Eqn. A2 can be used to obtain more refined and detailed models of dendritic growth . This work was supported by Grants NS17107 and RR00167 from the National Institutes of Health. Wisconsin Reg ional Primate Research Center publication No. 24-013 . I Hollingsworth, T . and Berry , M., Network a na lysis of dendrit ic fields of pyramidal cells in neo co rtex and Purk inje cells in th e cerebellum of the rat, Phil. Trans . Roy. So c. La nd . B, 270 (1975) 227-265. 2 Jones, E.G ., Development of conne ctivity in the cerebra l co rtex. In W.M . Cowan (Ed .), Stud ies in Developmental Neurobiology, Essays in Honor of Vikt or H amburger , Oxford University Press, Oxford, 1981, p. 375.
249 3 Koch, c., Poggio , T. and Torres, V., Retinal ganglion cells: a functional interpretation of dendritic morphology, Phil. Trans. Roy. Soc. Lond. Ser, B, 298 (1982) 227-264 . 4 Rubinow, 5.1., Mathematical problems in the biological sciences , Society for Industrial and Applied Mathematics , Philadelphia, J 973. 5 S.A .S . Users Gu ide, S.A.S. Institute, Cary, NC , U .S.A. , 1979, p. 317. 6 Ten Hoopen, M . and Reuver, H .A., Probabilistic analysis of dendritic branching patterns of cortical neurons, Kybernetik, 6 (1970) 176-188. 7 Ten Hoopen, M. and Reuver , H.A., Growth patterns of neuronal dendrites - an attempted probabilistic description, Kybernetik , 8 (1971) 234-239 . 8 Uernura, E. and Lopes, E. , A computer-assisted light micro scope for analys is of dendritic branches, Amer. Assoc . Vel. Anal. (1982) Abstract 29 . 9 Uylings , H .B.M ., Verwe r, R.W .H ., Van Pelt, J. and Parnavelas, J.G., Topological analysis of dendritic growth at var ious stages of cerebral development, Acta Stereo!. , 21 (1983) 55-62 . J 0 Van P elt, J. and Verwer, R.W .H ., The exact probabilities of branching patterns under terminal and segmental growth hypotheses, Bull. Math. BioI., 45 (1983) 269-285.