Brain Research, 109 (1976) 111-132
111
© ElsevierScientificPublishing Company, Amsterdam - Printed in The Netherlands
THE APPLICATION OF NETWORK ANALYSIS TO THE STUDY OF BRANCHING PATTERNS OF LARGE DENDRITIC FIELDS
M. BERRY AND P. M. BRADLEY
Department of Anatomy, University of Birmingham, Birmingham B15 2TJ (Great Britain) (Accepted October 28th, 1975)
SUMMARY
Network analysis of dendritic fields not only defines the topology and connectivity of segments of an arborescence, but offers a means of discovering how networks grow. An important theory has recently been formulated z9 suggesting that dendritic branching patterns may be established by synaptogenic interaction of dendritic growth cones with growing axons. This thesis may be verified through network analysis since the theory predicts that growth at pendant vertices will predominate in dendritic networks, that dendritic growth will be directed into areas of maximal synaptogenic activity and that arc lengths will be inversely related, and the order of branching at vertices directly related, to the magnitude of the synaptogenic activity operating about growing dendritic terminals. The possibility of a preponderance of terminal growth may be detected by comparing the topologies in an observed dendritic network with those of a series of hypothetical growth models. This paper provides the frequency tables for models grown by monochotomous, dichotomous and trichotomous branching on random pendant vertices and random arcs for large networks in which 'set theory' contingencies are included. The paper also describes a method of calculating branching probabilities from the measurement ofsegrnent lengths, which is a means of testing the last mentioned prediction of the synaptogenic theory of dendritic growth. The method of network analysis is then discussed in relation to probable dendritic growth patterns, the constancy of segment lengths and the interaction of extrinsic and intrinsic factors in determining branching probabilities.
INTRODUCTION
The branched arrangement of the dendritic fields of neurones probably functions to disperse, within the neuropil, a large surface area of postsynaptic membrane for engagement by presynaptic elements, whilst the actual pattern of branching of the
112 network m a y represent the morphological substrate for the integration of excitatory and inhibitory postsynaptic potentials whose summed effects generate spike potentials at the axon hillock in the soma~,16,19,zl,z2, 2~. Thus, the differential biophysical properties o f dendritic shaft and nodal membrane and the impedance o f the core o f dendrites may profoundly influence the flow o f current such that the pattern o f branching o f dendritic arrays may form the prime determinant o f the information processing capabilities o f a neurone. Despite their functional importance, detailed morphological analysis of neuronal networks has not so far been feasible because o f a lack of adequate quantitative m e t h o d o l o g y 5. Techniques have long been in use in geography lz, however, for studying the networks o f rivers, roadways and lines o f communication. These techniques are directly applicable to the quantitative definition o f dendritic arborizations 4 and some of this potential has been realised in studies on both the branching patterns and growth o f the dendritic fields o f neurones in the neocortex and cerebellum of the rat 14. The method o f network analysis requires an arborescence to be viewed as a plane graph, made up o f points, called vertices, interconnected by arcs. The outermost tips of the network are termed pendant vertices and the respective arcs, pendant arcs. The point o f origin o f the network is called the root vertex. Translated into dendritic terminology arcs are segments and pendant arcs the terminal segments o f the tree. All vertices are nodes, the root vertex is taken to be the axon hillock and the pendant vertices are the tipJ o f the terminal segments. Different forms o f branching can be defined by the n u m b e r o f arcs draining into each vertex and the order of magnitude of branching at a vertex may be described as dichotomous if it drains 3 arcs, trichotomous if it drains 4 arcs, and so on. Branching patterns can be established by a variety o f processes of growth. Terminal growth occurs when arcs are added to pre-existing pendant vertices and segmental growth when arcs are added to existing arcs including pendant arcs. Monochotomous, dichotomous and trichotomous growth patterns are generated by the addition o f one, 2 or 3 arcs, respectively, at newly established vertices. The connectivity o f dendritic trees is thus described by defining both the mode o f interconnection o f arcs in the network and the frequencies o f vertices o f differing orders o f magnitude o f branching.
Fig. 1. Diagrammatic illustration of simple extensory (A), dichotomous (B) and trichotomous growth (C and D) according to the theory of Vaughn et aL29. When growth occurs at the tips of dendrites, simple extension occurs if one filopodium is engaged by a presynaptic element (A); dichotomous branching if two filopodia are engaged simultaneously (B) and trichotomous branching if 3 filopodia are contacted. In the latter case a trichotomous node can be produced (C) or a structure (D) produced if growth subsequently occurs at the trichotomous node (see Berry et al.4 for further explanation). Examination of this diagram will explain how the direction of growth, the order of magnitude of branching at nodes and the length of segments will be related to synaptic potential. If the numbel s of presynaptic sites available is small the probability of two, or 3 filopodia being engaged at the same time is low and simple extension is thus more likely to occur than branching, moreover if a branch is formed it has a higher probability of being a dichotomous rather than a trichotomous branch. Thus the magnitude of synaptic potential will be inversely correlated with segment lengths and directly correlated with the order of branching at nodes. Since both filopodia and individual growth cones are more likely to be engaged by axons where synaptic potential is high dendrites will be directed into areas of neuropil where the probablity of filopodial synaptogenesis is highest.
113 SIMPLE EXTENSION
DICHOTOMOUSBRANCHING
B
TRICHOTOMOUSBRANCHING
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A
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115 Topological analysis can be used to study growth in both small and large dendritic trees, e.g. the small arrays of the basal dendrites of neocortical pyramids and the massive trees of Purkinje cells in the cerebellum 14. In the former case, the analysis is performed on the entire tree whilst, in the latter case, only the peripheral parts of the network, subtending a small number of pendant arcs, are analysed. This latter strategy is adopted because the numbers of topologically distinct patterns which are possible in networks with only small numbers of pendant arcs is so large that the number of examples needed for analysis becomes unmanageable 4. There is no a priori way of knowing how a given pattern of dendritic branching may have arisen during development, or how the environment may modify these patterns. However, Vaughn et al. 29 and Skoff and Hamburger ~4 have shown that, in the developing mouse and chick spinal cord, respectively, the first synapses are formed on growth cones, or on filopodia, and Vaughn et al. 29 have suggested that definitive dendritic branches may become established by the spontaneous formation of synapses on different filopodia of the same growth cone. Since growth cones can occur on dendritic shafts as well as on terminalsg,ls. 23, it follows that a complex number of branching patterns are likely to be formed if all types of branching occur during growth. Figs. 1 and 2 illustrate the types of branching patterns possible when one, 2 and 3 filopodia become synaptically engaged by axons at the same time. Topological types C in Fig. 1 ; C, D and F in Fig. 2 represent trichotomous, quadrichotomous and quinchotomous nodes and can be easily identified within a network. However, nodal extension occurs in patterns B and D in Fig. 1 and A, B and E in Fig. 2 and, as a result, all possess dichotomous nodes although their mode of formation is quite different in each case. The problem of how to distinguish these modes of growth from one another in systems exhibiting dichotomy throughout is approached through a topological analysis of the frequency of the branching patterns subtending from 4 to n pendant vertices (i.e. the 4th-nth p e n d a n t are series4). Since, for a given hypothesis of growth the frequencies of topological types in pendant arc series appear to be quite unique, similarities between the distribution of topologies in one particular model and that of the topologies in a sample of observed dendrites from a given population, would suggest that the underlying mode of growth in both cases was the same. There are, of course, an infinite number of growth hypotheses but perhaps the most useful formulations are stochastic models which highlight any deviation of growth from a purely random process. Random models also seem relevant in terms of growth cone/axon field interaction since filopodial synaptogenesis is likely to be a chance event as may be the number, length and stability of filopodia emanating from a growth cone at any one point in time. Fig. 2. Diagrammatic illustration of simple extension (A), dichtomous branching (B and C) and trichotomous branching (D-F) when growth occurs at cones located on segments. Branching structure B is formed from C if growth occurs at the nodes and similarly patterns E and F can be formed from the quinchotomous node D. The formation of all patterns occurs according to the hypothesis of Vaughn et al. 29 as explained in the legend to Fig. 1. One important difference between terminal and segmental growth is that segmental growth is unlikely to contribute significantly to overall dendritic growth since growth cones move from the shafts of segments to the tips of all newlyformed dendrites and thus, all subsequent growth will be at terminally located dendritic cones and will proceed as in Fig. 1.
116
Starting from the root vertex the method of generating specific topological branching patterns according to each hypothesis is illustrated in Fig. 3. it can be seen that by the third stage of branching the f~equency distl ibution of topological branching patterns discriminates between growth models. If the calculation for each hypothesis is continued, the distribution of topologies at subsequent stages of branching in large networks can be worked out. The complete series of topological types possible at each stage of branching is defined as an absolute pendant arc series. The task of calculating the frequencies of topological branching patterns for a given hypothesis of growth for large networks is, however, complicated by the mathematics of 'set theory'. a MONOCHOTOMOUS
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117 G r o w t h c a n m o s t c o n v e n i e n t l y be studied by subdividing large trees into subunits, categorised into 6 groups c o n t a i n i n g dendrites s u b t e n d i n g 4-9 p e n d a n t vertices and, accordingly, all small order dendrites will form parts of the higher order dendrites in p r o p o r t i o n s that could be represented by a Sven diagram. We have already suggested 4, a n d this paper provides the verification, that, in cases where growth produces complete consecutive p e n d a n t arc series, like b o t h m o n o c h o t o m o u s b r a n c h i n g on r a n d o m arcs and d i c h t o m o u s b r a n c h i n g on r a n d o m p e n d a n t vertices (Fig. 3a a n d 3b2 respectively), the frequencies of topological types in p e n d a n t arc series r e m a i n the same whether the analysis is performed o n small complete dendrites, like the basal dendrites o f neocortical pyramids, or on the subdivisions of larger dendritic trees such as those o f Purkinje cells 14. However, in the case of other forms of growth complete p e n d a n t arc series can be absent. F o r example, even n u m b e r e d absolute p e n d a n t arc
Fig. 3. Topological branching patterns formed by monochotomous, dichotomous, and trichotomous branching. The root point of each pattern is the uppermost pendant vertex, a: monochotomous branching can occur only on arcs. Beginningwith a single arc, the addition of another arc produces the topological type Z. Type Z is represented in the standard format used in this paper as A. The addition of an arc singly to any of the 3 arcs of A produces types Y, X and W which are resolvable into a standard type B. The addition of an arc singly to any of the 5 arcs of type B produces types U, V, T, S, and R which are resolvable into two standard types C and D. If growth occurs by the addition of an arc to any of the 5 segments of type B randomly then the distribution of types C and D in the population will be 4:1 lespectively. (A complete table of frequencies for the 4th-9th pendant arc series in large networks is given in Fig. 8). b: dichotomous branching can occur on arcs or pendant vertices. When this type of branching is constrained to arcs, the addition of one dichotomous branch to a single arc produces type Z. If subsequent growth occurs at the node this pattern may be transformed into type A. The addition of dic~aotomous branches singly to any of the 5 arcs of type Z produces types Y, X, W, V and U. If growth occurs at the nodes of these types, topological types B, C and D are formed. If growth occurs by the addition of a dichtomous branch to any of the 5 segments of type A, randomly, then the distribution of types B, C and D in the population will be 2:1:2 respectively. (A complete table of frequencies for the 4th-9th pendant arc series in large networks is given in Fig. 9). When dichotomous branching takes place on the pendant vertex of a single arc, type A is produced. The addition of dichotomous branches singly to the two pendant vertices of type A produces the patterns Z and Y which are resolvable into a standard type B. The addition of a dichotomous branch singly to the 3 pendant vertices of type B produces type X, W and V, which are resolvable into types C and D. If the addition of such branches is a random process types C and D will be distributed in a population in the ratio of 2:1 respectively. (A complete table of frequencies for the 4th-9th pendant arc series in large networks is given in Fig. 6). c: trichotomous branching may occur on arcs or pendant vertices. The addition of a trichotomous branch to a single arc produces type Z and if growth occurs at the node the standard type A is produced. The addition of a trichotomous branch singly to the 7 arcs contained in type A produces types Y, X, W, V, U, T and S and subsequent growth at the node produces the standard topological types B, C, D, E and F. If the addition of branches to arcs occurs randomly types B, C, D, E and F will be distributed in a population in the ratio of2:1:1 :1:2 respectively. (A complete table of frequencies for the 4th-9th pendant arc series in large networks is given in Fig. 10.) Trichotomous branching on the pendant vertex of a single arc produces type Z and subsequent growth at the node produces the standard type A: the addition of trichtomous branches produces types Y, X and W and growth at the nodes produces the standard topological types B and C. If branches are added to vertices randomly then the types B and C will be distributed in the population in the ratio of 1:2 respectively. (A complete table of frequencies for the 4th-9th pendant arc series in large networks is given in Fig. 7). Note that it is assumed that only the node giving rise to the new branch is active and that the shaft from which the node springs is passive. If however both shaft and branch are active branching patterns and frequencies will be different to those given here for trichotomous branching. (From Berry et al.4.)
118
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Fig. 4. A: illustration of the StrabJer method of ordering (numbers in parentheses). The tree can be ordered from the most peripheral branches inwards by assigning all terminal branches (pendant arcs) as order 1. Where two order 1 branches join, an order 2 branch is formed. Where two order 2 branches join, an order 3 branch is formed, etc. To reverse, the highest Strahler order branch draining into the root vertex, is assigned order 1 and all other orders are consecutively assigned order 2, 3 . . . . . etc. In this illustration the reversed Strahler order numbers occur to the side of each number (in parentheses). B: daughter branches of'n-l' Strahler order form parent branches of Strahler order 'n'. Collateral branches of 'n-l' Strahler order or less divide the parent branch into segments but do not change the ordex of the parent branch. The root vertex is arrowed and in Figs. 6-10 is the uppermost vertex in each topological branching pattern shown.
series are missing from the topological series of branching patterns generated by dichotomous branching on arcs (see Fig. 3bl). However, in the analysis of the sub-units of large networks established by this latter growth pattern, all pendant arc series do appear to be present, simply because the large dendritic fields can be subdivided into small dendrites, subtending 4, 5, 6 . . . . . etc. pendant vertices, irrespective of the mode of growth. The frequency of topological types in these series can most easily be worked out using a computer simulation method. Adopting this technique, the computer is programmed to generate large networks according to a given hypothesis of growth and the frequencies of topological branching patterns in the resulting network subtended by branches with from 4 to 9 pendant vertices are listed. Together with the topological analysis, the estimation of the relative occurrence of vertices of differing orders of magnitude (i.e. dichotomous, trichotomous, quadrichotomous, etc.) in a dendritic tree will give an indication of the relative contribution of various forms of branching to the underlying growth pattern. The hypothesis of Vaughn et al. 29, in addition to providing a possible sound morphological basis for the establishment of dendritic branching patterns in vivo, also allows several important predictions to be made about the formation of dendritic trees (see Figs. 1 and 2): firstly, terminal growth will predominate in dendritic systems, secondly, dendrites will be directed into areas of maximal synaptogenic aetivit), thirdly, the order of branching (dichotomy, trichotomy, etc.) will be directly correlated with synaptogenic activity and the number, length and stability of filopodia and fourthly, the magnitude of segment length will be inversely correlated withthe synaptogenie activity of the axon field in which the dendrites are growing. The first 3 of these predictions can be tested by a topological analysis of growing or mature dendritic
119 trees but the difficulties of quantifying segments and their connectivity are enormous and centre on choosing a method of assigning a ranking order to all arcs. Such methodologies are fraught with problems4,14,2s, but the method we have chosen, called the Strahler technique can be used to order the tree both centripetally and centrifugally (Fig. 4A) which does give this technique some advantages2,14,2s. The Strahler technique defines a branch as a series of arcs of identical Strahler order. Daughter branches of 'n - - 1' Strahler order create parent branches of Strahler order 'n' and collaterals draining into the parent branch are always 'n - - 1' Strahler order, or less (Fig. 4B). By estimating the relative frequencies of branching of each order the relationship between orders can be expressed in terms of the bifurcation ratio, i.e. the ratio between adjacent orders. The numbers of branches of different Strahler order in a given network tend to approximate to an inverse geometric series, defined by the mean or overall bifurcation ratio. The application of ordering methods to the study of dendritic connectivity is complicated by changes in the order of magnitude of branching at different vertices within the same tree and also by conceptual difficulties in the interpretation of the meaning of the bifurcation ratio parameter in the context of connectivity. In this latter respect, the greatest problems arise when different networks give the same overall bifurcation ratio 4. This lack of discrimination by the method comes about for two reasons; firstly, because the overall bifurcation ratio represents the slope of an inverse geometric series and thus is a measure of the size of an arborescence in terms of the maximal Strahler order attained by a given tree and secondly, Strahler ordering defines branches and not segments in the tree and thus gives only general information about connectivity4. These objections can be overcome either by choosing to quantify relationships between all arcs in terms of daughter and parent Strahler orders or by combining the topological analysis with the method of Strahler ordering and calculating absolute bifurcation ratios. This latter measure is obtained by computing the mean bifurcation ratios between successive Strahler orders for each topological branching pattern contained in an absolute pendant arc series for a given hypothesis of growth. Thus, absolute bifurcation ratios give information about both growth and connectivity in absolute and observed networks. If no extension occurs subsequent to the formation of a branch the probability of branching per unit length of dendrite can be worked out by an analysis of the frequency distribution of segment lengths. MATERIALS AND METHODS
Computer simulation of branching patterns The method of simulation of monochotomous, dichotomous and trichotomous growth on random arcs and random pendant vertices has been described by Berry et al. 4. Added to these programmes was a method of recording the topological types present in each pendant arc series.
Topological analysis Any branching pattern with 4 or more pendant arcs may be made up of 3 basic
120
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111
© ElsevierScientificPublishing Company, Amsterdam - Printed in The Netherlands
THE APPLICATION OF NETWORK ANALYSIS TO THE STUDY OF BRANCHING PATTERNS OF LARGE DENDRITIC FIELDS
M. BERRY AND P. M. BRADLEY
Department of Anatomy, University of Birmingham, Birmingham B15 2TJ (Great Britain) (Accepted October 28th, 1975)
SUMMARY
Network analysis of dendritic fields not only defines the topology and connectivity of segments of an arborescence, but offers a means of discovering how networks grow. An important theory has recently been formulated z9 suggesting that dendritic branching patterns may be established by synaptogenic interaction of dendritic growth cones with growing axons. This thesis may be verified through network analysis since the theory predicts that growth at pendant vertices will predominate in dendritic networks, that dendritic growth will be directed into areas of maximal synaptogenic activity and that arc lengths will be inversely related, and the order of branching at vertices directly related, to the magnitude of the synaptogenic activity operating about growing dendritic terminals. The possibility of a preponderance of terminal growth may be detected by comparing the topologies in an observed dendritic network with those of a series of hypothetical growth models. This paper provides the frequency tables for models grown by monochotomous, dichotomous and trichotomous branching on random pendant vertices and random arcs for large networks in which 'set theory' contingencies are included. The paper also describes a method of calculating branching probabilities from the measurement ofsegrnent lengths, which is a means of testing the last mentioned prediction of the synaptogenic theory of dendritic growth. The method of network analysis is then discussed in relation to probable dendritic growth patterns, the constancy of segment lengths and the interaction of extrinsic and intrinsic factors in determining branching probabilities.
INTRODUCTION
The branched arrangement of the dendritic fields of neurones probably functions to disperse, within the neuropil, a large surface area of postsynaptic membrane for engagement by presynaptic elements, whilst the actual pattern of branching of the
112 network m a y represent the morphological substrate for the integration of excitatory and inhibitory postsynaptic potentials whose summed effects generate spike potentials at the axon hillock in the soma~,16,19,zl,z2, 2~. Thus, the differential biophysical properties o f dendritic shaft and nodal membrane and the impedance o f the core o f dendrites may profoundly influence the flow o f current such that the pattern o f branching o f dendritic arrays may form the prime determinant o f the information processing capabilities o f a neurone. Despite their functional importance, detailed morphological analysis of neuronal networks has not so far been feasible because o f a lack of adequate quantitative m e t h o d o l o g y 5. Techniques have long been in use in geography lz, however, for studying the networks o f rivers, roadways and lines o f communication. These techniques are directly applicable to the quantitative definition o f dendritic arborizations 4 and some of this potential has been realised in studies on both the branching patterns and growth o f the dendritic fields o f neurones in the neocortex and cerebellum of the rat 14. The method o f network analysis requires an arborescence to be viewed as a plane graph, made up o f points, called vertices, interconnected by arcs. The outermost tips of the network are termed pendant vertices and the respective arcs, pendant arcs. The point o f origin o f the network is called the root vertex. Translated into dendritic terminology arcs are segments and pendant arcs the terminal segments o f the tree. All vertices are nodes, the root vertex is taken to be the axon hillock and the pendant vertices are the tipJ o f the terminal segments. Different forms o f branching can be defined by the n u m b e r o f arcs draining into each vertex and the order of magnitude of branching at a vertex may be described as dichotomous if it drains 3 arcs, trichotomous if it drains 4 arcs, and so on. Branching patterns can be established by a variety o f processes of growth. Terminal growth occurs when arcs are added to pre-existing pendant vertices and segmental growth when arcs are added to existing arcs including pendant arcs. Monochotomous, dichotomous and trichotomous growth patterns are generated by the addition o f one, 2 or 3 arcs, respectively, at newly established vertices. The connectivity o f dendritic trees is thus described by defining both the mode o f interconnection o f arcs in the network and the frequencies o f vertices o f differing orders o f magnitude o f branching.
Fig. 1. Diagrammatic illustration of simple extensory (A), dichotomous (B) and trichotomous growth (C and D) according to the theory of Vaughn et aL29. When growth occurs at the tips of dendrites, simple extension occurs if one filopodium is engaged by a presynaptic element (A); dichotomous branching if two filopodia are engaged simultaneously (B) and trichotomous branching if 3 filopodia are contacted. In the latter case a trichotomous node can be produced (C) or a structure (D) produced if growth subsequently occurs at the trichotomous node (see Berry et al.4 for further explanation). Examination of this diagram will explain how the direction of growth, the order of magnitude of branching at nodes and the length of segments will be related to synaptic potential. If the numbel s of presynaptic sites available is small the probability of two, or 3 filopodia being engaged at the same time is low and simple extension is thus more likely to occur than branching, moreover if a branch is formed it has a higher probability of being a dichotomous rather than a trichotomous branch. Thus the magnitude of synaptic potential will be inversely correlated with segment lengths and directly correlated with the order of branching at nodes. Since both filopodia and individual growth cones are more likely to be engaged by axons where synaptic potential is high dendrites will be directed into areas of neuropil where the probablity of filopodial synaptogenesis is highest.
113 SIMPLE EXTENSION
DICHOTOMOUSBRANCHING
B
TRICHOTOMOUSBRANCHING
Fig. 1.
A
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115 Topological analysis can be used to study growth in both small and large dendritic trees, e.g. the small arrays of the basal dendrites of neocortical pyramids and the massive trees of Purkinje cells in the cerebellum 14. In the former case, the analysis is performed on the entire tree whilst, in the latter case, only the peripheral parts of the network, subtending a small number of pendant arcs, are analysed. This latter strategy is adopted because the numbers of topologically distinct patterns which are possible in networks with only small numbers of pendant arcs is so large that the number of examples needed for analysis becomes unmanageable 4. There is no a priori way of knowing how a given pattern of dendritic branching may have arisen during development, or how the environment may modify these patterns. However, Vaughn et al. 29 and Skoff and Hamburger ~4 have shown that, in the developing mouse and chick spinal cord, respectively, the first synapses are formed on growth cones, or on filopodia, and Vaughn et al. 29 have suggested that definitive dendritic branches may become established by the spontaneous formation of synapses on different filopodia of the same growth cone. Since growth cones can occur on dendritic shafts as well as on terminalsg,ls. 23, it follows that a complex number of branching patterns are likely to be formed if all types of branching occur during growth. Figs. 1 and 2 illustrate the types of branching patterns possible when one, 2 and 3 filopodia become synaptically engaged by axons at the same time. Topological types C in Fig. 1 ; C, D and F in Fig. 2 represent trichotomous, quadrichotomous and quinchotomous nodes and can be easily identified within a network. However, nodal extension occurs in patterns B and D in Fig. 1 and A, B and E in Fig. 2 and, as a result, all possess dichotomous nodes although their mode of formation is quite different in each case. The problem of how to distinguish these modes of growth from one another in systems exhibiting dichotomy throughout is approached through a topological analysis of the frequency of the branching patterns subtending from 4 to n pendant vertices (i.e. the 4th-nth p e n d a n t are series4). Since, for a given hypothesis of growth the frequencies of topological types in pendant arc series appear to be quite unique, similarities between the distribution of topologies in one particular model and that of the topologies in a sample of observed dendrites from a given population, would suggest that the underlying mode of growth in both cases was the same. There are, of course, an infinite number of growth hypotheses but perhaps the most useful formulations are stochastic models which highlight any deviation of growth from a purely random process. Random models also seem relevant in terms of growth cone/axon field interaction since filopodial synaptogenesis is likely to be a chance event as may be the number, length and stability of filopodia emanating from a growth cone at any one point in time. Fig. 2. Diagrammatic illustration of simple extension (A), dichtomous branching (B and C) and trichotomous branching (D-F) when growth occurs at cones located on segments. Branching structure B is formed from C if growth occurs at the nodes and similarly patterns E and F can be formed from the quinchotomous node D. The formation of all patterns occurs according to the hypothesis of Vaughn et al. 29 as explained in the legend to Fig. 1. One important difference between terminal and segmental growth is that segmental growth is unlikely to contribute significantly to overall dendritic growth since growth cones move from the shafts of segments to the tips of all newlyformed dendrites and thus, all subsequent growth will be at terminally located dendritic cones and will proceed as in Fig. 1.
116
Starting from the root vertex the method of generating specific topological branching patterns according to each hypothesis is illustrated in Fig. 3. it can be seen that by the third stage of branching the f~equency distl ibution of topological branching patterns discriminates between growth models. If the calculation for each hypothesis is continued, the distribution of topologies at subsequent stages of branching in large networks can be worked out. The complete series of topological types possible at each stage of branching is defined as an absolute pendant arc series. The task of calculating the frequencies of topological branching patterns for a given hypothesis of growth for large networks is, however, complicated by the mathematics of 'set theory'. a MONOCHOTOMOUS
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117 G r o w t h c a n m o s t c o n v e n i e n t l y be studied by subdividing large trees into subunits, categorised into 6 groups c o n t a i n i n g dendrites s u b t e n d i n g 4-9 p e n d a n t vertices and, accordingly, all small order dendrites will form parts of the higher order dendrites in p r o p o r t i o n s that could be represented by a Sven diagram. We have already suggested 4, a n d this paper provides the verification, that, in cases where growth produces complete consecutive p e n d a n t arc series, like b o t h m o n o c h o t o m o u s b r a n c h i n g on r a n d o m arcs and d i c h t o m o u s b r a n c h i n g on r a n d o m p e n d a n t vertices (Fig. 3a a n d 3b2 respectively), the frequencies of topological types in p e n d a n t arc series r e m a i n the same whether the analysis is performed o n small complete dendrites, like the basal dendrites o f neocortical pyramids, or on the subdivisions of larger dendritic trees such as those o f Purkinje cells 14. However, in the case of other forms of growth complete p e n d a n t arc series can be absent. F o r example, even n u m b e r e d absolute p e n d a n t arc
Fig. 3. Topological branching patterns formed by monochotomous, dichotomous, and trichotomous branching. The root point of each pattern is the uppermost pendant vertex, a: monochotomous branching can occur only on arcs. Beginningwith a single arc, the addition of another arc produces the topological type Z. Type Z is represented in the standard format used in this paper as A. The addition of an arc singly to any of the 3 arcs of A produces types Y, X and W which are resolvable into a standard type B. The addition of an arc singly to any of the 5 arcs of type B produces types U, V, T, S, and R which are resolvable into two standard types C and D. If growth occurs by the addition of an arc to any of the 5 segments of type B randomly then the distribution of types C and D in the population will be 4:1 lespectively. (A complete table of frequencies for the 4th-9th pendant arc series in large networks is given in Fig. 8). b: dichotomous branching can occur on arcs or pendant vertices. When this type of branching is constrained to arcs, the addition of one dichotomous branch to a single arc produces type Z. If subsequent growth occurs at the node this pattern may be transformed into type A. The addition of dic~aotomous branches singly to any of the 5 arcs of type Z produces types Y, X, W, V and U. If growth occurs at the nodes of these types, topological types B, C and D are formed. If growth occurs by the addition of a dichtomous branch to any of the 5 segments of type A, randomly, then the distribution of types B, C and D in the population will be 2:1:2 respectively. (A complete table of frequencies for the 4th-9th pendant arc series in large networks is given in Fig. 9). When dichotomous branching takes place on the pendant vertex of a single arc, type A is produced. The addition of dichotomous branches singly to the two pendant vertices of type A produces the patterns Z and Y which are resolvable into a standard type B. The addition of a dichotomous branch singly to the 3 pendant vertices of type B produces type X, W and V, which are resolvable into types C and D. If the addition of such branches is a random process types C and D will be distributed in a population in the ratio of 2:1 respectively. (A complete table of frequencies for the 4th-9th pendant arc series in large networks is given in Fig. 6). c: trichotomous branching may occur on arcs or pendant vertices. The addition of a trichotomous branch to a single arc produces type Z and if growth occurs at the node the standard type A is produced. The addition of a trichotomous branch singly to the 7 arcs contained in type A produces types Y, X, W, V, U, T and S and subsequent growth at the node produces the standard topological types B, C, D, E and F. If the addition of branches to arcs occurs randomly types B, C, D, E and F will be distributed in a population in the ratio of2:1:1 :1:2 respectively. (A complete table of frequencies for the 4th-9th pendant arc series in large networks is given in Fig. 10.) Trichotomous branching on the pendant vertex of a single arc produces type Z and subsequent growth at the node produces the standard type A: the addition of trichtomous branches produces types Y, X and W and growth at the nodes produces the standard topological types B and C. If branches are added to vertices randomly then the types B and C will be distributed in the population in the ratio of 1:2 respectively. (A complete table of frequencies for the 4th-9th pendant arc series in large networks is given in Fig. 7). Note that it is assumed that only the node giving rise to the new branch is active and that the shaft from which the node springs is passive. If however both shaft and branch are active branching patterns and frequencies will be different to those given here for trichotomous branching. (From Berry et al.4.)
118
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ilnllilllll Daughter } llama Collateral Branches i Parent
Fig. 4. A: illustration of the StrabJer method of ordering (numbers in parentheses). The tree can be ordered from the most peripheral branches inwards by assigning all terminal branches (pendant arcs) as order 1. Where two order 1 branches join, an order 2 branch is formed. Where two order 2 branches join, an order 3 branch is formed, etc. To reverse, the highest Strahler order branch draining into the root vertex, is assigned order 1 and all other orders are consecutively assigned order 2, 3 . . . . . etc. In this illustration the reversed Strahler order numbers occur to the side of each number (in parentheses). B: daughter branches of'n-l' Strahler order form parent branches of Strahler order 'n'. Collateral branches of 'n-l' Strahler order or less divide the parent branch into segments but do not change the ordex of the parent branch. The root vertex is arrowed and in Figs. 6-10 is the uppermost vertex in each topological branching pattern shown.
series are missing from the topological series of branching patterns generated by dichotomous branching on arcs (see Fig. 3bl). However, in the analysis of the sub-units of large networks established by this latter growth pattern, all pendant arc series do appear to be present, simply because the large dendritic fields can be subdivided into small dendrites, subtending 4, 5, 6 . . . . . etc. pendant vertices, irrespective of the mode of growth. The frequency of topological types in these series can most easily be worked out using a computer simulation method. Adopting this technique, the computer is programmed to generate large networks according to a given hypothesis of growth and the frequencies of topological branching patterns in the resulting network subtended by branches with from 4 to 9 pendant vertices are listed. Together with the topological analysis, the estimation of the relative occurrence of vertices of differing orders of magnitude (i.e. dichotomous, trichotomous, quadrichotomous, etc.) in a dendritic tree will give an indication of the relative contribution of various forms of branching to the underlying growth pattern. The hypothesis of Vaughn et al. 29, in addition to providing a possible sound morphological basis for the establishment of dendritic branching patterns in vivo, also allows several important predictions to be made about the formation of dendritic trees (see Figs. 1 and 2): firstly, terminal growth will predominate in dendritic systems, secondly, dendrites will be directed into areas of maximal synaptogenic aetivit), thirdly, the order of branching (dichotomy, trichotomy, etc.) will be directly correlated with synaptogenic activity and the number, length and stability of filopodia and fourthly, the magnitude of segment length will be inversely correlated withthe synaptogenie activity of the axon field in which the dendrites are growing. The first 3 of these predictions can be tested by a topological analysis of growing or mature dendritic
119 trees but the difficulties of quantifying segments and their connectivity are enormous and centre on choosing a method of assigning a ranking order to all arcs. Such methodologies are fraught with problems4,14,2s, but the method we have chosen, called the Strahler technique can be used to order the tree both centripetally and centrifugally (Fig. 4A) which does give this technique some advantages2,14,2s. The Strahler technique defines a branch as a series of arcs of identical Strahler order. Daughter branches of 'n - - 1' Strahler order create parent branches of Strahler order 'n' and collaterals draining into the parent branch are always 'n - - 1' Strahler order, or less (Fig. 4B). By estimating the relative frequencies of branching of each order the relationship between orders can be expressed in terms of the bifurcation ratio, i.e. the ratio between adjacent orders. The numbers of branches of different Strahler order in a given network tend to approximate to an inverse geometric series, defined by the mean or overall bifurcation ratio. The application of ordering methods to the study of dendritic connectivity is complicated by changes in the order of magnitude of branching at different vertices within the same tree and also by conceptual difficulties in the interpretation of the meaning of the bifurcation ratio parameter in the context of connectivity. In this latter respect, the greatest problems arise when different networks give the same overall bifurcation ratio 4. This lack of discrimination by the method comes about for two reasons; firstly, because the overall bifurcation ratio represents the slope of an inverse geometric series and thus is a measure of the size of an arborescence in terms of the maximal Strahler order attained by a given tree and secondly, Strahler ordering defines branches and not segments in the tree and thus gives only general information about connectivity4. These objections can be overcome either by choosing to quantify relationships between all arcs in terms of daughter and parent Strahler orders or by combining the topological analysis with the method of Strahler ordering and calculating absolute bifurcation ratios. This latter measure is obtained by computing the mean bifurcation ratios between successive Strahler orders for each topological branching pattern contained in an absolute pendant arc series for a given hypothesis of growth. Thus, absolute bifurcation ratios give information about both growth and connectivity in absolute and observed networks. If no extension occurs subsequent to the formation of a branch the probability of branching per unit length of dendrite can be worked out by an analysis of the frequency distribution of segment lengths. MATERIALS AND METHODS
Computer simulation of branching patterns The method of simulation of monochotomous, dichotomous and trichotomous growth on random arcs and random pendant vertices has been described by Berry et al. 4. Added to these programmes was a method of recording the topological types present in each pendant arc series.
Topological analysis Any branching pattern with 4 or more pendant arcs may be made up of 3 basic
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