© European Neuroscience Association
European Journal of Neuroscience, Vol. 6, pp. 918-935, 1994
Computational Structure of Visual Callosal Axons Giorgio M. Innocenti 1, Patricia Lehmann 1 and Jean-Christophe Houzel2 1Institut d’Anatomie, 9 rue du Bugnon, 1005, Lausanne, Switzerland 2Laboratoire de Physiologie de la Perception et de l'Action, CNRS UMR9950, Collège de France, 11 Place Marcelin Berthelot 75231, Paris Cedex 05, France Key words: corpus callosum, visual cortex, axon, slmulatlon, neural computation, cat
Abstract We analysed the activation profiles obtained by simulating invasion of an orthodromic action potential in eleven anterogradely filled and serially reconstructed terminal arbors of callosal axons originating and terminating in areas 17 and 18 of the adult cat. This was done in order to understand how geometry relates to computational properties of axons. In the simulation, conduction from the callosal midline to the first bouton caused activation latencies of 0.9-3.2 ms, compatible with published electrophysiological values. Activation latencies of the total set of terminal boutons varied across arbors between 0.3 and 2.7 ms. Arbors distributed boutons in tangentially segregated terminal columns spanning one or, more often, several layers. Individual columns of one axon were frequently activated synchronously or else within a few hundred microseconds of each other. Synchronous activation of spatially separate columns is achieved by: (i) long primary or secondary branches of similar calibre running nearly parallel to each other for several millimetres; (ii) variations in the calibre of branches serially fed to separate columns by the same primary or secondary branch; (iii) exchange of high-order or preterminal branches across columns. The long, parallel branches blatantly violate principles of axonal economy. Simulated alterations of the axonal arbors indicate that similar spatiotemporal patterns of activity could, in principle, be obtained by less axon-costly architectures. The structure of axonal arbors, therefore, may not be determined solely by the type of spatiotemporal activation profiles it achieves in the cortex but also by other constraints, in particular those imposed by developmental mechanisms. Introduction Axons represent variable but often considerable fractions of the neuronal cytoplasm; presumably their production and maintenance engage the parent cell body in considerable energetic investments. One would therefore expect axonal geometry to conform to principles of economy. Indeed, a tendency towards the shortening of axons seems to have emerged in phylogenesis, for example with the evolution of invertebrate ganglia (Ramon y Cajal, 1911), the folding of the cerebral cortex in mammals (Innocenti, 1990) and the segregation of cortical areas and columns (Mitchison, 1992). Economy of axoplasm is not always the rule. In particular, some callosal axons in visual areas 17 and 18 of the cat (Houzel et al., 1994) form, in the white matter, branches of comparable length which run more or less parallel for several millimetres before they reach the same ‘terminal column’ (for definition see Houzel et al., 1994), or terminal columns separated by only a few hundred microns (axons with parallel architecture). In addition, terminal columns are often interconnected by what one might regard as superfluous collaterals. Other axons, however, appear to conform to principles of economy; their trunks and/or main branches sequentially supply roughly radial collaterals to separate cortical columns (axons with serial architecture). The architecture of axonal arbors may reflect the fact that they not only achieve topographically appropriate connections (spatial mapping), but also perform operations in the temporal domain. In the auditory
brainstem, for example, activation delays generated by appropriate axonal architectures are used for computing interaural delays and hence the spatial location of a stimulus (Carr and Konishi, 1990). In this light, the above-mentioned serial architecture seems adequate for generating delays in the activation of spatially separate cortical sites. In contrast, the seemingly wasteful parallel or convergent architectures could be appropriate for activating cortical sites in synchrony. Indeed, tangential corticocortical connections, including callosal ones, are involved in the synchronous activation of separate neuronal pools (T’so et al., 1986; Abeles, 1991; Engel et al., 1991; Nelson et al., 1992). To test these possibilities we have simulated the activation of callosal axons with different architectures. The results indicate that the geometry of axonal arbors could, in principle, operate transformations in the temporal domain, although somewhat differently from what we expected. Materials and methods Procedures for axonal visualization and 3-D reconstruction and software for axonal analysis and simulation are described in detail elsewhere (Tettoni et al., 1993, 1994; Houzel et al., 1994). Briefly, in adult cats, callosal axons originating from neurons located near the border between areas 17 and 18 were anterogradely filled with biocytin, visualized with an avidin-horseradish peroxidase complex and reconstructed from serial
Correspondence to G. M. Innocenti, as above Received 16 September 1993, revised 27 December 1993, accepted 2 February 1994
The authors apologize for the unsatisfactory reproduction of some Figures, in particular Figs 10 and 11.
Computational structure of visual callosal axons 919
F IG. 1. Example of 3-D reconstructed terminal arbor (A). An enlarged part of the same arbor with branches of six different diameters is shown in B; values of diameters (in µm) are medians of classes determined in Houzel et al. (1994); they are represented by 1 pixel increments and therefore do not correspond to the calibration bar. Position of boutons on part of the arbor is shown in C; circle encloses corresponding region in B and C.
sections using a computer-coupled light microscope and the Neurolucida software (Glaser et al., 1983; Glaser and Glaser, 1990). The Maxsim software (Tettoni et al., 1993; Houzel et al., 1994) allowed three operations: (i) visualization and 3-D rotation of the reconstructed axonal arbors; (ii) simulation of the progression of an action potential initiated at any chosen site in the arbor; and (iii) modification of the arbors by changing the thickness of the branches or their interconnections. Multiple axons separately reconstructed from the same brain could be displayed together. The program also tested the data files for the most common reconstruction errors, i.e. inconsistencies in the thickness measurements, abnormal branching patterns and boutons not connected to a terminal branch. A number of assumptions and simplifications underlie the study: (i) Axon diameters were measured light-microscopically using 100 X immersion objectives (final magnification of 1600 X or 2000 X) and a graded ocular. In order to test how completely the axon was actually filled with biocytin in its radial dimension we performed an electron microscopic analysis of labelled axons in the middle of the corpus callosum and near their site of termination. In these axons, electronopaque reaction product was found throughout the axoplasm (D. AggounZouaoui et al., unpublished data). Therefore, notwithstanding tissue deformation and shrinkage (see below), the measures obtained were considered to be good approximations of the real axon diameters. However, since biocytin did not diffuse into the myelin, only the axoplasm was visible in the light microscope. In order to estimate conduction velocity we assumed that our axons had a g ratio (ratio between axoplasm diameter and fibre diameter, the latter inclusive of myelin) of ~ 0.7, as have callosal axons in rabbits (Waxman and Swadlow, 1976). Sometimes axon diameter appeared to decrease slightly in the middle of the section, presumably owing to incomplete penetration
of the avidin - horseradish peroxidase complex used for the visualization of biocytin; therefore diameters were, as a rule, measured near the section surface. Axon diameter was initially recorded in 0.46 µm incremental classes. In a second series (axons 2lB, 21C, 22B, 22D and 22E) appropriate changes in the optics of the microscope and modifications in the software allowed the use of 0.26 µm wide classes, close to the limits of optical resolution. Diameter will be indicated by the central value of its class (with two decimal points, as used in the calculations), with the half-width of the class in brackets. Process thickness could be visualized by the Maxsim software by incremental multiples of pixel size (Fig. 1). (ii) We have assumed that conduction velocity increases linearly with fibre diameter i.e. 5.5 m x s-1 x diameter (µm) according to Waxman and Bennett (1972). This relation applies to myelinated axons, although some central nervous system axons may deviate from it (Carr and Konishi, 1990). Axons in the corpus callosum become myelinated above 0.2-0.3 µm in diameter (Berbel and Innocenti, 1988), which is at the limit of what could be visualized in the present light microscope study. We assumed axons to be myelinated up to their termination. This appears to be the case in all the axons thus far examined electron microscopically (D. Aggoun-Zouaoui et al., unpublished data). If the preterminal branches (~0.25 µm in diameter) were unmyelinated, conduction velocity would have been underestimated by only ~ 10% in the terminal portion of the arbor, since the conduction velocities of small myelinated and unmyelinated axons are very similar (Waxman and Bennett, 1972). From what was determined above one can calculate that the width of classes used for estimating axon diameters corresponds to speed ranges of ± 1.8 µm/µs for the 0.46 µm classes and to ±1 µm/µs for the classes of 0.26 µm. (iii) The Neurolucida software allows length measurements accurate
920 Computational structure of visual callosal axons to at least 1 µm along all axes. Mismatches of axon segments across neighbouring sections owing to local, presumably random distortions of the sections can increase or decrease length of a few process by maximally 10 µm. The histological material underwent two deformations, shrinkage and compression. Shrinkage refers to a volumetric decrease of the tissue, nearly entirely due to fixation. This was estimated to be 35 -40% and to be isotropic in the x, y and z axes (Houzel et al., 1994). No corrections were applied for shrinkage before the simulations because we found that if axon diameter and length shrink proportionally, the spatiotemporal activation patterns are not modified. However, estimates of conduction velocity were compensated for shrinkage (see Results). Compression is a reduction of section thickness to ~ l/3 of its value at cutting, due to dehydration and coverslipping. This type of deformation causes axons travelling along the z axis to take characteristic wavy trajectories, which were followed as faithfully as possible at tracing. This eliminated the need for further corrections in the calculation of action potential propagation. However, in the 3-D visualization of the axon we compensated for compression by a threefold expansion of the z axis. (iv) We have not tried to simulate saltatory conduction along the axons. We also assumed that conduction velocity depends only on the length and mean axon calibre of a branch and did not consider possible changes in conduction velocity at the branching points (Goldstein and Rall, 1974; Raymond and Lettvin, 1978; Waxman and Abakarov, 1978; Parnas, 1979; Parnas and Segev, 1979; Lüscher and Shiner, 1990a, b; Manor et al., 1991). The latter is justified by the fact that significant (>50 µs) delays at branching points are supposed to occur for geometrical ratios (GR = E dj3/2/da3/2, where Edj is the sum of the diameters of the daughter branches and da that of the parent branch) > 2 or < 0.5 (Manor et al., 1991; see also Discussion). In our sample, GR (measured on 1591 nodes in 11 axons) was 1.8 on average (range: 0.64-2). We also did not try to simulate the changes in conduction velocity related to the previous state of activation of the axon (Swadlow and Waxman, 1978). (v) Maxsim allows visualization of the state of activation of the arbor in temporal windows of different width, which can be moved along the time axis (hence along the arbor) following action potential initiation. Windows of different width were used. However, for the illustrations, windows of between 50 and 300 µs will be used, which are considerably shorter than the total action potential. This significantly improves the spatial resolution of the simulations. In addition, windows between 100 and 300 µs may be reasonably close to the time of Ca2+ channels opening at the synaptic bouton (Llinas, 1982), although the time of transmitter release and transmitter-triggered postsynaptic events may operate on a still different time scale [see Clements et al. (1992) for data and discussion]. (vi) Terminal columns were identified by 3-D rotation according to the procedure described elsewhere (Houzel et al., 1994).
Results Eleven axonal arbors were analysed; their morphology was described in detail by Houzel er al. 1994, who used the same codes for denoting animals, axons and terminal columns (see their Table 1). Interhemispheric conduction times Seven axons were reconstructed from the midline of the corpus callosum in three animals over lengths (to the site of entry into the grey matter) of 12.4 mm on average (range 11.7- 13.5 mm). The latencies to the activation of the first bouton in these axons were 1.84 ms on average (range 0.9-3.18 ms). These are close to the electrophysiologically
T ABLE 1. Parameters calculated for seven axons reconstructed and activated from the midline: diameter of trunk (median of class), curvilinear length from midline, activation latency (AT) of first bouton, conduction velocity derived from axon diameter (Waxman and Bennett, 1972), uncorrected (Vl) and corrected (V2) for shrinkage. Axon code
TtUlk diameter (µm)
Length (µm)
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V2 (m/s)
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1.57 2.02 0.64 1.16 0.64 1.68 1.42
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measured antidromic activation latencies in areas 17 and 18 of the cat following corpus callosum stimulation (McCourt et al., 1990) of 1.8 ms on average (range 0.9-4.2 ms), but they are slightly longer than half of the antidromic activation latencies following stimulation of the contralateral cortex [mean values are 2.73 ms in Innocenti (1980) and 2.9 ms in McCourt et al. (1990)]. The conduction velocities calculated from the measured axon diameters (see Materials and methods) were 10.2 m/s on average (range 5.0- 15.9; Table 1). However, when the diameters were corrected for tissue shrinkage (35%), velocities of 13.8 m/s (range 6.8-21.4; Table 1) were obtained. These values are close to those estimated electrophysiologically by Harvey (1980) of 13 m/s on average (range 1.4-27.0). The fact that values of conduction parameters similar to those found electrophysiologically were obtained in the simulation validates reconstruction and simulation procedures. Activation profiles As described in the accompanying paper (Houzel et al., 1994), callosal axons terminated with boutons often clustered into discrete, separate volumes of cortex (terminal columns), probably corresponding to orientation columns. The number of terminal columns established by one axon varied between one and seven. In most columns the majority of boutons was in both supra- and infragranular layers, more heavily in the former. In the present study axonal arbors were characterized by the spatial and temporal sequence of activation of boutons obtained in the simulation (activation profne). For a quantitative assessment of the results we generated histograms of the latencies of activation of boutons over the whole terminal arbor, as well as over separate layers and terminal columns. The activation of all the boutons of an arbor [total activation time (TAT)] required between 316 and 2690 µs (average 990 µs). The value of TAT depended on the geometry of the arbor, in particular, the diameter of its branches and the spatial dispersion of the boutons. Thus, the shortest TAT was found in the only axon (22E) with a simple arbor restricted to layer IV and the bottom of layer III. The longest TAT was found in an axon (16D) with separate branches, coursing tangentially through supra- and infragranular layers (see below; Fig. 9). Activation of all boutons in a column [columnar activation time (CAT)] was measured in 34 columns and was found to require between 55 and 1578 µs (mean ± SD, 461 ± 341). Higher values were found in columns with many widely distributed boutons; lower values were found in columns with few, narrowly distributed boutons. Examples of TAT and
Computational structure of visual callosal axons 921
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IV. Axon (numbers and capital letters) and column (small letters) codes are indicated for each histogram.
CAT histograms can be seen in Figures 2-7, 9, 11 and 12. BimodaJ and polymodal histograms usually indicated the existence of discontinuities in the spatial distribution of synaptic boutons, as in the case of axons with discrete clusters of boutons to supra- and infragranular layers or to distant terminal columns (see below). Columns which span both supra- and infragranular layers were invaded in an inside-out sequence (deep layers first; Fig. 2) when they were supplied by radially ascending branches (but see description of 16D below).
Synchronous activation of terminal columns In order to quantify the degree of synchronicity in the activation of different terminal columns of the same arbor two indexes were calculated. The first [total coactivation index (TCI)] was defined as the number of boutons activated simultaneously (with resolution 1 µs) in all columns as a percentage of the total number of boutons. TCI equals 0 when at least one of the columns is completely asynchronous with all the others. This index estimates the degree of global synchronous activation in an
arbor. The second index [paired coactivation index (PCI)] was calculated in the following way. For each column we determined the maximal number of boutons coactivated with boutons of another column; the numbers thus obtained throughout all columns were summed and expressed as a percentage of the total number of boutons. This index estimates the degree of partial synchronous activation in an arbor. The two indexes are uncorrelated but are identical for arbors with two columns. In the description below, the activation profiles will be analysed with respect to the different types of architecture defined by the branching patterns of callosal terminal arbors (Houzel et al., 1994). Axons 16E and 20C are examples of arbors with parallel architectureIn 16E (Fig. 3) two separate branches feed columns a (394 boutons) and b-c (89 and 46 boutons respectively). Columns a and c begin to fire within 15 µs of each other while b starts 200 µs later. During the next 120 µs all the columns become active, then activation stops in column c while it continues in a and b for another 400 µs. TCI is 53% while PC1 is 86%. In 20C (Fig. 4), column a (202 boutons) is fed by a thicker
922 Computational
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FIG.3. Activation profile of axon 16E (top views); simulated invasion by an orthodromically propagating action potential. Left-hand panels show the arbor, right-hand panels are enlarged views of the boutons (probably synaptic boutons) and histograms of their activation latencies for the total arbor (Total) and for individual terminal columns (separate clusters of boutons; small letters). Axon segments and boutons active within a 99 ps window (w) are shown in full black, otherwise in grey; ‘r’ is the time (in ps) elapsed from the onset of the action potential at the origin of the reconstructed portion of the axon. Notice the substantial temporal overlap in the activation of terminal columns achieved by the parallel architecture (see text for definition) of this arbor. (M = medial, A = anterior).
(2.02 f 0.23 am) and faster branch than column b (1.12 f 0.23 pm diameter; 80 boutons). Therefore, its boutons are also activated 150 ps earlier, but this concerns mainly infragranular layers (deeper part of the column in Fig. 4) which are not supplied in b. The supragranular layers
are coactivated in the two columns. Furthermore, the convergence of three parallel branches onto column a increases the synchronicity of its activation. In total, these two columns are coactivated for 210 ps. The TCI and PC1 are 60%. A closer inspection of these two columns shows
Computational structure of visual callosal axons 923
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FIG. 4. Activation profile of axon 20C. Arbors and enlarged views of boutons as in Figure 3. Top panels and bottom left-hand panels (first two columns on the left) are medial views. Inset shows top views of the arbor and boutons, and histograms of their activation latencies for the total arbor and for its individual terminal columns. (D = dorsal, A = anterior, L = lateral). All other conventions are as in Figure 3 but w = 50 µs. Notice the substantial temporal overlap in the activation of the terminal columns in this arbor, even though column b is supplied by a slower branch than column a.
that they are activated in an inside-out and lateral-to-medial sequence (see medial and top views in Fig. 4). Axon 22D (Fig. 5) has a mixed parallel and serial architecture but with a marked parallel component, and distributes to four terminal
columns, one of which may consist of two subunits. The two first-order branches of this axon have different diameters, 1.43 ± 0.13 and 0.65 ±0.13 pm respectively. Two of the columns (a and e) are fed through converging collaterals issued from both first-order branches; since the
924 Computational structure of visual callosal axons
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F IG. 5. Activation profile of axon 22D (top views). Left panels show the arbor; the diameter of branches is visualized as in Figure 1B. Bight panels are enlarged views of the boutons and histograms of their activation latcncies (M = medial, P = posterior). All other conventions as in Figures 3 and 4 but w = 300 Âľs. Notice the considerable temporal overlap in the activation of columns achieved by a substantially parallel architecture and, for columns a and e, converging collaterals of different diameters.
higher conduction velocity pathways to both columns are also longer, the converging â&#x20AC;&#x2DC;inputs are synchronous (see 16C, Fig. 8 for a similar arrangement). The other columns (b and c + d) are fed by the highvelocity primary branch, but through thin collaterals and preterminal branches. In conclusion, this geometry achieves a remarkably
synchronous activation of the four terminal columns (TCI = 63.6% ; PC1 = 94%). Although favoured by parallel architecture, coactivation of terminal columns can also be achieved by axons with serial architecture. Axon 16F (Fig. 6) serially distributes 63 boutons to the more lateral
Computational structure of visual callosal axons 925
16 F
FIG. 6. Activation profile of axon 16F (front views). Top left panels show the arbor; diameter of branches is visualized as in Figure 1B. Right panels are enlarged views of the boutons; bottom left panels are histograms of their activation latencies. All other conventions as in Figures 3 -5 but w = 100 µs. Notice that the reduction in the diameter of the branch supplying column a achieves considerable coactivation of columns in spite of the serial architecture of this arbor.
(and proximal) column a and 269 boutons to the more medially located (and distal) column b. In spite of the additional conduction distance, the activation reaches boutons in column b 10 µs earlier than boutons in column a. The two columns are coactivated during the CAT of b, i.e. a period of 323 µs. Column a continues to be activated for another 450 µs. The TCI and PCI are 85 % . The delayed and protracted activation of column a and the resulting synchronization of the two columns are
due to the fact that a is fed by a much thinner branch than b. Axon 16C (Figs 7 and 8; for complete morphology see fig. 13 in Houzel et al., 1994) exemplifies the synchronous activation of terminal columns, in spite of a partially serial architecture. This is achieved by a combination of the contrivances described above, i.e. an important parallel component in the architecture, diameter modulation, and convergence. Owing to the serial component in the architecture, the most
926 Computational structure of visual callosal axons
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Computational structure of visual callosal axons 927
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proximal column a is active 140 µs earlier that column b, which precedes column c by 250 µs. However, a drastic reduction in the diameter of the tibres feeding column a results in a slow and protracted activation of this column while the activation spreads through b, c and d. All columns, except e, are coactivated within a 210 µs period, which is the total activation time of column c, although other columns are coactivated for longer periods. Because of the asynchronous activation of column e, TCI in this arbor is 0 % but it increases to 53 % if column e is excluded. PC1 is the highest observed (95%). Converging coactivation through pathways of different length and conduction velocity (resembling axon 22D; see above) is illustrated for column c (Fig. 8). This column is fed directly through a main branch and indirectly through a thicker collateral from column d. The action potential arrives at c simultaneously through the direct and the indirect routes, because the different diameters compensate for the unequal lengths. In addition, the activation of column d is slowed down by the thin calibre of its preterminal branches, which favours its coactivation with column c. Asynchronous activation of terminal columns Terminal columns of a given axon are frequently coactivated, but exceptions occur, particularly in arbors with serial architecture. In axon 16D (Fig. 9) four columns are sequentially activated in their serial order along the axon. This produces a 2160 µs delay between the first (a) and the last column (d). The TCI is 0% and the PC1 is the lowest encountered
(34%). Notice that in column c the activation of supragranular layers precedes that of infragranular layers by ~ 270 µs (Fig. 2). This is opposite to what was most commonly seen (see above) and appears to be due to the fact that the infra- and supragranular layers are fed by separate branches running parallel to the cortical surface rather than by radially oriented branches. Similarly, in the axon 22B (Fig. 7; for morphology see Houzel et al., 1994, and Fig. 10), seven columns are asynchronously activated, giving a TCI of 0%; there is, however, considerable overlap in the activation of the different columns (PC1 68%). Table 2 summarizes the TCI and PC1 values found in the nine axons with at least two terminal columns. On the whole, the majority of callosal axons achieve synchronous activation of columns since five out of nine axons have TCIs of > 50%. PCIs are > 50% in seven of nine axons. One of the axons with parallel architecture (20B) is exceptional since it gives very low PC1 and TCI. Activation of multiple axons Figure 10 shows the activation profile of three axons with similar origin and destination in the same animal. The axons (B, D and E) differ in the diameter of their trunk. This is 0.65 ± 0.13 µm for B, 1.69 ± 0.13 µm for D and 1.43 ± 0.13 µm for E. The axons distribute to neighbouring, but minimally overlapping, terminal columns. In addition, axon E terminates in layer IV while axons B and E terminate supraand infragranularly. The axons were reconstructed from the callosal
928
Computational
structure of visual callosal axons
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FIG. 9. Activation profile of axon 16D (front views). Left panels show the arbor and histograms of activation latencies of boutons in the total arbor and in individual terminal columns. Right panels are enlarged views of the boutons. Terminal columns are denoted by interrupted lines and small letters. Conventions as in Figures 3 -6 but w = 104 ks. Notice the sequential lateral-to-medial activation of terminal columns; in column c activation of supragnmular layers precedes that of infragranular layers. For further explanations see text.
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Computational
structure of visual callosal axons
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midline. Therefore, in order to simulate the propagation of action potentials initiated in the contralateral hmisphere, we assumed axons to be activated at the midline with delays corresponding approximately
to tire time required to propagate an action potential from the injection site in layer III [where their cell body is most probably located (Innocenti, 1980)] to the midline. The first boutons are activated after a little more
Computational structure of visual callosal axons 931
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FIG. 12. Activation profile of axon 20C/F (medial views), modified from axon 20C (Fig. 4). Column b is now supplied by a substantially shorter branch (interrupted) than in 20C. All conventions as in Figure 4, but no top views are shown. Notice that the latency histogram for column b is shifted to the left by ~ 100 µs; the activation profile shows that different sets of boutons are coactivated in the original arbor and after the modification.
than 2 ms and are those of axon D, initially in infragranular layers and then supragranularly. Layer IV is activated 900 µs later by axon E. A silent period follows of ~ 2 ms until the boutons of axon B become active, infragranular layers first. The activity of axon E lasts ~ 350 µs. In total, synchronous spikes originating at the cell bodies of this set of three axons
initiate presynaptic activity for > 4 ms in the contralateral hemisphere. Although the three axons distribute mainly to different terminal columns, they have some degree of spatial overlap, compatible with the possibility that the same neurons may receive both from axon D and axon E, and may therefore become active twice with a delay of ~2.5 ms.
932 Computational structure of visual callosal axons Activation profiles after modifications of axonal geometry Arbors with parallel architecture appear to wastefully duplicate axonal segments of considerable length. In order to test if this might be required for the generation of specific activation profiles we replaced the ‘extravagant’ branches by shorter ones. The new segment maintained the diameter of the old segment and was grafted onto an existing proximal branch of identical or larger diameter. These rules were introduced in order not to create geometrical conditions which may be incompatible with either axoplasmic flow or the propagation of action potential. This experiment produced different results in different axons. Figure 11 shows the modified structure of axon 16E (16E/F; F stands for ‘fudged’). The total volume of 16E, extrapolated as far as the midline, was calculated to be 47 760 µm3. The total volume of 16E/F is 43 383 µm3. Therefore the fudged structure spares 9 % of the total axoplasm in the target hemisphere. TAT and CAT histograms, as well as TCI and PCI, are unaffected by the modification. We could also not detect any change in the spatiotemporal distribution of activated boutons. Similar results were obtained with the axon 22D (22D/F; not shown). The fudged structure spared 1% (545 µm3) of the total axoplasm in the target hemisphere. TAT and CAT histograms and the spatiotemporal distribution of activated boutons showed minimal changes. The TCI was unmodified and the PC1 increased from 94 to 98 % . Figure 12 shows the case of axon 20UF. This structure spares 2% (2062 µm3) of the total axoplasm in the target hemisphere. The activation of column b begins and ends 88 µs earlier than with the original structure. The TCI is increased from 60 to 81% . Different sets of boutons are coactivated compared with the original (compare with Fig. 4).
Discussion Computational structure of neurons Structural characteristics of neurons, and in particular of dendrites, appear to determine the fundamental properties of neural operations now recognized as computational in nature (for discussion of the concept of neural computation see Churchland and Sejnowski, 1992). Thus, for example, in the retina, ganglion cells differ in their dendritic geometry and this correlates with the size, structure and summation properties of their receptive fields (for review see Shapley and Perry, 1986). The analysis of how dendritic geometry may affect computational properties of neurons, however, is hampered by the high number of unknown biophysical parameters of dendritic membranes, including their possible local heterogeneity. In spite of this, realistic simulations of dendritic function (for review see De Schutter, 1992) and increasing knowledge of their biophysical properties are profoundly changing our views on elementary neuronal operations. That axonal structure may be critically involved in specific neural operations is suggested by a number of hints, including the fact that, in the visual system, retinofugal pathways involved in different aspects of vision use axons with different diameters and conduction velocities (Hubel and Livingstone, 1987; Merigan et al., 1991). Axons of parvoand magnocellular origin also form different terminal arbors in the lateral geniculate body (Lachica and Casagrande, 1988). Even more compelling is the fact that axons appear to implement the ‘delay lines’ involved in ‘coincidence detection’ in several brain structures (Braitenberg, 1967; Carr and Konishi, 1990; Meek, 1992). Indeed, in the nervous system, axonal geometry (in particular the architecture of terminal arbors and the variations in the diameter of branches) performs two kinds of computation. The first is spatial mapping: the distribution of synaptic boutons achieved by an arbor maps the spatial coordinates of one point, the cell body, onto a set of different spatial coordinates, corresponding to the locations of the postsynaptic
targets. Visual callosal axons probably interconnect neurons with the same orientation specificity but differ in the degree of divergence they achieve in the mapping, as discussed elsewhere (Houze1 et al., 1994). The second computation is in the temporal domain. On first approximation, each terminal bouton of an axonal arbor can be viewed as an element of a matrix, which multiplies the time of occurrence of an action potential at the cell body (with the value of 1) by the conduction delay introduced by the geometry of the arbor. The temporal-computational characteristics of callosal axons were analysed in this study by simulating the propagation of an action potential in arbors traced with histological techniques. The simulation was based on realistic assumptions (discussed in Materials and methods) and ought to reproduce the temporal aspects of the activation of these arbors reasonably accurately. Consistent with this view, the simulations produced interhemispheric latencies ,and conduction velocities very close to those found electrophysiologically. To what extent the simulation captured the fine temporal structure of action potential propagation in the arbor is difficult to assess. In principle, the velocity of spike propagation could be affected by changes in axon diameter as they occur at branching points or at en passant boutons (e.g. Goldstein and Rall, 1974; Parnas and Segev, 1979; Manor et al., 1991). According to the detailed simulations of Manor et al. (1991), GR values of 2 (the maximal values we found; see Materials and methods) might cause a delay of action potential propagation of ±50 µs; conversely, the minimal GR we found (0.6) might advance propagation of the action potential of ~0.30 µs. Thus, in our case, conduction velocity should be minimally affected by branching. The precise role played by branching points in the behaviour of action potentials remains difficult to assess since it may depend on factors other than the geometrical ratio, in particular temperature, concentration and clearance of extracellular K ions, and the density and kinetics of the Na channels (Pamas and Segev, 1979; Manor et al., 1991). An increased density of Na channels near the branch point could compensate for conduction velocity changes due to nonoptimal GR. Indeed, in myelinated axons, the density of Na channels increases near the node of Ranvier (Black et al., 1990), where branches usually originate. Thus the present simulation must be considered only a first, rough approximation to the computation that callosal axons may perform in the temporal domain. A more detailed analysis would require the use of different simulation procedures but also, unfortunately, assumptions on the biophysical properties of axons, in particular at the level of the terminal arbor, for which no information exists. In addition, callosal axons may do more than what the simulation tested, for example frequency filtering through selective blockade of spikes at nodes (Raymond and Lettvin, 1978; Waxman and Abakarov, 1978; Parnas, 1979; Lüscher and Shiner, 1990a, b). Furthermore the conduction velocity of an axon may be modified by the previous occurrence of an action potential (Swadlow and Waxman, 1978). Axonal geometry contributes to dynamic properties of cortical networks The present study suggests that the geometry of callosal axons, in particular the architecture of its terminal arbor and the distribution of fibre diameters within it, generates delays (D) in the activation latencies (Dta) of different synaptic boutons. Depending on the arbor, Dta between two boutons can range between 0 and 2-3 ms. The critical question is whether delays of this order of magnitude are relevant to neural function. There seems to be no universally accepted answer to this question (for discussion see Softky and Koch, 1993). Nevertheless, the possibility that small delays in the input to one neuron may carry relevant information has been raised for the cerebellar cortex (Braitenberg, 1967). Neurons, including cerebral cortical neurons, might
Computational structure of visual callosal axons 933 function as coincidence detectors (Braitenberg, 1967; Abeles, 1991; Bugman, 1991; Meek, 1992; Softky and Koch, 1993). Temporal disparities in the microsecond range are used in the nucleus laminaris of the owl to compute stimulus location in space (Carr and Konishi, 1990) and can be detected by neurons in the midbrain of the electric fish (Carr et al., 1986). Within the context of the present study, temporal transformations generated by the structure of callosal axons could play a significant role in (i) the interaction between an axon and its target and (ii) generating cooperative properties in interhemispheric cortical networks. The temporal dispersion introduced by the terminal arbor transforms an action potential generated at the cell body into a high-frequency burst of spikes, distributed over its synaptic boutons. Although spatially distributed, this burst may still reach, particularly within one column, dendrites of the same postsynaptic neuron. The instantaneous frequencies within the burst are determined by the Ata between pairs of boutons. If one takes the range of Ata in one terminal column (CAT; Table 2), the lower limits of instantaneous frequencies can be expected to be between 600 (for a CAT of 1578 µs) and 18 000 Hz (for a CAT of 55 µs). In general, however, in a column, the majority of boutons fire at even lower Dta and therefore generate instantaneous frequencies close to the second of the values above or higher. The temporal multiplication produced by the terminal arbor could amplify the signal generated at the cell body. One potential disadvantage is that the high frequencies generated by the geometry of the arbor add to those generated at the cell body. However, given the relatively low frequencies of the latter in cortical neurons (usually < 100 Hz), this should not interfere with the use of frequency codes (Richmond and Optican, 1992) in transferring neural information. On the other hand, if cortical neurons function as coincidence detectors, the temporal dispersion generated by the terminal arbors may set a limit to the precision of coincidence detection possible; alternatively only the highest instantaneous frequencies generated by the arbor (those close to the CAT peak) might be used for detection. The tangential widespread arbor of many callosal axons affects postsynaptic targets at considerable distances from each other and presumably, in the visual cortex, in separate columns coding for the same orientation as the cell body (for a discussion see Houzel et al., 1994). Synchronous input to the columns is generated by three structural arrangements. The first is equalization of pathway length to the different columns, through parallel architecture; the second is diameter adjustment for different pathway length, as found in some axons with serial architecture; the third is converging connections between cortical columns through short axon collaterals. It is amazing, and strongly supports the view that axonal structure may comply with precise computational requirements, that axonal designs similar to the first two have been implicated in the synchronization of motor commands to the electric organs in fish (Bennett, 1968). Furthermore, appropriate changes in the diameter of ganglion cells may be used to compensate for conduction distances in the retina (Stanford, 1987). The synchronous outputs generated by the kinds of geometries described here seem in principle to be adequate for sustaining synfire chains (Abeles, 1991) and/or synchronous activation with zero phaselag of spatially separate cortical columns (T’so et al., 1986; Gray and Singer, 1989). These properties are not by themselves able to explain the synchronous activation of area 17 and 18 neurons in the two hemispheres (Engel et al., 1991). For this, connections to corresponding points of both hemispheres from callosally projecting neurons located in areas 17 and 18 or in non-primary visual areas may be required. The first are known to exist since callosally projecting neurons also have abundant local collaterals (Innocenti, 1980) and there is evidence, albeit scarce, in favour of the second (Segraves and Innocenti, 1985). But it
T ABLE 2. Columnar (CAT) and total (TAT) activation times, and indexes of total (ICI) and paired (PCI) coactivations in axons of different architecture Axon code
Architecture
column
Activation times
Coactivation
(µs)
(%)
CAT
TAT
TCI
PC1
894 639 210 489 150
1227
0
95
535 299 1578 528
2690
0
34
16C
mixed
16D
mixed
16E
parallel
818 535 322
891
53
86
16ElF
-
878 535 312
898
53
86
16F
serial
772 323
772
85
85
20B
parallel
86 1074 308
1074
0
39
2oc
parallel
381 207
381
60
60
20ClF
-
381
81
81
21B
parallel
381 207 -
991
-
-
21c
parallel
292 552
627
63
63
22B
mixed
145 480 112 292 93 88 55
1038
0
68
22D
mixed
389 207 754 693
883
64
94
22DlF
-
709
64
98
22E
simple
292 207 580 609 -
316
-
-
Axon (numbers aad capital letters) and column (small letters) codes are indicated.
is not known if these axons have the required structure for activating sites in the two hemispheres in synchrony. The issue is complicated by the fact that reciprocally connected oscillators can also synchronize their activity, and this relatively independently of conduction delays (Konig and Schillen, 1991). It should be noticed that in a substantial number of cases axonal geometry introduces significant Dta (up to 2 ms in the present study; axon 16D) in the activation of separate cortical columns. The present material does not allow one to decide whether a relationship exists between the topographic distribution of an axon’s terminal columns and
934 Computational structure of visual callosal axons their degree of coactivation, although (i) arbors with asynchronous activation profiles also have the highest degree of topographical divergence, and (ii) in some cases the asynchronous columns are located at the periphery of the group of those which are synchronously activated (e.g. 16C, Fig. 8 and 16D, Fig. 9). In addition, substantially asynchronous input to neighbouring columns, and possibly to the same column, appears to be conveyed by different axons with different conduction velocity. Synchronous and asynchronous activation of cortical columns could have different roles in visual perception. While the former might be used to solve the problem of connecting features relevant for object identification (the so-called binding or segmentation problem; for recent discussions see Engel et al., 1992; von der Malsburg and Buhmann 1992), the asynchronous activations could be used to provide figure-background separation, or for computing other aspects of a stimulus, for example directionality or speed. Axonal geometry reflects more than computational constraints Although the geometry of an axon may implement computations in the spatial and temporal domains, similar computation can be implemented by different geometries. Thus, identical spatial mapping can in principle be performed with parallel or with serial architecture. Furthermore, although the first seems best suited for coactivating columns and the second for activating them with delays, diameter modulation can modify these relations. The result of fudging real arbors also indicates that different geometries can lead to similar patterns of spatiotemporal activation, sometimes with considerable saving of axoplasm. Thus there may be no unique correspondence between morphology and computational properties of an axon. Preliminary information (Aggoun-Zouaoui and Innocenti, 1994) suggests that the architecture of an axon is initially established by branches formed in the white matter before an axon has entered the cortex. Interestingly, however, at these early stages, many more white matter branches are established per axon than there will be in the adult. Therefore the same mechanisms which operate at the population level in the development of callosal connections, i.e. axon overproduction (exuberance) followed by selection (reviewed in Innocenti, 1991), seem to operate at the cellular level. Thus in determining the initial axonal pattern, developmental constraints and strategies may be more important than computational constraints. At later stages, it is likely that some of the factors involved in the maintenance/elimination of the juvenile callosal projections will be found to be relevant for the formation of the individual axonal arbors. Prominent among these factors is information coming from the retina. This, and the large body of evidence concerning the development of ocular dominance columns (Wiesel et al., 1982; Singer, 1987; Antonini and Striker, 1993), encourages one to think that computational constraints may be involved in shaping the final geometry of axonal arbors. The formation of arbors capable of coactivating cortical columns could be supervised effectively by temporal coactivation rules such as those expressed by a Hebbian-like mechanism in the phase of synaptic maintenance/elimination. Biochemical transformations of the cytoskeleton of callosal axons (Riederer and Innocenti, 1991) may play a role in setting the critical period for this kind of developmental plasticity.
Acknowledgements Supported by European Training Programme in Brain and Behavioural Research Twinning grant 9185 and Swiss National Science Foundation grant 3129948.90. We are grateful to Eric Bernardi for the illustrations and to Peter Clarke for useful comments on the manuscript.
Abbreviations CAT CC PCI TAT TCI
columnar activation time corpus callosum paired coactivation index total activation time total coactivation index
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