| Home | Genome | Blast / Blat | WormMart | Batch Sequences | Markers | Genetic Maps | Submit | Searches | Site Map |
In order for nerve cells to make connections with each other they must be
in physical contact. Therefore the physical arrangement of the neuropil is
an important part of its design. The C. elegans nerve ring is
essentially a large parallel bundle of fibres bent around the pharynx. A
typical transverse section through the C. elegans nerve ring shows
an apparently homogeneous group of process outlines on each side.
Bilaterally symmetric processes occupy approximately symmetrical positions
within the bundle, but there is local disorder on the scale of a few
process diameters so that in general it is impossible to identify processes
on the basis of their positions, even over fairly long stretches of
reconstruction (although characteristic diagnostic properties of certain
neurons do make them identifiable).
It is presumably unnecessary to specify the exact relative positions of all
the processes, but important for there to be reasonably tight control over
process position because processes do not branch, so the only way to make
contact is to lie next to each other in the bundle. How is position
controlled? There are essentially two different possible sources of order,
either from contact with other processes or from an external source of
information, such as a gradient (e.g. Bonhoeffer and Huf, 1982). The most
likely form of contact mediated information would be a mutual adhesivity
that kept two or more neurons together and therefore simplified the task of
specifying their positions. Such selective fasciculation has been proposed
as important in laying down other invertebrate nervous systems during
development (see chapter 1 for review) and there are indications that it is
important in process outgrowth in C. elegans (PVPL/PVPR/PVQL/PVQR
behaviour, discussed in chapter 5).
Figure 9.1
The distribution of adjacency in the database. The crosses connected by
the heavy line indicate the number of cell pairs in the database with a
particular adjacency. The fine lines are the corresponding numbers from
the outcome of the random mixing stochastic model, using three different
values of the parameter, p, and averaged over 10 runs to get smooth
results. The best overall fit to the distribution is given by p - 0.08.
This leaves two regions of misfit, X and Y, which are discussed in the
text. Note that the vertical axis in this graph is nonlinear.
Figure 9.2
An expansion of the region Y with three separate simulations of the random
mixing model with p = 0.08. The gap between the true data and the model
data is clearly significant. Since the vertical scale is linear in this
case the area of region Y corresponds to the number of "extra" high
adjacency contacts. This predicts around 400 extra persistent contacts, or
2.3 per neuron (178 neurons).
If selective fasciculation were important in organising the nerve ring, and
the adhesive forces remained after early development, then one would expect
to find pairs of processes with persistent contacts. These should be
detectable in the database as pairs of neurons with exceptionally high
adjacencies. If one looks at the distribution of all the adjacencies in
the database it would be the sum of two components, a random mixing
component, and a high adjacency component due to persistent contacts. The
distribution of adjacencies is shown in figure 9.1. There is a clear
change in slope at the curve at an adjacency of around 30.
In order to assess the significance of this shoulder, and to estimate its
size, and hence the average number of persistent contacts made by a neuron,
I produced a stochastic model of a collection of randomly mixing parallel
fibres. This operates by recording the positions of the fibres in a
hexagonal grid representing a slice through the process tract, and then
moving to the next slice and allowing neighbouring processes to exchange
positions with a certain probability. The adjacency of a pair of fibres is
then taken to be the number of slices in which they are neighbours. The
total number of slices was taken to be 75 to make the total adjacency (sum
of all its adjacencies) of each fibre the same as the average total
adjacency for the processes in the database. The second parameter, the
probability of a process switching, p, was chosen so as to best match the
model's distribution of adjacencies to that of the database. Thje best fit
is given by p = 0.08.
There are two regions of misfit that cannot be eliminated, denoted by X and
Y. Region X is due to a very large number of additional contacts of very
short duration, which probably arise from processes crossing at an angle in
the nerve ring. Such events are known to occur in the nerve ring but are
not considered by the computer model. Region Y is the shoulder that
includes longer contacts than predicted by the random model. Figure 9.2
shows an expansion of the shoulder region of the database distribution
together with data from 3 simulations of the model. The shoulder is
clearly significant beyond the variation in the simulations due to
randomness in the model. However it is fit quite well by the random model
with a low switching probability (p= 0.025, figure 9.1), which is not
surprising, because low switching probabilities for a subset of process
pairs are an approximation to specific adhesion between the processes,
which is the sort of feature that we predicted might give rise to a
shoulder beforehand.
It is possible to estimate the number of significantly persistent contacts
from the graph in figure 9.2 as about 400, and thus to arrive at a figure
of on average 2.3 persistent specific contacts per neuron. This is very
crude - there may be many specific contacts of shorter length - but it
gives an indication that there may be fascicles or bundles of mutually
adhesive processes in the C. elegans nerve ring. However if such
bundles are common then they cannot contain very many processes, because
the average number for long bundles must be only 3 or 4. A second test
suggests the same result. The average number of contacts made by a neuron
is 52.1, most of which are short. If we compare the adjacencies of all the
contacts with that of the longest contact then we see that on average 12.6
are longer then 25% of the maximum, only 4.8 are longer than 50% of the
maximum, but 2.4 are longer than 75% of the maximum. Thus it seems that a
very small number of contacts are comparatively consistent.
Figure 9.3
Clusters of neuronal classes obtained by hierarchical clustering of the
adjacency data. There are three thicknesses of line, corresponding to an
association measure of 25 or more for the thickest, 15 or more for the
intermediate one, and 8 or more for the thinnest one. All these clusters
were seen on both sides of the nervous system. In some cases the dorsal
and ventral members of the same class ended up reproducibly in different
clusters (e.g. CEP, IL1, SMB). The positions of
classes were moved as little as possible from those in figure 8.1, in order
to show the relationship between possible bundle assignments and circuitry.
The RMED/L/R/V class is ringed because the four RMED/L/R/V neurons form a tight bundle
with each other. The RIAL/R
and RMD classes are linked in a dashed cluster because there are a number
of RIAL/R/RMD
pairs that have associations just under 8.
It is hard to tell with individual process pairs whether their high
adjacency in accidental or not, but if several processes combined in a
bundle it should be objectively deducible from the adjacency information in
the database. A bundle will consist of a group of processes with the
property that all pairs in the group are highly adjacent, but no other
process is very adjacent to the group as a whole.
The number of possible groups goes up exponentially with the size of the
group, so it is not possible to try every one even with small groups.
However there is a branch of multivariate statistics called cluster
analysis that is specifically designed to handle this type of problem, and
a variant of a standard algorithm from this theory was used to extract
clusters of highly mutually adjacent processes that are likely candidates
for bundles. The details of this algorithm are given in the appendix, but
the final result is a hierarchial set of nested clusters with a measure of
the degree of association at each level, which corresponds to an average
internal adjacency. Any real clusters, such as the proposed bundles,
should stand out as having a high association measure at the level of the
group, but not combine well with an external process or group at the next
level down. Figure 9.3 shows the bundles detected by the algorithm in the
C. elegans database at associations measure cutoffs of 25, 15 and
8. In the case of contralateral homologues, either bundles were seen on
both sides, or the same bundle included both homologues.
In order to provide an objective significance criterion for the association
measure of a cluster, I used the same algorithm on data from a simulation
of the random mixing model described in the last section (with p = 0.08).
The maximum association measure obtained was 12.75 and less than 10% of the
values were greater than 7.5. Thus according to this criterion all the
bundles shown in figure 9.3 at an associational level of 15 are likely to
be significant, as are most of those at a level of 8, especially when they
occur on both sides of the animal.
Since the clustering method is hierarchial and continues to make larger and
larger clusters it does generate further amalgamations of the bundles seen
in figure 9.3. Although the association measure for such bundles falls
below our significance test level there is evidence that some of them are
real, primarily because the same groupings are seen for homologous bundles
on the two different sides of the same animal. The fact that they have a
low association measure implies that they are not true completely mixing
bundles, but they may be either super-bundles - bundles of bundles - or
cases where processes are shared by several bundles. The suggestion that
particular processes might belong to more than one bundle is taken further
in the discussion section.
There is some evidence in the database for the presence of
reproducible persistent contacts between nerve fibres, both between
pairs of neurons and between groups of three or more processes that
run together round the ring as sub-bundles within the complete
process tract. The largest grouping of neurons which all had
fairly high adjacency to each other contained seven cell types
(figure 9.3) but most of the likely bundles generated by cluster
analysis of the neighbourhood information contained only two or
three cell types. The average number of high adjacency contacts
per neuron was also small (2.3).
The analysis presented
here suffers from its reliance on identifying specific contacts by
unusually high adjacencies. It would therefore miss any important
short term contacts, and would also be confused by processes that
for half their length are in one part of the neuropil, and for the
other half in another part. There is a clear example of such
behaviour in the case of the interneuron AIBL/AIBR, which runs near AIAL/AIAR
in the proximal part of its trajectory, and near RIBL/RIBR in the distal
part (White, 1983). This is consistent with AIBL/AIBR's role as the
major linking interneuron between the amphid receptor circuitry and
the motor control circuitry (figure 8.2). Such switching of
bundles could be used by other processes that carry information
between sufficiently different groups of processes.
Another
example of the possible presence of sub-bundles in the nervous
system is provided by the motor neuron processes in the ventral
nerve cord. The VA and VB classes of motor neuron are both
bipolar, with an axonal process that produces neuromuscular output
for part of its length (the other part neither makes nor receives
connections) and also receives some input, and a dendritic process
that is purely postsynaptic. All the dendrites run together in one
place, under the main motor neurons, while all the axons run in a
group against the basement membrane. Although these two groups of
processes are adjacent they rarely mix. In addition there are a
number of places where a motor neuron commissure cuts across the
entire nerve cord; when this happened the commissure usually runs
between the dendritic and axonal groups, separating one group from
the other, but splitting neither (7/13 cases; in 5/13 a VB dendrite
is on the wrong side - in only one case is the axonal bundle
split). In this case a general adhesion between like processes may
be useful in keeping all the dendrites near their source of
inervation, and keeping the axons near the basement membrane, where
neuromuscular junctions are made.
There is a strong
relationship between the proposed groupings of the neurons into
bundles and the circuitry. Figure 9.3 has been organised so as to
show the extent to which the bundles are formed from neurons that
are near in the processing diagram in figure 8.1, which was
obtained purely from connectivity data. However it is by no means
true that all persistent pairwise contacts are between neurons that
are connected, either by chemical synapses, or by gap junctions
(e.g. CEP and URX, or the ventral cord motor neuron bundles). In
some cases bundles correspond to parts of the processing modules
defined previously on the basis of internal feedback, but they also
often contain vertical groupings of neuronal classes from the
directional ordering, sometimes with elements from two modules, one
of which feeds into the other. Such organisation is to be expected
if the main criterion for process placement is to maximise the
adjacency of symaptic partners, since the main flow of information
is down through the network, across the modules.
The
observation that there are a small number of persistent contacts
suggests that specific fasciculation mechanisms are significant in
the C. elegans nerve ring, and appears to rule out the
specification of process position by a general mechanism that acts
equivalently on all cells. This is perhaps not surprising in an
organism with such a small number of cells, almost all of which are
distinct, forming different sets of specific connections. The data
certainly do not allow the prediction of a set of hierarchical
forces that could determine position in the nerve ring. There is
also the problem present in all the analysis of the database of
trying to investigate the underlying mechanisms involved in
building a structure (the nerve ring) by looking at the finished
product. However, taken together with the evidence for the role of
specific fasciculation in embryonic neural outgrowth presented in
the first part of this thesis, there are strong grounds for
believing that the organisation of the nerve ring may make use of
small specific bundles to correctly position processes so that
synaptic connections can be made.