BACKGROUND
brain function depends on precisely specified patterns of wiring between neurons, and failures of wiring can compromise normal function  <cit> .
this wiring develops during early life as axons grow and navigate to find their appropriate targets, often over long distances.
a critical role in guiding axons to their targets during neural development is played by neuronal growth cones  <cit> .
growth cones have a remarkably complex and dynamic morphology, with their shape changing on the timescale of minutes  <cit> .
in vivo some of these shape changes appear related to the position of the growth cone along its trajectory, with more complex morphology at choice points  <cit>  suggesting that shape changes play an important role in guidance.
however, previous morphological analyses of growth cones have been largely driven by human judgement regarding important shape dimensions, rather than these dimensions being determined directly from the data.
the most prominent features of growth cone structure are filopodia and lamellipodia.
filopodia can be quantified in terms of their number, positions, angles and lengths, while a simple measure of lamellipodial extent is the total area of the growth cone.
one way of quantifying the shape of a growth cone at each moment is therefore to provide a list of these quantities, which for a typical growth cone with say five filopodia would consist of  <dig> numbers .
while such a quantification can be useful, it clearly has significant limitations.
first, it relies on time-lapse imaging of a resolution sufficient to resolve all individual filopodia, which can be difficult to achieve for dynamic growth cones for long periods of time, especially in vivo.
second, despite its length this list still ignores many obvious characteristics of growth cone shape, such as the shape of the lamellipodia.
third, the formulation of this list takes no account of the statistical structure of the actual data: it is driven by human intuition rather than an objective dissection of where the most variance of growth cone shape actually lies.
here we therefore take a different approach to quantifying growth cone shape.
the approach is based on principal component analysis , a well-known mathematical method for revealing the dimensions of a dataset which have the most variance.
the application of pca in the context of shapes is often called eigenshape analysis  <cit> .
each shape can be parameterised by the coordinates of a set of points, usually placed around the perimeter of the shape.
this vector of coordinates can be represented as a point in a high-dimensional space, so that a set of shapes is represented by a cloud of points in that space.
pca rotates the axes of this space so that the axes are now ordered in terms of the variance in the data they explain.
this reveals the directions  through this cloud of points along which there is maximum variance, i.e., along which the cloud of points is most spread out.
the first few principal components or eigenshapes then represent the most important shape dimensions, which can be seen as the fundamental building blocks of the set of shapes in question.
on a cellular scale, eigenshape analysis has previously revealed important information about the shapes of keratocytes  <cit>  and dictyostelium  <cit> .
it has also proved to be an extremely useful data analysis tool in domains as diverse as caenorhabditis elegans locomotion  <cit> , computer vision  <cit> , palaeontology  <cit> , botany  <cit>  and musical instrument design  <cit> .
here we use eigenshape analysis to reveal the basic building blocks of growth cone morphology, previously unknown properties of how growth cone shape evolves through time, and new insights into the relationships between growth cone shape, chemotactic responses and forward movement.
we then show that a simple computational model of shape changes based on dynamic microtubule instability can quantitatively reproduce the characteristic timescales present in the data.
RESULTS
growth cone eigenshapes
to generate a database of growth cone shapes we first made time-lapse movies of growth cones from neonatal rat superior cervical ganglion neurites  growing in vitro for  <dig> to  <dig> h  at  <dig> s to  <dig> min intervals .
from these we determined characteristic growth cone shapes using eigenshape analysis, i.e., pca in the space of shapes for the dataset  <cit>  .
the outline of each growth cone in each frame  was automatically extracted, and parameterised by  <dig> evenly spaced points.
the vector of  <dig> numbers  representing each outline can be represented as a point in a 500-dimensional space .
pca was then applied to extract the directions in the shape space that captured the largest proportion of the variance.
the bayesian information criterion   <cit> , which trades off variance explained versus model complexity , can be used to determine objectively the number of these dimensions that capture most of the variance of the set of shapes.
according to this criterion, the optimal model retained only the top five components, which captured 86% of the variance in growth cone shape .
we henceforth refer to these as the significant modes.summary of datasets used
cell type
time step
duration
movies
frames
in vivo
time step is the frequency of image capture.
each movie is for a different growth cone.
rgc, retinal ganglion cell axons ; scg, superior cervical ganglion axons .
five significant shape modes explain 86% of the variance in growth cone shape.
typical sequence of frames  from a time-lapse movie of an scg growth cone in vitro.
scale bar:  <dig> µm.
schematic of the steps involved in the eigenshape analysis to extract the shape dimensions that capture the most variance: outline capture, parameterisation of outline by  <dig> evenly spaced points, principal component analysis of the resulting 500-dimensional space.
scale bar:  <dig> µm.
variance explained as a function of number of mode shapes for the in vitro  dataset .
the significant modes and their variance explained, shown as the mean shape plus the shape one standard deviation in each direction along the shape axis.
our naming convention for each mode is that the letter represents the type of symmetry, while the number is used to distinguish between different r/s/m modes.
m <dig> and m <dig> approximately represent linear combinations of shapes r <dig> and s <dig> .
note that all fine details  occur with a fairly random distribution around the growth cone, and are thus smoothed out once the dataset of images is appropriately large.
higher-order shape modes and their variance explained.
it is remarkable that the split between r and s symmetry persists across many higher-order modes.
m <dig> could be arising here as an attempt to explain slight asymmetries in the underlying data.
m modes in pairs, such as m <dig> and m <dig> in , can sometimes be understood as a linear combination of an r mode and an s mode.
this occurs because when two modes have similar eigenvalues, any two orthogonal directions in that two-dimensional subspace can appear in the principal component decomposition.
illustration of shape reconstruction using different numbers of modes.
the red curve is the traced outline of a growth cone at one instant, and the blue curve is the best reconstruction of this shape given the specified number of eigenshape modes.
higher modes provide additional levels of detail for reproducing the true shape, but each individually only reproduces a tiny proportion of the variance across the full dataset.
m, mixed; r, reflective; s, symmetric; scg, superior cervical ganglion; sd, standard deviation; var, variance.
these shape modes split into three types, which we term reflective , symmetric  and mixed  modes .
for r modes, the shapes corresponding to equal positive and negative movements along the shape axis are approximately reflections of each other, with the principal r mode  representing bending of the growth cone.
s modes instead have approximate symmetry about their midline for all positions along the shape axis, with s <dig> representing spreading of the growth cone.
m modes have neither of these properties, but may represent a linear combination of r and s modes .
the split between these different types continues for the higher-order modes .
the number following r, s or m refers to the logical sequence for each type of mode.
this is most clear for the r modes: r <dig> displays one bend, r <dig>  displays two bends, r <dig> displays three bends, and so on.
the modes were robust to the number of points used to define the outline, provided this exceeded approximately  <dig> .
thus, just five characteristic shape modes capture most of the variance in this set of growth cones, and these modes describe distinct features of growth cone morphology.mode shape robustness to number of points used to parameterise the outline, and independent components analysis.
<dig> points: shapes and variance explained are the same as for figure  <dig> with  <dig> points .
using  <dig>  points also produced identical results .
<dig> points: shapes are the same, but variance explained is slightly different .
<dig> points: both shapes and variance explained are now noticeably different .
<dig> points: shapes and variance explained are now very different .
thus  <dig> points is a sufficiently detailed parameterisation such that no further changes in shape or number of significant components occur as this number is increased.
<dig> independent components were found for the in vitro  dataset using the fastica algorithm  <cit> .
ordering is arbitrary.
some of these components  appear to be trying to represent individual filopodia.
taken individually, none of these mode shapes explicitly represent filopodia.
no information has been lost overall, however: the axes of the space have simply been rotated, and adding together a sufficient number of mode shapes will perfectly reconstruct the original shape to any desired level of accuracy .
rather, the analysis has determined that choosing a few axes that represent individual filopodia explicitly does not capture the largest amounts of variance in the data.
it is easy to see intuitively why this would be the case.
to construct a filopodium at a particular location using the pca axes requires the addition of many higher-order modes, each of which by itself only explains a very small amount of variance across the whole dataset.
individual filopodia are sparse, in the mathematical sense that most of the time there is no filopodium at a particular position, but when it is present it is represented strongly.
we therefore also performed an independent component analysis  <cit> , which is well suited for extracting sparse structure in data.
as expected, this produced mode shapes hinting at filopodia-like structures .
however, these independent component modes are less useful for capturing general patterns of overall shape  <cit> , and we therefore focused on eigenshape analysis.
to maximise throughput, we used a plastic substrate and relatively low magnification imaging for our experiments.
could the significant eigenshapes derived from these data be missing key features of growth cone shape that would become apparent from analysis of higher-quality images?
to address this we also performed experiments on a glass substrate , which allowed, for instance, clearer visualisation of filopodia.
in this case there were seven significant mode shapes, but their form was similar to those observed on a plastic substrate .
from this dataset we also generated degraded growth cone outlines to match approximately the quality of outlines available using the plastic substrate.
the resulting modes shapes were very similar .
thus the leading shape modes are not very sensitive to the level of detail at which the growth cone outlines are captured, making this form of quantification suitable for a much broader range of imaging regimes than quantifications relying on precise identification of individual filopodia.mode shapes are similar on a glass substrate.
all other in vitro data used a plastic substrate.
to confirm that the higher level of image quality available using a glass substrate  did not affect our overall conclusions, we also performed time-lapse imaging of rat scg axons growing in glass-bottomed dishes .
in this case there were seven significant mode shapes, but their form was similar to those observed on a plastic substrate .
eigenshapes from the same dataset with artificially degraded image quality are very similar .
this is because all fine details  occur with a fairly random distribution around the growth cone, and are thus smoothed out once the dataset of images is appropriately large.
scg, superior cervical ganglion.
growth cones oscillate
each growth cone outline can be projected onto each shape mode to give a set of mode scores.
these measure the degree to which each shape is represented by each outline, i.e., the position of the growth cone shape along that shape axis.
the overall mode score frequency distributions are shown in figure  <dig>  as expected the r mode projections have roughly symmetric frequency distributions, while m modes have slightly asymmetric distributions, and s modes have highly asymmetric distributions.
there are no linear relationships between the distributions of pairs of mode scores since, by definition, pca dimensions are orthogonal.
a statistical test for nonlinear relationships  <cit>  showed some dependencies between modes, though plotting pairs of mode scores against each other did not reveal any obvious patterns .mode score distributions and relations across the
in vitro
dataset.
r  modes are symmetrically distributed around the mean shape, while s  and m  modes have skewed distributions.
this is because they contain a ‘thinness versus fatness’ component, which has a hard lower limit.
we also examined the joint distributions of pairs of modes.
no linear correlations were expected since the shape modes are by definition orthogonal, and no obvious structure was visually apparent.
however, a statistical test for nonlinear relationships  <cit>  showed some dependencies between modes, particularly s1/m <dig> and s1/m <dig> .
we then examined how mode scores varied through time for each individual growth cone .
these revealed strong hints of periodic behaviour.
for instance in the top left panel of figure 5a, the r <dig> mode score varies regularly between positive and negative values, indicating a periodic alternation in the growth cone shape between bending left and bending right.
to detect periodic patterns quantitatively, we used the autocorrelation and fourier power spectrum of mode scores.
oscillations in mode scores were common in all significant modes .
similar oscillations were seen on a glass substrate .
oscillation strength and frequency for each mode were quantified using a modified version of the method of  <cit>  .
the distributions of oscillation strengths  and frequencies for all significant modes  are shown in figure 6a,b.
the mean oscillation frequency was  <dig>  ±  <dig>  min − <dig> , corresponding to a mean period of  <dig> min.
the mean oscillation score was  <dig>  ±  <dig> .
r <dig> oscillations were on average significantly stronger than oscillations in other modes .
we used shuffled controls to demonstrate that these oscillations were not simply an artefact of our analysis methods .
the mean shuffled frequency was  <dig>  min − <dig>  and the mean shuffled score was  <dig> .
t-tests comparing the real and shuffled distributions gave p values of 10− <dig> for frequencies and 10− <dig> for scores.
different modes for the same growth cone sometimes oscillated at the same frequency, and sometimes at different frequencies , with the former case revealing a range of phase relationships .
overall, we conclude that growth cone shape oscillates on an average timescale of  <dig> min.mode shapes oscillate over time.
variations in mode scores over time for the top five modes for two example growth cones.
note the clear oscillations, particularly for r <dig>  in the top row and s <dig>  in the bottom row.
autocorrelation and fourier power spectra for all significant modes from the above two growth cones.
analysis of oscillations for another example growth cone, this time showing much longer-period oscillations in r <dig>   shuffle control for the growth cone from panel .
randomly shuffling the positions of all frames within the movie destroys oscillations.
this is seen most clearly by the presence of very weak power, spread across many frequencies, in the fourier power spectrum.
we also shuffled all frames across the entire dataset  <dig>  times, and calculated the mean total power and peak power.
in all  <dig>  cases these mean power values were less than for the unshuffled data.
autocorr, autocorrelation.
properties of oscillations.
oscillations are predictive of movement.
distribution of strongest oscillation frequencies  and scores  across the in vitro dataset .
also shown are the corresponding distributions for the shuffled controls .
the mean of the oscillation scores is  <dig> , while for the shuffled controls it is  <dig>  .
phase relationships between shape mode oscillations.
frequency distributions of phase relationships are shown for pairs that showed consistent relationships for the five significant modes in the in vitro  dataset.
the lack of consistent phase between r  and s  modes is not surprising, due to their fundamentally different symmetry properties.
if, for instance, r <dig> and s <dig> were consistently in phase, it would mean that growth cones bending left were consistently fatter  than growth cones bending right.
average step length is correlated with mode s <dig> oscillation strength   and mode r <dig> oscillation frequency .
strengths and frequencies of all significant modes together are predictive of average step length.
see tables  <dig>   <dig> and  <dig>
correlations between mode scores
mode score
unsigned mode score
r1
s1
m1
m2
r3
r1
s1
m1
m2
r3
r
p
r
p
r
p
the data are for each moment and growth cone movement over the following minute for the in vitro  dataset.
r is the spearman correlation.
correlations for both signed and unsigned  mode scores are shown.
correlations between oscillation strength and period for the average step length
oscillation strength
oscillation period
r1
s1
m1
m2
r3
r1
s1
m1
m2
r3
r
p
this is for the in vitro  dataset.
average growth rate  is most correlated with s <dig> oscillation strength and r <dig> oscillation frequency.
no significant results were found for signed or unsigned bearing change .
the oscillation strength measure was designed to be accurate on narrow bands of frequency, and is known to be affected by large variations in frequency as a result of non-linear scalings in the computed power spectrum  <cit> .
we therefore only considered oscillations with a period under  <dig> min .
regression of oscillation strengths and periods for all significant modes and average step length
oscillation
oscillation
oscillation
strength
period
strength
and period
r
p
this is for the in vitro  dataset.
r is the coefficient of multiple correlation, the multidimensional generalisation of the correlation coefficient.
relationship between oscillations and movement
given the variability of oscillation strengths and frequencies about their means, we were interested in whether there were any predictive relationships between mode scores, or oscillation strengths and frequencies, and growth cone movement between frames, characterised by step lengths  and bearing changes  between consecutive frames.
step length from time t to t+ <dig> was weakly positively correlated with s <dig> mode score at time t , meaning that wider growth cones had a small tendency to take larger steps.
other mode scores were almost uncorrelated with the movement of the growth cone , and in particular the correlation of the r <dig> mode score with subsequent bearing change, although significant, was very small.
thus the periodic alternation in shape of bending left versus bending right did not translate into zigzags in overall axon trajectory.
rather, a periodic sweeping left and right of the growth cone was superimposed on a steady forward motion.
r <dig> oscillations could be seen as a way by which growth cones might systematically probe or ‘sniff’ the environment for molecular or topographical cues, sweeping out more area than they would without these oscillations.
however, oscillation strength and frequency for each movie were correlated with the movement of the growth cone averaged over all frames .
average growth rate  was correlated with s <dig> oscillation strength  and r <dig> oscillation frequency .
performing a linear regression of average step length against the ten variables of oscillation score and frequency for the five significant modes produced a good prediction of average growth rates .
thus, growth cones that are oscillating strongly and rapidly tend to make forward progress faster.
oscillations during chemotaxis
an important mechanism by which axons are guided in vivo is chemotaxis.
we therefore applied eigenshape analysis to determine the characteristic behaviour of a new set of growth cones as they underwent chemotactic movement in the growth cone turning  assay for  <dig> h  <cit>  .
this assay produces a gradient steepness of approximately 10% across the growth cone  <cit> .
previously it was found that axons growing in gradients of steepness less than 1% show a bell-shaped chemotactic sensitivity curve predicted well by a bayesian model of the chemotactic response  <cit> .
we confirmed that a similar curve holds for steeper gradients , as theoretically predicted  <cit> .
by reducing pka activity in the growth cone via addition of kt <dig>  <cit> , we also found that the chemotactic response for repulsion was approximately the mirror image of that for attraction .similar modes and oscillations are seen during chemotaxis.
example of growth cone moving in response to gradient produced from a pipette .
scale bar:  <dig> µm.
chemotactic sensitivity curves  for both attractive and repulsive conditions.
significant modes for pipette assay movies.
the two m  modes from figure 1c have now separated more clearly into r  and s  modes, but otherwise the mode shapes, and their variance explained, are very similar to the in vitro  dataset.
oscillations in a representative pipette movie.
ngf, nerve growth factor; sd, standard deviation.
summary of the pipette assay experiments
ngf only
ngf + kt5720
movies
frames
movies
frames
refers to ngf concentration in the pipette.
there were also  <dig> control movies, totalling  <dig>  frames.
ngf, nerve growth factor.
in this dataset there were six significant eigenshape components, explaining 87% of the variance .
these were very similar to those we observed for the original in vitro  dataset, but more clearly illustrated the split into r and s modes.
these modes oscillated similarly , though the short assay duration prevented observation of longer oscillations.
there were no significant differences between oscillation strength and period between different gradient conditions, and no relationship was observed between final turning angle and oscillations.
however, mean mode projection scores  varied systematically across gradient conditions, demonstrating a direct effect of chemotactic cues on growth cone shape .
for instance, the mean r <dig> mode scores roughly followed the same shape as the chemotactic sensitivity curves.average mode scores as a function of ngf concentration for the pipette assay data.
these show how average growth cone shape is affected by concentration and whether the gradient is attractive or repulsive.
r <dig>  roughly follows the chemotactic sensitivity curve , while the other modes show more complex relationships.
ngf, nerve growth factor.
significance values for shape differences between attraction and repulsion
ngf concentration
mode
<dig>
2
5
10
20
30
p values with holm–bonferroni correction for mean shape mode projections for ngf versus ngf plus kt <dig> pipette assays .
ngf, nerve growth factor.
in a further set of turning assay experiments , we imaged growth cones for  <dig> h without a gradient, then applied a gradient for  <dig> h, to determine whether the presence of the gradient would change the properties of the oscillations for each growth cone.
however, we found no significant differences.
eigenshapes and oscillations in vivo
to investigate whether similar behaviour is observed in vivo, we performed time-lapse imaging, image segmentation and eigenshape analysis as before on mgfp-labelled growth cones  of zebrafish retinal ganglion cell axons as they navigated across the optic tectum for periods of  <dig> to  <dig> h    <cit> .
this revealed six significant shape modes that were very similar to those seen for the in vitro no gradient and pipette datasets .
oscillations in mode shapes were also qualitatively similar .
the mean oscillation frequency was  <dig>  ±  <dig>  min − <dig>  the difference with the in vitro frequency is likely at least partly due to the longer duration of the movies, which allows lower frequency oscillations to be included in the mean .
the mean oscillation score was  <dig>  ±  <dig> , similar to the in vitro data.
correlations between oscillations and average step lengths were small , but this dataset is much smaller than the in vitro dataset.
thus, the eigenshapes and oscillations displayed by growth cones as they navigate in a molecularly and structurally complex in vivo environment are similar to those displayed in a much simpler in vitro environment, arguing that these behaviours represent fundamental characteristics of growth cones rather than an in vitro artefact or the response to a specific environment.
similar mode shapes and oscillations are also present
in vivo
.
typical sequence of frames  <dig> min apart from a time-lapse movie of a zebrafish retinal ganglion cell growth cone navigating across the tectum in vivo.
scale bar:  <dig> µm.
the significant shape modes derived from this dataset.
mode oscillations for a  <dig> h movie with frames every  <dig> min , and  <dig> h movies with frames every  <dig> min .
in the former case oscillations of similar period to those observed in vitro are seen, while in the latter cases oscillations with longer periods become apparent.
autocorr, autocorrelation; sd, standard deviation.
a computational model of oscillations
what events inside the growth cone could be driving the strong periodicity in shape dynamics we have observed?
can we explain both the mean period of  <dig> min, and the large variability about this average?
a critical component of the growth cone cytoskeleton is microtubules  <cit> .
these extend from the axon shaft into the body of the growth cone, and sometimes into individual filopodia  <cit> .
microtubule growth is characterised by dynamic instability, whereby phases of growth are followed by catastrophic collapse, and then a return to the growth phase  <cit> .
walker et al.
<cit>  constructed a computational model of this phenomenon, with parameter values constrained directly from experimental measurements.
janulevecius et al.
<cit>  then adapted this model to show that the small volumes of cells, and thus the limited supply of free tubulin, could significantly impact on microtubule dynamics.
as a minimal model of growth cone shape changes, here we consider two microtubules within a growth cone competing for the same limited supply of tubulin monomers .computational model of dynamic microtubule instability reproduces the periodicities seen experimentally.
the model consists of two microtubules  competing for a limited supply of tubulin monomers  within the growth cone.
lengths of both microtubules as a function of time for a typical simulation.
autocorrelation functions for three different simulations with identical parameters but different random seeds, illustrating strong periodicity but also different frequencies at which these can occur in the model.
distribution of oscillation scores  and frequencies  across  <dig> simulated growth cones.
mt, microtubule.
we assume that the extension of one of these microtubules corresponds to a shape deviation in one direction , while the extension of the other microtubule corresponds to a shape deviation in the opposite direction .
referring to the lengths of the two microtubules at time t as l <dig> and l <dig>  we take the normalised difference in lengths l=−l2)/+l2) as an analogue for the projection onto an eigenshape mode.
the model and parameters for each microtubule were taken directly from  <cit>  , with the growth cone volume assumed to be  <dig> µm  <dig> .
the only interaction between the microtubules was via their competition for the same pool of tubulin monomers.
growth cones  were simulated for the equivalent of approximately  <dig>  h .
l was then subsampled to one value every minute of simulated time, the time resolution of most of our in vitro experimental data.
as expected, l <dig> and l <dig> tended to compete with each other .
we found that l generally showed periodic oscillations over time, qualitatively resembling the experimental data .
to quantify this behaviour, oscillation scores and frequencies were calculated for l exactly as for the experimental data, producing distributions over the set of simulated growth cones.
oscillation scores were larger than for the experimental data, with a mean and standard deviation  <dig>  ±  <dig> , compared to  <dig>  ±  <dig>  for the experimental data .
this is perhaps not surprising, since there are many additional sources of noise in the data, which are not present in the model.
however, surprisingly given the simplicity of the model, there was an extremely close match between the oscillation frequencies of the model and the in vitro data:  <dig>  ±  <dig>  min − <dig> for the model, compared to  <dig>  ±  <dig>  for the data .
thus, the model implicates dynamic microtubule instability as the driving force behind oscillations in growth cone shape.
the model also allows an analysis of how the duration of simulated time affects mean frequencies.
intuitively, longer durations would lead to lower average frequencies, since more distant parts of the underlying frequency distribution are now included in the average.
simulating  <dig> growth cones for  <dig> × <dig> events  gave a mean frequency of  <dig>  ±  <dig>  min − <dig>  confirming this intuition.
this was not significantly different to the mean oscillation frequency observed in the in vivo data , suggesting that the lower mean frequency observed in our in vivo compared to in vitro data is mainly due to the longer average length of these movies, rather than any fundamental difference in oscillations between in vitro and in vivo settings.
this is consistent with the idea that oscillations are intrinsically driven.
discussion
neuronal growth cones have one of the most dynamic morphologies of any cellular system, and this is challenging to quantify.
here we have applied for the first time to growth cones a shape analysis technique that has proved useful in many other contexts, and shown that just a few basic shape primitives capture the vast majority of the variance in growth cone shape.
the form of the leading modes themselves is quite intuitive: r <dig> represents bending while s <dig> represents thinness versus fatness, whether these are a result of lamellipodial or filopodial outgrowth.
by reducing shape to just the list of numbers representing the projections onto the significant modes, it becomes possible to identify patterns in the data that are not otherwise apparent.
in particular, by examining how these projections evolve over time, we found that shape oscillations are a key organising principle of growth cone shape dynamics.
the forward movement of growth cones has previously been reported to follow cycles of pausing and growth  <cit> , and zigzag behaviour in gradients  <cit> .
however, our morphometric  <cit>  analysis reveals for the first time the shapes representing the most important degrees of variance in growth cone morphology , how the projections of these shapes vary through time , and how they are correlated with growth cone movement .
we observed the same general patterns of eigenshapes and oscillations in vitro and in vivo, suggesting that oscillations are a surprisingly robust and fundamental aspect of growth cone behaviour.
we also presented a computational model of growth cone shape changes based on dynamic microtubule instability.
this model is clearly a highly abstracted version of reality, and is intended as simply a minimal model of variation along a shape axis .
it is therefore remarkable that such a simple mechanism produces an excellent match to not only the mean frequency of oscillations in the data, but also the variance in this distribution.
this provides strong support for the hypothesis that the basic driver of the growth cone shape oscillations we have observed is dynamic microtubule instability in the context of competition for a limited supply of tubulin monomers.
clearly in reality many other components will be involved, most notably the actin cytoskeleton, and how these work together to determine oscillations remains to be determined.
we have examined growth cones in a relatively featureless in vitro environment, a chemotactic gradient, and traversing the optic tectum in vivo.
eigenshapes and oscillations are remarkably consistent between these different cases.
this argues that these properties are intrinsic and do not represent responses to the local environment, such as a way of navigating around local obstacles, consistent with an explanation in terms of intrinsic periodicity of microtubule growth.
the complete trajectories of growth cones in vivo often involve navigating through many stages and choice points, for instance retinal axons growing from the eye to the brain  <cit>  or callosal axons finding their targets in the opposite hemisphere  <cit> .
eigenshape analysis provides a method for re-examining more quantitatively exactly how shape changes over such trajectories, and how it is correlated with navigational function.
whether shape oscillations change their properties over complex trajectories and how such changes might be related to the environmental cues present at each moment remain to be determined.
our tracing of growth cone outlines was performed on relatively low-magnification phase contrast  or fluorescent  images, and thus fine details of some filopodia were inevitably lost.
however, we have demonstrated directly that the eigenshapes we found are quite robust to image quality .
filopodia are not ignored in our analysis: for instance the presence of a filopodium in the right-hand side of the growth cone will show its effect by increasing the score for mode r <dig> .
eigenshape analysis complements rather than replaces fine-scale analysis of filopodial dynamics, since it combines filopodial and lamellipodial structure to emphasise overall shape trends rather than fine-scale structure.
despite recent advances for fluorescently labelled growth cones  <cit> , fine analysis of filopodia is still limited by the difficulty in obtaining large datasets: fully automated image analysis techniques have difficulty with phase-contrast imaging at this level of detail, and hand-tracing is prohibitively time-intensive.
unfortunately manual observations are not scalable  <cit> , meaning that there is currently a growing mismatch between our ability to manipulate growth cones and to assess the effect of these manipulations on growth cone behaviour using manual techniques.
CONCLUSIONS
eigenshape analysis of growth cones has the potential to provide a novel quantitative understanding of the differences between growth cones in different conditions.
these include the effects of the environment as axons navigate towards their targets during development, differences between initial development and regeneration, and differences between mutants and wild type.
overall this work reveals a new dimension to the understanding of the dynamic morphology of growth cones, and potentially opens up novel directions for research into understanding the biological basis of developmental brain disorders.

 brain function depends on precisely specified patterns of wiring between neurons, and failures of wiring can compromise normal function. this wiring develops during early life as axons grow and navigate to find their appropriate targets, often over long distances. a critical role in guiding axons to their targets during neural development is played by neuronal growth cones. growth cones have a remarkably complex and dynamic morphology, with their shape changing on the timescale of minutes. in vivo some of these shape changes appear related to the position of the growth cone along its trajectory, with more complex morphology at choice points suggesting that shape changes play an important role in guidance. however, previous morphological analyses of growth cones have been largely driven by human judgement regarding important shape dimensions, rather than these dimensions being determined directly from the data. the most prominent features of growth cone structure are filopodia and lamellipodia. filopodia can be quantified in terms of their number, positions, angles and lengths, while a simple measure of lamellipodial extent is the total area of the growth cone. one way of quantifying the shape of a growth cone at each moment is therefore to provide a list of these quantities, which for a typical growth cone with say five filopodia would consist of numbers. while such a quantification can be useful, it clearly has significant limitations. first, it relies on time-lapse imaging of a resolution sufficient to resolve all individual filopodia, which can be difficult to achieve for dynamic growth cones for long periods of time, especially in vivo. second, despite its length this list still ignores many obvious characteristics of growth cone shape, such as the shape of the lamellipodia. third, the formulation of this list takes no account of the statistical structure of the actual data: it is driven by human intuition rather than an objective dissection of where the most variance of growth cone shape actually lies. here we therefore take a different approach to quantifying growth cone shape. the approach is based on principal component analysis, a well-known mathematical method for revealing the dimensions of a dataset which have the most variance. the application of pca in the context of shapes is often called eigenshape analysis. each shape can be parameterised by the coordinates of a set of points, usually placed around the perimeter of the shape. this vector of coordinates can be represented as a point in a high-dimensional space, so that a set of shapes is represented by a cloud of points in that space. pca rotates the axes of this space so that the axes are now ordered in terms of the variance in the data they explain. this reveals the directions through this cloud of points along which there is maximum variance, i.e., along which the cloud of points is most spread out. the first few principal components or eigenshapes then represent the most important shape dimensions, which can be seen as the fundamental building blocks of the set of shapes in question. on a cellular scale, eigenshape analysis has previously revealed important information about the shapes of keratocytes and dictyostelium. it has also proved to be an extremely useful data analysis tool in domains as diverse as caenorhabditis elegans locomotion, computer vision, palaeontology, botany and musical instrument design. here we use eigenshape analysis to reveal the basic building blocks of growth cone morphology, previously unknown properties of how growth cone shape evolves through time, and new insights into the relationships between growth cone shape, chemotactic responses and forward movement. we then show that a simple computational model of shape changes based on dynamic microtubule instability can quantitatively reproduce the characteristic timescales present in the data.  growth cone eigenshapes to generate a database of growth cone shapes we first made time-lapse movies of growth cones from neonatal rat superior cervical ganglion neurites growing in vitro for to h at s to min intervals. from these we determined characteristic growth cone shapes using eigenshape analysis, i.e., pca in the space of shapes for the dataset. the outline of each growth cone in each frame was automatically extracted, and parameterised by evenly spaced points. the vector of numbers representing each outline can be represented as a point in a 500-dimensional space. pca was then applied to extract the directions in the shape space that captured the largest proportion of the variance. the bayesian information criterion, which trades off variance explained versus model complexity, can be used to determine objectively the number of these dimensions that capture most of the variance of the set of shapes. according to this criterion, the optimal model retained only the top five components, which captured 86% of the variance in growth cone shape. we henceforth refer to these as the significant modes.summary of datasets used cell type time step duration movies frames in vivo time step is the frequency of image capture. each movie is for a different growth cone. rgc, retinal ganglion cell axons; scg, superior cervical ganglion axons. five significant shape modes explain 86% of the variance in growth cone shape. typical sequence of frames from a time-lapse movie of an scg growth cone in vitro. scale bar: µm. schematic of the steps involved in the eigenshape analysis to extract the shape dimensions that capture the most variance: outline capture, parameterisation of outline by evenly spaced points, principal component analysis of the resulting 500-dimensional space. scale bar: µm. variance explained as a function of number of mode shapes for the in vitro dataset. the significant modes and their variance explained, shown as the mean shape plus the shape one standard deviation in each direction along the shape axis. our naming convention for each mode is that the letter represents the type of symmetry, while the number is used to distinguish between different r/s/m modes. m and m approximately represent linear combinations of shapes r and s. note that all fine details occur with a fairly random distribution around the growth cone, and are thus smoothed out once the dataset of images is appropriately large. higher-order shape modes and their variance explained. it is remarkable that the split between r and s symmetry persists across many higher-order modes. m could be arising here as an attempt to explain slight asymmetries in the underlying data. m modes in pairs, such as m and m in, can sometimes be understood as a linear combination of an r mode and an s mode. this occurs because when two modes have similar eigenvalues, any two orthogonal directions in that two-dimensional subspace can appear in the principal component decomposition. illustration of shape reconstruction using different numbers of modes. the red curve is the traced outline of a growth cone at one instant, and the blue curve is the best reconstruction of this shape given the specified number of eigenshape modes. higher modes provide additional levels of detail for reproducing the true shape, but each individually only reproduces a tiny proportion of the variance across the full dataset. m, mixed; r, reflective; s, symmetric; scg, superior cervical ganglion; sd, standard deviation; var, variance. these shape modes split into three types, which we term reflective, symmetric and mixed modes. for r modes, the shapes corresponding to equal positive and negative movements along the shape axis are approximately reflections of each other, with the principal r mode representing bending of the growth cone. s modes instead have approximate symmetry about their midline for all positions along the shape axis, with s representing spreading of the growth cone. m modes have neither of these properties, but may represent a linear combination of r and s modes. the split between these different types continues for the higher-order modes. the number following r, s or m refers to the logical sequence for each type of mode. this is most clear for the r modes: r displays one bend, r displays two bends, r displays three bends, and so on. the modes were robust to the number of points used to define the outline, provided this exceeded approximately. thus, just five characteristic shape modes capture most of the variance in this set of growth cones, and these modes describe distinct features of growth cone morphology.mode shape robustness to number of points used to parameterise the outline, and independent components analysis. points: shapes and variance explained are the same as for figure with points. using points also produced identical results. points: shapes are the same, but variance explained is slightly different. points: both shapes and variance explained are now noticeably different. points: shapes and variance explained are now very different. thus points is a sufficiently detailed parameterisation such that no further changes in shape or number of significant components occur as this number is increased. independent components were found for the in vitro dataset using the fastica algorithm. ordering is arbitrary. some of these components appear to be trying to represent individual filopodia. taken individually, none of these mode shapes explicitly represent filopodia. no information has been lost overall, however: the axes of the space have simply been rotated, and adding together a sufficient number of mode shapes will perfectly reconstruct the original shape to any desired level of accuracy. rather, the analysis has determined that choosing a few axes that represent individual filopodia explicitly does not capture the largest amounts of variance in the data. it is easy to see intuitively why this would be the case. to construct a filopodium at a particular location using the pca axes requires the addition of many higher-order modes, each of which by itself only explains a very small amount of variance across the whole dataset. individual filopodia are sparse, in the mathematical sense that most of the time there is no filopodium at a particular position, but when it is present it is represented strongly. we therefore also performed an independent component analysis, which is well suited for extracting sparse structure in data. as expected, this produced mode shapes hinting at filopodia-like structures. however, these independent component modes are less useful for capturing general patterns of overall shape, and we therefore focused on eigenshape analysis. to maximise throughput, we used a plastic substrate and relatively low magnification imaging for our experiments. could the significant eigenshapes derived from these data be missing key features of growth cone shape that would become apparent from analysis of higher-quality images? to address this we also performed experiments on a glass substrate, which allowed, for instance, clearer visualisation of filopodia. in this case there were seven significant mode shapes, but their form was similar to those observed on a plastic substrate. from this dataset we also generated degraded growth cone outlines to match approximately the quality of outlines available using the plastic substrate. the resulting modes shapes were very similar. thus the leading shape modes are not very sensitive to the level of detail at which the growth cone outlines are captured, making this form of quantification suitable for a much broader range of imaging regimes than quantifications relying on precise identification of individual filopodia.mode shapes are similar on a glass substrate. all other in vitro data used a plastic substrate. to confirm that the higher level of image quality available using a glass substrate did not affect our overall conclusions, we also performed time-lapse imaging of rat scg axons growing in glass-bottomed dishes. in this case there were seven significant mode shapes, but their form was similar to those observed on a plastic substrate. eigenshapes from the same dataset with artificially degraded image quality are very similar. this is because all fine details occur with a fairly random distribution around the growth cone, and are thus smoothed out once the dataset of images is appropriately large. scg, superior cervical ganglion. growth cones oscillate each growth cone outline can be projected onto each shape mode to give a set of mode scores. these measure the degree to which each shape is represented by each outline, i.e., the position of the growth cone shape along that shape axis. the overall mode score frequency distributions are shown in figure as expected the r mode projections have roughly symmetric frequency distributions, while m modes have slightly asymmetric distributions, and s modes have highly asymmetric distributions. there are no linear relationships between the distributions of pairs of mode scores since, by definition, pca dimensions are orthogonal. a statistical test for nonlinear relationships showed some dependencies between modes, though plotting pairs of mode scores against each other did not reveal any obvious patterns.mode score distributions and relations across the in vitro dataset. r modes are symmetrically distributed around the mean shape, while s and m modes have skewed distributions. this is because they contain a ‘thinness versus fatness’ component, which has a hard lower limit. we also examined the joint distributions of pairs of modes. no linear correlations were expected since the shape modes are by definition orthogonal, and no obvious structure was visually apparent. however, a statistical test for nonlinear relationships showed some dependencies between modes, particularly s1/m and s1/m. we then examined how mode scores varied through time for each individual growth cone. these revealed strong hints of periodic behaviour. for instance in the top left panel of figure 5a, the r mode score varies regularly between positive and negative values, indicating a periodic alternation in the growth cone shape between bending left and bending right. to detect periodic patterns quantitatively, we used the autocorrelation and fourier power spectrum of mode scores. oscillations in mode scores were common in all significant modes. similar oscillations were seen on a glass substrate. oscillation strength and frequency for each mode were quantified using a modified version of the method of. the distributions of oscillation strengths and frequencies for all significant modes are shown in figure 6a,b. the mean oscillation frequency was ± min −, corresponding to a mean period of min. the mean oscillation score was ±. r oscillations were on average significantly stronger than oscillations in other modes. we used shuffled controls to demonstrate that these oscillations were not simply an artefact of our analysis methods. the mean shuffled frequency was min − and the mean shuffled score was. t-tests comparing the real and shuffled distributions gave p values of 10− for frequencies and 10− for scores. different modes for the same growth cone sometimes oscillated at the same frequency, and sometimes at different frequencies, with the former case revealing a range of phase relationships. overall, we conclude that growth cone shape oscillates on an average timescale of min.mode shapes oscillate over time. variations in mode scores over time for the top five modes for two example growth cones. note the clear oscillations, particularly for r in the top row and s in the bottom row. autocorrelation and fourier power spectra for all significant modes from the above two growth cones. analysis of oscillations for another example growth cone, this time showing much longer-period oscillations in r shuffle control for the growth cone from panel. randomly shuffling the positions of all frames within the movie destroys oscillations. this is seen most clearly by the presence of very weak power, spread across many frequencies, in the fourier power spectrum. we also shuffled all frames across the entire dataset times, and calculated the mean total power and peak power. in all cases these mean power values were less than for the unshuffled data. autocorr, autocorrelation. properties of oscillations. oscillations are predictive of movement. distribution of strongest oscillation frequencies and scores across the in vitro dataset. also shown are the corresponding distributions for the shuffled controls. the mean of the oscillation scores is, while for the shuffled controls it is. phase relationships between shape mode oscillations. frequency distributions of phase relationships are shown for pairs that showed consistent relationships for the five significant modes in the in vitro dataset. the lack of consistent phase between r and s modes is not surprising, due to their fundamentally different symmetry properties. if, for instance, r and s were consistently in phase, it would mean that growth cones bending left were consistently fatter than growth cones bending right. average step length is correlated with mode s oscillation strength and mode r oscillation frequency. strengths and frequencies of all significant modes together are predictive of average step length. see tables and correlations between mode scores mode score unsigned mode score r1 s1 m1 m2 r3 r1 s1 m1 m2 r3 r p r p r p the data are for each moment and growth cone movement over the following minute for the in vitro dataset. r is the spearman correlation. correlations for both signed and unsigned mode scores are shown. correlations between oscillation strength and period for the average step length oscillation strength oscillation period r1 s1 m1 m2 r3 r1 s1 m1 m2 r3 r p this is for the in vitro dataset. average growth rate is most correlated with s oscillation strength and r oscillation frequency. no significant results were found for signed or unsigned bearing change. the oscillation strength measure was designed to be accurate on narrow bands of frequency, and is known to be affected by large variations in frequency as a result of non-linear scalings in the computed power spectrum. we therefore only considered oscillations with a period under min. regression of oscillation strengths and periods for all significant modes and average step length oscillation oscillation oscillation strength period strength and period r p this is for the in vitro dataset. r is the coefficient of multiple correlation, the multidimensional generalisation of the correlation coefficient. relationship between oscillations and movement given the variability of oscillation strengths and frequencies about their means, we were interested in whether there were any predictive relationships between mode scores, or oscillation strengths and frequencies, and growth cone movement between frames, characterised by step lengths and bearing changes between consecutive frames. step length from time t to t+ was weakly positively correlated with s mode score at time t, meaning that wider growth cones had a small tendency to take larger steps. other mode scores were almost uncorrelated with the movement of the growth cone, and in particular the correlation of the r mode score with subsequent bearing change, although significant, was very small. thus the periodic alternation in shape of bending left versus bending right did not translate into zigzags in overall axon trajectory. rather, a periodic sweeping left and right of the growth cone was superimposed on a steady forward motion. r oscillations could be seen as a way by which growth cones might systematically probe or ‘sniff’ the environment for molecular or topographical cues, sweeping out more area than they would without these oscillations. however, oscillation strength and frequency for each movie were correlated with the movement of the growth cone averaged over all frames. average growth rate was correlated with s oscillation strength and r oscillation frequency. performing a linear regression of average step length against the ten variables of oscillation score and frequency for the five significant modes produced a good prediction of average growth rates. thus, growth cones that are oscillating strongly and rapidly tend to make forward progress faster. oscillations during chemotaxis an important mechanism by which axons are guided in vivo is chemotaxis. we therefore applied eigenshape analysis to determine the characteristic behaviour of a new set of growth cones as they underwent chemotactic movement in the growth cone turning assay for h. this assay produces a gradient steepness of approximately 10% across the growth cone. previously it was found that axons growing in gradients of steepness less than 1% show a bell-shaped chemotactic sensitivity curve predicted well by a bayesian model of the chemotactic response. we confirmed that a similar curve holds for steeper gradients, as theoretically predicted. by reducing pka activity in the growth cone via addition of kt, we also found that the chemotactic response for repulsion was approximately the mirror image of that for attraction.similar modes and oscillations are seen during chemotaxis. example of growth cone moving in response to gradient produced from a pipette. scale bar: µm. chemotactic sensitivity curves for both attractive and repulsive conditions. significant modes for pipette assay movies. the two m modes from figure 1c have now separated more clearly into r and s modes, but otherwise the mode shapes, and their variance explained, are very similar to the in vitro dataset. oscillations in a representative pipette movie. ngf, nerve growth factor; sd, standard deviation. summary of the pipette assay experiments ngf only ngf + kt5720 movies frames movies frames refers to ngf concentration in the pipette. there were also control movies, totalling frames. ngf, nerve growth factor. in this dataset there were six significant eigenshape components, explaining 87% of the variance. these were very similar to those we observed for the original in vitro dataset, but more clearly illustrated the split into r and s modes. these modes oscillated similarly, though the short assay duration prevented observation of longer oscillations. there were no significant differences between oscillation strength and period between different gradient conditions, and no relationship was observed between final turning angle and oscillations. however, mean mode projection scores varied systematically across gradient conditions, demonstrating a direct effect of chemotactic cues on growth cone shape. for instance, the mean r mode scores roughly followed the same shape as the chemotactic sensitivity curves.average mode scores as a function of ngf concentration for the pipette assay data. these show how average growth cone shape is affected by concentration and whether the gradient is attractive or repulsive. r roughly follows the chemotactic sensitivity curve, while the other modes show more complex relationships. ngf, nerve growth factor. significance values for shape differences between attraction and repulsion ngf concentration mode 2 5 10 20 30 p values with holm–bonferroni correction for mean shape mode projections for ngf versus ngf plus kt pipette assays. ngf, nerve growth factor. in a further set of turning assay experiments, we imaged growth cones for h without a gradient, then applied a gradient for h, to determine whether the presence of the gradient would change the properties of the oscillations for each growth cone. however, we found no significant differences. eigenshapes and oscillations in vivo to investigate whether similar behaviour is observed in vivo, we performed time-lapse imaging, image segmentation and eigenshape analysis as before on mgfp-labelled growth cones of zebrafish retinal ganglion cell axons as they navigated across the optic tectum for periods of to h. this revealed six significant shape modes that were very similar to those seen for the in vitro no gradient and pipette datasets. oscillations in mode shapes were also qualitatively similar. the mean oscillation frequency was ± min − the difference with the in vitro frequency is likely at least partly due to the longer duration of the movies, which allows lower frequency oscillations to be included in the mean. the mean oscillation score was ±, similar to the in vitro data. correlations between oscillations and average step lengths were small, but this dataset is much smaller than the in vitro dataset. thus, the eigenshapes and oscillations displayed by growth cones as they navigate in a molecularly and structurally complex in vivo environment are similar to those displayed in a much simpler in vitro environment, arguing that these behaviours represent fundamental characteristics of growth cones rather than an in vitro artefact or the response to a specific environment. similar mode shapes and oscillations are also present in vivo. typical sequence of frames min apart from a time-lapse movie of a zebrafish retinal ganglion cell growth cone navigating across the tectum in vivo. scale bar: µm. the significant shape modes derived from this dataset. mode oscillations for a h movie with frames every min, and h movies with frames every min. in the former case oscillations of similar period to those observed in vitro are seen, while in the latter cases oscillations with longer periods become apparent. autocorr, autocorrelation; sd, standard deviation. a computational model of oscillations what events inside the growth cone could be driving the strong periodicity in shape dynamics we have observed? can we explain both the mean period of min, and the large variability about this average? a critical component of the growth cone cytoskeleton is microtubules. these extend from the axon shaft into the body of the growth cone, and sometimes into individual filopodia. microtubule growth is characterised by dynamic instability, whereby phases of growth are followed by catastrophic collapse, and then a return to the growth phase. walker et al. constructed a computational model of this phenomenon, with parameter values constrained directly from experimental measurements. janulevecius et al. then adapted this model to show that the small volumes of cells, and thus the limited supply of free tubulin, could significantly impact on microtubule dynamics. as a minimal model of growth cone shape changes, here we consider two microtubules within a growth cone competing for the same limited supply of tubulin monomers.computational model of dynamic microtubule instability reproduces the periodicities seen experimentally. the model consists of two microtubules competing for a limited supply of tubulin monomers within the growth cone. lengths of both microtubules as a function of time for a typical simulation. autocorrelation functions for three different simulations with identical parameters but different random seeds, illustrating strong periodicity but also different frequencies at which these can occur in the model. distribution of oscillation scores and frequencies across simulated growth cones. mt, microtubule. we assume that the extension of one of these microtubules corresponds to a shape deviation in one direction, while the extension of the other microtubule corresponds to a shape deviation in the opposite direction. referring to the lengths of the two microtubules at time t as l and l we take the normalised difference in lengths l=−l2)/+l2) as an analogue for the projection onto an eigenshape mode. the model and parameters for each microtubule were taken directly from, with the growth cone volume assumed to be µm. the only interaction between the microtubules was via their competition for the same pool of tubulin monomers. growth cones were simulated for the equivalent of approximately h. l was then subsampled to one value every minute of simulated time, the time resolution of most of our in vitro experimental data. as expected, l and l tended to compete with each other. we found that l generally showed periodic oscillations over time, qualitatively resembling the experimental data. to quantify this behaviour, oscillation scores and frequencies were calculated for l exactly as for the experimental data, producing distributions over the set of simulated growth cones. oscillation scores were larger than for the experimental data, with a mean and standard deviation ±, compared to ± for the experimental data. this is perhaps not surprising, since there are many additional sources of noise in the data, which are not present in the model. however, surprisingly given the simplicity of the model, there was an extremely close match between the oscillation frequencies of the model and the in vitro data: ± min − for the model, compared to ± for the data. thus, the model implicates dynamic microtubule instability as the driving force behind oscillations in growth cone shape. the model also allows an analysis of how the duration of simulated time affects mean frequencies. intuitively, longer durations would lead to lower average frequencies, since more distant parts of the underlying frequency distribution are now included in the average. simulating growth cones for × events gave a mean frequency of ± min − confirming this intuition. this was not significantly different to the mean oscillation frequency observed in the in vivo data, suggesting that the lower mean frequency observed in our in vivo compared to in vitro data is mainly due to the longer average length of these movies, rather than any fundamental difference in oscillations between in vitro and in vivo settings. this is consistent with the idea that oscillations are intrinsically driven. discussion neuronal growth cones have one of the most dynamic morphologies of any cellular system, and this is challenging to quantify. here we have applied for the first time to growth cones a shape analysis technique that has proved useful in many other contexts, and shown that just a few basic shape primitives capture the vast majority of the variance in growth cone shape. the form of the leading modes themselves is quite intuitive: r represents bending while s represents thinness versus fatness, whether these are a result of lamellipodial or filopodial outgrowth. by reducing shape to just the list of numbers representing the projections onto the significant modes, it becomes possible to identify patterns in the data that are not otherwise apparent. in particular, by examining how these projections evolve over time, we found that shape oscillations are a key organising principle of growth cone shape dynamics. the forward movement of growth cones has previously been reported to follow cycles of pausing and growth, and zigzag behaviour in gradients. however, our morphometric analysis reveals for the first time the shapes representing the most important degrees of variance in growth cone morphology, how the projections of these shapes vary through time, and how they are correlated with growth cone movement. we observed the same general patterns of eigenshapes and oscillations in vitro and in vivo, suggesting that oscillations are a surprisingly robust and fundamental aspect of growth cone behaviour. we also presented a computational model of growth cone shape changes based on dynamic microtubule instability. this model is clearly a highly abstracted version of reality, and is intended as simply a minimal model of variation along a shape axis. it is therefore remarkable that such a simple mechanism produces an excellent match to not only the mean frequency of oscillations in the data, but also the variance in this distribution. this provides strong support for the hypothesis that the basic driver of the growth cone shape oscillations we have observed is dynamic microtubule instability in the context of competition for a limited supply of tubulin monomers. clearly in reality many other components will be involved, most notably the actin cytoskeleton, and how these work together to determine oscillations remains to be determined. we have examined growth cones in a relatively featureless in vitro environment, a chemotactic gradient, and traversing the optic tectum in vivo. eigenshapes and oscillations are remarkably consistent between these different cases. this argues that these properties are intrinsic and do not represent responses to the local environment, such as a way of navigating around local obstacles, consistent with an explanation in terms of intrinsic periodicity of microtubule growth. the complete trajectories of growth cones in vivo often involve navigating through many stages and choice points, for instance retinal axons growing from the eye to the brain or callosal axons finding their targets in the opposite hemisphere. eigenshape analysis provides a method for re-examining more quantitatively exactly how shape changes over such trajectories, and how it is correlated with navigational function. whether shape oscillations change their properties over complex trajectories and how such changes might be related to the environmental cues present at each moment remain to be determined. our tracing of growth cone outlines was performed on relatively low-magnification phase contrast or fluorescent images, and thus fine details of some filopodia were inevitably lost. however, we have demonstrated directly that the eigenshapes we found are quite robust to image quality. filopodia are not ignored in our analysis: for instance the presence of a filopodium in the right-hand side of the growth cone will show its effect by increasing the score for mode r. eigenshape analysis complements rather than replaces fine-scale analysis of filopodial dynamics, since it combines filopodial and lamellipodial structure to emphasise overall shape trends rather than fine-scale structure. despite recent advances for fluorescently labelled growth cones, fine analysis of filopodia is still limited by the difficulty in obtaining large datasets: fully automated image analysis techniques have difficulty with phase-contrast imaging at this level of detail, and hand-tracing is prohibitively time-intensive. unfortunately manual observations are not scalable, meaning that there is currently a growing mismatch between our ability to manipulate growth cones and to assess the effect of these manipulations on growth cone behaviour using manual techniques.  eigenshape analysis of growth cones has the potential to provide a novel quantitative understanding of the differences between growth cones in different conditions. these include the effects of the environment as axons navigate towards their targets during development, differences between initial development and regeneration, and differences between mutants and wild type. overall this work reveals a new dimension to the understanding of the dynamic morphology of growth cones, and potentially opens up novel directions for research into understanding the biological basis of developmental brain disorders. 
a critical role in guiding axons to their targets during neural development is played by neuronal growth cones. by analysing how the projections of growth cones onto these principal modes evolve in time, we found that growth cone shape oscillates with a mean period of min. the variability of oscillation periods and strengths between different growth cones was correlated with their forward movement, such that growth cones with strong, fast shape oscillations tended to extend faster. axon guidanceneurite growthneural developmenteigenshape analysisshape analysisbrain morphometryoscillationsmicrotubulesissue-copyright-statement© the author 2015 more generally, eigenshape analysis has the potential to provide new quantitative information about differences in growth cone behaviour in different conditions. a simple computational model of growth cone shape dynamics based on dynamic microtubule instability was able to reproduce quantitatively both the mean and variance of oscillation periods seen experimentally, suggesting that the principal driver of growth cone shape oscillations may be intrinsic periodicity in cytoskeletal rearrangements. here we use eigenshape analysis to uncover the set of five to six basic shape modes that capture the most variance in growth cone form. intrinsically driven shape oscillations are an important component of growth cone shape dynamics. these have a complex and rapidly changing morphology; however, a quantitative understanding of this morphology, its dynamics and how these are related to growth cone movement, is lacking.
 normal brain function depends on the development of appropriate patterns of neural connections. a critical role in guiding axons to their targets during neural development is played by neuronal growth cones. these have a complex and rapidly changing morphology; however, a quantitative understanding of this morphology, its dynamics and how these are related to growth cone movement, is lacking.  here we use eigenshape analysis to uncover the set of five to six basic shape modes that capture the most variance in growth cone form. by analysing how the projections of growth cones onto these principal modes evolve in time, we found that growth cone shape oscillates with a mean period of min. the variability of oscillation periods and strengths between different growth cones was correlated with their forward movement, such that growth cones with strong, fast shape oscillations tended to extend faster. a simple computational model of growth cone shape dynamics based on dynamic microtubule instability was able to reproduce quantitatively both the mean and variance of oscillation periods seen experimentally, suggesting that the principal driver of growth cone shape oscillations may be intrinsic periodicity in cytoskeletal rearrangements.  intrinsically driven shape oscillations are an important component of growth cone shape dynamics. more generally, eigenshape analysis has the potential to provide new quantitative information about differences in growth cone behaviour in different conditions. keywords axon guidanceneurite growthneural developmenteigenshape analysisshape analysisbrain morphometryoscillationsmicrotubulesissue-copyright-statement© the author 2015 