Working with individual neurons as graph structures

Introduction

Neurons can helpfully be treated as graphs (branching trees) in which nodes connected by edges define the morphology of the neuron. The nat package provides a number of built-in functions that allow you to analyse the branching structure of graphs

http://jefferis.github.io/nat/reference/index.html#section-neurons-as-graphs-

or use aspects of the branching structure to manipulate graphs. Examples of such functions include the strahler_order function which calculates the Strahler branch order for each node or segment in the neuron or the spine function that extracts the longest path across the neuron.

More sophisticated analysis and manipulations can be carried out by converting neuron objects into ngraph objects.

Neurons as Graphs

A neuron will typically be a graph in the form of a binary tree. We follow the convention of the Matlab trees toolbox by Hermann Cuntz and colleagues in treating the root (typically the soma) as the origin of the graph and then having directed edges leaving the root.

neuron-as-graph

Nodes will be one of a:

root
branch point
end point
continuation point

Each node will have a numeric identifier or label (an arbitrary integer stored the in the neuron’s PointNo field that may have come from an external source) as well an index (an integer starting at 1 and increasing without gaps). Although these two identifiers may often be the same, code should never rely on this being the case. In the wild, one frequently encounters cases where e.g. the numeric labels

have gaps because a neuron was edited
are not in the same order as the vertices
have some other significance e.g. are globally unique across all neurons in a database and therefore have large values (which may be challenging for R to to represent give its maxint of 2^31 - 1=2.147483610^{9})

library(nat)
n=Cell07PNs[[1]]
summary(n)

##   root nodes segments branchpoints endpoints cable.length
## 1    1   180       33           16        18     297.1763

We can extract the points as follows:

rootpoints(n)

## [1] 1

branchpoints(n)

##  [1]  34  48  51  75  78  95  98  99 108 109 115 119 135 143 160 169

endpoints(n)

##  [1]   1  42  59  62  80  85  96 100 102 112 117 121 134 148 154 165 172
## [18] 180

Segments are unbranched connected sequences of nodes that terminate in a branch point or end point.

Built-in neuron graph functions

We will give a few examples of the use of the built-in functions that treat neurons as graphs.

Strahler Order

The branching structure of a neuron is commonly summarised by calculating the Strahler Order.

n=Cell07PNs[[1]]
so=strahler_order(n)
orders=1:max(so$points)
for (i in orders) {
  plot(subset(n, so$points==i), col=i, add = i!=1, boundingbox = boundingbox(n))
}

Note the use of multiple calls to plot.neuron using the add=FALSE argument for all but the first plot. Note also the use of the boundingbox argument/function in order to ensure that the plot is set up with appropriate axes for the whole neuron even if only part of it is plotted in the first call to plot.

Spine

You can find the longest path across a neuron using the spine function.

n=Cell07PNs[[1]]
sp=spine(n)
plot(n, col='grey')
plot(sp, add=T, col='blue')

spine has a variety of options that you can use to control the results.

Segment graph

You can use the segmentgraph function to make a simplified representation of the branching structure of the neuron. In this object (which has class igraph) each unbranched segment in the original neuron (which might have contained many vertices) is collapsed to a single edge joining the branch points (which are retained).

sg=segmentgraph(Cell07PNs[[1]])
plot(sg)

ngraph objects

The nat package provides a bridge to a rich cross-platform graph library, igraph. We provide a class ngraph that provides a thin wrapper for the igraph class. This looks after things that we might need to know about a neuron (like the 3D coordinates of each node) while still giving access to all of the graph functions in the igraph package.

g=as.ngraph(Cell07PNs[[1]])
class(g)

## [1] "ngraph" "igraph"

## IGRAPH 5fdf829 DN-- 180 179 -- 
## + attr: name (v/n), X (v/n), Y (v/n), Z (v/n), diam (v/n)
## + edges from 5fdf829 (vertex names):
##  [1]  1-> 2  2-> 3  3-> 4  4-> 5  5-> 6  6-> 7  7-> 8  8-> 9  9->10 10->11
## [11] 11->12 12->13 13->14 14->15 15->16 16->17 17->18 18->19 19->20 20->21
## [21] 21->22 22->23 23->24 24->25 25->26 26->27 27->28 28->29 29->30 30->31
## [31] 31->32 32->33 33->34 34->35 35->36 36->37 37->38 38->39 39->40 40->41
## [41] 41->42 34->43 43->44 44->45 45->46 46->47 47->48 48->49 49->50 50->51
## [51] 51->52 52->53 53->54 54->55 55->56 56->57 57->58 58->59 51->60 60->61
## [61] 61->62 48->63 63->64 64->65 65->66 66->67 67->68 68->69 69->70 70->71
## [71] 71->72 72->73 73->74 74->75 75->76 76->77 77->78 78->79 79->80 78->81
## + ... omitted several edges

You can use functions such as

igraph::diameter(g)

## [1] 89

to find the length of the longest path across the neuron. This is defined in terms of the number of intervening nodes. You can also make a graph in which the edge weights are the euclidean distance between the connected 3D nodes:

gw=as.ngraph(Cell07PNs[[1]], weights=TRUE)
igraph::diameter(gw)

## [1] 186.0859

This gives you the longest path length (geodesic) across the graph in units of µm in this case.

Note that although you can do library(ngraph), it adds a lot of functions to the search path, some of which have name clashes, so I often just use the package name (igraph::) prepended to the function that I want to call.

walking along ngraph objects

You can use the graph representation of neurons e.g. to find the path between nodes.

g=as.ngraph(Cell07PNs[[1]], weights=TRUE)
eg=endpoints(g)

p=igraph::shortest_paths(g, from=1, to=180)
p$vpath[[1]]

## + 90/180 vertices, named, from 6ef429c:
##  [1] 1   2   3   4   5   6   7   8   9   10  11  12  13  14  15  16  17 
## [18] 18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34 
## [35] 43  44  45  46  47  48  63  64  65  66  67  68  69  70  71  72  73 
## [52] 74  75  86  87  88  89  90  91  92  93  94  95  97  98  103 104 105
## [69] 106 107 108 135 155 156 157 158 159 160 166 167 168 169 173 174 175
## [86] 176 177 178 179 180

# fails
p2=igraph::shortest_paths(g, from=180, to=1)

## Warning in igraph::shortest_paths(g, from = 180, to = 1): At
## structural_properties.c:4597 :Couldn't reach some vertices

p2$vpath[[1]]

## + 1/180 vertex, named, from 6ef429c:
## [1] 1

# mode all will find the path irrespective of direction of links, which are
# directed from the soma
p3=igraph::shortest_paths(g, from=180, to=1, mode = 'all')
p3$vpath[[1]]

## + 90/180 vertices, named, from 6ef429c:
##  [1] 180 179 178 177 176 175 174 173 169 168 167 166 160 159 158 157 156
## [18] 155 135 108 107 106 105 104 103 98  97  95  94  93  92  91  90  89 
## [35] 88  87  86  75  74  73  72  71  70  69  68  67  66  65  64  63  48 
## [52] 47  46  45  44  43  34  33  32  31  30  29  28  27  26  25  24  23 
## [69] 22  21  20  19  18  17  16  15  14  13  12  11  10  9   8   7   6  
## [86] 5   4   3   2   1

all.equal(p$vpath[[1]], rev(p3$vpath[[1]]))

## [1] TRUE

# just the distances - by default uses mode=all
igraph::distances(g, v=1, to=180)

##        180
## 1 186.0859

igraph::distances(g, v=180, to=1)

##            1
## 180 186.0859

You can also ask for nodes upstream or downstream of a given starting node. For example the neurons in the Cell07PNs set have a tag called AxonLHEP that defines the entry point of the axon into the lateral horn neuropil of the fly brain. Here we defined

n=Cell07PNs[[1]]
g=as.ngraph(n)
# find the nodes distal to this point
# nb you must set unreachable=F if you only want to get downstream nodes
igraph::dfs(g, neimode='out', unreachable = FALSE, root=n$AxonLHEP)

## $root
## [1] 71
## 
## $neimode
## [1] "out"
## 
## $order
## + 180/180 vertices, named, from 2866093:
##   [1] 72   73   74   75   76   77   78   79   80   81   82   83   84   85  
##  [15] 86   87   88   89   90   91   92   93   94   95   96   97   98   99  
##  [29] 100  101  102  103  104  105  106  107  108  109  110  111  112  113 
##  [43] 114  115  116  117  118  119  120  121  122  123  124  125  126  127 
##  [57] 128  129  130  131  132  133  134  135  136  137  138  139  140  141 
##  [71] 142  143  144  145  146  147  148  149  150  151  152  153  154  155 
##  [85] 156  157  158  159  160  161  162  163  164  165  166  167  168  169 
##  [99] 170  171  172  173  174  175  176  177  178  179  180  <NA> <NA> <NA>
## [113] <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA>
## [127] <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA>
## + ... omitted several vertices
## 
## $order.out
## NULL
## 
## $father
## NULL
## 
## $dist
## NULL

# the proximal nodes back to the soma (including any branches)
igraph::dfs(g, neimode='in', unreachable = FALSE, root=n$AxonLHEP)

## $root
## [1] 71
## 
## $neimode
## [1] "in"
## 
## $order
## + 180/180 vertices, named, from 2866093:
##   [1] 72   71   70   69   68   67   66   65   64   63   48   47   46   45  
##  [15] 44   43   34   33   32   31   30   29   28   27   26   25   24   23  
##  [29] 22   21   20   19   18   17   16   15   14   13   12   11   10   9   
##  [43] 8    7    6    5    4    3    2    1    <NA> <NA> <NA> <NA> <NA> <NA>
##  [57] <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA>
##  [71] <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA>
##  [85] <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA>
##  [99] <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA>
## [113] <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA>
## [127] <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA> <NA>
## + ... omitted several vertices
## 
## $order.out
## NULL
## 
## $father
## NULL
## 
## $dist
## NULL

Note that dfs (depth first search) provides a good way to visit all the nodes of the neuron

Let’s use this to make a function that prunes neurons downstream of this axon entry point:

prune_from_lhep <- function(n, ...) {
  g=as.ngraph(n)
  downstream_indices=igraph::dfs(g, root = n$AxonLHEP, unreachable = FALSE)$order
  prune_vertices(n, verticestoprune = downstream_indices, invert = TRUE)
}
pruned=nlapply(Cell07PNs[1:3], prune_from_lhep)
plot(Cell07PNs[1:3], col='grey')
plot(pruned, lwd=2, add = T)

The pruned neurons show up in red, green, and blue in the above plot.

Gregory Jefferis

2019-01-10