Elastically deformable path to path matching based neuron categorization.

1. Elastic Path2Path:
Automated morphological classification
of neurons by elastic path matching
Tamal Batabyal, Scott T. Acton
1

2. Quick look: Elastic deformation between
neurons
Objective Classification
Automated
Features
Unsupervised
clustering
Graph theory
Category Neuromorphology
Tree matching
2

3. Why Neuromorphology?
To explore structure-function relationship.
Ramon y Cajal (1899)
To explain the effect of structure on
spike timing, spike propagation
delay, ionic channel health, spike
backpropagation etc.
To quantify and assess structural
degeneration of neurons during
neurological disorders, such as
Alzheimer’s disease.
To answer the key anatomical
changes during synaptic
plasticity, long term potentiation
(LTP), STP.
To formulate the mechanism for
synaptic integration, which is
believed to be Bayesian but lacks
theoretical justification.
To apply the tools on the
morphology of Glia
(Lymphatic system).
3

4. Dataset :: Neuromorpho.org
Ascoli, Giorgio A., Duncan E. Donohue, and Maryam Halavi. "NeuroMorpho. Org: a central resource for neuronal morphologies." Journal of Neuroscience 27.35 (2007): 9247-9251.
Neuron (stained)
Confocal microscopy
Registration
Tracing
SWC files
4

5. Motivation of our work
5
We assume that ALL neurons
form a dense manifold
In the ‘SHAPE’ space one can reach from one
neuron to the other via a process. In our
case, the process is continuous morphing.
This process aids in Visualization.
Feature extraction,
Probability density extraction,
Kernel base approaches,
ML based approaches
Path based approaches
(Path2Path, Tree2Tree)
Visualization in the implicit (kernel) or explicit
high-dimensional feature space is impossible.
Visualization possible. ML
using the features on paths for
classification. Visualization
through morphing the paths.
How to ensure that the
intermediate neurons
generated via morphing have
“meaningful” structures?

6. Our model :: Path based
Between a pair of nodes/vertices, there exists one and only one path
because a neuron is modeled as a tree graph.
SOMA
DENDRITE
DENDRITE
6

7. Path2Path by Basu et al.
Rooted path features
(5 tuples)
Hierarchy
Concurrence
3D locations of vertices
Basu, Saurav, Barry Condron, and Scott T. Acton. "Path2Path: Hierarchical path-based analysis for neuron matching." Biomedical Imaging: From Nano to Macro, 2011 IEEE International
Symposium on.
Extract for each
path of a neuron
Not limited to the three
features. One can
extract tree asymmetry,
fragmentation etc.
7

8. Path2Path concurrence, C
How many paths are there following the node when one traverses from the root ? Concurrence value
𝟑
𝟑
𝟑
𝟑
𝟑
𝟑
𝟑
𝟑
𝟐
𝟏
𝟑
𝟑
𝟑
𝟑
𝟐
𝟏
𝟏
8

9. Path2Path hierarchy, H
Depth of the node when one traverses from the root = Hierarchy value
𝟏
𝟏
𝟏
𝟏
𝟏
𝟏
𝟏
𝟏
𝟐
𝟐
𝟏
𝟏
𝟏
𝟏
𝟐
𝟑
𝟑
9

10. Path2Path metric, D
D( ) =
𝟑, 𝟏
𝟑, 𝟏
𝟑, 𝟏
𝟑, 𝟏
𝟑, 𝟏
𝟑, 𝟏
𝟑, 𝟏
𝟑, 𝟏
2, 𝟐
2, 𝟐
1, 𝟏1, 𝟏
0
1
|𝐶1 𝑡 − 𝐶2(𝑡)||𝑙𝑜𝑐1(𝑡) − 𝑙𝑜𝑐2(𝑡)||2
𝜃 + 𝐻1(𝑡)𝐻2(𝑡)
𝑑𝑡
𝜃 = 0.001 (𝑠𝑚𝑎𝑙𝑙 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
=
1
𝑛
𝑖=1
𝑛
|𝐶1 𝑖 − 𝐶2(𝑖)||𝑙𝑜𝑐1(𝑖) − 𝑙𝑜𝑐2(𝑖)||2
𝜃 + 𝐻1(𝑖)𝐻2(𝑖)Same number of samples per pair of paths.
If they do not have, then RESAMPLE
10
If C1 t = C2 t ∀𝑡, 𝐷 = 0.
No relevance of H and the locations. Scale
invariance. D is NOT a distance metric

11. Path2Path flow chart
Neuron
.swc files
Decomposition
to Rooted paths
Concurrence
& hierarchy
Path descriptor
(5 tuples to
each location)
Feature
generation
Neuron 1
Neuron 2
Feature generation
Feature
generation
Pathi
Pathk
Resampling
ResamplingRegistration
Hierarchy +
Concurrence
Interpolation
Hierarchy +
Concurrence
Interpolation
DISTANCE
11

12. Path2Path
Neuron 1 Neuron 2
Path11, Path12, …
Path1j
Path21, Path22, …
Path2k
Greedy,
Many-one
Matching of
Paths
Distance Matrix
Path11
Path12
Path1N
Path21 Path22 Path2M
12

13. Many-one mapping
Neuron 1 Neuron 2
Path11, Path12, …
Path1j
Path21, Path22, …
Path2k
13
>> Lets assume:
No of paths (Neuron1) <= No of paths (Neuron2)
>> Pick a path “Path11” (in Neuron1)
>> Find the closest (minimum D) path from
Neuron2.
>> Append to the list of Path correspondence
Drawback: All the paths (Neuron1) matched to only one or very few paths (Neuron2)

14. Why Path2Path?
It is global in the sense that
each path starts from the
root (soma) and ends up in a
dendritic terminal. So each
path is a “Neuronal atom”. In
summary, path2path is an
atomic decomposition.
Apart from hierarchy and
concurrence, many
features such as
fragmentation count of a
path, can be incorporated.
It identifies the
‘Caulescence’ (the
extent to which a tree
exhibits its main path)*
*Brown, Kerry M., Todd A. Gillette, and Giorgio A. Ascoli. "Quantifying neuronal size: summing up trees and splitting the branch difference." Seminars in cell & developmental biology.
Vol. 19. No. 6. Academic Press, 2008.
It gets rid of spurious
edges and leave nodes
by selecting suitable
distance measure.
14

15. Elastic Path2Path
Resampling is big problem,
especially if someone uses the
resample routine in MATLAB.
It will affect the hierarchy and
concurrence interpolation.
Look at the metric,
0
1
|C1 t − C2(t)||loc1(t) − loc2(t)||2
θ + H1(t)H2(t)
dt
Here, ||loc1(t) − loc2(t)||2 is the
Euclidean distance. But open curves
(rooted path) are not on the
Euclidean manifold
The matching is many to one
and greedy. There might be
cases in which all paths of
neuron 1 are matched to
only one path in neuron2
(degeneration).
Mid-point based
iterative
resampling
Square Root
Velocity function
Hungarian
algorithm
Elastic Path2Path
15

16. Why Elastic Path2Path?
Elasticity has an enormous impact, because it
helps provide a framework to morph one
neuron into another. Morphing is in the
“Physical domain” rather than in the “high-
dimensional” manifold.
It provides visualization as well as classification.
If a neuron is misclassified, we can visually inspect the
path correspondence. On the other hand, we can also
relate the improvement of accuracy to the path
correspondence.
The intermediate deformations
between two neurons do not deviate
from “Meaningful” structures.
*Brown, Kerry M., Todd A. Gillette, and Giorgio A. Ascoli. "Quantifying neuronal size: summing up trees and splitting the branch difference." Seminars in cell & developmental biology.
Vol. 19. No. 6. Academic Press, 2008. 16
No sophisticated classifier is used.
So the accuracy depends on the
choice of distance metric and “very
good” path correspondence.

17. Elastic deformation
Srivastava, Anuj, et al. "Shape analysis of elastic curves in Euclidean spaces." IEEE Transactions on Pattern Analysis and Machine Intelligence 33.7 (2011): 1415-1428.
17

18. Elastic curve: SRVF
The manifold consisting of
open curves is non-Euclidean
The manifold consisting of
SRVF open curves is Euclidean
fi(0)
fi(𝑡1)
fi(𝑡2)
fi(𝑡3)
fi(1)
Tangent / gradient
𝑞𝑖 0 =
𝑓𝑖(0)
|| 𝑓𝑖(0)||
𝑞𝑖 𝑡1 =
𝑓𝑖(𝑡1)
|| 𝑓𝑖(𝑡1)||
𝑞𝑖 𝑡2 =
𝑓𝑖(𝑡2)
|| 𝑓𝑖(𝑡2)||
𝑞𝑖 𝑡3 =
𝑓𝑖(𝑡3)
|| 𝑓𝑖(𝑡3)||
𝑞𝑖 1 =
𝑓𝑖(1)
|| 𝑓𝑖(1)||
fi 0
fi 𝑡1
fi 1
fi 𝑡2
fi 𝑡3
18

19. 𝐟𝐢(𝟎)
𝐟𝐢(𝒕 𝟏)
𝐟𝐢(𝒕 𝟐)
𝐟𝐢(𝒕 𝟑)
𝐟𝐢(𝟏)
𝒒𝒊 𝟎 =
𝒇𝒊(𝟎)
|| 𝒇𝒊(𝟎)||
𝒒𝒊 𝒕 𝟏 =
𝒇𝒊(𝒕 𝟏)
|| 𝒇𝒊(𝒕 𝟏)||
𝒒𝒊 𝒕 𝟐 =
𝒇𝒊(𝒕 𝟐)
|| 𝒇𝒊(𝒕 𝟐)||
𝒒𝒊 𝒕 𝟑 =
𝒇𝒊(𝒕 𝟑)
|| 𝒇𝒊(𝒕 𝟑)||
𝒒𝒊 𝟏 =
𝒇𝒊(𝟏)
|| 𝒇𝒊(𝟏)||
𝐟𝐢 𝟎
𝐟𝐢 𝒕 𝟏
𝐟𝐢 𝟏
𝐟𝐢 𝒕 𝟐
𝐟𝐢 𝒕 𝟑
19

20. Intermediate elastic deformation
The manifold consisting of
open curves is non-Euclidean
0
1
|C1 t − C2(t)||loc1(t) − loc2(t)||2
θ + H1(t)H2(t)
dt
A rooted path is an open curve, 𝑓𝑖 𝑡 ; 𝑡 ∈ [0,1]
Apply SRVF, 𝑞𝑖 𝑡 =
𝑓 𝑖(𝑡)
|| 𝑓 𝑖(𝑡)||
The manifold consisting of
transformed open curves is Euclidean
Intermediate morphing
between two curves
𝑓𝑖 𝑡 𝑔 𝑘 𝑡
(Neuron 1) (Neuron 2)
𝑞𝑖 𝑡 𝑞 𝑘 𝑡
𝑞 𝑗𝑘 𝑡 =
𝜏 𝑞𝑖 𝑡 + (1 − 𝜏)𝑞 𝑘 𝑡
Back to curve
Srivastava, Anuj, et al. "Shape analysis of elastic curves in Euclidean spaces." IEEE Transactions on Pattern Analysis and Machine Intelligence 33.7 (2011): 1415-1428. 20

21. One-One matching
21
Path11
Path12
Path1N
Path21 Path22 Path2M
C =
N <= M
Neuron1
Neuron 2
𝐜𝐢,𝐣
Find 𝑘1, … , 𝑘 𝑁 such that 𝑐1,𝑘1
+ ⋯ + 𝑐 𝑁,𝑘 𝑁
is minimum
Hungarian algorithm
For N< M, one needs to
add dummy rows with 0s

22. Results (Elastic Path2Path)
22

23. Thank You
23

24. More
results
24

25. ElasticPath2Path and Path2Path can not ans..
• Is the morphing a true physiological process
• How to take care of the huge dissimilarity in the number of paths in
different neurons.
• What about bifurcation angle, branch diameter, diameter tapering,
etc.
• What is degeneration of neuronal arbor? How to identify a healthy
neuron with its degenerated version using Path2Path?
• What is mean/average arborial shape?
25

26. Results (Elastic Path2Path)
26

27. Technical drawbacks of Path2Path
Registration
dependent
Distance is not
converging with the
number of samples
after resample
No rationale behind
the selection of the
metric D.
Resampling is big problem,
especially if someone uses the
resample routine in MATLAB.
It will affect the hierarchy and
concurrence interpolation.
Look at the metric,
0
1
|C1 t − C2(t)||loc1(t) − loc2(t)||2
θ + H1(t)H2(t)
dt
Here, ||loc1(t) − loc2(t)||2 is the
Euclidean distance. But open curves
(rooted path) are not on the
Euclidean manifold
If C1 t = C2 t ∀𝑡, 𝐷 = 0.
No relevance of H and the locations.
Scale invariance.
The matching is many to one
and greedy. There might be
cases in which all paths of
neuron 1 are matched to
only one path in neuron2
(degeneration).
What if neuron1 has 10 and neuron2 has 10000 paths. Is Path2Path justified?
Cardinality
dependent
27

28. Resampling
• The order is not
maintained. It is difficult
to sort in more than
one D. Hard to
interpolate concurrence
and hierarchy.
Problem with
MATLAB resample
How to compare?
They have different
number of samples
• Actual coordinates are
changed. We have to apply
threshold to find the actual
corresponding locations in the
.swc file.
Our solution: Mid-
point based.
What is the problem then?
28