Information Technology Reference
In-Depth Information
where
!
X
X
X
1
A
1
S
1
M
ij
1
A
A
i`
+
A
ij
Q
ij
=
S
ik
+
M
ir
(11.6)
`
k
r
being
ij
the elements of the identity matrix, and
A =
1
2
(A + A
T
) :
(11.7)
Note that the symmetrization of matrix A is not an initial constraint but a conse-
quence of the optimization process.
It is convenient to dene also the following constants for the determination of
the parameters:
X
X
1
2N
A
A
=
A
ij
;
(11.8)
i
j
X
X
1
2N
A
S
=
S
ik
;
(11.9)
i
k
X
X
1
2N
A
M
=
M
ir
:
(11.10)
i
r
The actual neuronal layout of C. elegans is represented in Fig. 11.2. In the plot
we have separated neurons at dierent areas and also dierentiated them from sensor
neurons and muscles. The rst interesting observation is that the connectivity is
clearly biased towards the head of the animal, where more sensorial connections are
established. The muscular connectivity is also more dense in the anterior part of
the body although not as dense than for the sensors. Realizing that our constraints
are the positions of sensor neurons and muscles, we expect a signicant scatter of
the predictions in the region between the middle and the posterior part of the body.
Moreover, the scatter will result in an under-prediction of the position of neurons,
the predicted positions will be biased towards the anterior part.
11.4. Results and Discussion
Equation (11.5) gives the position of the neurons in the abstracted optimization
model. To measure the success of this method, the mean absolute dierence between
the actual (x) and predicted (x
0
) neuron positions is used:
X
1
N
A
jx
i
x
0
i
j:
E =
(11.11)
i
We have prepared several experiments in the scope of the current optimization
problem, to check the reliability of the current approach to get support on the
hypothesis of wiring optimization in the neuronal layout of C. elegans. Here we
expose the set up and results of each experiment. Finally a summary is presented
in Table 11.1.
