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to the variation of the parameters, and statistically signicant. Here we follow
the general formulation presented in [2, 3]. The optimization problem, is stated
as a cost function that must be minimized. The cost is separated in two dierent
contributions: a connection cost between neurons, and a connection cost between
neurons and sensorial organs or muscles. This dierence arises from the dierent
connectivity patterns and also because its mathematical convenience, note that the
position of sensors and muscles is taken as input data, constraining then the opti-
mization problem. We will work in the scope of the dedicated-wire model (following
the terminology in [3]) which simply means that every synapse has its own wire.
At dierence, the shared-wire model introduced also in [3] considers a neuron as a
wire (segment) with multiple synapses; we do not include its analysis here because
it does not provide betters results.
Mathematically, the general formulation of wiring cost is:
C tot = C int + C ext
(11.1)
where
X
X
1
2 A
C int =
A ij (x i x j ) 2 ;
(11.2)
i
j
X
X
X
X
1
S
1
M
C ext =
S ik (x i s k ) 2 +
M ir (x i m r ) 2
(11.3)
r
i
k
i
and
A = (A ij ) neuron-neuron connectivity matrix (N A N A )
S = (S ik ) neuron-sensor connectivity matrix (N A N S )
M = (M ir ) neuron-motor connectivity matrix (N A N M )
x = (x 1 ; : : : ; x N A ) T neurons positions vector
s = (s 1 ; : : : ; s N S ) T sensors positions vector
m = (m 1 ; : : : ; m N M ) T motors positions vector
A , S , and M are parameters to be determined by normalization con-
straints
Everything will be treated as input data except the position of the neurons x.
This quadratic expression of the cost is just a convention, other exponents have been
considered in [3] with no improvement on the optimization results. Moreover, it is
mathematically convenient because in this form the system is analytically solvable.
The optimization of the total cost (11.1) is obtained by imposing
@C
@x a
= 0 ;
a = 1; : : : ; N A :
(11.4)
They yield a system of linear equations whose solution, in matrix form, reads
1
S Ss +
1
M Mm
x = Q 1
(11.5)
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