Biomedical Engineering Reference
In-Depth Information
and Dress (43) present a simple formalism of this problem and demonstrate that
one important selection pressure in favor of compositional signals is a need to
become robust against errors in signal perception (18). As a result of space limi-
tations, I shall only demonstrate the nature of the signalling problem, and omit
the full solution.
Assume that a language
L
employs
n
signals to communicate about
n
ob-
jects. When two individuals communicate, they obtain a payoff:
n
=
.
F
a
[25]
i
i
=
1
If all objects have the same value, then the total payoff is simply
F
=
kn
. In real-
ity, communication is error prone. Denote the probability of mistaking a signal
i
for a signal
ju
ij
. The error matrix
U
is a row stochastic error matrix. The diago-
nal values
u
ii
give the probability of correct communication. Hence,
n
=
.
F
a u
[26]
i i
i
=
1
The error matrix can be defined in terms of similarity between any two signals
i
and
j
:
s
ij
. Similarity is a value between 0 and 1, and hence
u
ij
=
.
s
/
n
k
s
ij
ik
=
1
This enables us to write the payoff in terms of signal similarity:
¬
-
n
a
-
-
F
=
i
.
[27]
-
-
n
-
-
s
-
i
=
1
®
ij
j
=
1
Signals are embedded in some metric space
X
and
d
ij
denotes the distance be-
tween
i
and
j
. Assume that similarity is a monotonically decreasing function of
distance,
s
ij
=
f
(
d
ij
). One choice of function is
s
ij
= exp(-B
d
ij
), where the parame-
ter B is a measure of the resolution of perception.
For a given number of objects we wish to find the optimum configuration of
sounds
x
1
, ...,
x
n
in a sound continuum that maximize the payoff function:
¬
-
n
1
-
-
F
=
.
[28]
-
-
n
-
exp(
B
|
xx
|)
-
i
=
1
®
i
j
j
=
1
It can be proved that the maximum value of
F
, as
n
tends to infinity, converges
to
