Biomedical Engineering Reference
In-Depth Information
A bias for one peak over the other has to be introduced to make a unique choice.
This bias is derived from the prior distribution for the model parameter,
θ
, which we
model as a Gaussian distribution of the form
1
2
2
−
qs
/(
2
)
p
()
q
=
e
,
(4.6)
p
2
ps
p
where
σ
p
= 23. 3
°
is the standard deviation. In this case, the mean is zero which
relects the owl's strong bias for sound sources at the front, while tending to ignore
possible sources from the sides. The prior distribution is shown in Fig.
4.2c
as the
grey curve. Note that
θ
is a valid angle on the unit circle only when
θ
∈ ( − 180
°
, 180
°
],
and we make the approximation that the probability mass is negligible outside these
bounds. This prior is consistent with the known interactions of barn owls and their
potential preys (Edut and Eilam
2004
).
Applying Eqs. (
4.5
) and (
4.6
) to Eq. (
4.2
), we can compute the shape of the pos-
terior distribution, shown as the grey area curve in Fig.
4.2c
. With the posterior
distribution, the owl can ask how probable the various source directions
θ
are given
a measured ITD and use this information to make a behaviourally relevant choice.
Making this choice involves using a decision rule to reduce a distribution over
source directions
p
(
θ
| ITD) to a single estimated source direction
q
. One decision
rule that is consistent with the behavioural data is to take an average of unit vectors
weighted by their posterior probability of the form
ˆ
(
=
∫
u
q
ITD
)
()(| ),
q
p
q
ITD
d
q
(4.7)
where
u
(
θ
) is the two-dimensional unit vector for each angle (Fig.
4.2d
, inset) and
the integral is taken over the unit circle. This is also referred to as the circular mean.
The result of this estimation procedure for a number of azimuth directions is given
by the black points in Fig.
4.2d
. As a reference, the black dashed line represents the
perfect estimation. Note that the algorithm exhibits a bias towards central positions,
that is, it tends to underestimate the true azimuth direction when the source is posi-
tioned at eccentric positions. This bias has been shown to exist in the barn owl by
means of behavioural experiments. The good agreement between Eq. (
4.7
) and
experimental data suggests that this equation may be implemented neurally, a topic
we address in the following section.
4.4
Neural Encoding and Population Vector Decoding
We next ask how sound source localization in the barn owl is implemented by a
population of neurons. One approach consists in building a model of the encoding
process and then decoding the resulting neural activity using a population vector
(PV). The population vector decoding method was pioneered more than 20 years
