Biomedical Engineering Reference
In-Depth Information
4.6
Correlated Tuning Curves
In actual neural networks, the trial by trial iring rates of neurons may be correlated
with each other (Averbeck et al.
2006
, Ecker et al.
2010
). To model these correla-
tions, we assume that our
N
neurons have the same tuning curves and distributions
of preferred directions as in Eqs. (
4.9
) and (
4.10
). In addition, we assume that their
iring rates on a single trial are drawn from the multinormal probability distribution
of the form
1
′ −
1
−−
(
kr
(
ITD
))
S
(
kr
−
(
ITD
)) /
2
p
(|
k
ITD
)
() |
=
e
(4.14)
g
12
/(
N
)
12
/
2
p
S
|
g
(Anderson
2003
). In this equation,
r
(
is the mean ir-
ing rate of the
N
neurons given the ITD, and
v
′ denotes the transpose of vector
v
.
The covariance matrix is represented by
Σ
g
and its determinant by |
Σ
g
| . If the cova-
riance matrix is given by
Σ
g
= (
Σ
ij
), with
ITDITD
) (( ), ,( ))
= …
′
r
r
N
ITD
1
Σ
ij
=
r
(
ITDITD
d
)
r
(
)
,
ij
,
= …
1
,
,
N
,
i
j
ij
and
δ
ij
= 1 when
i
=
j
, while
δ
ij
= 0 when
i
≠
j
, we get an uncorrelated multinormal
distribution. Because each neuron's iring rate variance is proportional to its mean
iring rate, this formulation is close to that of Sect.
4.4
using Poisson neurons (mean
equal to variance). If the covariance matrix elements have the form
Σ
ij
=
r
(
ITDITD
drd
)
r
(
)(
+
(
1
−
)),
(4.15)
i
j
ij
ij
where
ρ
∈ [0, 1) is the correlation coeficient, we have introduced correlations of
magnitude
ρ
into all pairs of neurons' iring rates.
When applying the PV, we see no difference in sound source estimates from
independent neurons. This is illustrated in Fig.
4.3
d by the black circled curve,
which is exactly overlapping with the grey crossed curve obtained from indepen-
dent neurons, in spite of sizable correlations between single neurons' iring rates
(
ρ
= 0. 5). Thus, neuronal correlations do not affect the results exposed in the previ-
ous sections. Intuitively, this may be understood from the fact that the direction of
the PV will not be changed by correlated noise, if the noise scales uniformly with
the mean iring rate of the neurons, as implemented by Eq. (
4.15
). In conclusion,
Bayesian statistical modelling is a computational analysis technique that can
provide insight in the coding of sensory information from a neuroethological per-
spective, as illustrated in this chapter.
