Biomedical Engineering Reference
In-Depth Information
13.2.2
Mutual information and sampling bias
The mutual information quantifies how well an ideal observer of neuronal responses
can, on average, discriminate which stimulus occurred based on a response observed
on a single trial (Shannon, 1948):
)=
Â
n
s
log
2
P
(
n
|
s
)
I
(
S
,
R
P
(
n
|
s
)
(13.1)
P
(
n
)
Here
S
is the set of responses;
n
can be a sin-
gle cell response, or a population response and either a spike count or a spike se-
quence.
P
=
{
s
}
is the set of stimuli,
R
=
{
n
}
(
n
|
s
)
is the posterior probability of a response
n
given stimulus
s
;
P
(
n
)
is
the stimulus-average response probability and
P
is the prior stimulus probability.
Mutual information quantifies
diversity
in the set of probabilities
P
(
s
)
(
n
|
s
)
.Ifthese
probabilities are all equal, for a given response
n
, and hence equal to
P
,thear-
gument of the logarithm is one and the response
n
contributes nothing to
I
(
n
)
(
S
,
R
)
.
Conversely, the more that the
P
(
n
|
s
)
differ, the greater the contribution of response
n
to the information.
In principle, to obtain an estimate of mutual information, we simply measure these
probabilities from recorded data and substitute them into Equation (13.1). The prob-
lem is that it is difficult to estimate the conditional probabilities accurately, given the
number of stimulus repetitions presented in a typical physiological experiment. The
consequent fluctuations in the estimated conditional probabilities lead to spurious
diversity that mimics the effect of genuine stimulus-coding responses. Hence, the
effect of limited sampling is an
upward
bias in the estimate of the mutual informa-
tion [38]. Since this sampling problem worsens as the number of response categories
increases, the bias is intrinsically greater for timing codes compared to count codes.
Hence it is important to exercise considerable care when assessing the information
content of complex codes (many response categories) compared to simple ones (few
response categories).
One way to approach this issue is to use bias correction procedures [14, 38]. The
basis for these methods is the fact that, provided the number of trials is not too small,
the information bias depends on the data in a surprisingly simple way. Essentially,
the bias is proportional to the number of response categories divided by the number
of trials. More precisely [23],
1
2
N
ln 2
(
Â
Bias
=
−
R
−
1
−
s
(
R
s
−
1
))
(13.2)
This expression depends only on the total number of trials
N
and on the number of
relevant
response categories
. A given stimulus will evoke different responses
with different probabilities: some will have high probability, some low probability,
and others will have zero probability. A response is
relevant
if its probability, condi-
tional to the stimulus, is non-zero.
R
s
is the number of responses that are relevant to
stimulus
s
;
R
is the number of responses that are relevant considering all the stimuli
together. In the simplest case,
R
is simply the total number of response categories
(
R
,
R
s
)
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