Biomedical Engineering Reference
In-Depth Information
(e.g., 2
L
for a single neuron with
L
binary time bins). For a review of bias correction,
including more sophisticated methods, see [29].
In principle, the expected bias of Equation (13.2) can be subtracted from the naive
information estimate of Equation (13.1) to yield a bias-free estimate. In practice, of
course, we need to know how many trials is
not too small
. Extensive experiments
with simulated data indicate that the number of trials should be at least a factor of
two times greater than the number of response categories [23, 30]. With 50 trials, for
example, codes with up to about 25 response categories can typically be accurately
evaluated. For a single neuron, this implies at most 4 time bins (each containing 0 or
1 spikes); for a pair of neurons, at most 2 time bins each.
Hence, even with bias correction, it is difficult to evaluate spike timing codes,
and extremely difficult to evaluate spike timing population codes, using so-called
brute force
application of Equation (13.1). As a consequence, it is usually difficult to
study neural coding in a truly systematic way. However, progress has recently been
made in the important special case of neurons that fire small numbers of spikes; the
theoretical foundation for which is described next.
13.2.3
Series expansion approach to information estimation
The variety of possible spike sequences, and hence the potential complexity of the
neural code, increases rapidly with the number of spikes emitted per trial; conversely,
low firing rates limit the complexity. Since typical firing rates in the barrel cor-
tex are just 0-3 spikes per whisker deflection, the mutual information can be well-
approximated by a second order power series expansion in the time window T, that
depends only on PSTHs and pair-wise correlations between spikes at different times
[18, 21, 33]. These quantities are far easier to estimate from limited experimental
data than are the full conditional probabilities required by the direct method. Pro-
vided that (i) the number of spikes in the response window is
1 (averaged over
stimuli) and (ii) spikes are not locked to one another with infinite time precision, the
mutual information can be approximated by a Taylor series expansion in the duration
of the response window
T
[21, 22]. To second order:
<
T
3
I
(
S
,
R
)=
I
t
+
I
tta
+
I
ttb
+
I
ttc
+
O
(
)
(13.3)
Here
I
t
is first order in
T
and
I
tta
,
I
ttb
and
I
ttc
are second order. The first order term
depends only on the PSTH of each neuron; the second order term depend also on
pairwise correlations. Provided that the approximation is accurate, mutual informa-
tion can thus be estimated from knowledge of only first and second order statistics - it
is not necessary to measure the full conditional probabilities demanded by Equation
(13.1). This property makes the series expansion approach much less prone to sam-
pling bias than the brute force approach. In practice, this means that coding can be
studied to significantly better temporal precision than would otherwise be possible.
An important feature of the method is that the contribution of individual spikes
(
I
t
and
I
tta
) is evaluated separately from that of correlated spike patterns (
I
ttb
and
I
ttc
). The amount of information that a neuronal population conveys by the timing
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