Fig. 2. S values of conditional entropies for binary alphabet. The y-axis represents the S measures of the conditional entropies Sh
n
= h
n orig
− h
n surr
s
surr
. The Sh
n
gives the number of ss separating the value of the measurement obtained with the original data and the mean of its surrogates. The x-axis represents the different selected spike-train forms for the respective spontaneous Sp
and stimulated St portions. Each spike-train form is identified by a different pattern of the bars. The z-axis identifies the Sh
n
s for the conditional entropies h
n
from n = 1, … , 5. In all spike forms, during the spontaneous condition Sp the Sh
n
values were always near zero. In contrast, the sequences recorded during the stimulated portions of the intermittent INT and walk-through W-T
forms, differed considerably from their surrogates i.e. the number of ss from the expectation value of the null hypothesis was large for all h
n
s.
145.97 vs. 148.52 ms. However, most intervals were lengthened and, the CV was similar to that
observed during irregular stimulation CV = 0.17. The ISIHs for the walk-through and locked
spike forms were qualitatively similar. During stimulation, intervals were shortened and their
distributions were Gaussian-like. However, the width was much narrower in the locked form,
reflecting the consistency of the interval values.
3. Results
3
.
1
. Entropy and conditional entropies We analyzed the four arbitrarily chosen individ-
ual spike-train forms that were encoded in a binary alphabet. In each case, we compared the
original digitized signals obtained during stimula- tion with their corresponding surrogates. We used
100 different random shuffles. The randomly shuffled surrogates examine a specific null hypoth-
esis: the structure of the spike-train is indistin- guishable from uncorrelated noise that has the
same distribution. We calculated the S-measure of the conditional entropies Sh
n
of the different spike-train forms, coded in a binary alphabet, for
both spontaneous Sp and stimulated St condi- tions Fig. 2. From these calculations, several
important observations can be made. In all spike forms, during the spontaneous condition the Sh
n
values were always near zero. Hence, the original and the mean surrogate h
n
values were similar. The similarity of the original and surrogate se-
quences was also observed for the stimulated por- tions of the irregular IRR and locked 11
forms. Thus, the original and surrogate sequences were similar in both spontaneous and stimulated
portions. In contrast, the sequences recorded dur- ing the stimulated portions of the intermittent
INT and walk-through W-T forms, were con- siderably different from their surrogates i.e. the
number of ss from the expectation value of the null hypothesis was large for all h
n
. The only exception was the Sh
1
value in the W-T case. Furthermore, the Sh
n
values were obviously dif- ferent from those of their respective spontaneous
portions. Hence, the interval sequences of these spike-trains had a considerable degree of inner
structure that was destroyed by the shuffling.
S-measures were also calculated with four and eight letter alphabets. With the four-letter alpha-
bet, results for intermittent and walk-through forms were, in general, similar to those observed
with the binary alphabet. The exception, was the lack of significant difference for the Sh
4
value in the W-T form in the stimulated portion. With the
eight-letter alphabet, results were similar to those obtained with the binary alphabet except for the
locked form.
3
.
2
. Binary grammar-complexity We now address the following question: is the
pattern of the message i.e. the temporal order of the spikes a significant contributor of the com-
plexity or is this simply an exotic way of measur- ing its firing rate? This question can be addressed
with very simple calculations. In all the sponta- neous conditions, the variation between the com-
plexities of the original sequences and their surrogates was not significant. Hence, we will
describe the results observed during the stimu- lated portions. Consider the irregular form as a
specific example. This sequence was found to have a complexity, in the binary representation C
2
= 153, almost identical with the average complexity
of 100 random shuffles C
2
= 152.9. The distribu-
tion and average firing rates were identical for the original and surrogate data sets. The structure of
the spike-train was indistinguishable from uncor- related noise with the same distribution. This
result is confirmed by the low value of the S-mea- sure: SC
2
= 0.25 Fig. 3. Thus, the rate and not
the internal structure of the message, which was destroyed by the shuffle, determined the C
2
ob- tained with the original data. Now let us consider
the intermittent condition. This data set was found to have a C
2
value of 65. This sequence was subjected to 100 random shuffles, and the average
complexity of the shuffled sequences was 84.64. Although the original and the randomly shuffled
sequences had the same distribution and average firing rate, their complexity values were quite
different. The SC
2
was 9.57 in this case. Because S is the number of ss from the expectation value of
Fig. 3. S measures of complexity for the different alphabets. The y-axis represents the S-measures of the complexity
SC
l
= C
l orig
− C
l surr
s
surr
. In this case, l is the number of symbols in the alphabet that was used. The SC
l
gives the number of ss separating the value of the measurement ob-
tained with the original data from the mean of its surrogates. The x-axis identifies the alphabet used to reduce the corre-
sponding spontaneous Sp and stimulated St portions of the selected spike-train forms. These are identified by a different
pattern of the bars that are located sequentially on the z-axis. In all the spontaneous conditions, the variation between the
complexities of the original sequences and their surrogates was minimal. In the stimulated condition, the intermittent and
walk-trough spike forms showed significant differences be- tween the complexity values of the original and surrogate data
sequences. In contrast, in the irregular and locked forms the variation of the original sequences with respect to their respec-
tive surrogates was not significant.
the null hypothesis, the intermittent neural spike form had a degree of structure that was rate-inde-
pendent. The C
2
value increased with the random shuffle. The increase in disorder that occurred was
what one might have expected. The SC
2
for the W-T form was 6.01. This also reflects the rate-in-
dependent inner structure of this form. In con- trast, the SC
2
value of the locked form was 0.52 i.e. the variation of the original sequence with
respect to its surrogates was minimal. The signifi- cant differences between original and surrogate
data sets for the intermittent and walk-through forms were also observed with the four and eight-
letter alphabets Fig. 3. Therefore, the complexity measures gave consistent results, regardless of the
alphabet employed.
4. Discussion
