methods to study spike-trains generated by the isolated neuron of the slowly adapting stretch
receptor organ SAO. Our objective was to cor- relate the informational and algorithmic measures
of complexity with different known neural behav- iors i.e. individual spike-train forms. Segundo et
al., 1987 observed the influence of regularly and irregularly arriving stimuli. These were pulse-like
lengthenings applied to the muscle element of the SAO. The SAO neuron is a pacemaker cell that
responds to isolated stimuli much like other cells respond to the arrival of EPSPs Bryant et al.,
1973; Diez-Martı´nez and Segundo, 1983. The effects of stimuli include shortening of the inter-
vals in which they occur, i.e. they excite. Diez- Martı´nez et al. 1988 studied the consequences of
periodically applied stimuli in pacemaker SAO neurons. Two individual spike-train forms are
pervasive and straightforward: i locking is char- acterized by almost fixed phases; ii intermittency
is distinguished by discharges that shift irregu- larly between prolonged epochs where spike
phases barely change, and brief bursts with marked
variations. As
stimulus frequencies
change, locking alternates with intermittency. Locked domains have simple, rational spike to
tug ratios e.g. 1:1, 2:1, etc.. This approach, based upon Dynamical System Theory, could not
be applied as conveniently to study the effect of irregular perturbations. Since situations where
pacemaker neurons are influenced by irregular inputs must be common in nature, other methods
of analysis are needed. Furthermore, particularly with small amplitudes, other spike-train forms
appear which are less clear. These include Se- gundo et al., 1998: i phase walk-through, i.e.
phases vary cyclically, increasing or decreasing respectively depending on whether stimulus fre-
quency is larger or smaller than the natural one; ii ‘messy’, i.e. difficult to describe succinctly.
Furthermore, with prolonged stimulation periods, irregular transitions between behaviors could oc-
cur. Thus, many important facets of periodic driving remain puzzling; indeed, whereas behav-
iors at some frequencies are summarized and un- derstood clearly, others are unclear. Therefore, to
improve the comprehension of these ill-defined issues, we complement standard statistical meth-
ods and non-linear analytical techniques, em- ployed in former publications, with informational
and algorithmic
complexity mea-
sures.
2. Materials and methods
2
.
1
. Representations of sequences A time series consists of a sequence of real
numbers. Symbol sequences are composed of symbols letters from an alphabet of l letters
e.g. for l = 4, {A, C, G, T} is the DNA alpha- bet; for l = 2, {0, 1} is the binary alphabet, etc..
By digitization, time series may be converted into symbol sequences. The usefulness of subsequent
complexity calculations depends crucially on the procedure used to partition the data among a
finite alphabet of symbols. One alternative, sug- gested in Rapp et al., 1994, is partitioning about
the median. It is considered a good choice be- cause it most effectively reveals the randomness
of the data. All binary sequences reported in this article correspond to interspike interval data con-
verted to symbols by partitioning about the me- dian.
Two sections formed each test: a spontaneous discharge pre-stimulus control condition and one
during which trains of stimuli were applied. The median was calculated separately for each of
these portions. If the data values were less than the median, they were assigned the symbol 0.
Otherwise, symbol 1 was assigned. The reduction of the data to a sequence of symbols was not
limited to the binary alphabet. We applied the procedure for partitioning about the median to
construct sequences composed of larger symbol sets four and eight. In each case, an equal
number of interspike intervals were assigned to each symbol.
2
.
2
. Entropy-like measures of sequence structure The determination of block-entropies Ebeling
et al., 1987 is a well established method for the investigation of discrete data Herzel et al., 1995.
We give a short review of the definition of entropies and other informational concepts. Substrings of n
letters are termed n-words. If stationarity is as- sumed, any word i can be expected at any arbitrary
site to occur with a well-defined probability p
i
. For n = 1 we get the usual Shannon entropy Shannon,
1948: H
1
= −
l i = 1
p
i
·log
2
p
i
1 For n \ 1 we obtain the so-called n-word entropies
block-entropies or higher-order entropies which are defined by
H
n
= −
i
p
i n
· log
2
p
i n
2 where the summation has to be carried out over all
n-words with p
i n
\ 0. The conditional entropies,
h
n
= H
n + 1
− H
n
3 give the new information contained in the n + 1th
symbol given the preceding n symbols. The maxi- mal value of H
1
is log l, which occurs when the letters are independent and have the same probabil-
ity 1l. The decay of the h
n
, as n increases, measures correlation within the sequence. Hence, the condi-
tional entropies are good candidates to detect structure in symbolic sequences because they re-
spond to any deviation from statistical indepen- dence. To validate the results, we constructed
surrogates. In the standard random shuffle called algorithm zero in Rapp et al., 1994, the original
data sequence is shuffled. This conserves the fre- quency of the letters but destroys all correlations.
Following Theiler et al. 1992, we will take as our measure of significance the S-measure, defined by
the difference between the original value and the mean surrogate value of a measurement, divided by
the standard deviation of the surrogate values:
S = M
orig
− M
surr
s
surr
4 S gives the number of standard deviations ss
separating the value of the measurement obtained with the original data and its surrogates. In this
paper, M will alternatively represent the values of the calculations of interest; i.e. conditional en-
tropies and grammar complexities see below.
2
.
3
. Context-free grammatical complexity The Grammar-complexity is an attempt to deter-
mine the algorithmic complexity of a sequence Ebeling and Jime´nez-Montan˜o, 1980, by means of
a context-free grammar Rayward-Smith, 1983. The essence of the concept is to compress a
sequence by introducing new variables. The length of the compressed sequence is taken as a measure
of the complexity of the sequence. Briefly, our procedure is the following: all subwords of length
two are formed to make, with each one, a search over the whole string to determine the most fre-
quent one Jime´nez-Montan˜o et al., 1997. The most frequent pattern is substituted by a non-termi-
nal symbol, also called variable or syntactic cate- gory Rayward-Smith, 1983, in all its appearances
in the sequence, with the condition that it is repeated more than twice. This operation is per-
formed, as many times as possible, until there are no more strings of length two, which occur more
than two times. Then, one searches for strings of length equal or greater than three, that appear at
least two times, substituting the longest one by a non-terminal symbol. In this way a context-free
grammar which generates the original sequence is obtained. If the sequence is coded in a l-letter
alphabet, the corresponding complexity is denoted by C
l .
. In the following we shall calculate C
l
for l =
2, 4, 8. A detailed description of the procedure and appropriate examples may be found in a
former publication Rapp et al., 1994.
2
.
4
. Experimental methods We studied 32 spike-train sequences time series
and digitized versions obtained in previous exper- iments Diez-Martı´nez et al., 1988. Experiments
were performed on abdominal segments of Pro- cambarus clarkii or P. bou6ieri. Dissection, record-
ing and data storage procedures are explained in detail elsewhere Segundo et al., 1987. SAO spikes
were recorded extracellularly from the dorsal nerve. Usually, trains of 100 or more stimuli were applied
at
invariant average
stimulus frequencies.
Many frequencies were examined in random order.
In some
experiments, stimuli
were applied irregularly at equivalent frequencies. We
analyzed four arbitrarily chosen individual spike- train forms. Every spike-train analyzed here, se-
lected from a broader set, was typical of a particular form.
Interspike interval histograms ISIHs were constructed to compare the corresponding control
conditions with the effects of stimulus application Fig. 1. The SAO spontaneously discharged as a
noisy pacemaker. Its interval distributions were Gaussian
with relatively
narrow histograms,
which reflected the small CVs. Hence, interval fluctuations during the spontaneous firings were
slight e.g., CV = 0.02. During stimulation, the interval lengths differed significantly between
spike forms since their distributions were deter- mined by the stimulus parameters i.e. regularity
and frequency. When irregular stimulation was applied, the spike-train form displayed a bimodal
pattern. Some intervals were shortened with re- spect to the spontaneous ones; the average fre-
quency of the intervals was lower than during the spontaneous condition 138.65 vs. 150.03 ms.
However, some intervals were lengthened and, as expected, the overall variation of intervals in-
creased considerably CV = 0.16. In the intermit- tent spike form, the interval distribution was
complex. Although some intervals were shortened with respect to the spontaneous ones, these were
infrequent. The average frequency of the intervals was only slightly lower than during the control
Fig. 1. First-order interspike interval histograms ISIHs. These plots consider the set of time intervals between adjacent spikes in the train. Ordinates represent the relative frequency i.e. number of intervals that occur in analysis periods that lasted one ms bins
divided by the total number of events in the train. The abscissae plot the duration of the intervals themselves. Each panel shows the interval distribution for the spontaneous black and stimulated grey conditions for the selected spike-train forms. For each
data sample, basic statistics were calculated: n represents the number of intervals, N is the average of these and CV is the coefficient of variation.
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
