The effect of gallamine triethiodide on the activity of gamma motor neurons in the acute spinal cat

Entropy as an index of the informational state of neurons

Techniques were developed for using the classical information theory descriptor, entropy, to quantify the "uncertainty" present in neuronal spike trains. Entropy was calculated on the basis of a method that describes the relative relationships of serially ordered interspike intervals by encoding the intervals as a series of symbols, each of which depicts the relative duration of two adjacent spike intervals. Each symbol, or set of symbols has a specific fractional entropy value, derived from its probability of occurrence; moreover, fractional entropy can describe the relative amount of "information" that is associated with the relative location of a given symbol in a string of symbols. Using spike trains from 12 single neurons in the cerebellar cortex of rats, we determined: (1) the mean and S.D. of information content of each symbol in each specific position in a group of symbols (2-4 symbols/group, based on 3-5 adjacent intervals), (2) the 4-symbol groups which had the least and the most average fractional entropy, (3) that the 4-symbol groups with both low and high fractional entropy had significant positive correlations with the probability of occurrence of those groups after a drug treatment (ethanol), and (4) that the degree of drug-induced change in the incidence of both low- and high-fractional entropy groups did not correlate with predrug entropy. Thus, the entropy of clusters of 3-5 adjacent spike intervals, when computed in this particular way, seems to be a useful measure or index of the informational state of neurons.

Serial ordering in spike trains: what's it "trying to tell us"?

Summarized herein is the evidence that supports the hypothesis that neuronal action potentials (spike trains) are coded not only in terms of simple discharge rate but also can be serially coded by certain patterns of spike intervals. Based on the relative interval description method of Sherry and Marczynski (1972), our analyses of single-unit activity from cerebellar cortex neurons of rats seem to support three principal categories of conclusions; (1) Serial dependence of intervals does exist. This has been demonstrated with a variety of conventional statistical tests. These serial dependencies have also been shown to be independent of the (nonsequential) interval distribution variability. (2) Information theory is appropriate for evaluating spike trains. We have developed and tested methods for computing a fractional entropy for a given number of adjacent intervals, for assessing the relative fractional entropy of any one interval in a set of intervals, for computing for a group of neurons the mean and standard deviation of fractional entropy for specified clusters or intervals, and for transforming these values so that interval clusters of differing number can all be compared on the same numerical entropy scale (percentage maximum fractional entropy). In addition to the descriptive and quantitative value of such measures, we have also demonstrated their utility in testing hypotheses and in making empirical correlations. (3) The nervous system seems to process spike train intervals in "bytes", not "hits", of adjacent, serially ordered intervals. Among the several lines of evidence for this conclusion is the demonstration that drug-induced (ethanol) changes in fractional entropy of specific interval clusters seem to involve a "linked" combination of certain interval clusters, some which increase and others which decrease in incidence. Also, by using n-dimensional Chi-Square methodology, we have demonstrated that the relationships of adjacent intervals represent a Markovian process in which the duration of a given interval is partially determined by the duration of as many as four immediately preceding intervals. Finally, we showed that the relative fractional entropy (% maximum) of interval clusters of different numbers does not have a Gaussian distribution but rather is distributed in surprising ways by the specific number and relative durations of adjacent intervals. So just what is the serial ordering and information content of spike train intervals trying to tell us?. Perhaps it is trying to say that the nervous system processes information on a moment-by-moment basis in terms of "bytes" of short sequences of spikes with specific patterns of relative interspike durations. If so, we should be able to identify and characterize those "bytes". Much further testing must be done before we can claim that "neural codes" operate on the principles described herein. Nonetheless, we have made the issues explicit, and in our opinion have provided enough evidence to warrant further investigation.

The statistical relationship between the "entropy" of a neuronal signal and its variability

We used a serial order nonparametric analysis technique to analyze trains of neuronal action potential intervals in terms of classical information theory. We observed a marked correlation between the information theory descriptor, entropy, and several common measures of variability (median interval length, range of interval length, and skewness of the probability density function). The correlation between variability and entropy was accounted for in the following decreasing order: range of intervals, median interval, and skewness. These data suggest a significant relationship between the signal and its variability and entropy, when entropy is calculated by our serial-order pattern detection method.