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

Dendritic excitability modulates dendritic information processing in a purkinje cell model

Using an electrophysiological compartmental model of a Purkinje cell we quantified the contribution of individual active dendritic currents to processing of synaptic activity from granule cells. We used mutual information as a measure to quantify the information from the total excitatory input current (I_Glu) encoded in each dendritic current. In this context, each active current was considered an information channel. Our analyses showed that most of the information was encoded by the calcium (I_CaP) and calcium activated potassium (I_Kc) currents. Mutual information between I_Glu and I_CaP and I_Kc was sensitive to different levels of excitatory and inhibitory synaptic activity that, at the same time, resulted in the same firing rate at the soma. Since dendritic excitability could be a mechanism to regulate information processing in neurons we quantified the changes in mutual information between I_Glu and all Purkinje cell currents as a function of the density of dendritic Ca (g_CaP) and Kca (g_Kc) conductances. We extended our analysis to determine the window of temporal integration of I_Glu by I_CaP and I_Kc as a function of channel density and synaptic activity. The window of information integration has a stronger dependence on increasing values of g_Kc than on g_CaP, but at high levels of synaptic stimulation information integration is reduced to a few milliseconds. Overall, our results show that different dendritic conductances differentially encode synaptic activity and that dendritic excitability and the level of synaptic activity regulate the flow of information in dendrites.

Correlated neuronal variability in monkey visual cortex revealed by a multi-microelectrode

Recordings from the visual cortex of anaesthetized monkeys taken with a 30-fold multi-microelectrode demonstrate that the neuronal variability, defined as the change in response strength over time spans of a few seconds to several minutes, is highly correlated within groups of neurones. Several such groups exhibiting independent variability between groups, coexist within the area recorded. This within-group covariance suggests that a major part of neuronal variability is due not to a noise process in the cells, but rather to additional inputs to the neurones, which are not under control of the experimenter.

What is the meaningful measure of neuronal spike train activity?

Neuronal spike train activity is conventionally viewed in terms of certain measures of central tendency (mean or median of intervals between discrete events or mean rate of such events) and their associated measures of variability (S.D., skew, etc.). Less commonly, investigators will compute the probabilities of occurrence in an Information Theory context. Another rarely considered measure of spike trains is serial ordering. Seldom is more than one of these approaches applied to the same spike train, and we are unaware of any case where all 3 have been applied to the same train. In order to test the inter-relationships among these approaches, we examined the same spike trains with all 3 analytic methods. Measures of central tendency were taken from the original absolute interval values, whereas entropy and serial order (Markov order) were computed from the sequences of patterns of adjacent intervals, expressed in the same non-parametric format. We found that entropy (measure of uncertainty) did not correlate well with the degree of serial ordering (Markov order). Entropy also did not correlate well with measures of central tendency (median or mean interval, or impulses/s) nor with the variability of such measures. An inverse correlation was seen between Markov order and several measures of central tendency (mean interval and rate) as well as with several measures of variability. The implications of these analyses extend beyond the analysis of spike trains to most all biological and physical time series. For neurophysiologists, these analyses may challenge our common assumptions about the most appropriate way to describe and interpret neuronal spike train activity.(ABSTRACT TRUNCATED AT 250 WORDS)

Do neurons process information by relative intervals in spike trains?

We suggest the possibility that neurons process information in terms of the relative duration of clusters of adjacent and successive inter-action potential intervals ("bytes" of intervals). If this concept is plausible, as is supported by research from several laboratories which have specifically addressed this possibility, one should be able to see evidence for such patterning in the published illustrations from studies in which this concept was not considered. We present some of this evidence here, along with some illustrations from the original publications. Byte patterns are evident in these examples, even though they went unrecognized by authors and readers alike. It is true that interval patterns are not obvious in all published illustrations of spike trains, and we suggest that this can be explained by one or more of the following: (1) some neurons may operate with an interval-pattern code while others do not, (2) a given neuron may use an interval-pattern code only under certain conditions, and (3) even when such a code exists, it may be difficult to detect for identifiable technical reasons. Therefore, we believe that the relative-internal-pattern concept is a valid scientific hypothesis which merits specific testing of its validity and range of applicability.

Serial dependencies and Markov properties of neuronal interspike intervals from rat cerebellum

Using three different approaches, each with different theoretical assumptions, we showed that mammalian neuronal spike trains contain serial ordering. We demonstrated that: (1) when intervals are categorized according to whether their durations are short, medium, or long, sequential groupings of adjacent interval categories exhibit Markov dependencies, extending to at least the 4th order; (2) the observed incidence of specified patterns of these groups of adjacent interval categories differs from the independent case, based on Chi square goodness-of-fit tests, and by using similar procedures; (3) there is divergence from independence when adjacent interval patterns are described in terms of relative lengths of adjacent intervals. The statistical indicators of serial dependence were significantly greater when applied to the original data than when applied to the same data after shuffling. Each of these approaches leads to the notion that "information" is carried in clusters of adjacent intervals ("bytes" or "words") and moreover, we can identify which specific patterns of interspike intervals contribute most to the statistical significance (i.e., those clusters that are potential candidates for "information carriers"). In most of the ten neurons, the "memory" of the system appears to be at least 36--45 msecs.

Entropy measures of signal in the presence of noise: evidence for 'byte' versus 'bit' processing in the nervous system

A method for detecting signal in the presence of noise in a highly specific way is described. Using action potential interval data from 12 neurons in rat cerebellum, we have demonstrated that the sequential ordering of spike intervals contains both noise and signal. We have identified and quantified the magnitude of relative entropy (uncertainty) of specified sets of interval patterns, ranging in length from 3-5 successive intervals. Some of these sets have exceptionally low entropy and thus seem to be especially meaningful as a set ('word') to the brain.

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.

A comparison of two techniques for analyzing neuronal interspike intervals: autocorrelation and relative interval coding

Only two theoretical approaches for serial order analysis of spike trains seem to be practical and to preserve a degree of the sequential ordering of the intervals. We herein compared these approaches, employing the two common techniques for autocorrelation and a group of relative interval coding techniques, using 4 short, idealized trains of spike intervals. The relative interval coding approach seemed to be more sensitive for accurately specifying and analyzing serial order. The autocorrelation methods create ambiguity because of inherent "partitioning" effects in the computation and because the interrelations among peaks in the correlogram can be deceptive.