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Department of Anatomy, University of Cambridge, Downing Street, Cambridge CB2 3DY, UK
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(Received 11 August 2003;
accepted after revision 6 November 2003;
first published online 7 November 2003)
Corresponding author R. E. J. Dyball: Department of Anatomy, University of Cambridge, Downing Street, Cambridge CB2 3DY, UK. Email: red1000{at}cam.ac.uk
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Introduction |
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At present there is no way of describing spike coding that is universally accepted as reflecting underlying physiology. In the presence of a stimulus, changes in spike rates over repeated sweeps are represented by the peri- (or post-) stimulus time histogram (Gerstein & Kiang, 1960). Neurophysiologists working in motor and sensory systems have contributed to a substantial body of literature in quantifying neural responses to stimuli (Optican & Richmond, 1987; Bialek et al. 1991; Rieke et al. 1999). It is of particular interest that two spikes in close temporal proximity carry far more than double the information conveyed by a single spike (Brenner et al. 2000). More recent work has acknowledged the importance of short interspike intervals by adopting a log time scale to describe changes in the interval histogram in the context of a rapidly changing stimulus (Fairhall et al. 2001).
This paper focuses on ‘spontaneous activity’, which may be defined as activity in the absence of any specific stimuli. Such activity is of particular interest in the hypothalamus where modulatory changes may occur over longer time periods than seen in motor and sensory systems. Spontaneous activity has been used to identify hypothalamic cells with a specific phenotype where cells that release particular neurohormones display different firing patterns (Poulain & Wakerley, 1982). Although such methods of identification can be useful in discriminating different cell types, they often rely on ad hoc approaches rather than using objective numerical techniques. In this paper, existing measures of spontaneous activity are compared and their limitations are considered. We introduce novel measures of spontaneous activity in an attempt to address the shortcomings of current methods and illustrate their use by applying the techniques to cells of the supraoptic nucleus recorded in vitro and in vivo.
Statistical descriptions of spontaneous activity broadly fall into two general categories. The more common method is to count spike numbers over certain periods of time to provide frequency measures of activity. Changes in mean spike frequency have been used in the past to assess substantial changes in the activity of specific groups of cells under different physiological conditions, such as under different osmotic environments (Brimble & Dyball, 1977). By contrast, interval methods focus on the intervals between spikes rather than spikes themselves, for example an interspike interval histogram (ISIH) represents the distribution of intervals of different lengths (Gerstein & Mandelbrot, 1964). An exponential function has been fitted to the right-hand tail of the interspike interval histogram in an attempt to characterize the modulation of synaptic inputs in a number of different cells types, including the magnocellular neurones of the supraoptic nucleus (Leng et al. 2001). Similarly, a number of different functions have been used to describe various interspike interval distributions, including the gamma, inverse Gaussian, the Weibull (Tuckwell, 1988), and the lognormal (Burns & Webb, 1976) distributions. To date, however, no single distribution has been put forward as a general working function to describe interval distributions for the spontaneous activities of different cell types. In this paper, a simple bimodal lognormal probability density function is proposed as a candidate that is flexible enough to accommodate the different characteristics of the processes that underlie the activity of different cells. We illustrate its ability to model very different interval distributions equally well by applying it to different cells recorded in the supraoptic nucleus in vivo.
Measurement of the irregularity of spike activity is important since it represents the coding capacity per spike. For example, an ideal metronome cannot convey any more information than its overall rate. Variance measures are second-order statistics and thus are limited because they neglect different irregularities arising from changes in higher orders (moments). Entropies are similar to variances in that they quantify the extent of uncertainty but they are not restricted to second-order moments. Existing methods of calculating the entropy of the spike train based on spike counts are limited since they are sensitive to the time scale used to quantize the spike train relative to the mean spike frequency (Mackay & McCulloch, 1952); for example, using previous entropy measures, the entropy per spike for a cell firing at 10 Hz using 1 s bins would be identical to the entropy per spike for a cell firing at 5 Hz using 2 s bins because the average spike count per bin is the same. In this paper, the entropy of the log intervals is proposed as a novel measure of the irregularity of activity in a way that is independent of units and avoids having to count spikes over arbitrary time periods. Since the irregularity of the spike train determines how much uncertainty is associated with each interval, we propose that our entropy measure quantifies objectively the maximum amount of information that a single action potential can possibly encode.
A limitation of interval histograms is that they do not represent the importance of the order in which the intervals occur, and such patterning is likely to be of considerable physiological importance. By plotting each interval against its predecessor, any association between adjacent intervals can be represented graphically by the resulting interspike interval scattergram; the interval scattergram can be used to construct a two-dimensional joint interval histogram that can be normalized to represent a joint probability mass distribution for the adjacent intervals (Rodieck et al. 1962). In this paper, the mutual information between adjacent log intervals is proposed as a measure of dependence between adjacent intervals as an alternative to correlation coefficients (Perkel et al. 1967), that are limited in their general applicability due to the non-Gaussian distribution of the linear or log intervals. Since the mutual information per spike quantifies the amount of information shared by the two intervals on either side of each spike, we propose that the mutual information is a useful measure of spike patterning that requires no assumptions regarding the underlying interval distribution.
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Methods |
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Recordings in vivo
Rats weighing between 250 and 300 g were anaesthetized with urethane at a dose of 1.1 g kg-1I.P. The jugular vein was cannulated and a tracheostomy was performed. Using the ventral surgical approach (Leng & Dyball, 1991), the supraoptic nucleus and pituitary stalk were exposed. Extracellular recordings were made using glass electrodes with a tip diameter of approximately 0.5 µm, filled with 0.5 M sodium acetate to give a tip resistance of approximately 10 M
. Neurosecretory magnocellular cells were identified by antidromic activation from the pituitary stalk, (Yagi et al. 1966) confirmed by the two-shock collision test. Oxytocin- and vasopressin-secreting cells were distinguished by their continuous and ‘phasic’ firing patterns, respectively (Poulain & Wakerley, 1982). At the end of the experiment, the animals were killed by an overdose of pentobarbitone sodium.
Recording in vitro
Rats weighing between 75 and 150 g were anaesthetized with urethane at a dose of 1.1 g kg-1I.P. The animals were decapitated, the brains excised, and a Vibratome (Series 1000; Horwell Instruments, St Louis, MO, USA) was used to cut coronal slices of thickness 500 µm through the supraoptic nucleus. Slices were transferred to the recording chamber and perfused with artificial cerebrospinal fluid (aCSF) oxygenated with 95% oxygen and 5% carbon dioxide. The composition of the aCSF was (mM): NaCl 124, KCl 3, KH2PO4 1.24, CaCl2 2.1, MgSO4 1.3, NaHCO3 24, and glucose 10 (Cui et al. 1997). The recording chamber was maintained at 36.8°C. Extracellular recordings were made using glass electrodes with a tip diameter of approximately 0.5 µm, filled with 0.9% NaCl to give a tip resistance of approximately 10 M. Although oxytocin and vasopressin cells can be identified by their characteristic firing patterns in vivo (Poulain & Wakerley, 1982), their distinction in the slice is not so clear, thus all cells recorded in vitro were treated as a single group for data analysis since the histological features of almost all the cells of the supraoptic nucleus are magnocellular (Dyball & Kemplay, 1982).
Data acquisition
Extracellular spikes were recorded from single cells and the signal was preamplified then filtered using a 50 Hz noise eliminator (Hum Bug; Quest Scientific, North Vancouver). The signal was then amplified and passed through an interface device (1401 Plus; Cambridge Electronic Design, Cambridge, UK) to a computer. Spike 2 V4 (Cambridge Electronic Design, Cambridge, UK) software was used to collect the amplified signal. For some recordings, an on-line hardware discriminator was used (Digitimer D 130) but recordings were excluded if interspike intervals of less than 2 ms suggested inadequate discrimination. For most cells, the recording was sampled at 20 kHz and offline discrimination of spike events was undertaken using custom software (Dyball & Bhumbra, 2003). Recordings of at least 500 spikes were collected from every cell for data analysis during a period at least 5 min. The choice of the minimum values was made to balance the need to acquire a sufficient number of intervals for data analysis with the requirement for it to be reasonable to treat each excerpt of activity as stationary. For the vasopressin cell group in vivo, the minimum number of spikes collected was 833, and the minimum number of cycles of phasic activity was 2, where each cycle consisted of a period of spike activity followed by a period of relative silence. However, these numbers were usually much greater, with a range of 833–4299 spikes and 2–14 cycles. While such criteria require a degree of judgement, there is no generally accepted method of establishing stationarity in all cell types. Nevertheless, cells with obvious increases or decreases in ratemeter activity were excluded so that the periods of recording used for analysis could be treated as stationary.
Data analysis
Two distinct comparisons were made when analysing the results. The first comparison was between all the cells recorded in vivo with those recorded in vitro. Within the in vivo group, the oxytocin cell and vasopressin cell subgroups were also compared. MATLAB 6.1 (MathWorks) software was used to perform the data analysis. Simple statistics were first collected (for at least 300 s): the mean spike frequency, the coefficient of variation of the interspike intervals, the log interval mean, the log interval standard deviation, and the median interspike interval. Using the interspike intervals wi in milliseconds, the coefficient of variation (CV) was calculated from the mean 
and standard deviation s.
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| (1) |
1 and 2, where:
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| (2) |
1, and 2. The Kolmogorov-Smirnov test statistic D was used to determine how well the data conformed to a bimodal function, where D represents the maximum distance between the expected cumulative density and the observed cumulative probability. The null hypothesis of a conforming distribution was rejected if P(D)
0.05, otherwise goodness of fit was accepted. The expected cumulative density function f(xix) was calculated from the bimodal probability density function f(x) using integration.
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Using Shannon's formula (Shannon & Weaver, 1949), the absolute disorder represented by the distribution of either marginal probability distribution was quantified by the entropy:
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A previous method of evaluating entropies from intervals was based on the arbitrary classification of interval lengths as ‘short’, ‘medium’ and ‘long’ (Sherry & Klemm, 1984). By contrast, the use of a logarithmic scale allowed the entropy per spike to be calculated on a continuous scale in a way that is not affected by units of time. The joint entropy of the joint probability mass distribution P(xi,yj) was calculated by applying the formula in two dimensions:
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Results |
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Comparison of three example cells recorded in vivo
For the purposes of illustration and comparison, spontaneous recordings of oxytocin- and vasopressin-secreting cells are represented in Fig. 1, and a third cell recorded from the perinuclear zone just dorsolateral to supraoptic nucleus is also shown. The mean spikes per second and corresponding standard deviations for the oxytocin, vasopressin and perinuclear zone cells shown in Fig. 1 are 1.21 ± 1.08, 0.61 ± 1.26 and 1.15 ± 1.09 Hz, respectively. Although the mean spike frequency of the vasopressin cell is lower than that of the other two cells, it reaches the greatest spike count per second (8 Hz) as shown in Fig. 1E. Amongst other aspects of activity, this information is neglected if the mean spike frequency is considered alone. To account for the variation in the spike frequency, it is useful to consider the standard deviation. However, the standard deviation of the spikes per second for the oxytocin and perinuclear zone cells shown in Fig. 1D and F, respectively, are very similar despite their clear differences in activity. Thus neither the frequency mean nor standard deviation are able to clearly distinguish between the very different types of activities of the three cells. Although these limitations can be mitigated by adjusting the bin width used for counting spikes, for example with the Fano factor (Rieke et al. 1999), such solutions are very sensitive to an arbitrary selection of the duration of the bin width rather than genuine physiology.
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A total of 20 cells were recorded in vitro and 51 cells were recorded in vivo. Examples of cells recorded in vitro and in vivo are shown in Fig. 5 to illustrate the types of activity that have different values for entropy and mutual information. The entropies of the continuously firing cells recorded in vitro and in vivo were 6.71 and 7.71 bits spike-1, respectively. The lower entropy in vitro reflects the greater uniformity in interval length which can be seen in trace A. Since entropies operate on a logarithmic scale to base 2, a difference of 1 shows that the irregularity of the activity of the cell recorded in vivo is precisely twice that seen for the cell recorded in vitro. In both cases, the mutual information between adjacent log intervals was zero. This means that there were no patterns or ‘motifs’ in activity that would allow any improvement in predicting the length of any one interval by knowing the duration of its predecessor. By contrast, the ‘phasic’ activity of the vasopressin cells recorded in vivo showed significant associations between adjacent intervals. The mutual information for the cell with the short bursts was 0.02 bits spike-1 and for the cell with the long bursts was 0.08 bits spike-1, reflecting a greater extent of patterning. The difference in entropy seen in traces (A) and (B), neither of which show mutual information, and the lower entropy in trace (D), that has a higher mutual information than trace (C), illustrates how entropy and mutual information can vary independently of one another.
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Within the in vivo group, 29 oxytocin cells and 22 vasopressin cells were recorded and the summary results are shown in Fig. 7. The data representation and statistical tests are identical to those described above. There were no significant differences in mean spike frequency (z= 1.20, P= 0.231). The coefficient of variation was significantly greater in the vasopressin cell group (z= 6.07, P < 0.001), suggesting a lower variability of activity in the oxytocin cell group. By contrast, there were no significant differences in the log interval standard deviation (t=–0.57, P= 0.574) or in the entropy (t=–0.04, P= 0.970). Despite the lack of a significant difference in mean spike frequency, the mean log interval was less in the vasopressin cell group than in the oxytocin cell group (z=–2.57, P= 0.010), and a corresponding difference was seen in the median interspike interval (z=– 2.40, P= 0.016). Patterning was significantly greater for the vasopressin cells than for the oxytocin cells as measured by mutual information (z= 5.15, P < 0.001).
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2= 17.03, P < 0.001). For both cell groups, trends between mean spike frequency and mutual information were determined using Spearman's rank correlation coefficients. For the oxytocin cells, there was a positive correlation (r = 0.5210, P= 0.0038), and for the vasopressin cells, the correlation was negative (r =–0.6059, P= 0.0055).
The non-Gaussian nature of the interval histogram means that a feature of the coefficient of variation is its sensitivity to asymmetries in the interval distribution due to the presence of very long intervals. To investigate the extent to which the increased coefficient of variation in the vasopressin cells over the oxytocin cells was a reflection of skew, the coefficient of skewness (CS) of the interspike intervals was calculated.
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is the mean, and s is the standard deviation. The coefficient of variation was plotted against the coefficient of skewness in Fig. 8 for both the oxytocin and vasopressin cells. A visual inspection of the scatter distribution strongly suggests that the oxytocin cells and vasopressin cells segregate in a very similar way for the two coefficients. Since the data were not normally distributed, Spearman's rank coefficient was used to test for correlation. The test statistic (r = 0.886; P < 0.001) confirmed a strong positive trend.
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Discussion |
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Oxytocin-secreting cells display a pattern of activity that can be described as continuous (Poulain & Wakerley, 1982). In the case of the example oxytocin cell, the log interval distribution was normally distributed. Thus when the bimodal model was applied, the two modes of the bimodal model virtually superimposed. This suggests that the log intervals belong to one population and that the interval histogram can be adequately described by a single mode. By contrast, it has long been established that some neurosecretory cells are capable of a ‘phasic’ firing pattern (Wakerley & Lincoln, 1971), where periods of activity are ‘interrupted’ by long periods of relative silence. Such activity is characteristic of vasopressin-secreting cells (Poulain & Wakerley, 1982). The vasopressin cell illustrated in Fig. 3 displayed a pattern of activity where the log interval histogram could be described by two Gaussian functions. Since the small proportion of long intervals could only be described using an additional mode, they can be regarded as a population that is separate and distinct from the population of shorter intervals. For vasopressin cells therefore, a bimodal model is necessary to describe the interval histogram adequately.
Cells in the perinuclear zone recorded from the lateral hypothalamus are known to show ‘burster’ activity that is very different from the bursting activity of oxytocin cells seen during the milk ejection reflex (Dyball & Leng, 1986). In the case of the perinuclear zone cell used as an example, the highly irregular nature of the firing patterns is reflected in the interval histogram. The ISIH shows two populations of intervals that are either very short or long, and this is more obvious on a log time scale as shown in Fig. 3F. Intracellular recordings in vitro have shown that a characteristic feature of perinuclear zone cells is a low threshold depolarizing ‘hump’ when they are depolarized from a hyperpolarized membrane potential (Armstrong & Stern, 1997). Similar humps have been described in other regions of the central nervous system, such as those associated with the low threshold calcium potentials seen in the region of the paraventricular nucleus (Tasker & Dudek, 1991). A depolarizing hump results in a ‘window of opportunity’ shortly after an action potential when the cell is likely to spike again thereby terminating an interval of a particularly short duration. An advantage of a bimodal model is that it allows for two windows of opportunity for the spike that ends the interval. Doublet or triplet spike motifs may be important neuronal encoding strategies. Thus a model that accommodates a bimodal function is likely to be generally applicable to the firing of many different cell types. It therefore constitutes a valuable method to describe spontaneous activity in a way that reflects underlying physiological properties.
Measures of irregularity
The irregularity of a spike train is related to the variability of the interspike intervals. For the three types of cell used as examples, the very different types of spike activity are likely to reflect important differences in their physiological properties. Figure 6A shows that activity in vitro and in vivo cannot be discriminated by mean spike frequency. Similarly, the differences in firing patterns between oxytocin and vasopressin cells in vivo are not evident from a comparison of mean spike frequency as shown in Fig. 7A. While mean spike frequency is a straightforward measure of spike activity, there are clearly aspects of spontaneous activity that are not quantified by frequencies despite their common usage in hypothalamic electrophysiology. We introduce an entropy measure that shows numerically that for the three example cells, the oxytocin cell shows the least irregularity in activity and that the greatest irregularity is seen in the perinuclear zone cell.
The coefficient of variation has been adopted in the past to quantify variability in activity. The markedly elevated coefficient of variation in the vasopressin cell group accounts for the increased values in the in vivo group illustrated in Fig. 6B. Since the log interval standard deviation and entropy are measures of spike train irregularity, the similarity of the parameters in oxytocin and vasopressin cells may seem odd in the light of the significant differences in spike train irregularity measured by the coefficient of variation. However, a direct comparison between the coefficient of variation and log interval standard deviation or entropy is misleading. Its strong correlation with the coefficient of skewness shows that the coefficient of variation is strongly sensitive to the skew of the interval distribution. Figure 8 shows that the coefficient of skewness was never zero or negative, indicating that the asymmetry was always in the direction of a positive skew. Both coefficients are greater in the vasopressin cell group because they are strongly influenced by the long intervals that are characteristic of phasic firing patterns. In this situation therefore, the coefficient of variation is effectively a measure of the asymmetry of the interval distribution. Since interval histograms are rarely negatively skewed, the coefficient of variation strongly weights long intervals at the expense of the shorter intervals.
The similarities between the oxytocin and vasopressin cell groups in the log interval standard deviation and in the entropy is consistent with the similar appearances of their interval histograms for all but the longest intervals. The log scale reduces the effect of long intervals on the standard deviation. Similarly, the entropy is rendered resistant to such effects because the subpopulation of long intervals is treated as a distinct subset of the intervals that represent the entire ‘repertoire’ of the cell, regardless of how far the values are from the main group. Unlike the coefficient of variation, neither the log interval standard deviation nor entropy weights the long intervals at the expense of the shorter intervals. Any variability in the log interval standard deviation and entropy is likely to be of particular biological interest because of the effect that short intervals might have on the availability of calcium in the nerve terminals.
Although the log interval mean was lower in vivo that in vitro, it does not necessarily follow that the mean spike frequency in vitro should be expected to be significantly lower. The reason for the difference is that the exponential of the log interval mean does not necessarily correspond to the arithmetic mean interval. The expected value E(
) of a lognormal probability density function f() for the variable is not only a function of the mean µ but also the variance 2 such that 
(Aitchison & Brown, 1963). Thus the increased log interval standard deviation in vivo masks the reduced log interval mean when a comparison is made of the differences in mean spike frequency alone. An overall ‘shift’ of the interval histogram to the left in the in vivo group is confirmed by the reduced median interspike interval, notwithstanding the differences in ISIH dispersion.
Previously adopted methods of calculating the entropy of a spike train are sensitive to the spike frequency relative to the time scale used to quantize the spike train (Mackay & McCulloch, 1952). A potential limitation of past methods is that their frequency dependence on entropy determination overestimates the entropy of an ideal metronome. If the entropy is regarded as the information content per ‘tick’, any suitable estimate of the irregularity of a perfect clock should give a value of zero. As a result, the novel method of calculating the entropy per spike was devised to quantify the irregularity of a spike train in a way that avoided counting spikes over bins of arbitrary time widths (Bhumbra et al. 2003). Since the method is based on the interval histogram rather than the spike counts, it would give a zero value ‘per beat’ for the entropy of a perfect metronome, independent of the frequency, because all the intervals would be the same. Since an increased entropy is associated with a greater coding capacity (Jaynes, 2003), the log interval entropy quantifies objectively the maximum amount of information that a single spike can possibly encode.
The mean entropy for the oxytocin and vasopressin cell groups recorded in vivo had the same value: 7.46 bits spike-1. However, without some familiarity with information theory, it is perhaps not immediately obvious what precisely is meant by ‘one bit’ as a unit of entropy. One bit is the measure of uncertainty associated with two equiprobable outcomes, such as a single toss of a fair coin. By applying the methods to artificial spike trains generated from random simulations, we have calculated entropy measures to provide a ‘calibration scale’ for a temporal bin width of 0.02 loge(time). The entropy of a perfectly regular spike train, such as that of an ideal metronome is 0 bits spike-1. For a Poisson process of a constant rate, the entropy is 7.95 bits spike-1. Both these measures are completely independent of the frequency or mean interval, and this was confirmed by testing over the frequency range of 1 Hz to 1000 Hz using a million artificially generated spikes.
Figure 6 illustrates that the activity of cells in vitro is dissimilar to that seen in vivo in the supraoptic nucleus. The information parameters calculated for the in vitro and in vivo groups highlight important differences between the firing in the two experimental situations. A lower entropy in vitro shows that that the coding capacity per spike in the slice is less than in the intact brain. The greater mutual information in vivo illustrates that spike patterning is also lower in the slice. It seems likely that the reduced complexity of the coding in vitro results from the deafferentation associated with the preparation of the slices (Zaborszky et al. 1975). There is considerable interconnectivity within and just local to the supraoptic nucleus (Leranth et al. 1975). The altered coding in vitro may thus result from a disruption of the intranuclear axon collateral network or the local interneurone circuitry, as well as from the severing of more distant inputs. Experimentally, a reduced complexity of coding may be useful to simplify the changes caused, for example, when investigating the overall effects of pharmacological agents on cells recorded in vitro.
Measures of patterning
The patterning of a spike train is related to the associations in the interspike interval sequence due to the repeated patterns of activity. For the comparison in vivo, the interspike interval scattergrams of a majority of vasopressin cells showed significant mutual information but the opposite was true for the oxytocin cells. This suggests that spontaneous activity of oxytocin cells is usually adequately described by the interval histogram alone whereas the permutation of intervals in vasopressin cells are arranged in a manner that shows patterned coding at a higher level. The phasic firing pattern seen in vasopressin cells is consistent with this observation. Action potentials in vasopressin cells are followed by a slow depolarizing after-potential. A possible mechanism of initiating phasic bursts is through temporal summation of depolarizing after-potentials as a result of consecutive spikes in close succession (Andrew & Dudek, 1984). The termination of phasic bursts may be mediated by a repolarization resulting from an activity-dependent inactivation of the conductance underlying the depolarizing after-potential (Andrew & Dudek, 1984). Results from intracellular studies suggest that neurotransmitter regulation of the depolarizing after-potentials is a possible mechanism by which phasic firing could be modulated synaptically (Ghamari-Langroudi & Bourque, 1998). A consequence of slow modulation is that intervals of similar lengths tend to follow each other. The appearance of the interspike interval scattergram of the vasopressin cell in Fig. 4B implies such a positive trend (r = 0.088, P < 0.02) between adjacent intervals. By contrast, the negative trend (r =–0.555, P < 0.001) in the interspike interval scattergram of the perinuclear zone cell illustrated in Fig. 4C suggests that intervals of dissimilar lengths tend occur side by side. Thus the distribution of the interspike interval scattergram is likely to reflect different underlying mechanisms governing the coding strategies adopted by different cell types.
Analysis of the interval scattergrams for the three example cells showed no ordered interval patterning in the oxytocin cell but there was patterned activity in the vasopressin and perinuclear zone cells. The mutual information was much greater for the perinuclear zone cell than for the vasopressin cell demonstrating a stronger association between intervals. A visual inspection of the scatter distributions in Fig. 4B and C suggests such a pattern. For the vasopressin cell, it can be seen that the shortest and longest of intervals never occurred side by side since there are no scatter points in the top left and bottom right of the scatter. The absence of any scatter points in the lower left quadrant of the interval scattergram for the perinuclear zone cell shows that two short intervals never occurred together. For both the vasopressin and perinuclear zone cells, there are clearly two distinct populations of intervals. It is thus inappropriate to measure the extent of the association between adjacent intervals using autocorrelation coefficients since the strength of their relationship cannot be evaluated simply in terms of monotonic linear trends that assume a single normal distribution of the log intervals. However, the substantial differences in the extent of the association between adjacent intervals for the vasopressin and perinuclear zone cells are quantitatively described by the large difference in the values for their mutual information. Calculation of the mutual information makes no assumptions concerning the underlying interval distribution. The differences in the direction of the trends between mutual information and mean spike frequencies for the oxytocin and vasopressin groups in vivo show that any dependency between mutual information and firing rate is a consequence of cell type rather than the method of calculating the mutual information.
The increased capacity of the parameters described here to distinguish the patterns of activity displayed by the same cell types in different situations and by different cell types in the same situation has been illustrated by recordings from cells in the supraoptic nucleus. Activity recorded in vitro shows less variability than that recorded in vivo and is consistent with what might have been expected after partial deafferentation. Spike activity of vasopressin cells, that has an obvious pattern discernible from the ratemeter record, is significantly more patterned than in oxytocin cells. The differences support the use of appropriate information parameters to provide biologically valuable measures of spike coding. Using such parameters that are numeric, objective, and can be compared statistically, firing patterns that cannot be discriminated by mean spike frequency can be distinguished. The study highlights the potential limitations of using frequency measures alone to quantify neural coding and offers alternatives.
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Appendix |
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The choice of a bin width of 0.02 loge(time) was a result of limitations in computer memory to accommodate the firing patterns of cells that showed interval lengths of both extremes in the same recording. Although the entropy of the log interval histogram could be satisfactorily calculated using much smaller bin widths, the calculation of the Kullback-Leibler divergence between independent and dependent joint probability distributions of the interval scattergram required two-dimensional analysis that necessitated the use of coarser bin widths. The absolute value of the entropy is affected by the bin width such that smaller bins results in greater values.
However, to make valid comparisons between different cells, it is necessary to adopt a constant bin width. Differences between entropies are resistant to bin widths so long as the bin width is sufficiently small to model the profile of the probability mass distribution accurately. This can be seen in Fig. 9 by comparing the entropies calculated for the oxytocin, vasopressin and perinuclear zone cells that were used as examples in the main text. The log–linear relationship between the bin width and entropy can be seen for all three cells. For the most part, the three lines are parallel, illustrating that the differences in the entropies calculated for the three cells are not substantially affected by the bin width. Where the bin width exceeds 0.08 loge(time), it can be seen that the constant differences in entropies are not maintained. This reflects a distortion in the profiles of the probability mass distributions resulting from an unduly coarse bin width. Fortunately, modern computers are of sufficient speed and memory capacity to allow the use of smaller bin widths.
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x, the discrete entropy S(X) and differential entropy s(X) can be interrelated (Cover & Thomas, 1991).
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x
0 log2x is a quantization reference term that is dependent only on the logarithm of the bin width. Since halving the bin width x would decrease the value of the quantization reference term by one, the discreet entropy would simply be incremented by 1 bit as long as the original bin width x was small. This can be seen in the data represented in Fig. 9. For example, the first five discreet entropies for the vasopressin cell, when doubling the bin width from 0.005 loge(time), were 9.99, 8.99, 7.99, 6.99 and 5.99 bits spike-1. An advantage of the discreet entropy measure over its continuous counterpart is that it makes no assumptions concerning the underlying interval distribution. However, in both the continuous and discreet cases, the entropy measure is affected neither by mean spike frequency nor the units in which time is measured since a logarithmic temporal axis treats the interspike intervals on a relative rather than an absolute scale. Use of Monte-Carlo methods in calculating the mutual information
The Kullback-Leibler divergence between the dependent and independent joint interval probability mass distributions is an approximation of the mutual information between adjacent intervals. However, the value is likely to be an overestimate due to random coincidences that would occur in a finite data set simply as a result of the way in which the log interval scattergram was discretized. It is thus necessary to establish first that the relative entropy is significantly greater than zero. Where significance is confirmed, it is then necessary to adjust the relative entropy for the finite nature of the data set in order to calculate the corrected mutual information. Monte-Carlo methods use random simulation techniques and are used here to address both these aims.
Let X and Y denote the data sets of preceding log intervals xi and succeeding log intervals yj. For the perinuclear zone cell illustrated in Fig. 4F, the Kullback-Leibler divergence between the dependent and independent distributions D(X,Y||XY) was 0.37 bits spike-1. If R was employed as a random permutation transformation, RY denotes a shuffled version of Y. The Kullback-Leibler divergence D(X,RY||XRY) may be determined by subtracting the entropy of the joint probability distribution from the sum of the marginal entropies.
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can be evaluated numerically. For the perinuclear zone cell illustrated in Fig. 4F, the mean value 
following 100 random shuffles was 0.01 bits spike-1. The maximum value of the relative entropy for the randomly permutated trials was 0.02 bits spike-1. Since a relative entropy of 0.37 bits spike-1 for the ordered data D(X,Y||XY) is greater than the maximal value for randomly permutated data D(X,RY||XY), the large value for D(X,Y||XY) is unlikely to have arisen purely by chance occurrences in the order of intervals. If after 100 random permutation transformations, no greater value for the relative entropy was ever computed, it is reasonable to suggest that the probability of this occurring by chance is less than the reciprocal of 100, or P < 0.01.
Having evaluated the mean relative entropy 
for the randomly permutated data, a ‘zero’ value in relative entropy for which there is no significant association between adjacent intervals has been established. The mutual information I(X; Y) can be evaluated as the difference between the relative entropy for the original data as initially ordered D(X,Y||XY) and the mean relative entropy for the randomized data 
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) and vasopressin (
) cells recorded in vivo the coefficient of variation (CV) is plotted against the coefficient of skewness (CS) with both axes on logarithmic scales. Spearman's rank coefficient for the combined data shows a strong positive correlation: r= 0.886 (P < 0.001).
, dotted line) as shown in 
