Measuring spike coding in the rat supraoptic nucleus

Abstract

Measuring spike coding objectively is essential to establish whether activity recorded under one set of conditions is truly different from that recorded under another set of conditions. However, there is no generally accepted method for making such comparisons. Measuring firing frequency alone only partially reflects spike patterning. In this paper, novel quantities based on the logarithmic interspike intervals are proposed as useful measures of spontaneous activity. We illustrate the methods by comparing extracellular recordings from magnocellular cells of the rat supraoptic nucleus in vivo and in vitro and between oxytocin and vasopressin cells in vivo. A bimodal Gaussian function fitted to the log interspike interval histogram accurately described the distribution profile for very different types of activity. We introduce the entropy of the log interval distribution as a novel quantity that measures the capacity of a cell to encode information other than a constant instantaneous frequency. Unlike existing entropy measures that are based on spike counts, it quantifies the variability in the interval distribution. In addition, the mutual information between adjacent log intervals is proposed as an objective measure of patterned activity. For cells recorded in vivo and in vitro, there was no significant difference in mean spike frequencies but there were differences in the log interval entropy (t = -4.97, P < 0.001) and the mutual information (z = -2.64, P < 0.01). The differences may result from the disruption of connections in the slice preparation. When a comparison was made between the spike activity of oxytocin and vasopressin cells recorded in vivo, there was a difference in mutual information (z = 5.15, P < 0.001) but not in mean spike frequency. Both comparisons highlight the potential limitations of using mean spike frequency alone as a measure of spike coding. We propose that our novel parameters based on interval analysis constitute informative measures of spontaneous activity under different physiological conditions.

Figure 9. Differences between entropies are resistant to bin widths

Although the absolute value of the entropy is affected by the bin width, differences between entropies are resistant to bin widths so long as they are sufficiently small to model the probability mass distribution accurately. The graph shows the entropy of the log interval histogram for the same oxytocin (○, continuous line), vasopressin (▿, dashed line), and perinuclear zone cell (□, dotted line) as shown in Figs 1–Fig 4. A log scale is used for the bin width axis to show its log–linear relationship with the entropy. The points plotted for each cell correspond to temporal bin widths of 0.005, 0.01, 0.02, 0.04, 0.08, 0.16 and 0.32 log_e(time).

Figure 10. Accurate calculation of mutual information requires a minimum size of data set

If the number of intervals is very small there is a negative bias in the calculated mutual information, but the bias asymptotes to zero as the number of intervals is increased. The graph shows the mutual information, for the same perinuclear zone cell shown in Fig. 1C, calculated from different sample sizes.

Figure 1. Mean spike frequency may only partially reflect spike patterns

The top traces show 30 s excerpts of extracellular recordings made in vivo from an oxytocin cell (A), a vasopressin cell (B) and a perinuclear zone cell (C) of a female rat; plots D, E and F show their respective ratemeter records using 1 s bins over longer time periods. A ratemeter record is a histogram made up bars of a width that is specified by the bin duration, which for example may be 1 s. The height of each bar simply represents the number of spikes that fell within the time period occupied by its bin width. For trace C, a part of the recording has been expanded to show that many pairs of spikes occur in close temporal proximity in perinuclear zone cells. In E, an excerpt of the ratemeter record of the vasopressin cell has been expanded to show its phasic pattern of activity.

Figure 2. Interspike interval histograms provide a more complete description of spontaneous activity than mean spike frequencies

The top three graphs show interval histograms of bin width 10 ms and truncated to 1000 ms to show the activities of the cells represented in Fig. 1, whereas the bottom three graphs show the same histograms using a logarithmic time axis for the entire interval range. In each case, 700 interspike intervals were used to construct the histograms. The activity of the oxytocin cell is shown in A and D, the vasopressin cell activity is shown in B and E, and the activity of the perinuclear zone cell is shown in C and F. Inset histograms are shown in B and C to represent the longer intervals.

Figure 3. A bimodal lognormal model can be readily applied as a general method to describe the profiles of different interspike interval histograms accurately

From left to right, the activities of the same oxytocin, vasopressin and perinuclear zone cells as shown in Figs 1 and 2 are represented. The top three panels show the same interval histograms as in Fig. 2 with a bin width of 10 ms and truncated to 1000 ms. In C, the y-axis has been truncated from 258 to 10 counts to allow the section of histogram profile that reflects intervals of intermediate duration to be seen more readily. D–F show the histograms of the log interval distributions to base e, using a bin width of 0.1 log_e(time). For both the top and middle panels, a dotted line is overlaid to show the bimodal lognormal fit. G–I plot the observed cumulative probability (Observed Cum. Probability) against the expected cumulative density (Expected Cum. Density) given by the bimodal fit so that a perfect fit would follow a straight path as indicated by the dashed line. Goodness of fit of the bimodal lognormal model was confirmed by the Kolmogorov-Smirnov statistic for the oxytocin cell (D= 0.0194, P= 0.953), vasopressin cell (D= 0.0245, P= 0.791) and the perinuclear zone cell (D= 0.0257, P= 0.738).

Figure 4. Information theory can be used to describe the strength of patterning between adjacent interspike intervals without making any assumptions concerning their distributions

From left to right, the activities of the same oxytocin, vasopressin and perinuclear zone cells as shown in the previous figures are represented. A–C show the log interval scattergrams to base e, plotting the succeeding intervals (Succ. ISI) against the preceding intervals (Prec. ISI). Using a Gaussian kernel, the scattergrams were convolved to construct joint probability mass distributions shown in panels D–F, using a bin width of 0.02 log_e(time). A lighter shading indicates a contour of increased probability, where each contour step represents a change of 2 × 10⁶ units in probability mass. G–I overlay the waveforms discriminated as spikes that were used to construct the interval scattergrams.

Figure 5. Information measures can be used to distinguish different types of activity in an objective way

The entropy of the log intervals (Ent.) measures the irregularity of spike activity, and the mutual information (MI) between adjacent log intervals quantifies the patterning. Continuously firing activity shows less irregularity for the neurone illustrated in vitro (A) than for a representative oxytocin cell recorded in vivo (B); 60 s excerpts are expanded to show the differences in the activity: the more uniform intervals in vitro mean that the entropy is lower than that seen in vivo. The lack of mutual information in both cases reflects the absence of patterned motifs. By contrast, the ‘phasic’ activity in vasopressin cells recorded in vivo shows patterning illustrated by traces C and D: the cell with short bursts C shows less patterning than the cell with long bursts D despite a higher entropy.

Figure 6. The activity of supraoptic neurones in vitro was not the same as the activity in vivo despite the lack of a significant difference between their mean spike frequencies

Where the distributions passed the normality test, means and standard errors of the means are represented; otherwise box and whiskers plots are shown. The different panels represent the summary statistics of mean spike frequency (MSF; A), coefficient of variation (CV; B), log interval mean (Log ISI Mean; C), log interval standard deviation (Log ISI S.D.; D), interval median (ISI Median; E), and interval entropy (F). To visualize the distribution of the mutual information (MI) more clearly, separate histograms have been constructed for the in vitro (G) and in vivo groups (H). (A box and whiskers plot represents the median as a single line with a box to indicate the interquartile range; the whiskers represent the furthest data values within one-and-a-half times the interquartile range away from the lower and upper quartile, and outliers are marked as crosses.) For the normally distributed data, Student's t test statistics were used to test for significant differences, otherwise Wilcoxon ranksum test statistics were used. **P < 0.01 and ***P < 0.001, respectively.

Figure 7. Significant differences in the activities of oxytocin (OT) and vasopressin (VP) cells recorded in vivo in the supraoptic nucleus can be seen using different measures of activity with a number of exceptions, notably mean spike frequency

Where the normality test was passed, means and standard errors of the mean are represented; otherwise box and whiskers plots are shown. The different graphs represent the summary statistics of mean spike frequency (MSF; A), coefficient of variation (CV; B), log interval mean (Log ISI Mean; C), log interval standard deviation (Log ISI S.D.; D), interval median (ISI Median; E), and interval entropy (F). To visualize the distribution of mutual information (MI) more easily, separate histograms have been constructed for the oxytocin (G) and vasopressin cell groups (H). An explanation of the box and whiskers plot is provided in the legend of Fig. 6. For the normally distributed data, Student's t test statistics were used to test for significant differences, otherwise Wilcoxon ranksum test statistics were used. *P < 0.02 and ***P < 0.001, respectively.

Figure 8. The coefficient of variation strongly relates to the skewness of the interval distribution

For both the oxytocin (○) 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).