And Autocorrelograms

The previously mentioned expression of Synchronicity served as a simple means of visualizing entrainment and the consistency of spike occurrences as a function of stimulus phase. However, it quantified only the distribution of discharges over the vibratory cycle, and left other spike-train features out. Additional information can be obtained from examining interspike intervals (ISIs). If, as previously done for response phase, the interspike intervals are plotted as a function of behavioral time, a time-dependent representation of the spike train periodicity can be constructed. This form of visualization has several benefits. As with the phase plots described above, shifting trial centering points so that they align with behaviorally significant events does little to degrade the image one sees. If the spike train contains ISIs of about the same duration, these will be viewed as a horizontal cluster of dots. The width of the cluster, then, is an indication of the consistency of the ISIs around a central tendency. If the stimulus eliciting the spikes is itself periodic, as are flutter and/or vibratory stimuli, then values on the ordinate can also be scaled in stimulus cycles. Another benefit of this form of data representation is that it does not necessarily make assumptions about the presence of periodicities at a given frequency in the spike trains. Moreover, it can reveal intrinsic periodicities present in spike trains even without an external periodic input. For example, plotting ISIs of a neuron that is intrinsically rhythmic or is driven by a secure driver at a frequency not previously known will result in a horizontal band of dots representing those ISIs. Thus, this form of data representation can allow for a crude but sometimes adequate assessment of the spike train's frequency spectrum.

Some basic variants of ISI plots can provide additional information about the temporal characteristics of spike trains. Joint interval scattergrams26-40 provide a means of analyzing the serial dependencies of the ISIs. Simply, in these plots for a given spike occurrence (n), the plotted point represents the ISI between it and the next spike (n+1) on the abscissa. The ISI involving spike n+1 and spike n+2 forms the value of the ordinate. Clusters of dots at regular intervals on both axes are often seen.40 Those equidistant along both axes represent ISIs that have occurred at the primary frequency component of the spike train. Clusters represented asymmetrically along the axes represent either multiple spikes per cycle or missing spikes (instances where a beat has been missed). Diagonal lines indicate regions of unentrained spikes (see Reference 40 for discussion). By this method, unentrained spike trains or those without intrinsic rhythmicity are represented without clusters and with dense groups of points near either axis.

Another way to visualize the consistency of spike occurrences is to construct expectation density (ED) histograms that illustrate the autocorrelation func-tion.1'7'39'41-43 Conventionally, an interval is chosen over which all spike occurrences from a reference point are compiled into a histogram with binwidths that may or may not have any relationship to the expected periodicity of the spikes. For example, one may choose to plot the time of occurrence of all spikes for several hundred milliseconds after the occurrence of each spike or after the occurrence of some other event (see Reference 39 for our implementation). If the ISIs are relatively consistent, the ED histogram will contain multiple peaks at intervals that thus give an indication of the firing pattern. The difference in the heights of the peaks in the ED histogram as a function of time gives an indication of this consistency as well. Variations in ISIs are correlated with progressively decreasing heights of peaks further from the point of reference. If the ISIs are quite consistent, the heights of the peaks will be approximately the same. Renewal density (RD) histograms can be used to determine the serial dependency in spike trains. One may think of a renewal process as if there is a clock whose countdown time is fixed and that is constantly reset by the occurrence of a spike. If the ISIs are shuffled or randomized, thus destroying their serial order, nonrenewal processes will have quite different ED and RD histograms. If, on the other hand, the occurrence of spikes takes on the properties of a renewal process, there may be little difference in the ED and RD histograms. This comparison can be very important since it has been argued that, in the case of externally driven activity, a peak in the RD histogram is smaller than the corresponding peak in the ED histogram.1'7 However, neurons that are entrained to peripheral stimuli, that are themselves periodic, often exhibit little difference when ISIs are shuffled to construct RD histograms.39 Histogram plots of ED and RD have a disadvantage since they represent large intervals and thus do not give much information about instantaneous changes in firing pattern consistency.

Returning to basic ISI plots, if, instead of simply plotting the first ISI on the ordinate for any given point on the abscissa, one plots all ISIs subsequent to a given point, for the extent of the range of the axis, a sort of running autocorrelogram is constructed. When these plots are examined closely, neuronal activity across several trials is compressed into a single representation. Rhythmic or entrained activity is represented as a series of horizontal bands. The width of these gives an indication of the degree of entrainment; the narrower the band, the better the entrainment. The consistency of the entrainment, in part, is indicated by the number of horizontal bands apparent in the plot. The number of bands is dependent on the extent of the axis and the frequency to which the spike train is entrained. This form of data representation, however, does little to convey the strength of the entrainment other than by the width of the bands. Figure 10.4C presents an example of this type of plot for the cortical neuron that has been used to illustrate the other types of data representation. A sinusoid which set the manipulandum in motion at time zero caused periodic spike occurrences in virtually all trials, as seen in Figure 10.4B. The multiple horizontal bands result from the security with which the stimulus drives the neural activity. Despite the continuing presence of the stimulus, the periodic response pattern is disrupted at about 325-500 ms after stimulus onset. This is about the time at which the monkey initiated a hand movement in response to the stimulus. The disruption in the temporal properties of the response pattern was probably due to receptive field surface shear as the movement began. Despite this, remnants of the temporal pattern are still visible in Panel C until about 500-650 ms post-stimulus. At this point, on average, the stimulus was turned off.

One way to achieve a representation of entrainment strength is to bin the data and encode the count in each bin using either gray or color scales. When this is done the resultant plot, in essence, retains those features of a running autocorrelo-gram described above, but also adds some information about the relative strength of the correlation between the driving stimulus and the spike trains associated with it. As such, these plots contribute additional information over and above the traditional expectation density histograms. To our knowledge, we are the only ones to have employed joint expectation density histograms of a spike train plotted against themselves to show preservation of rhythmic activity but alteration of intrinsic frequency during the course of a behavioral trial.44 Extrapolating from this, it is not hard to imagine shuffling the ISIs as in a renewal density histogram (RD; References 1,7, and 41-42) to determine if the rhythmicity observed in the ED is preserved in the RD. If so, the driving force is probably intrinsic or extremely secure and extrinsic. If the periodicity breaks down with shuffling, the driving force is most likely from an extrinsic source (see Reference 39 for conceptual review).

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