A measure of local variation of inter-spike intervals
Introduction
A purpose of neurophysiological study is to discover systematic changes in neuronal activity correlated to the animal's behavior. The site that contains a number of neurons that exhibit significant changes in activity in some particular behavioral context is regarded as the functional area. Though such a phenomenological understanding is interesting by itself, it is desirable to grasp the neuronal activity in the context of causal relation between anatomical circuitry and physiological function. For the purpose of obtaining such information with regard to circuitry, the information that identifies the type and location of the neuron would be useful.
Recorded data have some aspects useful for obtaining such information. It is conceived that a typical firing rate of an interneuron is higher than that of a pyramidal neuron (Ranck, 1973, Buzsaki et al., 1983). It is also conceived that there is a significant difference between the action potential waveforms of interneurons and pyramidal neurons (Csicsvari et al., 1998, Constantinidis et al., 2002, Takahashi et al., 2003).
In addition to those cues that distinguish between pyramidal neurons and interneurons, we have suggested that an analysis of the sequence of inter-spike intervals (ISIs) helps to classify the pyramidal neurons into distinct groups. In our previous study, we characterized the spike sequences with the estimated values of the coefficient of variation, CV, the skewness coefficient, and the correlation coefficient of consecutive intervals, and found that their distributions depend strongly on the recording site (Shinomoto et al., 2002a, Shinomoto et al., 2002b). In a more recent study, we introduced a measure of local variation of inter-spike intervals, LV, which characterizes the spiking randomness, and found that a distribution of LV values for neurons in the medial frontal cortex is distinctly bimodal (Shinomoto et al., 2003). Two groups of neurons classified according to the values of LV were found to exhibit different responses to the same stimulus. This fact suggests that neurons in the same area can be classified into different groups possessing unique spiking characteristics and corresponding functional properties.
In the present paper, we obtain values of the local variation LV and the coefficient of variation CV for a variety of mathematical point processes. First, we examine the doubly stochastic Poisson process to see how the values of LV and CV undergo changes by the rate fluctuation of the Poisson process. It is found that for the doubly stochastic Poisson process, the value of the local variation LV stays near unity even if the rate fluctuation is comparable to the mean rate, while the value of the coefficient of variation CV largely deviates from unity according to such rate fluctuation. Second, we obtain the range of the values of LV and CV for the Ornstein–Uhlenbeck process, which represents the conditions that the leaky integrate-and-fire neuron is subjected to independent inputs of many other neurons. Third, the values of LV and CV are obtained for the gamma process analytically, and for the rate modulated gamma process numerically. It is also confirmed that for the rate modulated gamma process, the value of LV does not undergo large changes according to the rate fluctuation, but are mainly determined by the form of intrinsic ISI distribution parameterized by the order of a gamma distribution, while the value of CV undergoes large changes by the rate fluctuation.
Section snippets
CV and LV
From a sequence of consecutive inter-spike intervals, {T1, T2, …, Tn}, we first estimate the coefficient of variation (see Cox and Lewis, 1966). An asymptotically unbiased estimator, which gives a theoretical value of the coefficient of variation in the limit of n→∞ could bewhere ΔT is the sample standard deviation and is the sample mean interval. Throughout this paper, we will refer to the above estimator as CV. For a series of intervals, which are independently
Values of CV and LV for model point processes
As has been shown in the preceding section, with LV we were capable of categorizing neurons into two groups, while with CV we were not. In order to have a grasp of the characteristics of these measures, we wish to obtain their values for a variety of mathematical point processes.
Discussion
In the present paper, we have obtained values of the measure of local variation LV and the conventional coefficient of variation CV for typical mathematical point processes.
First, we examined the doubly stochastic Poisson process, and found that LV stays near unity even if the amplitude of rate fluctuation is comparable to the mean rate, while CV deviates very largely according to such rate fluctuation. Second, we examined the Ornstein–Uhlenbeck process, which represents the conditions that the
Acknowledgments
We are grateful to Keisetsu Shima and Jun Tanji for providing the precious experimental data, which were used in preparing Fig. 1. Thanks are also due to an anonymous referee for the highly suggestive and constructive comments, with which the original manuscript was significantly improved by removing misleading parts. This study is supported in part by Grants-in-Aid for Scientific Research (No. 12680382 and No. 1516059) to S.S. by the Ministry of Education, Culture, Sports, Science and
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