Gamma-distributed Intervals Research Articles

Synchronization of uncoupled oscillators by common gamma impulses: From phase locking to noise-induced synchronization

Nonlinear oscillators can mutually synchronize when they are driven by common external impulses. Two important scenarios are (i) synchronization resulting from phase locking of each oscillator to regular periodic impulses and (ii) noise-induced synchronization caused by the Poisson random impulses, but their difference has not been fully quantified. Here, we analyze a pair of uncoupled oscillators subject to common random impulses with gamma-distributed intervals, which can be smoothly interpolated between the regular periodic and the random Poisson impulses. Their dynamics are characterized by phase distributions, frequency detuning, Lyapunov exponents, and information-theoretic measures, which clearly reveal the differences between the two synchronization scenarios.

Spontaneous Activity of Auditory Nerve Fibers in the Barn Owl (Tyto alba): Analyses of Interspike Interval Distributions

In vertebrate auditory systems, the conversion from graded receptor potentials across the hair-cell membrane into stochastic spike trains of the auditory nerve (AN) fibers is performed by ribbon synapses. The statistics underlying this process constrain auditory coding but are not precisely known. Here, we examine the distributions of interspike intervals (ISIs) from spontaneous activity of AN fibers of the barn owl (Tyto alba), a nocturnal avian predator whose auditory system is specialized for precise temporal coding. The spontaneous activity of AN fibers, with the exception of those showing preferred intervals, is commonly thought to result from excitatory events generated by a homogeneous Poisson point process, which lead to spikes unless the fiber is refractory. We show that the ISI distributions in the owl are better explained as resulting from the action of a brief refractory period ( approximately 0.5 ms) on excitatory events generated by a homogeneous stochastic process where the distribution of interevent intervals is a mixture of an exponential and a gamma distribution with shape factor 2, both with the same scaling parameter. The same model was previously shown to apply to AN fibers in the cat. However, the mean proportions of exponentially versus gamma-distributed intervals in the mixture were different for cat and owl. Furthermore, those proportions were constant across fibers in the cat, whereas they covaried with mean spontaneous rate and with characteristic frequency in the owl. We hypothesize that in birds, unlike in mammals, more than one ribbon may provide excitation to most fibers, accounting for the different proportions, and that variation in the number of ribbons may underlie the variation in the proportions.

Interactions between brainstem and trigeminal neurons detected by cross-spectral analysis

Cells in the nucleus raphe magnus that are inhibited by noxious skin stimuli (off-cells) have been postulated to suppress pain by continuously inhibiting spinal and trigeminal nociceptive neurons. To test this hypothesis, spontaneous activity was simultaneously recorded from off-cells ( n=15) and wide-dynamic range cells ( n=27) of the trigeminal complex (subnucleus interpolaris), in rats anesthetized with pentobarbital. Most off-cells ( n=14) had rhythmic interspike intervals, their modes averaging 106 ms. No trigeminal cell fired rhythmically. Rhythmic firing was defined quantitatively: the autospectrum's peak power had to exceed 1.75 times its asymptote. This formula was obtained by generalizing from a natural cut-off in the theoretical autospectrum for serially uncorrelated, gamma-distributed intervals, whose firing can be varied from Poissonian to highly regular by adjusting one parameter. It encompasses the qualitative judgement of autocorrelograms commonly made in neurophysiology. Cross-correlograms ( n=29) appeared noisy and otherwise featureless. However, their power spectra (cross-periodograms) sometimes showed significant peaks, compared with simulated non-interactive distributions. The latter were generated by interchanging the raphe interval sequences at one random point (as in cutting a deck of cards), thus retaining most of their serial correlation. Of 29 cross-periodograms, 21 were significant at 1 to 13 frequencies (100 points, 0.4 to 39 Hz). These frequencies were often near the peak raphe power, and sometimes near its harmonics too. Furthermore, cross-spectral phase angles at peak power were non-uniform, most falling between 0 and 180 degrees (unit vector sum 60°, n=20). To understand why the frequency domain gave better detection, cross-spectra and cross-correlations were modeled theoretically by convolving idealized input autocorrelations and synaptic response functions. This demonstrated that rhythmic firing is insufficient for better frequency-domain detection, and that serially correlated input intervals or non-additive synaptic responses are necessary. The conclusion was confirmed by stochastic simulation of a simple non-additive synapse, that required successful input spikes to fall within a specified interval of the preceding spike. Experimentally, serial correlation was found in 12 of the 15 raphe cells, and in 20 of the 27 trigeminal cells. It is proposed that the weak experimental cross-correlograms arise because many asynchronous raphe inputs converge on each trigeminal cell, possibly to optimize the resting suppression of pain. The distribution of cross-spectral phase angles at peak raphe power suggested that raphe spikes arriving at the synapses' preferred interval cause a fall in trigeminal activity. In general, cross-spectral analysis can sometimes uncover influences hidden in cross-correlograms, but the firing of one neuron must be rhythmic and non-renewal, or else certain input intervals must be favored in synaptic transmission.

Steady discharges of macaque retinal ganglion cells.

Steady discharges were collected from ganglion cells of the magnocellular (MC) and parvocellular (PC) pathways of the macaque while their receptive fields were uniformly illuminated with a 4.7-deg steady yellow light of photopic illuminance. The mean rates, coefficients of variation, interval distributions, serial correlation coefficients, and power spectra of these discharges were determined. The results presented permit one to estimate the noise power in the discharges of macaque ganglion cells and hence determine how visual signals of different amplitudes will be affected by the noise resident in their discharges. Although there was some small serial correlation in the discharges of both MC- and PC-pathway cells, their discharges can be considered to result from renewal processes with reasonable accuracy. As with the discharges of cat ganglion cells, macaque ganglion cell discharges can be considered to have approximately gamma-distributed intervals. Steady discharges of MC- and PC-pathway cells show considerable overlap in their statistics, although small but significant differences are present.

Steady discharges of X and Y retinal ganglion cells of cat under photopic illuminance.

The discharges of ON- and OFF-center X and Y retinal ganglion cells in the presence of stationary patterns or of a uniform field of photopic luminance were recorded from urethane-anesthetized adult cats. The interval statistics and power spectra of these discharges were determined from these discharge records. The patterned stimuli were selected and positioned with respect to a cell's receptive field so as to generate steady discharges that were different in mean discharge rate from that cell's discharge for the diffuse field. The interval statistics of discharges recorded for diffuse or patterned illumination for all cell types can be modeled, approximately, as coming from renewal processes with gamma-distributed intervals. The gamma order of the interval distributions was found to be nearly proportional to the mean discharge rate for X cells, but not for Y cells. Typical values for the gamma orders and their dependence on mean rate for different cell types are given. The same model of a renewal process with gamma-distributed intervals is used to model the measured power spectra and performs well. When the gamma order is proportional to mean rate, the power spectral density at low temporal frequencies is independent of discharge rate. Gamma order was proportional to mean rate for X cells but not for Y cells. Nonetheless, the power spectral densities of both cell types at low frequencies were approximately independent of discharge rate. Hence, noise in this band of frequencies can be considered additive. The consequences of departures from the renewal process and of the gamma order not being proportional to mean rate are considered. The significance of different rates of discharge for signaling is discussed.

Nature of the maintained discharge of Q, X, and Y retinal ganglion cells of the cat.

Cat retinal ganglion cells with center-surround receptive fields have an irregular discharge whose rate is altered by visual stimulation. In assessing the detectability of stimulus-induced changes in the discharge, a consideration of the power spectral density of the discharge is helpful. The power spectral density of Q, X, and Y cells is flat at low frequencies, rises to a peak at the mean frequency of firing, and then decays away at higher frequencies in an oscillatory manner to an asymptotic level equal to the mean rate of discharge. Measured spectra correspond closely with spectra predicted by a renewal-point process with gamma-distributed intervals. When the rate of the discharge is altered by visual stimulation, the spectral density at low frequencies remains roughly constant. Assuming that it is the noise power at these frequencies that is effective in limiting the detectability of visual stimuli, it appears that at the retinal level the irregularity of the discharge can be treated as an additive noise.

Some results on tests for Poisson processes

SUMMARY It is shown that a test proposed by Barnard for Poisson processes, using certain distributionfree statistics, is not consistent against renewal alternatives. However, empirical evidence is given which suggests that a modification of this test due to Durbin results in relatively powerful tests of the Poisson hypothesis. A simple test based on Durbin's modification is described, and its asymptotic relative efficiency with respect to the asymptotically most powerful test against the alternative of a renewal process with gamma-distributed intervals is given. A central problem in the statistical analysis of stationary series of events (point processes) is to test whether an observed series is a realization of a Poisson process. Denote by A the rate parameter of the Poisson process. The test is for a composite null hypothesis, A being a nuisance parameter. In a Poisson process the numbers of events in non-overlapping intervals are independent Poisson variates, and moreover the intervals between events are independent random variables with the simple exponential distribution. Consequently tests of the Poisson hypothesis can be devised for various alternative hypotheses. Epstein (1960) has surveyed these tests. We will be concerned here with tests which are useful against rather vaguely specified alternatives. There are many such tests and we will try to assess the relative utility of the most important ones. The main source of these tests is an idea of Barnard's (1953). Assume first of all that observation of the series is for a fixed time period (0, t) and that n events occur at times t1, t2..., t., measured from the origin. The idea is to test, conditionally on the number of events in (0, t), N, being equal to n, that the variables Ui = Tilt (i = 1, 2,..., n), are independent variates with distribution 0 (u < 0),