Serial dependencies and markov properties of neuronal interspike intervals from rat cerebellum

Using three different approaches, each with different theoretical assumptions, we showed that mammalian neuronal spike trains contain serial ordering. We demonstrated that: (1) when intervals are categorized according to whether their durations are short, medium, or long, sequential groupings of adjacent interval categories exhibit Markov dependencies, extending to at least the 4th order; (2) the observed incidence of specified patterns of these groups of adjacent interval categories differs from the independent case, based on Chi square goodness-of-fit tests, and by using similar procedures; (3) there is divergence from independence when adjacent interval patterns are described in terms of relative lengths of adjacent intervals. The statistical indicators of serial dependence were significantly greater when applied to the original data than when applied to the same data after shuffling. Each of these approaches leads to the notion that “information” is carried in clusters of adjacent intervals (“bytes” or “words”) and moreover, we can identify which specific patterns of interspike intervals contribute most to the statistical significance (i.e., those clusters that are potential candidates for “information carriers”). In most of the ten neurons, the “memory” of the system appears to be at least 36–45 msecs.     

Assessment of spike activity in the supraoptic nucleus

Novel approaches to the characterization of coding carried by spike trains are discussed. Measuring firing frequency alone may only partially reflect spike patterning, and can only quantify changes of the most obvious kind. We have devised a method that combines probabilistic and information approaches to quantify the variability of the interspike intervals in a way that is independent of spike frequency. To illustrate the technique, the firing of an oxytocin cell and a vasopressin cell were compared before and after osmotic stimulation. A bimodal lognormal function was fitted to the interspike interval histograms. The entropy of the log interval histogram was used to measure the variability of intervals and to reflect the coding capacity of the cell per spike. A perfect metronome shows no variability in interval and thus has no greater coding capacity than is conveyed by its frequency, whereas the variability of intervals of magnocellular neurones means that their irregular activity has greater potential for coding. While the mean spike frequency increased in both the oxytocin and vasopressin cells in response to osmotic stimulation, the changes in their irregularity showed differences. Osmotic stimulation reduced the entropy of the oxytocin cell, reflecting an increase in the regularity of its spike activity. Conversely, osmotic stimulation had little effect on the entropy of the vasopressin cell. Such differences are not evident from a simple inspection of ratemeter activity. The comparison highlights the limitations of mean spike frequency as a measure of spike coding. Parameters based on the interspike intervals constitute informative measures of spike activity that allow objective comparisons to be made between the activity under different physiological conditions.     

Stochastic description of complex and simple spike firing in cerebellar Purkinje cells

Cerebellar Purkinje cells generate two distinct types of spikes, complex and simple spikes, both of which have conventionally been considered to be highly irregular, suggestive of certain types of stochastic processes as underlying mechanisms. Interestingly, however, the interspike interval structures of complex spikes have not been carefully studied so far. We showed in a previous study that simple spike trains are actually composed of regular patterns and single interspike intervals, a mixture that could not be explained by a simple rate-modulated Poisson process. In the present study, we systematically investigated the interspike interval structures of separated complex and simple spike trains recorded in anaesthetized rats, and derived an appropriate stochastic model. We found that: (i) complex spike trains do not exhibit any serial correlations, so they can effectively be generated by a renewal process, (ii) the distribution of intervals between complex spikes exhibits two narrow bands, possibly caused by two oscillatory bands (0.5-1 and 4-8 Hz) in the input to Purkinje cells and (iii) the regularity of regular patterns and single interspike intervals in simple spike trains can be represented by gamma processes of orders, which themselves are drawn from gamma distributions, suggesting that multiple sources modulate the regularity of simple spike trains.     

Testing a neural coding hypothesis using surrogate data

Determining how a particular neuron, or population of neurons, encodes information in their spike trains is not a trivial problem, because multiple coding schemes exist and are not necessarily mutually exclusive. Coding schemes generally fall into one of two broad categories, which we refer to as rate and temporal coding. In rate coding schemes, information is encoded in the variations of the average firing rate of the spike train. In contrast, in temporal coding schemes, information is encoded in the specific timing of the individual spikes that comprise the train. Here, we describe a method for testing the presence of temporal encoding of information. Suppose that a set of original spike trains is given. First, surrogate spike trains are generated by randomizing each of the original spike trains subject to the following constraints: the local average firing rate is approximately preserved, while the overall average firing rate and the distribution of primary interspike intervals are perfectly preserved. These constraints ensure that any rate coding of information present in the original spike trains is preserved in the members of the surrogate population. The null-hypothesis is rejected when additional information is found to be present in the original spike trains, implying that temporal coding is present. The method is validated using artificial data, and then demonstrated using real neuronal data.     

The temporal structure of spike trains in the primate basal ganglia: afferent regulation of bursting demonstrated with precentral cerebral cortical ablation

We studied the temporal pattern of discharge of single units in the basal ganglia of awake primates sitting quietly. Bursting was studied with a procedure that identified individual bursts in a spike train, quantifying burst intensity (surprise), bursts per 1000 spikes, spikes per burst and burst length. Autocorrelation techniques were used to assess the dependencies of spike trains on the temporal order of intervals. Striatal units had a greater tendency to burst (79% of units) than pallidal units (50%). The caudate nucleus and putamen had nearly identical burst properties on all measures. In the pallidum, bursting was more prevalent in the external segment and bursts were more intense and more frequent than in the internal segment. The autocorrelation analysis revealed that the temporal structure of the spike train was more dependent on the order of intervals in the striatum than in the pallidum. Bursting units had an increased probability of discharge after each spike and the relative refractory period was shorter in bursting units than units without bursts. Very few units exhibited cyclic discharge properties. Ablations of areas 4 and 6 in the precentral cortex demonstrated that striatal bursting was under afferent control. The putamen, which receives more cortical afferents from areas 4 and 6 than the caudate nucleus, had fewer and less intense bursts after the afferents were lesioned. Burst intensity did not change in the pallidum after the lesion. The findings indicate that bursting properties contribute to discharge variability in the basal ganglia and suggest that information transfer in the striatum may utilize bursts. In contrast, rate coding may be a more important mechanism for units in the pallidum.     

Do neurons process information by relative intervals in spike trains?

We suggest the possibility that neurons process information in terms of the relative duration of clusters of adjacent and successive inter-action potential intervals (“bytes” of intervals). If this concept is plausible, as is supported by research from several laboratories which have specifically addressed this posibility, one should be able to see evidence for such patterning in the published illustrations from studies in which this concept was not considered. We present some of this evidence here, along with some illustrations from the original publications. Byte patterns are evident in these examples, even though they often went unrecognized by authors and readers alike. It is true that interval patterns are not obvious in all published illustrations of spike trains, and we suggest that this can be explained by one or more of the following: (1) some neurons may operate with an interval-pattern code while others do not, (2) a given neuron may use an interval-pattern code only under certain conditions, and (3) even when such a code exists, it may be difficult to detect for identifiable technical reasons. Therefore, we believe that the relative-interval-pattern concept is a valid scientific hypothesis which merits specific testing of its validity and range of applicability.     

Properties of favored patterns in spontaneous spike trains and responses of favored patterns to electroacupuncture in evoked trains.

By using 'the modified detection method', our previous study has shown that all spontaneous spike trains recorded from several areas of brain and spinal cord have favored patterns (FPs). The present study further shows that: (1) all newly detected spike trains from substantia nigra zona compacta, nucleus periventricularis hypothalami and nucleus hypothalamicus posterior also have FPs, and some spike trains from neurons in the same nucleus have a common favored pattern (CF, i.e. they share the same FP), indicating that FP and CF in spike trains are common phenomena; (2) all serial correlation coefficients of FP repetitions (in serial order) in different spike trains detected are less than 0.3 (close to 0), revealing that the repetition of FPs is a renewal process; (3) in different periods of the spike trains evoked by electroacupuncture (EA), the number of different FPs and the number of repetitions of the same representative FP either increase or decrease along with the change of firing rate. The tendencies of these changes are very similar, but after EA the repetitions of different FPs in the same spike trains change differently, showing that different (hidden) responses exist at the same time. The above results suggest that the FPs in spike trains may represent various neural codes, and 'the modified detection method of FP' can pick up more information from spike trains than the firing rate analysis, hence it is a very useful tool for the study of neural coding.     

A pattern grouping algorithm for analysis of spatiotemporal patterns in neuronal spike trains. 1. Detection of repeated patterns

The existence of precise temporal relations in sequences of spike intervals, referred to as 'spatiotemporal patterns', is suggested by brain theories that emphasize the role of temporal coding. Specific analytical methods able to assess the significance of such patterned activity are extremely important to establish its function for information processing in the brain. This study proposes a new method called 'pattern grouping algorithm' (PGA), designed to identify and evaluate the statistical significance of patterns which differ from each other by a defined and small jitter in spike timing of the order of few ms. The algorithm performs a pre-selection of template patterns with a fast computational approach, optimizes the jitter for each spike in the template and evaluates the statistical significance of the pattern group using three complementary statistical approaches. Simulated data sets characterized by various types of known non stationarities are used for validation of PGA and for comparison of its performance to other methods. Applications of PGA to experimental data sets of simultaneously recorded spike trains are described in a companion paper (Tetko IV, Villa AEP. A pattern grouping algorithm for analysis of spatiotemporal patterns in neuronal spike trains. 2. Application to simultaneous single unit recordings. J Neurosci Method 2000; accompanying article).     

Neuronal coding and spiking randomness

Fast information transfer in neuronal systems rests on series of action potentials, the spike trains, conducted along axons. Methods that compare spike trains are crucial for characterizing different neuronal coding schemes. In this paper we review recent results on the notion of spiking randomness, and discuss its properties with respect to the rate and temporal coding schemes. This method is compared with other widely used characteristics of spiking activity, namely the variability of interspike intervals, and it is shown that randomness and variability provide two distinct views. We demonstrate that estimation of spiking randomness from simulated and experimental data is capable of capturing characteristics that would otherwise be difficult to obtain with conventional methods.     

Do neurons process information by relative intervals in spike trains?

We suggest the possibility that neurons process information in terms of the relative duration of clusters of adjacent and successive inter-action potential intervals (“bytes” of intervals). If this concept is plausible, as is supported by research from several laboratories which have specifically addressed this posibility, one should be able to see evidence for such patterning in the published illustrations from studies in which this concept was not considered. We present some of this evidence here, along with some illustrations from the original publications. Byte patterns are evident in these examples, even though they often went unrecognized by authors and readers alike. It is true that interval patterns are not obvious in all published illustrations of spike trains, and we suggest that this can be explained by one or more of the following: (1) some neurons may operate with an interval-pattern code while others do not, (2) a given neuron may use an interval-pattern code only under certain conditions, and (3) even when such a code exists, it may be difficult to detect for identifiable technical reasons. Therefore, we believe that the relative-interval-pattern concept is a valid scientific hypothesis which merits specific testing of its validity and range of applicability.     

A nonparametric approach for detection of bursts in spike trains

In spike-train data, bursts are considered as a unit of neural information and are of potential interest in studies of responses to any sensory stimulus. Consequently, burst detection appears to be a critical problem for which the Poisson-surprise (PS) method has been widely used for 20 years. However, this method has faced some recurrent criticism about the underlying assumptions regarding the interspike interval (ISI) distributions. In this paper, we avoid such assumptions by using a nonparametric approach for burst detection based on the ranks of ISI in the entire spike train. Similar to the PS statistic, a “Rank surprise” (RS) statistic is extracted. A new algorithm performing an exhaustive search of bursts in the spike trains is also presented. Compared to the performances of the PS method on realizations of gamma renewal processes and spike trains recorded in cat auditory cortex, we show that the RS method is very robust for any type of ISI distribution and is based on an elementary formalization of the definition of a burst. It presents an alternative to the PS method for non-Poisson spike trains and is simple to implement.     

Slow rhythmic activity of caudate neurons in the cat: Statistical analysis of caudate neuronal spike trains

Spike trains of caudate neurons initially having mean interspike intervals of less than 4 ms were analyzed with progressive administration of pentobarbital (5 to 20 mg/kg). Among the neurons investigated, 77% (N = 79) showed evidence of a rhythmic basis of their activity in first-order interspike interval histograms and/or autocorrelation histograms in the course of becoming silent due to progressive administration of pentobarbital. Although the rhythmicies of given units varied depending on the level of anesthesia the most prominent cycle was almost always within the range of 200 to 320 ms; the majority were not discernable on visual inspection of the spike trains. Cortical stimuli reset the cycle. Cross-correlation histograms constructed from pairs of caudate neurons provided some evidence that their spontaneous firing was mutually inhibited. The possibility that the rhythmicities might arise from such mutual inhibition of spontaneously firing caudate neurons is discussed.     

Measuring spike train synchrony

Estimating the degree of synchrony or reliability between two or more spike trains is a frequent task in both experimental and computational neuroscience. In recent years, many different methods have been proposed that typically compare the timing of spikes on a certain time scale to be optimized by the analyst. Here, we propose the ISI-distance, a simple complementary approach that extracts information from the interspike intervals by evaluating the ratio of the instantaneous firing rates. The method is parameter free, time scale independent and easy to visualize as illustrated by an application to real neuronal spike trains obtained in vitro from rat slices. In a comparison with existing approaches on spike trains extracted from a simulated Hindemarsh-Rose network, the ISI-distance performs as well as the best time-scale-optimized measure based on spike timing.     

Multiple spike-train analysis using mutual interval matrix

The principal-component approach is applied to the analysis of sequences of neuronal action potentials (spike trains). Multiple spike trains are represented as a sequence of vectors of mutual interspike intervals and are considered to be part of the trajectory of a dynamic system. The trajectory matrix is decomposed into a number of ‘basic spike patterns’ and their relative magnitudes by singular-value decomposition. The representation provides a convenient framework for analysis of dynamic relations and cooperation between neurons in an observed network. Examples of applications to simulated and cerebellar data are presented.     

Comparison of the recruitment and discharge properties of motor units in human brachial biceps and adductor pollicis during isometric contractions

Experiments were designed to assess the relative contribution of rate coding and motor unit recruitment to force production in two muscles of different fiber composition and function. Single motor unit action potentials were recorded during steady isometric contraction in biceps brachii, a large proximal limb muscle of mixed fiber composition, and adductor pollicis, a small hand muscle comprised mainly of type I muscle fibers. Action potential spike trains were obtained over the entire force range in each muscle. The results suggest that these two muscles are controlled in different ways. In biceps brachii, recruitment was observed from 0 to 88% maximum voluntary contraction (MVC). In adductor pollicis, no motor unit was observed to be recruited at forces greater than 50% MVC, with the majority of recruitment occuring below 30% MVC. On the average, motor units in adductor pollicis discharged at higher rates, with less regularity, and with a greater frequency of occurrence of short interspike intervals (intervals ≤ 20 msec) than those in biceps brachii. Such findings suggest that rate coding plays a more prominent role in force modulation in adductor pollicis, while recruitment plays a more important role throughout the contractile force range in biceps brachii.     

Favored patterns in spontaneous spike trains.

By using the modified detection method, favored patterns can be detected in a total of 44 spontaneous spike trains. Among these the 'periodical burst' discharge of one sympathetic preganglionic neuron and the 'fast-slow' alternative discharge of some hypothalamic neurons have visible characteristics, hence we use them to test the reliability of our method by comparing the detected patterns with the non-sequential interval histograms and oscillograms of the spike trains. The comparisons show that our method is reliable. The spike trains of nucleus raphe magnus (NRM) and the locus coeruleus (LC) have no visible characteristics; from these the following results have been observed: (1) all spike trains have one or more favored patterns; (2) some spike trains from neurons in the same nucleus have common fragments of favored patterns; (3) the favored patterns in spike trains recorded from different nuclei are different from each other; (4) some favored patterns in spike trains of the NRM neurons remain unchanged from beginning to end in 35-min records and their repetitions are relatively stable; and (5) microinjection of normal saline or normal serum into the LC has no significant influence on the occurrence of favored patterns in 35-min records of spike trains of the LC neurons. The above results indicate that the favored patterns in spike trains are objective and regular phenomena with relative stability. It seems likely that favored pattern may be used (as an index of the neuronal activity) in combination with the microinjection technique, etc., for various studies including studies on neural coding.     

Editing trains of action potentials from multi-electrode arrays

When recording from multi-electrode arrays, only a short period around the time of a threshold crossing is generally saved for later analysis. Then, waveforms are often sorted automatically to identify templates of spikes from individual neurons near an electrode. As spikes sum from different neurons and noise is present, some spikes may be missed and others erroneously accepted. This paper describes methods for identifying and correcting errors in recorded spike trains to recover the pattern of spikes from each neuron as faithfully as possible. These methods are complementary to, but distinct from methods to reconstruct waveforms that arise from summation of individual templates that overlap one another. Our methods are based on the local statistics of the firing rates or inter-spike intervals and the methods work best for neurons that fire regularly (small standard deviation relative to the mean interval). First, we test whether accepting more spikes, whose waveforms are close to the templates that have been identified, will increase the regularity or smoothness of the firing rates. Then, after accepting spikes that increase regularity, we test whether individual intervals are sufficiently longer (or shorter) than their neighbors to identify spikes that have been omitted (or accepted) erroneously. The methods are tested on simulated spike trains, where spikes have been inserted or deleted at random, and on spike trains recorded from multi-electrode arrays in dorsal root ganglia of cats walking on a treadmill.