Mathematical analysis of the possibility of detecting correlation between spike trains of weakly interacting neurons

The number of spikes which must be recorded in order to detect significant correlation between spike trains of two synaptically connected neurons was estimated by a mathematical model. Dependence of this number of spikes on importance of interneuronal connection (measured as the amplitude of the EPSP evoked by a single spike of the input neuron in the output cell) and on the intensity of total spontaneous excitatory influences on the output neuron and on its own parameters was studied. For cells which corresponded in the weight of connections between them, their intrinsic parameters, and characteristics of spontaneous activity to real spinal neurons, the necessary number of spikes was 10⁷–10⁸. An increase in amplitude of the single EPSP and also a decrease in the intensity of the input spontaneous spike train and parameters of after-hyperpolarization of the postsynaptic neuron led to a decrease in the number of spikes necessary for the detection of significant correlation. On the basis of the results of this and previous investigations the possible principles for construction of a spinal locomotor generator are discussed.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 12, No. 3, pp. 290–296, May–June, 1980.     

Transmission of spike trains by nerve fibers after near-threshold stimulation

Spike trains in group A nerve fibers were studied in anesthetized cats in response to stimulation of the hind-limb nerves by random (Poisson) and regular sequences of stimuli. In response to above-threshold stimulation spike trains in the nerve fibers were shown to differ from the stimulating trains purely in the absence of intervals less than 1–1.5 msec in duration, as a result of the presence of a refractory period. With near-threshold stimulation with an average frequency of over 10 per second, spike trains differed significantly from the stimulating trains, as reflected in histograms of interspike intervals, the shape of the intensity function, and the magnitude of the coefficient of correlation for successive intervals. It is postulated that changes in the structure of the spike trains conveyed by a nerve fiber are attributable to the presence of after-activity.A. A. Bogomolets Institute of Physiology, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 8, No. 1, pp. 91–98, January–February, 1976.     

Neuronal spike train analysis in likelihood space

Background

Conventional methods for spike train analysis are predominantly based on the rate function. Additionally, many experiments have utilized a temporal coding mechanism. Several techniques have been used for analyzing these two sources of information separately, but using both sources in a single framework remains a challenging problem. Here, an innovative technique is proposed for spike train analysis that considers both rate and temporal information.

Methodology/Principal Findings

Point process modeling approach is used to estimate the stimulus conditional distribution, based on observation of repeated trials. The extended Kalman filter is applied for estimation of the parameters in a parametric model. The marked point process strategy is used in order to extend this model from a single neuron to an entire neuronal population. Each spike train is transformed into a binary vector and then projected from the observation space onto the likelihood space. This projection generates a newly structured space that integrates temporal and rate information, thus improving performance of distribution-based classifiers. In this space, the stimulus-specific information is used as a distance metric between two stimuli. To illustrate the advantages of the proposed technique, spiking activity of inferior temporal cortex neurons in the macaque monkey are analyzed in both the observation and likelihood spaces. Based on goodness-of-fit, performance of the estimation method is demonstrated and the results are subsequently compared with the firing rate-based framework.

Conclusions/Significance

From both rate and temporal information integration and improvement in the neural discrimination of stimuli, it may be concluded that the likelihood space generates a more accurate representation of stimulus space. Further, an understanding of the neuronal mechanism devoted to visual object categorization may be addressed in this framework as well.     

Spectral analysis of input spike trains by spike-timing-dependent plasticity

Spike-timing-dependent plasticity (STDP) has been observed in many brain areas such as sensory cortices, where it is hypothesized to structure synaptic connections between neurons. Previous studies have demonstrated how STDP can capture spiking information at short timescales using specific input configurations, such as coincident spiking, spike patterns and oscillatory spike trains. However, the corresponding computation in the case of arbitrary input signals is still unclear. This paper provides an overarching picture of the algorithm inherent to STDP, tying together many previous results for commonly used models of pairwise STDP. For a single neuron with plastic excitatory synapses, we show how STDP performs a spectral analysis on the temporal cross-correlograms between its afferent spike trains. The postsynaptic responses and STDP learning window determine kernel functions that specify how the neuron "sees" the input correlations. We thus denote this unsupervised learning scheme as 'kernel spectral component analysis' (kSCA). In particular, the whole input correlation structure must be considered since all plastic synapses compete with each other. We find that kSCA is enhanced when weight-dependent STDP induces gradual synaptic competition. For a spiking neuron with a "linear" response and pairwise STDP alone, we find that kSCA resembles principal component analysis (PCA). However, plain STDP does not isolate correlation sources in general, e.g., when they are mixed among the input spike trains. In other words, it does not perform independent component analysis (ICA). Tuning the neuron to a single correlation source can be achieved when STDP is paired with a homeostatic mechanism that reinforces the competition between synaptic inputs. Our results suggest that neuronal networks equipped with STDP can process signals encoded in the transient spiking activity at the timescales of tens of milliseconds for usual STDP.     

Sequential interval histogram analysis of non-stationary neuronal spike trains

The spike interval histogram, a commonly used tool for the analysis of neuronal spike trains, is evaluated as a statistical estimator of the probability density function (pdf) ofinterspike intervals. Using a mean square error criterion, it is concluded that a Parzen convolution estimate of the pdf is superior to the conventional histogram procedure. The Parzen estimate using a Gaussian weighting function reduces the number of intervals required to achieve a given error by a factor of 5–10. The Parzen estimation procedure has been implemented in the sequential interval histogram (SQIH) procedure for analysis of non-stationary spike trains. Segments of the spike train are defined using a moving window and the pdf for each segment is estimated sequentially. The procedure which we have found most practical is interactive with the user and utlizes the theoretical results of the error analysis as guidelines for the evolution of an estimation strategy. The SQIH procedure appears useful both as a criterion for stationarity and as a means to characterize non-stationary activity.Portions of this work were presented at the Symposium on Computer Technology in Neuroscience Research, West Virginia University Medical Center, Morgantown, West Virginia, USA, April, 1975.     

Neuronal spike trains and stochastic point processes. II. Simultaneous spike trains

The statistical analysis of two simultaneously observed trains of neuronal spikes is described, using as a conceptual framework the theory of stochastic point processes.The first statistical question that arises is whether the observed trains are independent; statistical techniques for testing independence are developed around the notion that, under the null hypothesis, the times of spike occurrence in one train represent random instants in time with respect to the other. If the null hypothesis is rejected—if dependence is attributed to the trains—the problem then becomes that of characterizing the nature and source of the observed dependencies. Statistical signs of various classes of dependencies, including direct interaction and shared input, are discussed and illustrated through computer simulations of interacting neurons. The effects of nonstationarities on the statistical measures for simultaneous spike trains are also discussed. For two-train comparisons of irregularly discharging nerve cells, moderate nonstationarities are shown to have little effect on the detection of interactions.Combining repetitive stimulation and simultaneous recording of spike trains from two (or more) neurons yields additional clues as to possible modes of interaction among the monitored neurons; the theory presented is illustrated by an application to experimentally obtained data from auditory neurons.A companion paper covers the analysis of single spike trains.     

Model based decoding of spike trains

Reliably decoding neuronal responses requires knowing what aspects of neuronal responses are stimulus related, and which aspects act as noise. Recent work shows that spike trains can be viewed as stochastic samples from the rate variation function, as estimated by the time dependent spike density function (or normalized peristimulus time histogram). Such spike trains are exactly described by order statistics, and can be decoded millisecond-by-millisecond by iterative application of order statistics.     

Alias-free sampling of neuronal spike trains

Summary Spectral analysis provides powerful techniques for describing the lower order moments of a stochastic process and interactions between two or more stochastic processes. A major problem in the application of spectral analysis to neuronal spike trains is how to obtain equispaced samples of the spike trains which will give unbiased and alias-free spectral estimates. Various sampling methods, which treat the spike train as a continuous signal, a point process and as a series of Dirac delta-functions, are reviewed and their limitations discussed. A new sampling technique, which gives unbiased and alias-free estimates, is described. This technique treats the spike train as a series of delta functions and generates samples by digital filtering. Implementation of this technique on a small computer is simple and virtually on-line.     

Recurrence plots of neuronal spike trains

The recently developed qualitative method of diagnosis of dynamical systems — recurrence plots has been applied to the analysis of dynamics of neuronal spike trains recorded from cerebellum and red nucleus of anesthetized cats. Recurrence plots revealed robust and common changes in the similarity structure of interspike interval sequences as well as significant deviations from randomness in serial ordering of intervals. Recurring episodes of alike, quasi-deterministic firing patterns suggest the spontaneous modulation of the dynamical complexity of the trajectories of observed neurons. These modulations are associated with changing dynamical properties of a neuronal spike-train-generating system. Their existence is compatible with the information processing paradigm of attractor neural networks.     

Use of `relative-phase' analysis to assess correlation between neuronal spike trains

 We propose a new method of studying the correlation between neuronal spike trains. This technique is based on the analysis of relative phase between two point processes. Relative phase here is defined as the relative timing difference between two spike trains normalized by the associated interspike interval of one cell. This phase measurement is intended to reveal the relative timing relationship between spike trains atdifferent firing rates. We apply this method to a numerical example and an example from two cerebellar neuronal spike trains of a behaving rat. The results are compared with classical cross-correlation analysis. We show that the technique can avoid some of the limitations of cross-correlation methods, reveal certain statistical dependencies that cannot be shown by cross correlation, and provide information as to the direction of influence between two spike trains. Received: 8 November 2001 / Accepted: 30 September 2002 / Published online: 24 January 2003 Correspondence to: Y. Chen (e-mail: chen@nsi.edu, Fax: + 1-858-626-2099) Acknowledgements. Research for this paper was supported by the Alafi Family Foundation and the Neurosciences Research Foundation.     

Spatial and temporal correlations of spike trains in frog retinal ganglion cells

For a neuron, firing activity can be in synchrony with that of others, which results in spatial correlation; on the other hand, spike events within each individual spike train may also correlate with each other, which results in temporal correlation. In order to investigate the relationship between these two phenomena, population neurons’ activities of frog retinal ganglion cells in response to binary pseudo-random checker-board flickering were recorded via a multi-electrode recording system. The spatial correlation index (SCI) and temporal correlation index (TCI) were calculated for the investigated neurons. Statistical results showed that, for a single neuron, the SCI and TCI values were highly related—a neuron with a high SCI value generally had a high TCI value, and these two indices were both associated with burst activities in spike train of the investigated neuron. These results may suggest that spatial and temporal correlations of single neuron’s spiking activities could be mutually modulated; and that burst activities could play a role in the modulation. We also applied models to test the contribution of spatial and temporal correlations for visual information processing. We show that a model considering spatial and temporal correlations could predict spikes more accurately than a model does not include any correlation.     

Reading spike timing without a clock: intrinsic decoding of spike trains

The precise timing of action potentials of sensory neurons relative to the time of stimulus presentation carries substantial sensory information that is lost or degraded when these responses are summed over longer time windows. However, it is unclear whether and how downstream networks can access information in precise time-varying neural responses. Here, we review approaches to test the hypothesis that the activity of neural populations provides the temporal reference frames needed to decode temporal spike patterns. These approaches are based on comparing the single-trial stimulus discriminability obtained from neural codes defined with respect to network-intrinsic reference frames to the discriminability obtained from codes defined relative to the experimenter''s computer clock. Application of this formalism to auditory, visual and somatosensory data shows that information carried by millisecond-scale spike times can be decoded robustly even with little or no independent external knowledge of stimulus time. In cortex, key components of such intrinsic temporal reference frames include dedicated neural populations that signal stimulus onset with reliable and precise latencies, and low-frequency oscillations that can serve as reference for partitioning extended neuronal responses into informative spike patterns.     

Recurring discharge patterns in multiple spike trains

Neural networks are parallel processing structures that provide the capability to perform various pattern recognition tasks. A network is typically trained over a set of exemplars by adjusting the weights of the interconnections using a back propagation algorithm. This gradient search converges to locally optimal solutions which may be far removed from the global optimum. In this paper, evolutionary programming is analyzed as a technique for training a general neural network. This approach can yield faster, more efficient yet robust training procedures that accommodate arbitrary interconnections and neurons possessing additional processing capabilities.     

Recurring discharge patterns in multiple spike trains

Simultaneously recorded spike trains were obtained using microwire bundles from unrestrained, drug-free cats during different sleep-waking states in forebrain areas associated with cardiac and respiratory activity. Cardiac and respiratory activity was simultaneously recorded with the spike trains. We applied the recurring discharge patterns detection procedure described in a companion paper (Frostig et al. 1990) to the spike and cardiorespiratory trains. The pattern detection procedure was applied to detect only precise (in time and structure) recurring patterns. Recurring discharge patterns were detected in all simultaneously recorded groups. Recurring discharge patterns were composed of up to ten spikes per pattern and involved up to four simultaneously recorded spike trains. Fourty-two percent of the recurring patterns contained cardiac and/or respiratory events in addition to neuronal spikes. When patterns were compared over different sleep-waking states it was found the the same units produced different patterns in different states, that patterns were significantly more compact in time during quiet sleep, and that changes in the discharge rates accompanying changes in sleep-waking states were not correlated with changes in pattern rate.     

Maximum likelihood for parasitologists

In quantitative biology, observed data are fitted to a model that captures the essence of the system under investigation in order to obtain estimates of the parameters of the model, as well as their standard errors and interactions. The fitting is best done by the method of maximum likelihood, though least-squares fits are often used as an approximation because the calculations are perceived to be simpler. Here Brian Williams and Chris Dye argue that the method of maximum likelihood is generally preferable to least squares giving the best estimates of the parameters for data with any given error distribution, and the calculations are no more difficult than for least-squares fitting. They offer a relatively simple explanation of the methods and describe its implementation using examples from leishmaniasis epidemiology.     

Visualisation of synchronous firing in multi-dimensional spike trains

The gravity transform algorithm is used to study the dependencies in firing of multi-dimensional spike trains. The pros and cons of this algorithm are discussed and the necessity for improved representation of output data is demonstrated. Parallel coordinates are introduced to visualise the results of the gravity transform and principal component analysis (PCA) is used to reduce the quantity of data represented whilst minimising loss of information.     

Chaos game representation of human pallidal spike trains

Many studies have demonstrated the presence of scale invariance and long-range correlation in animal and human neuronal spike trains. The methodologies to extract the fractal or scale-invariant properties, however, do not address the issue as to the existence within the train of fine temporal structures embedded in the global fractal organisation. The present study addresses this question in human spike trains by the chaos game representation (CGR) approach, a graphical analysis with which specific temporal sequences reveal themselves as geometric structures in the graphical representation. The neuronal spike train data were obtained from patients whilst undergoing pallidotomy. Using this approach, we observed highly structured regions in the representation, indicating the presence of specific preferred sequences of interspike intervals within the train. Furthermore, we observed that for a given spike train, the higher the magnitude of its scaling exponent, the more pronounced the geometric patterns in the representation and, hence, higher probability of occurrence of specific subsequences. Given its ability to detect and specify in detail the preferred sequences of interspike intervals, we believe that CGR is a useful adjunct to the existing set of methodologies for spike train analysis.