Single-trial estimation of neuronal firing rates: from single-neuron spike trains to population activity

We present a method to estimate the neuronal firing rate from single-trial spike trains. The method, based on convolution of the spike train with a fixed kernel function, is calibrated by means of simulated spike trains for a representative selection of realistic dynamic rate functions. We derive rules for the optimized use and performance of the kernel method, specifically with respect to an effective choice of the shape and width of the kernel functions. An application of our technique to the on-line, single-trial reconstruction of arm movement trajectories from multiple single-unit spike trains using dynamic population vectors illustrates a possible use of the proposed method.     

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.     

Towards statistical summaries of spike train data

Statistical inference has an important role in analysis of neural spike trains. While current approaches are mostly model-based, and designed for capturing the temporal evolution of the underlying stochastic processes, we focus on a data-driven approach where statistics are defined and computed in function spaces where individual spike trains are viewed as points. The first contribution of this paper is to endow spike train space with a parameterized family of metrics that takes into account different time warpings and generalizes several currently used metrics. These metrics are essentially penalized L(p) norms, involving appropriate functions of spike trains, with penalties associated with time-warpings. The second contribution of this paper is to derive a notion of a mean spike train in the case when p=2. We present an efficient recursive algorithm, termed Matching-Minimization algorithm, to compute the sample mean of a set of spike trains. The proposed metrics as well as the mean computations are demonstrated using an experimental recording from the motor cortex.     

Nonlinear modeling of neural population dynamics for hippocampal prostheses

Developing a neural prosthesis for the damaged hippocampus requires restoring the transformation of population neural activities performed by the hippocampal circuitry. To bypass a damaged region, output spike trains need to be predicted from the input spike trains and then reinstated through stimulation. We formulate a multiple-input, multiple-output (MIMO) nonlinear dynamic model for the input–output transformation of spike trains. In this approach, a MIMO model comprises a series of physiologically-plausible multiple-input, single-output (MISO) neuron models that consist of five components each: (1) feedforward Volterra kernels transforming the input spike trains into the synaptic potential, (2) a feedback kernel transforming the output spikes into the spike-triggered after-potential, (3) a noise term capturing the system uncertainty, (4) an adder generating the pre-threshold potential, and (5) a threshold function generating output spikes. It is shown that this model is equivalent to a generalized linear model with a probit link function. To reduce model complexity and avoid overfitting, statistical model selection and cross-validation methods are employed to choose the significant inputs and interactions between inputs. The model is applied successfully to the hippocampal CA3–CA1 population dynamics. Such a model can serve as a computational basis for the development of hippocampal prostheses.     

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.     

Statistical technique for analysing functional connectivity of multiple spike trains

A new statistical technique, the Cox method, used for analysing functional connectivity of simultaneously recorded multiple spike trains is presented. This method is based on the theory of modulated renewal processes and it estimates a vector of influence strengths from multiple spike trains (called reference trains) to the selected (target) spike train. Selecting another target spike train and repeating the calculation of the influence strengths from the reference spike trains enables researchers to find all functional connections among multiple spike trains. In order to study functional connectivity an "influence function" is identified. This function recognises the specificity of neuronal interactions and reflects the dynamics of postsynaptic potential. In comparison to existing techniques, the Cox method has the following advantages: it does not use bins (binless method); it is applicable to cases where the sample size is small; it is sufficiently sensitive such that it estimates weak influences; it supports the simultaneous analysis of multiple influences; it is able to identify a correct connectivity scheme in difficult cases of "common source" or "indirect" connectivity. The Cox method has been thoroughly tested using multiple sets of data generated by the neural network model of the leaky integrate and fire neurons with a prescribed architecture of connections. The results suggest that this method is highly successful for analysing functional connectivity of simultaneously recorded multiple spike trains.     

Wavelet-based processing of neuronal spike trains prior to discriminant analysis

Investigations of neural coding in many brain systems have focused on the role of spike rate and timing as two means of encoding information within a spike train. Recently, statistical pattern recognition methods, such as linear discriminant analysis (LDA), have emerged as a standard approach for examining neural codes. These methods work well when data sets are over-determined (i.e., there are more observations than predictor variables). But this is not always the case in many experimental data sets. One way to reduce the number of predictor variables is to preprocess data prior to classification. Here, a wavelet-based method is described for preprocessing spike trains. The method is based on the discriminant pursuit (DP) algorithm of Buckheit and Donoho [Proc. SPIE 2569 (1995) 540-51]. DP extracts a reduced set of features that are well localized in the time and frequency domains and that can be subsequently analyzed with statistical classifiers. DP is illustrated using neuronal spike trains recorded in the motor cortex of an awake, behaving rat [Laubach et al. Nature 405 (2000) 567-71]. In addition, simulated spike trains that differed only in the timing of spikes are used to show that DP outperforms another method for preprocessing spike trains, principal component analysis (PCA) [Richmond and Optican J. Neurophysiol. 57 (1987) 147-61].     

Measurement of variability dynamics in cortical spike trains

We propose a method for the time-resolved joint analysis of two related aspects of single neuron variability, the spiking irregularity measured by the squared coefficient of variation (CV(2)) of the ISIs and the trial-by-trial variability of the spike count measured by the Fano factor (FF). We provide a calibration of both estimators using the theory of renewal processes, and verify it for spike trains recorded in vitro. Both estimators exhibit a considerable bias for short observations that count less than about 5-10 spikes on average. The practical difficulty of measuring the CV(2) in rate modulated data can be overcome by a simple procedure of spike train demodulation which was tested in numerical simulations and in real spike trains. We propose to test neuronal spike trains for deviations from the null-hypothesis FF=CV(2). We show that cortical pyramidal neurons, recorded under controlled stationary input conditions in vitro, comply with this assumption. Performing a time-resolved joint analysis of CV(2) and FF of a single unit recording from the motor cortex of a behaving monkey we demonstrate how the dynamic change of their quantitative relation can be interpreted with respect to neuron intrinsic and extrinsic factors that influence cortical variability in vivo. Finally, we discuss the effect of several additional factors such as serial interval correlation and refractory period on the empiric relation of FF and CV(2).     

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.     

Detection of synchrony in the discharge of a population of neurons. II. Implementation and sensitivity of a synchronization index

This report describes the use of a synchronization index (Is; Hamm et al., 1985a) and its sensitivity to various forms and degrees of synchrony between spike trains. The dependence of the Is on signal-to-noise ratio, the number of synchronized spike trains and their degree of synchrony is shown in analog and digital simulations. These simulations and a comparison with peristimulus time histograms under conditions of induced synchrony reveal that the Is is a sensitive measure of synchronization in a population of spike trains.     

Short-term plasticity optimizes synaptic information transmission

Short-term synaptic plasticity (STP) is widely thought to play an important role in information processing. This major function of STP has recently been challenged, however, by several computational studies indicating that transmission of information by dynamic synapses is broadband, i.e., frequency independent. Here we developed an analytical approach to quantify time- and rate-dependent synaptic information transfer during arbitrary spike trains using a realistic model of synaptic dynamics in excitatory hippocampal synapses. We found that STP indeed increases information transfer in a wide range of input rates, which corresponds well to the naturally occurring spike frequencies at these synapses. This increased information transfer is observed both during Poisson-distributed spike trains with a constant rate and during naturalistic spike trains recorded in hippocampal place cells in exploring rodents. Interestingly, we found that the presence of STP in low release probability excitatory synapses leads to optimization of information transfer specifically for short high-frequency bursts, which are indeed commonly observed in many excitatory hippocampal neurons. In contrast, more reliable high release probability synapses that express dominant short-term depression are predicted to have optimal information transmission for single spikes rather than bursts. This prediction is verified in analyses of experimental recordings from high release probability inhibitory synapses in mouse hippocampal slices and fits well with the observation that inhibitory hippocampal interneurons do not commonly fire spike bursts. We conclude that STP indeed contributes significantly to synaptic information transfer and may serve to maximize information transfer for specific firing patterns of the corresponding neurons.     

Information trains. The technique and its uses in spike train and network analysis, with examples taken from the nucleus parabrachialis medialis during sleep-waking states

We describe an analytical procedure for assessing functional interactions between neuronal spike trains based on the outcome of cross-correlation procedures. Subsets of a reference cell spike train in a two-train recording are extracted, based on their time-locked relationship to spikes in the dependent train. Such timing relationships comprise the significant primary structures in the cross-correlogram. Different subsets can be extracted for different primary structures in the same correlogram (i.e. a subset responsible for an interaction effect, a subset responsible for a shared input effect, etc.) These new spike trains represent an information transfer process across synapses. These ‘information trains’ may be compared and correlated to different cells of the network across different functional conditions such as sleep-waking states, and may also be subjected to conventional spike train analysis techniques such as rate histogram, auto-correlation and cross-correlation procedures. We illustrate the information train procedures with a network analysis of a set of cells recorded in the nucleus parabrachialis medialis during different sleep-waking states.     

A simple indicator of nonstationarity of firing rate in spike trains

The assessment of stationarity of firing rate in neural spike trains is important but is often performed only visually. Facing the growing amount of neural data generated by multi-electrode recording, there is a need for an automatic method to identify and disqualify spike trains with highly nonstationary firing rates. In this report, we propose a simple test of nonstationarity, associated with an indicator quantifying the degree of nonstationary in a spike train. This method is compared to the Mann-Kendall test of trend detection and the Runs test on simulated and real spike trains.     

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).     

Analysis of phase-locked oscillations in multi-channel single-unit spike activity with wavelet cross-spectrum

Electrophysiological measures of neural activity frequently display oscillatory patterns at various frequencies. Furthermore, these oscillatory patterns can become dynamically synchronized across a wide region of the brain in a task-dependent manner. In this study, phase-locked oscillations in simultaneously recorded spike trains were analyzed using the wavelet cross-spectrum. Adaptation of the existent methods of calculating wavelet cross-spectrum to spike train data was straightforward. In contrast, new methods were needed for evaluating the statistical significance of the cross-spectrum. Although a permutation test based on a large number of re-sampled cross-spectra can provide a reliable estimate of statistical significance, this was quite time-consuming. As an alternative, statistical significance was determined with a normal probability density function estimated from a small number of re-sampled cross-spectra. When applied to neuron pairs recorded in the primate supplementary motor area, the re-sampling procedure produced a reliable outcome even when it was based on as few as ten re-sampled cross-spectra. These results suggest that the wavelet analysis in combination with a re-sampling procedure provides a useful tool to examine the dynamic patterns of temporal correlation in cortical spike trains.     

A comparison of two techniques for analyzing neuronal interspike intervals: Autocorrelation and relative interval coding

Only two theoretical approaches for serial order analysis of spike trains seem to be practical and to preserve a degree of the sequential ordering of the intervals. We herein compared these approaches, employing the two common techniques for autocorrelation and a group of relative interval coding techniques, using 4 short, idealized trains of spike intervals. The relative interval coding approach seemed to be more sensitive for accurately specifying and analyzing serial order. The autocorrelation methods create ambiguity because of inherent “partitioning” effects in the computation and because the interrelations among peaks in the correlogram can be deceptive.     

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.     

Computer analysis of vestibular responses to putative neurotransmitters

An inexpensive microcomputer (Commodore 64K) based system was developed for the analysis of neural spike trains. The trains were recorded from single ampullary units in response to mechanical stimulation of the isolated semicircular canal of the bullfrog (Rana catesbeiana). A BASIC program provided a number of options while machine language subroutines generated interstimulus interval (ISI) and peristimulus time (PST) histograms. Up to thirty 5-s spike trains could be combined for analysis (0.1 ms resolution ISI, 100 ms bin width PST). Histograms and summary statistics were saved on floppy disks. The cost of adding this computer system to an existing neurophysiology laboratory is less than US $600 (printed, tape, and disk versions of these programs are available). The system was used to measure vestibular responses to putative vestibular neurotransmitters such as carbachol (an acetylcholine mimic) (Rossi et al., 1980) and gamma-aminobutyric acid (GABA) (Flock and Lam, 1974).     

Estimating the correlation between bursty spike trains and local field potentials

To further understand rhythmic neuronal synchronization, an increasingly useful method is to determine the relationship between the spiking activity of individual neurons and the local field potentials (LFPs) of neural ensembles. Spike field coherence (SFC) is a widely used method for measuring the synchronization between spike trains and LFPs. However, due to the strong dependency of SFC on the burst index, it is not suitable for analyzing the relationship between bursty spike trains and LFPs, particularly in high frequency bands. To address this issue, we developed a method called weighted spike field correlation (WSFC), which uses the first spike in each burst multiple times to estimate the relationship. In the calculation, the number of times that the first spike is used is equal to the spike count per burst. The performance of this method was demonstrated using simulated bursty spike trains and LFPs, which comprised sinusoids with different frequencies, amplitudes, and phases. This method was also used to estimate the correlation between pyramidal cells in the hippocampus and gamma oscillations in rats performing behaviors. Analyses using simulated and real data demonstrated that the WSFC method is a promising measure for estimating the correlation between bursty spike trains and high frequency LFPs.