A characterization of the time-rescaled gamma process as a model for spike trains

Stimulus presentation can enhance spiking irregularity across subcortical and cortical regions

Stimulus presentation is believed to quench neural response variability as measured by fano-factor (FF). However, the relative contributions of within-trial spike irregularity and trial-to-trial rate variability to FF fluctuations have remained elusive. Here, we introduce a principled approach for accurate estimation of spiking irregularity and rate variability in time for doubly stochastic point processes. Consistent with previous evidence, analysis showed stimulus-induced reduction in rate variability across multiple cortical and subcortical areas. However, unlike what was previously thought, spiking irregularity, was not constant in time but could be enhanced due to factors such as bursting abating the quench in the post-stimulus FF. Simulations confirmed plausibility of a time varying spiking irregularity arising from within and between pool correlations of excitatory and inhibitory neural inputs. By accurate parsing of neural variability, our approach reveals previously unnoticed changes in neural response variability and constrains candidate mechanisms that give rise to observed rate variability and spiking irregularity within brain regions.

Mounting evidence suggest neural response variability to be important for understanding and constraining the underlying neural mechanisms in a given brain area. Here, by analyzing responses across multiple brain areas and by using a principled method for parsing variability components into rate variability and spiking irregularity, we show that unlike what was previously thought, event-related quench of variability is not a brain-wide phenomenon and that point process variability and nonrenewal bursting can enhance post-stimulus spiking irregularity across certain cortical and subcortical regions. We propose possible presynaptic mechanisms that may underlie the observed heterogeneities in spiking variability across the brain.

Fano Factor: A Potentially Useful Information

The Fano factor, defined as the variance-to-mean ratio of spike counts in a time window, is often used to measure the variability of neuronal spike trains. However, despite its transparent definition, careless use of the Fano factor can easily lead to distorted or even wrong results. One of the problems is the unclear dependence of the Fano factor on the spiking rate, which is often neglected or handled insufficiently. In this paper we aim to explore this problem in more detail and to study the possible solution, which is to evaluate the Fano factor in the operational time. We use equilibrium renewal and Markov renewal processes as spike train models to describe the method in detail, and we provide an illustration on experimental data.

Review: Methods of firing rate estimation

Neuronal firing rate is traditionally defined as the number of spikes per time window. The concept is essential for the rate coding hypothesis, which is still the most commonly investigated scenario in neuronal activity analysis. The estimation of dynamically changing firing rate from neural data can be challenging due to the variability of spike times, even under identical external conditions; hence a wide range of statistical measures have been employed to solve this particular problem. In this paper, we review established firing rate estimation methods, briefly summarize the technical aspects of each approach and discuss their practical applications.

Estimation of neuronal firing rate using Bayesian Adaptive Kernel Smoother (BAKS)

Neurons use sequences of action potentials (spikes) to convey information across neuronal networks. In neurophysiology experiments, information about external stimuli or behavioral tasks has been frequently characterized in term of neuronal firing rate. The firing rate is conventionally estimated by averaging spiking responses across multiple similar experiments (or trials). However, there exist a number of applications in neuroscience research that require firing rate to be estimated on a single trial basis. Estimating firing rate from a single trial is a challenging problem and current state-of-the-art methods do not perform well. To address this issue, we develop a new method for estimating firing rate based on a kernel smoothing technique that considers the bandwidth as a random variable with prior distribution that is adaptively updated under an empirical Bayesian framework. By carefully selecting the prior distribution together with Gaussian kernel function, an analytical expression can be achieved for the kernel bandwidth. We refer to the proposed method as Bayesian Adaptive Kernel Smoother (BAKS). We evaluate the performance of BAKS using synthetic spike train data generated by biologically plausible models: inhomogeneous Gamma (IG) and inhomogeneous inverse Gaussian (IIG). We also apply BAKS to real spike train data from non-human primate (NHP) motor and visual cortex. We benchmark the proposed method against established and previously reported methods. These include: optimized kernel smoother (OKS), variable kernel smoother (VKS), local polynomial fit (Locfit), and Bayesian adaptive regression splines (BARS). Results using both synthetic and real data demonstrate that the proposed method achieves better performance compared to competing methods. This suggests that the proposed method could be useful for understanding the encoding mechanism of neurons in cognitive-related tasks. The proposed method could also potentially improve the performance of brain-machine interface (BMI) decoder that relies on estimated firing rate as the input.

Detecting joint pausiness in parallel spike trains

BACKGROUND

Transient periods with reduced neuronal discharge - called 'pauses' - have recently gained increasing attention. In dopamine neurons, pauses are considered important teaching signals, encoding negative reward prediction errors. Particularly simultaneous pauses are likely to have increased impact on information processing.

COMPARISON WITH EXISTING METHODS

Available methods for detecting joint pausing analyze temporal overlap of pauses across spike trains. Such techniques are threshold dependent and can fail to identify joint pauses that are easily detectable by eye, particularly in spike trains with different firing rates.

NEW METHOD

We introduce a new statistic called pausiness that measures the degree of synchronous pausing in spike train pairs and avoids threshold-dependent identification of specific pauses. A new graphic termed the cross-pauseogram compares the joint pausiness of two spike trains with its time shifted analogue, such that a (pausiness) peak indicates joint pausing. When assessing significance of pausiness peaks, we use a stochastic model with synchronous spikes to disentangle joint pausiness arising from synchronous spikes from additional 'joint excess pausiness' (JEP). Parameter estimates are obtained from auto- and cross-correlograms, and statistical significance is assessed by comparison to simulated cross-pauseograms.

RESULTS

Our new method was applied to dopamine neuron pairs recorded in the ventral tegmental area of awake behaving mice. Significant JEP was detected in about 20% of the pairs.

CONCLUSION

Given the neurophysiological importance of pauses and the fact that neurons integrate multiple inputs, our findings suggest that the analysis of JEP can reveal interesting aspects in the activity of simultaneously recorded neurons.

The Gamma renewal process as an output of the diffusion leaky integrate-and-fire neuronal model

Statistical properties of spike trains as well as other neurophysiological data suggest a number of mathematical models of neurons. These models range from entirely descriptive ones to those deduced from the properties of the real neurons. One of them, the diffusion leaky integrate-and-fire neuronal model, which is based on the Ornstein-Uhlenbeck (OU) stochastic process that is restricted by an absorbing barrier, can describe a wide range of neuronal activity in terms of its parameters. These parameters are readily associated with known physiological mechanisms. The other model is descriptive, Gamma renewal process, and its parameters only reflect the observed experimental data or assumed theoretical properties. Both of these commonly used models are related here. We show under which conditions the Gamma model is an output from the diffusion OU model. In some cases, we can see that the Gamma distribution is unrealistic to be achieved for the employed parameters of the OU process.

Rapid Modulation of Axon Initial Segment Length Influences Repetitive Spike Firing

Neurons implement a variety of plasticity mechanisms to alter their function over timescales ranging from seconds to days. One powerful means of controlling excitability is to directly modulate the site of spike initiation, the axon initial segment (AIS). However, all plastic structural AIS changes reported thus far have been slow, involving days of neuronal activity perturbation. Here, we show that AIS plasticity can be induced much more rapidly. Just 3 hr of elevated activity significantly shortened the AIS of dentate granule cells in a calcineurin-dependent manner. The functional effects of rapid AIS shortening were offset by dephosphorylation of voltage-gated sodium channels, another calcineurin-dependent mechanism. However, pharmacological separation of these phenomena revealed a significant relationship between AIS length and repetitive firing. The AIS can therefore undergo a rapid form of structural change over timescales that enable interactions with other forms of activity-dependent plasticity in the dynamic control of neuronal excitability.

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Structural plasticity at the axon initial segment can occur within hours

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Ankyrin-G and sodium channel distributions shorten after 3 hr of elevated activity

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Rapid plasticity depends on calcineurin signaling opposed by CDK5

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All else being equal, AIS shortening correlates with lowered neuronal excitability

Stein's neuronal model with pooled renewal input

The input of Stein's model of a single neuron is usually described by using a Poisson process, which is assumed to represent the behaviour of spikes pooled from a large number of presynaptic spike trains. However, such a description of the input is not always appropriate as the variability cannot be separated from the intensity. Therefore, we create and study Stein's model with a more general input, a sum of equilibrium renewal processes. The mean and variance of the membrane potential are derived for this model. Using these formulas and numerical simulations, the model is analyzed to study the influence of the input variability on the properties of the membrane potential and the output spike trains. The generalized Stein's model is compared with the original Stein's model with Poissonian input using the relative difference of variances of membrane potential at steady state and the integral square error of output interspike intervals. Both of the criteria show large differences between the models for input with high variability.

Estimating individual firing frequencies in a multiple spike train record

Neuronal activity of several neurons is commonly recorded by a single electrode and then the individual spike trains are separated. If the separation is difficult or fails, then as a minimal result of the experiment, the individual firing rates are of interest. The proposed method solves the problem of their identification. This is possible under the condition that the recorded neurons are independent in their activities. The number of the neurons in the multi-unit record needs to be given (known or assumed) prior the calculation. The proposed method is based on the presence of the refractory period in neuronal firing, however, its precise value is not required. In addition to the determination of the individual firing rates the method can be used for an inference about the refractory period itself.

Identification of noisy response latency

In many physical systems there is a time delay before an applied input (stimulation) has an impact on the output (response), and the quantification of this delay is of paramount interest. If the response can only be observed on top of an indistinguishable background signal, the estimation can be highly unreliable, unless the background signal is accounted for in the analysis. In fact, if the background signal is ignored, however small it is compared to the response and however large the delay is, the estimate of the time delay will go to zero for any reasonable estimator when increasing the number of observations. Here we propose a unified concept of response latency identification in event data corrupted by a background signal. It is done in the context of information transfer within a neural system, more specifically on spike trains from single neurons. The estimators are compared on simulated data and the most suitable for specific situations are recommended.

Analysis and modelling of variability and covariability of population spike trains across multiple time scales

As multi-electrode and imaging technology begin to provide us with simultaneous recordings of large neuronal populations, new methods for modelling such data must also be developed. We present a model of responses to repeated trials of a sensory stimulus based on thresholded Gaussian processes that allows for analysis and modelling of variability and covariability of population spike trains across multiple time scales. The model framework can be used to specify the values of many different variability measures including spike timing precision across trials, coefficient of variation of the interspike interval distribution, and Fano factor of spike counts for individual neurons, as well as signal and noise correlations and correlations of spike counts across multiple neurons. Using both simulated data and data from different stages of the mammalian auditory pathway, we demonstrate the range of possible independent manipulations of different variability measures, and explore how this range depends on the sensory stimulus. The model provides a powerful framework for the study of experimental and surrogate data and for analyzing dependencies between different statistical properties of neuronal populations.

Optimizing Time Histograms for Non-Poissonian Spike Trains

The time histogram is a fundamental tool for representing the inhomogeneous density of event occurrences such as neuronal firings. The shape of a histogram critically depends on the size of the bins that partition the time axis. In most neurophysiological studies, however, researchers have arbitrarily selected the bin size when analyzing fluctuations in neuronal activity. A rigorous method for selecting the appropriate bin size was recently derived so that the mean integrated squared error between the time histogram and the unknown underlying rate is minimized (Shimazaki & Shinomoto, 2007 ). This derivation assumes that spikes are independently drawn from a given rate. However, in practice, biological neurons express non-Poissonian features in their firing patterns, such that the spike occurrence depends on the preceding spikes, which inevitably deteriorate the optimization. In this letter, we revise the method for selecting the bin size by considering the possible non-Poissonian features. Improvement in the goodness of fit of the time histogram is assessed and confirmed by numerically simulated non-Poissonian spike trains derived from the given fluctuating rate. For some experimental data, the revised algorithm transforms the shape of the time histogram from the Poissonian optimization method.

Firing variability is higher than deduced from the empirical coefficient of variation

A convenient and often used summary measure to quantify the firing variability in neurons is the coefficient of variation (CV), defined as the standard deviation divided by the mean. It is therefore important to find an estimator that gives reliable results from experimental data, that is, the estimator should be unbiased and have low estimation variance. When the CV is evaluated in the standard way (empirical standard deviation of interspike intervals divided by their average), then the estimator is biased, underestimating the true CV, especially if the distribution of the interspike intervals is positively skewed. Moreover, the estimator has a large variance for commonly used distributions. The aim of this letter is to quantify the bias and propose alternative estimation methods. If the distribution is assumed known or can be determined from data, parametric estimators are proposed, which not only remove the bias but also decrease the estimation errors. If no distribution is assumed and the data are very positively skewed, we propose to correct the standard estimator. When defining the corrected estimator, we simply use that it is more stable to work on the log scale for positively skewed distributions. The estimators are evaluated through simulations and applied to experimental data from olfactory receptor neurons in rats.

Estimating Neuronal Information: Logarithmic Binning of Neuronal Inter-Spike Intervals

Neurons communicate via the relative timing of all-or-none biophysical signals called spikes. For statistical analysis, the time between spikes can be accumulated into inter-spike interval histograms. Information theoretic measures have been estimated from these histograms to assess how information varies across organisms, neural systems, and disease conditions. Because neurons are computational units that, to the extent they process time, work not by discrete clock ticks but by the exponential decays of numerous intrinsic variables, we propose that neuronal information measures scale more naturally with the logarithm of time. For the types of inter-spike interval distributions that best describe neuronal activity, the logarithm of time enables fewer bins to capture the salient features of the distributions. Thus, discretizing the logarithm of inter-spike intervals, as compared to the inter-spike intervals themselves, yields histograms that enable more accurate entropy and information estimates for fewer bins and less data. Additionally, as distribution parameters vary, the entropy and information calculated from the logarithm of the inter-spike intervals are substantially better behaved, e.g., entropy is independent of mean rate, and information is equally affected by rate gains and divisions. Thus, when compiling neuronal data for subsequent information analysis, the logarithm of the inter-spike intervals is preferred, over the untransformed inter-spike intervals, because it yields better information estimates and is likely more similar to the construction used by nature herself.

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.