Estimating instantaneous irregularity of neuronal firing

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.

Similarity in Neuronal Firing Regimes across Mammalian Species

The architectonic subdivisions of the brain are believed to be functional modules, each processing parts of global functions. Previously, we showed that neurons in different regions operate in different firing regimes in monkeys. It is possible that firing regimes reflect differences in underlying information processing, and consequently the firing regimes in homologous regions across animal species might be similar. We analyzed neuronal spike trains recorded from behaving mice, rats, cats, and monkeys. The firing regularity differed systematically, with differences across regions in one species being greater than the differences in similar areas across species. Neuronal firing was consistently most regular in motor areas, nearly random in visual and prefrontal/medial prefrontal cortical areas, and bursting in the hippocampus in all animals examined. This suggests that firing regularity (or irregularity) plays a key role in neural computation in each functional subdivision, depending on the types of information being carried.

SIGNIFICANCE STATEMENT

By analyzing neuronal spike trains recorded from mice, rats, cats, and monkeys, we found that different brain regions have intrinsically different firing regimes that are more similar in homologous areas across species than across areas in one species. Because different regions in the brain are specialized for different functions, the present finding suggests that the different activity regimes of neurons are important for supporting different functions, so that appropriate neuronal codes can be used for different modalities.

Ventral Tegmental Area Dopamine Neurons Firing Model Reveals Prenatal Nicotine Induced Alterations

The dopamine (DA) neurons found in the ventral tegmental area (VTA) are widely involved in the addiction and natural reward circuitry of the brain. Their firing patterns were shown to be important modulators of dopamine release and repetitive burst-like firing activity was highlighted as a major firing pattern of DA neurons in the VTA. In the present study we use a state space model to characterize the DA neurons firing patterns, and trace transitions of neural activity through bursting and non-bursting states. The hidden semi-Markov model (HSMM) framework, which we use, offers a statistically principled inference of bursting states and considers VTA DA firing patterns to be generated according to a Gamma process. Additionally, the explicit Gamma-based modeling of state durations allows efficient decoding of underlying neural information. Consequently, we decode and segment our single unit recordings from DA neurons in VTA according to the sequence of statistically discriminated HSMM states. The segmentation is used to study bursting state characteristics in data recorded from rats prenatally exposed to nicotine (6 mg/kg/day starting with gestational day 3) and rats from saline treated dams. Our results indicate that prenatal nicotine exposure significantly alters burst firing patterns of a subset of DA neurons in adolescent rats, suggesting nicotine exposure during gestation may induce severe effects on the neural networks involved in addiction and reward.

Lognormal firing rate distribution reveals prominent fluctuation-driven regime in spinal motor networks

When spinal circuits generate rhythmic movements it is important that the neuronal activity remains within stable bounds to avoid saturation and to preserve responsiveness. Here, we simultaneously record from hundreds of neurons in lumbar spinal circuits of turtles and establish the neuronal fraction that operates within either a 'mean-driven' or a 'fluctuation-driven' regime. Fluctuation-driven neurons have a 'supralinear' input-output curve, which enhances sensitivity, whereas the mean-driven regime reduces sensitivity. We find a rich diversity of firing rates across the neuronal population as reflected in a lognormal distribution and demonstrate that half of the neurons spend at least 50 % of the time in the 'fluctuation-driven' regime regardless of behavior. Because of the disparity in input-output properties for these two regimes, this fraction may reflect a fine trade-off between stability and sensitivity in order to maintain flexibility across behaviors.

A Model for Single Neuron Activity With Refractory Effects and Spike Rate Estimation Techniques

The use of random point processes as models for neural spike trains allows the derivation of powerful statistical estimation techniques for time varying firing rates. Frequently, however, such estimators are based on the assumption that spike sequences follow a Poisson point process. Because of the bio-physical properties of neuronal action potentials, spike trains are affected by the refractory phenomenon that induces history dependency, and hence contradicts the Poisson assumption. In this work we present a neural spiking model, and a Maximum Likelihood (ML) estimation framework for time varying firing rates, that account for history dependencies in spike trains. Our framework is based on an exponential of polynomial model for the excitation function (stimulus), that generates a self exciting point process representing spike trains with absolute as well as relative refractory effects. Using this framework we employ techniques based on non-convex optimization and model order selection to derive ML estimators for neuronal firing rates. Results on simulated data with a refractory period show an improvement in accuracy when our estimation technique, that accounts for the complete refractory phenomenon, is used. Employing this estimation method for measured neuronal data shows an improvement in goodness of fit over estimators that do not account for the refractory effect, and also over other commonly used techniques.

Reconstruction of neuronal input through modeling single-neuron dynamics and computations

Mathematical models provide a mathematical description of neuron activity, which can better understand and quantify neural computations and corresponding biophysical mechanisms evoked by stimulus. In this paper, based on the output spike train evoked by the acupuncture mechanical stimulus, we present two different levels of models to describe the input-output system to achieve the reconstruction of neuronal input. The reconstruction process is divided into two steps: First, considering the neuronal spiking event as a Gamma stochastic process. The scale parameter and the shape parameter of Gamma process are, respectively, defined as two spiking characteristics, which are estimated by a state-space method. Then, leaky integrate-and-fire (LIF) model is used to mimic the response system and the estimated spiking characteristics are transformed into two temporal input parameters of LIF model, through two conversion formulas. We test this reconstruction method by three different groups of simulation data. All three groups of estimates reconstruct input parameters with fairly high accuracy. We then use this reconstruction method to estimate the non-measurable acupuncture input parameters. Results show that under three different frequencies of acupuncture stimulus conditions, estimated input parameters have an obvious difference. The higher the frequency of the acupuncture stimulus is, the higher the accuracy of reconstruction is.

ANNarchy: a code generation approach to neural simulations on parallel hardware

Many modern neural simulators focus on the simulation of networks of spiking neurons on parallel hardware. Another important framework in computational neuroscience, rate-coded neural networks, is mostly difficult or impossible to implement using these simulators. We present here the ANNarchy (Artificial Neural Networks architect) neural simulator, which allows to easily define and simulate rate-coded and spiking networks, as well as combinations of both. The interface in Python has been designed to be close to the PyNN interface, while the definition of neuron and synapse models can be specified using an equation-oriented mathematical description similar to the Brian neural simulator. This information is used to generate C++ code that will efficiently perform the simulation on the chosen parallel hardware (multi-core system or graphical processing unit). Several numerical methods are available to transform ordinary differential equations into an efficient C++code. We compare the parallel performance of the simulator to existing solutions.

Probability distributions of the electroencephalogram envelope of preterm infants

OBJECTIVE

To determine the stationary characteristics of electroencephalogram (EEG) envelopes for prematurely born (preterm) infants and investigate the intrinsic characteristics of early brain development in preterm infants.

METHODS

Twenty neurologically normal sets of EEGs recorded in infants with a post-conceptional age (PCA) range of 26-44 weeks (mean 37.5 ± 5.0 weeks) were analyzed. Hilbert transform was applied to extract the envelope. We determined the suitable probability distribution of the envelope and performed a statistical analysis.

RESULTS

It was found that (i) the probability distributions for preterm EEG envelopes were best fitted by lognormal distributions at 38 weeks PCA or less, and by gamma distributions at 44 weeks PCA; (ii) the scale parameter of the lognormal distribution had positive correlations with PCA as well as a strong negative correlation with the percentage of low-voltage activity; (iii) the shape parameter of the lognormal distribution had significant positive correlations with PCA; (iv) the statistics of mode showed significant linear relationships with PCA, and, therefore, it was considered a useful index in PCA prediction.

CONCLUSION

These statistics, including the scale parameter of the lognormal distribution and the skewness and mode derived from a suitable probability distribution, may be good indexes for estimating stationary nature in developing brain activity in preterm infants.

SIGNIFICANCE

The stationary characteristics, such as discontinuity, asymmetry, and unimodality, of preterm EEGs are well indicated by the statistics estimated from the probability distribution of the preterm EEG envelopes.

Estimating latency from inhibitory input

Stimulus response latency is the time period between the presentation of a stimulus and the occurrence of a change in the neural firing evoked by the stimulation. The response latency has been explored and estimation methods proposed mostly for excitatory stimuli, which means that the neuron reacts to the stimulus by an increase in the firing rate. We focus on the estimation of the response latency in the case of inhibitory stimuli. Models used in this paper represent two different descriptions of response latency. We consider either the latency to be constant across trials or to be a random variable. In the case of random latency, special attention is given to models with selective interaction. The aim is to propose methods for estimation of the latency or the parameters of its distribution. Parameters are estimated by four different methods: method of moments, maximum-likelihood method, a method comparing an empirical and a theoretical cumulative distribution function and a method based on the Laplace transform of a probability density function. All four methods are applied on simulated data and compared.

Spike generation estimated from stationary spike trains in a variety of neurons in vivo

To any model of brain function, the variability of neuronal spike firing is a problem that needs to be taken into account. Whereas the synaptic integration can be described in terms of the original Hodgkin-Huxley (H-H) formulations of conductance-based electrical signaling, the transformation of the resulting membrane potential into patterns of spike output is subjected to stochasticity that may not be captured with standard single neuron H-H models. The dynamics of the spike output is dependent on the normal background synaptic noise present in vivo, but the neuronal spike firing variability in vivo is not well studied. In the present study, we made long-term whole cell patch clamp recordings of stationary spike firing states across a range of membrane potentials from a variety of subcortical neurons in the non-anesthetized, decerebrated state in vivo. Based on the data, we formulated a simple, phenomenological model of the properties of the spike generation in each neuron that accurately captured the stationary spike firing statistics across all membrane potentials. The model consists of a parametric relationship between the mean and standard deviation of the inter-spike intervals, where the parameter is linearly related to the injected current over the membrane. This enabled it to generate accurate approximations of spike firing also under inhomogeneous conditions with input that varies over time. The parameters describing the spike firing statistics for different neuron types overlapped extensively, suggesting that the spike generation had similar properties across neurons.

Influence of external inputs and asymmetry of connections on information-geometric measures involving up to ten neuronal interactions

The investigation of neural interactions is crucial for understanding information processing in the brain. Recently an analysis method based on information geometry (IG) has gained increased attention, and the property of the pairwise IG measure has been studied extensively in relation to the two-neuron interaction. However, little is known about the property of IG measures involving more neuronal interactions. In this study, we systematically investigated the influence of external inputs and the asymmetry of connections on the IG measures in cases ranging from 1-neuron to 10-neuron interactions. First, the analytical relationship between the IG measures and external inputs was derived for a network of 10 neurons with uniform connections. Our results confirmed that the single and pairwise IG measures were good estimators of the mean background input and of the sum of the connection weights, respectively. For the IG measures involving 3 to 10 neuronal interactions, we found that the influence of external inputs was highly nonlinear. Second, by computer simulation, we extended our analytical results to asymmetric connections. For a network of 10 neurons, the simulation showed that the behavior of the IG measures in relation to external inputs was similar to the analytical solution obtained for a uniformly connected network. When the network size was increased to 1000 neurons, the influence of external inputs almost disappeared. This result suggests that all IG measures from 1-neuron to 10-neuron interactions are robust against the influence of external inputs. In addition, we investigated how the strength of asymmetry influenced the IG measures. Computer simulation of a 1000-neuron network showed that all the IG measures were robust against the modulation of the asymmetry of connections. Our results provide further support for an information-geometric approach and will provide useful insights when these IG measures are applied to real experimental spike data.

Information-geometric measures estimate neural interactions during oscillatory brain states

The characterization of functional network structures among multiple neurons is essential to understanding neural information processing. Information geometry (IG), a theory developed for investigating a space of probability distributions has recently been applied to spike-train analysis and has provided robust estimations of neural interactions. Although neural firing in the equilibrium state is often assumed in these studies, in reality, neural activity is non-stationary. The brain exhibits various oscillations depending on cognitive demands or when an animal is asleep. Therefore, the investigation of the IG measures during oscillatory network states is important for testing how the IG method can be applied to real neural data. Using model networks of binary neurons or more realistic spiking neurons, we studied how the single- and pairwise-IG measures were influenced by oscillatory neural activity. Two general oscillatory mechanisms, externally driven oscillations and internally induced oscillations, were considered. In both mechanisms, we found that the single-IG measure was linearly related to the magnitude of the external input, and that the pairwise-IG measure was linearly related to the sum of connection strengths between two neurons. We also observed that the pairwise-IG measure was not dependent on the oscillation frequency. These results are consistent with the previous findings that were obtained under the equilibrium conditions. Therefore, we demonstrate that the IG method provides useful insights into neural interactions under the oscillatory condition that can often be observed in the real brain.

Estimating nonstationary inputs from a single spike train based on a neuron model with adaptation

Because every spike of a neuron is determined by input signals, a train of spikes may contain information about the dynamics of unobserved neurons. A state-space method based on the leaky integrate-and-fire model, describing neuronal transformation from input signals to a spike train has been proposed for tracking input parameters represented by their mean and fluctuation [11]. In the present paper, we propose to make the estimation more realistic by adopting an LIF model augmented with an adaptive moving threshold. Moreover, because the direct state-space method is computationally infeasible for a data set comprising thousands of spikes, we further develop a practical method for transforming instantaneous firing characteristics back to input parameters. The instantaneous firing characteristics, represented by the firing rate and non-Poisson irregularity, can be estimated using a computationally feasible algorithm. We applied our proposed methods to synthetic data to clarify that they perform well.

Fano factor estimation

Fano factor is one of the most widely used measures of variability of spike trains. Its standard estimator is the ratio of sample variance to sample mean of spike counts observed in a time window and the quality of the estimator strongly depends on the length of the window. We investigate this dependence under the assumption that the spike train behaves as an equilibrium renewal process. It is shown what characteristics of the spike train have large effect on the estimator bias. Namely, the effect of refractory period is analytically evaluated. Next, we create an approximate asymptotic formula for the mean square error of the estimator, which can also be used to find minimum of the error in estimation from single spike trains. The accuracy of the Fano factor estimator is compared with the accuracy of the estimator based on the squared coefficient of variation. All the results are illustrated for spike trains with gamma and inverse Gaussian probability distributions of interspike intervals. Finally, we discuss possibilities of how to select a suitable observation window for the Fano factor estimation.

Sudden synchrony leaps accompanied by frequency multiplications in neuronal activity

A classical view of neural coding relies on temporal firing synchrony among functional groups of neurons, however, the underlying mechanism remains an enigma. Here we experimentally demonstrate a mechanism where time-lags among neuronal spiking leap from several tens of milliseconds to nearly zero-lag synchrony. It also allows sudden leaps out of synchrony, hence forming short epochs of synchrony. Our results are based on an experimental procedure where conditioned stimulations were enforced on circuits of neurons embedded within a large-scale network of cortical cells in vitro and are corroborated by simulations of neuronal populations. The underlying biological mechanisms are the unavoidable increase of the neuronal response latency to ongoing stimulations and temporal or spatial summation required to generate evoked spikes. These sudden leaps in and out of synchrony may be accompanied by multiplications of the neuronal firing frequency, hence offering reliable information-bearing indicators which may bridge between the two principal neuronal coding paradigms.

Estimating inputs and an internal neuronal parameter from a single spike train

Because neurons are integrating input signals and translating them into timed output spikes, examining spike timing may reveal information about inputs, such as population activities of excitatory and inhibitory presynaptic neurons. Here we construct a state-space method for estimating not only such extrinsic parameters, but also an intrinsic neuronal parameter such as the membrane time constant from a single spike train.

Information transmission using non-poisson regular firing

In many cortical areas, neural spike trains do not follow a Poisson process. In this study, we investigate a possible benefit of non-Poisson spiking for information transmission by studying the minimal rate fluctuation that can be detected by a Bayesian estimator. The idea is that an inhomogeneous Poisson process may make it difficult for downstream decoders to resolve subtle changes in rate fluctuation, but by using a more regular non-Poisson process, the nervous system can make rate fluctuations easier to detect. We evaluate the degree to which regular firing reduces the rate fluctuation detection threshold. We find that the threshold for detection is reduced in proportion to the coefficient of variation of interspike intervals.

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.

Estimating nonstationary input signals from a single neuronal spike train

Neurons temporally integrate input signals, translating them into timed output spikes. Because neurons nonperiodically emit spikes, examining spike timing can reveal information about input signals, which are determined by activities in the populations of excitatory and inhibitory presynaptic neurons. Although a number of mathematical methods have been developed to estimate such input parameters as the mean and fluctuation of the input current, these techniques are based on the unrealistic assumption that presynaptic activity is constant over time. Here, we propose tracking temporal variations in input parameters with a two-step analysis method. First, nonstationary firing characteristics comprising the firing rate and non-Poisson irregularity are estimated from a spike train using a computationally feasible state-space algorithm. Then, information about the firing characteristics is converted into likely input parameters over time using a transformation formula, which was constructed by inverting the neuronal forward transformation of the input current to output spikes. By analyzing spike trains recorded in vivo, we found that neuronal input parameters are similar in the primary visual cortex V1 and middle temporal area, whereas parameters in the lateral geniculate nucleus of the thalamus were markedly different.

Information-geometric measures for estimation of connection weight under correlated inputs

The brain processes information in a highly parallel manner. Determination of the relationship between neural spikes and synaptic connections plays a key role in the analysis of electrophysiological data. Information geometry (IG) has been proposed as a powerful analysis tool for multiple spike data, providing useful insights into the statistical interactions within a population of neurons. Previous work has demonstrated that IG measures can be used to infer the connection weight between two neurons in a neural network. This property is useful in neuroscience because it provides a way to estimate learning-induced changes in synaptic strengths from extracellular neuronal recordings. A previous study has shown, however, that this property would hold only when inputs to neurons are not correlated. Since neurons in the brain often receive common inputs, this would hinder the application of the IG method to real data. We investigated the two-neuron-IG measures in higher-order log-linear models to overcome this limitation. First, we mathematically showed that the estimation of uniformly connected synaptic weight can be improved by taking into account higher-order log-linear models. Second, we numerically showed that the estimation can be improved for more general asymmetrically connected networks. Considering the estimated number of the synaptic connections in the brain, we showed that the two-neuron IG measure calculated by the fourth- or fifth-order log-linear model would provide an accurate estimation of connection strength within approximately a 10% error. These studies suggest that the two-neuron IG measure with higher-order log-linear expansion is a robust estimator of connection weight even under correlated inputs, providing a useful analytical tool for real multineuronal spike data.

Behavior-related pauses in simple-spike activity of mouse Purkinje cells are linked to spike rate modulation

Purkinje cells (PCs) in the mammalian cerebellum express high-frequency spontaneous activity with average spike rates between 30 and 200 Hz. Cerebellar nuclear (CN) neurons receive converging input from many PCs, resulting in a continuous barrage of inhibitory inputs. It has been hypothesized that pauses in PC activity trigger increases in CN spiking activity. A prediction derived from this hypothesis is that pauses in PC simple-spike activity represent relevant behavioral or sensory events. Here, we asked whether pauses in the simple-spike activity of PCs related to either fluid licking or respiration, play a special role in representing information about behavior. Both behaviors are widely represented in cerebellar PC simple-spike activity. We recorded PC activity in the vermis and lobus simplex of head-fixed mice while monitoring licking and respiratory behavior. Using cross-correlation and Granger causality analysis, we examined whether short interspike intervals (ISIs) had a different temporal relationship to behavior than long ISIs or pauses. Behavior-related simple-spike pauses occurred during low-rate simple-spike activity in both licking- and breathing-related PCs. Granger causality analysis revealed causal relationships between simple-spike pauses and behavior. However, the same results were obtained from an analysis of surrogate spike trains with gamma ISI distributions constructed to match rate modulations of behavior-related Purkinje cells. Our results therefore suggest that the occurrence of pauses in simple-spike activity does not represent additional information about behavioral or sensory events that goes beyond the simple-spike rate modulations.

Mapping of visual receptive fields by tomographic reconstruction

The moving bar experiment is a classic paradigm for characterizing the receptive field (RF) properties of neurons in primary visual cortex (V1). Current approaches for analyzing neural spiking activity recorded from these experiments do not take into account the point-process nature of these data and the circular geometry of the stimulus presentation. We present a novel analysis approach to mapping V1 receptive fields that combines point-process generalized linear models (PPGLM) with tomographic reconstruction computed by filtered-back projection. We use the method to map the RF sizes and orientations of 251 V1 neurons recorded from two macaque monkeys during a moving bar experiment. Our cross-validated goodness-of-fit analyses show that the PPGLM provides a more accurate characterization of spike train data than analyses based on rate functions computed by the methods of spike-triggered averages or first-order Wiener-Volterra kernel. Our analysis leads to a new definition of RF size as the spatial area over which the spiking activity is significantly greater than baseline activity. Our approach yields larger RF sizes and sharper orientation tuning estimates. The tomographic reconstruction paradigm further suggests an efficient approach to choosing the number of directions and the number of trials per direction in designing moving bar experiments. Our results demonstrate that standard tomographic principles for image reconstruction can be adapted to characterize V1 RFs and that two fundamental properties, size and orientation, may be substantially different from what is currently reported.

Power-law inter-spike interval distributions infer a conditional maximization of entropy in cortical neurons

The brain is considered to use a relatively small amount of energy for its efficient information processing. Under a severe restriction on the energy consumption, the maximization of mutual information (MMI), which is adequate for designing artificial processing machines, may not suit for the brain. The MMI attempts to send information as accurate as possible and this usually requires a sufficient energy supply for establishing clearly discretized communication bands. Here, we derive an alternative hypothesis for neural code from the neuronal activities recorded juxtacellularly in the sensorimotor cortex of behaving rats. Our hypothesis states that in vivo cortical neurons maximize the entropy of neuronal firing under two constraints, one limiting the energy consumption (as assumed previously) and one restricting the uncertainty in output spike sequences at given firing rate. Thus, the conditional maximization of firing-rate entropy (CMFE) solves a tradeoff between the energy cost and noise in neuronal response. In short, the CMFE sends a rich variety of information through broader communication bands (i.e., widely distributed firing rates) at the cost of accuracy. We demonstrate that the CMFE is reflected in the long-tailed, typically power law, distributions of inter-spike intervals obtained for the majority of recorded neurons. In other words, the power-law tails are more consistent with the CMFE rather than the MMI. Thus, we propose the mathematical principle by which cortical neurons may represent information about synaptic input into their output spike trains.

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.

Estimation of Time-Dependent Input from Neuronal Membrane Potential

The set of firing rates of the presynaptic excitatory and inhibitory neurons constitutes the input signal to the postsynaptic neuron. Estimation of the time-varying input rates from intracellularly recorded membrane potential is investigated here. For that purpose, the membrane potential dynamics must be specified. We consider the Ornstein-Uhlenbeck stochastic process, one of the most common single-neuron models, with time-dependent mean and variance. Assuming the slow variation of these two moments, it is possible to formulate the estimation problem by using a state-space model. We develop an algorithm that estimates the paths of the mean and variance of the input current by using the empirical Bayes approach. Then the input firing rates are directly available from the moments. The proposed method is applied to three simulated data examples: constant signal, sinusoidally modulated signal, and constant signal with a jump. For the constant signal, the estimation performance of the method is comparable to that of the traditionally applied maximum likelihood method. Further, the proposed method accurately estimates both continuous and discontinuous time-variable signals. In the case of the signal with a jump, which does not satisfy the assumption of slow variability, the robustness of the method is verified. It can be concluded that the method provides reliable estimates of the total input firing rates, which are not experimentally measurable.

Measuring burstiness and regularity in oscillatory spike trains

The ability of neurons to emit different firing patterns such as bursts or oscillations is important for information processing in the brain. In dopaminergic neurons, prominent patterns include repetitive, oscillatory bursts, regular pacemakers, and irregular spike trains with nonstationary properties. In order to describe and measure the variability of these patterns, we describe burstiness and regularity in a single model framework. We present a doubly stochastic spike train model in which a background oscillation with independent and normally distributed intervals gives rise to either single spikes or bursty spike events with Gaussian firing intensities. Five easily interpretable parameters allow a classification into bursty or single spike and irregularly or regularly oscillating firing patterns. This classification is based primarily on features of the autocorrelation histogram which are usually studied qualitatively by visual inspection. The present model provides a quantitative and objective classification scheme and relates these features directly to the underlying processes. In addition, confidence intervals visualize the uncertainty of parameter estimation and classification precision. We apply the model to a data set obtained from single dopaminergic substantia nigra neurons recorded extracellularly in vivo. The model is able to represent a high variety of discharge patterns observed empirically, and the classification agrees closely with visual inspection. In addition, changes in the parameters can be studied quantitatively, including also the properties related to bursting behavior. Thus, the proposed model can be used for the description of neuronal firing patterns and the investigation of their dynamical changes with cellular and experimental conditions.

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.

Applying the multivariate time-rescaling theorem to neural population models

Statistical models of neural activity are integral to modern neuroscience. Recently interest has grown in modeling the spiking activity of populations of simultaneously recorded neurons to study the effects of correlations and functional connectivity on neural information processing. However, any statistical model must be validated by an appropriate goodness-of-fit test. Kolmogorov-Smirnov tests based on the time-rescaling theorem have proven to be useful for evaluating point-process-based statistical models of single-neuron spike trains. Here we discuss the extension of the time-rescaling theorem to the multivariate (neural population) case. We show that even in the presence of strong correlations between spike trains, models that neglect couplings between neurons can be erroneously passed by the univariate time-rescaling test. We present the multivariate version of the time-rescaling theorem and provide a practical step-by-step procedure for applying it to testing the sufficiency of neural population models. Using several simple analytically tractable models and more complex simulated and real data sets, we demonstrate that important features of the population activity can be detected only using the multivariate extension of the test.

Distribution of interspike intervals estimated from multiple spike trains observed in a short time window

Several nonparametric estimators of the probability distribution of interspike intervals are introduced. The methods are suitable for simultaneous spike trains observed in a time window of length comparable with the mean interspike interval. This reflects the situation in which a high number of input spike trains converge to a single cortical neuron that has to react in a relatively short time. The simulation study is performed to compare the estimators. For that purpose, several types of stationary point processes are considered as the models of neuronal activity. The methods permit one to estimate the distribution of interspike intervals even if practically none of them are observed. The Kaplan-Meier estimator seems to be the most flexible and reliable among all studied methods, but no direct conclusions as to how real neurons work can be deduced from it.

Estimating network parameters from combined dynamics of firing rate and irregularity of single neurons

High firing irregularity is a hallmark of cortical neurons in vivo, and modeling studies suggest a balance of excitation and inhibition is necessary to explain this high irregularity. Such a balance must be generated, at least partly, from local interconnected networks of excitatory and inhibitory neurons, but the details of the local network structure are largely unknown. The dynamics of the neural activity depends on the local network structure; this in turn suggests the possibility of estimating network structure from the dynamics of the firing statistics. Here we report a new method to estimate properties of the local cortical network from the instantaneous firing rate and irregularity (CV(2)) under the assumption that recorded neurons are a part of a randomly connected sparse network. The firing irregularity, measured in monkey motor cortex, exhibits two features; many neurons show relatively stable firing irregularity in time and across different task conditions; the time-averaged CV(2) is widely distributed from quasi-regular to irregular (CV(2) = 0.3-1.0). For each recorded neuron, we estimate the three parameters of a local network [balance of local excitation-inhibition, number of recurrent connections per neuron, and excitatory postsynaptic potential (EPSP) size] that best describe the dynamics of the measured firing rates and irregularities. Our analysis shows that optimal parameter sets form a two-dimensional manifold in the three-dimensional parameter space that is confined for most of the neurons to the inhibition-dominated region. High irregularity neurons tend to be more strongly connected to the local network, either in terms of larger EPSP and inhibitory PSP size or larger number of recurrent connections, compared with the low irregularity neurons, for a given excitatory/inhibitory balance. Incorporating either synaptic short-term depression or conductance-based synapses leads many low CV(2) neurons to move to the excitation-dominated region as well as to an increase of EPSP size.

Kernel bandwidth optimization in spike rate estimation

Kernel smoother and a time-histogram are classical tools for estimating an instantaneous rate of spike occurrences. We recently established a method for selecting the bin width of the time-histogram, based on the principle of minimizing the mean integrated square error (MISE) between the estimated rate and unknown underlying rate. Here we apply the same optimization principle to the kernel density estimation in selecting the width or “bandwidth” of the kernel, and further extend the algorithm to allow a variable bandwidth, in conformity with data. The variable kernel has the potential to accurately grasp non-stationary phenomena, such as abrupt changes in the firing rate, which we often encounter in neuroscience. In order to avoid possible overfitting that may take place due to excessive freedom, we introduced a stiffness constant for bandwidth variability. Our method automatically adjusts the stiffness constant, thereby adapting to the entire set of spike data. It is revealed that the classical kernel smoother may exhibit goodness-of-fit comparable to, or even better than, that of modern sophisticated rate estimation methods, provided that the bandwidth is selected properly for a given set of spike data, according to the optimization methods presented here.

Relating neuronal firing patterns to functional differentiation of cerebral cortex

It has been empirically established that the cerebral cortical areas defined by Brodmann one hundred years ago solely on the basis of cellular organization are closely correlated to their function, such as sensation, association, and motion. Cytoarchitectonically distinct cortical areas have different densities and types of neurons. Thus, signaling patterns may also vary among cytoarchitectonically unique cortical areas. To examine how neuronal signaling patterns are related to innate cortical functions, we detected intrinsic features of cortical firing by devising a metric that efficiently isolates non-Poisson irregular characteristics, independent of spike rate fluctuations that are caused extrinsically by ever-changing behavioral conditions. Using the new metric, we analyzed spike trains from over 1,000 neurons in 15 cortical areas sampled by eight independent neurophysiological laboratories. Analysis of firing-pattern dissimilarities across cortical areas revealed a gradient of firing regularity that corresponded closely to the functional category of the cortical area; neuronal spiking patterns are regular in motor areas, random in the visual areas, and bursty in the prefrontal area. Thus, signaling patterns may play an important role in function-specific cerebral cortical computation.

Neurons, or nerve cells in the brain, communicate with each other using stereotyped electric pulses, called spikes. It is believed that neurons convey information mainly through the frequency of the transmitted spikes, called the firing rate. In addition, neurons may communicate some information through the finer temporal patterns of the spikes. Neuronal firing patterns may depend on cellular organization, which varies among the regions of the brain, according to the roles they play, such as sensation, association, and motion. In order to examine the relationship among signals, structure, and function, we devised a metric to detect firing irregularity intrinsic and specific to individual neurons and analyzed spike sequences from over 1,000 neurons in 15 different cortical areas. Here we report two results of this study. First, we found that neurons exhibit stable firing patterns that can be characterized as “regular”, “random”, and “bursty”. Second, we observed a strong correlation between the type of signaling pattern exhibited by neurons in a given area and the function of that area. This suggests that, in addition to reflecting the cellular organization of the brain, neuronal signaling patterns may also play a role in specific types of neuronal computations.