Interspike Interval Distribution Research Articles

On modeling the neuronal activity in movement disorder patients by using the Ornstein Uhlenbeck process.

Mathematical models of the neuronal activity in the affected brain regions of Essential Tremor (ET) and Parkinson's Disease (PD) patients could shed light into the underlying pathophysiology of these diseases, which in turn could help develop personalized treatments including adaptive Deep Brain Stimulation (DBS). In this paper, we use an Ornstein Uhlenbeck Process (OUP) to model the neuronal spiking activity recorded from the brain of ET and PD patients during DBS stereotactic surgery. The parameters of the OUP are estimated based on Inter Spike Interval (ISI) measurements, i.e., the time interval between two consecutive neuronal firings, by means of the Fortet Integral Equation (FIE). The OUP model parameters identified with the FIE method (OUP-FIE) are then used to simulate the ISI distribution resulting from the OUP. Other widely used neuronal activity models, such as the Poisson Process (PP), the Brownian Motion (BM), and the OUP whose parameters are extracted by matching the first two moments of the ISI (OUP-MOM), are also considered. To quantify how close the simulated ISI distribution is to the measured ISI distribution, the Integral Square Error (ISE) criterion is adopted. Amongst all considered stochastic processes, the ISI distribution generated by the OUP-FIE method is shown to produce the least ISE. Finally, a directional Wilcoxon signed rank test is used to show statistically significant reduction in the ISE value obtained from the OUP-FIE compared to the other stochastic processes.

Correlation between spike statistics and T-type calcium channel activation in simulated subthalamic nucleus neurons

Firing patterns of subthalamic nucleus (STN) neurons are closely related to behavioral functions such as motor controls. In the patients of the Parkinson disease, which is a representative of basal ganglia disorders, STN neurons show a distinctive firing pattern, rebound bursts. To understand the relation between the STN firing patterns and their functional roles, it is useful to characterize the firing patterns, especially the burst firings. Although several methods have been proposed so far, an objective definition of burst firings is still unclear. On the other hand, the mechanism for the rebound bursts is attributed to the low-voltage-activated calcium channels, also known as T-type calcium channels [1]. The channels are activated by strong inhibitory synaptic inputs from globus pallidal neurons that have reciprocal connections with STN. If we could observed directly T-type calcium currents, the observation could be a criterion to determine whether the firing activity is a burst firing or not. Here we investigated to what degree spike statistics correlate with an activation level of the T-type calcium currents by simulating an STN neuron model [2]. We employed the following spike statistics of interspike intervals (ISIs) of STN neural activities: (i) coefficient of variation (Cv), (ii) time-local variation (Lv) [3], (iii) chisquared test statistic between a spike count histogram and a Poisson distribution [4], and (iv)-(vi) KolmogorovSmirnov (K-S) statistics in K-S test between an observed and hypothesized ISI distributions (Gaussian, exponential, and gamma distributions). By comparing multiple correlation coefficients between these statistics and temporally-averaged conductances of the T-type calcium channels under different conditions including burst-preferred parameter sets, we found that the K-S statistic between the observed ISI distribution and the hypothesized exponential one (v) showed the highest correlation with the averaged T-type calcium currents among them under any condition. This would provide a more objective criterion, which is linked to the generation mechanism, to define rebound bursts observed in pathological STN.

Astrocytic theory of working memory

Working memory (WM) is the ability to transiently hold and manipulate goal-related information to forthcoming actions. In primates performing delayed-response tasks, the neural correlate of WM is the emergence of selective persistent activity in the prefrontal cortex (PFC), during the delay period – i.e., the increase of neuronal firing rates compared to baseline in response of specific stimulatory cues. Neuronal firing during such persistent activity can be highly irregular, with a coefficient of variation (CV) of the interspike interval distribution larger than one. The origin of this increased irregularity is a matter of debate, as it is not known whether it stems from inherent properties of neurons or the plasticity of their synaptic connections or both. Astrocytes, the main type of glial cells in the cortex, have recently emerged as potential active players in synaptic plasticity due to their proposed ability to regulate synaptic neurotransmitter release in response to neuronal activity. Because available WM models do not take into account astrocytes, we investigated the possible role of astrocyte regulation of synaptic plasticity in the emergence of persistent neuronal firing in WM and its statistics. Our modeling study suggests that selective persistent activity could indeed emerge from astrocyte-mediated short-term synaptic facilitation triggered by the brief presentation of a stimulatory cue (Figures 1A,B). Bifurcation analysis revealed that astrocyte-mediated facilitation could outlast the cue and switch to a self-sustained mode where it is maintained by the ongoing synaptic activity, consistent with the existence of bistability of synaptic release (Figures 1C,D). In parallel, mean field analysis showed that such bistable synaptic release resulted in bistability of neural firing rates, with persistent “UP” states characterized by larger irregularity (Figures 1E,F). Altogether, these results suggest a novel astrocyte-based mechanism for WM providing experimentally testable hypotheses for the possible involvement of astrocytes in WM-related cognitive tasks.

Noise-induced speed up in repetitively firing neurons occurs far from spike threshold

Injecting white noise currents has been shown to increase mean firing rate in several types of neuron. This speed up is generally attributed to the fact that depolarizing currents on the tails of the noise distribution push cells past spike threshold sooner than they would have otherwise. As part of a larger effort to understand the dynamic properties of rat subthalamic (STN) neurons, we have performed perforated patch recordings of these tonically active neurons in brain slices, exposing them to contiguous sequences of brief current pulses drawn from a Gaussian distribution. Pulsed noise disrupted firing pattern with no significant change in rate over a wide range of noise amplitudes, so long as the pulse durations were brief (<2 ms). With longer pulse durations, neurons increased their firing rate with noise, with greater increases for pulses of larger magnitude and longer duration. However, individual neurons displayed varying degrees of noise-induced speed up. We used phase resetting methods to understand the noise-induced changes in the mean and variance of the recorded interspike interval distributions. For each cell, a phase response curves (PRC) was derived by using linear interpolation to estimate the phase of the neuron between spikes, followed by a linear regression to estimate the sensitivity of the cell to input perturbations at each phase. We then compared Monte Carlo simulations of a one-dimensional phase model to predict the response patterns of individual neurons. Surprisingly, simulation results produced a good match to experimental mean and variances even at noise levels leading to autocorrelation histograms with no visible periodicity (CVs ~ 0.4). We then investigated simulations results in detail to understand the dynamic mechanism that underlies the noise-induced speed up in these neurons. Plots of the mean phase as a function of time revealed that the noise-induced phase advance occurred early in an interspike interval, well before trajectories approach the time of the next spike. This phase advance can be understood as ‘noise-induced drift’ in the Stratanovich interpretation of stochastic differential equations, with possible supralinear contributions for pulses of finite duration. Current research uses artificially constructed PRCs to examine how the increasing spread of the phase distribution during each intervals leads to an asymmetric sampling of the noise-induced drift, preventing the early noise-induced speed up from being canceled by an expected noise-induced slowing as the PRC approaches zero near the time of the next spike.

One month in the life of a neuron: longitudinal single-unit electrophysiology in the monkey visual system.

Conventional recording methods generally preclude following the activity of the same neurons in awake animals across days. This limits our ability to systematically investigate the principles of neuronal specialization, or to study phenomena that evolve over multiple days such as experience-dependent plasticity. To redress this shortcoming, we developed a drivable, chronically implanted microwire recording preparation that allowed us to follow visual responses in inferotemporal (IT) cortex in awake behaving monkeys across multiple days, and in many cases across months. The microwire bundle and other implanted components were MRI compatible and thus permitted in the same animals both functional imaging and long-term recording from multiple neurons in deep structures within a region the approximate size of one voxel (<1 mm). The distinct patterns of stimulus selectivity observed in IT neurons, together with stable features in spike waveforms and interspike interval distributions, allowed us to track individual neurons across weeks and sometimes months. The long-term consistency of visual responses shown here permits large-scale mappings of neuronal properties using massive image libraries presented over the course of days. We demonstrate this possibility by screening the visual responses of single neurons to a set of 10,000 stimuli.

Ghost stochastic resonance in FitzHugh–Nagumo circuit

The response of a neural circuit submitted to a bi-chromatic stimulus and corrupted by noise is investigated. In the presence of noise, when the spike firing of the circuit is analysed, a frequency not present at the circuit input appears. For a given range of noise intensities, it is shown that this ghost frequency is almost exclusively present in the interspike interval distribution. This phenomenon is for the first time shown experimentally in a FitzHugh-Nagumo circuit.

Temperature-driven coherence resonance and stochastic resonance in a thermochemical system.

We perform the stochastic analysis of a thermochemical system using a master equation which describes a chemical reaction and includes discrete and continuous temperature jumps. We study the time evolution of the system selecting the temperature of the thermostat as an easily tunable control parameter. Depending on the thermostat temperature, the system can be in an excitable, oscillatory, or stationary regime. Stochastic time series for the system temperature are generated and the distributions of interspike intervals are analyzed in the three dynamical regimes separated by a homoclinic bifurcation and a Hopf bifurcation. Different constructive roles of internal fluctuations are exhibited. A noise-induced transition is observed in the vicinity of the Hopf bifurcation. Coherence resonance and stochastic resonance are found in the oscillatory regime. In a range of thermostat temperatures, a nontrivial behavior of the highly nonlinear system is revealed by the existence of both a minimum and a maximum in the scaled standard deviation of interspike intervals as a function of particle number. This high sensitivity to system size illustrates that controlling dynamics in nanoreactors may remain a difficult task.

Spiking Irregularity and Frequency Modulate the Behavioral Report of Single-Neuron Stimulation

The action potential activity of single cortical neurons can evoke measurable sensory effects, but it is not known how spiking parameters and neuronal subtypes affect the evoked sensations. Here, we examined the effects of spike train irregularity, spike frequency, and spike number on the detectability of single-neuron stimulation in rat somatosensory cortex. For regular-spiking, putative excitatory neurons, detectability increased with spike train irregularity and decreasing spike frequencies but was not affected by spike number. Stimulation of single, fast-spiking, putative inhibitory neurons led to a larger sensory effect compared to regular-spiking neurons, and the effect size depended only on spike irregularity. An ideal-observer analysis suggests that, under our experimental conditions, rats were using integration windows of a few hundred milliseconds or more. Our data imply that the behaving animal is sensitive to single neurons' spikes and even to their temporal patterning.

Beyond GLMs: A Generative Mixture Modeling Approach to Neural System Identification

Generalized linear models (GLMs) represent a popular choice for the probabilistic characterization of neural spike responses. While GLMs are attractive for their computational tractability, they also impose strong assumptions and thus only allow for a limited range of stimulus-response relationships to be discovered. Alternative approaches exist that make only very weak assumptions but scale poorly to high-dimensional stimulus spaces. Here we seek an approach which can gracefully interpolate between the two extremes. We extend two frequently used special cases of the GLM—a linear and a quadratic model—by assuming that the spike-triggered and non-spike-triggered distributions can be adequately represented using Gaussian mixtures. Because we derive the model from a generative perspective, its components are easy to interpret as they correspond to, for example, the spike-triggered distribution and the interspike interval distribution. The model is able to capture complex dependencies on high-dimensional stimuli with far fewer parameters than other approaches such as histogram-based methods. The added flexibility comes at the cost of a non-concave log-likelihood. We show that in practice this does not have to be an issue and the mixture-based model is able to outperform generalized linear and quadratic models.

Single-Neuron Correlates of Atypical Face Processing in Autism

People with autism spectrum disorder (ASD) show abnormal processing of faces. A range of morphometric, histological, and neuroimaging studies suggest the hypothesis that this abnormality may be linked to the amygdala. We recorded data from single neurons within the amygdalae of two rare neurosurgical patients with ASD. While basic electrophysiological response parameters were normal, there were specific and striking abnormalities in how individual facial features drove neuronal response. Compared to control patients, a population of neurons in the two ASD patients responded significantly more to the mouth, but less to the eyes. Moreover, we found a second class of face-responsive neurons for which responses to faces appeared normal. The findings confirm the amygdala's pivotal role in abnormal face processing by people with ASD at the cellular level and suggest that dysfunction may be traced to a specific subpopulation of neurons with altered selectivity for the features of faces.

The emergence of subcellular pacemaker sites for calcium waves and oscillations

Calcium (Ca(2+)) waves generating oscillatory Ca(2+) signals are widely observed in biological cells. Experimental studies have shown that under certain conditions, initiation of Ca(2+) waves is random in space and time, while under other conditions, waves occur repetitively from preferred locations (pacemaker sites) from which they entrain the whole cell. In this study, we use computer simulations to investigate the self-organization of Ca(2+) sparks into pacemaker sites generating Ca(2+) oscillations. In both ventricular myocyte experiments and computer simulations of a heterogeneous Ca(2+) release unit (CRU) network model, we show that Ca(2+) waves occur randomly in space and time when the Ca(2+) level is low, but as the Ca(2+) level increases, waves occur repetitively from the same sites. Our analysis indicates that this transition to entrainment can be attributed to the fact that random Ca(2+) sparks self-organize into Ca(2+) oscillations differently at low and high Ca(2+) levels. At low Ca(2+), the whole cell Ca(2+) oscillation frequency of the coupled CRU system is much slower than that of an isolated single CRU. Compared to a single CRU, the distribution of interspike intervals (ISIs) of the coupled CRU network exhibits a greater variation, and its ISI distribution is asymmetric with respect to the peak, exhibiting a fat tail. At high Ca(2+), however, the coupled CRU network has a faster frequency and lesser ISI variation compared to an individual CRU. The ISI distribution of the coupled network no longer exhibits a fat tail and is well-approximated by a Gaussian distribution. This same Ca(2+) oscillation behaviour can also be achieved by varying the number of ryanodine receptors per CRU or the distance between CRUs. Using these results, we develop a theory for the entrainment of random oscillators which provides a unified explanation for the experimental observations underlying the emergence of pacemaker sites and Ca(2+) oscillations.

On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model

An Ornstein-Uhlenbeck diffusion process is considered as a model for the membrane potential activity of a single neuron. We assume that the neuron is subject to a sequence of inhibitory and excitatory post-synaptic potentials that occur with time-dependent rates. The resulting process is characterized by time-dependent drift. For this model, we construct the return process describing the membrane potential. It is a non homogeneous Ornstein-Uhlenbeck process with jumps on which the effect of random refractoriness is introduced. An asymptotic analysis of the process modeling the number of firings and the distribution of interspike intervals is performed under the assumption of exponential distribution for the firing time. Some numerical evaluations are performed to provide quantitative information on the role of the parameters.

Cooperative behavior in a jump diffusion model for a simple network of spiking neurons

The distribution of time intervals between successive spikes generated by a neuronal cell --the interspike intervals (ISI)-- may reveal interesting features of the underlying dynamics. In this study we analyze the ISI sequence --the spike train-- generated by a simple network of neurons whose output activity is modeled by a jump-diffusion process. We prove that, when specific ranges of the involved parameters are chosen, it is possible to observe multimodal ISI distributions which reveal that the modeled network fires with more than one single preferred time interval. Furthermore, the system exhibits resonance behavior, with modulation of the spike timings by the noise intensity. We also show that inhibition helps the signal transmission between the units of the simple network.

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.

Information gain on variable neuronal firing rate

The question of how much information can be theoretically gained from variable neuronal firing rate with respect to constant mean firing rate is investigated. For this purpose, we employ the Kullback-Leibler divergence as a measure of information gain. We first give a statistical interpretation of this information in terms of detectability of rate variation: the lower bound of detectable rate variation, below which the temporal variation of firing rate is undetectable with a Bayesian decoder, is entirely determined by this information [1]. We derive a formula for the lower bound, which tells how much information is necessary for the rate variation to be detected from spike trains. For instance, if a spike, on average, is expected to be observed in the characteristic timescale of the rate variation, it is necessary for the spike train to carry more than 0.36 bits per spike information so that the underlying rate variation is detectable. We show that the information depends not only of the variation of firing rates (i.e., signals), but also significantly on the dispersion properties of neuronal firing described by the shape of interspike interval (ISI) distribution (i.e., noise properties). It is shown that under certain condition, the gamma distribution attains the theoretical lower bound of the information among all ISI distributions when the coefficient of variation of ISIs is given [2]. We also give a useful formula for the (approximate) information, which provides a theoretical prediction for the range of the information gain. With the basis of the theoretical investigations, we propose a practical procedure for estimating the information from spike trains, and apply this method to biological spike data recorded from a cortical area. The estimated information ranges up to 0.8 bits/spike, which roughly matches the theoretical prediction. Acknowledgements The author was supported by JSPS KAKENHI Grant Number 24700287.

Going beyond Poisson processes: a new statistical framework in neuronal modeling and data analysis

Cortical spike trains in vivo often show a high irregularity, reflected by a coefficient of variation (CV) close to one [1]. Such irregular single neuron spiking has consequently been associated with Poisson processes, and many models involving neuronal spike trains inherited this ideology. This viewpoint is further supported by the Palm-Khintchine theorem, which states that a superposition of a large number of renewal processes with very small intensity behaves like a Poisson process. It was demonstrated, however, that this theorem doesn't always apply to the superposition of neuronal spike trains [2-4]. Moreover, Poisson processes lack the temporal properties observed in population responses to input modulation either via a stimulus in vivo [5] or via electrical stimulation in vitro [6]. In this study, we report on new techniques to deal with non-Possonian aspects of stationary neuronal spike trains, as well as non-equilibrium population responses [7], based on Markov point processes (MPP), commonly known as continuous time Markov chains (CTMC). We compute the interspike interval (ISI) distribution by algebraically solving the first passage time problem for MPP neuron models, and compute the transient population responses with a similar technique. The same technique is used to compute exact cross-correlation functions for a shared input paradigm [8]. We advertise MPPs as a new powerful framework in neural network modeling and neural data analysis with many possible applications.

Decision-making out of neural events: from discrimination information to psychometric power laws

The strength/intensity of the stimulus in the random dot motion task (RDMT) [1] is determined by the percentage of dots in the kinematogram moving towards a saccadic target, a. Due to the uncertainty in the stimuli, neurons in sensory systems have evolved to transform environmental information, comprising evidence upon which a decision can be made (e.g. saccading to a). The neurons in the middle-temporal area (MT) appear to produce such evidence during the RDMT, given their tuning to a 'preferred' direction of visual motion. If the dots move predominantly in the preferred direction of an MT neuron, it generates inter-spike intervals (ISI) supporting a saccade to a. These ISIs seem randomly sampled from a distribution, fa, with mean, μa. Otherwise, the ISIs follow another distribution, fb, with mean, μb, where μb is larger than μa and this difference increases with stimulus strength. The accuracy vs motion-strength function of an ideal observer provided with empirical distributions like fa and fb, from a single MT neuron, approximates the subject's psychometric function (at the behavioral level) [1]. The distributions fa and fb are non-negative, positively skewed and have a mode larger than 0 (figure ​(figure1A),1A), as is typical for neural events recorded in many brain areas. Here we investigate why this is advantageous for decision-making. As theoretical decision-making units, we produced 5 new instantiations of the multi-hypothesis sequential probability ratio test (MSPRT) [2]. Each unit assumes its stream of input evidence to follow 1 of 5 probability density functions (PDF) whose compatibility with the empirical distribution of ISIs varies (figure ​(figure1A).1A). These include the Inverse Gaussian, Lognormal, Gamma, Inverse-Gamma and Exponential PDFs (the latter is the distribution of the inter-event times in the oft-used Poisson process). Under equal and appropriate conditions, we then compared their mean decision sample with that of an MSPRT instantiation that assumes Gaussian inputs proposed in [3] and discussed in general in [4], as exemplified in figure ​figure1B.1B. The mean decision sample is the mean number of observations required by a unit to identify which of N parallel information sources supports saccading to a, with a given accuracy. This decision sample is a model of the 'neural decision time'; the psychophysical reaction time also includes sensory and motor delays. The pattern of our results is explicable using a measure of the discrimination information between fa and fb, i.e. the Kullback-Leibler divergence (KLD). We found that, the mean decision sample decreases with increasing fa to fb KLD and, crucially, this follows a power law (figure ​(figure1C).1C). At the behavioral level, Pieron [5] reported the mean reaction time to the presentation of a stimulus (go/no-go decision-making) being shorter for more intense stimuli, and that a power law relates these measures. The universality of Pieron's law indicates that it can inform us of something fundamental about sensorimotor decision-making. Our results suggest that its explanation could lie in the power law relationship between the mean neural decision time and the discrimination information (KLD) among the distributions of sensory evidence. Figure 1 (A) PDFs of interest fitted to ISIs (grey bars) recorded in [1]from the MT during the RDMT. (B) mean decision sample for each MSPRT realisation (accuracy 95% over 1000 trials). (C) the values in (B) vs the fa to fb KLD. The dashed line is a fitted power ...

Maximum penalized likelihood estimation of interspike interval distribution

We address the problem of non-parametric estimation of the recently proposed measures of statistical dispersion of positive continuous random variables. The measures are based on the concepts of differential entropy and Fisher information and describe the spread or variability of the random variable from a different point of view than the ubiquitously used concept of standard deviation. The maximum penalized likelihood (MPL) estimation [1,2] of the probability density function is applied and a methodology how to estimate the dispersion measures is presented. We illustrate the approach on three standard statistical models describing the neuronal activity. Frequently, the dispersion of measured data needs to be described. Although standard deviation is used ubiquitously for quantification of variability, such approach has limitations. For example highly variable data might not be random at all if it consists only of extremely small and extremely large measurements. Although the probability density function, or its estimate provide a complete view, quantitative methods are needed in order to compare different models or experimental results. In a series of recent studies, some alternative measures of dispersion were proposed. However, the estimation of these coefficients from data is more problematic. We present the obtained estimations of the dispersion coefficients based on the MPL method, which extends our previous study [3].

Method for analyzing spike patterns with Markov transition matrices and Kullback-Leibler divergence

We describe a novel method for analyzing neural data that uses a combination of Markov transition matrices and Kullback-Leibler divergence to characterize spike history and spike patterns. For this method, the interspike intervals (ISIs) are divided into bins by quantiles, and a Markov transition matrix is computed for the ISI sequence. This Markov transition matrix is then compared to another Markov transition matrix constructed under the assumption of independent spiking. We then compute the Kullback-Leibler divergence between the distributions of ISI transition probabilities for each bin. The purpose of this method is to quantify the effects of spike history on neuron output and to help better characterize the flaws associated with assuming independent spiking. We test this method on both simulated data and experimental data from primary visual cortex of cats, publicly available through CRCNS [1-3]. Interspike intervals are simulated based on exponential and gamma distributions fit to the corresponding experimental ISI distributions.

Spike-Based Probabilistic Inference in Analog Graphical Models Using Interspike-Interval Coding

Temporal spike codes play a crucial role in neural information processing. In particular, there is strong experimental evidence that interspike intervals (ISIs) are used for stimulus representation in neural systems. However, very few algorithmic principles exploit the benefits of such temporal codes for probabilistic inference of stimuli or decisions. Here, we describe and rigorously prove the functional properties of a spike-based processor that uses ISI distributions to perform probabilistic inference. The abstract processor architecture serves as a building block for more concrete, neural implementations of the belief-propagation (BP) algorithm in arbitrary graphical models (e.g., Bayesian networks and factor graphs). The distributed nature of graphical models matches well with the architectural and functional constraints imposed by biology. In our model, ISI distributions represent the BP messages exchanged between factor nodes, leading to the interpretation of a single spike as a random sample that follows such a distribution. We verify the abstract processor model by numerical simulation in full graphs, and demonstrate that it can be applied even in the presence of analog variables. As a particular example, we also show results of a concrete, neural implementation of the processor, although in principle our approach is more flexible and allows different neurobiological interpretations. Furthermore, electrophysiological data from area LIP during behavioral experiments are assessed in light of ISI coding, leading to concrete testable, quantitative predictions and a more accurate description of these data compared to hitherto existing models.