Interspike Interval Sequences Research Articles

Characterizing the spike timing of a chaotic laser by using ordinal analysis and machine learning.

Semiconductor lasers with optical feedback are well-known nonlinear dynamical systems. Under appropriate feedback conditions, these lasers emit optical pulses that resemble neural spikes. Influenced by feedback delay and various noise sources, including quantum spontaneous emission noise, the dynamics are highly stochastic. A good understanding of the spike timing statistics is needed to develop photonic systems capable of using the fast-spiking laser output for novel applications, such as information processing or random number generation. Here we analyze experimental sequences of inter-spike intervals (ISIs) recorded when a sinusoidal signal was applied to the laser current. Different combinations of the DC value and frequency of the signal applied to the laser lead to ISI sequences with distinct statistical properties. This variability prompts an investigation into the relationship between experimental parameters and ISI sequence statistics, aiming to uncover potential encoding methods for optical spikes, since this can open a new way of encoding and decoding information in sequences of optical spikes. By using ordinal analysis and machine learning, we show that the ISI sequences have statistical ordinal properties that are similar to Flicker noise signals, characterized by a parameter α that varies with the signal that was applied to the laser current when the ISIs were recorded. We also show that for this dataset, the (α, permutation entropy) plane is more informative than the (complexity, permutation entropy) plane because it allows better differentiation of ISI sequences recorded under different experimental conditions, as well as better differentiation of original and surrogate ISI sequences.

An integrate-and-fire approach to Ca2+ signaling. Part II: Cumulative refractoriness

Inositol 1,4,5-trisphosphate-induced Ca2+ signaling is a second messenger system used by almost all eukaryotic cells. The agonist concentration stimulating Ca2+ signals is encoded in the frequency of a Ca2+ concentration spike sequence. When a cell is stimulated, the interspike intervals (ISIs) often show a distinct transient during which they gradually increase, a system property we refer to as cumulative refractoriness. We extend a previously published stochastic model to include the Ca2+ concentration in the intracellular Ca2+ store as a slow adaptation variable. This model can reproduce both stationary and transient statistics of experimentally observed ISI sequences. We derive approximate expressions for the mean and coefficient of variation of the stationary ISIs. We also consider the response to the onset of a constant stimulus and estimate the length of the transient and the strength of the adaptation of the ISI. We show that the adaptation sets the coefficient of variation in agreement with current ideas derived from experiments. Moreover, we explain why, despite a pronounced transient behavior, ISI correlations can be weak, as often observed in experiments. Finally, we fit our model to reproduce the transient statistics of experimentally observed ISI sequences in stimulated HEK cells. The fitted model is able to qualitatively reproduce the relationship between the stationary interval correlations and the number of transient intervals, as well as the strength of the ISI adaptation. We also find positive correlations in the experimental sequence that cannot be explained by our model.

Hippocampal and Medial Prefrontal Cortex Fractal Spiking Patterns Encode Episodes and Rules

A central question in neuroscience is how the brain represents and processes information to guide behavior. The principles that organize brain computations are not fully known, and could include scale-free, or fractal patterns of neuronal activity. Scale-free brain activity may be a natural consequence of the relatively small subsets of neuronal populations that respond to task features, i.e., sparse coding. The size of the active subsets constrains the possible sequences of inter-spike intervals (ISI), and selecting from this limited set may produce firing patterns across wide-ranging timescales that form fractal spiking patterns. To investigate the extent to which fractal spiking patterns corresponded with task features, we analyzed ISIs in simultaneously recorded populations of CA1 and medial prefrontal cortical (mPFC) neurons in rats performing a spatial memory task that required both structures. CA1 and mPFC ISI sequences formed fractal patterns that predicted memory performance. CA1 pattern duration, but not length or content, varied with learning speed and memory performance whereas mPFC patterns did not. The most common CA1 and mPFC patterns corresponded with each region's cognitive function: CA1 patterns encoded behavioral episodes which linked the start, choice, and goal of paths through the maze whereas mPFC patterns encoded behavioral “rules” which guided goal selection. mPFC patterns predicted changing CA1 spike patterns only as animals learned new rules. Together, the results suggest that CA1 and mPFC population activity may predict choice outcomes by using fractal ISI patterns to compute task features.

Basal ganglia neurons in healthy and parkinsonian primates generate recurring sequences of spikes.

The spiking activity of basal ganglia neurons can be characterized by summary statistics such as the average firing rate, or by measures of firing patterns, such as burst discharges, or oscillatory fluctuations of firing rates. Many of these features are altered by the presence of parkinsonism. This study examined another distinct attribute of firing activity, i.e., the occurrence of repeating sequences of interspike intervals (ISIs). We studied this feature in extracellular electrophysiological recordings that were made in the basal ganglia of rhesus monkeys, before and after they had been rendered parkinsonian by treatment with the neurotoxin 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine. Neurons in both pallidal segments and in the subthalamic nucleus tended to fire in repeating sequences, typically two ISIs long (i.e., involving three spikes). In recordings that were 5,000 interspike intervals long, 20%-40% of spikes participated in one of many sequences with each ISI replicating the sequence pattern with a timing error of ≤1%. Compared with similar analyses in shuffled representations of the same data, sequences were more common in the original representation of ISIs in all of the tested structures. Induction of parkinsonism reduced the proportion of sequence spikes in the external pallidum but increased it in the subthalamic nucleus. We found no relation between the sequence generation and the firing rate of neurons, and, at most, a weak correlation between sequence generation and the incidence of bursts. We conclude that basal ganglia neurons fire in recognizable sequences of ISIs, whose incidence is influenced by the induction of parkinsonism.NEW & NOTEWORTHY Previous work has shown that the timing of the electrical activity of basal ganglia neurons has nonstochastic properties, resulting in oscillatory firing patterns, or bursting. This article describes another such property in the monkey brain; a surprisingly large proportion of action potentials generated by cells in the extrastriatal basal ganglia are part of precisely timed recurring sequences of spiking events. We also found that the generation of these sequences changes substantially in the parkinsonian state.

Enhanced Signal Detection by Adaptive Decorrelation of Interspike Intervals.

Spike trains with negative interspike interval (ISI) correlations, in which long/short ISIs are more likely followed by short/long ISIs, are common in many neurons. They can be described by stochastic models with a spike-triggered adaptation variable. We analyze a phenomenon in these models where such statistically dependent ISI sequences arise in tandem with quasi-statistically independent and identically distributed (quasi-IID) adaptation variable sequences. The sequences of adaptation states and resulting ISIs are linked by a nonlinear decorrelating transformation. We establish general conditions on a family of stochastic spiking models that guarantee this quasi-IID property and establish bounds on the resulting baseline ISI correlations. Inputs that elicit weak firing rate changes in samples with many spikes are known to be more detectible when negative ISI correlations are present because they reduce spike count variance; this defines a variance-reduced firing rate coding benchmark. We performed a Fisher information analysis on these adapting models exhibiting ISI correlations to show that a spike pattern code based on the quasi-IID property achieves the upper bound of detection performance, surpassing rate codes with the same mean rate-including the variance-reduced rate code benchmark-by 20% to 30%. The information loss in rate codes arises because the benefits of reduced spike count variance cannot compensate for the lower firing rate gain due to adaptation. Since adaptation states have similar dynamics to synaptic responses, the quasi-IID decorrelation transformation of the spike train is plausibly implemented by downstream neurons through matched postsynaptic kinetics. This provides an explanation for observed coding performance in sensory systems that cannot be accounted for by rate coding, for example, at the detection threshold where rate changes can be insignificant.

MODELING REFRACTORINESS FOR STOCHASTICALLY DRIVEN SINGLE NEURONS

A description of the sequence of interspike intervals and of the subsequent firing times for single neurons is performed by means of an instantaneous return pro- cess in the presence of refractoriness. Every interspike interval consists of an absolute refractory period of fixed duration followed by a period of relative refractoriness whose duration is described by the first-passage time of the modeling diffusion process through a generally time-dependent threshold. In the cases of Wiener and Ornstein-Uhlenbeck processes, the interspike probability density functions and some of its statistical fea- tures are explicitly obtained for special monotonically non-increasing thresholds. 1 Introduction Stochastic models for neuronal firing in the presence of refractoriness have been the object of various investigations. The first attempt to study the effect of refractoriness in a point process is made in (22) and in (25) in which the authors consider the role played by the dead time in determining the distribution of the output when the input obeys a given distribution. Successively, an instantaneous return process, constructed on a diffusion, has been considered aiming to a quantitative description of neuron's membrane potential behavior. Within such a context, the presence of refractoriness has been included in two different ways. The first way assumes that the firing threshold acts as an elastic barrier that is partially transparent, i.e. such that its behavior is intermediate between total absorption and total reflection (cf. (3), (4), (5), (20)). Alternatively, the return process paradigm for the description of the time course of the membrane potential is analyzed by assuming that the neuronal refractoriness period is a random variable with a pre-assigned probability density (cf. (1), (10), (11), (15), (23)). Recently, in (2) and (16), the Wiener neuronal model in the presence of constant and of exponentially distributed refractoriness has been considered, and expressions for output distributions and for interspike interval densities have been obtained in closed form. Customarily, the firing threshold has been viewed in the literature as a constant which may not be appropriate, especially for rapidly firing cells (cf. (7), (13), (14), (26)). Indeed, when a neuron releases an action potential, it becomes temporarily incapable of responding to further input signals. In fact, for a period of time, of the order of one or two milliseconds, the neuron is unable to respond to any stimuli (absolute refractory period). After that, for several successive milliseconds its sensitivity to the incoming stimuli is normally reduced, in some cases increasing successively. This type of after-firing behavior (after potentials) may last up to about 100 msec. In the present context we focus our attention on a constant absolute refractory period followed by a period of relative refractoriness that we model as a random variable. Hence, after a spike release, we assume that the neuron is unable to fire again during the absolute refractory period, while the firing threshold is assumed to decrease progressively as the inhibitory effect of the previous spike fades away.

Characterizing signal encoding and transmission in class I and class II neurons via ordinal time-series analysis.

Neurons encode and transmit information in spike sequences. However, despite the effort devoted to understand the encoding and transmission of information, the mechanisms underlying the neuronal encoding are not yet fully understood. Here, we use a nonlinear method of time-series analysis (known as ordinal analysis) to compare the statistics of spike sequences generated by applying an input signal to the neuronal model of Morris-Lecar. In particular, we consider two different regimes for the neurons which lead to two classes of excitability: class I, where the frequency-current curve is continuous and class II, where the frequency-current curve is discontinuous. By applying ordinal analysis to sequences of inter-spike-intervals (ISIs) our goals are (1) to investigate if different neuron types can generate spike sequences which have similar symbolic properties; (2) to get deeper understanding on the effects that electrical (diffusive) and excitatory chemical (i.e., excitatory synapse) couplings have; and (3) to compare, when a small-amplitude periodic signal is applied to one of the neurons, how the signal features (amplitude and frequency) are encoded and transmitted in the generated ISI sequences for both class I and class II type neurons and electrical or chemical couplings. We find that depending on the frequency, specific combinations of neuron/class and coupling-type allow a more effective encoding, or a more effective transmission of the signal.

Sensitivity to Stimulus Irregularity Is Inherent in Neural Networks

Behavior is controlled by complex neural networks in which neurons process thousands of inputs. However, even short spike trains evoked in a single cortical neuron were demonstrated to be sufficient to influence behavior in vivo. Specifically, irregular sequences of interspike intervals (ISIs) had a more reliable influence on behavior despite their resemblance to stochastic activity. Similarly, irregular tactile stimulation led to higher rates of behavioral responses. In this study, we identify the mechanisms enabling this sensitivity to stimulus irregularity (SSI) on the neuronal and network levels using simulated spiking neural networks. Matching in vivo experiments, we find that irregular stimulation elicits more detectable network events (bursts) than regular stimulation. Dissecting the stimuli, we identify short ISIs-occurring more frequently in irregular stimulations-as the main drivers of SSI rather than complex irregularity per se. In addition, we find that short-term plasticity modulates SSI. We subsequently eliminate the different mechanisms in turn to assess their role in generating SSI. Removing inhibitory interneurons, we find that SSI is retained, suggesting that SSI is not dependent on inhibition. Removing recurrency, we find that SSI is retained due to the ability of individual neurons to integrate activity over short timescales ("cell memory"). Removing single-neuron dynamics, we find that SSI is retained based on the short-term retention of activity within the recurrent network structure ("network memory"). Finally, using a further simplified probabilistic model, we find that local network structure is not required for SSI. Hence, SSI is identified as a general property that we hypothesize to be ubiquitous in neural networks with different structures and biophysical properties. Irregular sequences contain shorter ISIs, which are the main drivers underlying SSI. The experimentally observed SSI should thus generalize to other systems, suggesting a functional role for irregular activity in cortex.

Study on intervention mechanism of acupuncture for neuron abnormal neural coding in visual cortex area 17 of monocular deprivation rats

To investigate the electrophysiological mechanism of acupuncture for the visual cortex plasticity during the sensitive period. Fifty 2-week-old Wistar rats were randomly divided into a blank control group, a model group, an early-stage acupuncture group, a middle-stage acupuncture group and a late-stage acupuncture group, 10 rats in each group. Monocular deprivation amblyopia models were prepared in rats except those in the blank control group by unilateral eyelid suture. After successful modeling, no treatment was applied to the rats in the model group. Rats in each acupuncture group were treated with acupuncture at bilateral Jingming (BL 1), Cuanzhu (BL 2), Fengchi (GB 20) and Guangming (GB 37), started from the 3rd day, 12th day or 21st day after modeling separately, once a day, for a total of 9 d treatment. The neuronal discharge frequency and action potential inter-spike interval (ISI) in the rat visual cortex area 17 of each group were measured by multi-channel microelectrode array nerve signal technique. The discharge number of neurons in the visual cortex of the model group was significantly lower than that in the blank control group (P 0.05). The discharge number in the middle-stage acupuncture group was lower than that in the early-stage acupuncture group (P<0.05), and the discharge number of the late-stage acupuncture group was lower than that in the middle-stage acupuncture group (P<0.05). The ISI sequences of the visual cortex neurons were mainly distributed under 0.3 s in the blank control group, under 15 s in the model group, under 1 s in the early-stage acupuncture group, under 4 s in the middle-stage acupuncture group, and under 10 s in the late-stage acupuncture group, divergent in each group. The neuronal coding appears abnormality in the visual cortex area 17 of monocular deprivation rats, indicating that there is a plasticity change in the visual cortex neurons during the sensitive period. Acupuncture has a significant effect on the abnormal neural coding. The therapeutic efficacy is closely related to the stage to start the treatment. Early stage treatment in the sensitive period is the key to achieving the good efficacy.

Determining the largest Lyapunov exponent of chaotic dynamics from sequences of interspike intervals contaminated by noise

We discuss abilities of quantifying low-dimensional chaotic oscillations at the input of two threshold models from the output sequences of interspike intervals in the presence of noise. We propose a modification of the standard approach for computing the largest Lyapunov exponent from a time series that verifies the performed estimations for noisy data. We consider features of its application to different types of point processes.

Emergence of spike correlations in periodically forced excitable systems.

In sensory neurons the presence of noise can facilitate the detection of weak information-carrying signals, which are encoded and transmitted via correlated sequences of spikes. Here we investigate the relative temporal order in spike sequences induced by a subthreshold periodic input in the presence of white Gaussian noise. To simulate the spikes, we use the FitzHugh-Nagumo model and to investigate the output sequence of interspike intervals (ISIs), we use the symbolic method of ordinal analysis. We find different types of relative temporal order in the form of preferred ordinal patterns that depend on both the strength of the noise and the period of the input signal. We also demonstrate a resonancelike behavior, as certain periods and noise levels enhance temporal ordering in the ISI sequence, maximizing the probability of the preferred patterns. Our findings could be relevant for understanding the mechanisms underlying temporal coding, by which single sensory neurons represent in spike sequences the information about weak periodic stimuli.

Stochastic slowly adapting ionic currents may provide a decorrelation mechanism for neural oscillators by causing wander in the intrinsic period.

Oscillatory neurons integrate their synaptic inputs in fundamentally different ways than normally quiescent neurons. We show that the oscillation period of invertebrate endogenous pacemaker neurons wanders, producing random fluctuations in the interspike intervals (ISI) on a time scale of seconds to minutes, which decorrelates pairs of neurons in hybrid circuits constructed using the dynamic clamp. The autocorrelation of the ISI sequence remained high for many ISIs, but the autocorrelation of the ΔISI series had on average a single nonzero value, which was negative at a lag of one interval. We reproduced these results using a simple integrate and fire (IF) model with a stochastic population of channels carrying an adaptation current with a stochastic component that was integrated with a slow time scale, suggesting that a similar population of channels underlies the observed wander in the period. Using autoregressive integrated moving average (ARIMA) models, we found that a single integrator and a single moving average with a negative coefficient could simulate both the experimental data and the IF model. Feeding white noise into an integrator with a slow time constant is sufficient to produce the autocorrelation structure of the ISI series. Moreover, the moving average clearly accounted for the autocorrelation structure of the ΔISI series and is biophysically implemented in the IF model using slow stochastic adaptation. The observed autocorrelation structure may be a neural signature of slow stochastic adaptation, and wander generated in this manner may be a general mechanism for limiting episodes of synchronized activity in the nervous system.

Bifurcation and Spike Adding Transition in Chay–Keizer Model

Electrical bursting is an activity which is universal in excitable cells such as neurons and various endocrine cells, and it encodes rich physiological information. As burst delay identifies that the signal integration has reached the threshold at which it can generate an action potential, the number of spikes in a burst may have essential physiological implications, and the transition of bursting in excitable cells is associated with the bifurcation phenomenon closely. In this paper, we focus on the transition of the spike count per burst of the pancreatic [Formula: see text]-cells within a mathematical model and bifurcation phenomenon in the Chay–Keizer model, which is utilized to simulate the pancreatic [Formula: see text]-cells. By the fast–slow dynamical bifurcation analysis and the bi-parameter bifurcation analysis, the local dynamics of the Chay–Keizer system around the Bogdanov–Takens bifurcation is illustrated. Then the variety of the number of spikes per burst is discussed by changing the settings of a single parameter and bi-parameter. Moreover, results on the number of spikes within a burst are summarized in ISIs (interspike intervals) sequence diagrams, maximum and minimum, and the number of spikes under bi-parameter value changes.

Distinguishing cognitive state with multifractal complexity of hippocampal interspike interval sequences.

Fractality, represented as self-similar repeating patterns, is ubiquitous in nature and the brain. Dynamic patterns of hippocampal spike trains are known to exhibit multifractal properties during working memory processing; however, it is unclear whether the multifractal properties inherent to hippocampal spike trains reflect active cognitive processing. To examine this possibility, hippocampal neuronal ensembles were recorded from rats before, during and after a spatial working memory task following administration of tetrahydrocannabinol (THC), a memory-impairing component of cannabis. Multifractal detrended fluctuation analysis was performed on hippocampal interspike interval sequences to determine characteristics of monofractal long-range temporal correlations (LRTCs), quantified by the Hurst exponent, and the degree/magnitude of multifractal complexity, quantified by the width of the singularity spectrum. Our results demonstrate that multifractal firing patterns of hippocampal spike trains are a marker of functional memory processing, as they are more complex during the working memory task and significantly reduced following administration of memory impairing THC doses. Conversely, LRTCs are largest during resting state recordings, therefore reflecting different information compared to multifractality. In order to deepen conceptual understanding of multifractal complexity and LRTCs, these measures were compared to classical methods using hippocampal frequency content and firing variability measures. These results showed that LRTCs, multifractality, and theta rhythm represent independent processes, while delta rhythm correlated with multifractality. Taken together, these results provide a novel perspective on memory function by demonstrating that the multifractal nature of spike trains reflects hippocampal microcircuit activity that can be used to detect and quantify cognitive, physiological, and pathological states.

Effects of periodic forcing on the temporally correlated spikes of a semiconductor laser with feedback.

Optical excitable devices that mimic neuronal behavior can be building-blocks of novel, brain-inspired information processing systems. A relevant issue is to understand how such systems represent, via correlated spikes, the information of a weak external input. Semiconductor lasers with optical feedback operating in the low frequency fluctuations regime have been shown to display optical spikes with intrinsic temporal correlations similar to those of biological neurons. Here we investigate how the spiking laser output represents a weak periodic input that is implemented via direct modulation of the laser pump current. We focus on understanding the influence of the modulation frequency. Experimental sequences of inter-spike-intervals (ISIs) are recorded and analyzed by using the ordinal symbolic methodology that identifies and characterizes serial correlations in datasets. The change in the statistics of the various symbols with the modulation frequency is empirically shown to be related to specific changes in the ISI distribution, which arise due to different phase-locking regimes. A good qualitative agreement is also found between simulations of the Lang and Kobayashi model and observations. This methodology is an efficient way to detect subtle changes in noisy correlated ISI sequences and may be applied to investigate other optical excitable devices.

Quantifying chaotic dynamics from integrate-and-fire processes

Characterizing chaotic dynamics from integrate-and-fire (IF) interspike intervals (ISIs) is relatively easy performed at high firing rates. When the firing rate is low, a correct estimation of Lyapunov exponents (LEs) describing dynamical features of complex oscillations reflected in the IF ISI sequences becomes more complicated. In this work we discuss peculiarities and limitations of quantifying chaotic dynamics from IF point processes. We consider main factors leading to underestimated LEs and demonstrate a way of improving numerical determining of LEs from IF ISI sequences. We show that estimations of the two largest LEs can be performed using around 400 mean periods of chaotic oscillations in the regime of phase-coherent chaos. Application to real data is discussed.

Multifractal analysis of information processing in hippocampal neural ensembles during working memory under Δ9-tetrahydrocannabinol administration

BackgroundMultifractal analysis quantifies the time-scale-invariant properties in data by describing the structure of variability over time. By applying this analysis to hippocampal interspike interval sequences recorded during performance of a working memory task, a measure of long-range temporal correlations and multifractal dynamics can reveal single neuron correlates of information processing. New methodWavelet leaders-based multifractal analysis (WLMA) was applied to hippocampal interspike intervals recorded during a working memory task. WLMA can be used to identify neurons likely to exhibit information processing relevant to operation of brain–computer interfaces and nonlinear neuronal models. ResultsNeurons involved in memory processing (“Functional Cell Types” or FCTs) showed a greater degree of multifractal firing properties than neurons without task-relevant firing characteristics. In addition, previously unidentified FCTs were revealed because multifractal analysis suggested further functional classification. The cannabinoid type-1 receptor (CB1R) partial agonist, tetrahydrocannabinol (THC), selectively reduced multifractal dynamics in FCT neurons compared to non-FCT neurons. Comparison with existing methodsWLMA is an objective tool for quantifying the memory-correlated complexity represented by FCTs that reveals additional information compared to classification of FCTs using traditional z-scores to identify neuronal correlates of behavioral events. Conclusionz-Score-based FCT classification provides limited information about the dynamical range of neuronal activity characterized by WLMA. Increased complexity, as measured with multifractal analysis, may be a marker of functional involvement in memory processing. The level of multifractal attributes can be used to differentially emphasize neural signals to improve computational models and algorithms underlying brain–computer interfaces.

Phase description of the Huber-Braun neuron model for mammalian cold receptors

The spiking activity of mammalian cold receptors is described by the Huber-Braun neuron model. Sweeping temperature as a control parameter across a biologically relevant range this model exhibits a complex bifurcation structure seen in the sequence of interspike intervals. The model’s distinctive feature is the interaction between a fast spike generating dynamics and a slow subthreshold oscillation. Viewing the spike generation as a cycle, the dynamics may also be modeled phenomenologically by two phases, one for the spike cycle and the second for the slow subthreshold oscillation. In fact, a phase model of temperature-dependent mammalian cold receptors was already proposed by Roper et al. (2000). Here we follow their approach and investigate to what extent this model is able to reproduce the bifurcation patterns of the Huber-Braun model. Special attention is paid to the tonic firing to bursting transition observed in the low temperature range.

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.

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.