Interspike Interval Statistics Research Articles

Comparison of modulation efficiency between normal and degenerated primate retina.

With electrical stimulation, retinal prostheses bypass dysfunctional photoreceptors and activate the surviving bipolar or retinal ganglion cells (RGCs). Therefore, the effective modulation of RGCs is crucial for developing retinal prostheses. Substantial research has been performed on the ability of an electrical stimulus to generate a reliable RGC response. However, different experimental conditions show varying levels of how well the electrical stimulation evokes RGC spikes. Therefore, in this study, we attempted to extract an indicator to understand how the electrical stimulation effectively evokes RGC spikes. Six cynomolgus monkeys were used: three as controls and three as an N-methyl-N-nitrosourea (MNU)-induced retinal degeneration model. The retinal recordings were performed using 8 × 8 multi-electrode arrays (MEAs). Electrical stimulation consisted of symmetrical biphasic pulses of varying amplitudes and durations. The number of stimulation conditions that resulted in significantly higher post-stimulation firing rates than pre-stimulus firing rates was defined as the modulation efficiency ratio (MER). The MER was significantly lower in degenerated retinas than in normal retinas. We investigated the relationship between the variables and the MER in normal and degenerated primate RGCs. External variables, such as duration and inter-electrode distance, and internal variables, such as average firing rates and statistics (mean, standard deviation, and coefficient of variation [CV]) of inter-spike intervals (ISIs) of spontaneous spikes, were used. External variables had similar effects on MER in normal and degenerated RGCs. In contrast, internal variables affected MER differently in normal and degenerated RGCs. While in normal RGCs, they were not related to MER, in degenerated RGCs, the mean ISIs were positively correlated with MER, and the CV of ISIs was negatively correlated with MER. The most important variable affecting MER was the mean ISI. A shorter ISI indicates hyperactive firing in the degenerated retina, which prevents electrical stimulation from evoking more RGCs. We believe that this hyperactivity in degenerated retinas results in a lower MER than that in the normal retina. Our findings can be used to optimize the selection of stimulation channels for in vitro MEA experiments and practical calibration methods to achieve higher efficiency when testing retinal prostheses.

Phase-Dependent Response to Electrical Stimulation of Cortical Networks during Recurrent Epileptiform Short Discharge Generation In Vitro.

The closed-loop control of pathological brain activity is a challenging task. In this study, we investigated the sensitivity of continuous epileptiform short discharge generation to electrical stimulation applied at different phases between the discharges using an in vitro 4-AP-based model of epilepsy in rat hippocampal slices. As a measure of stimulation effectiveness, we introduced a sensitivity function, which we then measured in experiments and analyzed with different biophysical and abstract mathematical models, namely, (i) the two-order subsystem of our previous Epileptor-2 model, describing short discharge generation governed by synaptic resource dynamics; (ii) a similar model governed by shunting conductance dynamics (Epileptor-2B); (iii) the stochastic leaky integrate-and-fire (LIF)-like model applied for the network; (iv) the LIF model with potassium M-channels (LIF+KM), belonging to Class II of excitability; and (v) the Epileptor-2B model with after-spike depolarization. A semi-analytic method was proposed for calculating the interspike interval (ISI) distribution and the sensitivity function in LIF and LIF+KM models, which provided parametric analysis. Sensitivity was found to increase with phase for all models except the last one. The Epileptor-2B model is favored over other models for subthreshold oscillations in the presence of large noise, based on the comparison of ISI statistics and sensitivity functions with experimental data. This study also emphasizes the stochastic nature of epileptiform discharge generation and the greater effectiveness of closed-loop stimulation in later phases of ISIs.

Exploring Filter Banks and Spike Interval Statistics of Level-Crossing ADCs for Fault Diagnosis of Rolling Element Bearings

Nowadays, lots of data are generated in industries using vibration sensors to evaluate the equipment’s working condition and identify faults. A significant challenge is that only a small fraction of data can be transmitted for intelligent fault diagnosis and storage. The edge processing capacity is often insufficient for advanced analysis due to time and resource constraints. The neuromorphic signal encoding scheme efficiently reduces the data rate by encoding relevant signal changes into spike trains while discarding redundant information and noise, enabling energy-efficient neuromorphic processing. Due to the presence of dominant operational features and noise in the original measurements, signal pre-processing is required to extract the relevant features before spike coding and processing. The work investigates the effects of different filter banks (pre-processing methods) on the spike encodings for vibration measurements from bearings. This also includes bearing fault features diagnosis based on statistical analysis of generated spikes. The comparative analysis is made for benchmarking different signal pre-processing methods (e.g., envelope, empirical mode decomposition (EMD), and gammatone filter) on bearing vibration datasets. An event-triggered scheme, i.e., Level-crossing analog-to-digital converters (LC-ADCs) is applied to encode the vibration measurement to spikes. Inter-spike intervals (ISIs) statistics are analysed for fault diagnosis of bearings. The results obtained for CWRU bearing databases indicate a possible fault detection and diagnosis with significant data rate reduction and an opportunity for improved computational efficiency. With the developed approach, the envelope filter is found to be the most efficient of all. This work enables a new approach to improve the energy efficiency of condition monitoring systems and further sets a new course of research development in this area using neuromorphic technologies.

Stochastic generation of oscillations in the 3D-Model of cool-flame combustion of a hydrocarbon mixture

The stochastic model of cool-flame combustion of a hydrocarbon mixture is considered. For this 3D model, noise excitement of large-amplitude oscillations in the parametric zone, where the deterministic model has a single equilibrium attractor, is studied. By statistics of interspike intervals, a phenomenon of anti-coherence resonance is revealed. To estimate threshold intensities of noise that causes excitement, stochastic sensitivity technique and method of principal directions are applied.

Modeling IP[formula omitted]-induced Ca[formula omitted] signaling based on its interspike interval statistics

Inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ signaling is a second messenger system used by almost all eukaryotic cells. Recent research demonstrated randomness of Ca2+ signaling on all structural levels. We compile eight general properties of Ca2+ spiking common to all cell types investigated and suggest a theory of Ca2+ spiking starting from the random behavior of IP3 receptor channel clusters mediating the release of Ca2+ from the endoplasmic reticulum capturing all general properties and pathway-specific behavior. Spike generation begins after the absolute refractory period of the previous spike. According to its hierarchical spreading from initiating channel openings to cell level, we describe it as a first passage process from none to all clusters open while the cell recovers from the inhibition which terminated the previous spike. Our theory reproduces the exponential stimulation response relation of the average interspike interval Tav and its robustness properties, random spike timing with a linear moment relation between Tav and the interspike interval SD and its robustness properties, sensitive dependency of Tav on diffusion properties, and nonoscillatory local dynamics. We explain large cell variability of Tav observed in experiments by variability of channel cluster coupling by Ca2+-induced Ca2+ release, the number of clusters, and IP3 pathway component expression levels. We predict the relation between puff probability and agonist concentration and [IP3] and agonist concentration. Differences of spike behavior between cell types and stimulating agonists are explained by the different types of negative feedback terminating spikes. In summary, the hierarchical random character of spike generation explains all of the identified general properties.

Stochastic Bifurcations and Excitement in the ZS-Model of a Thermochemical Reaction

The Zeldovich–Semenov model of the continuous stirred tank reactor with parametric random disturbances in temperature is considered. We study a phenomenon of noise-induced transformation of the equilibrium mode into the mixed-mode oscillatory stochastic regime with alternations between small and large amplitudes. In the parametric analysis of the stochastic excitement, we use the analytical method of confidence domains based on the stochastic sensitivity technique. Analyzing a mutual arrangement of the confidence ellipses and separatrices, we estimate the critical intensity of the noise that causes the excitation. The phenomena of stochastic P-bifurcations and coherence resonances are discovered and studied by probability density functions and the statistics of interspike intervals.

Variability of complex oscillatory regimes in the stochastic model of cold-flame combustion of a hydrocarbon mixture.

Processes of the cold-flame combustion of a mixture of two hydrocarbons are studied on the base of a three-dimensional nonlinear dynamical model. Bifurcation analysis of the deterministic model reveals mono- and bistability parameter zones with equilibrium and oscillatory attractors. For this model, effects of random disturbances in the bistability parameter zone are studied. We show that random forcing causes transitions between coexisting stable equilibria and limit cycles with the formation of complex stochastic mixed-mode oscillations. Properties of these oscillatory regimes are studied by means of statistics of interspike intervals. A phenomenon of anti-coherence resonance is discussed. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

Temporal Correlates to Monaural Edge Pitch in the Distribution of Interspike Interval Statistics in the Auditory Nerve.

Pitch is a perceptual attribute enabling perception of melody. There is no consensus regarding the fundamental nature of pitch and its underlying neural code. A stimulus which has received much interest in psychophysical and computational studies is noise with a sharp spectral edge. High-pass (HP) or low-pass (LP) noise gives rise to a pitch near the edge frequency (monaural edge pitch; MEP). The simplicity of this stimulus, combined with its spectral and autocorrelation properties, make it an interesting stimulus to examine spectral versus temporal cues that could underly its pitch. We recorded responses of single auditory nerve (AN) fibers in chinchilla to MEP-stimuli varying in edge frequency. Temporal cues were examined with shuffled autocorrelogram (SAC) analysis. Correspondence between the population’s dominant interspike interval and reported pitch estimates was poor. A fuller analysis of the population interspike interval distribution, which incorporates not only the dominant but all intervals, results in good matches with behavioral results, but not for the entire range of edge frequencies that generates pitch. Finally, we also examined temporal structure over a slower time scale, intermediate between average firing rate and interspike intervals, by studying the SAC envelope. We found that, in response to a given MEP stimulus, this feature also systematically varies with edge frequency, across fibers with different characteristic frequency (CF). Because neural mechanisms to extract envelope cues are well established, and because this cue is not limited by coding of stimulus fine-structure, this newly identified slower temporal cue is a more plausible basis for pitch than cues based on fine-structure.

Statistical moments of the interspike intervals for a neuron model driven by trichotomous noise.

The influence of a colored three-level input noise (trichotomous noise) on the spike generation of a perfect integrate-and-fire (PIF) model of neurons is studied. Using a first-passage-time formulation, exact expressions for the Laplace transform of the output interspike interval (ISI) density and for the statistical moments of the ISIs (such as the coefficient of variation, the skewness, the serial correlation coefficient, and the Fano factor) are derived. To model the anomalous subdiffusion that can arise from, e.g., the trapping properties of dendritic spines, the model is extended by including a random operational time in the form of an inverse strictly increasing Lévy-type subordinator, and exact formulas for ISI statistics are given for this case as well. Particularly, it is shown that at some parameter regimes, the ISI density exhibits a three-modal structure. The results for the extended model show that the ISI serial correlation coefficient and the Fano factor are nonmonotonic with respect to the input current, which indicates that at an intermediate value of the input current the variability of the output spike trains is minimal. Similarities and differences between the behavior of the presented models and the previously investigated PIF models driven by dichotomous noise are also discussed.

Mixed-mode self-oscillations, stochastic excitability, and coherence resonance in flows of highly concentrated suspensions

Complex mixed-mode oscillatory regimes in flows of concentrated suspensions are studied on the base of the nonlinear dynamical model. A mechanism of the generation of such regimes is associated with the N-shaped flow curve which reflects the dependence of the shear rate on the shear stress. In the zone of the unstable equilibria, amplitude and frequency characteristics of mixed-mode self-sustained oscillations are investigated. A transformation of quasi-harmonic oscillations to non-harmonic ones under increase of the stiffness of the flow curve is discussed. In the zone of the stable equilibria, a phenomenon of stochastic excitement of spike oscillations is studied by the analytical stochastic sensitivity method along with direct numerical simulations. For these mixed-mode stochastic oscillations, a parametric analysis of statistics of interspike intervals is carried out and a phenomenon of coherence resonance is discussed.

Stochastic sensitivity analysis of volcanic activity

In the present paper, we study the stochastically induced behavior of a nonlinear volcanic model containing three prognostic variables: the plug velocity u, the pressure p under the plug, and the conduit volume V. The nouvelle phenomena of noise‐induced transitions from the equilibrium to the cycle in the bistability parametric zone and noise‐induced excitement with the generation of spike oscillations in the monostability zone are found in the presence of N‐shaped friction force. To study these phenomena numerically, we used the computations of random solutions, the phase trajectories and time series, the statistics of interspike intervals, and the mean square variations. To study these phenomena analytically, we applied the stochastic sensitivity function technique and the confidence domains method. This approach is used to predict the noise‐induced transition from a “dormant volcano” state to the “active volcano” mode. From the physical point of view, the volcano is capable to become active under the influence of external noises in the friction force, which model various compositions and properties of volcanic rocks. What is more, the volcanic plug can pop out when its slipping becomes heavy, and the volcano can erupt.

Effects of transient subordinators on the firing statistics of a neuron model driven by dichotomous noise.

The behavior of a stochastic perfect integrate-and-fire (PIF) model of neurons is considered. The effect of temporally correlated random activity of synaptic inputs is modeled as a combination of an asymmetric dichotomous noise and a random operation time in the form of an inverse strictly increasing Lévy-type subordinator. Using a first-passage-time formulation, we find exact expressions for the output interspike interval (ISI) statistics. Particularly, it is shown that at some parameter regimes the ISI density exhibits a multimodal structure. Moreover, it is demonstrated that the coefficient of variation, the serial correlation coefficient, and the Fano factor display a nonmonotonic dependence on the mean input current μ, i.e., the ISI's regularity is maximized at an intermediate value of μ. The features of spike statistics, analytically revealed in our study, are compared with previously obtained results for a perfect integrate-and-fire neuron model driven by dichotomous noise (without subordination).

Stochastic sensitivity analysis of noise-induced transitions in a biochemical model with birhythmicity

A mathematical model of the biochemical reaction with nonlinear recycling of the product into substrate is considered. The main control parameter of this model is the substrate injection rate, which is subject to random disturbances. We study noise-induced oscillations in the parameter zones where the system is bistable and exhibit a coexistence of the equilibrium and limit cycle, or two stable cycles (birhythmicity). Phenomena of the generation of noisy birhythmicity and ‘stochastic preference’ of attractors are studied on the basis of the stochastic sensitivity function technique, confidence domains method, and statistics of interspike intervals.

Stochastic Sensitivity and Method of Principal Directions in Excitability Analysis of the Hodgkin–Huxley Model

We study the probabilistic behavior of the Hodgkin–Huxley neuron model in the presence of random forcing of the external current parameter. The stochastic excitement in the zone of stable equilibria is illustrated by the statistics of interspike intervals and probabilistic distributions of mixed-mode oscillations. For the parametric analysis of this phenomenon, a constructive method for stochastic sensitivity and confidence ellipsoids is suggested. It is shown how to simplify this analysis using the principal direction approach. A constructive application of this technique is demonstrated by analyzing the stochastic excitement in the Hodgkin–Huxley model.

Noise-induced toroidal excitability in neuron model

Abstract We study the stochastic Hindmarsh–Rose neuron model in the torus canards region of the parameter space. We show that noise can transform the torus canard into the large amplitude torus. This corresponds to the noise-induced transition from amplitude-modulated spiking regime to the bursting one. This phenomenon is confirmed by qualitative changes of both amplitude and frequency characteristics. We investigate this phenomenon with the help of interspike intervals statistics. The onset of noise-induced spiking–bursting transition is accompanied by the increase of the mean values of interspike intervals. Moreover, the anti-coherence resonance reflecting the growth of variability of interspike intervals is observed. On the contrary, the distribution of burst duration values for noise-generated bursting oscillations shows the coherence resonance. We suggest an explanation for the observed stochastic phenomena by the specificity of the deterministic portrait and the high excitability of the system in the zone of torus canards.

Fluctuation Scaling of Neuronal Firing and Bursting in Spontaneously Active Brain Circuits

We employed high-density microelectrode arrays to investigate spontaneous firing patterns of neurons in brain circuits of the primary somatosensory cortex (S1) in mice. We recorded from over150 neurons for10min in each of eight different experiments, identified their location in S1, sorted their action potentials (spikes), and computed their power spectra and inter-spike interval (ISI) statistics. Of all persistently active neurons, 92% fired with a single dominant frequency- regularly firing neurons (RNs)- from1 to8Hz while 8% fired in burst with two dominant frequencies- bursting neurons (BNs)- corresponding to the inter-burst (2-6Hz) and intra-burst intervals (20-160Hz). RNs were predominantly located in layers2/3 and5/6 while BNs localized to layers4 and5. Across neurons, the standard deviation of ISI was a power law of its mean, a property known as fluctuation scaling, with a power law exponent of1 for RNs and1.25 for BNs. The power law implies that firing and bursting patterns are scale invariant: the firing pattern of a given RN or BN resembles that of another RN or BN, respectively, after a time contraction or dilation. An explanation for this scale invariance is discussed in the context of previous computational studies as well as its potential role in information processing.

Noise-induced spiking-bursting transition in the neuron model with the blue sky catastrophe.

We study a special variant of the noise-induced transition between spiking and bursting regimes associated with the blue sky catastrophe bifurcation in the Hindmarsh-Rose neuron model. We show that in the parameter region close to the bifurcation value, where the only attractor of the system is the limit cycle of tonic spiking type, noise can transform the spiking oscillatory regime to the bursting one. This phenomenon is studied by means of power spectral density and interspike intervals statistics. We show that noise shifts the bifurcation value, so that bursting activity can be observed for a wider parameter range. Moreover, we reveal that the stochastic spiking-bursting transitions in this system are accompanied by the change in sign of the Lyapunov exponent. We perform a detailed quantitative analysis of these phenomena with an approach that uses a concept of the stochastic sensitivity function, the confidence domains method, and Mahalanobis metrics.

The stochastic Fitzhugh\u2013Nagumo neuron model in the excitable regime embeds a leaky integrate-and-fire model

In this paper, we provide a complete mathematical construction for a stochastic leaky-integrate-and-fire model (LIF) mimicking the interspike interval (ISI) statistics of a stochastic FitzHugh–Nagumo neuron model (FHN) in the excitable regime, where the unique fixed point is stable. Under specific types of noises, we prove that there exists a global random attractor for the stochastic FHN system. The linearization method is then applied to estimate the firing time and to derive the associated radial equation representing a LIF equation. This result confirms the previous prediction in Ditlevsen and Greenwood (J Math Biol 67(2):239–259, 2013) for the Morris-Lecar neuron model in the bistability regime consisting of a stable fixed point and a stable limit cycle.

Statistics of inverse interspike intervals: The instantaneous firing rate revisited.

The rate coding hypothesis is the oldest and still one of the most accepted and investigated scenarios in neuronal activity analyses. However, the actual neuronal firing rate, while informally understood, can be mathematically defined in several different ways. These definitions yield distinct results; even their average values may differ dramatically for the simplest neuronal models. Such an inconsistency, together with the importance of "firing rate," motivates us to revisit the classical concept of the instantaneous firing rate. We confirm that different notions of firing rate can in fact be compatible, at least in terms of their averages, by carefully discerning the time instant at which the neuronal activity is observed. Two general cases are distinguished: either the inspection time is synchronised with a reference time or with the neuronal spiking. The statistical properties of the instantaneous firing rate, including parameter estimation, are analyzed, and compatibility with the intuitively understood concept is demonstrated.

Noise-induced torus bursting in the stochastic Hindmarsh-Rose neuron model.

We study the phenomenon of noise-induced torus bursting on the base of the three-dimensional Hindmarsh-Rose neuron model forced by additive noise. We show that in the parametric zone close to the Neimark-Sacker bifurcation, where the deterministic system exhibits rapid tonic spiking oscillations, random disturbances can turn tonic spiking into bursting, which is characterized by the formation of a peculiar dynamical structure resembling that of a torus. This phenomenon is confirmed by the changes in dispersion of random trajectories as well as the power spectral density and interspike intervals statistics. In particular, we show that as noise increases, the system undergoes P and D bifurcations, transitioning from order to chaos. We ultimately characterize the transition from stochastic (tonic) spiking to bursting by stochastic sensitivity functions.