Interspike Interval Series Research Articles

Spectrum dependency to rate and spike timing in neuronal spike trains

BackgroundSpike trains are series of interspike intervals in a specific order that can be characterized by their probability distributions and order in time which refer to the concepts of rate and spike timing features. Periodic structure in the spike train can be reflected in oscillatory activities. Thus, there is a direct link between oscillator activities and the spike train. The proposed methods are to investigate the dependency of emerging oscillatory activities to the rate and the spike timing features. MethodFirst, the circular statistics methods were compared to Fast Fourier Transform for best estimation of spectra. Second, two statistical tests were introduced to help make decisions regarding the dependency of spectrum, or individual frequencies, onto rate and spike timing. Third, the methodology is applied to in-vivo recordings of basal ganglia neurons in mouse, primate, and human. Finally, this novel framework is shown to allow the investigation of subsets of spikes contributing to individual oscillators. ResultsUse of circular statistical methods, in comparison to FFT, minimizes spectral leakage. Using virtual spike trains, the Rate versus Timing Dependency Spectrum Test (or RTDs-Test) permits identifying spectral spike trains solely dependent on the rate feature from those that are also dependent on the spike timing feature. Similarly, the Rate versus Timing Dependency Frequency Test (or RTDf-Test), allows to identify individual oscillators with partial dependency on spike timing. Dependency on spike timing was found for all in-vivo recordings but only in few frequencies. The mapping in frequency and time of dependencies showed a dynamical process that may be organizing the basal ganglia function. ConclusionsThe methodology may improve our understanding of the emergence of oscillatory activities and, possibly, the relation between oscillatory activities and circuitry functions.

Coherent or stochastic bi-resonance induced by conductance disturbance of chemical autapse

The autaptic structure of neurons has the function of self-feedback, which is easily disturbed due to the quantum characteristics of neurotransmitter release. This paper focuses on the effect of conductance disturbance of chemical autapse on the electrophysiological activities of FHN neuron. First, the frequency encoding of FHN neuron to periodic excitation signals exhibits a nonlinear change characteristic, and the FHN neuron without autapse has chaotic discharge behavior according to the maximum Lyapunov exponent and the sampled time series. Secondly, the chemical autaptic function can change the dynamics of FHN neuronal system, and appropriate autaptic parameters can cause the dynamic bifurcation, which corresponds to the transition between different periodic spiking modes. In particular, the self-feedback function of chemical autapse can induce a transition from a chaotic discharge state to a periodic spiking or a quasi-periodic bursting discharge state. Finally, based on the quantum characteristics of neurotransmitter release, the effect of random disturbance from autaptic conductance on the firing activities is quantitatively studied with the help of the discharge frequency and the coefficient of variation of inter-spike interval series. The numerical results show that the disturbance of autaptic conductance can change the activity of ion channels under the action of self-feedback, which not only improves the encoding efficiency of FHN neuron to external excitation signals, but also changes the regularity of neuronal firing activities and induces significant coherent or stochastic bi-resonance. The coherent or stochastic bi-resonance phenomenon is closely related to the dynamic bifurcation of FitzHugh-Nagumo(FHN) neuronal system, and its underlying mechanism is that the disturbance of autaptic conductance leads to the unstable dynamic behavior of neuronal system, and the corresponding neuronal firing activity may transit between the resting state, the single-cycle and the multicycle spike states, thereby providing the occurring possibility for coherent or stochastic bi-resonance. This study further reveals the self-regulatory effect of the autaptic structure on neuronal firing activities, and could provide theoretical guidance for physiological manipulation of autapses. In addition, according to the pronounced self-feedback function of autaptic structure, a recurrent spiking neural network with local self-feedback can be constructed to improve the performance of machine learning by applying a synaptic plasticity rule.

Effect of stochastic electromagnetic disturbances on autapse neuronal systems**Project supported by the National Natural Science Foundation of China (Grant No. 11672233), the Fundamental Research Funds for the Central Universities, China (Grant No. 3102017AX008), and the Seed Foundation of Innovation and Creation for Graduate Student in Northwestern Polytechnical University, China (Grant No. ZZ2018173).

With the help of a magnetic flux variable, the effects of stochastic electromagnetic disturbances on autapse Hodgkin–Huxley neuronal systems are studied systematically. Firstly, owing to the autaptic function, the inter-spike interval series of an autapse neuron not only bifurcates, but also presents a quasi-periodic characteristic. Secondly, an irregular mixed-mode oscillation induced by a specific electromagnetic disturbance is analyzed using the coefficient of variation of inter-spike intervals. It is shown that the neuronal discharge activity has certain selectivity to the noise intensity, and the appropriate noise intensity can induce the significant mixed-mode oscillations. Finally, the modulation effects of electromagnetic disturbances on a ring field-coupled neuronal network with autaptic structures are explored quantitatively using the average spiking frequency and the average coefficient of variation. The electromagnetic disturbances can not only destroy the continuous and synchronous discharge state, but also induce the resting neurons to generate the intermittent discharge mode and realize the transmission of neural signals in the neuronal network. The studies can provide some theoretical guidance for applying electromagnetic disturbances to effectively control the propagation of neural signals and treat mental illness.

Estimating the predictability time of noisy chaotic dynamics from point sequences

A method for increasing the accuracy of estimation of the predictability time of noisy chaotic dynamics from system-related point sequences is proposed. General laws observed in the application of this method to interspike interval series of model threshold devices of two types operating in the regime of phase-coherent chaos are illustrated.

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.

Fractal analysis reveals subclasses of neurons and suggests an explanation of their spontaneous activity

The present work used fractal time series analysis (detrended fluctuation analysis; DFA) to examine the spontaneous activity of single neurons in an anesthetized animal model, specifically, the mitral cells in the rat main olfactory bulb. DFA bolstered previous research in suggesting two subclasses of mitral cells. Although there was no difference in the fractal scaling of the interspike interval series at the shorter timescales, there was a significant difference at longer timescales. Neurons in Group B exhibited fractal, power-law scaled interspike intervals, whereas neurons in Group A exhibited random variation. These results raise questions about the role of these different cells within the olfactory bulb and potential explanations of their dynamics. Specifically, self-organized criticality has been proposed as an explanation of fractal scaling in many natural systems, including neural systems. However, this theory is based on certain assumptions that do not clearly hold in the case of spontaneous neural activity, which likely reflects intrinsic cell dynamics rather than activity driven by external stimulation. Moreover, it is unclear how self-organized criticality might account for the random dynamics observed in Group A, and how these random dynamics might serve some functional role when embedded in the typical activity of the olfactory bulb. These theoretical considerations provide direction for additional experimental work.

Cross-correlations between pairs of neurons in cerebellar cortex in vivo

In the present paper we apply a new neurophysiological technique to make single-electrode, dual loose-patch recordings from pairs of neuronal elements in the cerebellar cortex in vivo. The analyzed cell pairs consisted of an inhibitory molecular layer interneuron and a Purkinje cell (PC) or a Golgi cell and a granule cell, respectively. To detect the magnitude of the unitary inhibitory synaptic inputs we used histograms of the spike activity of the target cell, triggered by the spikes of the inhibitory cell. Using this analysis, we found that single interneurons had no detectable effect on PC firing, which could be explained by an expected very low synaptic weight of individual interneuron–PC connections. However, interneurons did have a weak delaying effect on the overall series of interspike intervals of PCs. Due to the very high number of inhibitory synapses on each PC, a concerted activation of the interneurons could still achieve potent PC inhibition as previously shown. In contrast, in the histograms of the Golgi cell–granule cell pairs, we found a weak inhibitory effect on the granule cell but only at the time period defined as the temporal domain of the slow IPSP previously described for this connection. Surprisingly, the average granule cell firing frequency sampled at one second was strongly modulated with a negative correlation to the overall firing level of the Golgi cell when the latter was modified through current injection via the patch pipette. These findings are compatible with that tonic inhibition is the dominant form of Golgi cell–granule cell inhibition in the adult cerebellum in vivo.

COHERENCE RESONANCE–INDUCED STOCHASTIC NEURAL FIRING AT A SADDLE-NODE BIFURCATION

Coherence resonance at a saddle-node bifurcation point and the corresponding stochastic firing patterns are simulated in a theoretical neuronal model. The characteristics of noise-induced neural firing pattern, such as exponential decay in histogram of interspike interval (ISI) series, independence and stochasticity within ISI series are identified. Firing pattern similar to the simulated results was discovered in biological experiment on a neural pacemaker. The difference between this firing and integer multiple firing generated at a Hopf bifurcation point is also given. The results not only revealed the stochastic dynamics near a saddle-node bifurcation, but also gave practical approaches to identify the saddle-node bifurcation and to distinguish it from the Hopf bifurcation in neuronal system. In addition, many previously observed firing patterns can be attribute to stochastic firing pattern near such a saddle-node bifurcation.

Exponential decay characteristics of the stochastic integer multiple neural firing patterns

Integer multiple neural firing patterns exhibit multi-peaks in inter-spike interval (ISI) histogram (ISIH) and exponential decay in amplitude of peaks, which results from their stochastic mechanisms. But in previous experimental observation that the decay in ISIH frequently shows obvious bias from exponential law. This paper studied three typical cases of the decay, by transforming ISI series of the firing to discrete binary chain and calculating the probabilities or frequencies of symbols over the whole chain. The first case is the exponential decay without bias. An example of this case was discovered on hippocampal CA1 pyramidal neuron stimulated by external signal. Probability calculation shows that this decay without bias results from a stochastic renewal process, in which the successive spikes are independent. The second case is the exponential decay with a higher first peak, while the third case is that with a lower first peak. An example of the second case was discovered in experiment on a neural pacemaker. Simulation and calculation of the second and third cases indicate that the dependency in successive spikes of the firing leads to the bias seen in decay of ISIH peaks. The quantitative expression of the decay slope of three cases of firing patterns, as well as the excitatory effect in the second case of firing pattern and the inhibitory effect in the third case of firing pattern are identified. The results clearly reveal the mechanism of the exponential decay in ISIH peaks of a number of important neural firing patterns and provide new understanding for typical bias from the exponential decay law.

Nonlinear temporal organization of neuronal discharge in the basal ganglia of Parkinson's disease patients

Previous electrophysiological studies of the basal ganglia in Parkinson's disease (PD) patients have utilized linear analyses in time-or-frequency domains to characterize neuronal discharge patterns. However, these measures do not fully describe the non-linear features of discharge rates and oscillatory activities of basal ganglia neurons.In this original research, we investigate whether non-linear temporal organizations exist in the inter-spike interval series of neurons recorded in the globus pallidus or the subthalamic nucleus in PD patients undergoing surgery for the implantation of deep brain stimulating electrodes.Our data indicate that in approximately 80% of globus pallidus and subthalamic neurons, the raw inter-spike interval sequences have lower entropy values than those observed after shuffling of the original series. This is the first report establishing non-linear temporal organization as a common feature of neuronal discharge in the basal ganglia of PD patients.

IDENTIFYING DISTINCT STOCHASTIC DYNAMICS FROM CHAOS: A STUDY ON MULTIMODAL NEURAL FIRING PATTERNS

Some chaotic and a series of stochastic neural firings are multimodal. Stochastic multimodal firing patterns are of special importance because they indicate a possible utility of noise. A number of previous studies confused the dynamics of chaotic and stochastic multimodal firing patterns. The confusion resulted partly from inappropriate interpretations of estimations of nonlinear time series measures. With deliberately chosen examples the present paper introduces strategies and methods of identification of stochastic firing patterns from chaotic ones. Aided by theoretical simulation we show that the stochastic multimodal firing patterns result from the effects of noise on neuronal systems near to a bifurcation between two simpler attractors, such as a point attractor and a limit cycle attractor or two limit cycle attractors. In contrast, the multimodal chaotic firing trains are generated by the dynamics of a specific strange attractor. Three systems were carefully chosen to elucidate these two mechanisms. An experimental neural pacemaker model and the Chay mathematical model were used to show the stochastic dynamics, while the deterministic Wang model was used to show the deterministic dynamics. The usage and interpretation of nonlinear time series measures were systematically tested by applying them to firing trains generated by the three systems. We successfully identified the distinct differences between stochastic and chaotic multimodal firing patterns and showed the dynamics underlying two categories of stochastic firing patterns. The first category results from the effects of noise on the neuronal system near a Hopf bifurcation. The second category results from the effects of noise on the period-adding bifurcation between two limit cycles. Although direct application of nonlinear measures to interspike interval series of these firing trains misleadingly implies chaotic properties, definition of eigen events based on more appropriate judgments of the underlying dynamics leads to accurate identifications of the stochastic properties.

Detecting Determinism in Firing Activities of Retinal Ganglion Cells during Response to Complex Stimuli

Complex stimuli are used to probe the response properties of the chicken's retinal ganglion cells (GCs). The correlation dimension method and the nonlinear forecasting method are applied to detect the determinism in the firing activities of the retinal GCs during response to complex stimuli. The inter-spike interval (ISI) series and the first difference of the ISI (DISI) series are analysed. Two conclusions are drawn. Firstly, the first difference operation of the ISI series makes it comparatively easier for determinism detection in the firing activities of retinal GCs. Secondly, the nonlinear forecasting method is more efficient and reliable than the correlation dimension method for determinism detection.

Wavelet transform analysis of time series generated by the stimulated neuronal activity

We have studied the stimulated discharge dynamics of fusimotor neurons by applying the wavelet transform technique and by adopting that the neuronal discharge dynamics is manifested by the random time series of interspike intervals. We found two different power-law type behaviors along interspike intervals (ISI) time scale (which implies existence of two different types of neuronal noise), which are separated by a crossover region. Our results reveal that complex neuronal dynamics, in the presence of external stimulation, is manifested with long-range correlated noise in the region before the crossover, on the ISI time scale.

Nonlinear analysis of discharge patterns in monkey basal ganglia

Spontaneous discharge of basal ganglia neurons is often analyzed with time- or frequency-domain methods. However, it has been shown that sequences of inter-spike interval series are not fully described by such linear procedures. We therefore carried out a characterization of the nonlinear features of spontaneous discharge of neurons in the primate basal ganglia. We studied the spontaneous activity of neurons in the subthalamic nucleus (22 cells), as well as neurons in the external and internal pallidal segments (53 and 39 cells, respectively), recorded with standard extracellular recording methods in two awake Rhesus monkeys. As a measure of the statistical irregularity of neuronal discharge, we compared the approximate entropy of inter-spike interval sequences with that of shuffled representations of the same data. In all three basal ganglia structures, approximately 95% of the original data showed lower approximate entropy values than the shuffled data, suggesting a temporal organization in the original sequence. Fano factor analysis confirmed the presence of a temporal organization of inter-spike interval sequences, and indicated the presence of self-similarity in the great majority of them. In addition, Hurst exponent analysis showed that the inter-spike interval series are persistent. Hurst exponents often differ between short and long scaling ranges. Subsequent principal component analyses allowed us to identify three distinct patterns of the temporal evolution of inter-spike interval sequences in the phase space. These types were found in varying distributions in all three nuclei. Our analyses demonstrate that the discharge of most neurons in the basal ganglia of awake monkeys has nonlinear features that may be important for information coding in the basal ganglia.

Dynamical Basis of Irregular Spiking in NMDA-Driven Prefrontal Cortex Neurons

Slow N-Methyl-D-aspartic acid (NMDA) synaptic currents are assumed to strongly contribute to the persistently elevated firing rates observed in prefrontal cortex (PFC) during working memory. During persistent activity, spiking of many neurons is highly irregular. Here we report that highly irregular firing can be induced through a combination of NMDA- and dopamine D1 receptor agonists applied to adult PFC neurons in vitro. The highest interspike-interval (ISI) variability occurred in a transition regime where the subthreshold membrane potential distribution shifts from mono- to bimodality, while neurons with clearly mono- or bimodal distributions fired much more regularly. Predictability within irregular ISI series was significantly higher than expected from a noise-driven linear process, indicating that it might best be described through complex (potentially chaotic) nonlinear deterministic processes. Accordingly, the phenomena observed in vitro could be reproduced in purely deterministic biophysical model neurons. High spiking irregularity in these models emerged within a chaotic, close-to-bifurcation regime characterized by a shift of the membrane potential distribution from mono- to bimodality and by similar ISI return maps as observed in vitro. The nonlinearity of NMDA conductances was crucial for inducing this regime. NMDA-induced irregular dynamics may have important implications for computational processes during working memory and neural coding.

Detecting Long‐Range Correlations in Time Series of Dorsal Horn Neuron Discharges

We have studied the discharge dynamics of dorsal horn neurons by applying the detrended fluctuation analysis and the wavelet transform technique. We have adopted that the neuronal discharge dynamics is manifested by the random time series of interspike intervals. In all cases studied, we found two different power-law type behaviors across interspike intervals enumeration scale, that are separated by crossover regions (which implies existence of two different types of neuronal noise). Our results reveal that complex neuronal dynamics may change in the presence of external stimulation, manifested by changing of the noise characteristics that appear before crossover.

Relation between responsiveness to neurotransmitters and complexity of epileptiform activity in rat hippocampal CA1 neurons.

Our previous works suggested that sensitivity of neurons with chaotic firing patterns to stimuli is significantly greater than that in neurons with periodic firing patterns, which shows that responsiveness of neurons may depend on the complexity of the firing series. This study was performed to determine the relation between responsiveness of the hippocampal CA1 neurons with epileptiform activity (EA) to neurotransmitters and their complexity of firing series. Firing series of CA1 neurons were recorded extracellularly in rat hippocampal slice. Approximate entropy was used to describe the complexity of the interspike interval (ISI) series. EA was induced by local application of penicillin (1,000 IU/ml). The change of firing rate induced by neurotransmitters (glutamate and gamma-aminobutyric acid) was compared with that of the degree of complexity of ISI series in the process of EA. The excitatory responses to glutamate and the inhibitory responses to gamma-aminobutyric acid in CA1 neurons appeared to be decreased during the process of penicillin-induced EA. However, during this same process, the approximate entropy of the ISI series also was decreased significantly. The results suggest that the reduced responses to neurotransmitters of the CA1 neurons appear to be closely related to the onset of EA. Furthermore, these neurons show that the changes in responsiveness are closely parallel to the decrease of degree of complexity of firing series during penicillin epileptogenesis.

Detection of deterministic behavior within the tissue injury-induced persistent firing of nociceptive neurons in the dorsal horn of the rat spinal cord.

To unravel the temporal features of the peripheral tissue injury induced persistent nociceptive discharge, single wide dynamic range (WDR) unit activity was recorded extracellularly in lumbar dorsal horn of anesthetized rats and interspike interval (ISI) series were obtained. Subcutaneous (s.c.) bee venom (BV) injection induced persistent discharge of spinal WDR neurons and has been well established to be a good model in evaluation of tissue injury induced pain. By applying a more novel approach, i.e., the unstable periodic orbit (UPO) identification method, we detected a family of significant separate UPOs (period-1, 2 and 3 orbits) within the ISI series of BV-induced nociceptive discharge, but not spontaneous background activity of spinal WDR neuron. Furthermore, temporally dynamic changes of UPOs at lower period-1, 2 and 3 for 4 successive time segments within 1 h time course of WDR unit firing showed temporally dynamic changes, i.e., new orbits with longer ISIs emerged and those with shorter ISIs vanished with time change. By using this method we suggest that BV-induced nociceptive discharge of spinal WDR neuron be a kind of deterministic activity and various UPOs may play some role in temporal coding of sensory information.

Different classifications of UPOs in the parametrically different chaotic ISI series of a neural pacemaker.

Various interspike intervals (ISIs) in a series of spontaneous discharges of experimental neural pacemakers were identified as chaotic and as lying between stable period 2 and stable period 3. The method of So et al. for detection of unstable periodic orbits (UPOs) was applied to analyze two chaotic ISI series generated under the action of [Ca2+]o in different concentrations. In the chaotic ISI series near the stable period 2, an unstable period 2 orbit was detected. In the chaotic ISI series near the stable period 3, both an unstable period 2 orbit and an unstable period 3 orbit were detected. The location of the unstable period 2 orbit was close to that of the stable period 2 in the return map, while the location of the unstable period 3 orbit was close to that of the stable period 3. The results not only revealed the structures of various chaotic ISI series, but also indicated that the classification of UPOs could reflect the experimental control parameters at which the chaotic ISI series were generated. The previously discovered period adding bifurcation in the ISI series generated by experimental neural pacemakers [12] was further described in terms of the evolution of the period orbits.

Chaotic dynamics from interspike intervals.

Considering two different mathematical models describing chaotic spiking phenomena, namely, an integrate-and-fire and a threshold-crossing model, we discuss the problem of extracting dynamics from interspike intervals (ISIs) and show that the possibilities of computing the largest Lyapunov exponent (LE) from point processes differ between the two models. We also consider the problem of estimating the second LE and the possibility to diagnose hyperchaotic behavior by processing spike trains. Since the second exponent is quite sensitive to the structure of the ISI series, we investigate the problem of its computation.