Interspike Interval Distribution Research Articles

Neuronal Synchronization Can Control the Energy Efficiency of Inter-Spike Interval Coding

The role of synchronous firing in sensory coding and cognition remains controversial. While studies, focusing on its mechanistic consequences in attentional tasks, suggest that synchronization dynamically boosts sensory processing, others failed to find significant synchronization levels in such tasks. We attempt to understand both lines of evidence within a coherent theoretical framework. We conceptualize synchronization as an independent control parameter to study how the postsynaptic neuron transmits the average firing activity of a presynaptic population, in the presence of synchronization. We apply the Berger-Levy theory of energy efficient information transmission to interpret simulations of a Hodgkin-Huxley-type postsynaptic neuron model, where we varied the firing rate and synchronization level in the presynaptic population independently. We find that for a fixed presynaptic firing rate the simulated postsynaptic interspike interval distribution depends on the synchronization level and is well-described by a generalized extreme value distribution. For synchronization levels of 15% to 50%, we find that the optimal distribution of presynaptic firing rate, maximizing the mutual information per unit cost, is maximized at ~30% synchronization level. These results suggest that the statistics and energy efficiency of neuronal communication channels, through which the input rate is communicated, can be dynamically adapted by the synchronization level.

Delay-induced stochastic bursting in excitable noisy systems

We show that a cumulative action of noise and delayed feedback on an excitable theta-neuron leads to rather coherent stochastic bursting. An idealized point process, valid if the characteristic time scales in the problem are well-separated, is used to describe statistical properties such as the power spectrum and the interspike interval distribution. We show how the main parameters of the point process, the spontaneous excitation rate and the probability to induce a spike during the delay action, can be calculated from the solutions of a stationary and a forced Fokker-Planck equation.

Mathematical Modeling and Analyses of Interspike-Intervals of Spontaneous Activity in Afferent Neurons of the Zebrafish Lateral Line

Without stimuli, hair cells spontaneously release neurotransmitter leading to spontaneous generation of action potentials (spikes) in innervating afferent neurons. We analyzed spontaneous spike patterns recorded from the lateral line of zebrafish and found that distributions of interspike intervals (ISIs) either have an exponential shape or an “L” shape that is characterized by a sharp decay but wide tail. ISI data were fitted to renewal-process models that accounted for the neuron refractory periods and hair-cell synaptic release. Modeling the timing of synaptic release using a mixture of two exponential distributions yielded the best fit for our ISI data. Additionally, lateral line ISIs displayed positive serial correlation and appeared to exhibit switching between faster and slower modes of spike generation. This pattern contrasts with previous findings from the auditory system where ISIs tended to have negative serial correlation due to synaptic depletion. We propose that afferent neuron innervation with multiple and heterogenous hair-cells synapses, each influenced by changes in calcium domains, can serve as a mechanism for the random switching behavior. Overall, our analyses provide evidence of how physiological similarities and differences between synapses and innervation patterns in the auditory, vestibular, and lateral line systems can lead to variations in spontaneous activity.

Transition rate dependence of interspike interval distributions of model neurons driven by two-state fluctuating input

Transition rate dependence of interspike interval distributions of model neurons driven by two-state fluctuating input

Stochastic Generation and Deformation of Toroidal Oscillations in Neuron Model

The stochastic Hindmarsh–Rose neuron model in the parametric regions of complex nonlinear dynamics with quiescent and torus bursting regimes is studied. We show that in the zone where the deterministic system exhibits the quiescence, noise can lead to the generation of toroidal dynamical structure with the transition to the bursting regime. The studies of dispersion of random trajectories as well as power spectral density and interspike intervals distribution confirm these changes in system dynamics. Moreover, the stochastic emergence of torus is followed by the noise-induced chaotization. We show that the generation of the torus bursting oscillations is associated with the specificity of the arrangement of deterministic trajectories in the vicinity of the equilibrium as well as its stochastic sensitivity. A probabilistic mechanism of the stochastic generation of torus is investigated on the basis of the analysis of the stochastic sensitivity and geometry of confidence regions. We also show that in the parametric range of the torus bursting, noise can lead to the increase of number of bursts per time interval and to the reduction of spiking and quiescent phase duration.

Suppression of Multimodality in Inter-Spike Interval Distribution: Role of External Damped Oscillatory Input.

A theoretical model based on the generalized neuronal model with distributed delay (GNMDD) is proposed to generate multimodal/bimodal inter-spike interval (ISI) distribution. The distributed delay is assumed to be general in the sense that it is a linear combination of gamma distributed weak and strong memory kernels. It is found that the expected membrane potential in the subthreshold regime exhibits damped oscillatory behavior. This causes the ISI pattern to become bimodal/multimodal. Furthermore, the effect of external damped oscillatory current in the GNMDD model is investigated. Based on extensive simulation study of a set of coupled stochastic differential equations in the presence of a threshold, a vast amount of spiking data are generated. Using the non-parametric modeling approach, the optimal bin width of the histogram is obtained by using the Freedman-Diaconis rule. It is to be noted that, with the increasing amplitude of the damped oscillatory current, the multimodal ISI distribution changes to unimodal ISI distribution when the magnitude of external current reaches a critical value. It is found that the entropy also shows a sudden transition around the critical point. This phenomenon is akin to a phase transition.

Classification of Neurons in the Primate Reticular Formation and Changes after Recovery from Pyramidal Tract Lesion.

The reticular formation is important in primate motor control, both in health and during recovery after brain damage. Little is known about the different neurons present in the reticular nuclei. Here we recorded extracellular spikes from the reticular formation in five healthy female awake behaving monkeys (193 cells), and in two female monkeys 1 year after recovery from a unilateral pyramidal tract lesion (125 cells). Analysis of spike shape and four measures derived from the interspike interval distribution identified four clusters of neurons in control animals. Cluster 1 cells had a slow firing rate. Cluster 2 cells had narrow spikes and irregular firing, which often included high-frequency bursts. Cluster 3 cells were highly rhythmic and fast firing. Cluster 4 cells showed negative spikes. A separate population of 42 cells was antidromically identified as reticulospinal neurons in five anesthetized female monkeys. The distribution of spike width in these cells closely overlaid the distribution for cluster 2, leading us tentatively to suggest that cluster 2 included neurons with reticulospinal projections. In animals after corticospinal lesion, cells could be identified in all four clusters. The firing rate of cells in clusters 1 and 2 was increased in lesioned animals relative to control animals (by 52% and 60%, respectively); cells in cluster 2 were also more regular and more bursting in the lesioned animals. We suggest that changes in both membrane properties and local circuits within the reticular formation occur following lesioning, potentially increasing reticulospinal output to help compensate for lost corticospinal descending drive.SIGNIFICANCE STATEMENT This work is the first to subclassify neurons in the reticular formation, providing insights into the local circuitry of this important but little understood structure. The approach developed can be applied to any extracellular recording from this region, allowing future studies to place their data within our current framework of four neural types. Changes in reticular neurons may be important to subserve functional recovery after damage in human patients, such as after stroke or spinal cord injury.

The effect of an exogenous magnetic fieldonneural coding in deep spikingneuralnetworks.

A ten-layer feed forward network was constructed in the presence of an exogenous alternating magnetic field. Results indicate that for rate coding, the firing rate is increased in the presence of an exogenous alternating magnetic field and particularly with increasing enhancement of the alternating magnetic field amplitude. For temporal coding, in the presence of alternating magnetic field, the interspike intervals of the spiking sequence are decreased and the distribution of interspike intervals tends to be uniform.

Characterizing ISI and sub-threshold membrane potential distributions: Ensemble of IF neurons with random squared-noise intensity

A theoretical investigation is presented that characterizes the emerging sub-threshold membrane potential and inter-spike interval (ISI) distributions of an ensemble of IF neurons that group together and fire together. The squared-noise intensity σ2 of the ensemble of neurons is treated as a random variable to account for the electrophysiological variations across population of nearly identical neurons. Employing superstatistical framework, both ISI distribution and sub-threshold membrane potential distribution of neuronal ensemble are obtained in terms of generalized K-distribution. The resulting distributions exhibit asymptotic behavior akin to stretched exponential family. Extensive simulations of the underlying SDE with random σ2 are carried out. The results are found to be in excellent agreement with the analytical results. The analysis has been extended to cover the case corresponding to independent random fluctuations in drift in addition to random squared-noise intensity. The novelty of the proposed analytical investigation for the ensemble of IF neurons is that it yields closed form expressions of probability distributions in terms of generalized K-distribution. Based on a record of spiking activity of thousands of neurons, the findings of the proposed model are validated. The squared-noise intensity σ2 of identified neurons from the data is found to follow gamma distribution. The proposed generalized K-distribution is found to be in excellent agreement with that of empirically obtained ISI distribution of neuronal ensemble.

Pitch and timbre discrimination at wave-to-spike transition in the cochlea

A new definition of musical pitch is proposed. A Finite-Difference Time Domain (FDTM) model of the cochlea is used to calculate spike trains caused by tone complexes and by a recorded classical guitar tone. All harmonic tone complexes, musical notes, show a narrow-band Interspike Interval (ISI) pattern at the respective fundamental frequency of the tone complex. Still this fundamental frequency is not only present at the bark band holding the respective best frequency of this fundamental frequency, but rather at all bark bands driven by the tone complex partials. This is caused by drop-outs in the basically regular, periodic spike train in the respective bands. These drop-outs are caused by the energy distribution in the wave form, where time spans of low energy are not able to drive spikes. The presence of the fundamental periodicity in all bark bands can be interpreted as pitch. Contrary to pitch, timbre is represented as a wide distribution of different ISIs over bark bands. The definition of pitch is shown to also works with residue pitches. The spike drop-outs in times of low energy of the wave form also cause undertones, integer multiple subdivisions in periodicity, but in no case overtones can appear. This might explain the musical minor scale, which was proposed to be caused by undertones already in 1880 by Hugo Riemann, still until now without knowledge about any physical realization of such undertones.

Methods of Stochastic Analysis of Complex Regimes in the 3D Hindmarsh–Rose Neuron Model

A problem of the stochastic nonlinear analysis of neuronal activity is studied by the example of the Hindmarsh–Rose (HR) model. For the parametric region of tonic spiking oscillations, it is shown that random noise transforms the spiking dynamic regime into the bursting one. This stochastic phenomenon is specified by qualitative changes in distributions of random trajectories and interspike intervals (ISIs). For a quantitative analysis of the noise-induced bursting, we suggest a constructive semi-analytical approach based on the stochastic sensitivity function (SSF) technique and the method of confidence domains that allows us to describe geometrically a distribution of random states around the deterministic attractors. Using this approach, we develop a new algorithm for estimation of critical values for the noise intensity corresponding to the qualitative changes in stochastic dynamics. We show that the obtained estimations are in good agreement with the numerical results. An interplay between noise-induced bursting and transitions from order to chaos is discussed.

Memory-induced resonancelike suppression of spike generation in a resonate-and-fire neuron model.

The behavior of a stochastic resonate-and-fire neuron model based on a reduction of a fractional noise-driven generalized Langevin equation (GLE) with a power-law memory kernel is considered. The effect of temporally correlated random activity of synaptic inputs, which arise from other neurons forming local and distant networks, is modeled as an additive fractional Gaussian noise in the GLE. Using a first-passage-time formulation, in certain system parameter domains exact expressions for the output interspike interval (ISI) density and for the survival probability (the probability that a spike is not generated) are derived and their dependence on input parameters, especially on the memory exponent, is analyzed. In the case of external white noise, it is shown that at intermediate values of the memory exponent the survival probability is significantly enhanced in comparison with the cases of strong and weak memory, which causes a resonancelike suppression of the probability of spike generation as a function of the memory exponent. Moreover, an examination of the dependence of multimodality in the ISI distribution on input parameters shows that there exists a critical memory exponent α_{c}≈0.402, which marks a dynamical transition in the behavior of the system. That phenomenon is illustrated by a phase diagram describing the emergence of three qualitatively different structures of the ISI distribution. Similarities and differences between the behavior of the model at internal and external noises are also discussed.

Motor unit discharge characteristics and walking performance of individuals with multiple sclerosis.

Walking performance of persons with multiple sclerosis (MS) is strongly influenced by the activation signals received by lower leg muscles. We examined the associations between force steadiness and motor unit discharge characteristics of lower leg muscles during submaximal isometric contractions with tests of walking performance and disability status in individuals who self-reported walking difficulties due to MS. We expected that worse walking performance would be associated with weaker plantar flexor muscles, worse force steadiness, and slower motor unit discharge times. Twenty-three individuals with relapsing-remitting MS (56 ± 7 yr) participated in the study. Participants completed one to three evaluation sessions that involved two walking tests (25-ft walk and 6-min walk), a manual dexterity test (grooved pegboard), health-related questionnaires, and measurement of strength, force steadiness, and motor unit discharge characteristics of lower leg muscles. Multiple regression analyses were used to construct models to explain the variance in measures of walking performance. There were statistically significant differences (effect sizes: 0.21-0.60) between the three muscles in mean interspike interval (ISI) and ISI distributions during steady submaximal contractions with the plantar flexor and dorsiflexor muscles. The regression models explained 40% of the variance in 6-min walk distance and 47% of the variance in 25-ft walk time with two or three variables that included mean ISI for one of the plantar flexor muscles, dorsiflexor strength, and force steadiness. Walking speed and endurance in persons with relapsing-remitting MS were reduced in individuals with longer ISIs, weaker dorsiflexors, and worse plantar flexor force steadiness. NEW & NOTEWORTHY The walking endurance and gait speed of persons with relapsing-remitting multiple sclerosis (MS) were worse in individuals who had weaker dorsiflexor muscles and greater force fluctuations and longer times between action potentials discharged by motor units in plantar flexor muscles during steady isometric contractions. These findings indicate that the control of motor unit activity in lower leg muscles of individuals with MS is associated with their walking ability.

Coding of time-dependent stimuli in homogeneous and heterogeneous neural populations.

We compare the information transmission of a time-dependent signal by two types of uncoupled neuron populations that differ in their sources of variability: i) a homogeneous population whose units receive independent noise and ii) a deterministic heterogeneous population, where each unit exhibits a different baseline firing rate ('disorder'). Our criterion for making both sources of variability quantitatively comparable is that the interspike-interval distributions are identical for both systems. Numerical simulations using leaky integrate-and-fire neurons unveil that a non-zero amount of both noise or disorder maximizes the encoding efficiency of the homogeneous and heterogeneous system, respectively, as a particular case of suprathreshold stochastic resonance. Our findings thus illustrate that heterogeneity can render similarly profitable effects for neuronal populations as dynamic noise. The optimal noise/disorder depends on the system size and the properties of the stimulus such as its intensity or cutoff frequency. We find that weak stimuli are better encoded by a noiseless heterogeneous population, whereas for strong stimuli a homogeneous population outperforms an equivalent heterogeneous system up to a moderate noise level. Furthermore, we derive analytical expressions of the coherence function for the cases of very strong noise and of vanishing intrinsic noise or heterogeneity, which predict the existence of an optimal noise intensity. Our results show that, depending on the type of signal, noise as well as heterogeneity can enhance the encoding performance of neuronal populations.

A simple model of the inner-hair-cell ribbon synapse accounts for mammalian auditory-nerve-fiber spontaneous spike times

The initial neural encoding of acoustic information occurs by means of spikes in primary auditory afferents. Each mammalian primary auditory afferent (type-I auditory-nerve fiber; ANF) is associated with only one ribbon synapse in one receptor cell (inner hair cell; IHC). The properties of ANF spike trains therefore provide an indirect view of the operation of individual IHC synapses. We showed previously that a point process model of presynaptic vesicle pool depletion and deterministic exponential replenishment, combined with short postsynaptic neural refractoriness, accounts for the interspike interval (ISI) distributions, serial ISI correlations, and spike-count statistics of a population of cat-ANF spontaneous spike trains. Here, we demonstrate that this previous synapse model produces unrealistic properties when spike rates are high and show that this problem can be resolved if the replenishment of each release site is stochastic and independent. We assume that the depletion probability varies between synapses to produce differences in spontaneous rate and that the other model parameters are constant across synapses. We find that this model fits best with only four release sites per IHC synapse, a mean replenishment time of 17 ms, and absolute and mean relative refractory periods of 0.6 ms each. This model accounts for ANF spontaneous spike timing better than two influential, comprehensive models of the auditory periphery. It also reproduces ISI distributions from spontaneous and steady-state driven activity from other studies and other mammalian species. Adding fractal noise to the rate of depletion of each release site can yield long-range correlations as typically observed in long spike trains. We also examine two model variants having more complex vesicle cycles, but neither variant yields a markedly improved fit or a different estimate of the number of release sites. In addition, we examine a model variant having both short and long relative refractory components and find that it cannot account for all aspects of the data. These model results will be beneficial for understanding ribbon synapses and ANF responses to acoustic stimulation.

Automated long-term recording and analysis of neural activity in behaving animals.

Addressing how neural circuits underlie behavior is routinely done by measuring electrical activity from single neurons in experimental sessions. While such recordings yield snapshots of neural dynamics during specified tasks, they are ill-suited for tracking single-unit activity over longer timescales relevant for most developmental and learning processes, or for capturing neural dynamics across different behavioral states. Here we describe an automated platform for continuous long-term recordings of neural activity and behavior in freely moving rodents. An unsupervised algorithm identifies and tracks the activity of single units over weeks of recording, dramatically simplifying the analysis of large datasets. Months-long recordings from motor cortex and striatum made and analyzed with our system revealed remarkable stability in basic neuronal properties, such as firing rates and inter-spike interval distributions. Interneuronal correlations and the representation of different movements and behaviors were similarly stable. This establishes the feasibility of high-throughput long-term extracellular recordings in behaving animals.

Temperature manipulation of neuronal dynamics in a forebrain motor control nucleus.

Different neuronal types within brain motor areas contribute to the generation of complex motor behaviors. A widely studied songbird forebrain nucleus (HVC) has been recognized as fundamental in shaping the precise timing characteristics of birdsong. This is based, among other evidence, on the stretching and the “breaking” of song structure when HVC is cooled. However, little is known about the temperature effects that take place in its neurons. To address this, we investigated the dynamics of HVC both experimentally and computationally. We developed a technique where simultaneous electrophysiological recordings were performed during temperature manipulation of HVC. We recorded spontaneous activity and found three effects: widening of the spike shape, decrease of the firing rate and change in the interspike interval distribution. All these effects could be explained with a detailed conductance based model of all the neurons present in HVC. Temperature dependence of the ionic channel time constants explained the first effect, while the second was based in the changes of the maximal conductance using single synaptic excitatory inputs. The last phenomenon, only emerged after introducing a more realistic synaptic input to the inhibitory interneurons. Two timescales were present in the interspike distributions. The behavior of one timescale was reproduced with different input balances received form the excitatory neurons, whereas the other, which disappears with cooling, could not be found assuming poissonian synaptic inputs. Furthermore, the computational model shows that the bursting of the excitatory neurons arises naturally at normal brain temperature and that they have an intrinsic delay at low temperatures. The same effect occurs at single synapses, which may explain song stretching. These findings shed light on the temperature dependence of neuronal dynamics and present a comprehensive framework to study neuronal connectivity. This study, which is based on intrinsic neuronal characteristics, may help to understand emergent behavioral changes.

Entropy factor for randomness quantification in neuronal data

A novel measure of neural spike train randomness, an entropy factor, is proposed. It is based on the Shannon entropy of the number of spikes in a time window and can be seen as an analogy to the Fano factor. Theoretical properties of the new measure are studied for equilibrium renewal processes and further illustrated on gamma and inverse Gaussian probability distributions of interspike intervals. Finally, the entropy factor is evaluated from the experimental records of spontaneous activity in macaque primary visual cortex and compared to its theoretical behavior deduced for the renewal process models. Both theoretical and experimental results show substantial differences between the Fano and entropy factors. Rather paradoxically, an increase in the variability of spike count is often accompanied by an increase of its predictability, as evidenced by the entropy factor.

Spike patterning in oxytocin neurons: Capturing physiological behaviour with Hodgkin-Huxley and integrate-and-fire models.

Integrate-and-fire (IF) models can provide close matches to the discharge activity of neurons, but do they oversimplify the biophysical properties of the neurons? A single compartment Hodgkin-Huxley (HH) model of the oxytocin neuron has previously been developed, incorporating biophysical measurements of channel properties obtained in vitro. A simpler modified integrate-and-fire model has also been developed, which can match well the characteristic spike patterning of oxytocin neurons as observed in vivo. Here, we extended the HH model to incorporate synaptic input, to enable us to compare spike activity in the model with experimental data obtained in vivo. We refined the HH model parameters to closely match the data, and then matched the same experimental data with a modified IF model, using an evolutionary algorithm to optimise parameter matching. Finally we compared the properties of the modified HH model with those of the IF model to seek an explanation for differences between spike patterning in vitro and in vivo. We show that, with slight modifications, the original HH model, like the IF model, is able to closely match both the interspike interval (ISI) distributions of oxytocin neurons and the observed variability of spike firing rates in vivo and in vitro. This close match of both models to data depends on the presence of a slow activity-dependent hyperpolarisation (AHP); this is represented in both models and the parameters used in the HH model representation match well with optimal parameters of the IF model found by an evolutionary algorithm. The ability of both models to fit data closely also depends on a shorter hyperpolarising after potential (HAP); this is explicitly represented in the IF model, but in the HH model, it emerges from a combination of several components. The critical elements of this combination are identified.

Manifestation of peripherial coding in the effect of increasing loudness and enhanced discrimination of the intensity of tone bursts before and after tone burst noise

To find the possible reasons for the midlevel elevation of the Weber fraction in intensity discrimination of a tone burst, a comparison was performed for the complementary distributions of spike activity of an ensemble of space nerves, such as the distribution of time instants when spikes occur, the distribution of interspike intervals, and the autocorrelation function. The distribution properties were detected in a poststimulus histogram, an interspike interval histogram, and an autocorrelation histogram—all obtained from the reaction of an ensemble of model space nerves in response to an auditory noise burst–useful tone burst complex. Two configurations were used: in the first, the peak amplitude of the tone burst was varied and the noise amplitude was fixed; in the other, the tone burst amplitude was fixed and the noise amplitude was varied. Noise could precede or follow the tone burst. The noise and tone burst durations, as well as the interval between them, was 4 kHz and corresponded to the characteristic frequencies of the model space nerves. The profiles of all the mentioned histograms had two maxima. The values and the positions of the maxima in the poststimulus histogram corresponded to the amplitudes and mutual time position of the noise and the tone burst. The maximum that occurred in response to the tone burst action could be a basis for the formation of the loudness of the latter (explicit loudness). However, the positions of the maxima in the other two histograms did not depend on the positions of tone bursts and noise in the combinations. The first maximum fell in short intervals and united intervals corresponding to the noise and tone burst durations. The second maximum fell in intervals corresponding to a tone burst delay with respect to noise, and its value was proportional to the noise amplitude or tone burst amplitude that was smaller in the complex. An increase in tone burst or noise amplitudes was caused by nonlinear variations in the two maxima and the ratio between them. The size of the first maximum in the of interspike interval distribution could be the basis for the formation of the loudness of the masked tone burst (implicit loudness), and the size of the second maximum, for the formation of intensity in the periodicity pitch of the complex. The auditory effect of the midlevel enhancement of tone burst loudness could be the result of variations in the implicit tone burst loudness caused by variations in tone-burst or noise intensity. The reason for the enhancement of the Weber fraction could be competitive interaction between such subjective qualities as explicit and implicit tone-burst loudness and the intensity of the periodicity pitch of the complex.