Calculation of interspike intervals for integrate-and-fire neurons with poisson distribution of synaptic inputs

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.

Roles for Coincidence Detection in Coding Amplitude-Modulated Sounds

Many sensory neurons encode temporal information by detecting coincident arrivals of synaptic inputs. In the mammalian auditory brainstem, binaural neurons of the medial superior olive (MSO) are known to act as coincidence detectors, whereas in the lateral superior olive (LSO) roles of coincidence detection have remained unclear. LSO neurons receive excitatory and inhibitory inputs driven by ipsilateral and contralateral acoustic stimuli, respectively, and vary their output spike rates according to interaural level differences. In addition, LSO neurons are also sensitive to binaural phase differences of low-frequency tones and envelopes of amplitude-modulated (AM) sounds. Previous physiological recordings in vivo found considerable variations in monaural AM-tuning across neurons. To investigate the underlying mechanisms of the observed temporal tuning properties of LSO and their sources of variability, we used a simple coincidence counting model and examined how specific parameters of coincidence detection affect monaural and binaural AM coding. Spike rates and phase-locking of evoked excitatory and spontaneous inhibitory inputs had only minor effects on LSO output to monaural AM inputs. In contrast, the coincidence threshold of the model neuron affected both the overall spike rates and the half-peak positions of the AM-tuning curve, whereas the width of the coincidence window merely influenced the output spike rates. The duration of the refractory period affected only the low-frequency portion of the monaural AM-tuning curve. Unlike monaural AM coding, temporal factors, such as the coincidence window and the effective duration of inhibition, played a major role in determining the trough positions of simulated binaural phase-response curves. In addition, empirically-observed level-dependence of binaural phase-coding was reproduced in the framework of our minimalistic coincidence counting model. These modeling results suggest that coincidence detection of excitatory and inhibitory synaptic inputs is essential for LSO neurons to encode both monaural and binaural AM sounds.

Detecting coincident arrivals of synaptic inputs is a shared fundamental property of many sensory neurons. Such 'coincidence detection' usually refers to the detection of synchronized excitatory inputs only. Experimental evidence, however, indicated that some auditory neurons are also sensitive to the relative timing of excitatory and inhibitory synaptic inputs. This type of sensitivity is suggested to be important for coding temporal information of amplitude-modulated sounds, such as speech and other naturalistic sounds. In this study, we used a minimal model of coincidence detection to identify the key elements for temporal information processing. Our series of simulations demonstrated that (1) the threshold and time window for coincidence detection were the major factors for determining the response properties to excitatory inputs, and that (2) timed interactions between excitatory and inhibitory synaptic inputs are responsible for determining the temporal tuning properties of the neuron. These results suggest that coincidence detection is an essential function of neurons that detect the 'anti-coincidence' of excitatory and inhibitory inputs to encode temporal information of sounds.

Self-organization in Balanced State Networks by STDP and Homeostatic Plasticity

Structural inhomogeneities in synaptic efficacies have a strong impact on population response dynamics of cortical networks and are believed to play an important role in their functioning. However, little is known about how such inhomogeneities could evolve by means of synaptic plasticity. Here we present an adaptive model of a balanced neuronal network that combines two different types of plasticity, STDP and synaptic scaling. The plasticity rules yield both long-tailed distributions of synaptic weights and firing rates. Simultaneously, a highly connected subnetwork of driver neurons with strong synapses emerges. Coincident spiking activity of several driver cells can evoke population bursts and driver cells have similar dynamical properties as leader neurons found experimentally. Our model allows us to observe the delicate interplay between structural and dynamical properties of the emergent inhomogeneities. It is simple, robust to parameter changes and able to explain a multitude of different experimental findings in one basic network.

It is widely believed that the structure of neuronal circuits plays a major role in brain functioning. Although the full synaptic connectivity for larger populations is not yet assessable even by current experimental techniques, available data show that neither synaptic strengths nor the number of synapses per neuron are homogeneously distributed. Several studies have found long-tailed distributions of synaptic weights with many weak and a few exceptionally strong synaptic connections, as well as strongly connected cells and subnetworks that may play a decisive role for data processing in neural circuits. Little is known about how inhomogeneities could arise in the developing brain and we hypothesize that there is a self-organizing principle behind their appearance. In this study we show how structural inhomogeneities can emerge by simple synaptic plasticity mechanisms from an initially homogeneous network. We perform numerical simulations and show analytically how a small imbalance in the initial structure is amplified by the synaptic plasticities and their interplay. Our network can simultaneously explain several experimental observations that were previously not linked.

On Firing Rate Estimation for Dependent Interspike Intervals

If interspike intervals are dependent, the instantaneous firing rate does not catch important features of spike trains. In this case, the conditional instantaneous rate plays the role of the instantaneous firing rate for the case of samples of independent interspike intervals. If the conditional distribution of the interspikes intervals (ISIs) is unknown, it becomes difficult to evaluate the conditional firing rate. We propose a nonparametric estimator for the conditional instantaneous firing rate for Markov, stationary, and ergodic ISIs. An algorithm to check the reliability of the proposed estimator is introduced, and its consistency properties are proved. The method is applied to data obtained from a stochastic two-compartment model and to in vitro experimental data.

Neuronal spike-train responses in the presence of threshold noise

The variability of neuronal firing has been an intense topic of study for many years. From a modelling perspective it has often been studied in conductance based spiking models with the use of additive or multiplicative noise terms to represent channel fluctuations or the stochastic nature of neurotransmitter release. Here we propose an alternative approach using a simple leaky integrate-and-fire model with a noisy threshold. Initially, we develop a mathematical treatment of the neuronal response to periodic forcing using tools from linear response theory and use this to highlight how a noisy threshold can enhance downstream signal reconstruction. We further develop a more general framework for understanding the responses to large amplitude forcing based on a calculation of first passage times. This is ideally suited to understanding stochastic mode-locking, for which we numerically determine the Arnol'd tongue structure. An examination of data from regularly firing stellate neurons within the ventral cochlear nucleus, responding to sinusoidally amplitude modulated pure tones, shows tongue structures consistent with these predictions and highlights that stochastic, as opposed to deterministic, mode-locking is utilised at the level of the single stellate cell to faithfully encode periodic stimuli.

Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data

The Ornstein-Uhlenbeck process has been proposed as a model for the spontaneous activity of a neuron. In this model, the firing of the neuron corresponds to the first passage of the process to a constant boundary, or threshold. While the Laplace transform of the first-passage time distribution is available, the probability distribution function has not been obtained in any tractable form. We address the problem of estimating the parameters of the process when the only available data from a neuron are the interspike intervals, or the times between firings. In particular, we give an algorithm for computing maximum likelihood estimates and their corresponding confidence regions for the three identifiable (of the five model) parameters by numerically inverting the Laplace transform. A comparison of the two-parameter algorithm (where the time constant tau is known a priori) to the three-parameter algorithm shows that significantly more data is required in the latter case to achieve comparable parameter resolution as measured by 95% confidence intervals widths. The computational methods described here are a efficient alternative to other well known estimation techniques for leaky integrate-and-fire models. Moreover, it could serve as a template for performing parameter inference on more complex integrate-and-fire neuronal models.

Analysis of a stochastic neuronal model with excitatory inputs and state-dependent effects

We propose a stochastic model for the firing activity of a neuronal unit. It includes the decay effect of the membrane potential in absence of stimuli, and the occurrence of time-varying excitatory inputs governed by a Poisson process. The sample-paths of the membrane potential are piecewise exponentially decaying curves with jumps of random amplitudes occurring at the input times. An analysis of the probability distributions of the membrane potential and of the firing time is performed. In the special case of time-homogeneous stimuli the firing density is obtained in closed form, together with its mean and variance.

A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties

The integrate-and-fire neuron model describes the state of a neuron in terms of its membrane potential, which is determined by the synaptic inputs and the injected current that the neuron receives. When the membrane potential reaches a threshold, an action potential (spike) is generated. This review considers the model in which the synaptic input varies periodically and is described by an inhomogeneous Poisson process, with both current and conductance synapses. The focus is on the mathematical methods that allow the output spike distribution to be analyzed, including first passage time methods and the Fokker-Planck equation. Recent interest in the response of neurons to periodic input has in part arisen from the study of stochastic resonance, which is the noise-induced enhancement of the signal-to-noise ratio. Networks of integrate-and-fire neurons behave in a wide variety of ways and have been used to model a variety of neural, physiological, and psychological phenomena. The properties of the integrate-and-fire neuron model with synaptic input described as a temporally homogeneous Poisson process are reviewed in an accompanying paper (Burkitt in Biol Cybern, 2006).

A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input

The integrate-and-fire neuron model is one of the most widely used models for analyzing the behavior of neural systems. It describes the membrane potential of a neuron in terms of the synaptic inputs and the injected current that it receives. An action potential (spike) is generated when the membrane potential reaches a threshold, but the actual changes associated with the membrane voltage and conductances driving the action potential do not form part of the model. The synaptic inputs to the neuron are considered to be stochastic and are described as a temporally homogeneous Poisson process. Methods and results for both current synapses and conductance synapses are examined in the diffusion approximation, where the individual contributions to the postsynaptic potential are small. The focus of this review is upon the mathematical techniques that give the time distribution of output spikes, namely stochastic differential equations and the Fokker-Planck equation. The integrate-and-fire neuron model has become established as a canonical model for the description of spiking neurons because it is capable of being analyzed mathematically while at the same time being sufficiently complex to capture many of the essential features of neural processing. A number of variations of the model are discussed, together with the relationship with the Hodgkin-Huxley neuron model and the comparison with electrophysiological data. A brief overview is given of two issues in neural information processing that the integrate-and-fire neuron model has contributed to - the irregular nature of spiking in cortical neurons and neural gain modulation.

Frequency-dependent selection of alternative spinal pathways with common periodic sensory input

Electrical stimulation of the lumbar cord at distinct frequency ranges has been shown to evoke either rhythmical, step-like movements (25-50 Hz) or a sustained extension (5-15 Hz) of the paralysed lower limbs in complete spinal cord injured subjects. Frequency-dependent activation of previously "silent" spinal pathways was suggested to contribute to the differential responsiveness to distinct neuronal "codes" and the modifications in the electromyographic recordings during the actual implementation of the evoked motor tasks. In the present study we examine this suggestion by means of a simplified biology-based neuronal network. Involving two basic mechanisms, temporal summation of synaptic input and presynaptic inhibition, the model exhibits several patterns of mono- and/or oligo-synaptic motor output in response to different interstimulus intervals. It thus reproduces fundamental input-output features of the lumbar cord isolated from the brain. The results confirm frequency-dependent spinal pathway selection as a simple mechanism which enables the cord to respond to distinct neuronal codes with different motor behaviours and to control the actual performance of the latter.

Enhancement of GABA-related signalling is associated with increase of functional connectivity in human cortex

Structural or operational synchrony analysis with EEG was conducted in order to detect functional interaction between cortical areas during an enhanced inhibition induced by the GABAergic agonist lorazepam in a double-blind, randomized, placebo-controlled, cross-over study in eight healthy human subjects. Specifically, we investigated whether a neuronal inhibitory system in the brain mediates functional decoupling of cortical areas. Single-dose lorazepam administration resulted in a widespread increase in the inter-area functional connectivity and an increase in the strength of functional long-range and interhemispheric connections. These results suggest that inhibition can be an efficient mechanism for synchronization of large neuronal populations.

Mean Instantaneous Firing Frequency Is Always Higher Than the Firing Rate

Frequency coding is considered one of the most common coding strategies employed by neural systems. This fact leads, in experiments as well as in theoretical studies, to construction of so-called transfer functions, where the output firing frequency is plotted against the input intensity. The term firing frequency can be understood differently in different contexts. Basically, it means that the number of spikes over an interval of preselected length is counted and then divided by the length of the interval, but due to the obvious limitations, the length of observation cannot be arbitrarily long. Then firing frequency is defined as reciprocal to the mean interspike interval. In parallel, an instantaneous firing frequency can be defined as reciprocal to the length of current interspike interval, and by taking a mean of these, the definition can be extended to introduce the mean instantaneous firing frequency. All of these definitions of firing frequency are compared in an effort to contribute to a better understanding of the input-output properties of a neuron.

Amplitude and frequency dependence of spike timing: implications for dynamic regulation

The spike-time reliability of motoneurons in the Aplysia buccal motor ganglion was studied as a function of the frequency content and the relative amplitude of the fluctuations in the neuronal input, calculated as the coefficient of variation (CV). Measurements of spike-time reliability to sinusoidal and aperiodic inputs, as well as simulations of a noisy leaky integrate-and-fire neuron stimulated by spike trains drawn from a periodically modulated process, demonstrate that there are three qualitatively different CV-dependent mechanisms that determine reliability: noise-dominated (CV < 0.05 for Aplysia motoneurons) where spike timing is unreliable regardless of frequency content; resonance-dominated (CV approximately 0.05-0.25) where reliability is reduced by removal of input frequencies equal to motoneuron firing rate; and amplitude-dominated (CV >0.35) where reliability depends on input frequencies greater than motoneuron firing rate. In the resonance-dominated regime, changes in the activity of the presynaptic inhibitory interneuron B4/5 alter motoneuron spike-time reliability. The increases or decreases in reliability occur coincident with small changes in motoneuron spiking rate due to changes in interneuron activity. Injection of a hyperpolarizing current into the motoneuron reproduces the interneuron-induced changes in reliability. The rate-dependent changes in reliability can be understood from the phase-locking properties of regularly spiking motoneurons to periodic inputs. Our observations demonstrate that the ability of a neuron to support a spike-time code can be actively controlled by varying the properties of the neuron and its input.

Summation of spatiotemporal input patterns in leaky integrate-and-fire neurons: application to neurons in the cochlear nucleus receiving converging auditory nerve fiber input

The response of leaky integrate-and-fire neurons is analyzed for periodic inputs whose phases vary with their spatial location. The model gives the relationship between the spatial summation distance and the degree of phase locking of the output spikes (i.e., locking to the periodic stochastic inputs, measured by the synchronization index). The synaptic inputs are modeled as an inhomogeneous Poisson process, and the analysis is carried out in the Gaussian approximation. The model has been applied to globular bushy cells of the cochlear nucleus, which receive converging inputs from auditory nerve fibers that originate at neighboring sites in the cochlea. The model elucidates the roles played by spatial summation and coincidence detection, showing how synchronization decreases with an increase in both frequency and spatial spread of inputs. It also shows under what conditions an enhancement of synchronization of the output relative to the input takes place.

Synchronization of the neural response to noisy periodic synaptic input

The timing information contained in the response of a neuron to noisy periodic synaptic input is analyzed for the leaky integrate-and-fire neural model. We address the question of the relationship between the timing of the synaptic inputs and the output spikes. This requires an analysis of the interspike interval distribution of the output spikes, which is obtained in the gaussian approximation. The conditional output spike density in response to noisy periodic input is evaluated as a function of the initial phase of the inputs. This enables the phase transition matrix to be calculated, which relates the phase at which the output spike is generated to the initial phase of the inputs. The interspike interval histogram and the period histogram for the neural response to ongoing periodic input are then evaluated by using the leading eigenvector of this phase transition matrix. The synchronization index of the output spikes is found to increase sharply as the inputs become synchronized. This enhancement of synchronization is most pronounced for large numbers of inputs and lower frequencies of modulation and also for rates of input near the critical input rate. However, the mutual information between the input phase of the stimulus and the timing of output spikes is found to decrease at low input rates as the number of inputs increases. The results show close agreement with those obtained from numerical simulations for large numbers of inputs.

Gaussian process approach to spiking neurons for inhomogeneous Poisson inputs

This article presents a new theoretical framework to consider the dynamics of a stochastic spiking neuron model with general membrane response to input spike. We assume that the input spikes obey an inhomogeneous Poisson process. The stochastic process of the membrane potential then becomes a gaussian process. When a general type of the membrane response is assumed, the stochastic process becomes a Markov-gaussian process. We present a calculation method for the membrane potential density and the firing probability density. Our new formulation is the extension of the existing formulation based on diffusion approximation. Although the single Markov assumption of the diffusion approximation simplifies the stochastic process analysis, the calculation is inaccurate when the stochastic process involves a multiple Markov property. We find that the variation of the shape of the membrane response, which has often been ignored in existing stochastic process studies, significantly affects the firing probability. Our approach can consider the reset effect, which has been difficult to deal with by analysis based on the first passage time density.

Temporal processing from the auditory nerve to the medial nucleus of the trapezoid body in the rat

This investigation examines temporal processing through successive sites in the rat auditory pathway: auditory nerve (AN), anteroventral cochlear nucleus (AVCN) and the medial nucleus of the trapezoid body (MNTB). The degree of phase-locking, measured as vector strength, varied with intensity relative to the cell's threshold, and saturated at a value that depended upon stimulus frequency. A typical pattern showed decline in the saturated vector strength from approximately 0.8 at 400 Hz to about 0.3 at 2000 Hz, with similar profiles in units with a range of characteristic frequencies (480-32,000 Hz). A new expression for temporal dispersion indicates that this variation corresponds to a limiting degree of temporal imprecision, which is relatively consistent between different cells. From AN to AVCN, an increase in vector strength was seen for frequencies below 1000 Hz. At higher frequencies, a decrease in vector strength was observed. From AVCN to MNTB a tendency for temporal coding to be improved below 800 Hz and degraded further above 1500 Hz was seen. This change in temporal processing ability could be attributed to units classified as primary-like with notch (PL(N)). PL(N) MNTB units showed a similar vector strength distribution to PL(N) AVCN units. Our results suggest that AVCN PL(N) units, representing globular bushy cells, are specialised for enhancing the temporal code at low frequencies and relaying this information to principal cells of the MNTB.