Analysis of integrate-and-fire neurons: synchronization of synaptic input and spike output

A new technique for analyzing the probability distribution of output spikes for the integrate-and-fire model is presented. This technique enables us to investigate models with arbitrary synaptic response functions that incorporate both leakage across the membrane and a rise time of the postsynaptic potential. The results, which are compared with numerical simulations, are exact in the limit of a large number of small-amplitude inputs. This method is applied to the synchronization problem, in which we examine the relationship between the spread in arrival times of the inputs (the temporal jitter of the synaptic input) and the resultant spread in the times at which the output spikes are generated (output jitter). The results of previous studies, which indicated that the ration of the output jitter to the input jitter is consistently less than one and that it decreases for increasing numbers of inputs, are confirmed for three classes of the integrate-and-fire model. In addition to the previously identified factors of axonal propagation times and synaptic jitter, we identify the variation in the spike-generating thresholds of the neurons and the variation in the number of active inputs as being important factors that determine the timing jitter in layered networks. Previously observed phase differences between optimally and suboptimally stimulated neurons may be understood in terms of the relative time taken to reach threshold.     

Response properties of an integrate-and-fire model that receives subthreshold inputs

A computational technique is described for calculation of the interspike interval and poststimulus time histograms for the responses of an integrate-and-fire model to arbitrary inputs. The effects of the model parameters on the response statistics were studied systematically. Specifically, the probability distribution of the membrane potential was calculated as a function of time, and the mean interspike interval and PST histogram were calculated for arbitrary inputs. For stationary inputs, the regularity of the output was studied in detail for various model parameters. For nonstationary inputs, the effects of the model parameters on the output synchronization index were explored. The results show that enhanced synchronization in response to low-frequency stimuli required a large number (n > 25) of weak inputs. Irregular responses and a linear input-output rate relationship required strong (but subthreshold) inputs with a small time constant. A model cell with mixed-amplitude synaptic inputs can respond to stationary inputs irregularly and have enhanced synchronization to nonstationary inputs that are phase-locked to low-frequency inputs. Both of these response properties have been reported for some cells in the ventral cochlear nucleus in the auditory brainstem.     

Spike-timing-dependent plasticity: the relationship to rate-based learning for models with weight dynamics determined by a stable fixed point

Experimental evidence indicates that synaptic modification depends on the timing relationship between the presynaptic inputs and the output spikes that they generate. In this letter, results are presented for models of spike-timing-dependent plasticity (STDP) whose weight dynamics is determined by a stable fixed point. Four classes of STDP are identified on the basis of the time extent of their input-output interactions. The effect on the potentiation of synapses with different rates of input is investigated to elucidate the relationship of STDP with classical studies of long-term potentiation and depression and rate-based Hebbian learning. The selective potentiation of higher-rate synaptic inputs is found only for models where the time extent of the input-output interactions is input restricted (i.e., restricted to time domains delimited by adjacent synaptic inputs) and that have a time-asymmetric learning window with a longer time constant for depression than for potentiation. The analysis provides an account of learning dynamics determined by an input-selective stable fixed point. The effect of suppressive interspike interactions on STDP is also analyzed and shown to modify the synaptic dynamics.     

Information encoding and computation with spikes and bursts

Neurons compute and communicate by transforming synaptic input patterns into output spike trains. The nature of this transformation depends crucially on the properties of voltage-gated conductances in neuronal membranes. These intrinsic membrane conductances can enable neurons to generate different spike patterns including brief, high-frequency bursts that are commonly observed in a variety of brain regions. Here we examine how the membrane conductances that generate bursts affect neural computation and encoding. We simulated a bursting neuron model driven by random current input signal and superposed noise. We consider two issues: the timing reliability of different spike patterns and the computation performed by the neuron. Statistical analysis of the simulated spike trains shows that the timing of bursts is much more precise than the timing of single spikes. Furthermore, the number of spikes per burst is highly robust to noise. Next we considered the computation performed by the neuron: how different features of the input current are mapped into specific output spike patterns. Dimensional reduction and statistical classification techniques were used to determine the stimulus features triggering different firing patterns. Our main result is that spikes, and bursts of different durations, code for different stimulus features, which can be quantified without a priori assumptions about those features. These findings lead us to propose that the biophysical mechanisms of spike generation enables individual neurons to encode different stimulus features into distinct spike patterns.     

Weak, stochastic temporal correlation of large scale synaptic input is A major determinant of neuronal bandwidth

We determine the bandwidth of a model neurone to large-scale synaptic input by assessing the frequency response between the outputs of a two-cell simulation that share a percentage of the total synaptic input. For temporally uncorrelated inputs, a large percentage of common inputs are required before the output discharges of the two cells exhibit significant correlation. In contrast, a small percentage (5%) of the total synaptic input that involves stochastic spike trains that are weakly correlated over a broad range of frequencies exert a clear influence on the output discharge of both cells over this range of frequencies. Inputs that are weakly correlated at a single frequency induce correlation between the output discharges only at the frequency of correlation. The strength of temporal correlation required is sufficiently weak that analysis of a sample pair of input spike trains could fail to reveal the presence of correlated input. Weak temporal correlation between inputs is therefore a major determinant of the transmission to the output discharge of frequencies present in the spike discharges of presynaptic inputs, and therefore of neural bandwidth.     

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

We present a new technique for calculating the interspike intervals of integrate-and-fire neurons. There are two new components to this technique. First, the probability density of the summed potential is calculated by integrating over the distribution of arrival times of the afferent post-synaptic potentials (PSPs), rather than using conventional stochastic differential equation techniques. A general formulation of this technique is given in terms of the probability distribution of the inputs and the time course of the postsynaptic response. The expressions are evaluated in the gaussian approximation, which gives results that become more accurate for large numbers of small-amplitude PSPs. Second, the probability density of output spikes, which are generated when the potential reaches threshold, is given in terms of an integral involving a conditional probability density. This expression is a generalization of the renewal equation, but it holds for both leaky neurons and situations in which there is no time-translational invariance. The conditional probability density of the potential is calculated using the same technique of integrating over the distribution of arrival times of the afferent PSPs. For inputs with a Poisson distribution, the known analytic solutions for both the perfect integrator model and the Stein model (which incorporates membrane potential leakage) in the diffusion limit are obtained. The interspike interval distribution may also be calculated numerically for models that incorporate both membrane potential leakage and a finite rise time of the postsynaptic response. Plots of the relationship between input and output firing rates, as well as the coefficient of variation, are given, and inputs with varying rates and amplitudes, including inhibitory inputs, are analyzed. The results indicate that neurons functioning near their critical threshold, where the inputs are just sufficient to cause firing, display a large variability in their spike timings.     

Spatiotemporal spike encoding of a continuous external signal

Interspike intervals of spikes emitted from an integrator neuron model of sensory neurons can encode input information represented as a continuous signal from a deterministic system. If a real brain uses spike timing as a means of information processing, other neurons receiving spatiotemporal spikes from such sensory neurons must also be capable of treating information included in deterministic interspike intervals. In this article, we examine functions of neurons modeling cortical neurons receiving spatiotemporal spikes from many sensory neurons. We show that such neuron models can encode stimulus information passed from the sensory model neurons in the form of interspike intervals. Each sensory neuron connected to the cortical neuron contributes equally to the information collection by the cortical neuron. Although the incident spike train to the cortical neuron is a superimposition of spike trains from many sensory neurons, it need not be decomposed into spike trains according to the input neurons. These results are also preserved for generalizations of sensory neurons such as a small amount of leak, noise, inhomogeneity in firing rates, or biases introduced in the phase distributions.     

Ergodicity of spike trains: when does trial averaging make sense?

Neuronal information processing is often studied on the basis of spiking patterns. The relevant statistics such as firing rates calculated with the peri-stimulus time histogram are obtained by averaging spiking patterns over many experimental runs. However, animals should respond to one experimental stimulation in real situations, and what is available to the brain is not the trial statistics but the population statistics. Consequently, physiological ergodicity, namely, the consistency between trial averaging and population averaging, is implicitly assumed in the data analyses, although it does not trivially hold true. In this letter, we investigate how characteristics of noisy neural network models, such as single neuron properties, external stimuli, and synaptic inputs, affect the statistics of firing patterns. In particular, we show that how high membrane potential sensitivity to input fluctuations, inability of neurons to remember past inputs, external stimuli with large variability and temporally separated peaks, and relatively few contributions of synaptic inputs result in spike trains that are reproducible over many trials. The reproducibility of spike trains and synchronous firing are contrasted and related to the ergodicity issue. Several numerical calculations with neural network examples are carried out to support the theoretical results.     

Neurons tune to the earliest spikes through STDP

Spike timing-dependent plasticity (STDP) is a learning rule that modifies the strength of a neuron's synapses as a function of the precise temporal relations between input and output spikes. In many brains areas, temporal aspects of spike trains have been found to be highly reproducible. How will STDP affect a neuron's behavior when it is repeatedly presented with the same input spike pattern? We show in this theoretical study that repeated inputs systematically lead to a shaping of the neuron's selectivity, emphasizing its very first input spikes, while steadily decreasing the postsynaptic response latency. This was obtained under various conditions of background noise, and even under conditions where spiking latencies and firing rates, or synchrony, provided conflicting informations. The key role of first spikes demonstrated here provides further support for models using a single wave of spikes to implement rapid neural processing.     

Short-term synaptic plasticity can enhance weak signal detectability in nonrenewal spike trains

We study the encoding of weak signals in spike trains with interspike interval (ISI) correlations and the signals' subsequent detection in sensory neurons. Motivated by the observation of negative ISI correlations in auditory and electrosensory afferents, we assess the theoretical performance limits of an individual detector neuron receiving a weak signal distributed across multiple afferent inputs. We assess the functional role of ISI correlations in the detection process using statistical detection theory and derive two sequential likelihood ratio detector models: one for afferents with renewal statistics; the other for afferents with negatively correlated ISIs. We suggest a mechanism that might enable sensory neurons to implicitly compute conditional probabilities of presynaptic spikes by means of short-term synaptic plasticity. We demonstrate how this mechanism can enhance a postsynaptic neuron's sensitivity to weak signals by exploiting the correlation structure of the input spike trains. Our model not only captures fundamental aspects of early electrosensory signal processing in weakly electric fish, but may also bear relevance to the mammalian auditory system and other sensory modalities.     

Information transfer in entrained cortical neurons

Cortical interneurons connected by gap junctions can provide a synchronized inhibitory drive that can entrain pyramidal cells. This was studied in a single-compartment Hodgkin-Huxley-type model neuron that was entrained by periodic inhibitory inputs with low jitter in the input spike times (i.e. high precision), and a variable but large number of presynaptic spikes on each cycle. During entrainment the Shannon entropy of the output spike times was reduced sharply compared with its value outside entrainment. Surprisingly, however, the information transfer as measured by the mutual information between the number of inhibitory inputs in a cycle and the phase lag of the subsequent output spike was significantly increased during entrainment. This increase was due to the reduced contribution of the internal correlations to the output variability. These theoretical predictions were supported by experimental recordings from the rat neocortex and hippocampus in vitro.     

Hebbian imprinting and retrieval in oscillatory neural networks

We introduce a model of generalized Hebbian learning and retrieval in oscillatory neural networks modeling cortical areas such as hippocampus and olfactory cortex. Recent experiments have shown that synaptic plasticity depends on spike timing, especially on synapses from excitatory pyramidal cells, in hippocampus, and in sensory and cerebellar cortex. Here we study how such plasticity can be used to form memories and input representations when the neural dynamics are oscillatory, as is common in the brain (particularly in the hippocampus and olfactory cortex). Learning is assumed to occur in a phase of neural plasticity, in which the network is clamped to external teaching signals. By suitable manipulation of the nonlinearity of the neurons or the oscillation frequencies during learning, the model can be made, in a retrieval phase, either to categorize new inputs or to map them, in a continuous fashion, onto the space spanned by the imprinted patterns. We identify the first of these possibilities with the function of olfactory cortex and the second with the observed response characteristics of place cells in hippocampus. We investigate both kinds of networks analytically and by computer simulations, and we link the models with experimental findings, exploring, in particular, how the spike timing dependence of the synaptic plasticity constrains the computational function of the network and vice versa.     

Reinforcement learning through modulation of spike-timing-dependent synaptic plasticity

The persistent modification of synaptic efficacy as a function of the relative timing of pre- and postsynaptic spikes is a phenomenon known as spike-timing-dependent plasticity (STDP). Here we show that the modulation of STDP by a global reward signal leads to reinforcement learning. We first derive analytically learning rules involving reward-modulated spike-timing-dependent synaptic and intrinsic plasticity, by applying a reinforcement learning algorithm to the stochastic spike response model of spiking neurons. These rules have several features common to plasticity mechanisms experimentally found in the brain. We then demonstrate in simulations of networks of integrate-and-fire neurons the efficacy of two simple learning rules involving modulated STDP. One rule is a direct extension of the standard STDP model (modulated STDP), and the other one involves an eligibility trace stored at each synapse that keeps a decaying memory of the relationships between the recent pairs of pre- and postsynaptic spike pairs (modulated STDP with eligibility trace). This latter rule permits learning even if the reward signal is delayed. The proposed rules are able to solve the XOR problem with both rate coded and temporally coded input and to learn a target output firing-rate pattern. These learning rules are biologically plausible, may be used for training generic artificial spiking neural networks, regardless of the neural model used, and suggest the experimental investigation in animals of the existence of reward-modulated STDP.     

Spiking神经元输入脉冲扰动敏感性研究

脉冲神经网络是一种基于生物的网络模型,它的输入输出为具有时间特性的脉冲序列,其运行机制相比其他传统人工神经网络更加接近于生物神经网络。神经元之间通过脉冲序列传递信息,这些信息通过脉冲的激发时间编码能够更有效地发挥网络的学习性能。脉冲神经元的时间特性导致了其工作机制较为复杂,而spiking神经元的敏感性反映了当神经元输入发生扰动时输出的spike的变化情况,可以作为研究神经元内部工作机制的工具。不同于传统的神经网络,spiking神经元敏感性定义为输出脉冲的变化时刻个数与运行时间长度的比值,能直接反映出输入扰动对输出的影响程度。通过对不同形式的输入扰动敏感性的分析,可以看出spiking神经元的敏感性较为复杂,当全体突触发生扰动时,神经元为定值,而当部分突触发生扰动时,不同突触的扰动会导致不同大小的神经元敏感性。     

Mechanisms that modulate the transfer of spiking correlations

Correlations between neuronal spike trains affect network dynamics and population coding. Overlapping afferent populations and correlations between presynaptic spike trains introduce correlations between the inputs to downstream cells. To understand network activity and population coding, it is therefore important to understand how these input correlations are transferred to output correlations.Recent studies have addressed this question in the limit of many inputs with infinitesimal postsynaptic response amplitudes, where the total input can be approximated by gaussian noise. In contrast, we address the problem of correlation transfer by representing input spike trains as point processes, with each input spike eliciting a finite postsynaptic response. This approach allows us to naturally model synaptic noise and recurrent coupling and to treat excitatory and inhibitory inputs separately.We derive several new results that provide intuitive insights into the fundamental mechanisms that modulate the transfer of spiking correlations.     

Stochastic Resonance and Coincidence Detection in Single Neurons

We demonstrate that a realistic neuron model expressed by the Hodgkin-Huxley equations shows a stochastic resonance phenomenon, by computing cross-correlation between input and output spike timing when the neuron receives both aperiodic signal input of spike packets and background random noise of both excitatory and inhibitory spikes. We consider that such a signal detection is realized because the neuron with active properties is sensitive to fluctuation caused by a sharp increase just after a sudden dip of excitatory noise spikes and a gradual decrease of inhibitory noise spikes. We also show that the model generates highly irregular firing of output spikes on the basis of the modulation detecting property.     

Distortion of neural signals by spike coding

Analog neural signals must be converted into spike trains for transmission over electrically leaky axons. This spike encoding and subsequent decoding leads to distortion. We quantify this distortion by deriving approximate expressions for the mean square error between the inputs and outputs of a spiking link. We use integrate-and-fire and Poisson encoders to convert naturalistic stimuli into spike trains and spike count and inter-spike interval decoders to generate reconstructions of the stimulus. The distortion expressions enable us to compare these spike coding schemes over a large parameter space. We verify that the integrate-and-fire encoder is more effective than the Poisson encoder. The disparity between the two encoders diminishes as the stimulus coefficient of variation (CV) increases, at which point, the variability attributed to the stimulus overwhelms the variability attributed to Poisson statistics. When the stimulus CV is small, the interspike interval decoder is superior, as the distortion resulting from spike count decoding is dominated by a term that is attributed to the discrete nature of the spike count. In this regime, additive noise has a greater impact on the interspike interval decoder than the spike count decoder. When the stimulus CV is large, the average signal excursion is much larger than the quantization step size, and spike count decoding is superior.     

Phase transition and hysteresis in an ensemble of stochastic spiking neurons

An ensemble of stochastic nonleaky integrate-and-fire neurons with global, delayed, and excitatory coupling and a small refractory period is analyzed. Simulations with adiabatic changes of the coupling strength indicate the presence of a phase transition accompanied by a hysteresis around a critical coupling strength. Below the critical coupling production of spikes in the ensemble is governed by the stochastic dynamics, whereas for coupling greater than the critical value, the stochastic dynamics loses its influence and the units organize into several clusters with self-sustained activity. All units within one cluster spike in unison, and the clusters themselves are phase-locked. Theoretical analysis leads to upper and lower bounds for the average interspike interval of the ensemble valid for all possible coupling strengths. The bounds allow calculating the limit behavior for large ensembles and characterize the phase transition analytically. These results may be extensible to pulse-coupled oscillators.     

Impact of correlated inputs on the output of the integrate- and-fire model

For the integrate-and-fire model with or without reversal potentials, we consider how correlated inputs affect the variability of cellular output. For both models, the variability of efferent spike trains measured by coefficient of variation (CV) of the interspike interval is a nondecreasing function of input correlation. When the correlation coefficient is greater than 0.09, the CV of the integrate-and-fire model without reversal potentials is always above 0.5, no matter how strong the inhibitory inputs. When the correlation coefficient is greater than 0.05, CV for the integrate-and-fire model with reversal potentials is always above 0. 5, independent of the strength of the inhibitory inputs. Under a given condition on correlation coefficients, we find that correlated Poisson processes can be decomposed into independent Poisson processes. We also develop a novel method to estimate the distribution density of the first passage time of the integrate-and-fire model.