Influence of temporal correlation of synaptic input on the rate and variability of firing in neurons

The spike trains that transmit information between neurons are stochastic. We used the theory of random point processes and simulation methods to investigate the influence of temporal correlation of synaptic input current on firing statistics. The theory accounts for two sources for temporal correlation: synchrony between spikes in presynaptic input trains and the unitary synaptic current time course. Simulations show that slow temporal correlation of synaptic input leads to high variability in firing. In a leaky integrate-and-fire neuron model with spike afterhyperpolarization the theory accurately predicts the firing rate when the spike threshold is higher than two standard deviations of the membrane potential fluctuations. For lower thresholds the spike afterhyperpolarization reduces the firing rate below the theory's predicted level when the synaptic correlation decays rapidly. If the synaptic correlation decays slower than the spike afterhyperpolarization, spike bursts can occur during single broad peaks of input fluctuations, increasing the firing rate over the prediction. Spike bursts lead to a coefficient of variation for the interspike intervals that can exceed one, suggesting an explanation of high coefficient of variation for interspike intervals observed in vivo.     

Accuracy of neuronal interspike times calculated from a diffusion approximation

The theory of neuronal firing in Stein's model is outlined as well as the corresponding theory for a diffusion approximation which has the same first two infinitesimal moments. The diffusion approximation is derived from the discontinuous model in the limit of large input frequencies and small postsynaptic potential amplitudes. A comparison of the calculated mean interspike intervals is made for various values of the threshold for firing and various input frequencies. The diffusion approximation can underestimate the interspike interval by up to 100% or severely overestimate it, depending on the input frequencies and the threshold. A general relation between the predictions of the two models is deduced.     

Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process

Stochastic leaky integrate-and-fire models are popular due to their simplicity and statistical tractability. They have been widely applied to gain understanding of the underlying mechanisms for spike timing in neurons, and have served as building blocks for more elaborate models. Especially the Ornstein–Uhlenbeck process is popular to describe the stochastic fluctuations in the membrane potential of a neuron, but also other models like the square-root model or models with a non-linear drift are sometimes applied. Data that can be described by such models have to be stationary and thus, the simple models can only be applied over short time windows. However, experimental data show varying time constants, state dependent noise, a graded firing threshold and time-inhomogeneous input. In the present study we build a jump diffusion model that incorporates these features, and introduce a firing mechanism with a state dependent intensity. In addition, we suggest statistical methods to estimate all unknown quantities and apply these to analyze turtle motoneuron membrane potentials. Finally, simulated and real data are compared and discussed. We find that a square-root diffusion describes the data much better than an Ornstein–Uhlenbeck process with constant diffusion coefficient. Further, the membrane time constant decreases with increasing depolarization, as expected from the increase in synaptic conductance. The network activity, which the neuron is exposed to, can be reasonably estimated to be a threshold version of the nerve output from the network. Moreover, the spiking characteristics are well described by a Poisson spike train with an intensity depending exponentially on the membrane potential.     

Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions

We introduce fractional Nernst-Planck equations and derive fractional cable equations as macroscopic models for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding, crowding or trapping. The anomalous subdiffusion is modelled by replacing diffusion constants with time dependent operators parameterized by fractional order exponents. Solutions are obtained as functions of the scaling parameters for infinite cables and semi-infinite cables with instantaneous current injections. Voltage attenuation along dendrites in response to alpha function synaptic inputs is computed. Action potential firing rates are also derived based on simple integrate and fire versions of the models. Our results show that electrotonic properties and firing rates of nerve cells are altered by anomalous subdiffusion in these models. We have suggested electrophysiological experiments to calibrate and validate the models.      

Threshold behavior in the initiation of hippocampal population bursts

Hippocampal population discharges such as sharp waves, epileptiform firing, and GDPs recur at long and variable intervals. The mechanisms for their precise timing are not well understood. Here, we show that population bursts in the disinhibited CA3 region are initiated at a threshold level of population firing after recovery from a previous event. Each population discharge follows an active buildup period when synaptic traffic and cell firing increase to threshold levels. Single-cell firing can advance burst onset by increasing population firing to suprathreshold values. Population synchrony is suppressed when threshold frequencies cannot be reached due to reduced cellular excitability or synaptic efficacy. Reducing synaptic strength reveals partially synchronous population bursts that are curtailed by GABA(B)-mediated conductances. Excitatory glutamatergic transmission and delayed GABA(B)-mediated signals have opposing feedback effects on CA3 cell firing and so determine threshold behavior for population synchrony.     

Amplification of trial-to-trial response variability by neurons in visual cortex

The visual cortex responds to repeated presentations of the same stimulus with high variability. Because the firing mechanism is remarkably noiseless, the source of this variability is thought to lie in the membrane potential fluctuations that result from summated synaptic input. Here this hypothesis is tested through measurements of membrane potential during visual stimulation. Surprisingly, trial-to-trial variability of membrane potential is found to be low. The ratio of variance to mean is much lower for membrane potential than for firing rate. The high variability of firing rate is explained by the threshold present in the function that converts inputs into firing rates. Given an input with small, constant noise, this function produces a firing rate with a large variance that grows with the mean. This model is validated on responses recorded both intracellularly and extracellularly. In neurons of visual cortex, thus, a simple deterministic mechanism amplifies the low variability of summated synaptic inputs into the large variability of firing rate. The computational advantages provided by this amplification are not known.     

The interspike interval of a cable model neuron with white noise input

The firing time of a cable model neuron in response to white noise current injection is investigated with various methods. The Fourier decomposition of the depolarization leads to partial differential equations for the moments of the firing time. These are solved by perturbation and numerical methods, and the results obtained are in excellent agreement with those obtained by Monte Carlo simulation. The convergence of the random Fourier series is found to be very slow for small times so that when the firing time is small it is more efficient to simulate the solution of the stochastic cable equation directly using the two different representations of the Green's function, one which converges rapidly for small times and the other which converges rapidly for large times. The shape of the interspike interval density is found to depend strongly on input position. The various shapes obtained for different input positions resemble those for real neurons. The coefficient of variation of the interspike interval decreases monotonically as the distance between the input and trigger zone increases. A diffusion approximation for a nerve cell receiving Poisson input is considered and input/output frequency relations obtained for different input sites. The cases of multiple trigger zones and multiple input sites are briefly discussed.     

Mechanisms of association learning in the post-synaptic neurone

A general mechanism for association learning in neurones is presented based on the biochemistry and electrophysiology of synaptic receptor regions. The mechanism involves the voltage sensitive generation of second messengers in the post-synaptic neurone, and their subsequent influence on the sensitivity of the local receptor region. The mechanism is examined in detail using a computer model of the reaction dynamics of two important synaptic receptors; the adrenergic receptor and the NMDA-type glutamate receptor. The computer model simulates concentration changes of several molecules in the post-synaptic neurone following an input to a receptor. It also models changes in membrane potential caused by the cell firing, and allows for the impact of these potential changes on second messenger generation within the cell. Finally the model incorporates two possible mechanisms whereby high concentrations of second messenger can lead to long-lasting changes in receptor sensitivity. The model demonstrates an enhancement of synaptic response when previous inputs to the same receptor region have occurred when the cell was firing. Using the model, long term potentiation (LTP) of the glutamate response is demonstrated following a high frequency input to the glutamate receptor. A neurone with ten glutamate receptors is simulated and demonstrates that individual neurones should be capable of recognising patterns of input activity when those patterns have repeatedly occurred at times of major cellular activity.     

Direction selectivity of excitation and inhibition in simple cells of the cat primary visual cortex

Direction selectivity in simple cells of primary visual cortex, defined from their spike responses, cannot be predicted using linear models. It has been suggested that the shunting inhibition evoked by visual stimulation is responsible for the nonlinear component of direction selectivity. Cortical inhibition would suppress a neuron's firing when stimuli move in the nonpreferred direction, but would allow responses to stimuli in the preferred direction. Models of direction selectivity based solely on input from the lateral geniculate nucleus, however, propose that the nonlinear response is caused by spike threshold. By extracting excitatory and inhibitory components of synaptic inputs from intracellular records obtained in vivo, we demonstrate that excitation and inhibition are tuned for the same direction, but differ in relative timing. Further, membrane potential responses combine in a linear fashion. Spike threshold, however, quantitatively accounts for the nonlinear component of direction selectivity, amplifying the direction selectivity of spike output relative to that of synaptic inputs.     

Efficiency of neural transmission as a function of synaptic noise, threshold, and source characteristics

There has been a growing interest in the estimation of information carried by a single neuron and multiple single units or population of neurons to specific stimuli. In this paper we analyze, inspired by article of Levy and Baxter (2002), the efficiency of a neuronal communication by considering dendrosomatic summation as a Shannon-type channel (1948) and by considering such uncertain synaptic transmission as part of the dendrosomatic computation. Specifically, we study Mutual Information between input and output signals for different types of neuronal network architectures by applying efficient entropy estimators. We analyze the influence of the following quantities affecting transmission abilities of neurons: synaptic failure, activation threshold, firing rate and type of the input source. We observed a number of surprising non-intuitive effects. It turns out that, especially for lower activation thresholds, significant synaptic noise can lead even to twofold increase of the transmission efficiency. Moreover, the efficiency turns out to be a non-monotonic function of the activation threshold. We find a universal value of threshold for which a local maximum of Mutual Information is achieved for most of the neuronal architectures, regardless of the type of the source (correlated and non-correlated). Additionally, to reach the global maximum the optimal firing rates must increase with the threshold. This effect is particularly visible for lower firing rates. For higher firing rates the influence of synaptic noise on the transmission efficiency is more advantageous. Noise is an inherent component of communication in biological systems, hence, based on our analysis, we conjecture that the neuronal architecture was adjusted to make more effective use of this attribute.     

Instantaneous Non-Linear Processing by Pulse-Coupled Threshold Units

Contemporary theory of spiking neuronal networks is based on the linear response of the integrate-and-fire neuron model derived in the diffusion limit. We find that for non-zero synaptic weights, the response to transient inputs differs qualitatively from this approximation. The response is instantaneous rather than exhibiting low-pass characteristics, non-linearly dependent on the input amplitude, asymmetric for excitation and inhibition, and is promoted by a characteristic level of synaptic background noise. We show that at threshold the probability density of the potential drops to zero within the range of one synaptic weight and explain how this shapes the response. The novel mechanism is exhibited on the network level and is a generic property of pulse-coupled networks of threshold units.     

Simulations of a phase comparing neuron of the electric fish Eigenmannia

1. Investigations of the jamming avoidance response in the weakly electric fish Eigenmannia have shown that the 'small cell', an identified cell type in lamina 6 of the torus semicircularis, can distinguish a brief time difference between the phase of externally applied sinusoidally varying electrical fields. Computer simulations were performed to determine what physiological and anatomical features of the small cell could permit this. 2. The small cell shows apparent inhibition despite having no identified inhibitory synaptic input. This effect was produced in the model using voltage-sensitive membrane channels. These permit an initial sub-threshold excitatory post-synaptic potential (EPSP) to increase the threshold to firing, preventing a subsequent EPSP from firing the cell. 3. Noise and conduction failure appeared to improve the range of cell responsivity. The addition of noise permitted a gradual change in cell firing with change in time disparity, allowing the cell to signal more subtle changes in disparity at the expense of decreased selectivity for a specific absolute time disparity. 4. The cell's multiple dendrites permit a 47% reduction in jitter for incoming signals. Other aspects of dendrite organization did not appear to contribute to the cell's precision.     

Electrical properties of neurons recorded from the rat supraoptic nucleus in vitro

The electrical properties of neurons in the supraoptic nucleus (so.n.) have been studied in the hypothalamic slice preparation by intracellular and extracellular recording techniques, with Lucifer Yellow CH dye injection to mark the recording site as being the so.n. Intracellular recordings from so.n. neurons revealed them to have an average membrane potential of -67 +/- 0.8 mV (mean +/- s.e.m.), membrane resistance of 145 +/- 9 M omega with linear current-voltage relations from 40 mV in the hyperpolarizing direction to the level of spike threshold in the depolarizing direction. Average cell time constant was 14 +/- 2.2 ms. So.n. action potentials ranged in amplitude from 55 to 95 mV, with a mean of 76 +/- 2 mV, and a spike width of 2.6 +/- 0.5 ms at 30% of maximal spike height. Both single spikes and trains of spikes were followed by a strong, long-lasting hyperpolarization with a decay fitted by a single exponential having a time constant of 8.6 +/- 1.8 ms. Action potentials could be blocked by 10(-6) M tetrodotoxin. Spontaneously active so.n. neurons were characterized by synaptic input in the form of excitatory and inhibitory postsynaptic potentials, the latter being apparently blocked when 4 M KCl electrodes were used. Both forms of synaptic activity were blocked by application of divalent cations such as Mg2+, Mn2+ or Co2+. 74% of so.n. neurons fired spontaneously at rates exceeding 0.1 spikes per second, with a mean for all cells of 2.9 +/- 0.2 s-1. Of these cells, 21% fired slowly and continuously at 0.1 - 1.0 s-1, 45% fired continuously at greater than 1 Hz, and the remaining 34% fired phasically in bursts of activity followed by silence or low frequency firing. Spontaneously firing phasic cells showed a mean burst length of 16.7 +/- 4.5 s and a silent period of 28.2 +/- 4.2 s. Intracellular recordings revealed the presence of slow variations in membrane potential which modified the neuron's proximity to spike threshold, and controlled phasic firing. Variations in synaptic input were not observed to influence firing in phasic cells.     

Determination of the inter-spike times of neurons receiving randomly arriving post-synaptik potentials

Stein's model for a neuron receiving randomly arriving post-synaptic potentials is studied from an analytic viewpoint, using some recent results in the theory of first passage times for temporally homogeneous Markov processes. The case when the only input is excitatory can be treated exactly. It is shown that the moments of the firing time are guaranteed to be finite so that the differential-difference equation for the expectation (and higher moments) of the time for the membrane potential to first reach threshold from resting level can be written down. Analytic solutions are obtained in a number of cases with main emphasis on the case when the threshold is twice the epsp magnitude. An invariance principie is formulated wherein at a given mean input frequency and for a given decay parameter, the distribution of firing times depends only on the ratio of threshld to epsp magnitude. For the case where this ratio is two, the variation in the mean discharge rate is obtained as a function of mean input frequency. The results are compared with the experimental data for the Poisson monosynaptic excitation of cat motoneurons by Redmanet al. Agreement between theoretical and experimental values is excellent at input frequencies near 102 sec^-1, and theory underestimates the firing rate below that input frequency. Reasons for the discrepancy are discussed at length including the uncertainties in the neuronal parameters and the dependence of epsp magnitude on mean input frequency. The problem of including an inhibitory input process together with excitation is treated by an approximation procedure when the inhibition is considerably weaker than the excitation. At the input frequency investigated it is shown that when inhibition “half as weak” as the excitation occurs, the mean discharge frequency is approximately halved. In the final section a method of estimating neuronal parameters from the moments of the experimental inter-spike time distribution is outlined.     

神经元放电阈值的可变性及其意义

神经元能够将不同时空模式的突触输入转化为时序精确的动作电位输出，这种灵活、可靠的信息编码方式是神经集群在动态环境或特定任务下产生所需活动模式的重要基础。动作电位的产生遵循全或无规律，只有当细胞膜电压达到放电阈值时，神经元才产生动作电位。放电阈值在细胞内和细胞间具有高度可变性，具体动态依赖于刺激输入和放电历史。特别是，放电阈值对动作电位起始前的膜电压变化十分敏感，这种状态依赖性产生的生物物理根源包括Na⁺失活和K⁺激活。在绝大多数神经元中，动作电位的触发位置是轴突起始端，这个位置处的阈值可变性是决定神经元对时空输入转化规律的关键因素。但是，电生理实验中动作电位的记录位置却通常是胞体或近端树突，此处的阈值可变性高于轴突起始端，而其产生的重要根源是轴突动作电位的反向传播。基于胞体测量的相关研究显示，放电阈值动态能够增强神经元的时间编码、特征选择、增益调控和同时侦测能力。本文首先介绍放电阈值的概念及量化方法，然后详细梳理近年来国内外关于放电阈值可变性及产生根源的研究进展，在此基础上归纳总结放电阈值可变性对神经元编码的重要性，最后对未来放电阈值的研究方向进行展望。     

Some properties of auditory neuron’s model trained by firing caused by tones modulated by low-frequency noise

In many sensory systems the formation of burst firing can be observed along a way from the periphery to the central nuclei. We investigate the putative transformation of spontaneous activity in the auditory pathway using a neuron model trained by real firing recorded in the auditory nuclei of the frog. The model has 200 separate inputs (neuronal spines). It is supposed that every spine is a coincidence detector. Its output (synaptic potential) sharply increases at emergence of the precisely certain interpulse interval in an input pulse sequence. If the total synaptic potentials excess a threshold, the model generates output spike, which changes weight of all spines according to the simplified Hebb principle. The model was trained by real firing caused in the auditory nuclei of the frog by tones modulated by low-frequency noise in the frequency ranges of 0–15 Hz, 0–50 Hz or 0–150 Hz. After that training the synaptic weights of every spine essentially changed. Thus, along with some increase of weights of spines tuned to boundary frequencies of modulating noise, the most characteristic change was the emphasizing weights of spines tuned to short interpulse intervals. As a result the spontaneous activity passed through the trained model became much more bursting. Efficiency of a signal transmission in model was higher when input spontaneous activity of real cells contains bursts of spikes. Results of modeling are discussed in connection with modern physiological data demonstrating the functional advantage of bursting.     

A neuronal modeling paradigm in the presence of refractoriness

A mathematical characterization of the membrane potential as an instantaneous return process in the presence of refractoriness is investigated for diffusion models of single neuron's activity, assuming that the firing threshold acts as an elastic barrier. Steady-state probability densities and asymptotic moments of the neuronal membrane potential are explicitly obtained in a form that is suitable for quantitative evaluations. For the Ornstein-Uhlenbeck (OU) and Feller neuronal models, closed form expression are obtained for asymptotic mean and variance of the neuronal membrane potential and an analysis of the different features exhibited by the above mentioned models is performed.     

Nitella fluctuation and instability in the membrane potential near threshold.

Excitation of Nitella internodal cell was investigated as an example of the phase transition in an open system far more thermal equilibrium. The power density spectrum of the membrane potential fluctuation had a bulge in a frequency range lower than 1 Hz at the resting state and a peak at approximately 0.03 Hz at a depolarized state near the threshold. A critical oscillation in the membrane potential was observed when threshold was gradually approached from the resting state. Repetitive firing was observed under a step-current of the superthreshold value. The frequency of spectral peaking, critical oscillation, and repetitive firing agree well with each other. The result suggests that the hard-mode instability occurs in the Nitella internodal cell. The membrane impedance had no peak in the same frequency region as the peak of the voltage spectrum. The spectral peak may be ascribed to be electrogenic pump modulated by the metabolic feedback system in photosynthesis.     

A spatial stochastic neuronal model with Ornstein–Uhlenbeck input current

 We consider a spatial neuron model in which the membrane potential satisfies a linear cable equation with an input current which is a dynamical random process of the Ornstein–Uhlenbeck (OU) type. This form of current may represent an approximation to that resulting from the random opening and closing of ion channels on a neuron's surface or to randomly occurring synaptic input currents with exponential decay. We compare the results for the case of an OU input with those for a purely white-noise-driven cable model. The statistical properties, including mean, variance and covariance of the voltage response to an OU process input in the absence of a threshold are determined analytically. The mean and the variance are calculated as a function of time for various synaptic input locations and for values of the ratio of the time constant of decay of the input current to the time constant of decay of the membrane voltage in the physiological range for real neurons. The limiting case of a white-noise input current is obtained as the correlation time of the OU process approaches zero. The results obtained with an OU input current can be substantially different from those in the white-noise case. Using simulation of the terms in the series representation for the solution, we estimate the interspike interval distribution for various parameter values, and determine the effects of the introduction of correlation in the synaptic input stochastic process. Received: 5 March 2001 / Accepted in revised form: 7 August 2001     

A homeostatic model of neuronal firing governed by feedback signals from the extracellular matrix

Molecules of the extracellular matrix (ECM) can modulate the efficacy of synaptic transmission and neuronal excitability. These mechanisms are crucial for the homeostatic regulation of neuronal firing over extended timescales. In this study, we introduce a simple mathematical model of neuronal spiking balanced by the influence of the ECM. We consider a neuron receiving random synaptic input in the form of Poisson spike trains and the ECM, which is modeled by a phenomenological variable involved in two feedback mechanisms. One feedback mechanism scales the values of the input synaptic conductance to compensate for changes in firing rate. The second feedback accounts for slow fluctuations of the excitation threshold and depends on the ECM concentration. We show that the ECM-mediated feedback acts as a robust mechanism to provide a homeostatic adjustment of the average firing rate. Interestingly, the activation of feedback mechanisms may lead to a bistability in which two different stable levels of average firing rates can coexist in a spiking network. We discuss the mechanisms of the bistability and how they may be related to memory function.