Analytical integrate-and-fire neuron models with conductance-based dynamics for event-driven simulation strategies

Event-driven simulation strategies were proposed recently to simulate integrate-and-fire (IF) type neuronal models. These strategies can lead to computationally efficient algorithms for simulating large-scale networks of neurons; most important, such approaches are more precise than traditional clock-driven numerical integration approaches because the timing of spikes is treated exactly. The drawback of such event-driven methods is that in order to be efficient, the membrane equations must be solvable analytically, or at least provide simple analytic approximations for the state variables describing the system. This requirement prevents, in general, the use of conductance-based synaptic interactions within the framework of event-driven simulations and, thus, the investigation of network paradigms where synaptic conductances are important. We propose here a number of extensions of the classical leaky IF neuron model involving approximations of the membrane equation with conductance-based synaptic current, which lead to simple analytic expressions for the membrane state, and therefore can be used in the event-driven framework. These conductance-based IF (gIF) models are compared to commonly used models, such as the leaky IF model or biophysical models in which conductances are explicitly integrated. All models are compared with respect to various spiking response properties in the presence of synaptic activity, such as the spontaneous discharge statistics, the temporal precision in resolving synaptic inputs, and gain modulation under in vivo-like synaptic bombardment. Being based on the passive membrane equation with fixed-threshold spike generation, the proposed gIF models are situated in between leaky IF and biophysical models but are much closer to the latter with respect to their dynamic behavior and response characteristics, while still being nearly as computationally efficient as simple IF neuron models. gIF models should therefore provide a useful tool for efficient and precise simulation of large-scale neuronal networks with realistic, conductance-based synaptic interactions.     

Exact simulation of integrate-and-fire models with synaptic conductances

Computational neuroscience relies heavily on the simulation of large networks of neuron models. There are essentially two simulation strategies: (1) using an approximation method (e.g., Runge-Kutta) with spike times binned to the time step and (2) calculating spike times exactly in an event-driven fashion. In large networks, the computation time of the best algorithm for either strategy scales linearly with the number of synapses, but each strategy has its own assets and constraints: approximation methods can be applied to any model but are inexact; exact simulation avoids numerical artifacts but is limited to simple models. Previous work has focused on improving the accuracy of approximation methods. In this article, we extend the range of models that can be simulated exactly to a more realistic model: an integrate-and-fire model with exponential synaptic conductances.     

Modeling of inhibition/excitation firing in olfactory bulb through spiking neurons

Spiking neural systems are based on biologically inspired neural models of computation since they take into account the precise timing of spike events and therefore are suitable to analyze dynamical aspects of neuronal signal transmission. These systems gained increasing interest because they are more sophisticated than simple neuron models found in artificial neural systems; they are closer to biophysical models of neurons, synapses, and related elements and their synchronized firing of neuronal assemblies could serve the brain as a code for feature binding and pattern segmentation. The simulations are designed to exemplify certain properties of the olfactory bulb (OB) dynamics and are based on an extension of the integrate-and-fire (IF) neuron, and the idea of locally coupled excitation and inhibition cells. We introduce the background theory to making an appropriate choice of model parameters. The following two forms of connectivity offering certain computational and analytical advantages, either through symmetry or statistical properties in the study of OB dynamics have been used:

• all-to-all coupling,
• receptive field style coupling.

Our simulations showed that the inter-neuron transmission delay controls the size of spatial variations of the input and also smoothes the network response. Our IF extended model proves to be a useful basis from which we can study more sophisticated features as complex pattern formation, and global stability and chaos of OB dynamics.     

Distinguishing the causes of firing with the membrane potential slope

In this letter, we aim to measure the relative contribution of coincidence detection and temporal integration to the firing of spikes of a simple neuron model. To this end, we develop a method to infer the degree of synchrony in an ensemble of neurons whose firing drives a single postsynaptic cell. This is accomplished by studying the effects of synchronous inputs on the membrane potential slope of the neuron and estimating the degree of response-relevant input synchrony, which determines the neuron's operational mode. The measure is calculated using the normalized slope of the membrane potential prior to the spikes fired by a neuron, and we demonstrate that it is able to distinguish between the two operational modes. By applying this measure to the membrane potential time course of a leaky integrate-and-fire neuron with the partial somatic reset mechanism, which has been shown to be the most likely candidate to reflect the mechanism used in the brain for reproducing the highly irregular firing at high rates, we show that the partial reset model operates as a temporal integrator of incoming excitatory postsynaptic potentials and that coincidence detection is not necessary for producing such high irregular firing.     

Synaptic shot noise and conductance fluctuations affect the membrane voltage with equal significance

The subthreshold membrane voltage of a neuron in active cortical tissue is a fluctuating quantity with a distribution that reflects the firing statistics of the presynaptic population. It was recently found that conductance-based synaptic drive can lead to distributions with a significant skew. Here it is demonstrated that the underlying shot noise caused by Poissonian spike arrival also skews the membrane distribution, but in the opposite sense. Using a perturbative method, we analyze the effects of shot noise on the distribution of synaptic conductances and calculate the consequent voltage distribution. To first order in the perturbation theory, the voltage distribution is a gaussian modulated by a prefactor that captures the skew. The gaussian component is identical to distributions derived using current-based models with an effective membrane time constant. The well-known effective-time-constant approximation can therefore be identified as the leading-order solution to the full conductance-based model. The higher-order modulatory prefactor containing the skew comprises terms due to both shot noise and conductance fluctuations. The diffusion approximation misses these shot-noise effects implying that analytical approaches such as the Fokker-Planck equation or simulation with filtered white noise cannot be used to improve on the gaussian approximation. It is further demonstrated that quantities used for fitting theory to experiment, such as the voltage mean and variance, are robust against these non-Gaussian effects. The effective-time-constant approximation is therefore relevant to experiment and provides a simple analytic base on which other pertinent biological details may be added.     

Compact damping models for laterally moving microstructures withgas-rarefaction effects

Compact models for the viscous damping coefficient in narrow air gaps between laterally moving structures are reported. In the first part of the paper, a simple frequency-independent first-order slip-flow approximation for the damping coefficient is derived and compared with a more accurate expression obtained from the linearized Boltzmann equation. The simple approximation is slightly modified and fitted to match the accurate model. The resulting simple approximation has a maximum relative error of less than ±6% at arbitrary Knudsen numbers in viscous, transitional and free molecular regions. In the second part of the paper, dynamic models for the damping force are derived, considering again gas rarefaction, by applying various boundary conditions. The damping admittance of the first-order slip-flow model is implemented also as an electrical equivalent admittance, constructed of RC sections, to allow both frequency and time domain simulations with a circuit simulator. The dependence of the damping admittance on pressure and gap displacement is demonstrated with model simulations. The accuracy and validity range of the model are verified with comparative numerical simulations of the Navier-Stokes equation. Finally, the damping coefficient in a lateral resonator is calculated using the compact model and compared with measured data with good agreement     

Accelerating event-driven simulation of spiking neurons with multiple synaptic time constants

The simulation of spiking neural networks (SNNs) is known to be a very time-consuming task. This limits the size of SNN that can be simulated in reasonable time or forces users to overly limit the complexity of the neuron models. This is one of the driving forces behind much of the recent research on event-driven simulation strategies. Although event-driven simulation allows precise and efficient simulation of certain spiking neuron models, it is not straightforward to generalize the technique to more complex neuron models, mostly because the firing time of these neuron models is computationally expensive to evaluate. Most solutions proposed in literature concentrate on algorithms that can solve this problem efficiently. However, these solutions do not scale well when more state variables are involved in the neuron model, which is, for example, the case when multiple synaptic time constants for each neuron are used. In this letter, we show that an exact prediction of the firing time is not required in order to guarantee exact simulation results. Several techniques are presented that try to do the least possible amount of work to predict the firing times. We propose an elegant algorithm for the simulation of leaky integrate-and-fire (LIF) neurons with an arbitrary number of (unconstrained) synaptic time constants, which is able to combine these algorithmic techniques efficiently, resulting in very high simulation speed. Moreover, our algorithm is highly independent of the complexity (i.e., number of synaptic time constants) of the underlying neuron model.     

Higher-order statistics of input ensembles and the response of simple model neurons

Pairwise correlations among spike trains recorded in vivo have been frequently reported. It has been argued that correlated activity could play an important role in the brain, because it efficiently modulates the response of a postsynaptic neuron. We show here that a neuron's output firing rate critically depends on the higher-order statistics of the input ensemble. We constructed two statistical models of populations of spiking neurons that fired with the same rates and had identical pairwise correlations, but differed with regard to the higher-order interactions within the population. The first ensemble was characterized by clusters of spikes synchronized over the whole population. In the second ensemble, the size of spike clusters was, on average, proportional to the pairwise correlation. For both input models, we assessed the role of the size of the population, the firing rate, and the pairwise correlation on the output rate of two simple model neurons: a continuous firing-rate model and a conductance-based leaky integrate-and-fire neuron. An approximation to the mean output rate of the firing-rate neuron could be derived analytically with the help of shot noise theory. Interestingly, the essential features of the mean response of the two neuron models were similar. For both neuron models, the three input parameters played radically different roles with respect to the postsynaptic firing rate, depending on the interaction structure of the input. For instance, in the case of an ensemble with small and distributed spike clusters, the output firing rate was efficiently controlled by the size of the input population. In addition to the interaction structure, the ratio of inhibition to excitation was found to strongly modulate the effect of correlation on the postsynaptic firing rate.     

Minimal models of adapted neuronal response to in vivo-like input currents

Rate models are often used to study the behavior of large networks of spiking neurons. Here we propose a procedure to derive rate models that take into account the fluctuations of the input current and firing-rate adaptation, two ubiquitous features in the central nervous system that have been previously overlooked in constructing rate models. The procedure is general and applies to any model of firing unit. As examples, we apply it to the leaky integrate-and-fire (IF) neuron, the leaky IF neuron with reversal potentials, and to the quadratic IF neuron. Two mechanisms of adaptation are considered, one due to an afterhyperpolarization current and the other to an adapting threshold for spike emission. The parameters of these simple models can be tuned to match experimental data obtained from neocortical pyramidal neurons. Finally, we show how the stationary model can be used to predict the time-varying activity of a large population of adapting neurons.     

Dynamically Reconfigurable Silicon Array of Spiking Neurons With Conductance-Based Synapses

A mixed-signal very large scale integration (VLSI) chip for large scale emulation of spiking neural networks is presented. The chip contains 2400 silicon neurons with fully programmable and reconfigurable synaptic connectivity. Each neuron implements a discrete-time model of a single-compartment cell. The model allows for analog membrane dynamics and an arbitrary number of synaptic connections, each with tunable conductance and reversal potential. The array of silicon neurons functions as an address-event (AE) transceiver, with incoming and outgoing spikes communicated over an asynchronous event-driven digital bus. Address encoding and conflict resolution of spiking events are implemented via a randomized arbitration scheme that ensures balanced servicing of event requests across the array. Routing of events is implemented externally using dynamically programmable random-access memory that stores a postsynaptic address, the conductance, and the reversal potential of each synaptic connection. Here, we describe the silicon neuron circuits, present experimental data characterizing the 3 mm times 3 mm chip fabricated in 0.5-mum complementary metal-oxide-semiconductor (CMOS) technology, and demonstrate its utility by configuring the hardware to emulate a model of attractor dynamics and waves of neural activity during sleep in rat hippocampus     

Detecting scalar intermittent faults in linear stochastic dynamic systems

Intermittent faults (IFs) have properties such as intermittency, random magnitude and random duration time. Hence the detection of IFs means: (i) to detect not only all the appearing time but also all the disappearing time of IFs and (ii) to detect the appearing time of an IF before this IF disappears, and the disappearing time of an IF before the subsequent IF appears. Within a statistical framework, the detection of scalar IFs in continuous linear stochastic dynamic systems has been mainly studied. Based on the sliding window, an analytical residual is generated, and two hypothesis tests are implemented to detect the appearing and disappearing times of IFs. In addition, a necessary and sufficient condition for the detectability of IFs is obtained, and the detection speed can be fast enough. Theoretical analysis and numerical simulations fully verify that IFs can be successfully detected.     

On the performance of voltage stepping for the simulation of adaptive, nonlinear integrate-and-fire neuronal networks

In traditional event-driven strategies, spike timings are analytically given or calculated with arbitrary precision (up to machine precision). Exact computation is possible only for simplified neuron models, mainly the leaky integrate-and-fire model. In a recent paper, Zheng, Tonnelier, and Martinez (2009) introduced an approximate event-driven strategy, named voltage stepping, that allows the generic simulation of nonlinear spiking neurons. Promising results were achieved in the simulation of single quadratic integrate-and-fire neurons. Here, we assess the performance of voltage stepping in network simulations by considering more complex neurons (quadratic integrate-and-fire neurons with adaptation) coupled with multiple synapses. To handle the discrete nature of synaptic interactions, we recast voltage stepping in a general framework, the discrete event system specification. The efficiency of the method is assessed through simulations and comparisons with a modified time-stepping scheme of the Runge-Kutta type. We demonstrated numerically that the original order of voltage stepping is preserved when simulating connected spiking neurons, independent of the network activity and connectivity.     

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.     

Exact subthreshold integration with continuous spike times in discrete-time neural network simulations

Very large networks of spiking neurons can be simulated efficiently in parallel under the constraint that spike times are bound to an equidistant time grid. Within this scheme, the subthreshold dynamics of a wide class of integrate-and-fire-type neuron models can be integrated exactly from one grid point to the next. However, the loss in accuracy caused by restricting spike times to the grid can have undesirable consequences, which has led to interest in interpolating spike times between the grid points to retrieve an adequate representation of network dynamics. We demonstrate that the exact integration scheme can be combined naturally with off-grid spike events found by interpolation. We show that by exploiting the existence of a minimal synaptic propagation delay, the need for a central event queue is removed, so that the precision of event-driven simulation on the level of single neurons is combined with the efficiency of time-driven global scheduling. Further, for neuron models with linear subthreshold dynamics, even local event queuing can be avoided, resulting in much greater efficiency on the single-neuron level. These ideas are exemplified by two implementations of a widely used neuron model. We present a measure for the efficiency of network simulations in terms of their integration error and show that for a wide range of input spike rates, the novel techniques we present are both more accurate and faster than standard techniques.     

Theory of input spike auto- and cross-correlations and their effect on the response of spiking neurons

Spike correlations between neurons are ubiquitous in the cortex, but their role is not understood. Here we describe the firing response of a leaky integrate-and-fire neuron (LIF) when it receives a temporarily correlated input generated by presynaptic correlated neuronal populations. Input correlations are characterized in terms of the firing rates, Fano factors, correlation coefficients, and correlation timescale of the neurons driving the target neuron. We show that the sum of the presynaptic spike trains cannot be well described by a Poisson process. In fact, the total input current has a nontrivial two-point correlation function described by two main parameters: the correlation timescale (how precise the input correlations are in time) and the correlation magnitude (how strong they are). Therefore, the total current generated by the input spike trains is not well described by a white noise gaussian process. Instead, we model the total current as a colored gaussian process with the same mean and two-point correlation function, leading to the formulation of the problem in terms of a Fokker-Planck equation. Solutions of the output firing rate are found in the limit of short and long correlation timescales. The solutions described here expand and improve on our previous results (Moreno, de la Rocha, Renart, & Parga, 2002) by presenting new analytical expressions for the output firing rate for general IF neurons, extending the validity of the results for arbitrarily large correlation magnitude, and by describing the differential effect of correlations on the mean-driven or noise-dominated firing regimes. Also the details of this novel formalism are given here for the first time. We employ numerical simulations to confirm the analytical solutions and study the firing response to sudden changes in the input correlations. We expect this formalism to be useful for the study of correlations in neuronal networks and their role in neural processing and information transmission.     

事件驱动的动态服务组合策略在线自适应优化

针对复杂应用环境中网络新媒体服务系统的特点,提出一种事件驱动的动态服务组合策略及其在线优化算法,在保证各类业务服务质量（QoS）的同时,提高系统资源的利用率．通过定义不同类型的事件,驱动服务组合的动态调整,实现对各类业务Qos的保障和对业务需求变化的感知．构建基于半Markov切换空间控制过程的系统分析模型,利用模型的动态结构特点,提出一种结合随机逼近和策略迭代的在线优化算法．该算法不依赖系统参数信息,对环境具有良好的自适应性．仿真实验结果验证了算法的有效性．     

Training integrate-and-fire neurons with the Informax principle II

For pt I see J. Phys. A, vol. 35, p. 2379-94 (2002).We develop neuron learning rules using the Informax principle together with the input-output relationship of the integrate-and-fire (IF) model with Poisson inputs. The learning rule is then tested with constant inputs, time-varying inputs and images. For constant inputs, it is found that, under the Informax principle, a network of IF models with initially all positive weights tends to disconnect some connections between neurons. For time-varying inputs and images, we perform signal separation tasks called independent component analysis. Numerical simulations indicate that some number of inhibitory inputs improves the performance of the system in both biological and engineering senses.     

双论域上的广义直觉模糊概率粗糙集模型及其应用

已有的双论域直觉模糊概率粗糙集模型通过设置两个阈值${\lambda _1}$、${\lambda _2} $,讨论了经典集合在直觉模糊二元关系下的概率粗糙下上近似。该模型不能计算直觉模糊集合在直觉模糊二元关系下的概率粗糙下上近似,这在一定程度上限制了该模型的应用。首先给出了直觉模糊条件概率的定义。在直觉模糊概率空间下构造了双论域广义直觉模糊概率粗糙集模型,讨论了模型的主要性质。最后,将模型应用到临床诊断系统中。与其他模型相比,所提出的广义直觉模糊概率粗糙集模型进一步丰富了概率粗糙集理论,更适合于实际应用。     

Performance analysis and scheduling of stochastic fork-join jobs ina multicomputer system

The authors model a parallel processing system comprising several homogeneous computers interconnected by a communication network. Jobs arriving to this system have a linear fork-join structure. Each fork of the job gives rise to a random number of tasks that can be processed independently on any of the computers. Since exact analysis of fork-join models is known to be intractable, the authors resort to obtaining analytical bounds to the mean job response time of the fork-join job. For jobs with a single fork-join and, probabilistic allocation of tasks of the job to the N processors, they obtain upper and lower bounds to the mean job response time. Upper bounds are obtained using the concept of associated random variables and are found to be a good approximation to the mean job response time. A simple lower bound is obtained by neglecting queueing delays. They also find two lower bounds that include queueing delays. For multiple fork-join jobs, they study an approximation based on associated random variables. Finally, two versions of the join-the-shortest-queue (JSQ) allocation policy (i.e., JSQ by batch and JSQ by task) are studied and compared, via simulations and diffusion limits     

Critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling

Computational techniques within the population density function (PDF) framework have provided time-saving alternatives to classical Monte Carlo simulations of neural network activity. Efficiency of the PDF method is lost as the underlying neuron model is made more realistic and the number of state variables increases. In a detailed theoretical and computational study, we elucidate strengths and weaknesses of dimension reduction by a particular moment closure method (Cai, Tao, Shelley, & McLaughlin, 2004; Cai, Tao, Rangan, & McLaughlin, 2006) as applied to integrate-and-fire neurons that receive excitatory synaptic input only. When the unitary postsynaptic conductance event has a single-exponential time course, the evolution equation for the PDF is a partial differential integral equation in two state variables, voltage and excitatory conductance. In the moment closure method, one approximates the conditional kth centered moment of excitatory conductance given voltage by the corresponding unconditioned moment. The result is a system of k coupled partial differential equations with one state variable, voltage, and k coupled ordinary differential equations. Moment closure at k = 2 works well, and at k = 3 works even better, in the regime of high dynamically varying synaptic input rates. Both closures break down at lower synaptic input rates. Phase-plane analysis of the k = 2 problem with typical parameters proves, and reveals why, no steady-state solutions exist below a synaptic input rate that gives a firing rate of 59 s(1) in the full 2D problem. Closure at k = 3 fails for similar reasons. Low firing-rate solutions can be obtained only with parameters for the amplitude or kinetics (or both) of the unitary postsynaptic conductance event that are on the edge of the physiological range. We conclude that this dimension-reduction method gives ill-posed problems for a wide range of physiological parameters, and we suggest future directions.