A population density approach that facilitates large-scale modeling of neural networks: extension to slow inhibitory synapses

A previously developed method for efficiently simulating complex networks of integrate-and-fire neurons was specialized to the case in which the neurons have fast unitary postsynaptic conductances. However, inhibitory synaptic conductances are often slower than excitatory ones for cortical neurons, and this difference can have a profound effect on network dynamics that cannot be captured with neurons that have only fast synapses. We thus extend the model to include slow inhibitory synapses. In this model, neurons are grouped into large populations of similar neurons. For each population, we calculate the evolution of a probability density function (PDF), which describes the distribution of neurons over state-space. The population firing rate is given by the flux of probability across the threshold voltage for firing an action potential. In the case of fast synaptic conductances, the PDF was one-dimensional, as the state of a neuron was completely determined by its transmembrane voltage. An exact extension to slow inhibitory synapses increases the dimension of the PDF to two or three, as the state of a neuron now includes the state of its inhibitory synaptic conductance. However, by assuming that the expected value of a neuron's inhibitory conductance is independent of its voltage, we derive a reduction to a one-dimensional PDF and avoid increasing the computational complexity of the problem. We demonstrate that although this assumption is not strictly valid, the results of the reduced model are surprisingly accurate.     

Population density methods for stochastic neurons with realistic synaptic kinetics: firing rate dynamics and fast computational methods

An outstanding problem in computational neuroscience is how to use population density function (PDF) methods to model neural networks with realistic synaptic kinetics in a computationally efficient manner. We explore an application of two-dimensional (2-D) PDF methods to simulating electrical activity in networks of excitatory integrate-and-fire neurons.We formulate a pair of coupled partial differential-integral equations describing the evolution of PDFs for neurons in non-refractory and refractory pools. The population firing rate is given by the total flux of probability across the threshold voltage. We use an operator-splitting method to reduce computation time. We report on speed and accuracy of PDF results and compare them to those from direct, Monte-Carlo simulations.We compute temporal frequency response functions for the transduction from the rate of postsynaptic input to population firing rate, and examine its dependence on background synaptic input rate. The behaviors in the1-D and 2-D cases--corresponding to instantaneous and non-instantaneous synaptic kinetics, respectively--differ markedly from those for a somewhat different transduction: from injected current input to population firing rate output (Brunel et al. 2001; Fourcaud & Brunel 2002).We extend our method by adding inhibitory input, consider a 3-D to 2-D dimension reduction method, demonstrate its limitations, and suggest directions for future study.     

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.     

Dynamics of the firing probability of noisy integrate-and-fire neurons

Cortical neurons in vivo undergo a continuous bombardment due to synaptic activity, which acts as a major source of noise. Here, we investigate the effects of the noise filtering by synapses with various levels of realism on integrate-and-fire neuron dynamics. The noise input is modeled by white (for instantaneous synapses) or colored (for synapses with a finite relaxation time) noise. Analytical results for the modulation of firing probability in response to an oscillatory input current are obtained by expanding a Fokker-Planck equation for small parameters of the problem - when both the amplitude of the modulation is small compared to the background firing rate and the synaptic time constant is small compared to the membrane time constant. We report here the detailed calculations showing that if a synaptic decay time constant is included in the synaptic current model, the firing-rate modulation of the neuron due to an oscillatory input remains finite in the high-frequency limit with no phase lag. In addition, we characterize the low-frequency behavior and the behavior of the high-frequency limit for intermediate decay times. We also characterize the effects of introducing a rise time to the synaptic currents and the presence of several synaptic receptors with different kinetics. In both cases, we determine, using numerical simulations, an effective decay time constant that describes the neuronal response completely.     

Dynamics of strongly-coupled spiking neurons

We present a dynamical theory of integrate-and-fire neurons with strong synaptic coupling. We show how phase-locked states that are stable in the weak coupling regime can destabilize as the coupling is increased, leading to states characterized by spatiotemporal variations in the interspike intervals (ISIs). The dynamics is compared with that of a corresponding network of analog neurons in which the outputs of the neurons are taken to be mean firing rates. A fundamental result is that for slow interactions, there is good agreement between the two models (on an appropriately defined timescale). Various examples of desynchronization in the strong coupling regime are presented. First, a globally coupled network of identical neurons with strong inhibitory coupling is shown to exhibit oscillator death in which some of the neurons suppress the activity of others. However, the stability of the synchronous state persists for very large networks and fast synapses. Second, an asymmetric network with a mixture of excitation and inhibition is shown to exhibit periodic bursting patterns. Finally, a one-dimensional network of neurons with long-range interactions is shown to desynchronize to a state with a spatially periodic pattern of mean firing rates across the network. This is modulated by deterministic fluctuations of the instantaneous firing rate whose size is an increasing function of the speed of synaptic response.     

Reliability analysis of structures with complex limit state functions using probability density evolution method

In reliability-based structural analysis and design optimization, there exist some limit state functions exhibiting disjoint failure domains, multiple design points and discontinuous responses. This study addresses this type of challenging problem of reliability assessment of structures with complex limit state functions based on the probability density evolution method (PDEM). Probability density function (PDF) of stochastic structures under static and dynamic loads can be acquired, which is independent of the specific form of limit state functions. Numerical results of several typical examples illustrate that, the time-invariant and instantaneous PDF curves and failure probabilities of stochastic structures with disjoint failure domains, multiple design points and discontinuous responses are calculated effectively and accurately. Moreover, the PDEM is validated to be more efficient than the Monte Carlo simulation and the subset simulation, and is a feasible and general approach to tackle the reliability analysis of complicated problems. In addition, the influence of random design parameters of structures on uncertainty propagation is also scrutinized.     

基于状态概率转移的SIRS病毒传播模型及其临界值分析

针对SIRS(Susceptible-Infected-Removed-Susceptible)病毒传播模型,利用状态转移概率的方法,通过计算节点处于各个状态的概率来研究SIRS病毒传播过程。首先建立状态概率方程组,描述各个时刻各个节点处于易感染态、感染态、免疫态的概率,通过稳态分析理论推导网络的病毒传播临界值;然后利用蒙特卡罗方法,对均匀网络和非均匀网络的病毒传播临界值进行分析和仿真。结果表明,相对于传统的平均场方法,基于状态概率方程组模型求得的传播临界值更加接近真实蒙特卡罗值,并且与免疫丧失率无关。     

Maximum memory capacity on neural networks with short-term synaptic depression and facilitation

In this work, we study, analytically and employing Monte Carlo simulations, the influence of the competition between several activity-dependent synaptic processes, such as short-term synaptic facilitation and depression, on the maximum memory storage capacity in a neural network. In contrast to the case of synaptic depression, which drastically reduces the capacity of the network to store and retrieve "static" activity patterns, synaptic facilitation enhances the storage capacity in different contexts. In particular, we found optimal values of the relevant synaptic parameters (such as the neurotransmitter release probability or the characteristic facilitation time constant) for which the storage capacity can be maximal and similar to the one obtained with static synapses, that is, without activity-dependent processes. We conclude that depressing synapses with a certain level of facilitation allow recovering the good retrieval properties of networks with static synapses while maintaining the nonlinear characteristics of dynamic synapses, convenient for information processing and coding.     

A model for fast analog computation based on unreliable synapses

We investigate through theoretical analysis and computer simulations the consequences of unreliable synapses for fast analog computations in networks of spiking neurons, with analog variables encoded by the current firing activities of pools of spiking neurons. Our results suggest a possible functional role for the well-established unreliability of synaptic transmission on the network level. We also investigate computations on time series and Hebbian learning in this context of space-rate coding in networks of spiking neurons with unreliable synapses.     

Global dynamics of a network of stochastic neurons maximizes local mutual information

We define a stochastic neuron as an element that increases its internal state with probability p until a threshold value is reached; after that its internal state is set back to the initial value. We study the local information of a stochastic neuron between the message arriving from the input neurons and the response of the neuron. We study the dependence of the local information on the firing probability alpha of the synaptic inputs in a network of such stochastic neurons. The values of alpha obtained in the simulations are the same as those obtained theoretically by maximization of local mutual information. We conclude that the global dynamics maximizes the local mutual information of single units, which means that the self-selected parameter value of the population dynamics is such that each neuron behaves as an optimal encoder.     

含指数积分型阻尼和周期激励系统的稳态随机响应分析

运用基于短时高斯逼近的广义胞映射方法,研究了含指数积分型非粘性阻尼和周期激励系统在高斯白噪声作用下的稳态响应.首先介绍了方法的实施过程,并推导了系统的矩方程.然后给出了系统的稳态概率密度函数,分析了阻尼系数和松弛参数对稳态响应的影响,并通过直接Monte Carlo模拟的结果验证了广义胞映射方法的有效性.     

The firing of an excitable neuron in the presence of stochastic trains of strong synaptic inputs

We consider a fast-slow excitable system subject to a stochastic excitatory input train and show that under general conditions, its long-term behavior is captured by an irreducible Markov chain with a limiting distribution. This limiting distribution allows for the analytical calculation of the system's probability of firing in response to each input, the expected number of response failures between firings, and the distribution of slow variable values between firings. Moreover, using this approach, it is possible to understand why the system will not have a stationary distribution and why Monte Carlo simulations do not converge under certain conditions. The analytical calculations involved can be performed whenever the distribution of interexcitation intervals and the recovery dynamics of the slow variable are known. The method can be extended to other models that feature a single variable that builds up to a threshold where an instantaneous spike and reset occur. We also discuss how the Markov chain analysis generalizes to any pair of input trains, excitatory or inhibitory and synaptic or not, such that the frequencies of the two trains are sufficiently different from each other. We illustrate this analysis on a model thalamocortical (TC) cell subject to two example distributions of excitatory synaptic inputs in the cases of constant and rhythmic inhibition. The analysis shows a drastic drop in the likelihood of firing just after inhibitory onset in the case of rhythmic inhibition, relative even to the case of elevated but constant inhibition. This observation provides support for a possible mechanism for the induction of motor symptoms in Parkinson's disease and for their relief by deep brain stimulation, analyzed in Rubin and Terman (2004).     

Influences of synaptic location on the synchronization of rhythmic bursting neurons

We study how the location of synaptic input influences the stablex firing states in coupled model neurons bursting rhythmically at the gamma frequencies (20-70 Hz). The model neuron consists of two compartments and generates one, two, three or four spikes in each burst depending on the intensity of input current and the maximum conductance of M-type potassium current. If the somata are connected by reciprocal excitatory synapses, we find strong correlations between the changes in the bursting mode and those in the stable phase-locked states of the coupled neurons. The stability of the in-phase phase-locked state (synchronously firing state) tends to change when the individual neurons change their bursting patterns. If, however, the synaptic connections are terminated on the dendritic compartments, no such correlated changes occur. In this case, the coupled bursting neurons do not show the in-phase phase-locked state in any bursting mode. These results indicate that synchronization behaviour of bursting neurons significantly depends on the synaptic location, unlike a coupled system of regular spiking neurons.     

Synchrony of fast-spiking interneurons interconnected by GABAergic and electrical synapses

Fast-spiking (FS) interneurons have specific types (Kv3.1/3.2 type) of the delayed potassium channel, which differ from the conventional Hodgkin-Huxley (HH) type potassium channel (Kv1.3 type) in several aspects. In this study, we show dramatic effects of the Kv3.1/3.2 potassium channel on the synchronization of the FS interneurons. We show analytically that two identical electrically coupled FS interneurons modeled with Kv3.1/3.2 channel fire synchronously at arbitrary firing frequencies, unlike similarly coupled FS neurons modeled with Kv1.3 channel that show frequency-dependent synchronous and antisynchronous firing states. Introducing GABA(A) receptor-mediated synaptic connections into an FS neuron pair tends to induce an antisynchronous firing state, even if the chemical synapses are bidirectional. Accordingly, an FS neuron pair connected simultaneously by electrical and chemical synapses achieves both synchronous firing state and antisynchronous firing state in a physiologically plausible range of the conductance ratio between electrical and chemical synapses. Moreover, we find that a large-scale network of FS interneurons connected by gap junctions and bidirectional GABAergic synapses shows similar bistability in the range of gamma frequencies (30-70 Hz).     

Optimality model of unsupervised spike-timing-dependent plasticity: synaptic memory and weight distribution

We studied the hypothesis that synaptic dynamics is controlled by three basic principles: (1) synapses adapt their weights so that neurons can effectively transmit information, (2) homeostatic processes stabilize the mean firing rate of the postsynaptic neuron, and (3) weak synapses adapt more slowly than strong ones, while maintenance of strong synapses is costly. Our results show that a synaptic update rule derived from these principles shares features, with spike-timing-dependent plasticity, is sensitive to correlations in the input and is useful for synaptic memory. Moreover, input selectivity (sharply tuned receptive fields) of postsynaptic neurons develops only if stimuli with strong features are presented. Sharply tuned neurons can coexist with unselective ones, and the distribution of synaptic weights can be unimodal or bimodal. The formulation of synaptic dynamics through an optimality criterion provides a simple graphical argument for the stability of synapses, necessary for synaptic memory.     

Improved sampling techniques for the direct simulation Monte Carlo method

The direct simulation Monte Carlo (DSMC) method is a widely used approach for flow simulations having rarefied or nonequilibrium effects. It involves heavily to sample instantaneous values from prescribed distributions using random numbers. In this note, we briefly review the sampling techniques typically employed in the DSMC method and present two techniques to speedup related sampling processes. One technique is very efficient for sampling geometric locations of new particles and the other is useful for the Larsen-Borgnakke energy distribution.     

基于粒子滤波的小波特征跟踪方法研究

该文提出了基于粒子滤波的小波特征跟踪方法。粒子滤波基于蒙特卡罗模拟方法来实现递推贝叶斯滤波,是一种实用的后验概率求解方法。文中研究了目标的Gabor小波网络表示,用一定数量的小波构成一个集合来表示目标特征,各小波的参数由优化方法来确定。构建了基于粒子滤波的跟踪框架,每个粒子表示一种Gabor小波网络的可能形式,并计算与当前图像的相似度。粒子权值与相似度成正比,目标状态的后验概率由粒子加权表示。与传统的“峰值”跟踪方法不同,粒子滤波具有“多峰”的跟踪形式。并结合对光照、噪声不敏感的小波表示形式,具有较强的抗局部遮挡能力。