Dynamics of the Langevin model subjected to colored noise: Functional-integral method

We have discussed the dynamics of Langevin model subjected to colored noise, by using the functional-integral method (FIM) combined with equations of motion for mean and variance of the state variable. Two sets of colored noise have been investigated: (a) one additive and one multiplicative colored noise, and (b) one additive and two multiplicative colored noise. The case (b) is examined with relevance to a recent controversy on the stationary subthreshold voltage distribution of an integrate-and-fire model including stochastic excitatory and inhibitory synapses and a noisy input. We have studied the stationary probability distribution and dynamical responses to time-dependent (pulse and sinusoidal) inputs of the linear Langevin model. Model calculations have shown that results of the FIM are in good agreement with those of direct simulations (DSs). A comparison is made among various approximate analytic solutions such as the universal colored noise approximation (UCNA). It has been pointed out that dynamical responses to pulse and sinusoidal inputs calculated by the UCNA are rather different from those of DS and the FIM, although they yield the same stationary distribution.     

Propagation of firing rate in a feed-forward neuronal network

Propagation of the firing rate and synchronous firings in a 10-layer feed-forward neuronal network are studied. When neurons in layer 1 are subject to white noise, synchrony can be built up in deep layers and the firing rate can be propagated. A network with 6 layers is found to be enough for such behavior. A periodic signal with frequencies of 30-80 Hz can be selectively transmitted through the network. These abilities in information processing due to synchrony can be modulated by noise and the operating mode of neurons, and our results are relevant to experimental findings.     

Stationary and dynamical properties of finite N-unit Langevin models subjected to multiplicative noises

We have studied the finite N-unit Langevin model subjected to multiplicative noises, by using the augmented moment method (AMM), as a continuation of our previous paper [H. Hasegawa, J. Phys. Soc. Japan 75 (2006) 033001]. Effects of couplings on stationary and dynamical properties of the model have been investigated. The difference and similarity between the results of diffusive and sigmoid couplings are studied in details. Time dependences of average and fluctuations in local and global variables calculated by the AMM are in good agreement with those of direct simulations (DSs). We also discuss stationary distributions of local and global variables with the use of the Fokker–Planck equation (FPE) method and DSs. It is demonstrated that stationary distributions show much variety when multiplicative noise and external inputs are taken into account.     

Effects of correlated Gaussian noise on the mean firing rate and correlations of an electrically coupled neuronal network

In this paper, we examine the effects of correlated Gaussian noise on a two-dimensional neuronal network that is locally modeled by the Rulkov map. More precisely, we study the effects of the noise correlation on the variations of the mean firing rate and the correlations among neurons versus the noise intensity. Via numerical simulations, we show that the mean firing rate can always be optimized at an intermediate noise intensity, irrespective of the noise correlation. On the other hand, variations of the population coherence with respect to the noise intensity are strongly influenced by the ratio between local and global Gaussian noisy inputs. Biological implications of our findings are also discussed.     

Synchrony due to parametric averaging in neurons coupled by a shared signal

Gonadotropin-releasing hormone (GnRH) is a decapeptide secreted by GnRH neurons located in the hypothalamus. It is responsible for the onset of puberty and the regulation of hormone release from the pituitary. There is a strong evidence suggesting that GnRH exerts an autocrine regulation on its own release via three types of G-proteins [L.Z. Krsmanovic, N. Mores, C.E. Navarro, K.K. Arora, K.J. Catt, An agonist-induced switch in G protein coupling of the gonadotropin-releasing hormone receptor regulates pulsatile neuropeptide secretion, Proc. Natl. Acad. Sci. 100 (2003) 2969-2974]. A mathematical model based on this proposed mechanism has been developed and extended to explain the synchrony observed in GnRH neurons by incorporating the idea of a common pool of GnRH [A. Khadra, Y.X. Li, A model for the pulsatile secretion of gonadotropin-releasing hormone from synchronized hypothalamic neurons, Biophys. J. 91 (2006) 74-83]. This type of coupling led to a very robust synchrony between these neurons. We aim in this paper to reduce the one cell model to a two-variable model using quasi-steady state (QSS) analysis, to further examine its dynamics analytically and geometrically. The concept of synchrony of a heterogeneous population will be clearly defined and established for certain cases, while, for the general case, two different types of phases are introduced to gain more insight on how the model behaves. Bifurcation diagrams for certain parameters in the one cell model are also shown to explain some of the phenomena observed in a coupled population. A comparison between the population model and an averaged two-variable model is also conducted.     

Mechanism for propagation of rate signals through a 10-layer feedforward neuronal network

Using numerical simulations, we explore the mechanism for propagation of rate signals through a 10-layer feedforward network composed of Hodgkin--Huxley (HH) neurons with sparse connectivity. When white noise is afferent to the input layer, neuronal firing becomes progressively more synchronous in successive layers and synchrony is well developed in deeper layers owing to the feedforward connections between neighboring layers. The synchrony ensures the successful propagation of rate signals through the network when the synaptic conductance is weak. As the synaptic time constant τ_syn varies, coherence resonance is observed in the network activity due to the intrinsic property of HH neurons. This makes the output firing rate single-peaked as a function of τ_syn, suggesting that the signal propagation can be modulated by the synaptic time constant. These results are consistent with experimental results and advance our understanding of how information is processed in feedforward networks.     

Inverse stochastic resonance induced by synaptic background activity with unreliable synapses

Inverse stochastic resonance (ISR) is a recently pronounced phenomenon that is the minimum occurrence in mean firing rate of a rhythmically firing neuron as noise level varies. Here, by using a realistic modeling approach for the noise, we investigate the ISR with concrete biophysical mechanisms. It is shown that mean firing rate of a single neuron subjected to synaptic bombardment exhibits a minimum as the spike transmission probability varies. We also demonstrate that the occurrence of ISR strongly depends on the synaptic input regime, where it is most prominent in the balanced state of excitatory and inhibitory inputs.     

Coherence depression in stochastic excitable systems with two-frequency forcing

We study the response of two generic neuron models, the leaky integrate-and-fire (LIF) model and the leaky integrate-and-fire model with dynamic threshold (LIFDT) (i.e., with memory) to a stimulus consisting of two sinusoidal drives with incommensurate frequency, an amplitude modulation ("envelope") noise and a relatively weak additive noise. Spectral and coherence analysis of responses to such naturalistic stimuli reveals how the LIFDT model exhibits better correlation between modulation and spike train even in the presence of both noises. However, a resonance-induced synchrony, occurring when the beat frequency between the sinusoids is close to the intrinsic neuronal firing rate, decreases the coherence in the dynamic threshold case. Under suprathreshold conditions, the modulation noise simultaneously decreases the linear spectral coherence between the spikes and the whole stimulus, as well as between spikes and the stimulus envelope. Our study shows that the coefficient of variation of the envelope fluctuations is positively correlated with the degree of coherence depression. As the coherence function quantifies the linear information transmission, our findings indicate that under certain conditions, a transmission loss results when an excitable system with adaptive properties encodes a beat with frequency in the vicinity of its mean firing rate.     

Effect of a fluctuating parameter mismatch and the associated time-scales on coupled Rossler oscillators

We study the effect of parameter fluctuations and the resultant multiplicative noise on the synchronization of coupled chaotic systems. We introduce a new quantity, the fluctuation rate ϕ as the number of perturbations occurring to the parameter in unit time. It is shown that ϕ is the most significant quantity that determines the quality of synchronization. It is found that parameter fluctuations with high fluctuation rates do not destroy synchronization, irrespective of the statistical features of the fluctuations. We also present a quasi-analytic explanation to the relation between ϕ and the error in synchrony.      

Activity driven adaptive stochastic resonance

Cortical neurons in vivo show fluctuations in their membrane potential of the order of several milli-volts. Using simple and biophysically realistic models of a single neuron we demonstrate that noise induced fluctuations can be used to adaptively optimize the sensitivity of the neuron's output to ensembles of subthreshold inputs of different average strengths. Optimal information transfer is achieved by changing the strength of the noise such that the neuron's average firing rate remains constant. Adaptation is fast, because only crude estimates of the output rate are required at any time.     

Filtered interspike interval encoding by class II neurons

Dynamics of class II neurons, firing frequencies of which are strongly regulated by the inherent neuronal property, have been extensively studied since the formulation of the Hodgkin–Huxley model in 1952. However, how class II neurons process stimulus information and what kind of external information and internal structure firing patterns of neurons represent are vaguely understood in contrast to firing rate coding by class I neurons. Here we show that the FitzHugh–Nagumo class II neuron simultaneously filters inputs based on the input frequency and represent the signal strength by interspike intervals. In this sense, the class II neuron works as an AM processor that passes the information on the carrier and on the temporal waveform of signals.     

Effects of synaptic noise and filtering on the frequency response of spiking neurons

Noise can have a significant impact on the response dynamics of a nonlinear system. For neurons, the primary source of noise comes from background synaptic input activity. If this is approximated as white noise, the amplitude of the modulation of the firing rate in response to an input current oscillating at frequency omega decreases as 1/square root[omega] and lags the input by 45 degrees in phase. However, if filtering due to realistic synaptic dynamics is included, the firing rate is modulated by a finite amount even in the limit omega-->infinity and the phase lag is eliminated. Thus, through its effect on noise inputs, realistic synaptic dynamics can ensure unlagged neuronal responses to high-frequency inputs.     

动态突触、神经耦合与时间延迟对神经元发放的影响

神经元膜电位的受激发放在神经系统的信息传递中起着重要作用.基于一个受动态突触刺激的突触后神经元发放模型,采用数值模拟和傅里叶变换分析的方法研究了动态突触、神经耦合与时间延迟对突触后神经元发放的影响.结果发现:突触前神经元发放频率与Hodgkin-Huxley神经元的固有频率发生共振决定了突触后神经元发放的难易,特定频率范围内的电流刺激有利于神经元激发,动态突触输出的随机突触电流中这些电流刺激所占的比率在很大程度上影响了突触后神经元的发放次数;将突触后神经元换成神经网络后,网络中神经元之间的耦合可以促进神经元的发放,耦合中的时间延迟可以增强这种促进作用,但是不会改变神经耦合对神经元发放的促进模式.     

Bridging rate coding and temporal spike coding by effect of noise

It is controversial whether temporal spike coding or rate coding is dominant in the information processing of the brain. We show by a two-layered neural network model with noise that, when noise is small, cortical neurons fire synchronously and intervals of synchronous firing robustly encode the signal information, but that the neurons desynchronize with moderately strong noise to encode waveforms of the signal more accurately. Further increase of noise just deteriorates the encoding. A positive role of noise in the brain is suggested in a meaning different from stochastic resonance, coherence resonance, and deterministic chaos.     

Stochastic resonance of a periodically driven neuron under non-Gaussian noise

We investigate the first-passage-time statistics of the integrate-fire neuron model driven by a sub-threshold harmonic signal superposed with a non-Gaussian noise. Here, we considered the noise as the result of a random multiplicative process displaced from the origin by an additive term. Such a mechanism generates a power-law distributed noise whose characteristic decay exponent can be finely tuned. We performed numerical simulations to analyze the influence of the noise non-Gaussian character on the stochastic resonance condition. We found that when the noise deviates from Gaussian statistics, the resonance condition occurs at weaker noise intensities, achieving a minimum at a finite value of the distribution function decay exponent. We discuss the possible relevance of this feature to the efficiency of the firing dynamics of biological neurons, as the present result indicates that neurons would require a lower noise level to detect a sub-threshold signal when its statistics departs from Gaussian.     

Impact of colored cross-correlated noises on stochastic resonance and mean extinction rate for a metapopulation system with a multiplicative periodic signal

In this paper, we aim to explore the mean extinction rate and the phenomena of the stochastic resonance (SR) for a metapopulation system induced by a multiplicative periodic signal, colored cross-correlated multiplicative and additive Gaussian noises. By use of the fast descent method and the adiabatic approximation theory for the signal-to-noise ratio, we obtain the expression of the signal-to-noise ratio (SNR). Numerical results indicate that the various SR phenomena occur in the metapopulation system due to the variation of the noise terms and the correlation time. Specifically, the noise correlation always plays a critical role in motivating the SR phenomenon, while the multiplicative noise exerts the inhibition effect on the SR. Interestingly, the weak additive noise can stimulate the resonant peak of the SNR, while the further increase of the noise intensity will lead to the reduction of the SR effect. On the other hand, the noise correlation time τ plays antipodal roles in motivating the SR phenomenon under different circumstances. With regard to the mean extinction rate of the population from the boom state to the extinction one, by performing the numerical calculations, it is found that the additive noise always accelerate the extinction of the population, while the correlation noise will slow down the decline for the population. The role that the noise correlation time plays in the population extinction depends on the values that λ takes.     

Pure Multiplicative Noises Induced Population Extinction in an Anti-tumor Model under Immune Surveillance

The dynamical characters of a theoretical anti-tumor model under immune surveillance subjected to a pure multiplicative noise are investigated. The effects of pure multiplicative noise on the stationary probability distribution (SPD) and the mean first passage time (MFPT) are analysed based on the approximate Fokker-Planck equation of the system in detail. For the anti-tumor model, with the multiplicative noise intensity D increasing, the tumorpopulation move towards to extinction and the extinction rate can beenhanced. Numerical simulations are carried out to check the approximate theoretical results. Reasonably good agreement is obtained.     

Mean First Passage Time for System with Fluctuating Potential Barrier and Coupled Noise

In this paper we study the mean first passage time (MFPT) over a fluctuation potential barrier driven by a coupled noise. It is shown that the MFPT over the fluctuation potential barrier displays resonant activations as the function of the flipping rate of the fluctuation potential barrier, and as the function of the dichotomous noise transition rate.     

Effects of channel noise on firing coherence of small-world Hodgkin-Huxley neuronal networks

We investigate the effects of channel noise on firing coherence of Watts-Strogatz small-world networks consisting of biophysically realistic HH neurons having a fraction of blocked voltage-gated sodium and potassium ion channels embedded in their neuronal membranes. The intensity of channel noise is determined by the number of non-blocked ion channels, which depends on the fraction of working ion channels and the membrane patch size with the assumption of homogeneous ion channel density. We find that firing coherence of the neuronal network can be either enhanced or reduced depending on the source of channel noise. As shown in this paper, sodium channel noise reduces firing coherence of neuronal networks; in contrast, potassium channel noise enhances it. Furthermore, compared with potassium channel noise, sodium channel noise plays a dominant role in affecting firing coherence of the neuronal network. Moreover, we declare that the observed phenomena are independent of the rewiring probability.     

EFFECTS OF COLOURED NOISE IN A TWO-MODE LASER SYSTEM

A homogeneously broadened two-mode laser system with multiplicative coloured noise is investigated when the mode coupling constant ξ=2. An analytical result is obtained when two-dimensional decoupling theory is applied to the system. The intensity auto-correlation function and effective eigenvalue of the laser system are calculated when the difference of pump parameters, multiplicative noise strength and noise correlation time are varied. It is shown that the multiplicative noise can enhance the fluctuations while the noise colour can reduce the fluctuations in the laser system.