Stochastic resonance in neuron models: endogenous stimulation revisited

Multi-scale noise transfer and feature frequency detection in SSVEP based on FitzHugh-Nagumo neuron system

Objective. The steady-state visual evoked potential (SSVEP) is one of the most commonly used control signals for brain-computer interfaces (BCIs) due to its excellent interactive potential, such as high tolerance to noises and robust performance across users. In addition, it has a stable cycle, obvious characteristics and minimal training requirements. However, the SSVEP is extremely weak and companied with strong and multi-scale noise, resulting in a poor signal-to-noise ratio in practice. Common algorithms for classification are based on the principle of template matching and spatial filtering, which cannot obtain satisfied performance of SSVEP detection under the multi-scale noise. Therefore, using linear methods to extract SSVEP with obvious nonlinear and non-stationary characteristics, the useful signal will be attenuated or lost.Approach.To address this issue, two novel frameworks based on a two-dimensional nonlinear FitzHugh-Nagumo (FHN) neuron system are proposed to extract feature frequency of SSVEP.Results.In order to evaluate the effectiveness of the proposed methods, this research recruit 22 subjects to participate the experiment. Experimental results show that nonlinear FHN neuron model can force the energy of noise to be transferred into SSVEP and hence amplifying the amplitude of the target frequency. Compared with the traditional methods, the FHN and FHNCCA methods can achieve higher classification accuracy and faster processing speed, which effectively improves the information transmission rate of SSVEP-based BCI.

Response to a periodic stimulus in a perfect integrate-and-fire neuron model driven by colored noise

The output interspike interval statistics of a stochastic perfect integrate-and-fire neuron model driven by an additive exogenous periodic stimulus is considered. The effect of temporally correlated random activity of synaptic inputs is modeled by an additive symmetric dichotomous noise. Using a first-passage-time formulation, exact expressions for the output interspike interval density and for the serial correlation coefficient are derived in the nonsteady regime, and their dependence on input parameters (e.g., the noise correlation time and amplitude as well as the frequency of an input current) is analyzed. It is shown that an interplay of a periodic forcing and colored noise can cause a variety of nonequilibrium cooperation effects, such as sign reversals of the interspike interval correlations versus noise-switching rate as well as versus the frequency of periodic forcing, a power-law-like decay of oscillations of the serial correlation coefficients in the long-lag limit, amplification of the output signal modulation in the instantaneous firing rate of the neural response, etc. The features of spike statistics in the limits of slow and fast noises are also discussed.

The Gamma renewal process as an output of the diffusion leaky integrate-and-fire neuronal model

Statistical properties of spike trains as well as other neurophysiological data suggest a number of mathematical models of neurons. These models range from entirely descriptive ones to those deduced from the properties of the real neurons. One of them, the diffusion leaky integrate-and-fire neuronal model, which is based on the Ornstein-Uhlenbeck (OU) stochastic process that is restricted by an absorbing barrier, can describe a wide range of neuronal activity in terms of its parameters. These parameters are readily associated with known physiological mechanisms. The other model is descriptive, Gamma renewal process, and its parameters only reflect the observed experimental data or assumed theoretical properties. Both of these commonly used models are related here. We show under which conditions the Gamma model is an output from the diffusion OU model. In some cases, we can see that the Gamma distribution is unrealistic to be achieved for the employed parameters of the OU process.

Distribution of interspike intervals estimated from multiple spike trains observed in a short time window

Several nonparametric estimators of the probability distribution of interspike intervals are introduced. The methods are suitable for simultaneous spike trains observed in a time window of length comparable with the mean interspike interval. This reflects the situation in which a high number of input spike trains converge to a single cortical neuron that has to react in a relatively short time. The simulation study is performed to compare the estimators. For that purpose, several types of stationary point processes are considered as the models of neuronal activity. The methods permit one to estimate the distribution of interspike intervals even if practically none of them are observed. The Kaplan-Meier estimator seems to be the most flexible and reliable among all studied methods, but no direct conclusions as to how real neurons work can be deduced from it.

Spike train statistics for consonant and dissonant musical accords in a simple auditory sensory model

The phenomena of dissonance and consonance in a simple auditory sensory model composed of three neurons are considered. Two of them, here so-called sensory neurons, are driven by noise and subthreshold periodic signals with different ratio of frequencies, and its outputs plus noise are applied synaptically to a third neuron, so-called interneuron. We present a theoretical analysis with a probabilistic approach to investigate the interspike intervals statistics of the spike train generated by the interneuron. We find that tones with frequency ratios that are considered consonant by musicians produce at the third neuron inter-firing intervals statistics densities that are very distinctive from densities obtained using tones with ratios that are known to be dissonant. In other words, at the output of the interneuron, inharmonious signals give rise to blurry spike trains, while the harmonious signals produce more regular, less noisy, spike trains. Theoretical results are compared with numerical simulations.

Quasistatic approximation of the interspike interval distribution of neurons driven by time-dependent inputs

Variability in neural responses is a ubiquitous phenomenon in neurons, usually modeled with stochastic differential equations. In particular, stochastic integrate-and-fire models are widely used to simplify theoretical studies. The statistical properties of the generated spikes depend on the stimulating input current. Given that real sensory neurons are driven by time-dependent signals, here we study how the interspike interval distribution of integrate-and-fire neurons depends on the evolution of the stimulus in a quasistatic limit. We obtain a closed-form expression for this distribution, and we compare it to the one obtained with numerical simulations for several time-dependent currents. For slow inputs, the quasistatic distribution provides a very good description of the data. The results obtained for the integrate-and-fire model can be extended to other nonautonomous stochastic systems where the first passage time problem has an explicit solution.

On the evaluation of firing densities for periodically driven neuron models

The leaky integrate-and-fire model for neuronal spiking events driven by a periodic stimulus is studied by using the Fokker-Planck formulation. To this purpose, an essential use is made of the asymptotic behavior of the first-passage-time probability density function of a time homogeneous diffusion process through an asymptotically periodic threshold. Numerical comparisons with some recently published results derived by a different approach are performed. Use of a new asymptotic approximation is then made in order to design a numerical algorithm of predictor-corrector type to solve the integral equation in the unknown first-passage-time probability density function. Such algorithm, characterized by a reduced (linear) computation time, is seen to provide a high computation accuracy. Finally, it is shown that such an approach yields excellent approximations to the firing probability density function for a wide range of parameters, including the case of high stimulus frequencies.

A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties

The integrate-and-fire neuron model describes the state of a neuron in terms of its membrane potential, which is determined by the synaptic inputs and the injected current that the neuron receives. When the membrane potential reaches a threshold, an action potential (spike) is generated. This review considers the model in which the synaptic input varies periodically and is described by an inhomogeneous Poisson process, with both current and conductance synapses. The focus is on the mathematical methods that allow the output spike distribution to be analyzed, including first passage time methods and the Fokker-Planck equation. Recent interest in the response of neurons to periodic input has in part arisen from the study of stochastic resonance, which is the noise-induced enhancement of the signal-to-noise ratio. Networks of integrate-and-fire neurons behave in a wide variety of ways and have been used to model a variety of neural, physiological, and psychological phenomena. The properties of the integrate-and-fire neuron model with synaptic input described as a temporally homogeneous Poisson process are reviewed in an accompanying paper (Burkitt in Biol Cybern, 2006).

Stochastic resonance driven by two different kinds of colored noise in a bistable system

The phenomenon of stochastic resonance in a bistable nonlinear system is investigated when both the multiplicative noise and the coupling between additive and multiplicative noise are colored with different values of noise correlation time tau(1) and tau(2). Combining the functional analysis and unified colored noise approximation, the two different kinds of colored noise in the nonlinear system can be simplified. The signal-to-noise ratio is calculated when a weakly periodic signal is added to the system. It is found that there appears a transition between one peak and two peaks in the curve of the signal-to-noise ratio when either the noise correlation time tau(1) and tau(2) or the coupling strength lambda between additive and multiplicative noise is increased. The transition between one and two peaks depending on tau(1) and lambda is more complex than that depending on tau(2).

Input-output behaviour of a model neuron with alternating drift

The input-output behaviour of the Wiener neuronal model subject to alternating input is studied under the assumption that the effect of such an input is to make the drift itself of an alternating type. Firing densities and related statistics are obtained via simulations of the sample-paths of the process in the following three cases: the drift changes occur during random periods characterised by (i) exponential distribution, (ii) Erlang distribution with a preassigned shape parameter, and (iii) deterministic distribution. The obtained results are compared with those holding for the Wiener neuronal model subject to sinusoidal input.