Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics

Modeling spiking behavior of neurons with time-dependent Poisson processes

Three kinds of interval statistics, as represented by the coefficient of variation, the skewness coefficient, and the correlation coefficient of consecutive intervals, are evaluated for three kinds of time-dependent Poisson processes: pulse regulated, sinusoidally regulated, and doubly stochastic. Among these three processes, the sinusoidally regulated and doubly stochastic Poisson processes, in the case when the spike rate varies slowly compared with the mean interval between spikes, are found to be consistent with the three statistical coefficients exhibited by data recorded from neurons in the prefrontal cortex of monkeys.

Fluctuation-induced memory retrieval in a pulsed neural network storing sparse patterns with hierarchical correlations

An associative memory in a pulsed neural network composed of the FitzHugh-Nagumo models storing sparse patterns with hierarchical correlations is investigated. The memory patterns composed of 0/1 digits are represented by the synchronous periodic firings of the neurons. It is found that the target pattern and the OR pattern are retrieved individually by controlling the intensity of fluctuations in the system.

Theoretical analysis of array-enhanced stochastic resonance in the diffusively coupled FitzHugh-Nagumo equation

The array-enhanced stochastic resonance (AESR) in the diffusively coupled FitzHugh-Nagumo equation is investigated. The two properties of AESR, namely, the scaling of the optimal noise intensity and the enhancement of the maximum value of the correlation coefficient as a function of the coupling strength, are analyzed theoretically. By transforming the dynamics of N elements into that of the mean and the deviation from it, it is found that AESR is caused by the correlation between them. A low-dimensional model that reproduces the above properties is constructed.

Stochastic dynamics in a two-dimensional oscillator near a saddle-node bifurcation

We study the oscillator equations describing a particular class of nonlinear amplifier, exemplified in this work by a two-junction superconducting quantum interference device. This class of dynamic system is described by a potential energy function that can admit minima (corresponding to stable solutions of the dynamic equations), or "running states" wherein the system is biased so that the potential minima disappear and the solutions display spontaneous oscillations. Just beyond the onset of the spontaneous oscillations, the system is known to show significantly enhanced sensitivity to very weak magnetic signals. The global phase space structure allows us to apply a center manifold technique to approximate analytically the oscillatory behavior just past the (saddle-node) bifurcation and compute the oscillation period, which obeys standard scaling laws. In this regime, the dynamics can be represented by an "integrate-fire" model drawn from the computational neuroscience repertoire; in fact, we obtain an "interspike interval" probability density function and an associated power spectral density (computed via Renewal theory) that agree very well with the results obtained via numerical simulations. Notably, driving the system with one or more time sinusoids produces a noise-lowering injection locking effect and/or heterodyning.

Transmission of noise coded versus additive signals through a neuronal ensemble

Neuronal populations receive signals through temporally inhomogeneous spike trains which can be approximated by an input consisting of a time dependent mean value (additive signal) and noise with a time dependent intensity (noise coded signal). We compare the linear response of an ensemble of model neurons to these signals. Our analytical solution for the mean activity demonstrates the high efficiency of the transmission of a noise coded signal in a broad frequency band. For both kinds of signal we show that the transmission by the ensemble reveals stochastic resonance as well as a nonmonotonous dependence on the driving frequency.

Stochastic resonance in neuron models: endogenous stimulation revisited

The paradigm of stochastic resonance (SR)-the idea that signal detection and transmission may benefit from noise-has met with great interest in both physics and the neurosciences. We investigate here the consequences of reducing the dynamics of a periodically driven neuron to a renewal process (stimulation with reset or endogenous stimulation). This greatly simplifies the mathematical analysis, but we show that stochastic resonance as reported earlier occurs in this model only as a consequence of the reduced dynamics.

Shot noise in the leaky integrate-and-fire neuron

We study the influence of noise on the transmission of temporal information by a leaky integrate-and-fire neuron using the theory of shot noise. The model includes a finite number of synapses and has a membrane potential variance de facto modulated by the input signal. The phenomenon of stochastic resonance in spiking neurons is analytically exhibited using an inhomogeneous Poisson process model of the spike trains, and links with the traditional Ornstein-Uhlenbeck process obtained by a diffusion approximation are given. It is shown that the modulated membrane potential variance inherent to the model gives better signal processing capabilities than the diffusion approximation.

Dynamics of moments of FitzHugh-Nagumo neuronal models and stochastic bifurcations

For the study of the behavior of noisy neuronal models, Rodriguez and Tuckwell have introduced an elegant and systematic method which consists of replacing the system of stochastic differential equations with a system of deterministic equations representing the dynamics of the means, variances, and covariance of the state variables [R. Rodriguez and H.C. Tuckwell, Phys. Rev. E 54, 5585 (1996)]. In this work, we first report a modification of their method in the case of the FitzHugh-Nagumo model which enhances the accuracy of the approximation without including higher order moments. This method is then combined with a self-consistency argument in order to better characterize the behavior of the underlying stochastic processes through the computation of approximate auto- and cross-correlation functions of the state variables. Finally, we argue that the moments' equations can also reveal the existence of stochastic bifurcations, i.e., qualitative changes in the dynamics of stochastic systems.

Biocomplexity: adaptive behavior in complex stochastic dynamical systems

Existing methods of complexity research are capable of describing certain specifics of bio systems over a given narrow range of parameters but often they cannot account for the initial emergence of complex biological systems, their evolution, state changes and sometimes-abrupt state transitions. Chaos tools have the potential of reaching to the essential driving mechanisms that organize matter into living substances. Our basic thesis is that while established chaos tools are useful in describing complexity in physical systems, they lack the power of grasping the essence of the complexity of life. This thesis illustrates sensory perception of vertebrates and the operation of the vertebrate brain. The study of complexity, at the level of biological systems, cannot be completed by the analytical tools, which have been developed for non-living systems. We propose a new approach to chaos research that has the potential of characterizing biological complexity. Our study is biologically motivated and solidly based in the biodynamics of higher brain function. Our biocomplexity model has the following features, (1) it is high-dimensional, but the dimensionality is not rigid, rather it changes dynamically; (2) it is not autonomous and continuously interacts and communicates with individual environments that are selected by the model from the infinitely complex world; (3) as a result, it is adaptive and modifies its internal organization in response to environmental factors by changing them to meet its own goals; (4) it is a distributed object that evolves both in space and time towards goals that is continually re-shaping in the light of cumulative experience stored in memory; (5) it is driven and stabilized by noise of internal origin through self-organizing dynamics. The resulting theory of stochastic dynamical systems is a mathematical field at the interface of dynamical system theory and stochastic differential equations. This paper outlines several possible avenues to analyze these systems. Of special interest are input-induced and noise-generated, or spontaneous state-transitions and related stability issues.

Effect of spatial extension on noise-enhanced phase locking in a leaky integrate-and-fire model of a neuron

Signal transmission enhanced by noise has been recently investigated in detail on the single compartment, also referred to as single point, leaky integrate-and-fire model neuron under a subthreshold stimulation. In this paper we study how this phenomenon is influenced by taking into account the spatial characteristics of the neuron. A stochastic two-point leaky integrate-and-fire model, comprising a dendritic compartment and trigger zone, under periodic stimulation is studied. A method of how to measure synchronization between the signal and the output in both, experiments and models, is proposed. This method is based on a distance between the exact periodic spiking, as expected for sufficiently strong and noiseless stimulation, and neuronal activity evoked by a subthreshold signal corrupted by noise. It is shown that qualitatively the same phenomenon, phase-locking enhanced by the noise, as found in the spatially unstructured neuron is produced by the spatially complex neuron. However, quantitatively there are significant differences. Namely, the two-point model neuron is more robust against the noise and therefore its amplitude has to be higher to enhance the signal. Further, it is found that the range of the critical levels of noise is larger for the two-point model than for the single-point one. Finally, the enhancing effect at the optimal noise is more efficient in the single-point model and thus the firing patterns at their optimal noise levels are different in both models.

Stochastic resonance on a circle without excitation: physical investigation and peak frequency formula

In this article the existence of stochastic resonance (SR) without external force in a simplified circular system for different values of the control parameter b is considered. The average power spectra are calculated as well as the signal-to-noise ratio as a measure for stochastic resonance. It is shown that in the monostable and semistable (b<1 and b=1) cases coherent oscillations occur and SR exists. For the case b>1, the system is oscillatory and noise plays only a destructive role; therefore no SR occurs. The rotation number of the system is calculated and compared to the peak frequency of the power spectrum. Although the coincidence in the noisy case is not as good as that in the deterministic case, we can derive an empirical formula between the peak frequency of the power spectrum and the rotation number of the system, which is in good agreement with results of numerical simulations.

Models of post-translational protein translocation

Organellar Hsp-70 is required for post-translational translocation into the endoplasmic reticulum and mitochondria. The functional role played by Hsp-70 is unknown. However, two operating principles have been suggested. The power stroke model proposes that Hsp-70 undergoes a conformational change, which pulls the precursor protein through the translocation pore, whereas, in the Brownian ratchet model, the role of Hsp-70 is simply to block backsliding through the pore. A mathematical analysis of both mechanisms is presented and reveals that qualitative differences between the models occur in the behavior of the mean velocity and effective diffusion coefficient as a function of Hsp-70 concentration. An experimental method is proposed for measuring these two quantities that only relies on current experimental techniques.

A simple stochastic model of spatially complex neurons

A method for studying the coding properties of a multicompartmental integrate-and-fire neuron of arbitrary geometry is presented. Depolarization at each compartment evolves like a leaky integrator with an after-firing reset imposed only at the trigger zone. The frequency of firing at the steady-state regime is related to the properties of the multidimensional input. The decreasing variability of subthreshold depolarization from the dendritic tree to the trigger zone is shown for an input that is corrupted by a white noise. The role of a Poissonian noise is also investigated. The proposed method gives an estimate of the mean interspike interval that can be used to study the input output transfer function of the system. Both types of the stochastic inputs result in broadening the transfer function with respect to the deterministic case.

Stochastic resonance in a pulse neural network with a propagational time delay

Stochastic resonance in a coupled FitzHugh-Nagumo equation with a propagational time delay is investigated. With an appropriate set of parameter values. i.e. the frequency of the periodic input, the propagational time delay, and the coupling strength, a deterministic firing induced by additive noise is observed, and its dependence on the number of neurons is examined. It is also found that a network composed of two assemblies shows a competitive behavior under control of the noise intensity.

Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors

Weakly electric fish generate a periodic electric field as a carrier signal for active location and communication tasks. Highly sensitive P-type receptors on their surface fire in response to carrier amplitude modulations (AM's) in a noisy phase locked fashion. A simple generic model of receptor activity and signal encoding is presented. Its suprathreshold dynamics, memory and receptor noise reproduce observed firing interval distributions and correlations. The model ultimately explains how smooth responses to AM's are compatible with its nonlinear phase locking properties, and reveals how receptor noise can sometimes enhance the encoding of small yet suprathreshold AM's.

Associative memory retrieval induced by fluctuations in a pulsed neural network

An associative memory retrieval in a pulsed neural network composed of the FitzHugh-Nagumo neurons is investigated. The memory is represented in the spatio-temporal firing pattern of the neurons, and the memory retrieval is accomplished using the fluctuations in the system. The storage capacity of the network is investigated numerically. It is demonstrated that this pulsed neural network is capable of an alternate retrieval of two patterns.

Coherence and stochastic resonance in a two-state system

The subject of our study is a two-state dynamics driven by Gaussian white noise and a weak harmonic signal. The system resulting from a piecewise linear FizHugh-Nagumo model in the case of perfect time scale separation between fast and slow variables shows either bistable, excitable, or oscillatory behavior. Its output spectra as well as the spectral power amplification of the signal can be calculated for arbitrary noise strength and frequency, allowing characterization of the coherence resonance in the bistable and excitable regimes as well as quantification of nonadiabatic resonances with respect to the external signal in all regimes.

Response of coupled noisy excitable systems to weak stimulation

It is known that coupling can enhance the response of noisy bistable devices to weak periodic modulation. This work examines whether a similar phenomenon occurs in the active rotator model for excitable systems. We study the dynamics of assemblies of weakly periodically modulated active rotators. The addition of noise to these brings about a number of behaviors that have no counterpart in networks of bistable systems. The analysis of the dynamics of the solution of the Fokker-Planck equation of active rotator networks shows that these new behaviors are similar to generic responses of periodically forced autonomous oscillators. This is because noise alone, in the absence of other inputs, can regularize the dynamics of single active rotators through coherence resonance, and lead to regular synchronous activity at the level of networks. We argue that similar phenomena take place in a broad class of excitable systems.

Mean discharge frequency locking in the response of a noisy neuron model to subthreshold periodic stimulation

Leaky integrate-and-fire neuron models display stochastic resonance-like behavior when stimulated by subthreshold periodic signal and noise. Previous works have shown that matching between the time scales of the noise induced discharges and the modulation period can account for this phenomenon at low modulation amplitudes, but not large subthreshold modulation amplitude. In order to examine the discharge patterns of the model in this regime, we introduce a method for the computation of the power spectral density of the discharge train. Using this method, we clarify the role of the distribution of the input phase at discharge times. Finally, we argue that for large subthreshold inputs, mean discharge frequency locking accounts for the enhanced response.

Markov analysis of stochastic resonance in a periodically driven integrate-and-fire neuron

We model the dynamics of the leaky integrate-and-fire neuron under periodic stimulation as a Markov process with respect to the stimulus phase. This avoids the unrealistic assumption of a stimulus reset after each spike made in earlier papers and thus solves the long-standing reset problem. The neuron exhibits stochastic resonance, both with respect to input noise intensity and stimulus frequency. The latter resonance arises by matching the stimulus frequency to the refractory time of the neuron. The Markov approach can be generalized to other periodically driven stochastic processes containing a reset mechanism.

Collective behavior of networks with linear (VLSI) integrate-and-fire neurons

We analyze in detail the statistical properties of the spike emission process of a canonical integrate-and-fire neuron, with a linear integrator and a lower bound for the depolarization, as often used in VLSI implementations (Mead, 1989). The spike statistics of such neurons appear to be qualitatively similar to conventional (exponential) integrate-and-fire neurons, which exhibit a wide variety of characteristics observed in cortical recordings. We also show that, contrary to current opinion, the dynamics of a network composed of such neurons has two stable fixed points, even in the purely excitatory network, corresponding to two different states of reverberating activity. The analytical results are compared with numerical simulations and are found to be in good agreement.

Resonance effect for neural spike time reliability

The spike timing reliability of Aplysia motoneurons stimulated by repeated presentation of periodic or aperiodic input currents is investigated. Two properties of the input are varied, the frequency content and the relative amplitude of the fluctuations to the mean (expressed as the coefficient of variation: CV). It is shown that, for small relative amplitude fluctuations (CV approximately 0.05-0.15), the reliability of spike timing is enhanced if the input contains a resonant frequency equal to the firing rate of the neuron in response to the DC component of the input. This resonance-related enhancement in reliability decreases as the relative amplitude of the fluctuations increases (CV-->1). Similar results were obtained for a leaky integrate-and-fire neuronal model, suggesting that these effects are a general property of encoders that combine a threshold with a leaky integrator. These observations suggest that, when the magnitude of input fluctuations is small, changes in the power spectrum of the current fluctuations or in the spike discharge rate can have a pronounced effect on the ability of the neuron to encode a time-varying input with reliably timed spikes.