Spike-interval triggered averaging reveals a quasi-periodic spiking alternative for stochastic resonance in catfish electroreceptors

Catfish detect and identify invisible prey by sensing their ultra-weak electric fields with electroreceptors. Any neuron that deals with small-amplitude input has to overcome sensitivity limitations arising from inherent threshold non-linearities in spike-generation mechanisms. Many sensory cells solve this issue with stochastic resonance, in which a moderate amount of intrinsic noise causes irregular spontaneous spiking activity with a probability that is modulated by the input signal. Here we show that catfish electroreceptors have adopted a fundamentally different strategy. Using a reverse correlation technique in which we take spike interval durations into account, we show that the electroreceptors generate a supra-threshold bias current that results in quasi-periodically produced spikes. In this regime stimuli modulate the interval between successive spikes rather than the instantaneous probability for a spike. This alternative for stochastic resonance combines threshold-free sensitivity for weak stimuli with similar sensitivity for excitations and inhibitions based on single interspike intervals.     

Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron

We study a white-noise driven integrate-and-fire (IF) neuron with a time-dependent threshold. We give analytical expressions for mean and variance of the interspike interval assuming that the modification of the threshold value is small. It is shown that the variability of the interval can become both smaller or larger than in the case of constant threshold depending on the decay rate of threshold. We also show that the relative variability is minimal for a certain finite decay rate of the threshold. Furthermore, for slow threshold decay the leaky IF model shows a minimum in the coefficient of variation whenever the firing rate of the neuron matches the decay rate of the threshold. This novel effect can be seen if the firing rate is changed by varying the noise intensity or the mean input current.     

Model of discharge patterns of units in the cochlear nucleus in response to steady state and time-varying sounds

A model for the coding of sounds in the discharge pattern of single neurons in the auditory periphery is described. The model consists of a non-linear differential equation and a conventional diffusion neuron model. The same model describes 1) the unit interspike interval distribution in response to continuous stimulation, 2) the response to tone bursts, as well as 3) the responses to sinusoidally amplitude-modulated stimuli. The parameters of the model are determined quantitatively for each neuron. The responses of a certain unit can be described by 5 parameters.     

The parameters of the stochastic leaky integrate-and-fire neuronal model

Five parameters of one of the most common neuronal models, the diffusion leaky integrate-and-fire model, also known as the Ornstein-Uhlenbeck neuronal model, were estimated on the basis of intracellular recording. These parameters can be classified into two categories. Three of them (the membrane time constant, the resting potential and the firing threshold) characterize the neuron itself. The remaining two characterize the neuronal input. The intracellular data were collected during spontaneous firing, which in this case is characterized by a Poisson process of interspike intervals. Two methods for the estimation were applied, the regression method and the maximum-likelihood method. Both methods permit to estimate the input parameters and the membrane time constant in a short time window (a single interspike interval). We found that, at least in our example, the regression method gave more consistent results than the maximum-likelihood method. The estimates of the input parameters show the asymptotical normality, which can be further used for statistical testing, under the condition that the data are collected in different experimental situations. The model neuron, as deduced from the determined parameters, works in a subthreshold regimen. This result was confirmed by both applied methods. The subthreshold regimen for this model is characterized by the Poissonian firing. This is in a complete agreement with the observed interspike interval data. Action Editor: Nicolas Brunel     

Neuron firing in driven nonlinear integrate-and-fire models

Statistical properties of neuron firing are studied in the framework of a nonlinear leaky integrate-and-fire model that is driven by a slow periodic subthreshold signal. The firing events are characterized by first passage time densities. The experimentally better accessible interspike interval density generally depends on the sojourn times in a refractory state of the neuron. This aspect is not part of the integrate-and-fire model and must be modelled additionally. For a large class of refractory dynamics, a general expression for the interspike interval density is given and further evaluated for the two cases with an instantaneous resetting (i.e. no refractory state) and a refractory state possessing a deterministic lifetime. First passage time densities and interspike interval densities following from the proposed theory compare favorably with precise numerical simulations.     

The interspike interval of a cable model neuron with white noise input

The firing time of a cable model neuron in response to white noise current injection is investigated with various methods. The Fourier decomposition of the depolarization leads to partial differential equations for the moments of the firing time. These are solved by perturbation and numerical methods, and the results obtained are in excellent agreement with those obtained by Monte Carlo simulation. The convergence of the random Fourier series is found to be very slow for small times so that when the firing time is small it is more efficient to simulate the solution of the stochastic cable equation directly using the two different representations of the Green's function, one which converges rapidly for small times and the other which converges rapidly for large times. The shape of the interspike interval density is found to depend strongly on input position. The various shapes obtained for different input positions resemble those for real neurons. The coefficient of variation of the interspike interval decreases monotonically as the distance between the input and trigger zone increases. A diffusion approximation for a nerve cell receiving Poisson input is considered and input/output frequency relations obtained for different input sites. The cases of multiple trigger zones and multiple input sites are briefly discussed.     

New parameter relationships determined via stochastic ordering for spike activity in a reversal potential neural model

Purpose of this work is to study the dependence of interspike interval distribution on the model parameters when use is made of the Feller diffusion process to describe the subthreshold membrane potential of a neuron. To this aim we make use of a new approach, namely the ordering of first passage times. The functional dependence among the model parameters (e.g, membrane time constant, reversal potential, etc.) resulting from the ordering criteria employed and from the study of mean trajectory plots is analyzed into detail for four different scenario.     

A qualitative comparison of some diffusion models for neural activity via stochastic ordering

A number of diffusion processes have been proposed as a continuous analog of Stein's model for the subthreshold membrane potential of a neuron. Interspike intervals are then described as the first-passage-time of the corresponding diffusion model through a suitable threshold. Various biological considerations suggest the use of more sophisticated models in lieu of the Ornstein-Uhlenbeck model. However, the advantages of the additional complexity are not always clear. Comparisons among different models generally use numerical methods in specific examples without a general sensitivity analysis on the role of the model parameters. Here, we compare the distribution of interspike intervals from different models using the method of stochastic ordering. The qualitative comparison of the role of each parameter extends the results obtained from numerical simulations. One result on neurons with high positive net excitation is that the reversal potential models considered do not greatly differ from the Ornstein-Uhlenbeck model. For neurons with increased inhibition, the models give greater differences among the interspike interval distributions. In particular, when the mean trajectories are matched, the Feller model gives shorter times than the Ornstein-Uhlenbeck model but longer times than our double reversal potential model. Received: 5 August 1999 / Accepted in revised form: 8 May 2000     

State-Dependent Effects of Na Channel Noise on Neuronal Burst Generation

We explore the effects of stochastic sodium (Na) channel activation on the variability and dynamics of spiking and bursting in a model neuron. The complete model segregates Hodgin-Huxley-type currents into two compartments, and undergoes applied current-dependent bifurcations between regimes of periodic bursting, chaotic bursting, and tonic spiking. Noise is added to simulate variable, finite sizes of the population of Na channels in the fast spiking compartment.During tonic firing, Na channel noise causes variability in interspike intervals (ISIs). The variance, as well as the sensitivity to noise, depend on the model's biophysical complexity. They are smallest in an isolated spiking compartment; increase significantly upon coupling to a passive compartment; and increase again when the second compartment also includes slow-acting currents. In this full model, sufficient noise can convert tonic firing into bursting.During bursting, the actions of Na channel noise are state-dependent. The higher the noise level, the greater the jitter in spike timing within bursts. The noise makes the burst durations of periodic regimes variable, while decreasing burst length duration and variance in a chaotic regime. Na channel noise blurs the sharp transitions of spike time and burst length seen at the bifurcations of the noise-free model. Close to such a bifurcation, the burst behaviors of previously periodic and chaotic regimes become essentially indistinguishable.We discuss biophysical mechanisms, dynamical interpretations and physiological implications. We suggest that noise associated with finite populations of Na channels could evoke very different effects on the intrinsic variability of spiking and bursting discharges, depending on a biological neuron's complexity and applied current-dependent state. We find that simulated channel noise in the model neuron qualitatively replicates the observed variability in burst length and interburst interval in an isolated biological bursting neuron.     

Intrinsic coherence resonance in excitable membrane patches

The influence of intrinsic channel noise on the spiking activity of excitable membrane patches is studied by use of a stochastic generalization of the Hodgkin-Huxley model. Internal noise stemming from the stochastic dynamics of individual ion channels does affect the electric properties of the cell-membrane patches. There exists an optimal size of the membrane patch for which the internal noise alone can cause a nearly regular spontaneous generation of action potentials. We consider the influence of intrinsic channel noise in presence of a constant and an oscillatory current driving for both, the mean interspike interval and the phenomenon of coherence resonance for neuronal spiking. Given small membrane patches, implying that channel noise dominates the excitable dynamics, we find the phenomenon of intrinsic coherence resonance. In this case, the relatively regular spiking behavior becomes essentially independent of an applied stimulus. We observed, however, the occurrence of a skipping of supra-threshold input events due to channel noise for intermediate patch sizes. This effect consequently reduces the overall coherence of the spiking.     

Interspike interval densities of resonate and fire neurons

The subthreshold dynamics of a neuron can follow one of the two patterns: resonant neurons generate intrinsic subthreshold membrane potential oscillations, whereas in nonresonant neurons these oscillations are not observed. Here, we investigate how these subthreshold behaviors affect the suprathreshold response. Both types of neurons are described by a resonate and fire model, with the stable fixpoint being either a focus or a node. Using analytic expression for a linear oscillator model with threshold and reset, we calculate the multimodal interspike interval densities. We show that a change in model parameters induces qualitative changes in the interspike interval densities.     

A hidden Markov model approach to neuron firing patterns.

Analysis and characterization of neuronal discharge patterns are of interest to neurophysiologists and neuropharmacologists. In this paper we present a hidden Markov model approach to modeling single neuron electrical activity. Basically the model assumes that each interspike interval corresponds to one of several possible states of the neuron. Fitting the model to experimental series of interspike intervals by maximum likelihood allows estimation of the number of possible underlying neuron states, the probability density functions of interspike intervals corresponding to each state, and the transition probabilities between states. We present an application to the analysis of recordings of a locus coeruleus neuron under three pharmacological conditions. The model distinguishes two states during halothane anesthesia and during recovery from halothane anesthesia, and four states after administration of clonidine. The transition probabilities yield additional insights into the mechanisms of neuron firing.     

Nonlinear systems analysis of computer models of repetitive firing

The inhibitory influences of recurrent inhibition and afterhyperpolarization are studied theoretically insofar as they affect the density of the interspike interval and the frequency transfer characteristic. The methods employed involve exact results for excitation with decay and constant threshold, computer simulations for decaying thresholds representing afterhyperpolarization, and the diffusion approximation for excitation with inhibition and a constant threshold. Afterhyperpolarization tends to preserve the linearity of the frequency transfer characteristic and the lognormality of the interspike time. Recurrent inhibition which grows linearly with frequency of excitation can lead to frequency limiting. Some forms of nonlinear recurrent inhibition may lead to a filter type effect whereby the neuron responds significantly only over certain ranges of input intensity. A simple network model is analysed which exhibits recurrent inhibitory frequency growing linearly with frequency of excitation. An estimate of 10 to 50 is made for the number of Renshaw cells which connect with a spinal motoneuron. The frequency limiting of motoneurons is discussed and the stabilizing influence attributed to Renshaw cells is questioned. It is postulated that Renshaw recurrent inhibition is of functional significance at low levels of excitatory drive to motoneurons and that its effect is diminished by reciprocal inhibition at high excitatory input frequencies.     

The faithful copy neuron

Theoretical and experimental evidence is presented for the presence in nervous tissue of neurons whose firing rate faithfully follow their input stimulus. Such neurons are shown to deliver their spikes with minimum dissipation per spike. This optimal performance is likely accomplished by use of local circuitry that adjusts conductances to match input currents so that the neuron operates near the threshold for firing. This results in an unusual mechanism for neuronal firing that uses background noise to achieve the desired firing rate. This framework takes place dynamically, and the present deliberations apply under time varying conditions. It is shown that an analytically explicit probability distribution function, which depends on one dimensionless parameter, can account for the interspike interval statistics under general time varying conditions. An innovative analysis based on the unsteady firing rate fits data to the appropriate probability distribution function.     

Weak periodic signal detection by sine-Wiener-noise-induced resonance in the FitzHugh–Nagumo neuron

Based on the FitzHugh–Nagumo (FHN) neuron model subjected to sine-Wiener (SW) noise, impacts of SW noise on weak periodic signal detection are investigated by calculating response measure Q for characterizing synchronization between the input signal and the output temporal activities of the neuron. It is numerically demonstrated that the response measure Q can achieve the optimal value under appropriate and moderate intensity or correlation time of SW noise, suggesting the occurrence of SW-noise-induced stochastic resonance. Furthermore, the optimal value of Q is sensitive to correlation time. Consequently, the correlation time of SW noise has a great influence on the performance of signal detection in the FHN neuron.     

Dynamics of the noisy neural network

The dynamic features of interspike interval sequences and structures of spatiotemporal patterns of firing in a coupled noisy neural network are investigated. The system displays complex dynamics under periodic external stimuli. The dynamics is modulated by a periodic impulse-like synaptic current which relates to a global coupling interaction between neurons. The firings of the stimulated neurons are phase-locked to this current. In addition, the interspike interval histograms are studied for the case of frozen noise which does not change its value within a time interval once it has been distributed onto the network. It is found that the peaks in these histograms are located at integer multiples of the period of the external stimulus, and the heights of these peaks decay exponentially, which corresponds to the experimental results. Received: 12 April 1997 / Accepted in revised form: 7 May 1997     

Near Poisson-type firing produced by concurrent excitation and inhibition

The effect of inhibition on the firing variability is examined in this paper using the biologically-inspired temporal noisy-leaky integrator (TNLI) neuron model. The TNLI incorporates hyperpolarising inhibition with negative current pulses of controlled shapes and it also separates dendritic from somatic integration. The firing variability is observed by looking at the coefficient of variation (C(V)) (standard deviation/mean interspike interval) as a function of the mean interspike interval of firing (delta tM) and by comparing the results with the theoretical curve for random spike trains, as well as looking at the interspike interval (ISI) histogram distributions. The results show that with 80% inhibition, firing at high rates (up to 200 Hz) is nearly consistent with a Poisson-type variability, which complies with the analysis of cortical neuron firing recordings by Softky and Koch [1993, J. Neurosci. 13(1) 334-530]. We also demonstrate that the mechanism by which inhibition increases the C(V) values is by introducing more short intervals in the firing pattern as indicated by a small initial hump at the beginning of the ISI histogram distribution. The use of stochastic inputs and the separation of the dendritic and somatic integration which we model in TNLI, also affect the high firing, near Poisson-type (explained in the paper) variability produced. We have also found that partial dendritic reset increases slightly the firing variability especially at short ISIs.     

峰峰间期逐渐增大的动力学机制

在3或4个轻度结扎受损的大鼠坐骨神经上加入5mmol／L EGTA的无钙灌流液的神经生理实验中,可以观察到一种在活动相峰峰间期逐渐增大的周期阵发放电现象。从非线性动力学角度分析该现象产生的动力学机制对于理解神经元复杂的放电行为具有重要意义。通过Hindmarsh-Rose神经元模型的分析,对该现象产生的一种可能的机制进行了揭示,即鞍结分岔和鞍点同宿分岔支配着这种阵发放电形式,而且后者对峰峰阃期逐渐的增大起着更重要的作用。     

Analytical and Simulation Results for Stochastic Fitzhugh-Nagumo Neurons and Neural Networks

An analytical approach is presented for determining the response of a neuron or of the activity in a network of connected neurons, represented by systems of nonlinear ordinary stochastic differential equations—the Fitzhugh-Nagumo system with Gaussian white noise current. For a single neuron, five equations hold for the first- and second-order central moments of the voltage and recovery variables. From this system we obtain, under certain assumptions, five differential equations for the means, variances, and covariance of the two components. One may use these quantities to estimate the probability that a neuron is emitting an action potential at any given time. The differential equations are solved by numerical methods. We also perform simulations on the stochastic Fitzugh-Nagumo system and compare the results with those obtained from the differential equations for both sustained and intermittent deterministic current inputs withsuperimposed noise. For intermittent currents, which mimic synaptic input, the agreement between the analytical and simulation results for the moments is excellent. For sustained input, the analytical approximations perform well for small noise as there is excellent agreement for the moments. In addition, the probability that a neuron is spiking as obtained from the empirical distribution of the potential in the simulations gives a result almost identical to that obtained using the analytical approach. However, when there is sustained large-amplitude noise, the analytical method is only accurate for short time intervals. Using the simulation method, we study the distribution of the interspike interval directly from simulated sample paths. We confirm that noise extends the range of input currents over which (nonperiodic) spike trains may exist and investigate the dependence of such firing on the magnitude of the mean input current and the noise amplitude. For networks we find the differential equations for the means, variances, and covariances of the voltage and recovery variables and show how solving them leads to an expression for the probability that a given neuron, or given set of neurons, is firing at time t. Using such expressions one may implement dynamical rules for changing synaptic strengths directly without sampling. The present analytical method applies equally well to temporally nonhomogeneous input currents and is expected to be useful for computational studies of information processing in various nervous system centers.     

Aperiodic stochastic resonance in neural information processing with Gaussian colored noise

The aim of this paper is to explore the phenomenon of aperiodic stochastic resonance in neural systems with colored noise. For nonlinear dynamical systems driven by Gaussian colored noise, we prove that the stochastic sample trajectory can converge to the corresponding deterministic trajectory as noise intensity tends to zero in mean square, under global and local Lipschitz conditions, respectively. Then, following forbidden interval theorem we predict the phenomenon of aperiodic stochastic resonance in bistable and excitable neural systems. Two neuron models are further used to verify the theoretical prediction. Moreover, we disclose the phenomenon of aperiodic stochastic resonance induced by correlation time and this finding suggests that adjusting noise correlation might be a biologically more plausible mechanism in neural signal processing.