Phase-locked responses in the Limulus lateral eye. Theoretical and experimental investigation

In vivo conditions induce faithful encoding of stimuli by reducing nonlinear synchronization in vestibular sensory neurons

Previous studies have shown that neurons within the vestibular nuclei (VN) can faithfully encode the time course of sensory input through changes in firing rate in vivo. However, studies performed in vitro have shown that these same VN neurons often display nonlinear synchronization (i.e. phase locking) in their spiking activity to the local maxima of sensory input, thereby severely limiting their capacity for faithful encoding of said input through changes in firing rate. We investigated this apparent discrepancy by studying the effects of in vivo conditions on VN neuron activity in vitro using a simple, physiologically based, model of cellular dynamics. We found that membrane potential oscillations were evoked both in response to step and zap current injection for a wide range of channel conductance values. These oscillations gave rise to a resonance in the spiking activity that causes synchronization to sinusoidal current injection at frequencies below 25 Hz. We hypothesized that the apparent discrepancy between VN response dynamics measured in in vitro conditions (i.e., consistent with our modeling results) and the dynamics measured in vivo conditions could be explained by an increase in trial-to-trial variability under in vivo vs. in vitro conditions. Accordingly, we mimicked more physiologically realistic conditions in our model by introducing a noise current to match the levels of resting discharge variability seen in vivo as quantified by the coefficient of variation (CV). While low noise intensities corresponding to CV values in the range 0.04-0.24 only eliminated synchronization for low (<8 Hz) frequency stimulation but not high (>12 Hz) frequency stimulation, higher noise intensities corresponding to CV values in the range 0.5-0.7 almost completely eliminated synchronization for all frequencies. Our results thus predict that, under natural (i.e. in vivo) conditions, the vestibular system uses increased variability to promote fidelity of encoding by single neurons. This prediction can be tested experimentally in vitro.

Mode locking in a periodically forced resonate-and-fire neuron model

The resonate-and-fire (RF) model is a spiking neuron model which from a dynamical systems perspective is a piecewise smooth system (impact oscillator). We analyze the response of the RF neuron oscillator to periodic stimuli by expressing the firing events in terms of an implicit one-dimensional time map. Based on such a firing map, we describe mode-locked solutions and their stability, leading to the so-called Arnol'd tongues. The boundaries of these tongues correspond to either local bifurcations of the firing time map or grazing bifurcations of the discontinuity of the flow. Despite the fact that the periodically driven RF system shows periodic firing, its behavior may become chaotic when the forcing frequency is near the resonant frequency. We compare these results to numerical simulations of the model undergoing sinusoidal forcing. Furthermore, upon varying a system parameter, the RF system can be reduced to the integrate-and-fire system and in this case we show the consistency of the results on mode-locked solutions.

Muscarinic receptor activation tunes mouse stratum oriens interneurones to amplify spike reliability

Cholinergic activation of hippocampal targets can initiate and sustain network oscillations in vivo and in vitro, yet the impact of cholinergic modulation on the oscillatory properties of interneurones remains virtually unexplored. Using whole cell current clamp recordings in acute hippocampal slices, we investigated the influence of muscarinic receptor (mAChR) activation on the oscillatory properties of CA1 stratum oriens (SO) interneurones in vitro. In response to suprathreshold oscillatory input, mAChR activation increased spike reliability and precision, and extended the bandwidth that interneurone firing phase-locked. These suprathreshold effects were largest at theta frequencies, indicating that mAChR activation tunes active conductances to enhance firing reliability and precision to theta frequency input. Muscarinic tuning of the intrinsic oscillatory properties of interneurones is a novel mechanism that may be crucial for the genesis of the theta rhythm.

A Network of Electronic Neural Oscillators Reproduces the Dynamics of the Periodically Forced Pyloric Pacemaker Group

Low-dimensional oscillators are a valuable model for the neuronal activity of isolated neurons. When coupled, the self-sustained oscillations of individual free oscillators are replaced by a collective network dynamics. Here, dynamical features of such a network, consisting of three electronic implementations of the Hindmarsh-Rose mathematical model of bursting neurons, are compared to those of a biological neural motor system, specifically the pyloric CPG of the crustacean stomatogastric nervous system. We demonstrate that the network of electronic neurons exhibits realistic synchronized bursting behavior comparable to the biological system. Dynamical properties were analyzed by injecting sinusoidal currents into one of the oscillators. The temporal bursting structure of the electronic neurons in response to periodic stimulation is shown to bear a remarkable resemblance to that observed in the corresponding biological network. These findings provide strong evidence that coupled nonlinear oscillators realistically reproduce the network dynamics experimentally observed in assemblies of several neurons.

Reliability of spike timing is a general property of spiking model neurons

The responses of neurons to time-varying injected currents are reproducible on a trial-by-trial basis in vitro, but when a constant current is injected, small variances in interspike intervals across trials add up, eventually leading to a high variance in spike timing. It is unclear whether this difference is due to the nature of the input currents or the intrinsic properties of the neurons. Neuron responses can fail to be reproducible in two ways: dynamical noise can accumulate over time and lead to a desynchronization over trials, or several stable responses can exist, depending on the initial condition. Here we show, through simulations and theoretical considerations, that for a general class of spiking neuron models, which includes, in particular, the leaky integrate-and-fire model as well as nonlinear spiking models, aperiodic currents, contrary to periodic currents, induce reproducible responses, which are stable under noise, change in initial conditions and deterministic perturbations of the input. We provide a theoretical explanation for aperiodic currents that cross the threshold.

Periodically forced leaky integrate-and-fire model

The discharge pattern of periodically forced leaky integrate-and-fire models is studied. While previous analyses have been mainly concerned with the response of this model to sinusoidal stimulation, our results hold for arbitrary periodic inputs. It is shown that, for any periodic input, the map representing the relation between input phases at consecutive discharge times can be restricted to a piecewise continuous, orientation preserving circle map. This implies that (i) the rotation number is well defined and independent of the initial condition, and (ii) in the same way as for sinusoidal forcing, other forms of periodic stimuli can evoke only one of four types of response, namely, phase locking, quasiperiodic discharges, nonchaotic aperiodic firing, and termination of the discharge after a finite number of firings.

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.

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.

Dynamics of the neural discharge in snail neurons

Spike trains recorded under weak sinusoidal driving from central neurons of Lymnaea stagnalis appear quite irregular and envisage the possibility of an underlying chaotic process. Therefore, the sequences of interspike intervals are analyzed in the framework of non-linear dynamics. Since, for several reasons, these sequences are rather short, the analysis is performed by using methods of non-linear forecasting. To reject the null hypothesis that the original time series is a realization of a linear stochastic process with the same autocorrelation function, the results obtained on the original data are compared with those from surrogate data sets. Some 'non-linear' predictability occurs only in narrow regions of the space of stimulus parameters and the frequency of perturbation is critical in determining it. Moreover, it is shown that such behavior can be qualitatively mimicked by the FitzHugh-Nagumo model driven by a weak sinusoidal signal plus noise. It is argued that the narrowness of the non-linear predictability regions renders quite unlikely the detection of deterministic dynamics in the activity of these neurons.

Coding of odor intensity

A model for coding of odor intensity in the first two neuronal layers of olfactory systems is proposed. First, the occupation and activation by odorant molecules of receptor proteins of different types borne by the first order neurons are described as birth and death processes. The occupation (birth) rate depends on the concentration of the odorant, whereas the probability of activation of an occupied receptor depends on the type of the odorant. Second, the spike generation mechanism proposed for the first order neuron depends on the level of the generator potential evoked by the activated receptors and on a time-decaying threshold which is reset to infinity after each spike. The various resulting stochastic regimes of firing activity at different concentrations are described. Third, each second order neuron is influenced by excitation coming from numerous first order neurons, lateral inhibition from other second order neurons, and self-inhibition. All these incoming signals are integrated at the second order neuron. The firing activity of the first and second order neurons is modeled by a first passage time scheme. For both types of neuron the shapes of the curves predicted by the model for the mean firing frequency as a function of stimulus concentration are shown to be in accordance with available experimental results.

The effect of a random initial value in neural first-passage-time models

The effect of a random initial value is examined in several stochastic integrate-and-fire neural models with a constant threshold and a constant input. The three models considered are approximations of Stein's model, namely: (1) a leaky integrator with deterministic trajectories, (2) a Wiener process with drift, and (3) an Ornstein-Uhlenbeck process. For model 1, different distributions for the initial value lead to commonly observed interspike interval distributions. For model 2, a discrete and a uniform distribution for the initial value are examined along with some parameter estimation procedures. For model 3, with a truncated normal distribution for the initial value, the coefficient of variation is shown to be greater than 1, and as the threshold becomes large the first-passage-time distribution approaches an exponential distribution. The relationships among the models and between them and previous models are also discussed, along with the robustness of the model assumptions and methods of their verification. The effects of a random initial value are found to be most pronounced at high firing rates.

Effects of current pulses on the sustained discharge of visual cells of Limulus

Current pulses were used in the eccentric and retinular cells of the Limulus lateral eye to produce changes in the interspike interval of the discharge sustained by a constant light level. The effects on the interspike interval of hyperpolarizing and depolarizing perturbations, applied at various delays from the previous spike, were measured for different intensities and durations of the current pulse. The results show that when the perturbations were applied in the first part of the interval, effects contrary to what is normal were produced (i.e, hyperpolarizing pulses decreased the interspike interval instead of increasing it and vice versa for depolarizing pulses). Here we discuss briefly the implications on neural encoding models.

On approximations of Stein's neuronal model

Stein's model represents a commonly-used description of spontaneous neuronal activity. Substituting Stein's model by the Ornstein-Uhlenbeck diffusion process increases the model tractability. A diffusion approximation of Stein's model is summarized in the present paper. It is proved that the cumulative distribution functions of interspike intervals under Stein's model converge to the cumulative distribution function of interspike intervals which are generated in accordance with the limiting Ornstein-Uhlenbeck diffusion model. The approach used allows us to determine to what extent Stein's model modifications and generalizations affect the possibility of diffusion approximation. It can be seen that non-diffusion approximations exist and they are also studied here. The results achieved can be considered as complementary to the numerical study published recently.

Relevance of the single ommatidium performance in determining the oscillatory response of the Limulus retina

The response of a healthy lateral eye of Limulus to constant, uniform and sufficiently intense light stimulation, consists of a sustained oscillatory discharge, all ommatidia firing synchronously in bursts, at intervals of about 0.2s (Barlow and Fraioli, 1978). This response has been analysed by a computer simulation, where the performance of the single unit is described by encoder models of the integrate-and-fire type, already extensively investigated. The results obtained show that the occurrence and the time features of the oscillatory response depend on the neural models adopted.

Resonant response of a neural model and of Limulus ommatidia to double frequency stimulation

Signals which are the sum of two sinusoids of frequencies v1 and v2 are used to stimulate: i) an electronic analog of the leaky integrator neural model, ii) the visual neurons of the Limulus lateral eye. This makes it possible to investigate the resonant amplification of the impulse density modulation for v1 + v2 which approaches the free-run discharge rate; this resonance is predicted by the Volterra series representation of the leaky integrator (Poggio and Torre, 1977). The resonant responses obtained look very similar for the simulated discharge and for the experimental one.