Repetitive firing: a quantitative study of feedback in model encoders

Ca2+ requirements for cerebellar long-term synaptic depression: role for a postsynaptic leaky integrator

Photolysis of a caged Ca(2+) compound was used to characterize the dependence of cerebellar long-term synaptic depression (LTD) on postsynaptic Ca(2+) concentration ([Ca(2+)](i)). Elevating [Ca(2+)](i) was sufficient to induce LTD without requiring any of the other signals produced by synaptic activity. A sigmoidal relationship between [Ca(2+)](i) and LTD indicated a highly cooperative triggering of LTD by Ca(2+). The duration of the rise in [Ca(2+)](i) influenced the apparent Ca(2+) affinity of LTD, and this time-dependent behavior could be described by a leaky integrator process with a time constant of 0.6 s. A computational model, based on a positive-feedback cycle that includes protein kinase C and MAP kinase, was capable of simulating these properties of Ca(2+)-triggered LTD. Disrupting this cycle experimentally also produced the predicted changes in the Ca(2+) dependence of LTD. We conclude that LTD arises from a mechanism that integrates postsynaptic Ca(2+) signals and that this integration may be produced by the positive-feedback cycle.

Analysis and simulation of gain control and precision in crayfish visual interneurons

Impulse trains in sustaining and dimming fibers of crayfish optic lobe (in situ) were elicited with sinusoidal extrinsic current and sine-wave illumination. Extrinsic currents and currents derived from postsynaptic potentials (PSPs) were used to compute the time course of the spike train with an adaptive integrate-and-fire model. The neurons exhibit variations in gain and spike timing precision related to the frequency of stimulation. These phenomena are influenced by spike-frequency adaptation and nonlinearities in the PSP. Dimming fibers exhibit relatively strong spike-frequency adaptation and an associated increase in gain with the frequency of sinusoidal extrinsic current and sine-wave illumination. The dimming fiber IPSP promotes spike train rectification, and rectification contributes to spike timing precision. Sustaining fibers exhibit weaker spike-frequency adaptation and the gain of the current-elicited response is less sensitive to stimulus frequency. The sustaining fiber excitatory PSP, however, exhibits a strong frequency-dependent nonlinearity that influences the frequency response. Spike timing precision is a function of stimulus frequency in all cells and it is enhanced by rectification of the discharge and/or resonance. In rectified responses the jitter in spike times is closely related to the variance in the times the membrane potential reaches spike threshold. These gain and spike timing results are well approximated by the simulated responses. Because the nonlinearity of the sustaining fiber PSP entails a high rate of depolarization, the PSP can increase the precision of spike timing by 10- to 100-fold compared with the response to pure sine-wave stimuli. This enhanced precision has implications for crayfish oculomotor reflexes that are driven by sustaining fibers and highly sensitive to impulse timing during transient excitation.

Encoder adaptation modulates the visual responses of crayfish interneurons

The responses of sustaining and dimming fibers were characterized by the time varying firing rates elicited by extrinsic current and flashes of light. These data were simulated by an adaptive integrate-and-fire model. A postimpulse shunt conductance simulated spike-frequency adaptation. The correlation between observed and model current-elicited impulse rates was 0.94-0.98. However, except for a difference in input resistance (both measured and simulated), the voltage to impulse encoders of the two cell groups was similar and exhibited comparable degrees of spike-frequency adaptation (40 to 45%). The encoder model derived from current-elicited responses (with fixed parameters) was used to simulate visual responses elicited by light flashes. These simulations included a synaptic current derived from the time course of the postsynaptic potential (PSP). The sustaining fiber visual response consisted of a large excitatory PSP and high-frequency transient burst that adapted (by approximately 80%) to a low-frequency plateau discharge. The simulations indicated that spike-frequency adaptation had no effect on the transient discharge but reduced the plateau firing rate by approximately 60%. Encoder adaptation enhances the sustaining fiber response to the time derivative of the stimulus. In dimming fibers, the light flash elicits an inhibitory PSP that interrupts the "dark discharge" and an off response following the end of the flash. The simulations indicated that spike-frequency adaptation reduces the firing rate of both the dark discharge and the off response. Thus the model suggests that different effects of encoder adaptation on the two cell types arise from the same encoder mechanisms, but different actions are determined by differences in impulse rate and the time course of the discharge.

Predicting every spike: a model for the responses of visual neurons

In the early visual system, neuronal responses can be extremely precise. Under a wide range of stimuli, cells in the retina and thalamus fire spikes very reproducibly, often with millisecond precision on subsequent stimulus repeats. Here we develop a mathematical description of the firing process that, given the recent visual input, accurately predicts the timing of individual spikes. The formalism is successful in matching the spike trains from retinal ganglion cells in salamander, rabbit, and cat, as well as from lateral geniculate nucleus neurons in cat. It adapts to many different response types, from very precise to highly variable. The accuracy of the model allows a compact description of how these neurons encode the visual stimulus.

Dynamic properties of the action potential encoder in an insect mechanosensory neuron

A variety of sensory receptors show adaptation to dynamic stimuli that can be well characterized as fractional differentiation of the input signal. The cause of this behavior is unknown, but because it can be represented by linear systems theory, it has been assumed to arise during early linear processes of transduction or adaptation, rather than during the nonlinear process of action potential encoding. I measured the action potential encoding properties of an insect mechanoreceptor by direct electrical stimulation of the sensory cell axon and found a dynamic response that is identical to the response given by mechanical stimulation. This indicates that the fractional differentiation is a property of the encoder rather than the transducer.

White noise analysis of pace-maker-response interactions and non-linearities in slowly adapting crayfish stretch receptor

Input-output relations were investigated in the slowly adapting stretch receptor organ of crayfish using a Gaussian white noise length input with a 0.03-12.5 Hz band width and the resulting action potential output. The noise input was presented to the de-efferented receptor in situ, at three mean elongations and at four different amplitudes. The three mean elongations were set within the normal range in vivo, two at the extremes close to the minimum and maximum physiological lengths and the other in the mid-range. With white noise inputs there is a finite probability that the system will be tested in all possible conditions within the chosen band width because white noise has the advantage that it contains, with a finite probability, all possible stimulus wave forms at random. The analysis indicated similarities between the effects of the input variables, namely white noise amplitude and mean elongation. With low input variables the activity was periodic. With larger inputs, impulse rates were higher and irregular. The average length trajectories leading to a spike (i.e. the average stimulus) were either biphasic with high inputs or multiphasic and periodic with lower input variables. The frequency of periodicity increased with mean elongation. Although for a given length and noise amplitude a variety of individual length trajectories preceded spikes, the final biphasic shortening-lengthening average stimulus sequence before a spike was similar in all cases irrespective of the input variables. The number of possible trajectories decreased with increments in the input variables. The standard deviation of length values for each average stimulus was computed and displayed as a function of time relative to the spike. It was first constant, and decreased gradually to a minimum value at the spike reference. Standard deviation values were lower for higher white noise amplitudes and mean elongation. Simple, short-lasting stimulus wave forms in the white noise were isolated and the corresponding responses were computed. Responses were periodic at low input variables and aperiodic with larger inputs and also asymmetric, being larger and briefer for lengthenings than for shortenings. Asymmetries augmented with input increments. Investigation of the interactions between the pace-maker and the white noise effects showed that lengthenings were more effective when they occurred just after a spike while the effectiveness of shortenings was greater just before a spike.(ABSTRACT TRUNCATED AT 400 WORDS)

A model for impulse frequency modulation used in neural encoding

The impulse rate at the output of a neural encoder can be interpreted as the sum of the mean impulse rate plus a noise component. From literature models are known which describe the transient phenomena of the encoder as far as the mean impulse rate is concerned. In this paper in addition the noise phenomenon is treated by a model which is in agreement with results derived from measurements. This model consists of two parts, a multiplicative and an additive estimator. The first one is similar to the automatic gain control system known from literature. This system estimates the amplification of the impulse rate due to the step input of the neural encoder. Multiplying the impulse rate with the inverse of this factor inhibits the change of the impulse rate. The second estimator calculates the step size of the impulse rate which is subtracted from the output of the encoder. Again the change of the impulse rate is inhibited. The comparison of the impulse rates simulated by the model and given by published measurements shows a good agreement for the properties of the mean impulse rate and the variance of the imposed noise.

Adaptive neural encoder model with selfinhibition and threshold control

A general model for an adaptive neural encoder relating the output firing frequency to input stimulus is presented. The model is based on an "integrate and fire at threshold" scheme and includes cumulative inhibitory feedback as well as output rate dependent threshold control.

Excitation properties of the squid axon membrane and model systems with current stimulation. Statistical evaluation and comparison

The space-clamped squid axon membrane and two versions of the Hodgkin-Huxley model (the original, and a strongly adapting version) are subjected to a first order dynamic analysis. Stable, repetitive firing is induced by phase-locking nerve impulses to sinusoidal currents. The entrained impulses are then pulse position modulated by additional, small amplitude perturbation sinusoidal currents with respect to which the frequencies response of impulse density functions are measured. (Impulse density is defined as the number of impulses per unit time of an ensemble of membranes with each membrane subject to the same stimulus). Two categories of dynamic response are observed: one shows clear indications of a corner frequency, the other has the corner frequency obscured by dynamics associated with first order conductance perturbations in the interspike interval. The axon membrane responds with first order perturbations whereas the unmodified Hodgkin-Huxley model does not. Quantitative dynamic signatures suggest that the relaxation times of axonal recovery excitation variables are twice as long as those of the corresponding model variables. A number of other quantitative differences between axon and models, including the values of threshold stimuli are also observed.

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.

The spatiotemporal transfer function of the Limulus lateral eye

The dynamics of the Limulus retina may be well described by the spatiotemporal transfer function, which measures the response of the eye to moving sinusoidal gratings. We consider a model for this system, which incorporates an excitatory generator potential, and self- and lateral inhibitory processes. Procedures are described which allow estimation of parameters for the model consistent with the empirical transfer function data. Transfer functions calculated from the model show good agreement with laboratory measurements, and may be used to predict accurately the response of the eye to arbitrary moving stimuli. The model allows convenient interpretation of the transfer function measurements in terms of physiological processes which underly the response of the Limulus retina.

Repetitive firing: quantitative analysis of encoder behavior of slowly adapting stretch receptor of crayfish and eccentric cell of Limulus

Techniques developed for determining summed encoder feedback in conjunction with the leaky integrator and variable-gamma models for repetitive firing are applied to spike train data obtained from the slowly adapting crustacean stretch receptor and the eccentric cell of Limulus. Input stimuli were intracellularly applied currents. Analysis of data from cells stringently selected by reproducibility criteria gave a consistent picture for the dynamics of repetitive firing. The variable-gamma model with appropriate summed feedback was most accurate for describing encoding behavior of both cell types. The leaky integrator model, while useful for determining summed feedback parameters, was inadequate to account for underlying mechanisms of encoder activity. For the stretch receptor, two summed feedback processes were detected: one had a short time constant; the other, a long one. Appropriate tests indicated that the short time constant effect was from an electrogenic sodium pump, and the same is presumed for the long time constant summed feedback. Both feedbacks show seasonal and/or species variations. Short hyperpolarizing pulses inhibited the feedback from the long time constant process. The eccentric cell also showed two summed feedback processes: one is due to self inhibition, the other is postulated to be a short time constant electrogenic sodium pump similar to that described in the stretch receptor.