Repetitive firing behavior of mammalian muscle spindle

Changes in primary muscle spindle ending excitability induced by a ramp-and-hold stretch

36 primary (Ia) muscle spindle afferents from the tibial anterior muscle of the cat were subjected to a ramp-and-hold stretch (stretch rate 10 mm/s, stretch amplitude 5 or 8.5 mm) of the muscle, on which a sinusoidal stretch (50 Hz) of four different amplitudes (25, 50, 250 and 500 microns) was superimposed. In 54 discharge patterns a Ia afferent subjected to a ramp-and-hold stretch with a sinusoidal stretch superimposed responded only to the superimposed sinusoidal stretch. In 25 of the cases the Ia afferent responded with an one-to-one driven action potential (AP) and in 29 of the cases with two APs per sinusoidal stretch. For these 54 discharge patterns the phase of the sinusoidal cycle was determined at which each AP occurred. Where the Ia afferent responded with one AP per cycle an accelerating phase advance was observed during the ramp stage of the underlying ramp-and-hold stretch and a decelerating phase advance during the plateau. This phase shift means that the excitability of the site generating the AP increased during the ramp stage and decreased during the plateau. If the Ia afferent responded with two APs per superimposed cycle, the second AP per cycle evinced a decelerating phase advance during the ramp and an accelerating phase advance during the plateau. The phase of the second AP per cycle showed a second, contrary change in excitability at the AP generating site. The excitability decreased during the ramp and increased during the plateau. The first kind of excitability change is interpreted as a consequence of an inward current at the AP generating site. The second, contrary type of excitability points to an interplay between an inward and an outward current. An increasing outward current lowers the excitability for the second AP per cycle during the ramp. A decreasing outward current raises the excitability during the plateau.

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.

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.

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.

Temporal summation in rat prefrontal pyramidal cells. Differential effects of pre- and postsynaptic neurochemical manipulations

The influence of both reserpine-induced depletion of catecholamines and chlorpromazine-induced blockade of dopamine receptors on integrative properties of rats prefrontal pyramidal cells was studied by evocating threshold direct cortical responses with different trains of electrical pulses. Catecholamine depletion results in increased pulse train stimulating currents for eliciting threshold cortical responses, whereas chlorpromazine blockade of dopamine receptors results in higher time constants characterizing postsynaptic temporal summation. It is suggested that the leaky-integrator neuron model may be a discriminating paradigm for detecting electrophysiologically the pre- or postsynaptic level of occurrence of some synaptic disorders.

Effects of muscle model parameter dispersion and multi-loop segmental interaction on the neuromuscular system performance

The effects of parameter dispersion among motor units on the neuromuscular system performance as well as interaction between muscle segments and spinal cord mechanisms are investigated. Elementary components of the system are modeled to simulate with simple models their input-output characteristics. A leaky SS-IPFM encoder with a time-dependent threshold simulates the motor-neuron encoding characteristics. An amplitude and time dependent nonlinear model represent the motor unit mechanical output to neuronal input relationship. The dispersion of parameters in the components of the whole muscle control model is investigated in the open loop mode. It is shown that the dispersion of parameters in the multi-efferent channels converging on a common tendon provides a spatial filtration generating a smoother muscle force in addition to extending the linear dynamic range compared to a similar system having identical motor units. Muscle segmental interaction is investigated in this distributed model by closing the loop through a coupling matrix, representing afferent-motorneuron interaction on the spinal cord level. A diagonal matrix represents no segmental interaction and a uniform matrix represents a uniform interaction between segments through the muscle spindles and Golgi tendon feedback elements. The close loop simulation studied shows that (a). The type of segmental interaction has little effect on the overall system performance, i.e., range of linerity and stability, which is the result of having a muscle system with a large number of motor units. (b) There are only minor differences in results between the uniform and normal parameter distributions tested. (c) A loop gain of 4 divided by 8 in the distributed model can provide linearity through the full physiological force range. (d) Type of segmental interaction has significant effects on the individual segment. A uniform matrix provides a more stable segment due to the spatial filtration resulting from the segmental interaction, while the diagonal noninteracting matrix shows instabilities on the local segmental level despite global stability. The more realistic exponentially decaying spatial interaction matrix yields both global neuromuscular and local segmental stability with the same linear dynamic range generated with the uniform or diagonal matrices.

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.

Input-output relationship of the Leaky-integrator neuron model

This paper presents a method of calculating the spike sequence at the output of the Leaky-Integrator Neuron Model (LIM) in response to an arbitrary input stimulus. The calculations have revealed new properties of the initial transient behavior of the LIM, as well as new constraints upon necessary and sufficient conditions for the appearance of spikes with a fixed phase relation to a periodic input. It is also possible to infer what knowledge about the input stimulus can be obtained from a temporal sequence of spikes at the output of the LIM. In the Discussion, neuronal examples are considered which do not encode the transmitted information as a spike rate, but rather monitor the time of occurence of individual spikes by comparison with a reference signal. This relatively common case is not adequately treated by previous descriptions based on system theory; ways are suggested by which the formalism developed here can be used to describe completely, and understand more fully, the performance of such systems.

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

The 1:1 phase locking of the neural discharge to sinusoidally modulated stimuli was investigated both theoretically and experimentally. On the theoretical side, a neural encoder model, the self-inhibited leaky integrator, was considered, and the phase of the locked impulse was computed for each frequency in the locking range by imposing the condition that the "leaky integral" u(t) of the driving signal should reach the threshold for the first time one stimulus period after the preceding impulse. As u(t) can be a nonmonotonic function, this approach leads to results that sometimes differ from those reported in the literature. It turns out that the phase excursion is often much smaller than the values of about 180 degrees predicted from previous analysis. Moreover, our analysis shows a peculiar effect; the phase locking frequency range narrows when the input modulation depth increases. The theoretical predictions are then compared with phase-locked discharge patterns recorded from visual cells of the Limulus lateral eye, stimulated by sinusoidally modulated light or depolarizing current. The phases of the locked spikes at each of a number of modulation frequencies have been measured. The predictions offered by the model fit the experimental data, although there are some difficulties in determining the effective driving signal.

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.

Repetitive firing: a quantitative study of feedback in model encoders

Recognition of nonlinearities in the neuronal encoding of repetitive spike trains has generated a number of models to explain this behavior. Here we develop the mathematics and a set of tests for two such models: the leaky integrator and the variable-gamma model. Both of these are nearly sufficient to explain the dynamic behavior of a number of repetitively firing, sensory neurons. Model parameters can be related to possible underlying basic mechanisms. Summed and nonsummed, spike-locked negative feedback are examined in conjunction with the models. Transfer functions are formulated to predict responses to steady state, steps, and sinusoidally varying stimuli in which output data are the times of spike-train events only. An electrical analog model for the leaky integrator is tested to verify predicted responses. Curve fitting and parameter variation techniques are explored for the purpose of extracting basic model parameters from spike train data. Sinusoidal analysis of spike trains appear to be a very accurate method for determining spike-locked feedback parameters, and it is to a large extent a model independent method that may also be applied to neuronal responses.

Analysis of the dynamic responses of deefferented primary muscle spindle endings to ramp stretch

A detailed analysis of the complex dynamic response of 8 deefferented primary muscle spindle endings to ramp stretches of the extensor digitorum longus muscle (EDL) was made in anaesthetized cats. The analysis was based on Lennerstand's linear muscle spindle model, in which the dynamic peak, i.e. the peak frequency at the end of the dynamic phase of a ramp stretch, is assumed to consist of three components: a position response, a slow velocity response, and a quick velocity response. The components of the dynamic peaks analysed were fairly well correlated with the static behaviour of the primary muscle spindle endings, the latter, in turn, being regarded as an indicator of the spindles' location in the non-homogeneous extrafusal muscle. The results thus provide a satisfactory explanation of the high correlations between the static and gross dynamic behaviour of deefferented primary muscle spindle endings of the EDL previously reported from our laboratory.