Probability of neuronal spike initiation as a curve-crossing problem for Gaussian stochastic processes

A diverse pattern of the spike threshold changes in feline gastrocnemius-soleus motoneurons during stretch reflex activation

The aim of the study was to define the spike threshold changes in motoneurons during naturally evoked activation. Spike activity was evoked in the gastrocnemius-soleus motoneurons of decerebrate cats by controlled stretches of the homonymous muscles. The spike thresholds were defined by using the first derivative of the membrane potential and statistical boundaries of its change. The evoked firing was analyzed in terms of the spike thresholds and the membrane potential trajectories between spikes, which were dependent on the intensity of the firing. The main inference of the study is the absence of a single-valued dependence between the spike rates and thresholds; the last ones can both increase and decrease with a rise in the firing rate; moreover, opposite directions in the threshold changes are often observed at various phases of the stretch-evoked firing. The dependency of the spike thresholds on firing rates was checked in 42 of the total set of 57 motoneurons. In a group of cells (n = 27) showing statistically significant correlation (P < 0.05), this parameter was negative in 18 and positive in 9 motoneurons. The obtained results indicate that the threshold-crossing models with a single-valued dependency of the spike thresholds on the firing rates are not completely suitable to analyze the spike generation processes in the motor unit records.

Fluctuations in the zeros of differentiable Gaussian processes

The stochastic point processes formed by the zero crossings or extremal points of differentiable, stationary Gaussian processes are studied as a function of their autocorrelation function. The properties of these point processes are mapped to the space formed by the parameters appearing in the autocorrelation function, their adopted form being sensitive to the structure of the autocorrelation function principally in the vicinity of the origin. The distribution for the number of zeros occurring in an asymptotically large interval are approximately negative-binomial or binomial depending upon whether the relative variance or Fano factor is greater or less than unity. The correlation properties of the zeros are such that they are repelled from each other or are "antibunched" if the autocorrelation function of the Gaussian process is characterized by a single scale size, but occur in clusters if more than one characteristic scale size is present. The intervals between zeros can be interpreted in terms of the autocorrelation function of the zeros themselves. When bunching occurs the interval density becomes bimodal, indicating the interval sizes within and between the clusters. The interevent periods are statistically dependent on one another with densities whose asymptotic behavior is governed by that of the autocorrelation function of the Gaussian process at large delay times. Poisson distributed fluctuations of the zeros occur only exceptionally but never form a Poisson process.

On some computational results for single neurons' activity modeling

The classical Ornstein-Uhlenbeck diffusion neuronal model is generalized by inclusion of a time-dependent input whose strength exponentially decreases in time. The behavior of the membrane potential is consequently seen to be modeled by a process whose mean and covariance classify, it as Gaussian-Markov. The effect of the input on the neuron's firing characteristics is investigated by comparing the firing probability densities and distributions for such a process with the corresponding ones of the Ornstein-Uhlenbeck model. All numerical results are obtained by implementation of a recently developed computational method.

Relationship between the time course of the afterhyperpolarization and discharge variability in cat spinal motoneurones

1. We elicited repetitive discharges in cat spinal motoneurones by injecting noisy current waveforms through a microelectrode to study the relationship between the time course of the motoneurone's afterhyperpolarization (AHP) and the variability in its spike discharge. Interspike interval histograms were used to estimate the interval death rate, which is a measure of the instantaneous probability of spike occurrence as a function of the time since the preceding spike. It had been previously proposed that the death rate can be used to estimate the AHP trajectory. We tested the accuracy of this estimate by comparing the AHP trajectory predicted from discharge statistics to the measured AHP trajectory of the motoneurone. 2. The discharge statistics of noise-driven cat motoneurones shared a number of features with those previously reported for voluntarily activated human motoneurones. At low discharge rates, the interspike interval histograms were often positively skewed with an exponential tail. The standard deviation of the interspike intervals increased with the mean interval, and the plots of standard deviation versus the mean interspike interval generally showed an upward bend, the onset of which was related to the motoneurone's AHP duration. 3. The AHP trajectories predicted from the interval death rates were generally smaller in amplitude (i.e. less hyperpolarized) than the measured AHP trajectories. This discrepancy may result from the fact that spike threshold varies during the interspike interval, so that the distance to threshold at a given time depends upon both the membrane trajectory and the spike threshold trajectory. Nonetheless, since the interval death rate is likely to reflect the instantaneous distance to threshold during the interspike interval, it provides a functionally relevant measure of fluctuations in motoneurone excitability during repetitive discharge.

On a non-Markov neuronal model and its approximations

Single neuron's activity modeling is considered with reference to some earlier contributions in which a non-Markov Gaussian process is assumed to describe the time course of the neuron's membrane potential. After re-formulating the problem in a rigorous framework and pinpointing the limits of validity of such a model, the available results on the firing probability density are compared with those obtained by us by means of an ad hoc numerical algorithm implemented for the leaky integrator diffusion firing model and with some data constructed by a simulation procedure of non-Markov Gaussian processes with pre-assigned covariances. Throughout this paper, the notion of 'correlation time' plays a fundamental role for the neuronal coding process modeling.

Effects of neurochemically excited group III-IV muscle afferents on motoneuron afterhyperpolarization

When humans voluntarily and maximally contract a muscle under isometric conditions, the average firing rate of motor units decreases from an initially high value over several tens of seconds. The mechanisms underlying the rate reduction are probably manifold. One mechanism could involve changes in the motoneuron afterhyperpolarization, another reflex effects of group III-IV muscle afferents that are excited during developing muscle fatigue. It appears possible that changes in motoneuron afterhyperpolarization are mediated by these afferent inputs. We therefore studied effects on motoneuron afterhyperpolarization of small-diameter muscle afferents excited by intra-arterially injected metabolites such as bradykinin and serotonin. In decerebrate and mostly spinalized cats, lumbosacral alpha-motoneurons were recorded intracellularly. Current pulses were injected to test for input resistance and elicit action potentials and afterhyperpolarizations. Afterhyperpolarizations were averaged from c. 10 successive stimulus repetitions. Measurements were taken of afterhyperpolarization amplitude, half-width and area; and exponential functions were fitted to the afterhyperpolarization decay phase to determine afterhyperpolarization decay time-constants. In selected cases, the entire afterhyperpolarization trajectory was fitted with a sum of two exponentials to assess more precisely changes in afterhyperpolarization trajectory. Small catheters were inserted into side-branches of the sural artery and the accompanying vein to apply substances like bradykinin, serotonin and KCl to the calf muscles. Concentrations were in the range of those used by other workers. Intra-arterial injection of bradykinin and serotonin usually decreased blood pressure, which may at times have affected mean motoneuron membrane potentials. Afterhyperpolarization amplitude usually changed with membrane potential in a way expected from ensuing changes in driving potential. Whenever excitation of group III-IV muscle afferents caused moderate to strong increases in motoneuron synaptic noise, afterhyperpolarization amplitudes were reduced, usually in parallel to decreases in input resistance. Afterhyperpolarization half-widths were mostly unaffected, but occasionally decreased. There was a significant trend for afterhyperpolarization decay time-constants to increase during increased synaptic noise, this increase being inversely correlated with the reduction in afterhyperpolarization amplitude. The reduction in input resistance was associated with a decrease in the membrane time-constant, which could therefore not account for the prolongation of the afterhyperpolarization decay time-constant. The afterhyperpolarization area decreased, indicating that the reduction of afterhyperpolarization amplitude outweighed the prolongation of afterhyperpolarization decay time-constant. During a prolonged fatiguing muscle contraction group III-IV afferents become increasingly excited, produce augmenting synaptic inputs in motoneurons, and will change afterhyperpolarization properties. On average, these changes per se tend to diminish the effect of afterhyperpolarization on motoneuron discharge.

Discharge patterns of tonically firing human motoneurones

We have attempted to reconcile the different patterns of distribution of interspike intervals that are found in motoneurones made to discharge by intracellular injection of constant current in reduced animal preparations and by voluntary control in human subjects. We recorded long spike trains from single motor units in three human muscles made to discharge at constant mean frequencies with the help of auditory and visual feedback. The distribution of interspike intervals in each spike train was analysed quantitatively. We found that the different pattern of discharge of the human motor units could be accounted for when due allowance was made for the variability of the drive to the human motoneurone which arose because of the feedback process used to maintain the target frequency. A model testing this hypothesis gave results that were qualitatively consistent with the human data.

Identification and estimation algorithm for stochastic neural system

An algorithm for the estimation of stochastic processes in a neural system is presented. This process is defined here as the continuous stochastic process reflecting the dynamics of the neural system which has some inputs and generates output spike trains. The algorithm proposed here is to identify the system parameters and then estimate the stochastic process called neural system process here. These procedures carried out on the basis of the output spike trains which are supposed to be the data observed in the randomly missing way by the threshold time function in the neural system. The algorithm is constructed with the well-known Kalman filters and realizes the estimation of the neural system process by cooperating with the algorithm for the parameter estimation of the threshold time function presented previously (Nakao et al., 1983). The performance of the algorithm is examined by applying it to the various spike trains simulated by some artificial models and also to the neural spike trains recorded in cat's optic tract fibers. The results in these applications are thought to prove the effectiveness of the algorithm proposed here to some extent. Such attempts, we think, will serve to improve the characterizing and modelling techniques of the stochastic neural systems.

Parameter estimation of the threshold time function in the neural system

An algorithm for parameter estimation is presented for the neural system model. Because of its firing mechanism analogous to that of the model based on the first time crossing problem, this problem is solved numerically for our model according to the results of Kostyukov et al. (1981). We propose the algorithm that estimates the parameters of the model considering the equivalence between the probability density function of the 1st crossing time and that of the interspike interval, which is derived from the interspike interval histogram by making use of the spline function technique. The ability of the algorithm is ensured by the application to the simulated interspike interval data. The parameter estimation is carried out also for the practical neural data recorded in the cat's optic tract fibers in both the spontaneous and the stimulated cases. These applications will show the effectiveness of the algorithm in practical cases.