Variation Of Interspike Intervals Research Articles

Computation of first passage time moments for stochastic diffusion processes modelling nerve membrane depolarization

For further understanding of neural coding, stochastic variability of interspike intervals has been investigated by both experimental and theoretical neuroscientists. In stochastic neuronal models, the interspike interval corresponds to the time period during which the process imitating the membrane potential reaches a threshold for the first time from a reset depolarization. For neurons belonging to complex networks in the brain, stochastic diffusion processes are often used to approximate the time course of the membrane potential. The interspike interval is then viewed as the first passage time for the employed diffusion process. Due to a lack of analytical solution for the related first passage time problem for most diffusion neuronal models, a numerical integration method, which serves to compute first passage time moments on the basis of the Siegert recursive formula, is presented in this paper. For their neurobiological plausibility, the method here is associated with diffusion processes whose state spaces are restricted to finite intervals, but it can also be applied to other diffusion processes and in other (non-neuronal) contexts. The capability of the method is demonstrated in numerical examples and the relation between the integration step, accuracy of calculation and amount of computing time required is discussed.

Network Amplification of Local Fluctuations Causes High Spike Rate Variability, Fractal Firing Patterns and Oscillatory Local Field Potentials

We investigate a model for neural activity in a two-dimensional sheet of leaky integrate-and-fire neurons with feedback connectivity consisting of local excitation and surround inhibition. Each neuron receives stochastic input from an external source, independent in space and time. As recently suggested by Softky and Koch (1992, 1993), independent stochastic input alone cannot explain the high interspike interval variability exhibited by cortical neurons in behaving monkeys. We show that high variability can be obtained due to the amplification of correlated fluctuations in a recurrent network. Furthermore, the cross-correlation functions have a dual structure, with a sharp peak on top of a much broader hill. This is due to the inhibitory and excitatory feedback connections, which cause “hotspots” of neural activity to form within the network. These localized patterns of excitation appear as clusters or stripes that coalesce, disintegrate, or fluctuate in size while simultaneously moving in a random walk constrained by the interaction with other clusters. The synaptic current impinging upon a single neuron shows large fluctuations at many time scales, leading to a large coefficient of variation (CV) for the interspike interval statistics. The power spectrum associated with single units shows a 1/f decay for small frequencies and is flat at higher frequencies, while the power spectrum of the spiking activity averaged over many cells—equivalent to the local field potential—shows no 1/f decay but a prominent peak around 40 Hz, in agreement with data recorded from cat and monkey cortex (Gray et al. 1990; Eckhorn et al. 1993). Firing rates exhibit self-similarity between 20 and 800 msec, resulting in 1/f-like noise, consistent with the fractal nature of neural spike trains (Teich 1992).

Quantification and statistical verification of neuronal stimulus responses from noisy spike train data

Usually neuronal responses to short-lasting stimuli are displayed as peri-stimulus time histogram. The function estimated by such a histogram allows to obtain informations about stimulus-induced postsynaptic events as long as the interpretation is restricted to the first response component after the stimulus. The interpretation of secondary response components is much more difficult, as they may be either due to stimulus effects or represent an "echo" of the primary response. In the present paper two output functions are developed that do not show such an echoing of responses. The first one, the interspike interval change function, represents an ideal way to quantify a neuronal stimulus response as its amplitude was found to be almost independent of the stimulation strategy used during acquisition of the spike train data. The other function, the displaced impulses function, allows to verify the statistical significance of an observed response component. Both functions may be estimated from stimulus-correlated spike train data, even if the neuron under investigation shows considerable interspike-interval variability in the absence of stimulation. The concepts underlying these neuronal output functions are developed on simulated responses of a Hodgkin-Huxley-type model for a mammalian neuron at body temperature that is exposed to a transient excitatory conductance increase. Additionally, estimation of these output functions is also demonstrated on responses of human soleus motoneurons that were exposed to electrical stimuli of the tibial nerve in the popliteal fossa.

RESPONSE PROFILES OF HUMAN MUSCLE: AFFERENTS DURING ACTIVE FINGER MOVEMENTS

Twenty-five human muscle afferents from the extensor digitorum muscles of the forearm were studied with the microneurographic method. Single unit impulses were recorded while the subjects performed alternating movements of moderate speed at the appropriate metacarpophalangeal joint. For comparison, responses to imposed movements of similar amplitudes and velocities were also studied. Most spindle afferents (n = 17) provided a stretch response with both kinds of movement. However, the impulse rate was slightly higher and the interspike interval variability much larger during active movement. Two units provided deviating response profiles: a flat profile and a converse stretch response. Small and constant torque loads usually failed to modify the response profile but gave rise to a moderate increase of impulse rate in 50% of the spindle afferents. In one single unit, a converse stretch response appeared with opposing loads. Tendon organ afferents (n = 8) were totally unmodulated by imposed stretch in the relaxed muscle. In contrast, their impulse rate was highly modulated during active movements, often following the rectified EMG which resulted in a converse relationship to muscle length and velocity. The findings support the view that, in general, human muscle spindles monitor muscle length and velocity in routine movements of moderate speed as long as opposing loads are small, whereas Golgi tendon organs monitor the amount of muscle recruitment. The significance of the deviating response profiles from spindle afferents remains obscure.

Effects of excitotoxic striatal lesions on single unit activity in globus pallidus and entopeduncular nucleus of the cat

Striatal projections to the globus pallidus and entopeduncular nucleus are thought to be GABAergic and inhibitory. Thus, striatal lesions might be expected to increase the spontaneous discharge rate of neurons in these nuclei. To test this prediction, we recorded spontaneous single unit activity from awake cats sitting quietly before and 7–160 days after striatal lesions. Striatal lesions were produced by injecting ibotenic acid into the caudate nucleus and putamen. Median, standard deviation, mean, and coefficient of variation of the interspike intervals were calculated for each unit. In globus pallidus the striatal lesion resulted in a significant decrease in median interval length, i.e. an increase in the discharge rate. The prelesion median of 36 ms (S.E.M.= 2.3) decreased 11% to a postlesion value of 32 ms (S.E.M.= 2.1.). The lesion also resulted in a significant decrease in the variability of interspike intervals. The coefficient of variation, 1.31 (S.E.M.= 0.08) before the lesion, decreased 25% to 0.97 (S.E.M.= 0.06) after the lesion. In entopeduncular nucleus, the lesion had no statistically significant effect on the rate of activity, but a significant decrease in the variability of activity occurred. The median interval was 33 ms (S.E.M.= 3.3) before the lesion and decreased 2% to 32 ms (S.E.M.= 2.4). The coefficient of variation decreased 48% from 1.44 (S.E.M.= 0.1) to 0.73 (S.E.M.= 0.03). These observations support the hypothesis that loss of GABAergic inputs to the globus pallidus results in disinhibition. The discharge rate in entopeduncular nucleus was not affected by the striatal lesion, suggesting that striatal substance P or subthalamic excitatory inputs may have a role in regulating discharge rate in the entopeduncular nucleus.

A neurophysiological study of prepositus hypoglossi neurons projecting to oculomotor and preoculomotor nuclei in the alert cat

The activity of 62 antidromically identified prepositus hypoglossi neurons was recorded in 10 alert cats during spontaneous, vestibular or visually induced eye movements. Neurons were antidromically activated from stimulating electrodes implanted in the ipsilateral medial longitudinal fasciculus ( n = 24), the ipsilateral interstitial nucleus of Cajal ( n = 6). the ipsilateral parabigeminal nucleus ( n = 2). the contralateral superior colliculus ( n = 6) and the contralateral cerebellar posterior peduncle ( n = 24). Neurons were identified as eye-movement-related when their rate position and/or rate- velocity plots showed correlation coefficients ≥0.6. They were further classified as “position”, “position-velocity” and “velocity-position” according to their relative eye position and velocity coefficients. However, they seemed to be distributed as a continuum in which a progressive decrease of eye velocity sensitivity was accompanied by a proportional increase in eye position sensitivity. “Position velocity” neurons ( n = 9) were mainly horizontal type II neurons projecting to the vicinity of the oculomotor complex; two of these neurons with vertical sensitivity were also activated from the interstitial nucleus of Cajal. Mean position and velocity sensitivity of these neurons were 5.2 spikes/s per degree and 0.62 spikes/s per degree per second, respectively. Pure “position” neurons ( n = 7) also showed activation during ipsilateral eye fixations; their mean position gain was 7.3 spikes/s per degree and they projected to the ipsilateral oculomotor and Cajal nuclei, and to the contralateral superior colliculus. “Velocity position” neurons ( n = 18) were type I or II neurons with rather irregular tonic firing rates and a mean velocity gain of 0.75 spikes/s per degree per second. Type II “velocity position” neurons projected mainly to the oculomotor area, while type I neurons projected preferentially to the cerebellum. A special type of “pause” neuron ( n = 5), with very low firing rate and pausing mainly for contralateral saccades, was activated exclusively from the contralateral posterior peduncle. Many neurons with weak eye movement sensitivity ( n = 22) were activated mainly (73%) from the cerebellum. It can be concluded that the prepositus hyperglossi nucleus distributes specific eye movement related signals to motor and premotor brainstem and cerebellar structures. The variability of interspike intervals of representative prepositus hypoglossi neurons of each class was compared to the discharge variability of identified abducens motoneurons. The evolution of the eye velocity signal from reticular formation burst neurons to abducens motoneurons was also analyzed. Based on these analyses, we conclude that prepositus hypoglossi neurons could be organized in a “cascade” fashion in order to generate eye position signal from the velocity signal present in burst-like reticular formation neurons.

Control from the brainstem of synchrony of discharge between gamma motoneurones in the cat.

An assessment has been made of the effect of partial and complete section of the spinal cord on the discharges of gamma motoneurones to hind limb muscles in the decerebrated cat. The degree to which the discharges of pairs of individual gamma motoneurones exhibit short-term synchrony and the variability in interspike intervals of the discharge in individual neurones was measured. Variability of discharge was assessed as coefficient of variation of interspike intervals and degree of synchronization assessed from cross correlation analysis. The discharges of gamma motoneurones in the decerebrated cat with intact spinal cord are regular (low coefficient of variation) and show no tendency to synchrony for up to 24 h following decerebration. Section of the more medial part of the dorsolateral funiculus, on either side of the thoracic spinal cord, resulted in a less regular discharge and the development of short-term synchrony between gamma motoneurones. A dependence of synchrony on variability of discharge was established. The synchrony and increased variability in the spinal cat persisted for at least 24 h. Both were markedly reduced following administration of the precursors of monoamines (either L-Dopa or 5-HTP). We conclude that a bilateral, monoaminergic pathway descending in the dorsolateral funiculus from the brainstem controls synchrony of gamma motoneurone discharge in the decerebrated cat. The possibility is discussed that synchrony of discharge between alpha motoneurones may be controlled by a similar pathway.

Firing and contractile properties of human lower lip motor units during sustained isometric contractions

Human subjects were taught to generate constant forces with their lips by contracting across a pair of hooks which held the corners of the mouth fixed in place. The force was measured with a strain gauge attached to one of the hooks. The activity of single motor units was recorded during production of small to near maximal levels of force. The recruitment and firing patterns of each unit were observed, and in vivo contractile properties of each unit were estimated using spike-triggered averaging of the high-gain force record. Recruitment of units was observed at all levels of force and recruitment level varied from trial to trial for each unit. Interspike interval variability was high at all levels of force. The twitch tensions of the units were all less than 10 g; the contraction times ranged from 16 to 90 ms. It is hypothesized that the differences between lip and limb motor unit properties in this task reflect the mechanical, anatomic, and neurophysiologic differences between these systems.

Responses of neurons in nucleus supratrigeminalis to sinusoidal jaw movements in the cat

Single neurons in the mesencephalic nucleus of the fifth nerve and in the rostral brain stem of barbiturate-anesthetized cats were studied in terms of their response to sinusoidal jaw movements at different frequencies and amplitudes. Units were quantitatively characterized by magnitude and phase of their firing rate modulation as a function of frequency, the linearity of their response at 5 Hz, and the variability of the interpsike intervals. Units were qualitatively characterized in terms of their response to gentle palpation of jaw muscles. Neurons were found in the nucleus supratrigeminalis that were selectively driven by jaw movement and that responded to gentle palpation to one or more ipsilateral jaw muscles. The magnitude of the sensitivity of these neurons as a function of frequency was not significantly different from cells of the mesencephalic nucleus of the fifth nerve. The phase of the sensitivity was, however, significantly more advanced than were those cells at all but the highest frequencies. The typical interspike interval variability of cells in nucleus supratrigeminalis was about 40% of the mean interval, whereas that variability of neurons of the mesencephalic nucleus of the fifth nerve was only about 10% of the mean interval. Supratrigeminal cells could relay jaw proprioceptive information to the cerebellum or thalamus.

A stochastic afterhyperpolarization model of repetitive activity in vestibular afferents.

A stochastic version of Kernell's (1968, 1972) model with cumulative afterhyperpolarization (AHP) was simulated. A characteristic of the model is that the AHP is the result of an increased potassium conductance (g K) that is time-dependent but not voltage-dependent. Quantal synaptic inputs are assumed to be the only source of interspike interval variability. The model reproduces many features of the steady-state discharge of peripheral vestibular afferents, provided that firing rates are higher than 40 spikes/s. Among the results accounted for are the interspike interval statistics occurring during natural stimulation, their alteration by externally applied galvanic currents and the increase in the interspike interval following an interposed shock. Empirical studies show that some vestibular afferents have a regular spacing of action potentials, others an irregular spacing (Goldberg and Fernández 1971b; Fernández and Goldberg 1976). Irregularly discharging afferents have a higher sensitivity to externally applied galvanic currents than do regular afferents (Goldberg et al. 1984). To explain the relation between galvanic sensitivity and discharge regularity requires the assumption that neurons differ in both their synaptic noise (sigma v) and the slopes of their postspike voltage trajectories (d mu v/dt). The more irregular the neuron's discharge at a given firing frequency, the greater is sigma v and the smaller is d mu v/dt. Of the two factors, d mu v/dt is estimated to be four times more influential in determining discharge regularity across the afferent population. The shortcomings of the model are considered, as are possible remedies. Our conclusions are compared to previous discussions of mechanisms responsible for differences in the discharge regularity of vestibular afferents.

Activity dependence of action potential duration in rat supraoptic neurosecretory neurones recorded in vitro.

Action potential durations, measured at one-third peak amplitude, were examined during intracellular recordings in 134 supraoptic nucleus neurones maintained in vitro in perfused hypothalamic explants. Spike durations ranged between 1.2 and 3.9 ms and were dependent on firing frequency. Shortest measurements (1.74 +/- 0.03 ms; mean +/- S.E. of mean) were obtained during relative quiescence, i.e. less than or equal to 0.5 Hz. A gradual increase in firing frequency through continuous injection of depolarizing current prolonged spike duration, with maximum levels (2.68 +/- 0.05 ms) achieved at 20 Hz. When interspike interval variability was eliminated and firing was more precisely regulated by brief 15-20 ms intracellular current pulses given at pre-determined frequencies, a proportional relationship between increasing spike duration and firing frequency was retained but the change in spike duration at frequencies between 2 and 10 Hz was less pronounced. Once action potentials had achieved the long duration configuration, their return to the shorter duration took place gradually during any succeeding silent interval with a time constant of 4.9 s. Action potential broadening occurred progressively and was most pronounced at the onset of spontaneous or current-induced bursts. In thirty-six phasically active neurones, spike broadening at the onset of a burst was concurrent with the presence of 5-10 consecutive short (less than or equal to 100 ms) interspike intervals; thereafter, despite a greater than 50% reduction in firing frequency, action potential durations remained prolonged throughout the burst. In all of nineteen cells tested, frequency-dependent changes in spike duration were reversibly decreased or blocked by Cd2+, Co2+ and Mn2+, or when CaCl2 was exchanged for equimolar amounts of EGTA in the perfusion medium. These observations indicate that a Ca2+ conductance contributes to frequency- and firing-pattern-dependent changes in spike duration in rat supraoptic nucleus neurones.

Activity of mesencephalic dopamine and non-dopamine neurons across stages of sleep and waking in the rat

Single unit activity of dopamine and non-dopamine neurons in the substantia nigra and ventral tegmental area was recorded across stages of sleep and waking in the rat. These stages consisted of slow wave sleep (SWS), rapid eye movement (REM) sleep, awake-quiet (AQ) and awake-moving (AM). The dopamine neurons showed no change in mean firing rate across the stages of sleep or waking. During REM sleep, however, the dopamine cells fired with a more variable interspike interval than during SWS. In contrast, non-dopamine neurons in the substantia nigra and ventral tegmental area showed large increases in firing rate in REM compared to SWS, and in AM compared to AQ, without showing changes in interspike interval variability. In conclusion, whereas other monoaminergic neurons and various cortical and subcortical neurons exhibit marked changes in firing rate across the stages of sleep and waking, the dopamine neurons are unique in their lack of change in firing rate across stages.

Jaw muscle afferent firing during an isotonic jaw-positioning task in the monkey.

The activity of jaw muscle receptors was studied by recording neurons in the mesencephalic nucleus of the trigeminal nerve in monkeys trained to control the position and movement of their mandible. Jaw position was measured by a weighted lever resting on the mandibular incisors. The force required to maintain the position of the lever was varied; in most cases it was either 25 or 360 g. Firing rates of neurons were related to stationary mandibular positions and to the velocity of movements during intervals when the movement velocity was constant. Of 49 neurons studied in detail, 21 fired at rates that were consistently and linearly related to static incisal openings. This static position sensitivity was typically about 5 spikes/mm of incisal opening. Most position-sensitive neurons fired at higher rates during opening movements and at lower rates during closing movements than would be accounted for by their position sensitivity. This sensitivity to the velocity of movement was not linear, however; slow closing movements sometimes did not produce a decrease in firing rate, and an actual increase during muscle shortening was seen in a few instances. The position sensitivity of eight neurons was evaluated during different loading conditions; in no case did it change substantially. Of the remaining 28 neurons, 26 fired at high rates during all opening movements and either stopped firing or fired at low, sporadic rates during closing movements. The static position sensitivity of these neurons was weak and variable both within and between neurons. The velocity sensitivity of these stretch-sensitive neurons was very nonlinear. Except for a range of slow movements (+/- 5 mm/s), the firing rate was maximal (200 spikes/s or higher) for most opening movements and zero for most closing movements. Maximal firing rates were higher when the loads being moved were increased from 25 to 360 g. The majority of position-sensitive neurons exhibited a large interspike-interval variability at wide incisal opening. In most of these neurons, this interspike-interval variability was periodic, usually at a rate of about 10 periods/s, and took the form of "saw-tooth" modulation on a record of instantaneous firing rate. Neurons that exhibited this modulation in a very prominent form also exhibited, in many instances, a substantial increase in firing rate during closing jaw movements.

The variability of inter-spike intervals of human spindle afferents in relaxed muscles

The impulse discharge in single afferent units was recorded in waking human subjects. Tungsten needle electrodes were percutaneously inserted in the left radial nerve, and activity in muscle spindle afferents from the extensor muscles of the forearm were studied while the subjects remained relaxed. Thirty-six spontaneously discharging single units were studied, 16 primaries, 10 secondaries and 10 non-classified. The regularity of afferent discharge and the shape of the inter-spike interval histograms was assessed by off-line statistical analysis of stationary segments of afferent discharge. The mean discharge rate of both primary and secondary afferents was generally low. Secondaries predominantly fired with low variability, whereas primaries showed high as well as low variability, but the median values for the two groups differed significantly. However, the individual unit values of coefficient of variation showed a considerable overlap between the two groups. The size of the coefficient of variation in the human spindle afferents was between that of de-efferented and of intact afferents in decerebrated cats. Determination of the coefficient of variation may therefore help, as a complementary criterion, to differentiate between primary and secondary afferents in man.

Behavior of a single neuron in a recurrent excitatory loop.

A recurrent excitation loop was constructed by enabling each impulse from the slowly adapting stretch receptor organ SAO (crayfish) to trigger through an electronic circuit a brief stretch, or "tug," of the receptor. When applied independently, each tug influenced the discharge as would an EPSP. Recurrent excitation led to characteristic discharge timings; hence, even an isolated neuron can have intrinsic mechanisms that prevent positive feedback from freezing it in an extreme non-operational state. Such timings depended critically on the "phase", i.e., on the time elapsed between an SAO impulse and the tug. When the control discharge was stationary (because the SAO length remained invariant), phases of a few ms simply changed the pattern to one of doublets, and affected little the average rate. As the phase increased, bursts appeared, bursts and interburst intervals became more prolonged, and average rates increased. With the largest phases examined (40 ms), the discharge consisted of a slow alternation of high rate bursts, separated by long intervals. When the discharge was modulated (by 0.2/s sinusoidal length variation) with recurrent excitation, the peak-to-peak rate swing, i.e., the sensitivity, and the proportion of the cycle without afferent discharges increased, and the rate vs. length display was distorted even though remaining "loop-plus-extension." Changes were phase-dependent: for example, loops could have a sharp high peak at one phase and be flat-topped at another. When the interspike interval variability was exaggerated (by a length jitter superimposed upon either invariant or sinusoidally varying lengths), recurrent excitation exerted fewer, weaker and somewhat different effects: e.g., it reduced the overall intensity of the invariant cases and the peak-to-peak swing in the modulated one. The precise mechanisms of these results can only be conjectured at but are likely to involve an electrogenic pump, electromechanical interactions, topographical issues, as well as their interplays. The functional implications involve, for instance, the modulation of the intensity, duration and occurrence of the bursting patterns in oscillating functions (e.g., breathing, chewing, etc.).

Changes of activity patterns in slow and fast muscles during postnatal development

The motor activity patterns of the slow soleus and the fast extensor digitorum longus (EDL) muscles of the rat were studied by electromyography (EMG) in the awake unrestrained animal during postnatal development. In the adult soleus, tonic, low frequency activity of motor units was recorded at rest and during movement. In animals less than 12 days old the muscle was activated phasically, the motor unit firing frequency was slightly lower than in the adult, and tonic discharges were absent at rest. After this time, tonic postural activity gradually developed and resembled the adult pattern by 3 weeks of age. The adult EDL muscle was silent at rest and was activated phasically at high frequencies during movement. In animals of less than 12 days of age the muscle was also activated phasically but at lower frequencies. The firing frequency increased to adult values during the second and third week of postnatal development. The discharge properties of motor units from soleus and EDL were compared in rats of different ages. There was a greater variability of interspike intervals during firing in motor units from very young animals than in adults. Whereas in adult animals it was possible to discern a graded recruitment of motor units, animals less than 10 days old showed no regular recruitment order and some motor units appeared to be activated synchronously.

Types of activity of skeletal muscle motor units during cold tremor in cats

Discharges of 184 motor units of the sartorius muscle functioning during cold tremor were investigated in acute experiments on anesthetized cats. Units whose discharges correlated with respiration cycles and units functionally independent of the rhythm of respiration were discovered. Discharges of both types of motor units possessed the same mean frequency (4–12 spikes/sec) and the same low variability of interspike intervals. Additional temperature stimulation of the vascular temperature receptors and changes in the frequency and depth of the respiratory excursions during cold tremor evoked identical responses in the two types of units. It is concluded that both types of motor units have similar thresholds and are of the slow, phasic type.

Effects of Whole-Body γ Irradiation on the Activity of Rabbit Hippocampal Neurons

The hippocampal cellular activity was recorded by means of microelectrodes in unanesthetized immobilized rabbits, receiving 450 rad of whole-body -y irradiation. A statistical analysis of the data was performed by a computer program developed to process time interval series. A comparison was made between observations obtained in unirradiated and irradiated rabbits. The following results were obtained: (1) After irradiation the hippocampal cellular activity was highly disturbed. (2) The mean discharge rate was clearly slowed down and the variability of interspike intervals was increased. (3) The background activity of some cells was greatly reduced or almost suppressed. (4) The firing patterns of discharge were different from the control. The patterns which revealed a temporal organization of the action potentials in various types of bursts were more rarely observed. On the other hand, the occurrence of spikes without evident temporal organization was the predominant pattern of firing. (5) These changes set in early and continued for the whole recording session (12 hr after the end of irradiation). Because present knowledge in the various related fields is scanty, the interpretation of these results must remain hypothetical.

Dynamics of a model for the variability of the interspike intervals in a retinal neuron.

Recently, a model has been proposed to explain the statistics of the variability of interspike intervals of ganglion cells in the retina of goldfish under steady-state stimulation. In this note, it is shown that the dynamical behaviour of the model, both under steady-state and dynamical stimuli, is characterized by input-invariance of the output random process after a time transformation. This allows to study the signal-processing properties of the model, and to test its applicability by a more complete experimental analysis.

A model for the variability of interspike intervals during sustained firing of a retinal neuron

The statistics of the variability of interspike intervals of ganglion cells in the retina of goldfish are modeled by assuming the noise in an integrate-and-fire mechanism is proportional to the reciprocal of a normally distributed variable. This model meets the constraint that the coefficient of variation of the interspike. This does not change when the mean firing rate of the neuron changes. Alternative sources of variability of interspike intervals are discussed.