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

Abstract

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.

The irregular nature of neuronal discharge led to the development of a number of stochastic models of neuronal input-output behaviour (reviewed in Tuckwell, 1988). These models were in turn used to deduce the properties of a neurone and its synaptic input from records of its discharge (e.g. Tuckwell & Richter, 1978; Smith & Goldberg, 1986). Since the steady-state, repetitive discharge properties of motoneurones have been shown to be closely related to the magnitude and time course of the afterhyperpolarization (AHP) following an action potential (cf. Kernell, 1983), the discharge statistics of voluntarily activated human motoneurones have been used to infer the characteristics of their AHPs (Tokizane & Shimazu, 1964; Person & Kudina, 1972; Matthews, 1996; Piotrkiewicz, 1999). Specifically, if the distribution of interspike intervals (ISIs) of motor unit discharge is obtained at a number of different firing rates, the relative variance is generally low at high firing rates and increases at lower firing rates (Tokizane & Shimazu, 1964; Clamann, 1969; Person & Kudina, 1972; Piotrkiewicz, 1999). As a consequence, a plot of the standard deviation of ISIs as a function of the mean interval exhibits an upward bend, and the ISI at which this upward bend occurs has been taken as an indication of the duration of the AHP (Person & Kudina, 1972; Piotrkiewicz, 1999).

Matthews (1996) has recently proposed a method of analysing ISI histograms of human motoneurone discharge that reveals not only the duration of a motoneurone's AHP, but also much of its earlier time course. This estimate of the AHP time course depends on a calculation of the instantaneous probability of a spike occurring as a function of time since the preceding spike. This function has been variously termed the hazard function, the conditional probability, or the interval death rate (Moore et al. 1966; Matthews, 1996), and can be obtained directly from the probability density function (as estimated from the ISI histogram). Matthews (1996) predicted that the time course of the AHP of a voluntarily activated human motoneurone can be determined by calculating the death rate from its ISI histogram, and then converting the death rate to an estimate of the distance to threshold as a function of time since the last spike.

To test the validity of the estimate of the AHP using the analysis described by Matthews (1996), we set out to determine the actual relationship between synaptic noise, AHP time course and ISI statistics in cat lumbar motoneurones. We found that although the predicted AHP trajectories were often close to the directly measured AHPs, in most cases the amplitude of the predicted AHP was less than that of the measured AHP. We attribute this discrepancy to the systematic variation in spike threshold during the ISI (cf. Calvin & Stevens, 1968; Calvin, 1974; Powers & Binder, 1996), which confounds any estimate of the AHP magnitude from the interval statistics alone. However, since the interval death rate is likely to reflect the instantaneous distance to threshold during the ISI, it should provide a functionally relevant measure of fluctuations in motoneurone excitability during repetitive discharge. A preliminary account of some of these results has been presented (Powers et al. 1997).

METHODS

Experimental preparation

Intracellular recordings from lumbar motoneurones were obtained from 10 adult cats. The experiments were carried out in accordance with the animal welfare guidelines in place at the University of Washington School of Medicine. Anaesthesia was induced with an intraperitoneal injection of pentobarbitone sodium (40 mg kg⁻¹) and maintained at a deep level throughout the surgical and experimental procedures by supplementary intravenous doses (1–4 mg kg⁻¹). Following a conventional laminectomy from L4 to S1 and dissection of either the left sciatic nerve at the hip or the nerves to the left medial gastrocnemius and lateral gastrocnemius-soleus muscles, the animals were mounted in a rigid spinal cord recording frame, paralysed with gallamine triethiodide and mechanically ventilated. Subsequently, the depth of anaesthesia was adjusted to maintain the mean blood pressure (monitored with a cannula in the carotid artery) below 120 mm Hg and to minimize the level of synaptic noise in the intracellular recordings. The depth of anaesthesia was also checked by testing for withdrawal reflexes during periods of recovery from paralysis.

Potassium sulfate- or potassium chloride-filled microelectrodes were driven into the spinal cord to penetrate motoneurones, which were identified by antidromic activation from muscle or mixed nerves. Only motoneurones with stable resting potentials greater than −60 mV and action potentials with positive overshoots were studied. At the conclusion of the experiments, the animals were killed with a lethal dose of pentobaritone. Additional details of our animal maintenance and recording protocols can be found in a number of recent publications from this laboratory (Powers & Binder, 1996; Poliakov et al. 1996, 1997).

Experimental protocol

Upon successful impalement of a motoneurone, we first recorded a series of antidromic spikes, followed by a series of spikes directly elicited by 1 ms suprathreshold injected current pulses. In most cells, these direct spikes were elicited not only at rest but also at different membrane potentials produced by 0.5–1 s injected current steps. Input resistance was determined from the relationship between membrane potential and the amplitude of the injected current step, and the relationship between the amplitude of the postspike afterhyperpolarization (AHP) and membrane potential was used to estimate the reversal potential of the AHP conductance (cf. Viana et al. 1993). We also measured the motoneurone rheobase (I_rh; i.e. the current level of a 50 ms injected current step needed to elicit a spike ∼50 % of the time) and the slope of the steady-state frequency-current (f–I) relationship as determined from the mean firing rate obtained over the last 0.5 s of a series of 1 s current steps of different amplitude (Powers & Binder, 1996).

We recorded the responses of the motoneurone to a series of 42 s injected current waveforms consisting of the following components: (1) a series of 12, 1 ms, −10 nA pulses delivered at 12.5 pulses s⁻¹, (2) a 1–5 s delay, (3) a 34–38 s injected current step, (4) a 26.2 s random noise waveform superimposed on the current step, (5) a 1 s delay, and (6) a second series of hyperpolarizing current pulses. In a subset of the motoneurones (one of which is illustrated in Fig. 1A), we used a modified version of this waveform that included two series of eight, suprathreshold, 1 ms depolarizing pulses delivered at 4 pulses s⁻¹ preceding and following the noise waveform. The random noise and hyperpolarizing current pulse components of the waveform were stored as computer files and reproduced by a D/A converter with a sampling rate of 10 kHz. Three different types of random noise waveform were used and we stored four different versions of each type as computer files. The first type of noise was generated with a recursive algorithm designed to produce a signal approximating white noise (cf. Poliakov et al. 1997). The resultant current waveform had a flat frequency spectrum up to about 500 Hz, a mean amplitude of zero and a standard deviation of 2.5 nA. The second type of noise waveform was obtained by scaling the first waveform by a factor of two. The third type of noise waveform was a digitally filtered version of the second waveform, resulting in a stimulus more like that produced by real synaptic noise. (We used a Butterworth filter with a frequency cut-off of 159.2 Hz, and a time constant of 1 ms.)

Figure 1.

Measurement of the intrinsic properties of motoneurones and the characteristics of their noise-driven discharge

A, response of a motoneurone (upper trace) to an injected current waveform (lower trace) composed of: (1) a series of hyperpolarizing pulses, (2) a variable delay, (3) a long current step just subthreshold for repetitive discharge, (4) a 26.2 s zero mean random noise component, (5) another delay and (6) a second set of hyperpolarizing current pulses. Before and after the noise component a series of eight single spikes were evoked by 1 ms suprathreshold current pulses in order to measure the average postspike AHP (see C). B, average voltage decay following the offset of the hyperpolarizing current pulses (dots), with the best double exponential fit to the response (continuous line). The background membrane potential was subtracted from the response and the result multiplied by −1. C, average AHP following spikes evoked at a membrane potential just subthreshold for repetitive discharge. (The mean background membrane potential has been subtracted from the response.) AHPs were characterized in terms of their peak amplitude, total duration (time from the spike onset to the point at which the membrane potential returned to within 2 standard deviations of the mean background membrane potential) and duration at half-amplitude. D, values of consecutive ISIs during the noise-driven discharge shown in A. The dots correspond to individual ISIs, whereas the continuous line represents the mean of 10 consecutive intervals. This running mean was used to subdivide the ISI record into portions in which the running mean was between 70 and 90 ms (S1; lower and middle horizontal dashed lines) and 90 and 110 ms (S2; middle and upper horizontal dashed lines).

The entire injected current waveform was synthesized by summing the D/A output of the computer with one or more outputs of a programmable pulse generator. The motoneurone voltage responses to the waveforms were sampled at 10 kHz and stored on the computer. The current and membrane potential data were also recorded on videotape using a pulse code modulation (PCM) digitizing unit.

In order to accumulate a sufficient number of motoneurone spikes to yield reasonable estimates of interval death rates (see below), we generally applied from 8 to 16 different injected current waveforms to each motoneurone. The magnitude of the step component was varied from trial to trial to depolarize the membrane to just below the threshold for eliciting repetitive discharge in the absence of superimposed noise. Whenever possible, further sets of responses were obtained at different levels of mean depolarization, to generate different mean firing rates. In some of the motoneurones, a shorter (10 s) version of the injected current waveform was used, and the mean level of injected current was varied in a systematic fashion from trial to trial. Motoneurone I_rh and antidromic spike height were remeasured at periodic intervals throughout the recording period to assess the quality of the impalement. Our recordings were terminated when the quality of impalement had deteriorated significantly, as indicated by a decrease in the resting potential and spike height and an increase in spike width. At this point the electrode was withdrawn from the cell and the extracellular potential was recorded.

Data analysis

The computer files containing the membrane potential responses were analysed off-line. The responses to the initial set of stimuli were used to determine the input resistance, I_rh and steady-state f–I relation. Two different values of input resistance were calculated: (1) R_N,max, based on the slope of the relation between the magnitude of the injected current and the peak voltage response of the motoneurone, and (2) R_N,ss, based on the slope of the relation between the magnitude of the injected current and the steady-state voltage response (cf. Binder et al. 1998). The R_N,max value corresponds most closely to that typically reported (based on the responses to 50–100 ms current steps, e.g. Zengel et al. 1985), and the ratio of I_rh to R_N,max was used to classify the motoneurones according to putative type: slow (S: I_rh/R_N,max < 5.6), fast fatigue-resistant (FR, 5.6 < I_rh/R_N,max < 18) and fast fatigable (FF: I_rh/R_N,max > 18; cf. Zengel et al. 1985). The responses of the motoneurone to the series of 1 s suprathreshold current steps were used to determine the slope of its steady-state f–I relation, and also to determine the minimum steady discharge rate in the absence of noise.

The responses of the motoneurone to the long, composite current waveforms (cf. Fig. 1A) were used to determine the passive impulse response of the motoneurone, its average AHP trajectory and the ISIs obtained in response to the combination of the injected current step and the superimposed noise. The average voltage response to the 1 ms, −10 nA hyperpolarizing current pulses was used to calculate the passive impulse response of the motoneurone. After subtracting the background membrane potential, the result was multiplied by −1 and a double exponential fit (curve-fitting routine of Igor Pro; WaveMetrics, Inc., Lake Oswego, OR, USA) was obtained to the voltage trajectory following the offset of the current pulse (Fig. 1B). The passive impulse response was obtained by normalizing the amplitude of each exponential component to that expected for a 1 nA pulse with a width corresponding to our sampling interval of 0.1 ms (cf. D'Aguanno et al. 1986). Following the onset of the long, injected current step, but prior to the onset of the noise component, there were generally several action potentials either in response to background synaptic noise or in response to the 1 ms suprathreshold current pulses. Several of these spikes were averaged to obtain the AHP trajectory (Fig. 1C). We measured the duration of the AHP, its peak amplitude, and its duration at half-amplitude, as indicated in Fig. 1C and described in the figure legend. When the noise component was added to the injected current step, the motoneurone responded with a series of discharges. After the membrane potential records were corrected for electrode capacitance artifacts (Poliakov et al. 1996), the time of occurrence of each spike was determined from the point at which the spike first crossed a specified threshold level in the positive-going direction.

Figure 1D illustrates the series of ISIs obtained from the response of a motoneurone to an injected current step with superimposed noise. The dots represent successive ISIs derived from the motoneurone spike train illustrated in Fig. 1A (epoch 4). The continuous line shows the running mean ISI, calculated as the average of 10 intervals (5 preceding and 5 following a given interval). As was commonly observed, the mean ISI gradually increased during the 26 s discharge record. To remove the contribution of slow drifts in firing rate to ISI variability, the running mean interval was used to subdivide the ISI records into segments in which the mean interval fell within certain limits (cf. Tokizane & Shimazu, 1965; Matthews, 1996). In the example illustrated in Fig. 1D, the ISIs were divided into two populations: those whose running mean fell between 70 and 90 ms (S1, lower and middle horizontal dashed lines) and those whose running mean fell between 90 and 110 ms (S2, middle and upper dashed lines). By ‘slicing’ the ISI records from a number of trials in a similar fashion and combining subpopulations with similar mean intervals, it was possible to obtain a sufficient number of intervals (generally > 1000) to characterize the ISI statistics with a reasonable degree of precision.

Each aggregate set of ISIs was used to compile an ISI histogram and to compute the first four moments of the ISI distribution (i.e. mean, standard deviation, skew and kurtosis). A Kolmogorov-Smirnoff test was used to determine whether the measured histogram differed significantly from a normal distribution with the same mean and standard deviation. The ISI histogram represents an estimate of the underlying probability density function, f(t), and a number of equivalent functions can be derived from f(t), including the cumulative distribution function, F(t) = ∫f(t)dt, and the hazard function (also called the conditional probability function or the death rate), Φ(t) = f(t)/(1—F(t)) (Moore et al. 1966; Matthews, 1996). The interval death rate represents the instantaneous probability of spike occurrence as a function of time since the last spike. When finite bin widths are used for the ISI histogram, the death rate can be approximated by Φ(i) = (ln(N₀/N₁))/bin width, where N₁ is the sum of all the spikes occurring in the bins greater than i and N₀ is the same value plus the number of spikes in bin i (cf. Matthews, 1996). As the time since the previous spike increases, the estimate of the death rate becomes increasingly noisy (cf. Fig. 4B and D) because a decreasing proportion of spikes occur at long intervals, resulting in increasing bin-to-bin fluctuations in spike counts, and thus in the ratio N₀/N₁. To improve the reliability of our estimates of death rates, we generally resorted to using large bin widths (5 ms) and relaxing the criteria for ‘slicing’ the ISI records. To ensure that these procedures did not seriously distort the time course of the death rate function, we generally compared the form of the death rate functions obtained from a given set of ISI records using a number of different ‘slicing’ criteria and bin widths.

Figure 4.

ISI histograms and interval death rates

A, ISI histograms for two different motoneurones firing at about the same mean rate. The thick line represents the ISI histogram from a motoneurone with a short AHP, whereas the thin line is the ISI histogram from a motoneurone with a long AHP. The curved dashed line is an exponential fit to the tail of the ISI histogram for the short AHP motoneurone. B, interval death rates for the ISI histograms in A. The interval death rate for the motoneurone with the long AHP (○) is closely approximated by the interval death rate expected for a normally distributed process with the same mean and standard deviation (curved dotted line). In contrast, the interval death rate for the short AHP motoneurone (•) clearly deviates from the normal prediction (curved continuous line). Instead, the function reaches a constant level (dashed horizontal line) corresponding to the rate of exponential decay of the tail of the ISI histogram. C, ISI histograms for the same motoneurone firing at two different mean rates (11.4 impulses s⁻¹, thin line; 13.8 impulses s⁻¹, thick line). Both histograms decay exponentially at long intervals. D, corresponding interval death rates (○, low firing rate; •, high firing rate). See text for further details.

Figure 2 illustrates how the passive impulse response, the injected current noise and the death rate for a particular motoneurone were used to predict its AHP trajectory. The first step in this process was the calculation of the membrane voltage noise produced by the injected current. Because of the resistive and capacitative electrode artifacts, the noise-induced fluctuations in membrane potential were not measured directly but were estimated instead by convolving the 26.2 s noise component of the injected current waveform with the estimated passive impulse response of the motoneurone. This voltage noise waveform was then used as an input to a computer routine that compared each successive voltage value to a specified threshold value and noted the times at which the voltage exceeded the threshold. To ensure a sufficient number of long intervals, this threshold-crossing algorithm included a 10 ms refractory period. Figure 2A is a schematic representation of a portion of the algorithm's ‘spike’ output for the same voltage noise record and two different values of threshold. The upper spike train was obtained by setting the spike threshold (dashed line) to 1.5 mV above the mean level of the voltage noise, whereas the lower train was obtained with a threshold value of 1 mV (continuous thick line). Each of these spike trains was used to estimate the death rate as a function of time since the preceding spike. Figure 2B shows that following the 10 ms refractory period, the death rate quickly attained a constant level. Figure 2C plots the relation between the mean death rate and the distance to threshold for a series of different threshold levels. As described by Matthews (1996), a smooth curve (the sum of two exponentials) was fitted to these points and this fitted relation was used to predict the distance from threshold as function of time since the last spike from the ISIs obtained from a particular motoneurone at a given mean firing rate.

Figure 2.

Determination of the relation between distance to threshold and interval death rate

A, voltage noise (bottom trace) is used as the input to a threshold detector with a 10 ms refractory period. The top trace illustrates the output of the threshold detector when the threshold was set to a value that was 1.5 mV above the mean noise level (dashed horizontal line), whereas the second trace from the top is the output for a threshold level of 1 mV (continuous horizontal line). B, interval death rates calculated from the spike output of the threshold detector for a threshold level of 1 mV (upper trace) and 1.5 mV (lower trace). The mean death rates are indicated by the horizontal lines. C, relation between the mean distance to threshold and the mean interval death rate (•). The continuous line is the best double exponential fit to the points.

RESULTS

Motoneurone properties

The following analyses of the relation between the characteristics of motoneurone AHPs and the statistical properties of the ISIs recorded during noise-driven repetitive discharge are based on intracellular recordings from 20 cat lumbar motoneurones. The AHP durations of these motoneurones ranged from 46 to 239 ms (mean ± s.d., 90 ± 30 ms). The mean input resistance (R_N,max) of our motoneurone sample was 1.41 ± 0.43 MΩ (range, 0.71- 2.14 MΩ) and the mean I_rh was 7.5 ± 4.3 nA (range, 1.9–16.6 nA). The rheobase values are lower and the input resistance values are higher than those previously reported by ourselves (e.g. Binder et al. 1998) and others (Zengel et al. 1985), indicating that our sample was biased towards low threshold motoneurones. This view is also supported by the provisional motor unit classification (see Methods): 11/20 motoneurones were classified as type S, eight as type FR and only one as type FF. This sampling bias was in fact fortuitous, since our goal was to replicate experimental results on the discharge behaviour of voluntarily activated human motoneurones, in which a similar sampling bias exists (Binder et al. 1996).

All of these 20 motoneurones fired repetitively in response to 1 s injected current pulses. As the magnitude of the injected current step was increased above the rheobase value, most motoneurones responded with only a few spikes at the onset of the current step, until a current level was reached that elicited repetitive discharge throughout the duration of the current step. As previously reported (Kernell, 1965), this minimum steady discharge rate (f_min) was linearly related to the reciprocal of the AHP duration (f_min = 1.77 + 0.91/AHP duration, r = 0.774, P < 0.0001).

Characteristics of noise-driven motoneurone discharge

After determining the minimum steady discharge rate in the absence of noise, the amplitude of the current step was set at a value between rheobase and that required for steady repetitive discharge. When various random noise waveforms were superimposed on longer current steps of this magnitude, the motoneurones responded with a series of discharges whose statistical properties were similar to those obtained from records of voluntarily activated human motoneurones. As described in Methods, a large set of ISIs (mean number of intervals, 2340 ± 1690; range, 494–8014) was obtained at about the same mean firing rate by accumulating spikes from a number of trials, and then subdividing the trials according to the value of the running mean of the ISIs. Figure 3A shows the relation between the standard deviation and the mean ISI for 44 sets of ISIs obtained in this fashion from the 20 motoneurones that we studied. Although there is a fair degree of scatter in the relationship, the value of the standard deviation is relatively low for short ISIs and appears to rise at a greater than linear rate with increasing ISI, as previously reported for human motor unit discharge (e.g. Tokizane & Shimazu, 1964; Clamman, 1969; Person & Kudina, 1972). Part of the scatter in the relation illustrated in Fig. 3A results from pooling data obtained with different noise inputs (i.e. quasi-white noise of two different amplitudes and filtered noise, see Methods). However, Fig. 3B shows that a similar relation holds when only the responses to filtered noise are included.

Figure 3.

Relations between the standard deviation and the mean ISI

A, standard deviation versus mean ISI for interval sets obtained from a series of responses to 26.2 s injected current noise waveforms; •, data from motoneurones with short AHPs (half-duration < 50 ms); ○, data from motoneurones with long AHPs (half-duration > 50 ms). B, same as A, except that only interval data obtained from responses to filtered current noise are shown. C, standard deviation versus mean ISI for two individual motoneurones, obtained from a series of responses to 10 s of filtered current noise with different mean current levels. The AHP durations of the two motoneurones are indicated by the arrows on the time axis (data from one motoneurone indicated by filled circles and arrowhead, the other by open circles and arrowhead).

The relation between the standard deviation and mean ISI differed in motoneurones with different AHP durations. Because of the exponential time course of the latter part of the AHP (see Fig. 1B and C), we used the duration of the AHP at half its maximum amplitude (i.e. half-width) to divide motoneurones into those with long AHPs (half-duration > 50 ms) and those with short AHPs (half-duration < 50 ms). Motoneurones with short AHPs (Fig. 3, •) exhibited increases in discharge variability at lower mean intervals than did motoneurones with long AHPs (Fig. 3, ○).

Most of the points in Fig. 3A and B were derived from a series of trials over which the amplitude of the current step was only varied slightly. In six motoneurones, we examined the relation between the standard deviation and mean ISI in a more systematic fashion by superimposing filtered noise on shorter (10 s) current steps, and by varying the amplitude of the current step over a wide range in different trials. Figure 3C illustrates the results from two of these motoneurones: one with a long AHP (○; AHP duration, 107 ms; half-width, 68 ms), and one with a short AHP (•; AHP duration, 71 ms; half-width, 34 ms). In both motoneurones, the relation between standard deviation and mean interval is characterized by a relatively flat region over which standard deviation is less than 15 ms and shows relatively little dependence on the value of the mean interval, followed by a rapid increase in standard deviation with further increases in mean ISI. The straight lines in Fig. 3C illustrate linear fits to the two different portions of the standard deviation versus mean ISI relation for each motoneurone. The mean interval at which the slope of the relation changes is related to the AHP duration of each cell (indicated by arrows on the x-axis).

The standard deviation versus mean ISI plots for the two motoneurones of Fig. 3C and for the four other motoneurones studied in the same fashion provide partial support for the recent suggestion that these plots can be used to infer the AHP duration (Piotrkiewicz, 1999). In the illustrated examples, the ‘break point’ in the standard deviation versus mean ISI plot occurs at a slightly shorter interval than the AHP duration, as suggested by the simulation results of Piotrkiewicz (1999). Although similar results were obtained in the other four cells, the break point was not always as distinct as in the illustrated examples, suggesting that these standard deviation versus mean ISI plots are likely to provide a relatively crude estimate of AHP duration. However, the results for these six motoneurones do support the suggestion (Piotrkiewicz, 1999) that the ISI variability over the initial (short-interval) portion of the standard deviation versus mean ISI relation tends to be lower for motoneurones with short AHPs. This latter phenomenon is thought to result from the fact that the membrane potential rises more steeply towards threshold in motoneurones with short AHPs, so that a given amount of membrane variability about the mean trajectory gives rise to a smaller range of threshold crossings than in motoneurones with long, slowly rising AHPs. In fact, if spike threshold were constant and both the rate of rise of membrane potential and the standard deviation of the membrane potential fluctuation were to remain constant over a range of firing rates, the standard deviation of the ISIs should also be constant over this range (Stein, 1967; Kostyukov et al. 1981; Piotrkiewicz, 1999). However, in all six motoneurones, the standard deviation showed a significant linear dependence on the mean ISI even over the short-interval portion of the relation, suggesting that one or more of the conditions required for a constant standard deviation does not hold.

ISI histograms and interval death rates

ISI histograms and interval death rates were calculated from large sets of ISIs obtained by pooling data collected from several long epochs of repetitive firing. As reported by Matthews (1996), the shape of these functions could be very different for two different motoneurones firing at about the same mean rate, and this difference in shape can be related to the underlying AHP. Figure 4A shows the ISI histograms obtained from two motoneurones firing at about 13 impulses s⁻¹. The thin line represents the ISI histogram for a motoneurone with an AHP duration of 239 ms, whereas the thick line is the histogram for a motoneurone with an AHP duration of 70 ms. The distribution of ISIs for the motoneurone with the long AHP is symmetrical (skew, 0.06) and does not differ significantly from a normal distribution with the same mean and standard deviation. In contrast, the ISI distribution for the short AHP motoneurone is positively skewed (skew, 1.84) and differs significantly from a normal distribution (Kolmogorov-Smirnoff test, P < 0.01). As described in Methods, the ISI distribution can be used to estimate the interval death rate, which represents the instantaneous probability of an interval being terminated as a function of time from the last spike. Figure 4B shows the death rates estimated from the two histograms in Fig. 4A (long AHP motoneurone, ○; short AHP motoneurone, •). The death rates expected for normal distributions with the same mean and standard deviations as those of the corresponding ISI distributions are also plotted (long AHP motoneurone, dotted line; short AHP motoneurone, continuous line). For the motoneurone with the long AHP, the probability of an interval being terminated increases monotonically from the time of the previous spike, as is characteristic of a normally distributed process. In contrast, the death rate for the motoneurone with the short AHP reaches a constant value at around 70–80 ms, indicating that after this time, the instantaneous probability of a spike being triggered is constant. This constant death rate corresponds to an exponential rate of decay of the ISI histogram. The dashed line in Fig. 4A is an exponential fit to the tail of the ISI histogram for the short AHP motoneurone, and the horizontal dashed line in Fig. 4B is the constant death rate corresponding to the rate of exponential decay. Although the death rate estimates for both motoneurones become increasingly variable at long intervals, it is clear that death rate continues to rise for the long AHP motoneurone and is close to that predicted for a normally distributed process. In contrast, for the short AHP motoneurone, the death rate clearly deviates from the normal prediction (P < 0.01; Kolmogorov-Smirnoff test) and varies about a constant level.

The shapes of the ISI distributions also show characteristic changes for an individual motoneurone firing at two different mean rates. Figure 4C and D illustrate ISI histograms and death rates for a single motoneurone (AHP duration, 72 ms) firing at 11.4 impulses s⁻¹ (thin line in C, ○ in D) and 13.8 impulses s⁻¹ (thick line in C, • in D). The ISI histograms are positively skewed in both cases, although the skew is slightly greater at the lower rate (1.03 versus 0.72). Neither histogram deviated significantly from that predicted for a normally distributed process with the same mean and standard deviation. However, Fig. 4D shows that at both mean rates, the death rate functions clearly deviate from those predicted for a normal distribution (P < 0.01; Kolmogorov-Smirnoff test). The death rate functions tend to approach a constant level that can be predicted from exponential fits to the tails of the ISI histograms. The death rate estimates deviate from the normal prediction at post-spike times of around 80–100 ms, which is longer than the AHP duration for this motoneurone (72 ms), although the variability of the estimates at long intervals makes the exact point of departure difficult to predict.

The ISI histograms shown in Fig. 4C are quite representative. Although most (37/44) of the ISI histograms did not differ significantly from a normal distribution, nearly all (41/44) were positively skewed (mean skew, 0.85 ± 0.95; range, −0.2 to 4.5). The relative amount of skew was related to the difference between the mean ISI and the AHP duration, as shown in Fig. 5. For trials in which the motoneurone was firing with a mean interval that was 10 ms or more shorter than the AHP duration, the ISIs were normally distributed with skew values < 1 in all but one case. In contrast, when the motoneurones were firing with mean intervals near the AHP duration or longer, most of the interval distributions (11/16) exhibited skew values > 1, and 7/16 distributions differed significantly from a normal distribution. These results support the idea that when motoneurones are firing at relatively low rates, many of the spikes are triggered at intervals longer than the AHP duration, when the mean motoneurone membrane potential is a constant distance from the spike threshold and the interval death rate reaches a constant value (Matthews, 1996). This condition results in positively skewed interval histograms, and death rate functions that clearly deviate from those predicted for normal distributions.

Figure 5.

Relation between the degree of skew of the ISI histograms and the difference between the mean ISI and the AHP duration

The data represent ISI histograms that did (○) or did not (•) deviate significantly from a normal distribution.

Prediction of AHP trajectories from ISI distributions

If it is assumed that motoneurone spikes occur whenever the membrane potential crosses a specified threshold value, then the instantaneous probability of spike occurrence (i.e. the death rate) should be a monotonic function of the average distance between the membrane potential and threshold. There is no general analytical solution for the relation between the distance to threshold and death rate, as it depends on both the amplitude and the frequency content of the membrane voltage noise (cf. Kostyukov et al. 1981). However, it is possible to predict this relation by measuring the responses of a simple threshold detector (Matthews, 1996). As described in Methods, the predicted relation between the distance to threshold and the interval death rate is derived by first estimating the membrane potential noise based on a convolution of the injected current noise with the impulse response of the motoneurone. This estimated membrane potential noise is then used as the input to a threshold detector. Finally, the spike output for different constant distances between threshold and the mean noise level is measured.

Figure 6A illustrates the relations between the distance to threshold and the death rate obtained in our sample of motoneurones. The distance to threshold has been normalized in terms of the standard deviation of the noise, so that the differences in the curves represent the effects of the frequency content of the voltage noise, which is in turn a function of both the frequency content of the injected current noise and the filtering characteristics of the motoneurone. The thick continuous lines in Fig. 6A represent the relations obtained for filtered noise and the thin continous lines represent those obtained for unfiltered noise. For a given normalized distance to threshold, the death rate is lower when the membrane potential fluctuations are slower, since the longer temporal correlations between adjacent membrane potential values ensure that the membrane potential will cross the threshold less frequently within a given time interval. The effects of the frequency content of the current noise are most marked when the membrane potential is at intermediate distances from threshold (about 0.5–1.5 standard deviations). This effect is illustrated more clearly in Fig. 6B, which shows the distance from threshold versus the interval death rate obtained in the same motoneurone for filtered (thick line) and unfiltered (thin line) current noise. The two curves overlap when the membrane potential is more than 2 standard deviations from threshold and converge again when the mean membrane potential is slightly above threshold. Similar results were obtained in two other motoneurones in which both filtered and unfiltered noise were applied.

Figure 6.

Characteristics of voltage noise and its effects on the relation between distance to threshold and interval death rate

A, distance to threshold versus interval death rate relations for all of the motoneurones. Thin lines represent the predicted relations for unfiltered current noise, whereas thick lines correspond to the relations for filtered noise. Dashed line is the ‘4 ms transform’ used by Matthews (1996). B, relations for a single motoneurone subjected to both filtered (thick line) and unfiltered (thin line) current noise. C, autocorrelograms of voltage noise produced in the motoneurone of B by filtered (thick line) and unfiltered (thin line) current noise. The dotted line is the autocorrelogram of the background synaptic noise recorded in the same cell. The dashed line represents the voltage noise produced by an RC filter with a time constant of 5 ms in response to a white noise input. D, power spectra of the different voltage noise waveforms. (Different line types have the same meaning as in C.)

Since the spike-triggering efficacy of voltage noise clearly depends upon its temporal structure (cf. Poliakov et al. 1996), it is important to determine how closely our simulated voltage noise resembles actual synaptic noise. Figure 6C and D illustrates the temporal structure of several different voltage noise waveforms in both the time (Fig. 6C) and frequency domains (Fig. 6D). The continuous lines in Fig. 6C are autocorrelograms of the voltage noise obtained by convolving filtered (thick line) and unfiltered (thin line) current noise with the impulse response of the motoneurone of Fig. 6B. The dotted curve in Fig. 6C is the autocorrelogram of the actual synaptic noise recorded in the same motoneurone prior to current injection. Figure 6D shows the power spectra of the same waveforms. The relative amount of low frequency (< 100 Hz) power is similar in the simulated voltage noise waveforms and the actual synaptic noise. However, the power in the voltage noise produced by the filtered current noise falls off more steeply with increasing frequency than does the actual synaptic noise, whereas the power of the voltage noise produced by the unfiltered current noise falls off less steeply. This suggests that the temporal characteristics of the voltage fluctuations produced by our two types of current waveform ‘bracket’ those of actual synaptic noise. The characteristics of the real synaptic noise are more closely approximated by the output of a single RC filter with a time constant of 5 ms (dashed lines in Figs 6C and D).

The dashed line in Fig. 6A is the membrane potential versus death rate relation used by Matthews (1996) to predict the AHP trajectories in human motoneurones. It was obtained by using the voltage noise output of a single RC filter with a time constant of 4 ms, which was chosen to mimic the characteristics of synaptic noise in cat motoneurones reported by Calvin & Stevens (1968). (This membrane potential versus death rate relation was derived from simulations using 1 ms sampling intervals, and is similar to the relation obtained using a smaller sampling interval and a 5 ms time constant.) Since the output of an RC filter with a 5 ms time constant resembles actual synaptic noise, the predictions based on Matthews’ (1996)‘4 ms transform’ are quite reasonable.

For a given voltage noise input, the predicted distance to threshold versus death rate relations can be used together with the measured death rate functions to predict the distance to threshold as a function of time from the preceding spike. Figure 7A and B shows ISI histograms and interval death rates for two motoneurones firing at about the same mean rate. Although the mean ISIs are nearly identical (89.5 and 90.3 ms for the ISI histograms in Fig. 7A and B, respectively), the shapes of the ISI histograms are different, as can be seen more clearly from the interval death rates (connected, filled circles in Fig. 7A and B). The ISI histogram in Fig. 7B exhibits a greater positive skew than that in Fig. 7A, and it has a longer exponential tail, which results in a longer plateau in the death rate function. The open circles in Fig. 7E and F represent the predicted distance to threshold as a function of time from the preceding spike based on the death rate functions in Fig. 7A and B, and the relations between the distance to threshold and the death rate derived for each motoneurone as described earlier. The motoneurone of Fig. 7A had shorter membrane time constants than that of Fig. 7B (3.44 and 0.63 ms versus 7.39 and 0.80 ms, respectively), so the estimated membrane voltage noise had a higher frequency content. As discussed earlier, more rapid fluctuations in membrane voltage allow more spikes to be triggered at a greater distance from threshold. As a result, the predicted distance to threshold for the motoneurone in Fig. 7A is greater than that for the motoneurone in Fig. 7B for the same death rate value.

Figure 7.

Predicted and measured AHP trajectories for two different motoneurones firing at about the same mean rate

A and B, ISI histograms (thin lines) and interval death rates (•) for the two motoneurones. C and D, average AHPs for the two motoneurones compiled from a series of spikes elicited before (thin traces) and after (thick traces) each application of the noise waveform. E and F, predicted (○) and measured (•) AHP trajectories. See text for further details.

Figure 7C and D shows the average AHPs of the two motoneurones evoked by suprathreshold current pulses applied before (thin traces) and after (thick traces) the application of the noise waveform. Since the spikes are evoked at the same mean level of depolarization as the noise-evoked spikes (see Methods), the average AHPs should provide a reasonable estimate of the average interspike membrane trajectory during the noise-evoked discharge. If spike threshold were constant during the ISI, then the distance to threshold should reflect this interspike membrane trajectory. The filled circles in Fig. 7E and F represent the average of the pre- and postnoise AHPs shown in Fig. 7C and D. In order to be comparable to the predicted distance to threshold, which is based on interval death rates calculated in 5 ms time bins, each filled circle in Fig. 7E and F represents the membrane potential averaged over 5 ms. The entire curve was then shifted vertically so that the plateaus in the predicted distance to threshold curves and the average membrane potential trajectories were aligned. For the motoneurone shown in Fig. 7E, the predicted distance to threshold and the average interspike membrane trajectory are nearly identical, suggesting that in this case the assumption of a constant spike threshold is reasonable. Thus, the predicted distance to threshold gives an accurate indication of the AHP time course. After the AHP is completed, the predicted membrane potential lies at a constant distance below threshold (1.45 mV, indicated by the double-headed arrow in Fig. 7E). In 31 of the 44 sets of interval records, the death rate function exhibited a clear plateau, indicating that a significant fraction of the spikes were evoked after the mean membrane potential had settled to a constant distance from threshold. The estimated distance to threshold following the AHP was on average −1.10 ± 0.65 mV (range, −2.56 to +0.54 mV). Normalized to the amplitude of the membrane noise, this value represented an average of −0.83 ± 0.44 noise standard deviations from threshold (range, −1.40 to +0.4). These parameters are roughly comparable to those estimated for voluntarily activated human motoneurones firing at low rates (Matthews, 1996), although the amplitude of the membrane noise fluctuations in the present experiments is somewhat smaller than that estimated for human motoneurones.

In most cases, however, the predicted membrane potential trajectory did not superimpose upon the measured AHP trajectory, but instead generated a shallower trajectory. Figure 7F shows a typical example of the cases in which the predicted and measured trajectories differed significantly. At postspike times longer than 110 ms, both the predicted and measured trajectories are flat, and the two curves necessarily superimpose, since the AHP trajectory has been shifted vertically to have the same final level as the predicted trajectory. However, the two curves diverge by an increasing amount at shorter times from the previous spike. At postspike times of 40 ms and less, the predicted AHP trajectory is undefined, since there were no ISIs < 40 ms. At postspike times between 40 and 60 ms, the predicted AHP is defined but relatively unreliable since it is based on a small proportion of the total number of ISIs (see ISI histogram in Fig. 7B). A conservative estimate of the difference between the predicted and measured membrane trajectories was obtained by measuring the difference at a single time point corresponding to the 25th percentile of the cumulative interval histogram. At this point there was relatively little noise in either the ISI histogram or the interval death rate, and the predicted and measured AHP trajectories also showed a clear divergence. The single-headed arrow in Fig. 7E and the left-most double-headed arrow in Fig. 7F show the points at which the differences between the predicted and measured AHP trajectories were obtained. For the AHP trajectories in Fig. 7E this difference was 0 mV, whereas a difference of 1 mV was obtained for the AHP trajectories in Fig. 7F. For the 31 sets of spike trains for which the predicted and measured AHP trajectories were compared, the mean difference at the 25th percentile point was 1.1 ± 1.2 mV (range, −0.3 to +6.0 mV). As illustrated in Fig. 7F, the maximum difference between the predicted and measured AHP trajectories is likely to be larger than that measured at the 25th percentile point.

The predicted AHP trajectories often exhibited a similar time course to the directly measured AHPs, so that the actual AHP trajectory could be closely approximated by a scaled version of the trajectory predicted from the interval death rate. After aligning the flat regions of the predicted and measured AHPs as described above, a least-squares fitting procedure was used to determine the amount by which the predicted AHP had to be scaled to most closely match the measured AHP. Figure 8A illustrates a typical example in which the predicted AHP trajectory (○) is much shallower than the measured AHP trajectory (•). However, if the predicted trajectory is multiplied by a factor of 2.4 (□), it matches the observed AHP trajectory quite well. On average, the measured AHP trajectories were 1.9 times larger than those predicted from the death rates (range, 1.0–3.9 times). The required scale factor tended to be larger in cases in which the response to unfiltered noise was used to predict the AHP trajectory (unfiltered: 2.20 ± 0.84; range, 1.53–3.79; filtered: 1.67 ± 0.52; range, 0.96–2.53). However, this difference failed to reach statistical significance (unpaired t test, t = 1.953, P = 0.06).

Figure 8.

Predicted and measured AHP trajectories for two different motoneurones

•, measured AHP trajectories; ○, trajectories estimated from the interval death rates; □, scaled versions of the estimated trajectories, with the scaling factor chosen to minimize the squared error between the scaled trajectories and the measured trajectories. A, example of a case in which the scaled trajectory provides a close fit to the measured trajectory. B, a case in which the scaled trajectory does not match the measured trajectory.

In about 30 % of the cases (9/31), the predicted AHP trajectory differed in both magnitude and time course from the measured trajectory. In these cases, an example of which is shown in Fig. 8B, the predicted AHP trajectory was flat over much of the last half of the AHP, so that even a scaled version of the predicted trajectory provided a poor fit to the measured AHP. It is possible that in these cases, the predicted AHP trajectory is distorted by mixing populations of intervals with different mean rates. We attempted to minimize this distortion by separating sets of ISIs according to the value of the running mean ISI (see Fig. 1D). If this ‘slicing’ procedure is not sufficiently stringent then the predicted trajectories derived from a ‘mixed’ population will exhibit a long flat region like that shown in Fig. 8B (compare to Fig. 6 of Matthews, 1996).

Potential contributions to differences between predicted and measured AHP trajectories

Since the predicted AHP trajectory is based upon a number of assumptions and estimates, systematic differences between predicted and measured AHPs could arise from either violation of the underlying assumptions or systematic errors in the estimates on which the prediction is based. There are three major assumptions underlying the method used by Matthews (1996) and in the present paper to predict the AHP trajectory. The first of these assumptions is that the membrane potential noise estimated by convolving the passive impulse response with the injected current noise has the same amplitude and frequency content as the voltage noise superimposed upon the average interspike membrane potential trajectory. The second assumption is that the probability of evoking a spike depends only on the voltage noise characteristics and the distance to threshold, but not on the rate at which threshold is approached. The third assumption is that the voltage threshold for spike initiation is constant during the ISI. It will be argued here that although the first two assumptions are not strictly met, the errors introduced by violating them are relatively minor. Consequently, violations of the third assumption, that the voltage threshold for spike initiation is constant, are likely to be the principal source of the differences between predicted and measured AHPs.

Differences between the estimated and actual voltage noise could arise from three sources: (1) errors in our estimates of the passive impulse response of the motoneurone, (2) variations in membrane conductance during the ISI and (3) the presence of significant background synaptic noise. Our estimates of the passive impulse response were based on a double exponential fit to the average voltage decay following the offset of a hyperpolarizing current pulse (see Fig. 1B). This fit may be somewhat in error due to noise in the average voltage response and residual electrode capacitative artifact. However, these errors are unlikely to account for the average difference between the predicted and measured AHP trajectories, which would require the actual voltage noise produced by the current injection to be almost twice as large as the estimated noise.

During repetitive discharge, interspike variations in membrane conductance occur, so that the conductance exceeds the resting value early in the ISI and declines towards the resting value or may even fall below it later in the ISI (Schwindt & Calvin, 1973; Mauritz et al. 1974). However, at low firing rates, the conductance is close to the resting value over much of the latter half of the ISI, so that the magnitude of the voltage noise predicted from the passive impulse response measured at rest may be close to that superimposed on the interspike trajectory during repetitive discharge. Simulations with a conductance-based threshold-crossing model (see Appendix) suggest that, for the range of firing rates considered here, the increased membrane conductance leads to relatively minor errors in the predicted AHP trajectory. Moreover, the differences between the observed and predicted AHP trajectories are in the wrong direction: the increase in membrane conductance early in the ISI causes the predicted trajectory to fall below the measured AHP trajectory (i.e. more hyperpolarized) rather than above it (i.e. more depolarized or closer to threshold), as was commonly found.

In contrast, the presence of appreciable background synaptic noise would increase the total amount of voltage noise superimposed upon the interspike trajectory, causing the predicted trajectory to fall above the measured AHP trajectory. However, the variance in membrane potential due to background synaptic noise generally represented a relatively small proportion of the variance caused by the injected current noise. We measured the characteristics of background synaptic noise during the variable delay period prior to the onset of the injected current step. The variance of the background membrane potential noise was on average 11.7 ± 12.0 % of the membrane potential variance produced by the injected current waveform (range, 0–35.9 %). In about half of the cases, the variance contributed by the background synaptic noise was less than 10 % of the total variance, and in all but two cases, it was less than 20 %. Moreover, there was no correlation between the discrepancy between the predicted and actual AHP trajectories (i.e. the amount by which the predicted trajectory needed to be scaled to match the actual trajectory) and the relative amount of background synaptic noise (r² = 0.014; P > 0.1). For example, in the motoneurone shown in Fig. 7F, the variance of the background synaptic noise was only 1 % of that due to the injected current noise, and yet the predicted AHP trajectory clearly deviates from the measured trajectory. These results suggest that the commonly observed differences between the predicted and measured trajectories cannot be attributed to background synaptic noise.

Our estimate of the AHP trajectory is based on the assumption that the probability of evoking a spike depends only upon the absolute distance to threshold, whereas the rate at which threshold is approached could be important as well. However, simulations based on a threshold-crossing model suggest that the rate at which the AHP approaches threshold has little effect on the estimated trajectory (see Appendix).

The remaining potential contributing factor to the differences between the actual and predicted AHP trajectories is variation in the spike threshold during the ISI. There is convincing evidence for significant variations in spike threshold during the ISI, at least in some motoneurones (Calvin & Stevens, 1968; Calvin, 1974; Powers & Binder, 1996). Spike threshold is relatively low near the point at which the membrane potential reaches its minimum value during its interspike trajectory and then rises with membrane potential towards the end of the ISI (Calvin, 1974; Powers & Binder, 1996). Although we did not attempt to measure interspike variations in spike threshold in the present study, the differences between predicted and measured AHP trajectories are consistent with our previously measured interspike variations in spike threshold (Powers & Binder, 1996). The range of differences between predicted and measured AHP trajectories suggests that variations in spike threshold may be prominent in some motoneurones and relatively minor in others, as previously suggested (Calvin & Stevens, 1968).

The variations in spike threshold during the ISI preclude a direct measure of the AHP trajectory from the ISI statistics. However, we will argue in Discussion that the interval death rate provides a functionally relevant measure of the interspike variations in the relative excitability of the motoneurone during repetitive discharge.

DISCUSSION

We have studied the relationship between synaptic noise, the time course of the postspike AHP and the ISI statistics of repetitively discharging motoneurones. Repetitive discharge was elicited in cat lumbar motoneurones by injecting a current waveform composed of long step with superimposed noise. Our ability to control the characteristics of the simulated synaptic noise and to measure directly the time course of the AHP allowed us to examine the factors controlling the statistical features of motoneurone discharge in more detail than previously possible.

Comparison of the discharge statistics of noise-driven cat motoneurones and voluntarily activated human motoneurones

The statistics of the ISIs we obtained exhibited a number of similarities to those reported in voluntarily activated human motoneurones. At relatively high firing rates (> 10 impulses s⁻¹), the distribution of ISIs tended to be symmetrical and the standard deviation was generally relatively low (< 20 ms). At lower firing rates (< 10 impulses s⁻¹), the standard deviation was higher and the distribution of ISIs often exhibited a positive skew. For a given motoneurone, the standard deviation exhibited a non-linear relationship to the mean ISI. Over a range of short ISIs, the standard deviation was low and changed relatively little as mean interval increased. At higher mean ISIs, the standard deviation increased rapidly with increasing mean interval. Similar standard deviation versus mean interval relations have been reported for human motor unit discharge (e.g. Tokizane & Shimazu, 1964; Clamman, 1969; Person & Kudina, 1972).

Inferring AHP characteristics from ISI statistics

Person & Kudina (1972) proposed that the upward bend in the standard deviation versus mean ISI plot reflected an increased proportion of ISIs that were longer than the AHP duration. Further, it has recently been proposed that the transition point between the region of low and high variability can be used to infer the AHP duration (Piotrkiewicz, 1999). In the present study, the point at which the standard deviation began to exhibit an increased dependence on the mean ISI was often close to the AHP duration of the motoneurone. However, the exact ‘break point’ between these two regions of the standard deviation versus mean interval relation was not always clear, suggesting that plots of standard deviation versus the mean interval are likely to provide a relatively crude estimate of AHP duration.

Alternatively, if certain assumptions are made regarding the process of spike triggering, the ISI death rate can be used to estimate the AHP trajectory (Matthews, 1996). As described in Results, we found that although the time courses of the estimated and measured AHP trajectories were similar, the estimated AHP amplitude was on average only about half that of the actual AHP. A number of factors could potentially contribute to this discrepancy, especially if they lead to an underestimate of the amount of voltage noise superimposed upon the average AHP trajectory. In particular, if the background synaptic noise is of comparable amplitude to the voltage noise induced by the noise component of the injected current waveform, then the total voltage noise will be appreciably larger than that predicted from the response to the injected current alone. However, the variance of the background synaptic noise was on average only about 12 % of the variance of the noise produced by the injected current, and large differences between the estimated and measured AHPs occurred even in cases in which the variance of the background synaptic noise was less than 5 % of that produced by the injected current.

We conclude that the most likely explanation for the differences between estimated and measured AHP trajectories is that the voltage threshold for spike initiation varies during the ISI. Previous experimental evidence suggests that spike threshold follows the time course of the interspike membrane potential trajectory, reaching a minimum early in the interval and then rising to its final value (Calvin, 1974; Powers & Binder, 1996). The interspike variations in threshold lead to a reduction in the distance to threshold at a given point within the ISI compared to the distance between the AHP trajectory at that point and a constant threshold set at the value reached at the end of the ISI. The triggering of spikes by a synaptic potential or by synaptic noise may also depend upon the rate of rise of membrane potential, with rapid increases in membrane potential being more effective in evoking spikes. This possibility is supported by our finding that the largest discrepancies between predicted and measured AHPs were obtained in the experimental trials using unfiltered current noise. The unfiltered current noise produced more rapid fluctuations in membrane potential than the filtered noise, which should have led to threshold crossings at more hyperpolarized membrane potentials.

Although the interval death rate does not provide an accurate measure of the AHP, it does provide an estimate of the instantaneous distance to threshold over much of the ISI. This distance is likely to depend upon a number of factors, including interspike variations in membrane potential, membrane conductance, the voltage threshold for spike initiation and the characteristics of the synaptic noise. As a result, the interval death rate may in fact reflect the amount of synaptic current required to bring the membrane to threshold. This ‘effective’ distance to threshold provides a functionally relevant measure of interspike fluctuations in excitability at different mean firing rates and may help to separate stimulus-related effects on discharge probability from those due to postspike refractoriness (cf. Johnson & Swami, 1983; Miller, 1985; Berry & Meister, 1998).

APPENDIX

Peter B. C. Matthews

Modelling the tonic firing elicited by noisy inputs

The present modelling extends that presented previously (Matthews, 1996) by testing the use of the transform in a more realistic model with membrane capacitance and conductance, with the synaptic input modelled either as conductance changes or as a noisy current. This was required to provide generality for application to real motoneurones and more particularly to establish that the present experimentally observed deviations between the predicted and measured AHPs did not arise merely from the earlier simplification of operating solely in terms of voltage. The original model had an exponential AHP decaying to a final level representing the mean level of excitation, with its trajectory disturbed by time-smoothed Gaussian noise. Firing occurred when the postspike trajectory crossed a fixed voltage threshold and the AHP was then reset. Both models behave very similarly to human motoneurones studied at low discharge rates, giving similar interval histograms, etc. (Matthews, 1996, 1999); the absolute firing rates of the present cat motoneurones were slightly higher, because their AHPs were of slightly shorter duration.

The expanded model has been recently described in detail (Matthews, 1999); the numerical values of its various parameters are given in the figure legends of this Appendix.

The expanded model had a resting leak conductance and an exponentially decaying AHP conductance. To provide generality, the synaptic input was normally represented by two steady conductances with different equilibrium levels, representing the mean levels of excitation and inhibition, together with corresponding noise conductances that varied randomly from bin to bin. It was also enacted in a restricted form, corresponding to the present experimental situation, with the synaptic input represented by a steady current with superimposed noise. The mean value of each noise conductance (or current) was zero and its amplitude distribution was Gaussian. The resulting voltage noise was smoothed by virtue of the model's membrane capacitance. The capacitance was set to yield a membrane time constant of 4 ms in the presence of synaptic input, thereby matching the value used in the voltage model to derive the original transform. The model's voltage noise was measured with the mean potential held at threshold by injecting current, and with the spiking and AHP inactivated. As in the earlier voltage model, the iterative calculations up-dating the membrane voltage were performed with a bin width of 1 ms, thereby effectively introducing some additional smoothing (Matthews, 1996). Spike initiation remained an all-or-none phenomenon, occurring whenever the membrane voltage exceeded a threshold value that remained constant throughout the ISI. The expanded model remains relatively simple; it has only a single compartment and omits the specific details of the spike- and AHP-generating ionic conductances (cf. Powers, 1993).

Effect of varying mean synaptic drive

Figure 9 displays the interval histograms generated by the model for firing elicited by three different levels of excitatory synaptic input, when the noise and inhibition were kept the same. Figure 9A shows the directly determined interval histograms; that for the highest firing rate is close to Gaussian, whereas that for the lowest firing rate is highly skewed with a prolonged tail that is approximately exponential. In Fig. 9B the three histograms have been transformed into plots of ‘interval death rate’versus time; the exponential tail for the lowest firing rate transmutes to a terminal plateau, while the death rate for the Gaussian-like distribution continues to rise throughout its course (note, however, that the plot terminates before reaching the start time of the plateau for the low firing rate). In Fig. 9C, the circles give the AHP voltages estimated by applying the previously described ‘4 ms transform’ (Matthews, 1996) to the death rate plots; the continuous lines give the actual values of the AHPs, each determined for the same mean level of synaptic drive in the absense of noise and with the spiking inactivated. The AHPs have been normalized by being scaled in terms of noise units, as given by the transform and corresponding to the standard deviation of the voltage noise; in this case, 1 noise unit was 0.67 mV. The actual AHP was measured directly in millivolts and then rescaled.

Figure 9.

Deduction of the AHP from the model's interval histograms

A, interval histograms, approximating to probability density functions, for three different firing rates generated by varying the model's mean excitatory conductance, while keeping the inhibitory, noise and AHP conductances constant. Each histogram is based on simulated firing for some 30 min yielding 24 000–54 000 intervals. B, the interval histograms transformed into plots of interval death rate versus time. These plots terminate after inclusion of 98 % of the intervals, as they then became excessively variable. C, the points give the estimated AHPs obtained by applying the ‘4 ms transform’ (Matthews, 1996) to the death rate plots. The continuous lines show the actual AHPs for the same mean input, determined by measuring the membrane voltage after removing the noise and eliminating spiking. The AHPs have been plotted in noise units (1 noise unit = 0.67 mV), corresponding to the standard deviation of the voltage noise. (The model parameter values were as follows: leak conductance, 0.5 μS, equilibrium potential = 0 mV; fixed spike threshold, +15 mV; excitatory conductances, 0.385, 0.400 and 0.430 μS, ±0.04 μS noise (s.d.), equilibrium potential = +70 mV; inhibitory conductance, 0.5 ± 0.05 μS, equilibrium potential = −15 mV; AHP initial conductance 0.4 μS decaying with a time constant of 30 ms, equilibrium potential = −15 mV; membrane capacitance, 5.5–5.7 nF, fine-tuned for each to bring the membrane time constant to 4 ms at final equilibrium.) No residual AHP conductance was carried forward to the subsequent interval. (See Matthews, 1999, for further details).

The efficacy of the transform determined with the voltage model in reproducing the AHP of the conductance model is not surprising, since the portion of the AHP studied lies within 2 mV of threshold. In this voltage region, the relation between the various conductance changes and resulting membrane voltage change is approximately linear and the AHP conductance has decayed to a low value. It bears emphasis, however, that what the transform recovers is a single AHP trajectory in the absence of repetitive discharge, rather than the mean interspike membrane trajectory during repetitive discharge. The mean trajectory tends to be more hyperpolarized than the AHP trajectory per se because there is a deficit of individual trajectories lying above the AHP (as compared to below), as some have been eliminated by the spiking (cf. Matthews, 1996, p. 623).

Effect of varying synaptic noise

The ‘4 ms transform’ (Matthews, 1996) provides the AHP scaled in noise units, as used in Fig. 9C. Likewise, at any time, it is the noise unit value rather than the absolute voltage of the membrane potential (in mV) that determines the probability of firing. Thus, for a given absolute AHP in millivolts, changing the noise level rescales the AHP as expressed in noise units and produces drastic changes in firing and the shape of the interval histogram. This point is illustrated in Fig. 10. Histograms D and H in Fig. 10A were obtained by running the model with the same level of drive and AHP conductance as the middle histogram of Fig. 9A for a firing rate of 10.9 Hz, but with the noise either doubled (D) or halved (H), increasing the firing rate to 16.8 Hz or reducing it to 7.5 Hz, respectively. However, transformation of these histograms yielded overlapping segments of the same absolute AHP, as expressed in millivolts. This is illustrated in Fig. 10B by the circles and the dots, while the continuous line is again the measured AHP (same as in Fig. 9C, but now plotted in mV). As the firing rate is increased, the precise segment of the AHP recovered by transforming the histogram becomes earlier and earlier. In Fig. 10, both excitatory and inhibitory noise conductances were either doubled or halved, producing corresponding changes in voltage noise. It bears emphasis that when they were altered independently, the net voltage noise across the membrane proved to be the important factor, rather than the particular values of the inhibitory and excitatory noise conductances per se.

Figure 10.

Estimation of the AHP using different amounts of noise

The interval histograms (A) and the AHPs (B) on halving (H), doubling (D), or tripling the noise (T). The mean synaptic excitation for plots H and D was the same as that for the middle histogram of Fig. 9 (10.9 Hz). Doubling the noise increased the firing rate to 16.8 Hz and halving the noise reduced it to 7.5 Hz, but the transform still gave a good estimate (circles, dots) of the invariant directly measured AHP (continuous line); all values are plotted in millivolts. The two noise levels provide estimates of different segments of the AHP, with only a slight overlap. When the noise was tripled (T) the mean synaptic drive was reduced to below the firing level for the original noise, but with the extra noise, firing was restored at 11.0 Hz. The resulting interval histogram (T) is much wider than the central histogram in Fig. 9A for virtually the same firing rate. The initial part of the AHP estimate (○) in B now shows a small but systematic deviation from the direct AHP (continuous line), presumably because of the non-linearities associated with a greater absolute deviation from threshold. (The excitatory conductance was reduced from 0.4 μS for H and D to 0.34 μS for T. The ‘standard’ noise of 0.67 mV of Fig. 1 was altered by changing the standard excitatory and inhibitory noise conductances by the same proportion. All other parameter values as in Fig. 1, except for a slightly smaller capacitance for T.)

Histogram T in Fig. 10A was obtained by tripling the original noise level of Fig. 9 and reducing the mean excitatory input to well below the earlier mean level for firing. The increased noise maintained the firing at the same mean rate as before (11.0 versus 10.9 Hz), but with a much wider spread of ISIs (standard deviation of interval histogram increased from 18.8 to 35.5 ms). Figure 10B shows the AHPs for T (circles, estimate; line, measured), and reiterates the finding that increasing the noise allowed firing to occur when the AHP was at a much greater absolute distance (in mV) from threshold than before. However, the initial part of the estimate provided by the transform now shows a slight but systematic deviation from the true AHP. This deviation may be attributed to the slight changes in the relation between current and conductance and the consequent voltage changes that occur as the mean voltage changes, but the matter has not been analysed and the precise mechanism remains uncertain. The errors would be expected to increase if firing occurred with yet larger absolute deviations (in mV) of the AHP from the present value of 15 mV. However, the transforms determined in results for each particular motoneurone should be applicable throughout the firing range studied experimentally (see Figs 7 and 8, which have very few values above 6 mV).

Most simply, the deviation might be due to the extra AHP current that flows when a noise-driven individual trajectory approaches threshold early in the AHP; from inspection of T in Fig. 10B it can be deduced that less firing occurs at a given mean AHP potential in the conductance model than in the earlier voltage model. In the voltage model used to determine the transform, the AHP, the noise and the membrane time constant were all independent of the mean voltage. In contrast, at the beginning of trace T in Fig. 10B, the mean AHP current was 20 % less than it would have been at threshold voltage and the membrane time constant was 10 % less than its final equilibrium value. However, the voltage noise proved to be within 1 % of the value used, namely that measured at threshold after completion of the AHP.

Excitation by noisy current

The transform proved equally applicable to a hybrid model, corresponding to the present experimental situation in which the motoneurones were excited by the intracellular injection of a noisy current rather than by synaptic inputs. This is illustrated in Fig. 11 which, as in Figs 9 and 10, compares the estimated and directly determined AHPs. Similar agreement was obtained on mixing a noisy current with noisy synaptic input.

Figure 11.

Responses with injected current

Agreement of estimated and measured AHPs in a reduced model, in which the synaptic conductances have been replaced by an injected current with noise. The AHP and leak conductances were retained, as for Fig. 9, and the membrane's time constant held at 4 ms by adjusting its capacitance. The excitation was provided by currents of 6.5, 7.1 and 8.4 nA each with ±2 nA current noise which produced 1.39 mV voltage noise; the membrane capacitance was 2 nF.

Potential further sources of error in application of transform

The good agreement between estimated and measured AHPs conceals two potential complications. These have proved to be numerically unimportant in the range presently studied experimentally, but might become significant for higher firing rates. The first is the effect of the initial slope of the membrane trajectory following the peak of the AHP. By analogy with the response to an EPSP, the amount of excitation associated with a given AHP voltage can be expected to increase with its slope (cf. Matthews, 1996, pp. 623–624). In consequence, the transform tends to displace the earlier part of the estimated AHP towards threshold, since the transform was determined under static conditions. This is shown by the deviation of the upper curve (slightly wiggly) in Fig. 12 from the middle (smooth) curve, indicating the estimated and measured AHPs, respectively. The effect has been emphasized by increasing the membrane time constant of the model from 4 ms to 8 ms, since the effect depended upon the relation between the slope of the AHP and the time structure of the noise. Even so, the effect remains too small to explain deviations like those in Fig. 7F. The effect could be compensated for by subtracting a correction from the estimated AHP; the correction was directly proportional to the value of the slope (in noise units ms⁻¹) and was determined by first estimating the slope over 5 ms, using the uncorrected transform, and then multiplying this value by a constant ‘slope correction factor’. Application of this correction to the estimated AHP gave the points which overlie the actual AHP (smooth curve), and which now provide a good fit to the measured results. The constant of proportionality was determined by generating AHPs that were linear rather than exponential and measuring the deviation between the actual trajectory and that given by the unadjusted transform; these plots also demonstrated that a linear correction for slope sufficed in the range studied.

Figure 12.

Insignificant action of potential complicating factors

AHPs from a full conductance model illustrating two minor complicating factors. The top line (slightly wiggly) shows the AHP estimated using the death rate transform. It now deviates slightly from the measured AHP, given by the smooth line just below, because the membrane time constant has been increased to 8 ms. This enhances the weak excitatory effect of the finite slope of the AHP. This effect can be readily allowed for when using the transform to estimate the AHP (slope correction, see text). The corrected estimate is given by the points, which lie close to the measured AHP. The bottom line (slightly irregular) shows the small but definite augmentation of the AHP (slope-corrected estimate) produced by allowing the residual AHP conductance that remains at the time of spike initiation to be carried over into the next ISI. Normally, the total AHP conductance was reset to a constant value. The excitatory conductance was 0.41 μS and the noise conductances were twice those of Fig. 1, giving a membrane voltage noise of 0.94 mV. The slope correction factor was 2.5 noise units for a slope of 1 noise unit ms⁻¹.

The second potential complication is the persistence of residual AHP conductance. For all the above modelling, any residual AHP conductance was eliminated by the occurrence of a spike, whereas in real motoneurones the AHP persists and summates with the ensuing AHP (Ito & Oshima, 1962; Baldissera & Gustafsson, 1974). The effect of allowing such summation is shown by the bottom line in Fig. 12, which indicates that the estimated AHP is now more hyperpolarized than the measured AHP, resulting in a reduction of the firing rate from 12.9 to 12.5 Hz. This is ‘physiologically’ correct since with the persistence of the residual conductance the average initial size of the AHP during firing will be larger than that of the one-shot AHP. The transform used to simulate firing with AHP persistence incorporated the slope correction, so that the effect of the persistent AHP conductance is fairly reproduced by the difference between the bottom and middle curves. This effect is again too small to explain the deviations of Figs 7 and 8. For the ‘4 ms’ model the slope effect became too small to matter, while the weak effect of carrying over the residual AHP persisted. Thus applying the transform to the discharge of a real motoneurone can be expected to give an estimated AHP that is very slightly larger than the one-shot AHP (as determined in Fig. 1A), with the deviation increasing with the firing rate. However, for low frequency firing, the effect should be quantitatively unimportant, as in Fig. 12.

In conclusion, the present extended modelling strongly supports the view that if a motoneurone's spike threshold were to remain constant throughout the ISI, then the transform should successfully recover its AHP, provided the firing rate is not too high or the noise level too great. However, the threshold is well known to vary continuously throughout the ISI (Calvin, 1974; Powers & Binder, 1996); the direction and time course of the deviation from its resting value approximately follows that of the AHP. Thus the ‘distance from threshold’ provided by the transform can be expected to approximate to a scaled down version of the AHP, rather than to its actual value. However, the trajectory provided by the transform should actually provide a better metric of motoneurone excitability than the actual AHP, since it incorporates the effect of the change in spike threshold.