Properties of human motoneurones and their synaptic noise deduced from motor unit recordings with the aid of computer modelling

J. Physiol. (Paris) 93 (1999) 135-145 0 Elsevier, Paris

Properties of human motoneurones and their synaptic noise deduced from motor unit recordings with the aid of computer modelling Peter B.C. Matthews* University Laboratory of Physiology, Parks Road, Oxford OXI 3PT, UK

paper reviews two new facets of the behaviour of human motoneurones; these were demonstrated by modelling combined with analysis of long periods of low-frequency tonic motor unit firing (sub-primary range). 1) A novel transformation of the interval histogram has shown that the effective part of the membrane’s post-spike voltage trajectory is a segment of an exponential (rather than linear), with most spikes being triggered by synaptic noise before the mean potential reaches threshold. The curvature of the motoneurone’s trajectory affects virtually all measures of its behaviour and response to stimulation. The ‘trajectory’ is measured from threshold, and so includes any changes in threshold during the interspike interval. 2) A novel rhythmic stimulus (amplitudemodulated pulsed vibration) has been used to show that the motoneurone produces appreciable phase-advance during sinusoidal excitation. At low frequencies, the advance increases with rising stimulus frequency but then, slightly below the motoneurones mean firing rate, it suddenly becomes smaller. The gain has a maximum for stimuli at the mean firing rate (the ‘carrier’). Such behaviour is functionally important since it affects the motoneurone’s response to any rhythmic input, whether generated peripherally by the receptors (as in tremor) or by the CNS (as with cortical oscillations). Low mean firing rates favour tremor, since the high gain and reduced phase advance at the ‘carrier’ reduce the stability of the stretch reflex. 0 Elsevier, Paris Abstract-This

motoneurone

I motor unit I synaptic noise I stretch reflex I tremor

1. Introduction Recordings from single human motor units are doubly important. First, of course, they have a special part to play in analysing motor performance by avoiding the inherent complexity of the gross EMG; indeed, almost any interpretation of summed EMG activity implicitly depends upon the accumulated knowledge about how single units behave. Second, they provide a unique example of the firing pattern of a human neurone, of any kind, while it is being activated physiologically by its synaptic input. Thus motor unit recording opens a window into the CNS, helping us to understand what determines the behaviour of an individual neurone, albeit a specialised one. The present paper gives an overview of two recent such studies of motor unit activity, dealing with the irregularity in their tonic firing and with their responsiveness to rhythmic inputs [ 16, 171. Computer analysis and modelling played an essential part in understanding the real life recordings. Both the recording and the modelling followed previous practice, but they have not been combined in this way before. The particular experimental feature was that the units were recorded with surface electrodes, thereby readily providing large numbers of spikes discharged by the same neurone. In addition, a novel stimulus was used to generate rhythmic motoneuronal excitation. The particular feature of the model motoneurone used was the introduction of appreciable synaptic noise, but it was otherwise maximally simplified. It was used to deduce a novel * Correspondence

and reprints

transform giving the relation between the membrane potential and the probability of a spike occurring in the next ms. Finally, it bears emphasis that the units studied were firing at low frequency, largely in what has been termed their sub-primary range [14, 211. Their behaviour then seems to differ from that found in their primary range, as studied by injecting noise-free current intracellularly in the cat.

2. Results 2.1. The interval histogram’s

exponential

tail

The starting point was the chance observation that when voluntarily activated human motor units are firing at the bottom of their frequency range, then the final part of their inter-spike interval distribution becomes a simple exponential. This seems to have hitherto escaped detection for the motor unit; entirely exponential interval histograms are well known for certain other neurones, with important theoretical implications. Figure I shows three different representations of the exponential tail for two different motor units (from abductor digiti minimi and soleus). The left hand plots show the standard interval histogram on linear axes, with the arrows marking the start of the exponential tail. The Y-axis of the centre plots is scaled logarithmically and thus emphasises the exponential relation. However, the value plotted is now the percentage of intervals surviving to the time in question, rather than the number of spikes per se. Its slope thus gives the ‘interval death rate’ (or hazard function),

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Inbxval Hfstogram

Surviving intervals

Intervaldeath rate

8-

Interval - ms

Time since spike - ms

Time since spike - ms

Figure 1. Interval histograms with exponential tails. Three different representations of the distribution of inter-spike intervals during slow tonic firing of two different motor units (soleus and abductor digiti minimi, both firing at the bottom of their frequency range). Left. Conventional interval histogram showing the percentage of intervals falling into a series of 5 ms bins. Centre. Semi-log plot of the percentage of intervals ‘surviving’ at the end of each 5 ms bin (i.e., number of intervals greater than the stated value, expressed as a percentage of the total number of intervals). Right. The percentage ‘death rate’ of the intervals which have survived up to a given time (the slope of the survivor plot, or approximately the number of spikes per bin divided by the number of survivors). Plots based on 9181 intervals for ADM and 32 101 for soleus (from 1161).

namely the probability that a given ongoing interval will be terminated in the next ms. This is shown in the right hand plots, with the horizontal plateau of constant probability corresponding to the exponential tail. Measurements of the ‘interval death rate’ are especially valuable since they can tell one about the membrane potential of the motoneurone itself. Given a noise free motoneurone excited by a constant current, the death rate plot would be a sharp transient occurring at the time when the ‘trajectory’ of the MN’s membrane potential reached threshold, with a similar peak in the interval histogram (following each spike the membrane is progressively depolarised as its afterhyperpolarisation decays). However, there will always be some synaptic noise, so the threshold will sometimes be crossed before the mean trajectory reaches it. Calvin and Stevens [5] showed that the resultant broadening of the interval histogram could be fully accounted for in terms of the measured characteristics of the synaptic noise (cat MNs excited by current injection). Under their conditions the interval death rate would have increased progressively with time up to the very end; this happens because the closer the mean trajectory gets to threshold, the more often will

a spike be triggered by noise of a given amplitude thereby progressively increasing the probability that the interval will be terminated. The first part of the present death rate plots show the expected steady increase, but they then run flat. This means that the mean membrane potential has become constant, somewhere below threshold, with the spikes triggered

solely by noise; if the noise fails to trigger a spike in any given 5-ms period (the binwidth), then it is equally likely to do so in the next. Thus for low firing rates in man, with the activation provided by continuous synaptic bombardment, a number of spikes are fired after the AHP has ended and the mean potential never reaches threshold. This situation should be readily reproducible in experimental animals, but hitherto attention has been concentrated on higher firing rates, with the excitation produced by the injection of noise-free current. 2.2. Determination

of the trajectory by modefling

A simplified model of the motoneurone was then used to estimate first, the amount by which the equilibrium potential for low firing rates lies below threshold and second, the voltage ‘trajectory’ leading up to this. The aim was to determine the transform that may be presumed to exist between the deviation of the mean membrane potential from threshold and the probability of an ongoing inter-spike interval being terminated by noise in the next ms (interval death rate). The model originally used [ 161 was deliberately maximally simplified and operated simply in terms of voltage; more complex models using conductances yield similar results (unpublished, with M.D. Binder and R.K. Powers). In the simple model a Gaussian noise voltage was added every ms to a steady depolarisation and their sum tested to see whether the

threshold

had been reached. The MN was not given

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Tonic firing of motor units analysed

Probabilityv Membrane Potential (Fitted with Gaussian

Membrane Potential v Probability (Fitted with two exponentials)

function)

-3.0 Membrane potential- Noise units

I 0

I

I

.2 Probability of spike in next ms

.l

Figure 2. The transform: probability-voltage

relations obtained by modelling the effect of noise. Each point was obtained from the model by setting its mean potential to a particular value, adding time smoothed noise (4 ms time constant), and repeatedly determining the time at which the noise triggered a spike by reaching threshold. The probability of its firing a spike during any given millisecond, when it had not yet done so, was then determined for each potential. The potential is scaled in terms of the SD. of the noise (1 noise unit); zero is at threshold, and negative values represent hyperpolarisation. A. The cause-and-effect relation of voltage to probability. B. Inverted axes, providing a calibration curve, which can be used to transform death rate plots, as injigure I, into an estimate of the trajectory of motoneurone’s membrane potential following a spike. The dashed lines show the effect of doubling the smoothing time constant (from [16]).

any capacitance but the series of noise voltages was exponentially smoothed with a time constant of 4 ms, representing that of the MN; this made the noise time structure similar to that of real MNs [5]. Interval histograms with their attendant death rates were then obtained for the noise-induced excitation using a range of values of mean membrane potential. The points (solid line) in the left hand part of figure 2 show the progressive increase in the probability of firing plotted against the voltage. Note, however, that the voltage is not given in absolute units (i.e., mV relative to zero), but is scaled in terms of the noise amplitude with zero corresponding to the spike threshold. The noise unit equals the standard deviation of the noise distribution in mV; using it provides automatic scaling for different amounts of noise. The right hand part offigure 2 shows the relation inverted, thereby providing a template which permits an experimentally observed ‘death rate’ to be converted into a membrane potential (albeit in NU and related to threshold). The solid line shows the arbitrary equation (the sum of two exponentials) which was fitted to the modelling data to provide a convenient algebraical transform to facilitate the computation of a trajectory for a real motoneurone. Figure 3 shows the family of trajectories, obtained with the transform, for a range of mean firing frequencies for two different MNs, one supplying biceps brachii and the other soleus. In both cases, the trajec-

tory for the lowest firing rate, corresponding to the weakest synaptic drive, terminates in a plateau about 1.5 NU below threshold. With increasing levels of synaptic drive the trajectories rise more rapidly, come closer to threshold, and lose their final plateau (the exponential tail of the interval histogram). This considerably expands on previous estimates of the shape of the trajectory in human MNs, based on probing their excitability with test stimuli delivered at a limited number of post-spike intervals [I, 9, 20,261 (n.b., such estimates typically equate ‘potential’ directly with firing probability, but on present showing the relation between them is unlikely to be linear). On expanding the model motoneurone to include a post-spike AHP (an exponentially decaying voltage) it gave interval histograms similar to those seen physiologically; moreover, the trajectories obtained by transforming the histograms for increasing degrees of mean excitation displayed systematic changes in form, as in &ure 3. For the model, these can be immediately recognised as successive segments of the decaying voltage used to mimic the AHP, with their vertical displacement determined by the voltage used to represent the mean synaptic drive. The underlying overall trajectory remains the same in every case (an exponential); however, the shape of the part displayed varies systematically, because it is being viewed through the window provided by the spike discharge,

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Biceps

Soleus 9.1

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Figure 3. Family of trajectories for different firing rates. The trajectory of the post-spike membrane voltage was estimated by applying the transform offigure 2 to a series of interval histograms for different mean firing rates (as marked), decreasing from left to right for each motor unit. In toto, 41 682 intervals were used for the biceps unit, and 47 629 for the soleus unit. The data for the different firing rates were obtained partly by deliberately varying the level of voluntary drive, and partly by selecting periods of discharge with a given mean rate from a long recording in which the mean rate varied ‘spontaneously’ (from [ 161).

so only the small portion close to threshold is visible. (When the trajectory is too far below threshold the interval death rate is very low and so no spikes are fired; when it rises above threshold the probability becomes high, but there are no spikes because all the intervals have already been terminated.) A much larger

proportion of the trajectory can be visualised by shifting the various individual trajectories vertically, so that their overlapping portions are superimposed; the shift required to superimpose a given pair of curves corresponds to the difference in their mean excitatory drive. Figure 4 does this for the real trajectories of figure 3, with the solid curves providing an estimate of the form of the underlying trajectory. The curves are simple exponentials with time constants of 29 and 41 ms for the biceps and soleus MNs respectively. This may be presumed to correspond to the time constants of their AHPs (more precisely, of the underlying change in potassium conductance), in reasonable accord with observations on cat MNs [2, 61. Thus the duration of a motoneurone’s AHP can be estimated in man simply by recording a motor unit’s interval histogram. A rough estimate could also be made of the amount of synaptic noise, measured in mV, developed across the motoneuronal membrane. This was done be extrapolating the AHP back to time zero to give a value of the initial potential in noise units. The value was found to differ between units, being higher for soleus MNs and lower for those supplying intrinsic hand muscles. On the assumption that the initial size of the AHP is about 15 mV above threshold for all MNs, as seen in the cat, the difference between MNs in the AHP

magnitude measured in NU reflects the differences in their noise level. Because the trajectories measured in the cat with current injection and intracellular recordings are fairly linear [24], a linear extrapolation was performed on the penultimate part of the present trajectories. (Exponential extrapolation assumes that the current generating the AHP is independent of the voltage, whereas the real current decreases as the voltage approaches the equilibrium potential of the conductance change responsible for the AHP.) On this basis the soleus noise had a standard deviation of about 1.5 mV, while two hand MNs had values of about 3 mV. Similar values have been directly recorded in lightly anaesthetised and decerebrate cats [5, 81

2.2.1. Limitations It must next be emphasised that all such estimates of the trajectory and its parameters are provisional and of limited numerical accuracy. To begin with, the modelling was deliberately simplified, so the transform might be somewhat in error. More seriously, there may be no unique transform applicable under all conditions; high frequency firing, in particular, is unlikely to be explicable in the same terms as low frequency firing since additional factors can then be expected to affect motoneuronal behaviour. Moreover, the threshold for spike generation was assumed to be constant, whereas it changes progressively over the course of the interspike interval [4, 231. The present estimates are thus of the ‘effective’ trajectory, namely the distance between the ongoing values of the AHP voltage and threshold, rather than of the absolute value of the membrane potential. This should be a second

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Tonic firing of motor units analysed

O-

Soleus

Biceps -l-

In z 3 -2$ $ -3-

I

-4-

3 2 -5E

_ -6-

-6 I 0

50

100

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trajectories showing more plots of fisure 3 were shifted vertically so as to reference (rationale in text). The data are fitted because of a potential artefact, their equilibrium (from [ 161).

250

0

I

50

100

I

150

I

200

,

250

Time since spike - ms

Time since spike - ms Figure 4. Compound

,

of the AHP, produced by combining the individual trajectories. The individual superimpose them in their regions of overlap, taking the lowest curve as the with an exponential, providing an estimate of the time constant of the AHP; value is deliberately slightly above the experimental points for long intervals

order difference for the low frequency firing currently studied (sub-primary range) since the reduction in threshold is linked to the amount of residual AHP, which will be small. In any case, the ‘effective’ trajectory is the more functionally relevant, since it is this which determines the motoneurone’s response to stimuli (i.e., it’s ‘excitability’). The major factor limiting the numerical accuracy of the trajectories obtained with low frequency firing is the uncertainty about the actual membrane time constant of any particular neurone studied. The precise values of the transform’s parameters depend crucially upon the time structure of the noise; this is illustrated in$gure 2, where the dashed lines show the effect of doubling the time constant to 8 ms. In addition, a correction is required because a transform determined under the static conditions of constant mean membrane potential is being applied under the dynamic conditions; both theory and experiment show that a neurone responds to the rate of change of an excitatory potential as well as to its actual level [ll, 121. However, modelling suggests that under present conditions such effects are rather small, since over the region studied the trajectory has a low slope. Yet further factors could influence the transform, and various experimental details might also affect the issue [ 161. Nonetheless, in spite of these numerical uncertainties it may be provisionally concluded that the transform yields trajectories which show a qualitatively meaningful pattern of behaviour. In particular, it can be securely concluded that noise plays a crucial part during tonic firing elicited by excitation by normal synaptic inputs, with

most spikes being discharged tory reaches threshold.

before the mean trajec-

2.2.2. Wider signijicance The present analysis has an immediate relevance to many of the standard measures of motor discharge. For example, indices of synchronisation between a pair of motor units will depend upon the slope of the region of the AHP as it approaches threshold. A fixed amount of common synaptic input will most readily synchronise a pair of neurones when they are both firing at low frequency with flat trajectories, since for much of the time both neurones remain potentially available to be excited with their trajectories close to threshold. In contrast, during high frequency firing their steep trajectories will only be jointly within range of threshold for a small proportion of the time. Thus the degree of synchronisation of a pair of neurones cannot, per se, provide a direct measure of the neurone’s common input. The standard post stimulus time histogram suffers from related defects which can be partially overcome by locking the time of stimulation to the occurrence of the immediately preceding spike, so that excitability is always tested at the same point of the trajectory. If the testing stimulus is applied before any ‘spontaneous’ spikes occur then expressing the number of responses in the histogram count as a percentage of the total number of trials is much the same as calculating the present ‘interval death rate’. The conversion of such a probability to the underlying excitatory depolarisation can be expected to involve a non-linear transform, as

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in jigwe 2, and to depend upon which point of the trajectory is tested (the rate of rise of the excitation should also matter). Inhibitory processes have been studied using the spike-locked PSTH by delivering the stimulus appropriately timed to prevent the discharge of the spikes that would normally have occurred at a particular post-spike interval; its strength of action is then typically expressed as a simple reduction in the number of counts in the relevant bins in the PSTH from their level without stimulation [ 181. This, however, provides a highly indirect measure of the depth of the inhibitory hyperpolarisation; it is also seriously misleading about its time course and underestimates its duration, especially when it is prolonged. This can be circumvented, without introducing the uncertainties of using a transform based on modelling, simply by calculating the present ‘interval death rate’ curves, both with and without the inhibitory stimulation. Inhibition is shown by a decreased death rate, with its completion marked by the return of the test curve to the normal level. In contrast, the number of spikes in a given bin of the test PSTH returns to normal while the death rate remains reduced, indicating that the membrane is still hyperpolarised. This divergence between the two measures occurs because the inhibition merely delays the occurrence of the spikes, and the ‘interval’ continues in being to terminate in a subsequent spike. Thus a given spike count in a subsequent bin represents a smaller proportion than normal of the spikes remaining to be discharged. Understanding of the underlying change in membrane potential is achieved by focussing attention on the ‘interval’ and what destroys it, rather than on the spike per se. 2.3. Motoneuronal excitation

phase shifts on sinusoidal

2.3.1. Model&g The motoneurones are often implicitly assumed to transmute their summed synaptic input signal into an output command to muscle without changing its dynamics. Indeed, under certain experimental conditions cat motoneurones appeared to introduce little or no phase shift when they were excited by sinusoidal muscle stretch [22]. The matter is of particular interest when considering the contribution of the stretch reflex to tremor, but also has a wider significance; for example, it will affect the temporal relation between the recently recognised low frequency oscillations at around 20 Hz in the motor cortex and their reflection in the EMG [19]. It was thus of interest to test the response of the noisy model MN to small sinusoidal inputs. As illustrated in jigure 5, for three different mean firing rates, the model then showed large swings in gain and phase as the stimulus frequency was changed. At the top, the frequency of the sinusoidal

stimulus is scaled in absolute units (Hz) and the peaks are staggered; at the bottom, they are brought into register by normalising the stimulus frequency to the MNs firing rate. Thus this neuronal model, like many previous ones [13, 251, shows ‘carrier dependence’ with the metrics of the response greatly affected by the interactions between the rhythmic stimulus and the rhythmic spiking frequency (the signal’s ‘carrier’). As expected, the extent of interaction and the magnitude of the effects increases with the regularity of the MN’s firing; in figure 5, the greatest effect is seen for the highest firing rate because the discharge was then at its most regular (see legend). It is especially noteworthy that the phase advance persists at all stimulus frequencies, including at just above the carrier, as not all neuronal models show this; moreover, while the gain peaked at the carrier frequency the phase advance had its maximum slightly below this. Thus the present noisy model suggests that the motoneurone produces major modifications to the signals it is transmitting. This encourages the testing of its behaviour in man, in the presence of synaptic noise. 2.3.2. A novel sinusoidal stimulus The easiest way of activating human motoneurones sinusoidally is via the stretch reflex. However, the Ia afferent discharge elicited by simply stretching a muscle sinusoidally is phase advanced by an unknown amount, making it difficult to disentangle the motoneurone’s contribution to the final reflex response. This was overcome by using a novel sinusoidal stimulus which by-passes the muscle spindle’s normal phase advance mechanism, and should produce Ia modulation in phase with the stimulus. The principle is to excite the muscle spindle by vibration and to modulate its amplitude sinusoidally at a lower frequency (avoiding sub-harmonics). The vibration’s frequency was usually 103 Hz and for greater conceptual clarity it consisted of a series of brief taps (7 ms duration), rather than having a sinusoidal waveform (figure 6). The mean amplitude must be kept very small so that any given spindle affferent only responds intermittently, with no possibility of 1: 1 driving [3]. The probability that any given tap will excite a given afferent will increase with its amplitude, so that when the tap amplitude is modulated sinusoidally then the summed discharge from a population of Ia afferents should be similarly modulated. The effective stimulus is now the sinusoid used to modulate the vibration. The spindle response will be in phase with the stimulus simply because the largest tap evokes the largest response. The validity of this line of reasoning was supported by control experiments showing a linear relation between the input sine and the reflex output on varying its amplitude, but it remains desirable to test it by la recording. There is no doubt, however, that the

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Gain

Phase

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stimulus

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30

;pHz)

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stimulus

frequency

(multiples of firing rate)

Figure 5. Frequency response of model motoneurone for different firing rates. The firing of the model (with AHP and noise) produced by a steady excitatory drive was modulated by a small sinusoidal input (0.25 NU); the gain and phase of its response were determined from cycle histograms relating the timing of its spikes to the stimulus. This was done for three increasing levels of excitatory drive, thereby progressively increasing the model’s mean firing rate (as labelled); the variability of discharge decreased as the firing rate increased (coefficients of variation, 28, 19 and 15%). Top. Stimulus frequency plotted in Hz. Bottom. Normalised to the mean firing rate. The different frequencies were tested one by one; combining them, as in a white noise analysis, might not have given identical plots (from [ 171).

modulated vibration produces a remarkably pure sinusoidal modulation of the firing of individual motor units (‘gure 7). Such cycle histograms are, of course, a statistical average and show the variation of probability of firing throughout the stimulus cycle; on stimulating at 35 Hz, the unit would only have actually fired in every third or so cycle [15]. It was also ‘responding’ to the vibration per se, showing deep modulation at 143 Hz (figure 7, right). 2.4. Carrier dependent

effects for motor units

Modulated vibration conventional frequency

was first used to determine a response for individual motor

units, concentrating on frequencies around their firing rate. Appropriate carrier-type effects on gain and phase were then seen, as found in the model. The interpretation of such findings is, however, confounded by the possibility that they might be partly due to a frequency sensitivity of intemeurones contributing to the reflex response, such as Renshaw cells, rather than to carrier dependence. This can be circumvented by changing the unit’s firing rate without altering the stimulus; this changes the stimulus frequency normalised in terms of the firing rate (figure 5), so any effects on gain or phase can be attributed to stimulus/carrier interactions. The left-hand part offigure 8 shows such carrier dependent effects on exciting the motor unit by stretch as well as

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103 Hz Vibration

Modulation

15 Hz

6 Hz

30 Hz

Figure 6. The novel stimulus:amplitude-modulated pulsed vibration. The waveform of the vibration (top) together with the sine (middle) that modulated the pulse train (bottom) supplied to the electro-mechanical ‘stretcher’. The modulation frequency increases from left to right, with the sweep speed increased in proportion. while the frequency of vibration was held constant. The timing of the pulses drifted continuously in relation to the modulating sine (from [17]).

by modulated dramatic

vibration. The effects on phase are more than those on the gain, with the increase in

firing rate from 8 to 12 Hz producing a phase lag of 60”. The plots also show that the muscle spindle produces a phase advance of about 75” when presented with its normal stimulus of stretch. The gain plot shows the requisite peak at the carrier, but it is much broader than those given by the model on changing the stimulus frequency (Jigure S).Such broadening and relative lack of prominence occurs also with the model when a plot for constant stimulus frequency is constructed from a family of curves for constant firing rate, as injgure 5; this is because these

A. Modulation

10 Hz

‘ol

latter are very assymetrical about their central peak. The right hand part of jigure 8 shows the more prominent effects obtained by combining data for different firing rates and different stimulation frequencies, using solely the modulated vibration (compound stretch data would include the unwanted effects of spindle dynamics). It can be concluded that the occurrence of such changes in responsiveness on changing the stimulus/carrier ratio demonstrates that human motoneurones show marked carrier dependent effects during their normal rhythmic firing, as seen in the model. This will help determine their response to any

. .w. . B. Modulation

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Figure 7. Single unit responses to amplitude-modulated vibration. The moment of firing of a single motor unit was related to the phase of the stimulus to give a cycle histogram, covering two full cycles of response. A, B. Spikes from two different biceps motor units related to the modulating sinusoid, with the points fitted by a sine; the ordinate gives the number of spikes in each bin, expressed as a percentage of the average number per cycle. The fit in A is better than in B because the number of spikes was greater (28807 versus 4810). C. A subset of 6887 spikes from A but now related to the vibration cycle, fitted with an arbitrary curve and with the phase converted to absolute time; the arrows mark the 35 ms delay found for the reflex response to an isolated tap. Both units were firing at 11 Hz. Pulses applied to biceps’ tendon at 143 Hz with a mean amplitude of 0.17 mm, modulated by f 20% (from [17]).

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Figure 8. Effect of changes in the firing rate of a single unit. A. Based on recordings obtained with the stimulus frequency fixed at 10 Hz, but while the unit’s firing rate was varying spontaneously around 10 Hz; this was done for muscle stretch as well as for modulated vibration. The recordings were sub-divided into sub-populations with different on-going mean firing rates and separate cycle histograms constructed for each. From these the gain and phase of the response was calculated and plotted against the stimulus frequency normalised in terms of the on-going firing rate. B. Similar data averaged from the same experiment on varying the stimulus frequency in steps of 1 Hz for 8 to 14 Hz (vibration only; error bars show S.E.M. for values with the same normalised stimulus frequency but different absolute frequencies. The phase measurements were adjusted for the phase lags introduced by simple conduction (from [17]).

rhythmic input, whether generated by higher centres.

at the periphery

or

2.5. Relation to tremor of human tremor remains an The genesis enigma [7], with too many potential mechanisms already known. Moreover, except when there is an abnormal set of ‘pacemaker’ neurones, tremor is an emergent form of behaviour arising from interactions between the various components of the neural networks involved. The stretch reflex provides an inescapable contribution, whether for better or worse, by subjecting every contractile or other mechanical irregularity to regulation by negative feedback. The

phase advance provided by the muscle spindles and motoneurones will partially compensate for the various reflex delays, and thus help prevent the tremor which would inevitably arise if the gain of the reflex was greater than unity at the frequency at which its phase lag reached 180”. Figure 9 continues the modelling to illustrate the potentially destabilising action of the large swings in a motoneurone’s gain and phase around its carrier frequency. This is accentuated by the peaks of the gain and phase plots being slightly out of register; the gain peaks at the carrier, while the phase maximum is appreciably lower. Figure 9A shows an expanded part of the normalised frequency response from figure 5 obtained with the MN firing at 13 Hz. When the stimulus frequency was increased above

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zone

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Figure 9. The danger zone for tremor in the frequency response of a motor unit. The gain and phase of the responses of the model motoneurone ofjgure 5 superimposed on an expanded frequency scale for frequencies of stimulation around its firing rate. The rapid changes in gain and phase interact so at to increase the liability to tremor at frequencies within the danger zone. A. The effect of changing the stimulation frequency while the MNs firing rate was fixed; the X-axis gives the stimulation frequency normalised in terms of the firing rate. B. The effect of changing its firing rate while the stimulation frequency was kept at 10 Hz; the frequency scale now shows the firing rate normalised in terms of the stimulation frequency; this demonstrates the need for an MN to have a minimum firing rate. B is not a simple ‘mirror image’ of A because changing the firing rate changes the variability of the discharge and with it the form of the frequency response, as in figure 5 (from [ 171).

about 0.8 of the carrier (i.e., 10 Hz) the MN’s gain increased by up to 400% while it’s phase advance rapidly diminished, with a final reduction of nearly 90”. This creates a ‘danger zone’ of frequencies for which the stretch reflex must become appreciably less stable, but frank tremor will normally be prevented by the concomitant rapid reduction in the gain of muscle as a mechanical effector (the muscle filter). Figure 9B shows the converse relation obtained with the stimulation frequency fixed at 10 Hz and the firing rate varied, with the X-axis now giving the firing rate. If the MN were initially firing at 12 Hz (i.e., 1.2 normalised) then any reduction in its firing frequency would push it into the ‘danger zone’ with rapid increase in gain accompanied by loss of phase advance. The low gain of muscle at 10 Hz (especially when operating into inertial loads) can again be expected to prevent tremor. However, if the stretching had been at 5 Hz with the MN initially firing at 6 Hz then the muscle filter might well have been unable to stop tremor occurring, as the firing rate was reduced into the danger zone. Such unwanted instability could be readily avoided by preventing the MN ever firing at such a low rate, with the particular value set by the speed of the muscle fibres supplied by the particular MN. This offers new insight into the desirable functional consequences of the fact that every motoneurone has a minimum rate below which it refuses to discharge, with fast motoneurones supplying fast muscle

fibres having motoneurones

higher minimal firing rates than slow with slow muscle fibres [IO].

3. Final discussion

and conclusions

Long periods of firing of single human motor units were recorded with surface electrodes during their ‘steady’ voluntary activation. The interval histograms were used to estimate the trajectory of motoneuronal membrane potential for the final part of the inter-spike interval during which firing occurred. This was based on a newly developed transform relating the membrane potential to the probability that a given interval would be terminated in the next msec (interval ‘death rate’ or hazard function). The potential was not given in mV, but in noise units equal to the standard deviation of the ongoing synaptic noise. Moreover, the value given was the potential relative to the ongoing value of the threshold, which itself changes progressively during the course of the interspike interval. This, the ‘effective’ trajectory is what determines the tonic firing of the motoneurone and its response to stimulation, rather than the absolute value of the membrane potential in mV. The transform was determined empirically from a simplified model of the motoneurone and its precise parameters varied with the time constant of the MN. This was unknown for any real MN studied, thus

145

Tonic firing of motor units analysed

limiting the numerical accuracy in deducing its trajectory. None the less, useful estimates could be made of the time constant of the MN’s post-spike afterhyperpolarisation and its synaptic noise in mV. It was notable that at low firing rates the mean trajectory never reached threshold, and the spikes were triggered by noise. This last conclusion is based directly on the data, and seems secures irrespective of the numerical details of the model and its underlying assumptions. When the model MN was excited by a small sinusoidal input it showed large changes in the gain and phase of its response (estimated from cycle histograms locked to the stimulus) on varying the stimulus frequency. As already known for other neuronal models, these effects depend upon the relation between the stimulus frequency and the firing rate (carrier dependence), and became less marked when the synaptic noise was increased. In life, such behaviour will condition the response to any rhythmic input, whether generated peripherally by the receptors or by higher neural centres (as with cortical oscillations). A particular feature of the present model MN was that, as usual, the gain peaked when the stimulus frequency was equal to the MN’s firing rate (the carrier), but the phase advance began to decrease rapidly as the stimulus frequency was increased to above 0.8 times the carrier. Human motoneurones behaved similarly, with important implications for the genesis of tremor and its relation to their minimum firing rate. The motoneurones were rhythmically activated by reflex action, produced either by simple muscle stretch or, more crucially, by the novel stimulus of amplitude modulated vibration which should eliminate phase changes from muscle spindles. It produced appreciably less phase advance in the motoneurone than simple stretch; the difference gives the spindle’s phase advance. References [II

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