Measurement of excitability of tonically firing neurones tested in a variable-threshold model motoneurone

Abstract

A new measure of excitability of tonically firing neurones termed the ‘estimated potential’ (EP) was tested on a model motoneurone (MN) with a voltage-dependent threshold; the threshold followed the noisy membrane potential with an exponential delay. First, the model MN's after-hyperpolarisation (AHP) was deduced from its interval histogram for tonic firing, using a recently described transform. This provided a ‘distance-to-threshold’ measure which underestimated the AHP's absolute size but had the same time course, thereby providing the time constant of the AHP's decay of conductance. The ‘estimated potential’ was then obtained from the classical ‘firing index’ (no. of responses/no. of stimuli) by using the estimated AHP to create a fixed threshold ‘daughter’ model MN to mimic its variable threshold parent and reproduce its input-output non-linearities. The EP gave a linear measure of the parent's stimulus-evoked depolarisation for firing indices up to about 60 %, corresponding to depolarisations of three to four times the noise s.d. The EP was scaled in units of voltage whose absolute value will usually be unknown for real neurones, since it depended upon the details of the parent model. The EP's virtue is that, within its range, combining stimuli gives arithmetical addition and subtraction, thereby improving on the firing index which scales sigmoidally with the input. Moreover, with weak stimuli, the EP for a given input did not change on varying the parent model's firing rate. The estimated ‘distance-to-threshold’ AHP did not, however, give an accurate measure of the recovery of excitability following a spike during tonic firing. Excitability then depends upon the ‘survivors’ trajectory’ giving the mean membrane potential, relative to threshold, of those intervals which have ‘survived’ up to the time in question rather than upon the AHP per se; the survivors’ mean is more hyperpolarised because spiking preferentially eliminates trajectories with noise-induced depolarisation.

Debate continues on the classical neurophysiological problem of how best to estimate the trajectory of a neurone's membrane potential from a spike train, with the situation complicated by a variety of non-linearities (Moore et al. 1966; Kirkwood, 1979; Koch, 1999; Powers & Binder,2001). Particular attention has been focussed on how to use the post-stimulus time histogram (PSTH) to deduce the underlying effect of an excitatory or inhibitory synaptic input since no single algebraical transform suffices for all conditions. The motoneurone (MN) has been studied intensely, partly because of its experimental convenience and partly to assist in the interpretation of human motor unit recordings. Like a variety of other neurones its spikes are initiated by the summing of synaptic potentials at a single site (the axon's initial segment), and so it provides a useful general model with its behaviour potentially widely applicable. One of its special features is its unusually prolonged post-spike after-hyperpolarisation (AHP) which plays an important part in setting its rate of firing for a maintained input and helps determine its moment-to-moment ‘excitability’ during tonic firing; the shorter recovery cycles of other neurones will play a similar role, so the MN differs quantitatively rather than qualitatively.

In man, using motor unit recording, the trajectory of the MN's AHP has typically been probed using spike-triggered stimulation (Ashby & Zilm, 1982), again requiring interpretation of the PSTH as with synaptic inputs. A new approach was provided by Matthews’ (1996) suggestion that the size and duration of the AHP could be deduced by applying a novel non-linear transform to its inter-spike interval histogram, determined for steady tonic firing induced by synaptic or other noisy input, as explained in Methods. This prediction was based on studying a simple model MN with a fixed threshold. It was subsequently tested experimentally by Powers & Binder (2000) for cat spinal motoneurones using intracellular stimulation and recording. They found that the AHP's time course was indeed usually predicted correctly, but the absolute size of the AHP measured in millivolts was almost always larger than the estimate (mean ratio = 1.9). This discrepancy in absolute size caused no surprise because the transform provides a ‘distance-to-threshold’ estimate of the AHP, relative to the ongoing threshold for firing, not to a fixed membrane potential. The threshold is known to vary continuously throughout the AHP so that spike triggering occurs with the membrane more hyperpolarised than normal (Calvin, 1974; Powers & Binder, 1996); moreover, the reduction in threshold mirrors the absolute AHP so the estimated AHP should approximate to a scaled down version of the actual AHP. Matthews’ (1996) false simplifying assumption of a fixed threshold thus proved serendipitous since it means that estimates of AHP, like his in man, incorporate the change of threshold and so should provide a closer measure of the ongoing excitability of the MN to synaptic inputs than would a knowledge of the absolute size of the AHP itself.

The present paper extends the earlier motoneuronal modelling (Matthews, 1996, 1999b, 2000) by making the model's firing threshold voltage dependent. The threshold was set to vary continuously with the prevailing membrane potential, rising and falling exponentially with its own time constant as it followed the declining AHP with its accompanying noise transients. Three issues relevant to the interpretation of the PSTH were given particular attention. (1) How closely does the estimate of the ‘distance-to-threshold’ AHP correspond to actuality as the threshold varies during the AHP? Contrasting with the constant threshold model, the agreement was no longer exact; but the estimate continued to provide a good measure of the AHP's duration. (2) How well does this estimate for an isolated AHP provide a moment-to-moment measure of excitability during tonic firing? Contrary to simple expectation, the noise-free AHP fails to provide an exact guide to the recovery of excitability during noisy tonic firing; this is due to the temporal structure of the noise combined with the ‘absorbing barrier’ provided by the threshold. (3) How far can the estimated trajectory be used to interpret the PSTH, obtained with randomly timed stimulation, in terms of the underlying change in membrane potential? A novel measure, termed the ‘estimated potential’ (EP) was derived from the PSTH by using the estimated ‘distance-to-threshold’ AHP to create a simplified constant-threshold ‘daughter’ MN to mimic the behaviour of its more complex ‘mother’ MN with its voltage-dependent threshold. The daughter's response to a range of excitatory inputs can then be studied to determine how much depolarisation is required to make its response, expressed as a firing index, the same as the mother's; the requisite depolarization is defined as the EP. The EP estimates the mother's stimulus-evoked depolarisation in units of voltage, but the scaling factor will be unknown for real MNs as it depends upon the mother's noise and the way in which the threshold varies. It improves on the conventional firing index (percentage of responses per stimulus), from which it is derived, by scaling linearly with the depolarisation; in contrast, the firing index varies sigmoidally so that the effect of combining two stimuli can swing from being larger to being smaller than their arithmetical sum. In essence, this defect is overcome by using the daughter to reproduce the mother's non-linearities so that they can be allowed for and thus cease to matter.

The new measure of excitability, the EP, should be readily determinable from the conventional firing index for real MNs; the only further knowledge required of the MN is the trajectory of its distance to threshold AHP for the particular firing rate studied, as derived from its interval histogram. This is approximated by an exponential and requires only three parameters, thereby bypassing the complexities of modelling the MN with a variety of individual ionic conductances with uncertain kinetics. The EP's efficacy should be readily verifiable using standard techniques. The EP can thus be hoped to provide a valuable tool in the armentarium for human neurophysiology, as in studying the interaction of various inputs converging upon the MN such as a reflex input and a cortically evoked descending volley. In addition, in modelling the behaviour of any group of neurones of a particular type (such as the MN), experimentally derived data could be used to create a realistic pool of ‘daughter’ neurones for inclusion in a simulated network. The important general principle is that measurements of excitability can be improved by using a neurone's own firing data to create a simplified daughter model with a similarly non-linear input-output relation.

METHODS

The two simple threshold-crossing models studied have both been used before. The very simple fixed-threshold ‘daughter’ model, operating solely in terms of voltage, had earlier provided the transform used to estimate the AHP (Matthews, 1996); it was currently used to mimic the behaviour of its more complex ‘mother’. The ‘mother’ had a ‘membrane’ with capacitance and conductance and was excited by injected current (Matthews, 1999b, 2000). The essential new feature was to make the conductance model's threshold for firing depend upon the pre-existing membrane depolarisation (voltage-dependent threshold); when the conductance model's threshold is fixed, transforming its interval histogram provides an accurate estimate of its AHP (Matthews, 2000). Both models have a single compartment and an exponentially decaying AHP; they use lumped values and ignore the variety of individual ion channels involved. The daughter's synaptic noise was a time-smoothed voltage; the mother's voltage noise was produced by exciting it with a noisy current, with the smoothing produced by the membrane capacitance. A fuller conductance model used earlier (Matthews, 1999b, 2000) had excitatory and inhibitory synaptic conductances to generate its noisy net current drive and then behaved very similarly to the present mother model with only leak and AHP conductances; both still behaved similarly when their threshold was made voltage dependent. The simpler conductance model was presently preferred as it has the advantage of bypassing the uncertainties arising from any boosting of the conductance-induced synaptic currents by further voltage-dependent ‘plateau potential’ type inward current (Lee & Heckman, 2000; Prather et al. 2001).

Threshold change

The model was given a variable threshold by making this depend on the membrane potential, measured slightly earlier in time. The time relation was made a simple exponential. For this, the model was given a set threshold for a very slowly increasing depolarisation (the equilibrium value). Any deviation of membrane potential from this set threshold evoked a threshold shift proportional to the deviation; following a change of potential the threshold rose or fell exponentially towards its new value. The calculation was performed for each successive millisecond; the last threshold value decayed to provide a residual contribution which was then summed with a new value set by the last value of membrane potential (it rose exponentially throughout the duration of the 1 ms bin). The ‘threshold (gain) factor’ was varied in the range 0.3-0.8. The ‘lag time constant’, setting the exponential delay between voltage and threshold, was varied in the range 2.5-7.5 ms.

Principle of estimation of AHP from interval histogram

This followed previous practice (Matthews, 1996,2000; Powers & Binder, 2000); Fig. 1 is a schema of the processing involved. The spike trains discharged by the noisy model MN under study are used to create a standard interval histogram (A). This is then converted into a plot of number of intervals that have ‘survived’ up to each post-spike time T shown in Fig. 1B (i.e. by eliminating shorter intervals, terminated by a spike occurring before T). This is used to calculate the interval death rate or hazard function, given in Fig. 1C, corresponding to the probability that a given interval that had survived up to time T would be terminated by a spike in the next millisecond (i.e. intervals discharged at T divided by remaining survivors; each point in Fig. 1A is divided by the corresponding value in B, with minor adjustments required to deal with finite binwidth). Finally the new transform is used to convert each point on the death rate plot into an estimate of the trajectory of the membrane potential during the AHP (Fig. 1D); this included the contribution provided by the steady depolarising drive. The trajectory is given as a voltage but it is scaled in noise units (NU), corresponding to the standard deviation of the noise, rather than millivolts; moreover, zero voltage corresponds to the threshold for spike initiation rather than absolute zero.

Figure 1. Procedure for estimating the AHP from the interval histogram.

Schema illustrating the stages involved in computing the distance-to-threshold AHP from the interval histogram (A, B, C, D) and in determining the transform used to achieve this (E, F). An explanation is given in the text. Potentials are scaled in noise units (NU) corresponding to the standard deviation in millivolts of the model's synaptic noise, with zero corresponding to the threshold for firing.

The transform depends upon the statistics of the time-smoothed noise and is derived by determining the steady-state relation between the amount of noise-induced firing and a constant sub-threshold depolarising drive (or current) applied to a threshold-crossing detector (i.e. a neurone without an AHP). Noise-induced transients periodically push the voltage above threshold and trigger a spike; increasing the sub-threshold drive brings the membrane voltage closer to threshold and increases the firing rate for a given noise level. Conversely, increasing the noise increases the firing for a given drive; this explains how the ‘noise unit’ arises as a useful ‘natural’ measure of membrane potential relative to threshold. The interval histogram for this artificial spike train decays exponentially, because the mean membrane potential is constant, giving a constant ‘death rate’ which is determined (intervals below 10 ms ignored to allow membrane potential to reach a steady state and to prevent any given noise transient exciting twice). Reiteration with a series of drives produces plot Fig. 1E giving the death rate against the depolarising drive. The axes are then inverted to give Fig. 1F in which the depolarisation is plotted against the death rate. This now provides a calibration curve which allows the death rate plot in Fig. 1C for the original spike train to be converted into an estimate of the AHP. For computational convenience it is best to use an arbitrary algebraic transform fitted to the curve of Fig. 1F (a good fit was obtained by summing two separate exponentials, each decaying with its own time constant and starting and finishing levels). Custom-built programs then allow an interval histogram to be rapidly converted into an estimated trajectory.

Details of AHP estimation

The model was run for 30–60 min of simulated time to yield over 10 000 spikes to create an interval histogram with 1 ms bins. This was then converted into a plot of interval death rate (hazard function, probability of an interval terminating or ‘dying’) against time since last spike, using the equation:

where P is the probability of a spike being discharged per unit time, averaged over the bin; N₀ and N₁ are the sum of all subsequent spikes in the interval histogram; N₀ starts with and includes the count for the bin whose P is being measured, and N₁ starts with the immediately subsequent bin; ln is the natural logarithm. The death rate was then transformed into a point-by-point estimate of the model MN's ‘distance-to-threshold’ AHP using Matthews’ (1996) transform consisting of the sum of two exponentials namely:

The voltage V is scaled in noise units (NU) equivalent to the s.d. of the noise voltage; the scale starts at threshold and negative values correspond to hyperpolarisation. For the present models with a membrane time constant of 4 ms, A, C and E are −1.054, −3.096 and +1.276 NU, respectively; B and D are 0.0120 and 0.2039 probability units, respectively.

Finally, the following points should be noted. (1) The time structure of the noise affects the precise form of the transform; this depends upon the smoothing introduced by the membrane capacitance and the time course of synaptic currents. Matthews (1996) assumed the noise was exponentially smoothed with an effective time constant of 5 ms; this was continued. (2) The static calibration of Fig. 1F was applied to the dynamic situation of D; however, due to the slowness of the AHP, the resulting errors are small (Matthews, 2000). (3) Matthews (1996) deduced the transform using a model in which all variables were expressed as voltage. It remains applicable to the presently used model with membrane capacitance and resting plus AHP conductances stimulated with a noisy current (Matthews, 2000). (4) The AHP provided by the transform corresponds to the underlying AHP occurring in the absence of noise and with a fixed threshold, rather than that obtained by averaging during maintained firing (see Results); this also applies to Powers and Binder's transforms for individually studied real MNs (Powers & Binder, 2000).

Direct determination of AHP

The AHP and related trajectories for the variable-threshold model were determined by direct measurement while running the model. The absolute AHP, dependent simply on the interaction of the conductances involved, was determined by removing the noise to prevent premature excitation; the same result would be achieved by averaging noisy trajectories after inactivating the spiking. The ‘distance-to-threshold’ AHP was obtained by calculating the voltage-dependent threshold shift from the absolute AHP, using the chosen paradigm, and then subtracting this from the AHP. The ‘survivors’ trajectory’ was determined by averaging the voltage of those trajectories which had survived up to each particular time; thus, each successive bin was based on a smaller sample. These values were initially measured in millivolts and then converted into noise units by inactivating spiking and measuring the voltage noise with the mean membrane voltage held at threshold by injecting current.

Stimulation and subsequent analysis

The stimulus consisted of a pulse occupying a single 1 ms bin; it may be equally thought of as due to synaptic action or to an extrinsic electrical stimulus. The interval between stimuli varied randomly in the range 300–400 ms giving a mean stimulation rate of 2.9 Hz; the timing of the stimulus was unrelated to that of the preceding spike, at any rate for the weak stimuli of present interest (cf. Matthews, 1999b) In the simple fixed-threshold voltage-operated daughter model the stimulus consisted of a voltage pulse in the single bin with no subsequent carry-over; its magnitude corresponded to the peak of the simulated EPSP. In the mother model, with conductance, a 1 ms current pulse was used; this charged the membrane capacitance over the course of the bin to give a step of voltage which then decayed. Except for the weakest stimuli, most spikes were then initiated at the end of the bin, on completion of the stimulus, corresponding to the peak of a motoneurone's ‘EPSP’. Any few on the stimulated EPSP's falling phase were ignored in the analysis presented; control simulations showed that similar results were obtained when any such delayed response was included (cf. Matthews, 1999b).

The magnitude of the EPSP produced by the current pulse was measured after the AHP conductance had decayed to zero; it was thus determined by the capacitance and leak resistance of the model's membrane. In control trials, the average value of the EPSP was determined during tonic firing at 8–16 Hz. It was then slightly reduced by shunting from the variable residual AHP conductance (reduction 2.5-4 %, increasing with firing rate as earlier stages of the AHP were on average probed by the stimulus).

The response to a stimulus was estimated from a standard post-stimulus time histogram (PSTH) with a binwidth of 1 ms and expressed as a firing index. This gave the percentage of stimuli which elicited a spike; it was calculated using the number of spikes in the stimulus bin minus those which would have occurred by chance for the given firing rate. The mean rate was sometimes determined in separate stimulus-free trials and sometimes with equivalent results from the 30 ms preceding the stimulus.

In a few control trials the duration of the mother's constant stimulating current was extended to 2 ms, thereby increasing the rise time of the resulting simulated EPSP from 1 to 2 ms; its peak value was measured as before. The response was then taken as the sum of the additional spikes occurring during the rise of the EPSP, corresponding to a cusum measured over 2 ms. The daughter's stimulating pulse was, however, kept at 1 ms and the response measured as before. This means that the daughter is used to estimate the peak size of the mother's EPSP independent of the duration of its rising phase.

Numerical values

As before (Matthews, 1999b), the various parameters of the standard model were chosen to make it reproduce the behaviour of human MNs, assuming a fixed threshold. They were as follows: leak conductance, 0.5 μS with an equilibrium potential of 0 mV; AHP conductance, 0.4 μS with an equilibrium potential of −15 mV and a time constant of 30 ms; standard threshold, +15 mV; tonic excitatory current, 6-9.5 nA, varied as required to produce the desired mean firing rate; Gaussian ‘synaptic’ current noise, mean 0, s.d. ± 2 nA producing ±1.4 mV resting voltage noise; membrane capacitance, 2 nF making the resting membrane time constant 4 ms. All values in the simple voltage model were expressed relative to threshold and measured in noise units corresponding to the standard deviation of the noise; this had a Gaussian amplitude distribution, and was exponentially time-smoothed with a 4 ms time constant. All calculations were done using a step size of 1 ms; this introduces some high-frequency filtering making the models behave as if they had a time constant of 5 ms rather than the stated 4 ms (Matthews, 1996). The exponential dependence of membrane voltage on current and threshold on voltage was computed as described by MacGregor (1987); this, so to speak, allows the membrane voltage to change exponentially throughout the course of 1 ms bin at a rate dependent on the total membrane current, though only its final value becomes available.

RESULTS

The ‘distance-to-threshold’ AHP

Voltage-independent variation

The previous studies (Matthews, 1996, 1999b, 2000) examined the behaviour of model MNs with a fixed threshold. The first modification was to vary the threshold after each spike along a preset time course, independent of the AHP and noise voltages and free of rapid transients. As before, the interval histogram for tonic firing induced by a noisy input was used to estimate the AHP, measured relative to the ongoing threshold and with its value scaled in noise units (NU) equalling the standard deviation of the voltage noise. This ‘distance-to-threshold’ estimate provided a good approximation to the value for a noise-free AHP (i.e. absolute AHP minus the varying threshold) and thereby inevitably mis-estimated the absolute size of the AHP relative to a fixed baseline.

Voltage-dependent threshold

Thereafter the threshold was always made to follow the ongoing noisy membrane potential with a slight delay (voltage-dependent threshold); for real MNs, hyperpolarisation facilitates spike triggering by reducing inactivation of Na⁺ channels. Accurate modelling can only be achieved when the kinetics of all the relevant channels are known. The simplified models were continued with the aim of establishing general principles and the threshold was made to lag behind the voltage with a distributed delay (cf. MacGregor, 1987). Following each change in potential, the threshold rose or fell exponentially from its existing value towards a new final equilibrium value, equalling K(V - V_o), with the values updated every millisecond; V_o was a fixed ‘resting’ threshold corresponding to the threshold for a very slow noise-free depolarisation and V was the ongoing voltage. The model's behaviour then depends upon the amount of delay (exponential time constant) and K the arbitrary ‘threshold factor’ (normally made 0.5). With an appreciable delay, the threshold largely follows the mean value of the AHP though lagging well behind it. With a short delay, the threshold also follows the moment-to-moment noise-induced swings in membrane potential, differing for every individual trial. The model MN was therefore routinely studied using two different delays, chosen to be ‘long’ (7.5 ms) or ‘short’ (2.5 ms) relative to the membrane time constant responsible for setting the time course of the noise (4 ms). A time lag of zero is equivalent to reducing the size of the whole of the AHP by the threshold factor, thereby providing a special case of the ‘voltage-independent variation’ model (see above). In the event, no striking differences emerged for the two delays, though larger currents were required to elicit comparable firing with the shorter delay.

Figure 2 shows the model's behaviour at low firing rates for which the interval histogram had an exponential tail and the estimated AHP approximately reached its final equilibrium. The ‘threshold factor’ was 0.5; this would have halved the effective AHP for a time lag of zero. Thus the ‘distance-to-threshold’ AHP determined from the interval histogram (upper points) grossly underestimated the size of the absolute AHP (continuous line, below); measured at equilibrium, the AHP was almost precisely twice the estimate for the 7.5 ms lag (left), but it was only 1.8 times larger for the 2.5 ms lag (right). The absolute AHP was determined following an isolated spike, with noise eliminated to prevent premature termination. Nonetheless, when suitably normalised (lower points), the estimated AHP proved to have the same time course as the absolute AHP. The normalisation followed that used by Powers & Binder (2000) for real MNs, for which agreement was again normally obtained (the tails of the AHPs are first aligned and then the rest of the AHP linearly rescaled, using the tail as the reference). The scaling factor was then 1.9 for the 7.5 ms lag and 1.4 for the 2.5 ms lag rather than 2.0. Thus the scaling factor depends upon the overall kinetics and so the values obtained for real MNs cannot be taken to provide a direct measure of the shift in threshold per unit change in membrane potential.

Figure 2. Comparison of actual and estimated AHPs in a model MN whose threshold followed the voltage with a time lag.

Plots of ‘membrane potential’ (in millivolts from threshold) against time-since-spike for variable-threshold model MNs with different exponential delays (A, B) when the firing rate was low; the AHP decayed to its final equilibrium and the interval histogram had a long exponential tail (not illustrated). Upper data points, the ‘distance-from-threshold AHP’ estimated by transforming the interval histogram for tonic firing. The dashed line shows the comparable ‘distance from threshold’ plot for a single AHP with both noise and spiking eliminated; it deviates initially, but provides an acceptable fit near equilibrium. The bottom continuous line gives the absolute AHP following a single spike (relative to the reference threshold with the voltage dependence inactivated). The original data points have now been shifted and then rescaled; the excellent fit shows that while reduced in size the estimated ‘distance-to-threshold’ AHP has the same time course as the absolute AHP. (The estimated AHP was first moved down to align its final tail with that of the absolute AHP; it was then linearly re-scaled, using the tail as baseline; shift, 1.5 and 1.4 NU, scaling factor, 1.9 and 1.4 for A and B, respectively. The ‘threshold factor’ relating threshold change to potential was 0.5 for both A and B; i.e. when the membrane was steadily hyperpolarised by 10 mV relative to the standard threshold the ‘distance-to-threshold’ was only 5 mV. The ‘delay’ is the time constant for the exponential change in threshold following a step change in membrane potential. Drive current 5.5 nA for A and 5.7 nA for B; both had 2 nA input noise producing 1.4 mV voltage noise.)

It had been hoped that the ‘distance-to-threshold’ estimate from the interval histogram would correspond to that for the isolated AHP; but this proved not to be so. In Fig. 2A the estimate (upper points) corresponds to the calculated value (dashed line) near equilibrium, but not at the start of the AHP. The difference is larger in Fig. 2B, and also occurs at equilibrium. Figure 3 (upper plots) shows the yet larger deviations found on using a higher firing rate to display an earlier, steeper, part of the AHP. Again, of course, the ‘distance-to-threshold’ estimate (upper points) was smaller than the absolute AHP (lower, continuous line). But renormalising the estimate (lower points) shows that it still has the same time course as the absolute AHP (continuous line); the scaling factor was reduced yet further below the notional value of 2.0 (see legend), emphasising the importance of the situation's dynamics.

Figure 3. Comparison of steeply rising AHPs seen with higher firing rates.

The drive has been increased from Fig. 2 so that the AHP comes within range of threshold earlier in its course; the interval histogram is then shifted to the left, so that the transform recovers an earlier part of the AHP where it is steeper. Analysis as in Fig. 1. Upper data points, estimated ‘distance-to-threshold’ AHP; dashed line, as measured without spiking. The curve now entirely fails to fit the points. Continuous line, absolute AHP following a single spike with the data points rescaled to fit. The ‘estimated’ AHP still has the same time course as the absolute AHP (shift −1.3 and −1.1 NU, scaling factor 1.7 and 1.3, drive current 7.1 and 7.4 nA, for A and B, respectively).

It must be concluded that the ‘distance-to-threshold’ AHP determined from the interval histogram provides only a rough measure of the excitability of noisy real MNs during the course of the AHP, since the kinetics of the factors regulating their threshold are unknown. In contrast, the AHP given by the transform has proved to provide a rather good estimate of the time course of the model's actual AHP, as also usually found for real MNs (Powers & Binder, 2000). Thus the interval histogram can be used to estimate the time constant of the AHP's exponential tail, and thereby that of the exponential decay of a real MN's underlying hyperpolarising conductance.

The ‘survivors’ trajectory’, not the AHP per se, gives the moment-to-moment excitability during tonic firing

In the absence of noise, the trajectory of the ‘distance-from-threshold’ AHP provides a good measure of a MN's ongoing excitability. For any given post-spike interval a just threshold stimulus is that which depolarises the membrane by the amount that the AHP has hyperpolarised it relative to the ongoing threshold; firing then invariably occurs. Noise complicates the situation in two ways. First, the ‘threshold stimulus’ becomes a statistical measure rather than an absolute one; the individual noisy trajectories are scattered about their mean leading to a sigmoid input-output relation as in Fig. 6 (cf. Matthews, 1999b). A useful reference value is then the stimulus that elicits firing on 50 % of trials (i.e. gives a firing index of 50 %). The second complication is that any individual trajectory which reaches threshold triggers a spike and is then eliminated from further consideration and stops contributing to the average; there is no comparable ‘absorption barrier’ for hyperpolarisation. Thus, relative to the noise-free trajectory, there will be an excess of hyperpolarised trajectories and a deficit of depolarised ones. In consequence, with noisy firing, the mean AHP of the survivors at any given time after the preceding spike will be hyperpolarised relative to the noise-free AHP, with consequent reduction of the MN's ongoing excitability to a test stimulus. Thus what may be termed the ‘survivors’ mean trajectory’ will provide a better measure of a MN's moment-to-moment excitability during noisy tonic firing than does its AHP determined in the absence of firing.

Figure 6. Determining the new index of excitation, the ‘estimated potential’ (EP), for the mother from the daughter.

A, input-output plots of firing index against magnitude of the depolarisation evoked by a 1 ms stimulus for mother and daughter models of Fig. 2A firing at 9 Hz. The dashed curve corresponds to the mother's value aligned with the daughter's curve by rescaling its voltage axis (each x-value multiplied by 1.1). B, similar data for the model of Fig. 3B (scaling factor, 1.33). After the rescaling the mother and daughter curves correspond along much of their length. The straight dashed lines show how to determine the new excitation index (the ‘estimated potential’ or EP) measuring the effectiveness of a given stimulus to the mother. It is defined as the excitatory depolarisation of the daughter that makes its firing index the same as that of the mother for the stimulus in question. The value of the EP for a given firing index depends on the details of the parent model; in this case a mother's firing index of 50 % corresponded to an EP of 2.9 in A and 3.45 in B. (As measured, the EP is expressed in the daughter's noise units, but from the point of view of the parent these are best termed effective noise units or ENU; the relation between ENU and the mother's NU depends on the mother's details.)

‘Survivors’ trajectory’ of constant-threshold model

Figure 4 illustrates these points using a simple ‘daughter type’ voltage model, with a fixed threshold and without conductance, to emphasise that they are an inevitable consequence of the statistics of threshold-crossing. Figure 4A shows two extreme individual trajectories obtained when the mean firing was 10 Hz. The prolonged trajectory would have been terminated earlier, by spiking, if the depolarising transients had been larger. Figure 4B shows that the mean value of the ‘survivors’ trajectories’ (thick line) falls progressively below the noise free AHP (thin line) as threshold is approached. The median value of the survivors’ trajectories lay very slightly above the mean, but imperceptibly so in terms of the present plot. The median (50 % percentile) is the more important value because half the trajectories lie closer to threshold than the median, so a stimulus corresponding to the median will give a firing index of 50 % (the mean and median were calculated excluding the survivor's final suprathreshold values, corresponding to the elimination of ‘spontaneous’ spikes from the firing index). The dashed lines give various other percentiles, corresponding to the stimuli required to elicit a firing index of the given value. At the beginning of the trajectory the percentiles lie symmetrically around the mean, and correspond to a Gaussian distribution. At the end of the trajectory the distribution becomes asymmetric, with the lower percentiles more closely bunched than the higher values. Thus the shape of the input-output plot (firing index vs. stimulus strength) for a spike-locked stimulus must change throughout the course of the interspike interval. The importance of knowing the survivors’ behaviour rather than the AHP is shown by the fact that at long intervals the 25 % percentile, corresponding to a firing index of 25 %, lies the same distance from threshold as the actual AHP; knowing just the AHP, the firing index might have been assumed to be 50 %. Figure 4C shows the relation between the ‘survivor's mean’ and the AHP. The difference between them increases as threshold is approached showing that the statistical effects of the ‘absorption barrier’ then become ever more important; this is presumably because a higher proportion of the surviving trajectories are eliminated every millisecond (i.e. the interval death rate increases; Matthews, 1996).

Figure 4. ‘The survivors’ trajectory’ compared with the noise-free AHP.

The simple voltage-operated model with a constant threshold used to demonstrate the statistical effects of threshold crossing, free of any conductance non-linearities. A, two extreme examples of noisy individual post-spike voltage trajectories during repetitive firing (thin lines) compared with the noise-free AHP following an isolated spike (thick line); potentials given in noise units above threshold. B, the thicker continuous line gives the mean value of those trajectories which survived up to each point in time (‘survivors’ mean trajectory’); as threshold was approached, the survivors’ mean fell below the noise-free AHP (thin continuous line). The dashed lines give the percentage of the surviving trajectories which lay above each curve (percentiles, values above threshold excluded from calculation); these correspond to plots of the stimulus-evoked depolarisation required to elicit the corresponding firing indices. C, the survivors’ mean voltage plotted against the AHP voltage; line of equality shown as a continuous line. (Firing rate, 10 Hz; membrane time constant, 4 ms; AHP time constant, 30 ms; AHP size, 30 NU; synaptic drive final equilibrium, −0.5 NU; in the later part of B, the 50 % percentile or median was just above the mean, but by less than the width of the line. The data in C was produced using an AHP that declined linearly, at 0.05 NU ms⁻¹, rather than exponentially; the plot agreed with that obtained on reducing the slope to 0.02 NU ms⁻¹ showing that it corresponded to the equilibrium relation, even though it was obtained dynamically.)

For similar reasons the survivors’ trajectories for the conductance model of Fig. 1 and Fig. 2 were found to differ both from the actual and the estimated AHPs, with all three measured relative to the ongoing threshold. The conclusion is that during firing estimates of post-spike excitability, whether of models or real MNs, should be based on the survivors’ trajectory rather than the ordinary AHP. This presents no problem when the threshold is constant because the survivors’ trajectory and its various percentiles can, in principle, be deduced from the interval histogram in the same way as the ordinary AHP by appropriately modifying the transform (i.e. the transform for the survivors’ mean would combine the usual transform of Fig. 1F with the relation of Fig. 4C). However this fails when the threshold varies, because the ordinary transform no longer provides an accurate estimate of the ‘distance-to-threshold’ AHP (Fig. 2 and Fig. 3); moreover, the relation between the survivors’ trajectory and the ‘distance-to-threshold’ estimate varies with the details of the model (see later). Thus accurate prediction of the excitability of real MNs throughout the course of the AHP during tonic firing must await detailed knowledge of their voltage-dependent threshold variation.

Creation of fixed-threshold ‘daughter’ model to assess response of variable-threshold ‘parent’ to random stimulation

The stimulus-response relation for the variable-threshold model MN exposed to randomly timed stimuli can be expected to be complex; that for the fixed-threshold model already depends upon a variety of factors, including especially the firing rate and the size of the stimulus in relation to the background noise level (Matthews, 1999b). Remarkably, over much of its course the relation for the complex model has proved to be similar to that for the simpler model, differing simply in scaling. This enables a response seen in the post stimulus time histogram (PSTH) of a ‘parent’ or ‘mother’ variable-threshold model to be interpreted by studying the behaviour of a fixed-threshold ‘daughter’ model, derived using the mother's estimated ‘distance-to-threshold’ AHP for the particular trajectory at issue. For any particular firing rate the relevant portion of the trajectory was approximated by an exponential with three parameters corresponding to the AHP's time constant and two notional membrane potentials, for its starting value and final equilibrium. Figure 5 introduces the workings of these related models.

Figure 5. Trajectories and interval histograms of fixed-threshold ‘daughter’ model MN correspond to those of its variable-threshold ‘parent’.

A, ‘distance-to-threshold’ AHP's estimated from the interval histograms shown in B; the values for mother and daughter overlap in both A and B (▪, parent model; +, daughter). The mother's values reproduce those in Figs 2 and 3 for the 7.5 ms lag but with the ordinate now scaled in noise units (NU), corresponding to the standard deviation of the voltage noise (1 NU = 1.4 mV). The continuous lines show exponential curves fitted to the mother's points; these were then used as the input AHP for the daughter. (The fixed-threshold daughter model operated solely in terms of voltage scaled in noise units; the AHPs started at −6.5 and −7.4 NU at time zero, came to equilibrium at −1.4 and +0.5 NU with time constants of 33 and 38 ms; constraining the fitted AHPs’ time constants to the parent's actual value of 30 ms slightly changed the other values without altering the daughter's behaviour.)

Figure 5A shows the identical ‘distance-to-threshold’ values for mother (▪) and daughter (+) AHP's determined point-by-point from the interval histograms in Fig. 5B for two different firing rates; the values are expressed in noise units (NU) corresponding to the s.d. of the voltage noise. The continuous lines in A are exponentials; each has been fitted to the mother's points, and then used as the input AHP for the daughter. Since the daughter has a constant threshold, its ‘distance-to-threshold’ AHP is the same as its absolute AHP (Matthews, 1996,2000). Further simplification is achieved by modelling the daughter solely in terms of voltage (Matthews, 1996). Figure 5B shows the equivalence of the mother's and daughter's interval histograms, obtained during tonic firing. This is inevitable; the histograms represent a direct transformation of the AHPs, and the daughter's AHP has been made the same as that of the mother. Thus the daughter provides a simplified model of the mother.

Their behaviour was not, however, identical in all respects. The daughter's response to a given brief stimulus-evoked depolarisation was always smaller than that of the mother, as shown by comparing their firing indices for random stimulation. The firing index (FI) is defined as the percentage of times a spike is discharged in association with the stimulus over and above that expected by chance. During the 9 Hz firing of Fig. 5, for example, the parent's firing index was 50 % for an ‘EPSP’ with a peak depolarisation of 2.5 NU while the daughter's value was only 41 %. Figure 6A systematises the findings for this mother-daughter pair firing at 9 Hz and Fig. 6B shows a larger difference for another pair, firing at 15 Hz. The daughters both required a greater depolarisation to produce any given firing index. As an experiment, therefore, the mothers’ curve was transformed by rescaling its x-values to align it with the daughter's curve. In both cases, the rescaled mother's curve (dashed) then agreed closely with the daughter's curve over the first half of their course. Thus within this range a given depolarisation of the mother had the same effect as an S times larger depolarisation of the daughter, with the value of the scaling factor depending upon the mother's details (S= 1.1 and 1.33 for Figs. 6A and B, respectively).

A new index - the ‘estimated potential’ (EP)

The efficacy of the rescaling opened the way for using the daughter's behaviour to assess the mother's response to a stimulus. Specifically, it encouraged the creation of a new index, the ‘estimated potential’ or EP, to represent the mother's stimulus-evoked depolarisation. The EP is defined as the amount of depolarisation of the daughter required to make its firing index the same as that of the mother for the given stimulus. The straight dashes in Fig. 6 illustrate the procedure for converting the firing index into an EP. In both Fig. 6A and B, the mothers had a firing index of 50 % for a stimulus-evoked depolarisation of about 2.5 NU; the daughters required appreciably more depolarisation to raise the firing index to the same value, giving EPs of 2.9 and 3.45 NU, respectively, measured in the daughter's noise units. As discussed below, its value should be directly proportional to the mother's depolarisation, measured in the mother's noise units; but, as in Fig. 6, the scaling factor needed for the conversion depends upon the situation studied and will often be unknown. The EP's units have thus been termed equivalent noise units (ENU), since the EP provides an estimate of the mother's depolarisation; the daughter merely provides the tool. ENU are units of voltage, but can only be converted into an absolute value in millivolts with considerable knowledge of the details of the parent model or MN (i.e. by determining both its noise level and the scaling factor of Fig. 6).

It must be emphasised that the EP is determined without knowing anything about the details of the variable-threshold parent to which it is applied, including its input-output relation. Only the parent's interval histogram is required, so its ‘distance-to-threshold’ AHP can be estimated and the daughter model created; this demands just the three parameters used to fit an exponential to the mother's AHP trajectory. The daughter's input-output relation can then be calculated and the EP determined for a given firing index. A computational alternative is to set up an iterative procedure varying the daughter's input until its firing index is equalised with that of the mother.

Confirmation of EP's linearity

Figure 7 examines the linearity of the ‘estimated potential’ (EP) as a measure of the stimulus-evoked depolarisation in a variable-threshold model MN. It replots the data of Fig. 6 as parent's depolarisation vs. daughter's depolarisation for a range of firing indices; the abscissa gives the estimated potential measured in equivalent noise units. For both lags, the relation remains highly linear for EPs of up to at least 4 ENU; thereafter, the estimate deviates slightly. With data of limited accuracy, as obtained physiologically from a smaller sample, little further error would be introduced by assuming a linear fit throughout. The line of equality is shown dashed, confirming that the numerical value of the EP in ENU overestimates the actual depolarisation in the parent in its own NU.

Figure 7. Testing the linearity between the EP (estimated potential) and the parent model's actual depolarisation.

Plots of the stimulus-evoked depolarisation of the parent model (ordinate) against the EP determined from the daughter model (abscissa); for each point the firing index is the same for mother and daughter (same data as in Fig. 6). Both plots are initially closely linear (continuous lines), but they lie well below the line of equality (dashed line). Values of EP up to 3–4 ENU provide a linear measure of the parent's depolarisation, with comparatively minor deviations immediately thereafter. But the numerical value of the EP measured in ENU (effective noise units, corresponding to the daughter's noise units) overestimates the parent's depolarisation measured in its own noise units.

Comparison of Fig. 6 and Fig. 7 shows why the EP improves on the firing index as a measure of responsiveness. Within its appreciable linear range the EP obeys the laws of simple arithmetic. When two stimuli are given simultaneously the resultant EP is simply the sum of the EPs found when each stimulus is given alone; even with larger stimuli the errors are relatively small. In contrast, the sigmoidicity of the firing index plots in Fig. 6 shows that this is far from true on summing firing indices (see also, Matthews, 1999b).

Reference back to Fig. 5A helps to explain why the linear prediction cannot be expected to hold throughout. The daughter's AHP is fitted to the mother's for values up to about 3 NU above threshold; over this range the trajectories are forced to correspond. But, above 3 NU the daughter's AHP is simply an extrapolation, obtained by fitting an exponential with three variables to the mother's limited data. The exponential's numerical values can be altered reciprocally while still retaining a reasonable fit to the mother's noisy trajectory; however, this strongly affects that part of the daughter's trajectory obtained by ‘extrapolation’ and it is this which provides the basis for deducing the daughter's response to large stimuli. The mother's trajectory is limited in range because at its extremes the interval histogram from which it is derived has too few spikes to be analysed reliably. It can, however, be expanded by combining measurements from different firing rates (Matthews, 1996).

Effect of the mean firing rate on the EP

Changing the firing rate has a complex effect on a fixed-threshold model's firing index for a given stimulus, as also occurs for real MNs (Matthews, 1999b). The variable-threshold model's firing index also varied with the firing rate, as illustrated in Fig. 8B. With the smallest input the firing index tripled as the firing rate increased, while with large inputs it slightly decreased (by up to 15 %). Figure 8A shows that with small inputs (up to 2.5 NU depolarisation) the EP behaved rather better than the FI and was approximately constant in the range studied; but with larger inputs it became unreliable below about 8 Hz, when its value for a given input fell by up to a third making it appreciably worse than the firing index. This was related to the deterioration in the goodness of fit of the linear relation between the EP and the mother's actual depolarisation, as in Fig. 7.

Figure 8. The effect of varying the model's firing rate on its response to a fixed stimulus compared for the two different indices.

A, plot of the EP (estimated potential, derived from daughter model) against mean firing rate for a parent model with a voltage-dependent threshold with 2.5 ms delay; the mother's stimulus-evoked depolarisation was varied from 0.5 to 4 NU. B, equivalent plots of parent's firing index for the same inputs. The EP provides a reasonably constant measure for inputs of 3 NU and below, but becomes unreliable at low firing rates for larger inputs. The FI also fails at low firing rates, falling by more than 50 % for the smallest input and rising by 10–15 % for the large inputs (model as in Fig. 2, with drive current varied from 4.7 to 7.4 nA. Data derived using an earlier procedure in which the mother's depolarisation was produced directly by injecting a voltage rather than a current; control trials confirmed that the two methods gave virtually the same numerical values with an identical pattern of frequency-dependent behaviour).

It is worth noting that when the firing rate is low the various curves for firing index progressively diverge. It follows that the shape of the underlying input-output curve depends on the firing rate, and that compensation cannot be achieved by rescaling the x-axis. Likewise, the input-output plots of the two mothers in Fig. 6 did not superimpose satisfactorily with rescaling. Thus the firing index cannot readily be standardised arithmetically to provide, of itself, a linear measure of stimulus efficacy or excitability; the essential problem is that the shape of its input-output plot depends upon the details of the model, including simply its firing rate.

Continued validity of EP on changing the model's membrane time constant

A further, encouraging, finding was that the EP remained an effective indicator when the membrane time constant of the parent model differed from that of its daughter. This was unexpected because the membrane time constant influences the frequency content of the membrane noise and this determines the choice of transform used to estimate the AHP from the interval histogram (Matthews, 1996; Powers & Binder, 2000). The variable-threshold mother was now given a membrane time constant of 8 ms, double the usual value. Its AHP, however, was estimated with the standard ‘4 ms’ transform, obtained from a model with a membrane time constant of 4 ms; thus the ‘distance-to-threshold’ estimate will have been more than usually in error (Fig. 13, Matthews, 1996). This mis-estimated AHP was none the less then used as the input for a fixed-threshold daughter; the daughter was given the usual 4 ms time constant, matching that underlying the transform. Figure 9A shows their input-output plots, with the mother's curve lying as usual above the daughter's. The points on the daughter's curve were obtained by re-scaling the mother's input as in Fig. 6 (each x-value multiplied by 1.82). The agreement justifies the continued determination of the EP in spite of the ‘wrong’ transform being used. Figure 9B demonstrates that the mother's actual stimulus-evoked depolarisation was again directly proportional to the estimated polarisation determined from the daughter, as in Fig. 7. This means that the EP can be usefully determined for a real MN even though its time constant is unknown.

Figure 9. The EP determined with the standard 4 ms transform continues to provide a linear measure when the model's membrane time constant is changed.

A, input-output plots of firing index vs. stimulus-evoked depolarisation for mother and daughter models. The points lying on the daughter's plot show the mother's data rescaled by multiplying each input x-value by 1.82. The mother's membrane time constant was 8 ms instead of the usual 4 ms, but its AHP was estimated using the standard transform obtained for a model MN with a time constant of 4 ms. This mis-estimated trajectory was then used as the input for a daughter model with a time constant of 4 ms. B, mother's input plotted against the new index, the ‘estimated potential’, obtained from the daughter. The EP still gives a linear measure of the actual input, even though the ‘wrong’ transform was used to estimate the mother's trajectory; the daughter's time constant matched that of the transform. As in Figs. 5-7, the mother's threshold was voltage dependent (lag, 7.5 ms; scale factor, 0.5), while the daughter's was constant. (Firing rate, 10.8 Hz; mother's drive current, 7 nA.)

Effect of increasing the duration of the simulated EPSP's rising phase

The standard model assumed that the EPSP's rising phase had a duration of 1 ms, measured its value at the end of the millisecond, and scored the response as the sum of the extra stimulus-induced spikes over this millisecond (the firing index). The validity of using the EP to estimate the peak size of EPSP's with different rising phases was tested by doubling the duration of the stimulating current input and halving its value. The resulting peak value of the EPSP was determined as usual at the end of 2 ms; it was slightly reduced from the value obtained by delivering the same charge over 1 ms, because some of the charge delivered in the first millisecond leaked away during the second millisecond. The response was now scored as the sum of the extra spikes occurring over the prolonged rising phase of 2 ms (corresponding to a cusum). The ‘2 ms EPSP’ then triggered fewer spikes than did a ‘1 ms EPSP’ with the same peak value; in other words, the excitatory effect of the EPSP depended upon its rate of rise and not just its absolute value, as typically occurs physiologically. The difference was more marked when the threshold followed the membrane potential with a short delay (2.5 vs. 7.5 ms); this occurred because the shorter time constant allowed the threshold to adapt more fully during the prolonged EPSP.

When a standard daughter model, created using an EPSP with a rising phase of 1 ms, was used to transmute the mother's firing index for a ‘2 ms EPSP’ into an estimated potential the EP continued to provide a linear measure of the peak value of the mother's depolarisation. Indeed, on otherwise using the same parameters as Fig. 6 the fit proved better than that illustrated there. However, the scaling factor relating mother to daughter was slightly smaller than that found when delivering a ‘1 ms EPSP’ to the same mother. Thus the EP continues to provide a linear measure of the mother's slowed EPSP, but in units (effective noise units) whose precise value will depend upon the EPSP's rise time. In other words, the EP remains an effective way of comparing the excitatory effects of EPSPs of similar rise time, even though the actual value is unknown. Modest changes of rise time as may occur in making physiological comparisons seem unlikely to vitiate the use of the EP to estimate peak depolarisation.

Effects of altering the model in various other ways

(1) Changing the parameters for threshold dependence

As to be expected, increasing the threshold (gain) factor, regulating the parent's threshold to the membrane potential, increased the value of the conversion factor relating the measured value of an EP to the parent's actual depolarisation. This conversion factor also depended upon the time constant of the exponential delay between membrane potential and consequential threshold shift (cf. Fig. 7).

(2) Changing the voltage-threshold relation

In the standard model, at equilibrium at a fixed membrane potential V, the change in threshold was directly proportional to any deviation of V from a reference value V_o (i.e. K(V - V_o)). In control trials, the linear relation was replaced either by the square root of (V - V_o) or by (V - V_o) raised to the power 1.5. With rescaling, as in Fig. 6, the input-output plots for mother and daughter again showed a good fit and the EP continued to provide a useful measure of the mother's stimulus-induced depolarisation. The daughter model was kept in its usual simple form. (The excitatory drive currents and firing rates were in the range used for Fig. 2 and Fig. 3, and both time lags were tested; when using the power relation the scaling factor K was reduced to 0.1 to prevent excessive firing.)

(3) Changing the threshold's temporal dependence

When the exponential temporal dependence of threshold on potential of the standard model was replaced by a fixed time delay the EP continued to provide an effective measure of the parent's underlying depolarisation. (The threshold was then made to depend simply upon the membrane at a fixed preceding point in time.)

(4) Allowing the residual AHP to persist

In further control trials the residual AHP conductance was carried over from one inter-spike interval to the next, as occurs in life (Ito & Oshima, 1962; Baldissera & Gustafsson, 1974); in the main modelling, the AHP was reset to a standard value after each spike. The daughter's AHP continued to be reset, and the daughter was derived using the standard transform which was derived without resetting. The EP still provided a linear measure of the mother's stimulus evoked depolarisation.

‘Survivors’ trajectory’ of variable-threshold parent model compared with its daughter

It was hoped that, as with random stimulation, the creation of a daughter model would enable the firing index obtained for spike-triggered stimulation to be transmuted into a linear measure of depolarisation. Given the statistical behaviour of threshold-crossing models as shown in Fig. 4 this requires the daughter model to have nearly the same ‘survivors’ trajectory’ as its parent, or one that can be made to agree by linear rescaling of the voltage axis. However, as illustrated in Fig. 10, such agreement was not found on testing the present variable-threshold model and so cannot be expected for real MNs. For both time lags, the daughter's survivors’ curve lay below the mother's, with the divergence increasing at short times. Thus the daughter's ‘excitability’ at any point in time on the AHP will be less than the mother's. Moreover, simple linear rescaling of the mother's mean (dashed) curve failed to make it correspond to the daughter's curve. The curves can be made to fit by first moving the mothers’ curve down to superimpose the equilibrium values, and then linearly rescaling the voltage values as was done on comparing AHPs (Fig. 2 and Fig. 3). This shows that, treated as exponentials, the two curves have the same time constant differing from that of the AHP per se (see Fig. 4B); but this transformation is invalid for determining the ongoing excitability which depends upon the absolute displacement of the curve from threshold (zero). Thus the input-output plot for spike-triggered stimulation of real MNs cannot be reliably deduced from the interval histogram and has to be determined directly.

Figure 10. Comparison of ‘survivors’ mean trajectory’ of fixed-threshold daughter model with that of its variable-threshold mother.

A and B, survivors’ means, for two different lags, giving distance-to-threshold relations for a variable-threshold parent model and its fixed-threshold daughter model. Ordinate, mean trajectory for survivors (ongoing threshold determined for every individual trajectory). Top curves, data for parent model; bottom curves, data for daughter model given an AHP corresponding to the mother's ‘distance-from-threshold’ estimate (not shown). Dashed curve, mother's values rescaled by a constant factor without further shift. The daughter's curve differs from the mother's, including after the rescaling; thus the daughter fails to reproduce the mother's pattern of excitability during the course of the AHP. (Means calculated as in Fig. 4. Drive current for mother: A, 6.5 nA; B, 6.7 nA. Daughter's AHP: −7.2 to −0.3 NU in A; −9.2 to −0.1 in B. Time constants: 34.7 and 36.9 ms, respectively.)

Inhibitory inputs

As with excitation, the magnitude of a randomly timed inhibitory input applied to the mother could be quantified by determining an ‘estimated potential’ (EP) from its daughter model, with consequent improvement of linearity over the firing index. This is illustrated by Fig. 11 which shows that the mother's input-output plot can again be superimposed upon the daughter's plot (cf. Fig. 6); for inhibition, however, no appreciable change of x-scaling was required. Both plots rapidly come to a maximum at a very low value, corresponding to the background level of firing without stimulation (dashed horizontal line - note scaling expanded from earlier). The dashed oblique line gives the first part of the mother's input-output plot for excitation, with the sign reversed for comparison. For infinitely small inputs the initial slopes for excitation and inhibition should be the same; but, for the finite signals presently used the slope is much greater for excitation. As already shown the excitatory slope progressively increases with input magnitude, whereas that for inhibition decreases (this is a result of the gross curvature of the plot of death rate against depolarisation, as in Fig. 1G).

Figure 11. Use of daughter model to study inhibitory hyperpolarisation.

Input-output plots for reduction of firing against magnitude of inhibitory hyperpolarisation for variable-threshold parent model and its simplified daughter. Dashed horizontal line shows the maximum possible response, namely total inhibition of the background firing at 15 Hz. Dashed oblique line shows the mother's response to excitation, reversed in sign. Same mother and daughter models as in Fig. 6B, but with the vertical scaling greatly increased.

It may be concluded that the same principles hold for inhibition as for excitation but that the range of inhibitory inputs that can be quantified is so limited that the direct determination of an inhibitory EP is not widely useful. This is simply because graded information ceases to be available the moment inhibition suffices to silence firing, as applies equally to any other measure. The situation might be improved by using higher firing rates, but at least for MNs it would not then be practicable to perform the large number of trials needed to produce a reliable PSTH. With low frequency tonic firing, inhibitory inputs which appreciably exceed the noise level are simply not measurable from the initial depth of inhibition in the PSTH. Moreover, attempts to obtain information about the later stages of the underlying IPSP from the PSTH's rising tail, as the inhibition decays and firing resumes, look unpromising because of the uncertainty about the way in which the threshold has been affected by the preceding inhibitory hyperpolarisation while firing ceased.

In spite of these problems, EPs can be readily derived for widely ranging amplitudes of peak hyperpolarisation of IPSPs by the classical expedient of using the inhibitory input to reduce the response to an excitatory input, while still leaving a net excitation; the timing of the inhibitory is set to give maximum effect. Because of the linearity between the EP and the net excitatory depolarisation (Fig. 7) the inhibitory EP is then simply given by the difference between the excitatory EP with and without the inhibitory conditioning.

DISCUSSION

The present modelling is aimed primarily at the experimentalist recording from single units, such as the human motor unit, and introduces a new analytical tool. It examines the behaviour of a noisy model MN whose firing threshold is voltage dependent, varying with the pre-existing level of membrane potential, including the AHP. As before (Matthews, 1996,2000) the underlying distance-to-threshold trajectory of the AHP for the particular firing frequency studied was estimated by transforming the unit's interval histogram. The AHP was then used to estimate a neurone's response to randomly timed stimuli, and produce a new index termed the EP. This improved on the standard measure, namely the firing index, by providing a linear rather than a sigmoidal measure of evoked MN depolarisation. Thus the underlying synaptic effects of different inputs can be directly compared, whether given separately or in combination, thereby aiding the study of reflex interaction. The analysis should also help in modelling the behaviour of networks of neurones whose discharge characteristics are known but with unknown underlying ionic conductances and kinetics. Transforming the interval histogram to give the AHP trajectory allows a simplified daughter model to be created with just three parameters, derived by fitting an exponential to the trajectory; this then behaves in much the same way as its complex parent model. A network of such daughter models could thus be used for a variety of simulations and provide greater realism than giving the neurones arbitrary parameters; moreover, using three-parameter daughter models greatly simplifies computation in comparison with using models incorporating the underlying conductances should these be known.

Absolute and estimated AHPs

The present analysis confirms that a motoneurone's interval histogram has the potential to tell one much about the MN. The difficulty lies in correctly interpreting the message, which requires enough pre-existing information to allow the MN to be modelled. If an MN's threshold stays the same throughout the inter-spike interval then transforming the interval histogram allows the size and time course of its AHP to be determined and also ‘the survivors’ trajectory’ giving its ongoing excitability during tonic noise-induced firing. However, when the threshold varies appreciably following a spike (Powers & Binder, 1996) the transform provides a ‘distance-to-threshold’ estimate of the AHP rather than its absolute value. Using intracellular recording, Powers & Binder (2000) found that with upwards rescaling the estimated AHP usually coincided with the absolute AHP, showing that the transform normally successfully predicts the time constant of the AHP's terminal phase of exponential decline. Matthews (1996) had already found that the value of the AHP's time constant varied appropriately with the type of MN studied. These findings are consolidated by the present modelling in which the threshold depended upon the membrane potential, following it with an exponential delay of 2.5-7.5 ms. The AHP's time course was then preserved, suggesting that Powers & Binder's exceptions (nearly 30 %) were due to non-stationary discharges, as they suspected (Powers & Binder, 2000).

However, the ‘distance-to-threshold’ estimate given by the transform only accurately predicted the corresponding value for a noise free AHP when the inter-spike interval was long. If the kinetics governing the threshold were known the situation could be modelled to fine tune the estimate for high firing rates. Since these remain unknown, it has to be concluded that the transform fails to provide an accurate, universally applicable, measure of a MN's ongoing excitability. Moreover, the more appropriate measure is the ‘survivors’ trajectory’, described in the Results, rather that any derivative of the noise-free AHP itself. Earlier, Matthews (1996) assumed that his transform would ‘simply lump any threshold change in with the AHP’ and would correctly predict ‘excitability’. This has proved to be so when the threshold varied along a predetermined time course, but not when it was being swung around by noise transients as occurs physiologically. Even so, the transform provides a better approximation than would seem to be otherwise available. Following this, Matthews’ (1996) estimates of the voltage noise in human MNs should be reduced from 2 to 1 mV, since Powers & Binder (2000) showed that on average the ‘estimated AHP’ was about half the absolute AHP.

The interval death rate

A related question is how far the intermediate transformation of the interval histogram into its hazard function (interval death rate) provides a useful measure of ‘excitability’ (Powers & Binder, 2000); the change in ‘death rate’ can be used to assess the response to a spike-locked stimulus as well as for charting the AHP. For a prolonged response, plotted bin by bin, the death rate should improve on simply measuring the number of additional spikes found in the PSTH, above background, whether expressed as a direct count or as a percentage increase. This is because the PSTH counts are strongly influenced by what has happened in the preceding bins (i.e. simple algebra shows that an excitatory excess of spikes in a given bin will reduce the number in its immediate successor even though the death rate remains the same, and vice versa for an inhibitory reduction). Fournier et al. (1986) recognised such problems when they pioneered the systematic use of the spike-locked PSTH for human reflex studies; but by conducting their statistical tests bin by bin irrespective of previous history they opened the way for subsequent error, especially for the belief that the duration of a response could be determined in this way (Meunier et al. 1994; Naito et al. 1998; see Matthews, 1999a). Unfortunately, however, judged by the behaviour of the present models the ‘death rate’ also has serious defects; using a fixed stimulus, its value varies with the delay from spike trigger to stimulus, and it does not behave arithmetically when two stimuli are superimposed (these imperfections stem from the curvature in the transform from membrane potential to death rate; Matthews, 1996). In the constant threshold model these non-linearities can potentially be overcome by using an appropriate transform to convert the death rate into a membrane depolarisation, whether of the AHP or of the ‘survivors’ trajectory’; stimuli of finite duration then require an iterative calculation for each successive bin (see below). But this approach becomes impossible when the threshold varies in an unknown manner.

‘Death rates’, though not so called, have previously been calculated for certain visual neurones and normalised to their exponential tail. This was done to provide a weighting factor (‘recovery function’) to calculate a ‘free firing rate’ expressing the excitatory effect of a stimulus, unconfounded by the effects of refractoriness in limiting the actual firing rate (Berry & Meister, 1998, eqn 17). Two matters limit the generality of using the death rate plot to resolve this classical problem. First, the plot can only be directly normalised when the firing rate is low so that the interval histogram has an exponential tail. Second, judging from present testing of the constant threshold model, the ‘recovery function’ measured at a low firing rate cannot be safely transferred to a higher firing rate; appreciable shifts of mid-position of the normalised curve were found on varying the model's firing rate, and these were accompanied by small changes in shape.

The estimated potential (the EP)

This was derived from the classical ‘firing index’ by creating a fixed-threshold daughter model to mimic the behaviour of its variable-threshold parent; the daughter was given the parent's ‘distance to threshold’ AHP, deduced from its interval histogram, without requiring any further knowledge of the parent's properties. The daughter was then used to determine the magnitude of the brief depolarisation required to make its firing index, derived from its PSTH, the same as that found for its parent in response to the test stimulus. This ‘estimated potential’ (EP) provides a measure of the efficacy of the stimulus in exciting the parent. The EP's particular virtue is that it provides a linear measure of the parent's depolarisation, provided the stimulus is not too large. Its disadvantage is that it is expressed in arbitrary units of voltage (ENU) rather than in millivolts; the value of the ENU in millivolts depends upon the parent model's properties and will be unknown for a real MN. The chief advantage of the EP over the firing index is that the effects of separate stimuli summate arithmetically. For weak stimuli the value of the EP was appreciably less affected by changes in the background firing rate, while for strong stimuli the firing index was the more constant. Importantly, the EP remained an effective measure irrespective of the temporal structure of the parent's noise; this will be unknown in human work and can affect a MN's behaviour (Matthews, 1996; Powers & Binder, 2000).

The present model used current as the input rather than a conductance change. This had the advantage that the EP relates to the net synaptic current reaching the soma, irrespective of whether the current was entirely generated at the synaptic knobs or was boosted by voltage-sensitive dendritic conductances (‘plateau potential’ currents). The disadvantage is that the model is only accurate for relatively small deviations from resting threshold, where the currents produced by a given conductance change are little affected by the change in driving potential. Earlier findings with a conductance input suggest that no serious error should occur within 5–6 mV of threshold (Matthews, 2000, Fig. 10). On the assumption that the synaptic noise is about 1 mV, the linear range of the EP of about 4 ENU should also correspond to about 5 mV. (The value in ENU has first to be scaled down to convert to the parent's noise units and then scaled up for conversion to millivolts to allow for the change in threshold.) The background firing rate also needs to below 15–20 Hz, as used to develop Matthews’ transform (Matthews, 1996,2000).

Use of daughter model to determine waveform of complex inputs

The present analysis has been restricted to deducing the peak size of an EPSP. The determination of the profile of the underlying depolarisation is a more complex problem, becoming particularly severe when a prolonged stimulus varies in amplitude. It has two major aspects, namely how to allow for the effects of refractoriness when multiple firing occurs (Gray, 1967; Berry & Meissner, 1998), and how to compensate for the rate of rise and fall of the stimulus, exemplified by the long-running debate on the algebraic equation relating the EPSP to the PSTH (Kirkwood, 1979; Kirkwood & Sears, 1991; also Midroni & Ashby, 1989, for ‘shadowing’ on reversal of EPSP slope). Given the inherent non-linearities, the solution will generally require an iterative calculation because the response at a given time depends on what happened before. In a fixed-threshold neurone the underlying depolarisation should be recoverable from the prolonged PSTH as follows. First, the interval histogram for the background firing is transformed to determine the AHP. The temporal structure of the voltage noise has to be assumed, to allow the correct transform to be used; it depends largely on the membrane time constant, modified by the structure of the synaptic currents (Matthews, 1996; Powers & Binder, 2000). The neurone can then be modelled, using a suitably short computational interval, and a stimulus-evoked depolarisation introduced into a single bin to determine the average response in the PSTH at the start of the stimulus; using an optimisation procedure the depolarisation is adjusted until the model's response corresponds to that of the neurone. This estimated depolarisation is then incorporated into the model and attention shifted to the next bin and the depolarisation again determined to match the neurone's PSTH. The serial computation is continued until the firing returns to background, when the waveform of the underlying stimulus-evoked depolarisation should have been recovered; cumulative errors could be reduced by optimising the cusum up to the time in question, rather than treating each bin in isolation. This procedure, however, would not deliver an accurate estimate for most real motoneurones, with their variable threshold. Nonetheless, over short periods of time, the creation of a fixed-threshold daughter model should still allow the waveform of the underlying excitation to be extracted from the PSTH, albeit now scaled in ‘effective noise units’ of unknown magnitude.

Conclusion

Testing the use of the new index (the EP) for characterising the response of a real MN to synaptic inputs should present no problem. It is readily derived by combining the conventional firing index with an estimate of the MN's AHP, without further knowledge of the MN; the requisite ‘distance-to-threshold’ trajectory can be rapidly computed by transforming the MN's interval histogram for tonic firing. Its arithmetical summation and frequency sensitivity can be studied using EMG recordings from single human motor units. More fundamentally, using intracellular recordings in animal MNs, the ‘estimated potential’ of the EP could be compared with that actually occurring and the EP's linearity tested. It would then also be interesting to determine the kinetics relating membrane potential to spike threshold; more realistic modelling should then allow the EP to be estimated in millivolts for direct comparison with the actual stimulus-evoked depolarisation. The EP's successful survival of such experimental scrutiny for the MN would greatly encourage its application to other types of neurone; this would also be helped if the EP could be validated theoretically as well as empirically. The essential principle, applicable to all types of neurone, is that detailed analysis of unitary responses to stimuli requires modelling, and this can be based on a simplified model of the neurone with the time course of its post-spike recovery from refractoriness derived from its own interval histogram.