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1 Neuroscience Graduate Program, Stanford University School of Medicine, Stanford, CA 94305-5404, USA2 Sir James Spence Institute, University of Newcastle, Royal Victoria Infirmary, Queen Victoria Road, Newcastle upon Tyne NE1 4LP, UK
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(Received 10 June 2003;
accepted after revision 6 January 2004;
first published online 14 January 2004)
Corresponding author S. N. Baker: Sir James Spence Institute, University of Newcastle, Royal Victoria Infirmary, Queen Victoria Road, Newcastle upon Tyne NE1 4LP, UK. Email: stuart.baker{at}ncl.ac.uk
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Introduction |
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TopAbstractIntroductionMethodsResultsRandom walk modelsDiscussionConclusionsReferences |
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Across the interspike interval (ISI) the form of the distance-to-threshold trajectory defines the neural firing pattern. For instance, a more excitable neurone will have a smaller average distance-to-threshold and will therefore require less excitatory input to fire. A steep afterhyperpolarization (AHP) that returns rapidly towards threshold will facilitate high-frequency firing more readily than a shallow AHP or one with a prominent negative-going ‘scoop’ that remains distant from threshold and therefore less excitable. AHPs have been characterized via intracellular recordings in terms of shape and duration (Schwindt & Calvin, 1972, 1973; Calvin & Schwindt, 1972; Calvin, 1974; Storm, 1989). Further studies across a range of mammalian cells have tested the susceptibility of post-spike membrane potentials to chemical compounds to determine the role of specific conductances (Wong & Prince, 1981; Yarom et al. 1985).
However, analysis of intracellular recordings has limitations in the description of the post-spike distance-to-threshold. Results may be altered by anaesthesia (in vivo studies) and disconnection from local and distant networks (in vitro studies). The shape of the trajectory is probably critically dependent on the pattern of inputs to the cell, so that in vitro intracellular recordings may not correctly portray post-spike excitability. In addition, intracellular recordings usually measure only the membrane potential, without also assessing changes in threshold. Both are required if distance-to-threshold, the most functionally important criterion for excitability, is to be assessed.
Matthews (1996) devised an indirect method for calculating the distance-to-threshold trajectory post-spike in human motor units. Recognizing that dense dendritic arborization provides a set of inputs that roughly approximates to normally distributed noise, Matthews estimated the probability of a spike occurring at any moment after the preceding spike as a function of the mean distance-to-threshold at that time. His transform relied exclusively on extracellularly recorded spike trains, allowing the use of data gathered in vivo. The results for human motor units suggested a distance-to-threshold trajectory that rises exponentially towards threshold. A similar trajectory had previously been suggested using a method based on the probability of response to a stimulus (Olivier et al. 1995).
In this paper, we have applied Matthews's method to the discharge of single neurones recorded from the primary motor cortex of an awake behaving monkey; some of these cells were antidromically identified as pyramidal tract neurones (PTNs; Baker et al. 1999a). Matthews's original method requires that the neurones analysed have relatively stable firing rates; this condition was not met for our recordings, requiring some modifications to the technique. The results show that many motor cortical neurones do not have a monotonically rising distance-to-threshold trajectory, but instead the trajectory peaks between 10 and 60 ms after the spike. These results were corroborated using a different method, based on analysis of the responses to nearby intracortical microstimulation. Motor cortical cells can show synchronized oscillatory discharge around 10–40 Hz (Murthy & Fetz, 1992; Sanes & Donoghue, 1993; Baker et al. 1997, 2001; Baker & Baker, 2003). This has been previously shown to depend partly on local circuit interactions, with GABAergic inhibitory interneurones playing a critical role (Pauluis et al. 1999; Baker & Baker, 2003). Our present results suggest that motor cortical neurones contain intrinsic cellular properties favouring regular discharge; this may be another important factor in the generation of synchronized network oscillations.
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Methods |
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A single female M. mulatta monkey was trained on a behavioural task that was a modified version of the precision grip task described in Baker et al. (2001). The animal was presented with two precision grip manipulanda, instructed to grip the left side, right side, or both, and cued to move after a 1 s delay period. Torque motors opposed these movements with forces that were proportional to lever displacement and hence simulated the action of springs. Incorrect movements, or premature release of the switches, resulted in a failure tone, and termination of that trial. The trial type (left, right or bimanual) was chosen at random.
After behavioural training was complete, the animal was prepared to record from the primary motor cortex (M1), using methods previously described in detail (Baker et al. 1999a, 2001). Under full general anaesthesia (isoflurane inhalation in 50:50 N2O:O2, 1.5–2.5% concentration) and aseptic conditions, the monkey was implanted with a stainless steel headpiece to allow subsequent atraumatic head fixation, a recording chamber placed above a craniotomy over M1, and two fine electrodes with tips located in the pyramidal tract for electrical stimulation. A full programme of postoperative analgesia (buprenorphine, 10 µg kg-1) and antibiotic care (coamoxyiclav 140/35, 1.75 mg kg-1 clavulanic acid, 7 mg kg-1 amoxycillin, Synulox, Pfizer Ltd) was provided. Once recordings were complete in one hemisphere, the recording chamber was moved to the other side and additional pyramidal tract electrodes were inserted in a further surgery under general anaesthesia.
Following recovery from each surgery, multiple single units were recorded simultaneously in daily recording sessions in the conscious state. These recordings used a microdrive capable of inserting up to 16 independently movable electrodes into the cortex. Some units were identified as PTNs by antidromic activation following electrical stimulation through the pyramidal tract electrodes. The antidromic nature of the activation was confirmed using a collision test (Lemon, 1984). Once sufficient cleanly discriminable units were located, stimulation was ceased and recordings were made of spontaneous activity whilst the animal performed the behavioural task. Spike waveform data were recorded continuously with a sampling rate of 25 kHz, and discriminated off-line into the times of occurrence of single units using cluster cutting software. Further analysis used only recordings that were clean single units, judged from consistency of wave shape and the presence of an absolute refractory period larger than 1 ms in the interspike interval histogram. Local field potential waveforms were similarly captured at 500 Hz for offline analysis.
At the end of the experiment, the animal was killed by an overdose of anaesthetic, and the brain processed for histology. All animal procedures accorded with UK legislation (Animals (Scientific Procedures) Act 1986) and were approved by the local ethical review panel.
General method of analysis
Matthews has developed (Matthews, 1996) and refined (Matthews, 1999, 2002; Powers & Binder, 2000) a method to calculate the distance-to-threshold trajectory during the period following a spike. Previously, this method has been exclusively applied to motoneurones, which are especially suited to this analysis because their firing rate can be tightly controlled within a narrow band.
By contrast, M1 cells during a behavioural task have more variable rates, with instantaneous firing frequency regularly varying from 5 Hz to more than 100 Hz. However, the high firing rates do confer some advantages for this type of analysis. Firstly, the presence of short interspike intervals permits the earliest part of the distance-to-threshold trajectory to be investigated. Secondly, the large number of recorded spikes permits the recording to be separated into many subfractions without excessive degradation in statistical power. A number of changes were made to Matthews's method to account for the firing properties of the cortical cells, as described below.
Matthews's method assumes that at a given time post-spike, the membrane potential adopts a particular mean value. It is this mean value as a function of time post-spike which defines the AHP trajectory. Superimposed on this trajectory is voltage noise, which may cross spiking threshold at a given time and initiate an action potential. The probability of spiking at a given post-spike latency will depend on the distance of the mean membrane potential from threshold. The post-spike distance-to-threshold trajectory represents the excitability of a cell and plays a central role in determining the firing patterns of its discharge.
The transformation from spike train to distance-to-threshold trajectory requires several manipulations of the raw spike train data. These are illustrated in Fig. 1 and described in detail below.
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The interspike interval (ISI) histogram is a commonly used method for illustrating the relative frequency of different interspike intervals (Fig. 1A). From this distribution, we produce a plot of surviving intervals (Fig. 1B). At each latency post-spike, the number of surviving intervals is equal to the sum of all intervals of that latency or longer. Next, we calculate the interval ‘death rate’ curve (Fig. 1C) as a function of post-spike time (an interval ‘dies’ when the next spike fires). The death rate is thus the ratio of the number of interspike intervals in a bin to the number of intervals surviving to that time (the curve in Fig. 1A divided by the curve in Fig. 1B). The interval death rate provides a simple metric for estimating the architecture of a cell's excitability post-spike.
Distance-to-threshold transform
The probability that an interval, having survived 50 ms, will ‘die’ due to a spike occurring before 51 ms is a direct approximation of the excitability of the membrane at that latency. Unfortunately, the interval death rate lacks the conceptual clarity of the more standard membrane potential trajectory. There is no simple frame of reference: it is unclear whether a death rate of, for example, 10% implies a particularly low or high level of excitability. To overcome this problem, the calculated interval death rate is transformed to represent a distance-to-threshold trajectory (Fig. 1D). The transform is determined by running a random walk model of a neurone. In this model, the membrane potential has a fixed mean distance from threshold; noise is superimposed on this trajectory, and the rate of threshold crossings found. The model is rerun with different mean distances-to-threshold, and a lookup curve is created which uniquely relates a death rate to a distance-to-threshold (inset to Fig. 1C).
The model was simulated in detail as follows. At a time t, the membrane potential V(t) was updated as a leaky integrator according to:
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| (1) |
t is the time step, 
(t) is Gaussian-distributed white noise (with appropriate adjustments for the variance of this input, see below), and 
is the membrane time constant. Threshold is taken to be at V= 0. In a given simulation, D was fixed as a constant. The time constant was taken as 12 ms, a value appropriate for a cortical pyramidal neurone (Kandel & Spencer, 1961a; Pauluis et al. 1999). In such a simulation, the random walk evolves as successive inputs of normally distributed noise decay towards the underlying distance-to-threshold, D, with the appropriate time constant. The model proceeds until V crosses the threshold. A spike is then deemed to occur, and the simulation is reset. In this way, a spike train is generated and a death rate readily determined. This can be repeated for different values of D, and a transform from death rate to distance-to-threshold constructed. Whereas the death rate is expressed in terms of a percentage of intervals dying per bin, the distance-to-threshold trajectory is scaled in units of the standard deviation of the membrane noise. Values of D are therefore given in ‘noise units’.
There is one further complexity which must be dealt with in simulating the random walk model used for calibrating the death rate to distance-to-threshold transform. At the start of the simulation of a particular interspike interval, the neurone has only been subjected to a single noisy input. As the interspike interval proceeds, the neurone receives more noisy inputs, which are integrated. Since the inputs are simulated as independent (white) noise, the variance of the sum of several inputs will be larger than the variance of a single input. The variance of the membrane potential therefore rises throughout the interspike interval, reaching a steady state only after several time constants. This is problematic for construction of the death rate to distance-to-threshold transform, which must scale the distance-to-threshold in terms of an assumed constant noise standard deviation. Because the variance is low at the beginning of the interval, including these bins in the estimate of interval death rate for a particular distance, D, from threshold would result in a lookup curve incorrectly shifted to lower death rates. Matthews (1996) addressed this issue by excluding spikes that fell within the first three time constants of his random walk model. Our method proceeded similarly. We did not impose the spike threshold process for the first 36 ms (=3 time constants) of the simulation, allowing the random walk model to integrate inputs and achieve stable noise variance. Following 36 ms of simulation, if the membrane potential V was above threshold, we discarded that simulation and began again. These trials, already being above threshold, could not be used to determine the likelihood of crossing threshold from below. However, if V was below threshold at 36 ms after the start of the simulation, we continued simulating until V exceeded threshold.
At each time point of the interval the same forces are acting upon the modelled cell – it receives Gaussian input and its potential decays towards the fixed distance-to-threshold with the 12 ms time constant. Therefore, at any point in the interval after the membrane noise variance has stabilized, the interval death rate is constant. We therefore estimated the death rate by averaging the death rate over intervals larger than 36 ms and shorter than a long latency cutoff at which there were too few intervals for a reliable death rate estimate. This mean death rate served as the estimated death rate for the distance-to-threshold, D. The standard deviation of the membrane voltage noise after the first 36 ms of simulation was also determined. The distance-to-threshold D was expressed relative to that value in all subsequent calculations.
This procedure was repeated with different mean distances-to-threshold, allowing the construction of a calibration curve relating death rate to distance-to-threshold. Following Matthews (1996), the transform determined by simulation was fitted by a double exponential curve which could be conveniently used as a lookup function.
Frequency selection
Matthews (1996) cautioned that the indirect estimate of the AHP would be erroneous if data containing a mixture of different underlying firing rates were used. One sign of such frequency mixing is that the reconstructed AHP has a gentle negative slope at long latencies. Matthews accounted for slow drifts in firing rate by dividing his data into subpopulations of intervals taken from similar firing rates, and then combined the resulting distance-to-threshold curves to form a composite curve. We used a similar approach in the current data (Fig. 1E).
Each interspike interval was assigned an instantaneous firing rate according to the mean of the four preceding and four succeeding intervals (Matthews, 1996) (Fig. 1E, inset). More complicated estimation methods, such as interpolation of firing rate according to linear or parabolic regression, were tried, but did not yield different results. The distribution of the estimated rate was displayed as a histogram, and the experimenter used interactive cursors to designate the high and low frequency tails of the distribution; intervals falling in the tails were not used for further analysis. The remaining intervals were assigned to a subpopulation based upon constant-width slices of the firing rate distribution. The number of subpopulations used was chosen individually for each cell depending on the range of its firing frequencies, as well as the total number of recorded spikes available.
We have quantified the reduction in variance that occurs due to binning intervals by firing rates. The mean coefficient of variation (CV) of all cells analysed for all intervals (1.01 ± 0.034; mean ±S.E.M.) is greater than the mean CV for frequency subpopulations (0.872 ± 0.033). Our frequency binning method reduces the CV of intervals (mean difference: 0.126 ± 0.022), suggesting that it successfully selects a more restricted set of intervals that occur during periods of similar synaptic drive.
Error bars
In order to carry out statistical tests on the shape of the distance-to-threshold trajectory, we determined which portion of the curve was reliable by assigning confidence limits to the trajectory estimates at each latency post-spike. This was achieved by firstly estimating confidence limits on the death rate calculation, and then transforming this range to a distance-to-threshold.
We assume that, at each interspike interval, there is a particular underlying death rate 
. This is the death rate which would be measured if an infinitely large dataset were available. However, in the experimental dataset, we saw m intervals die in this time bin out of n which had survived prior to the bin start. The problem is to find confidence limits on our estimate of given this experimental observation.
The rate at which intervals die is based upon a constant-probability () binomial process in which a cell either spikes or does not spike within a particular bin for a single interval. If the death rate is known to be , the probability of seeing m intervals die out of a possible n is given by the binomial probability distribution:
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However, we need to estimate the opposite probability: P(death rate =|m die from n). That is, we want to know the probability that the underlying death rate is given our observation of m spikes and n intervals. Bayes's rule applied to this problem gives the following (Sivia, 1996):
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) and P(m die from n) are known as the ‘prior distributions’ in Bayesian theory; they represent our prior expectation for a particular death rate or spike count, before observation of the data. In the Bayesian approach, it is often assumed that we have no prior information about such parameters, so that the relevant probability distributions are flat and P(death rate =) and P(m die from n) from the right hand side of eqn (3) are constants. Hence:
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| (4) |
In order to calculate the confidence limits on the death rate, we therefore proceed as follows. We calculate P(m die from n | death rate =) from eqn (2), on a finely spaced grid over a wide range of possible death rates . We then rescale these values so that the curve encloses unit area. This then gives P(death rate =| m die from n). The 2.5% and 97.5% points of this distribution provide the 95% confidence limits on the death rate estimate . The value of the death rate which has greatest probability – the maximum likelihood estimate – turns out to be m/n, the straightforward ratio estimate of the death rate.
Finally, the 95% confidence limits on the death rate can be propagated through the death rate to distance-to-threshold transformation, providing 95% confidence limits on the distance-to-threshold trajectory itself.
Composite trajectory formation
The interspike interval histogram was calculated separately for each firing rate subpopulation, and the interval death rate and distance-to-threshold trajectories found. Each trajectory estimate had low noise over only a small range of times post-spike; only parts of the trajectory that had 95% confidence limits narrower than a preset criterion level were therefore used. This criterion level was chosen with reference to the plots from each cell individually; it was normally set between 0.7 and 0.9 noise units. The region of overlap of the low noise sections of the AHPs determined from the highest frequency bin, and the next highest, were found, and the mean difference between the overlapping regions of these curves was calculated. Next, the lower frequency curve was shifted up by this amount. This process was then repeated for the AHP from the next lowest frequency band, and so on. Finally, the low noise parts of the shifted curves were averaged to generate the composite AHP (Fig. 1F–H).
Ninety-five per cent confidence limits for the composite AHP were produced as follows. At each time post-spike, we took the full probability distribution of the individual, frequency selected, distance-to-threshold estimates, computed as described by eqns (2)–(4). These were convolved to produce the probability distribution of the averaged distance-to-threshold estimate. The 2.5% and 97.5% points of this distribution were then used as the 95% confidence limits for the composite trajectory.
Simulation of realistically firing neurones
In order to test the validity of our method, we wished to simulate a random walk model of a cell using a distance-to-threshold T that varied in a known way following a spike. We also wished to arrange for the modelled neurone to show the same moment-by-moment modulation of its firing rate as seen in an experimentally recorded neurone. We could then test whether the analysis described above accurately recovered the time course of T. The firing rate profile from several single neurones – including the cell illustrated in Fig. 2 with a peaked distance-to-threshold trajectory – were used for all simulations in this section.
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| (5) |
The instantaneous firing probability I(t) of an experimentally recorded neurone was estimated by convolving the cell's spike train with a Gaussian kernel (Fig. 1I); the width parameter of the kernel was set equal to the cell's modal interspike interval (see Baker & Lemon, 2000). We then simulated the random walk model. However, the rate of firing of a cell is determined by all input occurring until the next spike, not just input at the time of the previous spike. To allow for this, whenever a spike occurred in the model cell at time t, we found the time t+ dt such that
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Using this method, we were able to generate a simulated spike train with similar rate modulation to an experimentally recorded cell, but with a known distance-to-threshold trajectory. The resulting spike times were analysed as described above; the recovered trajectory was compared to the T(t) used in the simulation.
To test the possibility that poor spike discrimination could alter the recovered trajectory, a proportion of spikes from the random walk spike train were randomly removed before distance-to-threshold analysis. By varying the proportion of ‘missed spikes’, we verified that missed spikes alone could not account for the peaked distance-to-threshold trajectories which we commonly observed. Likewise, to account for false positive spikes, a varying proportion of extra spikes were randomly added to the spike train.
Quantifying peaks in AHP trajectories
A common finding in the AHP trajectories was a peak. The statistical significance of such a peak was determined by comparing the lower bound of the 95% confidence limit at the peak with the points before and after the peak. Normally, we tested the 3rd to 10th points before and after the peak. These ranges were adjusted for early peaks that did not have 10 data points before the peak. We used a binomial distribution, assuming P(hit) = 0.05 to determine the minimum number of points that needed to be significantly less than the peak in order to declare a statistically significant peak. For example, if eight points were tested, the probability of two or more points lying below the lower 95% confidence limit on the peak is less than 0.05. If sufficient points both before and after the peak were significantly lower, the peak was considered to be statistically significant. In such cases, the peak height was calculated as the maximum difference between the peak and the post-peak trajectory.
Single-pulse intra-cortical micro-stimulation
At the end of some recording sessions, we further validated our results on the AHP trajectory shape using single-pulse intracortical microstimulation (ICMS). Single pulse stimuli at 20 µA were delivered with an interstimulus interval of 300 ms through one electrode, whilst the recordings on other channels were monitored. The stimulus intensity was chosen to produce a response probability of ca 0.25, and the response to several thousand stimuli was recorded. The response probability should be greater the closer a cell is to threshold. We therefore binned stimuli according to the time of the stimulus since the previous spontaneous spike, and measured the response probability as a function of spontaneous spike-stimulus interval. Error bars were determined for the response probabilities using a Bayesian approach similar to that described above for the death rate calculation. This provided an alternative estimate of the shape of the post-spike membrane potential trajectory for that cell.
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The results of each stage of the analysis are shown in detail for one PTN in Fig. 2. Figure 2A–C presents data from the whole population, with no attempt to separate different firing rates. The interval histogram (Fig. 2A) forms the basis for the interval death rate curve (Fig. 2B) which is subsequently transformed to create a distance-to-threshold trajectory (Fig. 2C). The grey shading in all figures corresponds to the 95% confidence range.
The most striking feature of the death rate and distance-to-threshold curves is the prominent peak at 37 ms post-spike. However, it is conceivable that such a peak could be generated by the highly variable firing rates present in this recording. Since membrane potentials at lower firing rates tend to be displaced further from threshold, Matthews (1996) described a negative slope at long latencies for AHP curves composed of mixed frequencies. This signature of frequency mixing contamination is clearly visible in Fig. 2C.
To mitigate the effects of frequency mixing, we controlled for firing rate by separately analysing intervals from periods of similar firing rates. For this PTN, the intervals were distributed to 30 subpopulations by taking slices from the histogram of estimated firing rates (Fig. 2E). The tails of the rate distribution were excluded from analysis (unshaded portion of Fig. 2E).
Each subpopulation of intervals underwent the identical set of transformations as described above for the full population. Three representative subpopulations indicate the general similarities and specific differences across frequency bins (Fig. 2F–H). The interval histograms of each subpopulation are similar in shape, with the distribution shifted to longer intervals for lower firing rates (Fig. 2F). The 95% confidence limits on the interval death rate curves (Fig. 2G) were used to determine which parts of each curve were estimated with sufficiently low noise; these points have been marked with dots in Figs 2G and H. Although the latency and sharpness of the peak varied across the different subpopulations, all curves showed evidence of a peak.
The subpopulation distance-to-threshold trajectories, are, by themselves, accurate representations of the post-spike excitability of a cell at a narrow range of mean firing rates. However, to capture the gross features of the distance-to-threshold trajectory across all firing rates, we followed Matthews (1996) in constructing a composite curve by combining the low-noise portions of the subpopulation trajectories (Fig. 2D).
Comparison of the composite distance-to-threshold curve (Fig. 2D) with the full population curve of Fig. 2C reveals a number of differences. Most importantly, the decline at long post-spike times is not seen in the composite trajectory, indicating that frequency mixing is probably having little effect. The peak is still seen, suggesting that this is a genuine feature of the recovery of the cell after a spike. The region of the composite trajectory (Fig. 2D) plotted is shorter than the population trajectory (Fig. 2C), since only the low noise parts of the individual subpopulation trajectories are used (Fig. 2H).
Three types of distance-to-threshold curves
The composite distance-to-threshold trajectories could be fitted into three categories. Results from cells that typify each of these shapes are shown in Fig. 3. Most cells (53/61) exhibited a statistically significant peak in their distance-to-threshold trajectory (Fig. 3B). Six cells had a simple exponential rise towards threshold, similar to the trajectories observed in motoneurones (Fig. 3C). Two cells had a ‘reverse exponential’ trajectory, in which the membrane potential declined away from threshold with greater times post-spike (Fig. 3A). There were both identified PTNs and UIDs with peaked (37 PTNs, 16 UIDs) and exponential (2 PTNs, 4 UIDs) trajectories, implying that these cells were probably pyramidal neurones. The two reverse exponential trajectories were both UIDs. Previous work has indicated that different classes of cortical neurones may be distinguished on the basis of their action potential widths (Swadlow et al. 1998; Nowak et al. 2003). The action potentials for the two cells with reverse exponential trajectories were relatively wide (0.72 ms and 0.76 ms), implying that they were probably pyramidal neurones or spiny stellate cells.
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Figure 4C shows that good reconstruction of an exponential trajectory was achieved using firing rate modulations derived from two other cells whose experimentally recorded discharge produced a peaked distance-to-threshold trajectory.
Although the general shape of the reconstructed trajectories shows a good match to the actual trajectories used in the simulations, in each case the reconstructed trajectories of Fig. 4 show a slight deviation from the underlying trajectory at membrane potentials far from threshold.
Separate analysis by power of network oscillations
Populations of neurones in macaque M1 have been shown to synchronize their firing in the 15–30 Hz range in a task-dependent manner (Baker et al. 1997; Baker et al. 2001). Local field potential (LFP) power at this frequency is highest during the hold phase of the task, whereas synchronous oscillations are abolished during active movement. Since the mean peak latency was 35.5 ± 12.1 ms (mean ±S.D.), it was possible that the observed peaks in the distance-to-threshold trajectory resulted exclusively from network synchronization rather than intrinsic cell properties.
In order to test this possibility, we analysed data from a single neurone with a clear distance-to-threshold peak as follows. Firstly, we computed the time-resolved power spectrum of the LFP averaged across all 12 electrodes available in this recording session. This was calculated for 3 s before and 1 s after the end of the hold period across all 361 trials performed with the contralateral hand (Fig. 5A). The computation used a wavelet method (see Baker & Baker, 2003 for details). Figure 5A shows that the LFP in this monkey desynchronized in the 15–20 Hz band during the period 2 s to 1 s before the end hold task marker (red bar in Fig. 5A), corresponding to the active movement of the levers into target, and synchronized in this band for the 1 s-long hold period (blue bar). We selected spikes from these two parts of the task and analysed them separately.
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Responses to single-pulse ICMS
Our multiple electrode recording system permitted yet another measurement of the excitability of a neurone during the interspike interval, using the responses to single-pulse ICMS. Figure 6 shows analysis for eight neurones in which we were able to record both a long period of spontaneous activity, and also the response to stimulation. For each cell, the interval histogram (example in Fig. 6A) for the spontaneous spikes produced a composite distance-to-threshold trajectory (Fig. 6B–I, thick lines). Following single-pulse ICMS through an electrode hundreds of micrometres away, these neurones responded with a short latency peak in their peri-stimulus time histogram (example in Fig. 6A, inset). Each stimulus was sorted into bins depending on the time between the last spontaneous spike and the stimulus delivery. For each spike-stimulus time, the unit's response probability was determined, i.e. the fraction of stimuli which resulted in a response spike at the latency of the PSTH peak. This response probability curve is shown in Fig. 6B–I (thin lines; grey shading marks 95% confidence limits). For all eight cells, the response probability across the ISI was well aligned with the composite distance-to-threshold trajectory. All three trajectory types were represented, though in different proportions to the cells analysed above: one peaked trajectory (Fig. 6A and B), five exponential trajectories (Fig. 6C–G), and two ‘reverse exponential’ trajectories (Fig. 6H–I). The consistency between response probability and distance-to-threshold trajectory across such a variety of trajectory shapes confirms that this method does accurately measure the post-spike excitability of a neurone. The results from ICMS cannot, however, by themselves determine whether the peaks in the distance-to-threshold trajectories are produced by synchronous network oscillations to which the cell is partially entrained, or by intrinsic cellular mechanisms.
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Composite distance-to-threshold curves represent general approximations of a neurone's excitability across a range of firing rates. However, the sharpness and latency of the peaks often varied across firing rate subpopulations. Figure 7 shows, for a single neurone, individual reconstructions of the distance to threshold trajectory for four different firing rate slices. At lower firing rates, the peak in the distance-to-threshold curve was less pronounced and appeared later (Fig. 7A). At higher firing rates, the peak tended to become clearer and to occur earlier (Fig. 7B–D). For the cell illustrated in Fig. 7 therefore the peak will have a more significant effect on excitability at higher firing rates.
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Figure 8 presents quantitative data on the properties of the peaks in the distance-to-threshold curves. Figure 8A shows that the peaks fell at a wide range of latencies (from 10 ms to nearly 60 ms). The majority of the peaks were of moderate height (less than 0.5 noise units), although some were considerably larger (Fig. 8B).
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Figure 8D plots the relationship between the peak latency and its distance-to-threshold; the latter has been measured at a frequency corresponding to the modal interspike interval. There was a weak, but statistically significant, correlation, showing that peaks at short latency tended to be closer to threshold than later ones; the least-squares regression had a slope of 0.37 ± 0.13 (95% confidence limit).
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Discussion |
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Adaptations to the distance-to-threshold transform
The greatest potential problem with this method is frequency mixing, and it was important to establish that the distance-to-threshold peaks were not simply artefacts of an unsatisfactory method for separating intervals by firing rate (Matthews, 1996). To control for variable levels of synaptic drive at different firing rates, it is imperative to analyse separately subfractions of intervals from periods of similar firing rates. Whereas Matthews used only a small number of bins, the highly variable firing rates of our cells necessitated – and the high spike counts allowed – dozens of subfractions. Our results confirm that such extensive frequency binning is both appropriate and necessary.
A second hazard of this analysis is the noisiness of interval death rate and distance-to-threshold trajectories at long latencies. With few remaining intervals, the interval death rate is highly dependent on the death of several – or even a single – interval. To account for low spike counts at long latencies, Matthews (1996) arbitrarily disregarded the longest 2% of intervals. However, applying such a strict standard to our data sometimes severely reduced the clarity of both the falling (hyperpolarizing) side of the peak and the subsequent levelling to equilibrium. To capture such details whilst maintaining statistical validity, we applied a binomial probability density function to the interval death rate, and via a Bayesian method calculated 95% confidence limits. These error bars permitted us to choose low noise portions of the death rate curves in a consistent, rigorous way, as well as to distinguish statistically significant peaks from exponential trajectories.
Matthews has emphasized that variation in the firing threshold post-spike (Powers & Binder, 1996) may prevent calculation of the precise membrane potential during the interspike interval (Matthews, 1996; Matthews, 2002). We also emphasize that our computed trajectories are distance-to-threshold estimates, rather than absolute reconstructions of post-spike voltage trajectories. However, it is distance-to-threshold which provides the functionally meaningful variable when considering the post-spike recovery in excitability of a cell.
As described in Methods, the distance-to-threshold we calculate is always scaled in units of the noise standard deviation. The noise variance increases following a spike due to synaptic integration. For motoneurones, this does not complicate interpretation of the calculated distance to threshold trajectories. Motoneurones fire at low rates, and have a short time constant (ca 4 ms, Matthews, 1996). This means that spikes will rarely be fired before the membrane noise variance has stabilized several time constants after a spike. The trajectories could therefore theoretically be converted to voltage trajectories simply by scaling by a constant (although usually unknown) factor. By contrast, the motor cortical cells which we have studied can fire at high rates (upwards of 100 Hz), and have longer time constants (12 ms). It is therefore common that spikes are fired during the period after the previous spike before the noise variance has stabilized. We have not attempted to make any correction for this. Instead, our curves of post-spike distance to threshold trajectory should be interpreted as being scaled, point-by-point, by the instantaneous noise standard deviation pertaining at that interspike interval. We have calculated that, with a 12 ms time constant, the noise standard deviation reaches 90% of its maximum value 20 ms after the preceding spike. Our basic finding, of peaks in distance-to-threshold trajectories which are mostly later than 20 ms (Fig. 8A), cannot therefore have been generated as an artefact of non-stationarities in post-spike noise variance.
Biophysical basis of peaks in distance-to-threshold trajectories
The presence of depolarizing after-potentials (DAPs) in pyramidal cells with latencies similar to those described here, as well as their functional importance for repetitive firing have been recognized for decades. Kandel & Spencer (1961a,b) identified DAPs in cat hippocampal pyramidal cells with an approximate latency of 30 ms and amplitude of 10 mV. In close agreement with these findings, Segal & Barker (1986) described an inward current in voltage-clamped hippocampal pyramidal cells that peaked between 25 and 50 ms after a spike. They estimated the reversal potential for this current to be 10 mV. In addition, solitary rat hippocampal cells exhibit a distinct DAP that lasts between 20 and 40 ms (Storm, 1987; Azouz et al. 1996; Jensen et al. 1996). Fetz et al. (2000) reported intracellular membrane potential trajectories that rebounded around 30 ms post-spike in monkey sensorimotor cortex.
However, as our results indicate, DAPs are a common but not ubiquitous feature of cortical neurones. It is possible that cells with and without a DAP can be distinguished both by firing characteristics and morphology: Yang et al. (1996) claim that layer V–VI prefrontal cortical cells identified as intrinsic bursting, repetitive oscillatory bursting and intermediate had both more dense dendritic arborization and DAPs, while regular spiking cells had more sparse dendritic arborization and lacked DAPs. Stafstrom et al. (1984) reported that roughly 30% of layer V neurones from cat sensorimotor cortex exhibit a DAP followed by a medium-latency AHP.
Due to their consistency and self-reproducibility, DAPs have been frequently implicated as contributors to regular firing and bursting. Some of the earliest intracellular recordings of pyramidal cells identified DAPs and recognized their importance for repetitive firing (Kandel & Spencer, 1961a,b; Calvin & Schwindt, 1972; Schwartzkroin et al. 1974; Calvin & Sypert, 1976; Schwartzkroin & Prince, 1976). More recent studies have used pharmacological agents to investigate the specific ionic conductances responsible. It does not appear that a single conductance can adequately describe the range of DAP latencies and amplitudes we have found (Fig. 8), as well as the described variability in the latency, amplitude, and sharpness of DAPs according to firing rate (Fig. 7).
By definition, the presence of a depolarizing peak requires both preceding and succeeding hyperpolarization of the membrane as well as an intermediate depolarizing current. Clearly, several subthreshold conductances – both depolarizing and hyperpolarizing – are involved at a range of relative strengths to account for the distinct DAP characteristics both across cells (Fig. 3) and at different firing rates for a single cell (Fig. 7). Furthermore, since our method describes distance-to-threshold, rather than membrane potential, it is also important to consider the kinetics of the threshold as it returns through the absolute and relative refractory periods to equilibrium.
Several studies have implicated a TTX-sensitive, slowly inactivating, persistent Na+ current (INaP) in DAP formation in pyramidal cells (Stafstrom et al. 1982; Azouz et al. 1996; Guatteo et al. 1996; Jensen et al. 1996; Yang et al. 1996; Crill, 1996) that shows a peak in current amplitude at intermediate, subthreshold voltages roughly 10 mV below spike threshold (Stafstrom et al. 1985; French et al. 1990; Brown et al. 1994; Chao & Alzheimer, 1995; Fleidervish & Gutnick, 1996). INaP has the capacity to bring a neurone closer to spike threshold without necessarily inducing a spike itself in the absence of further depolarizing input.
Although INaP is the most relevant subthreshold current for DAPs with latencies in the tens of milliseconds, other depolarizing currents may be involved. Several groups have shown that depolarization shoulders are abolished in pyramidal cells in the absence of Ca2+ (Wong & Prince, 1978; Wong & Prince, 1981; Yang et al. 1996).
Subthreshold depolarizing currents play a key role in modulating distance-to-threshold during the interspike interval. However, without pre- and post-peak hyperpolarizations, the depolarizations that form the peak would instead create a post-spike depolarizing shoulder or resting potential nearer to threshold. The pre-peak, fast AHP (fAHP) is generated by a Ca2+-activated K+ current (Ic) (Lancaster & Nicoll, 1987; Storm, 1987, 1989). In terms of post-peak hyperpolarization, there are several hyperpolarizing conductances active at latencies ranging from tens to hundreds of milliseconds post-spike that collectively form the medium AHP (mAHP). These include Ca2+-insensitive voltage-gated currents such as the transient K+ A-current (IA) (Bekkers, 2000; Mitterdorfer & Bean, 2002) and the small conductance Ca2+-activated K+ current (ISK) (Romey & Lazdunski, 1984; Blatz & Magleby, 1986; Lang & Ritchie, 1987). The time constant of ISK activation varies between 5 and 50 ms according to the concentration of free Ca2+ (Hirschberg et al. 1998; Sah & Faber, 2002), suggesting a possible role in variably shaping peaks at different firing rates. However, the finding that DAPs are longer but not abolished in the absence of Ca2+ (Azouz et al. 1996) implies that there are Ca2+-insensitive currents as well that play a role in forming the falling phase of afterpotential peaks.
Another possible cause of DAPs are dendritic depolarizations reverberating in a ‘ping-pong’ manner (Wang, 1999). According to this hypothesis, a somatic spike could activate dendritic depolarizations that are then passed back to the soma at consistent latencies due to the specific arborization and channel distribution of the dendrites.
Clarifying the causes of specific DAPs could have important clinical and theoretical implications: for instance, antiepileptic drugs such as phenytoin (Chao & Alzheimer, 1995; Lampl et al. 1998) and topiramate (Taverna et al. 1999) act by reducing INaP. From a theoretical perspective, a finer understanding of the ionic conductances underlying DAPs could provide important hints for understanding how cells modulate excitability.
Variation in peak size and latency with firing rate
The cells in this study exhibited a range of distance-to-threshold peak latencies and heights (Fig. 8). Since peak height was measured relative to the post-peak equilibrium distance-to-threshold, the interplay of the currents underlying the delayed depolarization and mAHP determine both the latency of the peak and its height.
The line fitted to the data of Fig. 8C indicates a strong correlation between peak latency and modal interval length, with a slope significantly less than 1. This means, unsurprisingly, that there is a relationship between the latency at which a cell is most excitable (the peak) and the modal interspike interval length. Why, then, does the peak latency lag behind the modal interval length? After the post-spike hyperpolarization, the cell depolarizes towards resting potential. The more time that passes before the distance-to-threshold trajectory begins its post-peak falling phase, the more time the membrane will spend nearer to threshold. As a result, random occurrences of large-amplitude excitatory input will cause more pre-peak spikes in cells with long-latency peaks. By the time the long latency peaks arrive, many intervals will have already died, thereby reducing the size of the modal interval. Therefore, even if two such cells have peaks equally distant from threshold, the earlier peak will be more important in determining the modal interval length.
A similar effect may underlie the sharpening of distance-to-threshold peaks and narrowing of interspike interval histograms at higher firing rates (Fig. 7). Since the membrane spends less time near enough threshold for noisy inputs to cause occasional spikes, most intervals last until the distance-to-threshold peak.
The sharpening of the distance-to-threshold peak at high firing rates could also have an underlying biophysical basis, due to increased hyperpolarizing conductances before and after the peak or higher amplitude depolarizing currents during the peak. Neurones during periods of high firing rates have higher levels of intracellular Ca2+ than at times of more sparse spiking. Increased calcium levels could potentiate Ca2+-activated K+ channels that act at short (Ic) (Storm, 1987) and medium latencies (Isk). Together, these hyperpolarizing currents could serve to sharpen a DAP. Taddese & Bean (2002) claim that such an effect would increase synchronous firing, a suggestion corroborated by more narrow ISI histogram peaks at high firing rates compared to lower ones (Fig. 2).
Azouz et al. (1996) showed that blocking Ic eliminated the notch between the falling phase of the action potential and the rising DAP, causing the DAP to begin at a more positive membrane potential and last 5–10 ms longer. Functionally, the reduced sharpness of the DAP could decrease the precision of spike timing.
Relationship of the three trajectory shapes to the ISI histogram
Neurones with clear distance-to-threshold peaks are distinguishable from those with exponential trajectories by the shape of their interval distributions (Fig. 3A1 and B1). By definition, a peak requires both a pre-peak rising phase and a post-peak falling phase – and it is the post-peak falling phase that distinguishes cells with exponential and peaked trajectories. A peaked trajectory is characterized by a decreased probability of spike firing in the milliseconds following the modal interspike interval as can be seen in the death rate and distance-to-threshold curves of Fig. 3B1–2. If there is a relative dearth of intervals slightly longer than the latency of the peak, it follows that the ISI histogram will fall more sharply to the right of the modal ISI for cells with peaked trajectories than for those with exponential ones (Fig. 3A1 and B1). It is this portion of the ISI histogram that distinguishes peaked and exponential trajectories.
While these two types of distance-to-threshold curves appear similar in some respects, the third type of trajectory, the ‘reverse exponential’, exhibits a markedly different firing profile. The interval histogram of a ‘reverse exponential’ cell (Fig. 3A1) shows a clear tendency to burst (short intervals). The interval death rate and distance-to-threshold curves for this cell type peak at very short latencies and decay for many milliseconds before reaching equilibrium quite distant from threshold: the probability of a spike firing becomes decreasingly likely as time passes until reaching a minimum, equilibrium probability around 100 ms post-spike (Fig. 3C2 and C3). Unfortunately, our model lacks the temporal precision to determine the distance-to-threshold trajectory in the few milliseconds after a spike. Presumably, however, there is a short latency DAP consistent with this type of bursting behaviour.
These distinctions among ISI histograms can be most simply quantified by producing an interval death rate curve. Although the transformation from ISI histogram to interval death rate is susceptible to frequency mixing, in our experience it almost always provides an accurate guide as to whether the post-spike excitability of a cell will be monotonically rising or show a peak.
Role of intrinsic cell membrane properties in network synchrony
The finding that M1 neurones often exhibited peaked distance-to-threshold trajectories is highly significant for the generation of oscillatory network synchrony. Motor cortical cell populations can enter a mode of weakly synchronized firing. This synchronization can have oscillatory components (Baker et al. 2001); such oscillations appear preferentially during steady contraction or rest, but are abolished during movement (see Baker et al. 1999b for review). The oscillations around 20 Hz appear to be generated at least in part by local circuit interactions within the cortex, with GABA-ergic inhibitory neurones playing an important role (Pauluis et al. 1999; Baker & Baker, 2003). However, the distance-to-threshold peaks which we report here will give motor cortical neurones an intrinsic tendency to fire rhythmically. The observed post-spike latency of the peaks (10–60 ms) would imply a preferred frequency of 16–100 Hz; however, most peaks were around 40 ms (mean peak latency ±S.D. of 35.5 ± 12.1 ms), implying a tendency to fire within the ‘beta’ band of 
25 Hz. Although such peaks could not produce the oscillatory synchrony across large populations of cells which is seen, they could act to strengthen oscillatory synchronization once it is established by other means. A similar effect has been reported in the visual cortex, where the tendency to rhythmic firing of ‘chattering’ cells is suggested to augment synchronous oscillations in the gamma frequency band (Gray & McCormick, 1996).
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We believe that our study of M1 cells confirms that, with suitable adaptations, the method of Matthews (1996) can be successfully applied to determine the mean excitability during the interspike interval in cells with a range of firing characteristics.
