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1Department of Biological Sciences, University at Buffalo, State University of New York, Buffalo, New York; and 2Department of Neurobiology, Harvard Medical School, Boston, Massachusetts
Submitted 20 December 2004; accepted in final form 14 June 2005
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ABSTRACT |
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INTRODUCTION |
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) and <1 µs in the electrosensory system (Guo and Kawasaki 1997; Rose and Heiligenberg 1985). This degree of precision is remarkable, particularly considering that the variability in spike timing ("jitter") in sensory afferents can be much greater (Carr 1993). The reduction in jitter observed at successive stages in some sensory pathways (Carr et al. 1986; Joris et al. 1994a,b; Paolini et al. 2001) is thought to involve the convergence of multiple inputs that ultimately produce an output with less jitter than the inputs.
Most of what is known regarding how synaptic convergence affects spike timing is based on analytical and computational models (Burkitt and Clark 1999; Joris et al. 1994a; Mar
álek et al. 1997; Rothman and Young 1996; Rothman et al. 1993). These models have established that the convergence of synaptic inputs can lead to considerable jitter reduction, and have provided insight into how the number and amplitude of inputs can influence both latency and jitter of response.
Although such models have provided an important framework for understanding jitter reduction, many aspects remain poorly understood. Computational models by necessity make simplifying assumptions, such as including a subset of cellular conductances or portraying neurons as integrate-and-fire. Active conductances in real neurons could play important roles in summing of inputs, controlling the refractory period, and regulating repetitive firing.
In addition, previous models have not clearly addressed how the temporal distribution of inputs affects the response. Evidence from in vivo experiments has indicated that the distribution of input times can vary depending on the stimulus (Johnson 1980; Joris et al. 1994a; Kiang 1965). In some cases, input timing is well-approximated by a Gaussian distribution, whereas in other cases there is a rapid onset to the input times and an alpha distribution is more appropriate. Although it seems reasonable to hypothesize that such large differences in the properties of input distributions could influence jitter reduction, models have either tested only a single distribution (usually Gaussian) or not explicitly identified the distribution used.
Here we use the dynamic-clamp technique (Robinson and Kawai 1993; Sharp et al. 1993) to test jitter reduction achieved through the convergence of multiple inputs in real cells. This approach allows us to take into account the complexity of real neurons. Dynamic-clamp conductances are based on synaptic conductances recorded in voltage-clamp experiments. In addition, this technique allows complete control over the size, number, and timing of inputs. The impact of each of these parameters can therefore be tested, without making any assumptions about cellular physiology. In addition, many trials can be rapidly collected, allowing precise measurements of latency and jitter. The synapse between auditory nerve (AN) fibers and bushy cells (BCs) was chosen for these studies because BCs exhibit jitter reduction in vivo (Joris et al. 1994a,b; Paolini et al. 2001). In addition BCs are electrically compact neurons that receive AN inputs directly onto their somata, which makes them well suited to the dynamic-clamp method.
Using this approach, we identified how the number of inputs (N), the total synaptic conductance (Gtot), and the temporal distribution of inputs each affect the latency and SD of spike timing. Jitter reduction was most effective with a large number of small synaptic inputs. The properties of the input distribution had important consequences for jitter reduction. For the Gaussian distribution, the greatest jitter reduction occurred with Gtot 2–3 times threshold, whereas for the alpha distribution, jitter in first spike latency continued to decline as Gtot was increased. However, when Gtot was increased to >5 times threshold, secondary spikes were triggered, which interfere with jitter reduction.
These results provide insight into how synaptic parameters, input distribution, and postsynaptic properties affect spike timing in real cells. In this paper, we examine the effects of convergence on postsynaptic spike timing when the number of inputs is not constrained. In the accompanying paper (Xu-Friedman and Regehr 2005), we consider jitter reduction when few synaptic inputs are available. Under these two conditions, jitter reduction is optimized using distinct strategies that depend on the properties of the inputs.
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METHODS |
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). These cell types are thought to be electrically very similar, and the cells we recorded from were consistent in their intrinsic membrane properties (Fig. 1A). Patch electrodes were pulled from borosilicate glass (1.5 OD, 0.86 ID, P-87 puller, Sutter Instruments, Novato, CA) with resistances of 1–2 M
. AN fibers were stimulated with an electrode (3–5 µm diameter) placed in the AVCN passing 
14 µA for 0.2 ms. Data were sampled at 50 kHz with an ITC-18 controlled by Igor Pro (Wavemetrics, Lake Oswego, OR) using a custom-written interface on an IBM computer.
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For dynamic-clamp trials, we mimicked AMPA (
-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid) conductances only, using a linear I–V relationship and a reversal potential of 0 mV. Dynamic-clamp trials were carried out using the drivers of the ITC-18 at a sampling rate of 50 kHz. For each trial, we first chose the times and amplitudes of the desired synaptic conductances, and convolved these with a normalized excitatory postsynaptic current (EPSC) recorded in voltage clamp. In most experiments, the peak conductance for each input was scaled relative to the threshold of the cell. Threshold was measured by applying a train of conductances (typically sweeping from 10 to 20 nS, in increments of 0.5 nS) and identifying the conductance that triggered a postsynaptic spike. During experiments, the cell's action potential threshold was monitored every 15–20 s. To determine threshold changes during 20-pulse trains, a conditioning conductance was selected that reliably triggered a spike for 19 pulses, followed by a test pulse whose amplitude was selected using an efficient search algorithm. This approach provided estimates of threshold to within 0.5 nS.
To detect BC spikes and measure their timing, the derivative of the membrane potential was used, and spikes were detected by thresholding. Spike timing was taken as the peak in the derivative, and latency for each trial was calculated relative to the mean timing of synaptic inputs averaged over all trials. When the synaptic conductances being tested were large, the derivative was inadequate to detect multiple, closely spaced spikes. To solve this problem, it was necessary to account for the passive response properties of the cell. This was done by estimating the input resistance and capacitance of each BC (25–40 M, 9–12 pF), calculating the response of a matched resistance–capacitance circuit to the synaptic conductance, and subtracting it from the actual response, before taking the derivative. This allowed detection of all spikes, even in the presence of highly dynamic conductance waveforms.
Jitter in the inputs was set to follow alpha or Gaussian distributions. The alpha distribution for a single input is described by a probability density function (pdf)
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. The SD is given by 
in = 
. To generate alpha-distributed random numbers, the rejection method (Leon-Garcia 1989) was used over the interval (0, 5in) on uniform random numbers generated using built-in functions in Igor Pro. To generate Gaussian-distributed random numbers, the built-in random number generator in Igor Pro was used. |
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RESULTS |
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). To achieve realistic synaptic conductances in dynamic clamp, we based their time course on EPSCs recorded in voltage clamp. BCs were identified in voltage clamp by their AN fiber inputs, which were large and fast, and showed paired-pulse depression at short intervals (Fig. 1B) (Isaacson and Walmsley 1996, 1995).
We mimicked the integration of multiple synaptic inputs as shown in Fig. 1, C–H. In this example, the timing of 10 inputs was randomly chosen and distributed according to the alpha distribution (Fig. 1C). This distribution has a rapid onset, similar to that seen in histograms of auditory nerve responses at low frequencies (Johnson 1980; Joris et al. 1994a). Each occurrence of an input (Fig. 1D, top) was convolved with the unitary synaptic conductance (Fig. 1D, inset) to produce the summed input waveform (Fig. 1D, bottom). Dynamic clamp was used to deliver this conductance waveform to the cell, and the resulting membrane potential was recorded (Fig. 1E). BC spikes were detected using the derivative of the recorded membrane potential (Fig. 1F; see METHODS). The BC spike latency was measured for many additional trials, where the timing of each input was chosen randomly. The variability in response is apparent when the BC spike latency is plotted as a raster (Fig. 1G) or as a histogram (Fig. 1H).
By comparing the histogram of BC response times with the original input distribution (Fig. 1H), two basic effects of convergence are apparent. First, the jitter of the BC response was much less than the jitter of the individual inputs. Second, the mean latency of the BC response was shifted relative to the latency of the inputs. Both of these effects constrain how temporal information in the BC can be used downstream.
These latency shifts and jitter reduction are important consequences of convergence. We examined how such effects are influenced by the size and number of synaptic inputs. In addition, we considered the distribution of input timing. In vivo, inputs show timing distributions that can be approximated by either alpha or Gaussian distributions (Fig. 2A), and the extent of jitter in the inputs can show a considerable range (Johnson 1980; Joris et al. 1994a; Kiang 1965). We therefore examined jitter reduction for both alpha and Gaussian distributions with a range of SDs.
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We first determined the influence of the size of synaptic inputs. For these experiments, we considered 10 inputs of various sizes with SD in their arrival times (SDin) of 0.5 ms. The total synaptic conductance (Gtot) was set relative to BC spike threshold to facilitate comparisons between different BCs. A representative experiment for a cell presented with alpha-distributed inputs is shown in Fig. 2A, left. In this BC, the threshold was 14 nS, so for Gtot = 2, the amplitude of each input would be 2/10 times threshold, or 2.8 nS. Changing Gtot influenced the number of spikes per trial, the mean latency, and the jitter of BC responses. For Gtot < 1, the BC showed no response. As Gtot increased, the magnitude of the BC response increased to one spike/trial, and for Gtot > 8, multiple spikes per trial were occasionally elicited (Fig. 2B, top, closed circles). In addition, as Gtot increased, the mean response latency became shorter (Fig. 2B, middle, closed circles). This likely happens for two reasons. First, fewer inputs are required to sum to threshold, so the BC need not wait for later inputs to arrive. Second, the amplitude of each input is greater, so action potentials are triggered more rapidly off the excitatory postsynaptic potential (EPSP). This second factor probably accounts for the continued reduction in latency for Gtot > 10 (i.e., after each input is suprathreshold). Furthermore, the jitter in BC spike latency decreased as Gtot increased (Fig. 2B, bottom, closed circles). This trend continued even with suprathreshold inputs when only the first spike was considered (Fig. 2A, black dots). However, for Gtot 
12 secondary spikes were elicited (Fig. 2A, gray dots) that led to an increase in the SD (Fig. 2B, closed gray circles).
When the input jitter followed a Gaussian rather than an alpha distribution, there were some similarities in the trends of BC response, but there were also important differences. The magnitude of the response and the generation of secondary spikes for large Gtot and the decrease in the mean latency were similar for both distributions (Fig. 2, A and B, top, middle). As Gtot increased to 4 times threshold, the amount of jitter decreased (Fig. 2A, right). However, as Gtot increased further, the SD of first spike latency increased (Fig. 2, A, right and B, bottom, open circles). Furthermore, jitter reduction with the Gaussian distribution was not as great as for alpha-distributed inputs over the entire range of Gtot tested.
Similar behavior was found in seven cells tested (Fig. 2B). The magnitude of BC responses reached nearly 100% for Gtot 1.6 times threshold. As Gtot increased >6 times threshold, some secondary spikes were elicited. However, secondary spikes were relatively rare. Even when Gtot = 24 (i.e., each input = 2.4 times threshold), the response magnitude was only 1.24 ± 0.07 and 1.29 ± 0.10 spikes/trial for alpha- and Gaussian-distributed inputs, respectively (n = 5 for this value). This indicates that after the first input fires the cell with 100% efficacy, each late input has an average efficacy of 2.7% (alpha) or 3.2% (Gaussian). In addition, there was a shift in mean latency of nearly 1.5 ms (i.e., 3 SDin) with both alpha- and Gaussian-distributed inputs. Most of this shift took place for Gtot < 6. In addition, jitter reduction was similar for alpha and Gaussian distributions for Gtot 2. Jitter reduction occurred at the very smallest values of Gtot. As Gtot increased, BC jitter increased slightly, but then decreased again for Gtot = 2. Strikingly, for Gtot > 3, BC spike jitter continued to decrease for the alpha distribution, whereas it increased for the Gaussian distribution. When secondary spikes were taken into account (gray symbols), jitter substantially increased, despite the fact that they occurred in only a fraction of trials. This indicates that estimates of jitter are highly sensitive to a few misplaced spikes.
These findings demonstrate that convergence has fundamentally different effects on jitter for different distributions. Considering just first spikes, it is clear that for alpha-distributed inputs, jitter is least for large inputs (Fig. 2B, bottom). By contrast, for Gaussian-distributed inputs, jitter increases with larger inputs, and instead is lowest for smaller inputs. Although there is some improvement with very small inputs (Gtot 
threshold), this is not an ideal regime for jitter reduction because the probability of responding is so low (Fig. 2B, top). More stable responses for Gaussian-distributed inputs occur over the range Gtot = 2–4 times threshold. Alpha-distributed inputs show least jitter for Gtot > 10 times threshold (i.e., each input is suprathreshold). Furthermore, over the whole range of Gtot, alpha-distributed inputs always showed significantly greater jitter reduction than Gaussian-distributed inputs. These results indicate that jitter reduction depends on the input distribution. This feature, which has not been documented previously, could have major consequences for the interpretation of in vivo data.
Latency depends on Gtot
In the experiments just described, we observed large shifts in mean latency in response to changing the total conductance. These effects could arise either because the total conductance was being changed or because the amplitude of individual inputs was being changed. To determine which of these alternatives was responsible, we compared the BC spike latency for 10 converging inputs with the extreme case of an infinite number of inputs while keeping the total conductance constant. For the infinite case, the conductance waveform is computed by convolving the unitary conductance with the input jitter distribution. Thus each input is infinitesimally small and there is no trial-to-trial variability in the conductance waveform. Sample conductance waveforms and their responses are shown in Fig. 3A for N = 10 and 
inputs with Gtot = 2 times threshold. For N = 10, the conductance waveform changes from trial to trial and the BC spike shows some jitter. By contrast, for N = , the conductance waveform is constant from trial to trial and the jitter in the BC is very small. However, the average latency was nearly identical for N = 10 or (n = 8 cells; Fig. 3B). This was true for both alpha- and Gaussian-distributed inputs, which indicates that it is the total synaptic conductance, and not the number and size of individual inputs, that controls the response latency.
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Although the experiments of Fig. 3 indicate that N does not determine response latency, they do show that N has a strong effect on jitter reduction. To obtain a more complete understanding of this effect, we varied the number of inputs while holding the total synaptic conductance constant at Gtot = 4 times threshold. In the representative experiment shown in Fig. 4A, the mean latency remained similar regardless of the number of inputs. In addition, jitter in BC spike timing decreased with increasing numbers of inputs and was highly precise for N = . This was true for both alpha- and Gaussian-distributed inputs, although as observed above, BC jitter with Gaussian-distributed inputs was typically greater than that with alpha-distributed inputs.
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, the applied conductance was constant from trial to trial, which should produce minimal variability in BC spike timing. In this case, the output jitter was 17 ± 2 and 24 ± 4 µs for alpha- and Gaussian-distributed inputs, respectively (n = 6 cells). These experiments establish the importance of the number of synaptic inputs in jitter reduction for inputs with either alpha or Gaussian distributions. In this example there was only a twofold jitter reduction with five inputs and more than fivefold improvement with 50 inputs. In the limit, with the integration of an extremely large number of very small synaptic inputs, there can be a reduction in jitter of >20-fold. This indicates that the postsynaptic cell contributes very little jitter during spike generation. The small, residual jitter is probably contributed to by slight variations in membrane potential or action potential threshold from trial to trial.
Spike timing and the rate of depolarization
Previous studies of the responses of motorneurons to synaptic inputs suggest that the rate of rise of synaptic inputs may be an important determinant of the time course of the spiking evoked in the postsynaptic cell (Cope et al. 1987; Fetz and Gustafsson 1983). We therefore performed experiments to test whether the time course of synaptic currents is an important factor in the jitter of the output cell. In these experiments a set of artificial, linearly increasing conductances was tested, which had different slopes but a constant integrated conductance (Fig. 5A, top). Brief conductances (e.g., 0.3 ms) reliably evoked a single, short-latency response that showed no detectable jitter (Fig, 5A, left). When the duration of the conductance was 3 ms, a response was still evoked, but a small amount of jitter was apparent (Fig. 5A, middle). When the duration of the conductance was increased to 10 ms, the cell did not depolarize sufficiently to evoke a spike (Fig. 5A, right).
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Thus even for conductances that last much longer than would be expected for BCs in vivo, the resulting jitter is small compared with the jitter inherent in the timing of their synaptic inputs. This indicates that for our experimental conditions the rise time of the synaptic conductance plays a minor role in postsynaptic jitter. Rather, the statistics of the timing of synaptic inputs is more important in determining postsynaptic jitter.
Consequences of variability in the number of active inputs
The experiments described so far indicate that jitter is affected by both Gtot and the number of active inputs, whereas the mean latency of response is determined primarily by Gtot and not the number of inputs. Under normal conditions in vivo, an individual input is typically active on only a subset of stimulus presentations. For example, auditory nerve inputs do not fire on each cycle of a tone stimulus. Therefore the number of active inputs per cycle varies and as a result both N and Gtot vary, which could affect jitter reduction.
We tested whether the variability in the number of active inputs that can occur in vivo affects jitter reduction by comparing trials with a fixed number of inputs (Nfix = 10) to trials with a variable number of inputs (Nvar = 10). For the variable trials, we mimicked 40 inputs total, which is the number of inputs to globular BCs in cats (Liberman 1991). To maintain the same average number of inputs per trial at 10, each of the 40 inputs had a 25% probability of being active and the number active on any given trial followed the binomial distribution (Fig. 6A, bottom left). To set the amplitude of inputs, it was not appropriate to hold Gtot constant for all trials as in Fig. 4 because that would require that synaptic inputs were not independent of each other. Instead, we held the peak conductance of individual inputs (Gpeak) constant at 0.4 times threshold, so that Gtot varied from trial to trial, but was on average 4 times threshold for both fixed and variable trials. The number of inputs was held constant for half the trials (Fig. 6, A and B, top), and varied for half the trials (Fig. 6, A and B, bottom). Even though the number of inputs varied considerably from trial to trial (Fig. 6A, bottom left), the distribution of BC spike latency was relatively unaffected (Fig. 6A, right).
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2. Perfect integrator model
To better understand the jitter reduction observed in our experiments, we compared our experimental results to theoretical expectations. We began by considering the simplest case, modeling the postsynaptic cell as a perfect integrator, which receives a total of N inputs, with n required to cross threshold and fire a postsynaptic spike (n N). The timing of input spikes is described by the probability density function fin(t) and cumulative density function Fin(t) = 
–t fin(
)d. To determine the distribution of postsynaptic spike timing, fout(t), we first derived the cumulative distribution
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In addition, we expanded the model to allow for the number of active inputs to vary between trials. On a given trial i, the number of active inputs, ni, is binomially distributed with probability p from a total population of N inputs. Only trials with ni n active inputs will produce a postsynaptic response. For those trials, the distribution of postsynaptic spike latency is given by substituting ni in place of N in Eq. 1
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Effect of SD
The similarity between theoretical and experimental results in Fig. 6 indicates that the BC behaves like a perfect integrator when SDin = 0.5 ms. This value is close to the intrinsic time constant of the BC (BC = 0.5–1 ms). To evaluate how well these results generalize to situations in which SDin is different from BC, we varied SDin from 0.2 to 5 ms (Fig. 8A) and examined the effect on the BC response. In these experiments, we also varied Gtot, while holding n = 10. In the representative experiment in Fig. 8B, the BC response for SDin = 0.2 ms (left column) was very similar to the responses for SDin = 0.5 ms (Fig. 8B, second column), except that there was an even lower likelihood of triggering secondary spikes for Gtot = 8. This is consistent with their being suppressed by the refractory period of the cell. However, for SDin 2 ms, the response was quite different. First the number of spikes/trial was lower for Gtot = 2, and there was no response at all for SDin = 5 ms. Second, there were many secondary spikes for Gtot = 8.
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2 ms, but not when SDin 0.5 ms (Fig. 8C). As SDin increased, there was a large decrease in BC response magnitude for Gtot = 2, but a large increase in BC response magnitude for Gtot = 8. For Gtot = 4, the response magnitude stayed relatively unaffected. BC jitter for SDin = 0.2 was very similar to when SDin = 0.5 ms, which suggests that the BC acts like a perfect integrator for these shorter SDin as well, as expected. However, as SDin increased, BC jitter also increased, for all values of Gtot tested. This increase was most dramatic for Gtot = 8 when secondary spikes were included (Fig. 8D, bottom), where jitter was only minimally reduced. This suggests that the refractory period is effective only over timescales of < 2 ms and plays a major role in improving timing in BCs. Spike timing during trains
Thus far we have examined how convergence affects spike timing in experiments where individual trials were isolated from each other. This has allowed us to understand some basic properties of spike timing, which appear to be relevant in general. However, under physiological conditions, neurons may be synaptically activated at high frequencies, which leads them to fire at high frequencies. This can activate and inactivate voltage- and calcium-dependent conductances of the postsynaptic cell, which will change the properties of synaptic integration. These effects are likely to be specific to the type of neuron being studied and the properties of its intrinsic conductances. For example, BCs have voltage-gated potassium conductances that cause the input resistance of the cell to drop on depolarization (Manis and Marx 1991; Oertel 1983; Rothman and Manis 2003a,b). These conductances activate and inactivate on the timescale of about 1 ms, and therefore may have an effect during repetitive activation.
To evaluate how postsynaptic activity affects subsequent convergence, we examined BC spike timing during trains of 20 cycles with 10 inputs per cycle. We tested different conditions for Gtot (2, 4, or 8 times threshold at rest), timing distribution (alpha or Gaussian, SD = 0.5 ms), and input firing rate (100, 200, and 333 Hz). To evaluate more clearly the contribution of changes in the postsynaptic cell, these presynaptic properties were held constant throughout a given train.
An example of such experiments is shown for inputs distributed according to an alpha distribution and activated at 100 or 333 Hz with Gtot = 2 and 8 times threshold (Fig. 9). For each condition the timing of the synaptic inputs is indicated by vertical bars; the corresponding conductance and the response of that cell are shown for a single trial. In addition a raster plot of the BC spikes is shown for many similar trials. When Gtot = 2 and the cell is activated at 100 Hz, each stimulus usually evokes a single, precisely timed spike, although there is an occasional failure (Fig. 9A). When the stimulus frequency is raised to 333 Hz and Gtot = 2, BC firing becomes unreliable and late in the train only every third or fourth stimulus evokes a response (Fig. 9B). This suggests that during the train the threshold of the postsynaptic cell becomes elevated. When the amplitude of the synaptic inputs is increased by a factor of 4 (Gtot = 8), for 100-Hz stimulation each synaptic input reliably evokes at least one spike (Fig. 9C). However, in some trials, two spikes may be evoked in one cycle (Fig. 9C, *), similar to what is seen in well-isolated trials (Fig. 2B, 7). When Gtot = 8 times threshold and the stimulus frequency is 333 Hz, each cycle also evokes at least one spike (Fig. 9D). Secondary spikes are more frequent early in the train (* in Fig. 9D), whereas late in the train it is rare for a stimulus to evoke more than a single spike.
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To determine how activity during trains affects spike timing, we compare BC responses during the first train cycle with those at the end of the train. Low-amplitude synaptic inputs have decreased efficacy during high-frequency activation (Fig. 10B, top, red trace). In addition, shifts in latency become larger with higher frequency (Fig. 10B, middle). These two findings are consistent with an increase in action potential threshold, particularly at higher frequency. Jitter is reduced compared with inputs throughout the trains at all frequencies examined (Fig. 10B, bottom). For Gtot = 2, jitter is similar at all frequencies, even at 333 Hz, despite the low reliability of evoking a BC spike. For Gtot = 4, jitter increases with the stimulus frequency. For Gtot = 8, first spike jitter is unaffected by high-frequency activation (Fig. 10B, bottom, closed green circles). By contrast, jitter of all spikes is reduced (Fig. 10B, bottom, open green circles) because secondary spikes are effectively eliminated during high-frequency activation.
The effects of high-frequency activation were similar when input timing followed a Gaussian rather than an alpha distribution (Fig. 10, C and D). As with isolated trials (e.g., Fig. 2), jitter reduction differed between alpha- and Gaussian-distributed inputs. However, response reliability and mean latency were similar and jitter was reduced for train frequencies of 333 Hz. In addition, secondary spikes were virtually eliminated during high-frequency stimulation, so that jitter of all evoked spikes was consequently reduced.
The results presented in Figs. 9 and 10 suggest that high rates of activity during trains lead to significant changes in action potential threshold. We examined this possibility by determining the minimum conductance to trigger a spike (threshold conductance, GTh) for isolated stimuli and after a train. For the example in Fig. 11A, GTh was 11.5 nS for isolated stimuli (left) and increased to 27.5 nS after a 333-Hz train of 19 suprathreshold stimuli Fig. 11A (right). This change in GTh was accompanied by an increase in action potential threshold (+2.6 mV) and a reduction in the baseline potential (–5.6 mV), which represented a total increase in the threshold depolarization (
VTh) from 9 mV for isolated stimuli to 17 mV after a 333-Hz train.
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VTh and decreases in the input resistance of the cell could both contribute to the increased GTh required to evoke a spike at the end of a train. If input resistance stays constant, then the threshold depolarization (VTh) should be directly proportional to GTh, which was the case for train frequencies 200 Hz (dashed line in Fig. 11C). After 333-Hz trains, VTh did not continue to increase, indicating that the input resistance decreased at high frequencies. We determined whether increases in threshold account for changes in latency and jitter during trains by using the predictions of the perfect integrator model (Eq. 1). To approximate the conditions at the end of the train, we scaled the number of inputs required to cross threshold (n) by the increase in GTh that we measured in Fig. 11B (top). Because the increases in GTh varied from cell to cell, we used values of GTh20/GTh1 ± 1 SD. For example, when Gtot = 4 times threshold with N = 10 total inputs, n at the start of the train would be 3 (i.e., 3 inputs x 0.4 per input >1), but at the end of a 333-Hz train n could range from 5 to 6 (i.e., 5 or 6 inputs x 0.4 per input > 1.9 ± 0.3).
The predictions of the perfect integrator are shown as shaded areas in Fig. 11D for alpha-distributed inputs, and in Fig. 11E for Gaussian-distributed inputs, for Gtot = 4 and 8 times threshold (blue and green, respectively). They are compared against the experimental data of Fig. 10. We did not consider the case where Gtot = 2, because BCs frequently failed to fire spikes late in the train (Fig. 10, B and D), which does not match the conditions we used to determine action potential threshold in Fig. 11B. There was good agreement between the predictions of Eq. 1 and the experimental data. The model predicts the increased latency with higher train frequency for both alpha- and Gaussian-distributed inputs. It also predicts that jitter increases for higher-frequency trains with alpha-distributed inputs. Minor discrepancies between the model and the experimental data are apparent. These may arise because GTh shifts during trains were measured in separate experiments from those in which spike timing was measured. However, the overall good match in Fig. 11, D and E indicates that BCs behave like perfect integrators both during relatively isolated trials and during high-frequency trains, provided activity-dependent shifts in action potential threshold are taken into account.
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DISCUSSION |
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The number of inputs and jitter reduction
As expected, increasing the number of inputs led to decreasing jitter (Fig. 4). This suggests that in systems where jitter reduction is of overriding importance, one would predict the convergence of large numbers of very small inputs (if they are available). In the ideal case, convergence of an infinite number of inputs leads to responses with jitter <25 µs in BCs.
Normally the number of inputs is limited by two factors. First, the converging inputs must encode similar information. For example, there are only a limited number of AN fibers that project from similar frequency regions of the cochlea. If too much of the cochlea were to converge on individual bushy cells in the AVCN, some of the inputs would be carrying information on different stimulus features (i.e., frequency), which would degrade jitter reduction. Similar limitations are undoubtedly at play in other brain regions. Second, only a subset of inputs is likely to be active at any given time, which further reduces the effective number converging. For example, with pure tones of low amplitude or high frequency, AN fibers fire on only a subset of cycles.
Because of these two factors, most systems are expected to have a limited number of active inputs. Therefore other features of the inputs may need to be optimized to achieve significant jitter reduction. One important factor is to have inputs with appropriate amplitudes. This is particularly important in the extreme case of very few inputs, as described in the accompanying paper (Xu-Friedman and Regehr 2005).
Differences between alpha and Gaussian distributions
Although inputs with alpha and Gaussian distributions have similar effects on the probability and mean latency of response, they have significantly different effects on jitter. For the same number and amplitude of inputs, alpha-distributed inputs were more effective than Gaussian-distributed inputs at reducing jitter (Fig. 2). In principle, the same levels of jitter reduction could be achieved with Gaussian-distributed inputs by increasing the number of inputs (Fig. 4). However, with a limited number of inputs, as is expected normally, the greatest jitter reduction for the two distributions was achieved in distinct ways. When the inputs were alpha distributed, increasing the total synaptic conductance (Gtot) always led to lower jitter in first spike latency (we explore further implications of this finding in Xu-Friedman and Regehr 2005). By contrast, for Gaussian-distributed inputs, a Gtot of 2–4 times threshold yielded the greatest jitter reduction (Fig. 2).
This has important implications in vivo. In the auditory system in cats, the distribution of inputs depends on the stimulus. For example, responses of AN fibers appear Gaussian distributed for high-frequency tone stimuli, but alpha distributed for low-frequency tones (Johnson 1980; Joris et al. 1994a; Kiang 1965). Thus to reduce jitter, BCs should use different strategies in different circumstances. For low frequencies, larger-amplitude inputs would yield lowest jitter, whereas at high frequencies, the input amplitude would have to be regulated to lie within a more restricted range.
Currently, there is little known about synaptic strength over different tonotopic regions of the AVCN. There is some evidence that spherical BCs, which predominantly encode low-frequency tones, receive few large endbulbs, whereas globular BCs, which also encode high-frequency tones, receive many small endbulbs (Brawer and Morest 1975; Liberman 1991; Smith et al. 1993). This anatomical organization suggests that these two cell types may use distinct strategies for reducing jitter because their inputs follow different distributions.
Influence of properties of the postsynaptic cell
The properties of the postsynaptic cell play three major roles in jitter reduction by convergence. First, the postsynaptic cell affects how convergent inputs summate to cross threshold. If the time constant of the cell is fast relative to the temporal dispersion of the inputs, then the inputs may not summate effectively, and the cell would not fire a spike. An illustration of this phenomenon is shown in Fig. 8, for small inputs (Gtot = 2) with SDin 2 ms. Under these conditions the reliability of the response was low and jitter reduction was relatively poor. However, with SDin 0.5 ms, BCs acted like perfect integrators, where the mean latency and jitter of the response could be predicted (Eq. 1). This is somewhat surprising because BCs express strongly rectifying potassium conductances that activate quickly and would be predicted to rapidly influence BC firing (Manis and Marx 1991; Rothman and Manis 2003a). However, our experiments suggest that these conductances do not interfere with summation for SDin 0.5 ms.
The second postsynaptic property that affects jitter reduction is the refractory period. The refractory period is important because it suppresses the effects of late inputs. If late inputs are able to trigger spikes, the overall jitter increases. An illustration of this is shown in Fig. 8 for large inputs (Gtot = 8) with SDin 2 ms. Under these conditions, secondary spikes were triggered, which interfered with jitter reduction. Secondary spikes were much less prominent with SDin 0.5 ms. This behavior is consistent with the presence of strongly rectifying potassium channels in BCs that suppress secondary spikes (Manis and Marx 1991; Rothman and Manis 2003a,b). This feature makes them particularly well suited for reducing jitter without interference from secondary spikes.
For cells whose refractory period is short relative to their inputs, the number of postsynaptic spikes is very sensitive to the amplitude of the inputs, suggesting that the amplitude of inputs would need to be tightly controlled. Without an effective refractory period, additional mechanisms would be required to suppress secondary spikes. For example, feedforward synaptic inhibition is well suited to this task. Alternatively, downstream neurons could have mechanisms to distinguish first spikes, such as synaptic depression, so that they would not respond to secondary spikes.
The third postsynaptic property that affects convergence is how the set of intrinsic conductances are affected during repetitive activation. In BCs, repetitive activation such as is expected in vivo increases the number of synaptic inputs needed to trigger a spike (Fig. 11). The depolarization needed to evoke an action potential increases during trains because the membrane potential for triggering spikes increases and because EPSPs occur during the afterhyperpolarization from preceding spikes. Sodium channel inactivation and potassium channel activation during trains likely contribute to the increase in conductance required to trigger spikes. The close agreement between simulations and experimental findings suggests that changes in threshold for action potential initiation in BCs accounts for most of the changes in spike timing seen in BCs over the course of a train (Figs. 9–11).
Although many of the experimental findings shown here are likely to generalize to other cell types, the behavior during trains may not. Threshold changes depend in part on the kinetics and extent of activation and inactivation of sodium and potassium conductances, which may be specialized in different cell types.
Latency
The latency of response was highly influenced by the total synaptic conductance (Gtot). Even small changes in Gtot could lead to shifts of >1 ms (Fig. 2), which is large considering that the SDin used here was 0.5 ms. Gtot is the product of the individual synaptic conductance (Gpeak) with the number of inputs (N). Both factors may show changes in vivo. We investigated cycle-to-cycle variability in N by setting it to be binomially distributed (Fig. 6), and found that it is not likely to have a significant impact on mean latency or jitter because the trials with large shifts are relatively rare. However, Gpeak could be affected by use-dependent changes in synaptic strength or by the presence of neuromodulators. For example, AN fibers in vivo exhibit a wide range of firing rates (Johnson 1980; Kiang 1965; Sachs and Abbas 1974) depending on the amplitude of the sound. High firing rates lead to synaptic depression (Bellingham and Walmsley 1999; Isaacson and Walmsley 1996; Oleskevich and Walmsley 2002), which significantly change Gpeak (Xu-Friedman and Regehr 2005) and consequently the postsynaptic response latency. Furthermore, neuromodulation can lead to changes in synaptic strength. There are several neuromodulatory inputs to the auditory brain stem, such as norepinephrine and acetylcholine (Klepper and Herbert 1991; Kössl et al. 1988; Kromer and Moore 1980; Yao and Godfrey 1995). Thus there are several ways that Gtot could change in vivo and thereby affect response latencies.
Shifts in latency can have important computational ramifications. For example, the outputs of BCs are involved in sound localization in the superior olive (Masterton and Imig 1984). It is thought that in cats the timing of sounds at the two ears is relayed by spherical BCs to the medial superior olive (MSO), where precise differences in timing are compared to determine azimuthal location (Smith et al. 1993; Yin and Chan 1990). These timing differences are on the order of tens of microseconds, so a shift in latency of 1 ms could have an adverse effect on accurate comparison and thus errors in sound localization. A similar situation may also exist for globular BCs, which in cats receive many more inputs than spherical BCs (Liberman 1991). The role of timing in the globular BC pathway is less well understood, but the characteristics of their calyceal terminals in the medial nucleus of the trapezoid body (MNTB) indicate precise timing is important (Oertel 1999). This suggests that the latency of BC spikes needs to be very tightly controlled and thus so must the effects of convergence.
Spike timing during trains of activity
When BCs were activated during trains, there were changes in the number of spikes elicited, mean latency, and jitter (Figs. 9 and 10). These changes were most pronounced at high frequencies. With low-amplitude inputs (Gtot = 2), activation during high-frequency trains led to low reliability of response, although the spikes that were elicited did show reduced jitter. For high-amplitude inputs (Gtot = 8), high-frequency activation led to elimination of secondary spikes. These spikes normally caused jitter to be high, so their elimination during the train reduced jitter considerably. We also observed that mean latency increased and jitter increased during trains. All these effects were consistent with changes in action potential threshold (Fig. 11).
Our exploration of the effects of trains on spike timing was necessarily simplified to consider postsynaptic processes. Many additional processes operate during high levels of activity that could affect spike timing. First, as discussed above, individual synaptic inputs that are repetitively activated depress over the course of the train, which would cause additional changes in latency and jitter. Second, individual inputs will not be active on every cycle. For example, if each input is active on half the cycles, then the effective number of inputs is cut in half, which will further decrease response probability, and increase jitter and mean latency. Third, the conditions that lead to increased firing rate in AN inputs are likely to also affect their timing distribution on a given cycle. Changes both in sound frequency and intensity affect not only firing rate, but also lead to changes in the SD and in the rate of onset of cycle histograms (Johnson 1980; Joris et al. 1994a; Rose et al. 1967). These factors will affect jitter reduction. Although we did not examine these processes directly, our experiments using more isolated trials make predictions about their effects.
Modeling latency and jitter
Our results indicate that the probability, latency, and jitter of postsynaptic spikes depend on the number, amplitude, and timing distribution of converging inputs. Over a specific range of SDin, these effects could be accounted for by modeling the postsynaptic cell as a perfect integrator of its inputs. This was also true during trains of activity, provided changes in spike threshold during the train were taken into account. However, for large SDin, there was significant deviation from the model's predictions, because of the time constant and refractory period of real cells. Thus when the distribution of spike arrival times is sufficiently narrow the postsynaptic cell behaves like a simple integrator. More detailed models of the postsynaptic cell are required to account for the occurrence and timing of secondary spikes and for spikes evoked by inputs with large SDs.
Our model differs from previous models. These models include Hodgkin–Huxley-type models (Marálek et al. 1997; Rothman and Young 1996) as well as integration models (Burkitt and Clark 1999; Marálek et al. 1997). However, the integration models arrived at a formulation of the general solution that differed from ours (Eq. 1). We found that only our formulation matched our dynamic-clamp results (Fig. 7) and also the results of Monte Carlo simulations of perfect integrators (data not shown). Compared with our model, Hodgkin–Huxley-type models are more complex and require detailed knowledge of the number, density, and type of currents in real cells. Although our model is greatly simplified compared with such models, it could account for the effects of convergence under many conditions. As a result of its simplicity, it provides a more intuitive understanding of how jitter reduction is accomplished. However, to deal with the additional complications when SDin is longer than the time constant and refractory period of the cell, or to explain the shifts in spike threshold during a train, a Hodgkin–Huxley-type model would be necessary.
In conclusion, our results make strong predictions about how the properties of converging inputs affect the timing of postsynaptic responses in vivo. To understand how jitter reduction takes place under physiological conditions, it is necessary to take into account the total synaptic conductance with respect to threshold, the number of inputs, and the temporal distribution of inputs. This last factor in particular has received little attention. Once these are sufficiently well characterized, it may be possible to account for postsynaptic latency and jitter using a simple model.
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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Address for reprint requests and other correspondence: M. Xu-Friedman, Department of Biological Sciences, 641 Cooke Hall, University at Buffalo, State University of New York, Buffalo, NY 14260 (E-mail: mx{at}buffalo.edu)
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INTRODUCTION
18–20) normalized by the number of spikes evoked in the first cycle (N1). B and D, middle: latency difference between the last 3 cycles (
18–20) and the first cycle (L1). B and D, bottom: SD over the last 3 train cycles normalized by the input SD ("jitter reduction", 
18–20) divided by the jitter reduction in the first cycle (J1), for first spikes per cycle only (closed symbols) and for all spikes (open symbols).