We first determine the parameters that are necessary for the dynamic spike-blocking model to produce realistic On-I PST histograms and entrainment. Then, we show that the model can also produce On-L discharge patterns for a different parameter range.
3.1. The Dynamic Spike-Blocking Model Gives Both Entrainment and On-I PST Histograms
shows the net synaptic current, the membrane voltage, the spiking threshold, and spike times for a high-frequency (6000 Hz) tone burst 50 dB above threshold for both the dynamic spike-blocking model and the constant-refractoriness model. Both models have the same number of inputs (N = 400) and net synaptic strength (N · Gα = 10), chosen such that entrainment extends up to 800 Hz. The net synaptic current in the two models for a single stimulus presentation is the same except for statistical variations. The current rises transiently to a maximum at the onset of the stimulus and then rapidly falls to a steady-state level. The fall to a lower steady-state level is due to adaptation of the discharge rate in the AN inputs. Fluctuations in the synaptic current are small because the random firings of AN fibers are averaged out by the summation of a large number of independent inputs.
Despite having the same synaptic input, spiking patterns for the two models are completely different. In , the membrane voltage for the constant-refractoriness model reaches threshold several times during the stimulus, causing the model to produce several spikes. The resulting PST histogram shows chopping as well as too much steady-state activity for an On neuron. In contrast, the dynamic spike-blocking model produces an On-I discharge pattern, with only one spike at onset (). After the membrane voltage reaches threshold and produces the first spike, the model enters the spike-blocking state indicated by elevation of the threshold to ∞. The spike-blocking state lasts until the membrane voltage repolarizes below the transition voltage Vt, an event that only happens at the end of the stimulus. As a result, the model produces only one spike at the stimulus onset and the PST histogram has an On-I shape. Thus, for the same synaptic current, the dynamic spike-blocking model produces an On-I PST histogram while the constant-refractoriness model produces a Sustained PST histogram with chopping.
Despite having completely different responses to high-frequency tone bursts, the responses of the two models to a low-frequency tone are similar. shows responses of the two models to a 700 Hz tone at 90 dB SPL. The net synaptic currents, which again are nearly identical, have oscillations at the frequency of the 700 Hz tone. These periodic oscillations occur because the discharges of AN fibers phase-lock to the low-frequency tone. The lower panel of B shows that the membrane voltage in the constant-refractoriness model reaches threshold and produces one spike for every stimulus cycle and hence entrains to the 700 Hz tone (EI = 1). The response of the dynamic spike-blocking model also entrains to the 700 Hz tone with an EI of 1. This is possible because, in contrast to the response to the high-frequency tone burst (), the membrane voltage falls below Vt on each cycle, allowing exit from the spike-blocking state, and firing on the next cycle.
Figure 3 Responses of the constant-refractoriness model and the dynamic spike-blocking model to 90 dB SPL low-frequency tones. A: A 20 millisecond segment of the response of the dynamic spike-blocking model to a 700 Hz tone, with Vt = 0.4. B: The same in the constant-refractoriness (more ...)
summarizes the entrainment in the two models for low-frequency tones presented at 90 dB SPL. In both models, the upper frequency limit of entrainment extends to 800 Hz. For frequencies below 300 Hz, the dynamic spike-blocking model continues to entrain while the constant-refractoriness model hyper-entrains (more than one spike per stimulus cycle). At these low frequencies, the membrane voltage in the constant-refractoriness model reaches spiking threshold more than once during each stimulus cycle. In contrast, the membrane voltage in the dynamic spike-blocking model reaches spiking threshold only once during each cycle, because the model enters the spike-blocking state after the first spike in a cycle and does not exit this state until the membrane voltage falls below Vt at the end of the cycle.
In summary, the dynamic spike-blocking model produces both entrainment to a broad range of frequencies and On-I
PST histograms for high-frequency tone bursts, while the constant-refractoriness model fails to produce any of these responses for comparable inputs. When N
are chosen such that the model entrains to tones up to 800 Hz, the constant-refractoriness model produces PST histograms with chopping, too much steady-state activity for On-I
neurons, and hyper-entrainment at very low frequencies. The short interspike intervals that are necessary to produce entrainment to 800 Hz also occur during high-frequency tone bursts (chopping) and very low-frequency tones (hyper-entrainment). We showed in Kalluri and Delgutte (2003)
that if Gα
is reduced so that there are no short interspike intervals (chopping) for high-frequency tone bursts, then the constant-refractoriness model fails to produce the short interspike intervals needed for entrainment. In contrast, the dynamic spike-blocking model produces both entrainment to low-frequency tones and On-I
PST histograms by preventing the “extra” spikes that lead to chopping and hyper-entrainment in the constant-refractoriness model. In the remainder of this paper, we focus on the dynamic spike-blocking model.
3.2. Model for On-I Discharge Patterns
3.2.1. Effect of the Transition Voltage
The transition voltage, Vt, determines when a transition occurs from the spike-blocking state to the integration state. Vt is critical for the discharge patterns produced by the model and it is sharply constrained by the PST histogram shape and the frequency range for entrainment.
shows how PST histograms produced by the model are affected by Vt, which has dimensionless units because it is normalized by the threshold; all other parameters are as in and . In PST histograms for a 6000 Hz tone burst at 50 dB above threshold, the steady-state rate increases while the onset rate is relatively unchanged as Vt increases from 0.1 to 0.4, 0.6, and 0.9 (). The PST histograms are On-I for Vt equal to 0.1 and 0.4, while they are On-L for Vt equal to 0.6 and 0.9. While the PST histogram for Vt = 0.9 appears to be Sustained, at lower levels the steady-state discharge rate is low and PST histograms meet our quantitative criteria for On-L responses (not shown). The steady-state rate for tone bursts presented at 50 dB increases steadily from 0 to 450 spikes/sec as Vt goes from 0.1 to 1 (), exceeding the range for On-I neurons when Vt exceeds 0.5. When Vt is small (<0.5), the steady-state rate is low because the membrane voltage rarely falls below Vt after a spike, causing the model to stay in the spike-blocking state throughout the stimulus. On the other hand, when Vt is high (>0.5) the steady-state discharge rate is high because fluctuations in the membrane voltage frequently fall below Vt, causing the model to reenter the integration state and fire again.
Figure 4 PST histograms in the dynamic spike-blocking model as a function of the transition voltage, Vt . A: PST histograms of model responses to 6000 Hz tone bursts at 50 dB above threshold for Vt equal to 0.1, 0.4, 0.6, and 0.9. B: Steady-state discharge rate (more ...)
The transition voltage also affects the frequency range of entrainment produced by the model. shows the synaptic current, membrane voltage, and threshold in the dynamic spike-blocking model with Vt = 0.4 for a 600 Hz tone and a 1000 Hz tone. The model fires on every cycle of the 600 Hz tone but fails to do so for the 1000 Hz tone. For the 1000 Hz tone, the troughs of the nearly periodic membrane voltage tend to be shallow and remain above Vt, so that the model fails to reenter the integration state on each stimulus cycle. The troughs are shallow for the 1000 Hz tone because their widths approach the duration of the synaptic conductance change (0.5 ms). In general, the troughs become progressively shallower with increasing frequency so that entrainment declines rapidly above a certain limit. For Vt = 0.4, this limit is 800 Hz (). Lowering Vt to 0.1 results in the upper frequency limit of entrainment falling to 500 Hz, while raising Vt to 0.6 results in the upper frequency limit of entrainment increasing slightly to 900 Hz. Further increasing Vt to 0.9 does not greatly change the upper frequency limit of entrainment, but results in hyper-entrainment at very low frequencies. For such values of Vt, the model behaves increasingly like the constant-refractoriness model because it almost always reenters the integration state shortly after each spike.
Figure 5 Entrainment in the dynamic spike-blocking model as a function of the transition voltage, Vt . A: Example traces of synaptic current, threshold, and membrane voltage for a 600 Hz tone and a 1000 Hz tone, with Vt = 0.4. B: EI versus frequency for Vt equal (more ...)
Entrainment by the model is summarized in , which shows EI as a function of Vt for 100 Hz and 800 Hz tones. The model entrains to 800 Hz tones if Vt > 0.3. On the other hand, it hyper-entrains to a 100 Hz tone if Vt ≥ 0.8. Thus, choosing Vt between 0.3 and 0.7 allows the model to entrain to tones from 100 Hz to 800 Hz.
Taken together, and show that Vt must be greater than 0.3 and less than 0.6 for the model to produce both On-I PST histograms and entrainment over a broad range of frequencies.
3.2.2. Effect of the Number of Inputs
While to this point the number of inputs to the model has been held at 400, we now examine the effect of this parameter. It turns out that the model cannot simulate On-I discharge patterns if it has many fewer inputs. shows PST histograms for 6000 Hz tone bursts at 20 dB above threshold for N equal to 40, 150, 250, and 350, with N · Gα still equal to 10. The transition voltage, Vt, was fixed to 0.4 based on the results of and . The PST histogram type is On for N equal to 250 and 350, but Sustained when N is lowered to 150 and 40. Furthermore, the steady-state discharge rate for 50 dB above threshold is below 10 spikes/sec—the range for On-I neurons—only if the model has at least 200 inputs ().
Figure 6 The effect of the number of inputs on PST histograms in the dynamic spike-blocking model. A: PST histograms of responses to 6000 Hz tone bursts presented at 20 dB above threshold for N equal to 40, 150, 250, and 350. The PST histograms for N equal to (more ...)
The effect of N
on PST histogram shape can be understood in terms of the size of membrane voltage fluctuations. In Kalluri and Delgutte (2003)
, we showed that with the net synaptic strength (N
) fixed, fluctuations of synaptic current and the resulting fluctuations of membrane voltage increase as N
decreases in the constant-refractoriness model. The same relationship holds in the dynamic spike-blocking model because the integration state is identical for the two models. When voltage fluctuations are large, the membrane voltage often falls below Vt
so that there are frequent transitions from the spike-blocking state to the integration state. The high rate of transitions into the integration state results in a relatively high steady-state discharge rate and precludes On-I
PST histograms. Thus, to produce On-I
PST histograms, the dynamic spike-blocking model must have the small fluctuations of membrane voltage that result from large N
The steady-state discharge rates in can be reduced by lowering Vt, so that the model can produce On-I PST histograms even when N is less than 200. However, shows that lowering Vt below 0.4 also reduces the upper frequency limit of entrainment. This constraint on Vt applies for a wide range of N because entrainment does not change greatly with N (not shown). Thus, taken together, the requirement of entraining to tones up to 800 Hz and producing On-I PST histograms for tone bursts at 50 dB above threshold constrain the number of inputs in the dynamic spike-blocking model to be at least 200 and Vt to be between 0.3 and 0.6.
3.2.3. Effect of Stimulus Level
The constraint on N
in the model can be further refined by requiring PST histograms to be On-I
over a wide range of levels. The change in the PST histogram shape with level is indirectly summarized by plots of average discharge rate and steady-state discharge rate versus level in . For N
= 400, the steady-state discharge rate is always 0 spikes/stimulus, indicating On-I
PST histograms at all levels. On the other hand, for N
= 250 and N
= 150, the steady-state discharge rate is nonzero for intermediate levels (due to underlying On-L
PST histograms), leading to nonmonotonic average rate versus level curves. Such nonmonotonic rate-level curves for CF tone bursts have not been observed in On-I
neurons (Rhode and Smith, 1986
; Winter and Palmer, 1995
). Furthermore, for N
= 150, the steady-state discharge rate at intermediate levels exceeds the 0.25 spikes/stimulus (10 spikes/sec) limit for On-I
responses and therefore PST histograms are On-L
. Thus, the problem with having a small N
is that steady-state discharge rate is non-zero at intermediate levels, leading to On-L
PST histograms over some range and to non-monotonic rate-level curves.
Figure 7 The effect of the number of inputs on responses at different levels. A: The average discharge rate (solid, computed over entire 25 ms duration of stimulus) and steady-state discharge rate (dashed, computed over last 12 ms of stimulus) as a function of (more ...)
The finding of nonmonotonic rate-level curves in a model having no synaptic inhibition can be understood by examining the synaptic current, membrane voltage, threshold, and Vt for tone bursts presented at two levels when N is 150 (). The steady-state rate is maximum at an intermediate level of 58 dB SPL because the fluctuating membrane voltage frequently falls below Vt . On the other hand, at a relatively high level of 88 dB SPL, the mean voltage is sufficiently high that its fluc-tuations rarely fall below Vt . If N is greater (e.g., N = 400), the fluctuations are small enough that the membrane voltage rarely falls below Vt at any level, so that steady-state discharge rate remains essentially zero.
The nonmonotonicity of the average rate versus level function indirectly reflects the ability of the model to produce On-I
PST histograms across a wide range of levels.2
For any given N,
there is a maximum transition voltage,
, for which the fluctuating membrane t
voltage is above Vt
at all suprathreshold levels, enabling the model to produce On-I
PST histograms and monotonic rate-level curves. This maximum exceeds 0.3, the minimum for getting entrainment up to 800 Hz, only if N
Thus, when the nonmonotonicity of rate-level curves is taken into account along with entrainment, our simulations show that the dynamic spike-blocking model must have at least 300 inputs to produce On-I PST histograms over a wide range of levels and entrainment over a broad range of frequencies. In summary, the model must have the following properties to produce On-I discharge patterns:
- – N ≥ 300
- – Vt between 0.3 and the lower of
3.3. Model for On-L Discharge Patterns
It would appear from and that one way to model On-L
discharge patterns is to raise Vt
near 0.6. With higher Vt,
the model makes a sufficient number of transitions out of the spike-blocking state for high-frequency tone bursts to produce a steady-state discharge rate in the range for On-L
neurons while also entraining to a broad range of tones without hyper-entraining to very low-frequency tones. However, for this range of Vt,
the model also produces nonmonotonic rate-level curves, which are not observed in On-L
neurons (Rhode and Smith, 1986
; Winter and Palmer, 1995
When N · Gα = 10, it turns out that the model cannot produce On-L PST histograms, monotonic rate-level curves, and entrainment because there is no N for which the range of Vt resulting in On-L PST histograms matches the range of Vt resulting in monotonic rate-level curves and entrainment. Thus, changing N alone is insufficient for producing On-L discharge patterns.
helps produce On-L
discharge patterns because the range of Vt
producing hyper-entrainment shrinks, thus enabling a choice of N
for which the model produces all three response properties. The range of Vt
for hyper-entrainment shrinks because the input is not sufficiently strong at low frequencies to cause multiple spikes in a stimulus period. However, N
cannot be lowered too much because the frequency range of entrainment would be too small (see Fig. 13 in Kalluri and Delgutte, 2003
With N · Gα lowered from 10 to 8.8, the model can produce On-L discharge patterns for many combinations of Vt and N . For example, with 400 inputs and Vt = 0.7, the dynamic spike-blocking model does produce On-L discharge patterns (). Based on the shape of the PST histogram at 20 dB above threshold (A), the PST type is On. Further, the PST histogram at 50 dB above threshold is On-L (B) and the rate-level curve (C) is nearly monotonic. Compared to when N · Gα = 10, the mean membrane voltage is lower and as a result, the range of Vt for which rate-level curves are non-monotonic moves to lower voltages. Furthermore, the voltage is low enough to avoid hyper-entrainment to very low-frequency tones. However, as a result of the lower mean membrane voltage, the frequency range of entrainment (D) decreases a little compared to that for the model of On-I neurons in . Entrainment by the On-L model is lower than entrainment by the On-I model at both the lower (100 Hz) and the upper (800 to 900 Hz) end of the frequency range.
Figure 8 On-L discharge patterns produced by the dynamic spike-blocking model, with N · Gα = 8.8. A: PST histograms for 6000 Hz tone bursts presented at 20 dB and 50 dB above threshold. B: Average discharge rate versus level of 6000 Hz tone bursts. (more ...)
summarizes the discharge patterns produced by the dynamic spike-blocking model for different combinations of N and Vt, with N · Gα fixed to 8.8. The PST histogram type (On-L, On-I, or Sustained) is coded by different symbols in A. The model produces On-L PST histograms when Vt is greater than 0.5 and N is greater than 200. On the other hand, it produces On-I PST histograms when Vt < 0.5 and Sustained PST histograms when N is small (<200). We have already noted these effects of N and Vt on the shape of PST histograms. The shaded area in the figure indicates the region for non-monotonic rate-level curves and shows that the model produces both On-L PST histograms and monotonic rate-level curves only for Vt > 0.6 and N > 200.
Figure 9 Response properties of the dynamic spike-blocking model for different N and Vt . A: PST histogram type for 6000 Hz tone bursts. The gray-shaded area indicates the N and Vt pairs that produced non-monotonic rate-level curves (NI ≥ 0.08). B: Entrainment (more ...)
Entrainment by the model, when N
is 8.8, is summarized in B. Combinations of Vt
that result in hyper-entrainment to 100 Hz tones are distinguished from combinations that do not produce hyper-entrainment by different plot symbols. A hyper-entraining response (EI >
1.1) occurs for small N
200) and large Vt
The gray-shaded area indicates the region for which the model fails to entrain to 800 Hz tones (EI <
0.8). This region encompasses values of Vt <
0.3 or N <
100. Together, the unshaded regions (entrainment to 800 Hz tones) and the diamond plot symbols (no hyper-entrainment to 100 Hz tones) show that the model entrains to tones up to 800 Hz without hyper-entraining to 100 Hz tones when it has 200 or more inputs and Vt
is greater than 0.3.
Finally, putting together reveals the conditions under which the dynamic spike-blocking model produces On-L PST histograms, monotonic rate-level curves, and entrainment:
- – 200 ≤ N ≤ 600
- – Vt roughly between 0.6 and 1.
- – 8.8 ≤ N . Gα < 10.
The model produces On-L discharge patterns over a broad range of N between 200 and 600. Although any N in this range results in On-L PST histograms at high levels, the responses to high-frequency tone bursts for large N increasingly resemble On-I responses. As N is increased toward 600, the PST histogram shape is On-I for an increasingly broader range of low stimulus levels above threshold, until for N more than 600, PST histograms are On-I at all stimulus levels.