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The Center for Hearing Science, 1 Department of Biomedical Engineering, The Johns Hopkins University School of Medicine, Baltimore, Maryland 21205; and 2Department of Otolaryngology/Head and Neck Surgery, and The Curriculum in Neurobiology, University of North Carolina, Chapel Hill, North Carolina 27599
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: MODEL EQUATIONSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: MODEL EQUATIONSACKNOWLEDGMENTSREFERENCES |
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,b). The measurements from these currents allowed us to create kinetic models of VCN neurons with a precision not previously available.
Several models of VCN stellate and bushy cells have already been described. Banks and Sachs (1991), for example, presented a stellate cell model consisting of an active somatic and axonal compartment coupled to a passive dendritic tree. Because little was known about the Na+ and K+ currents in VCN stellate cells before 1991, Banks and Sachs used modified versions of the Hodgkin and Huxley (HH) equations (1952), which included a fast Na+ current (INa) and a high-threshold K+ current (IHT). Wang and Sachs (1995) presented a modified version of the Banks and Sachs stellate cell model where, to account for a higher spike threshold observed in vitro, the activation curves of INa and IHT were shifted 10 mV positive. Arle and Kim (1991) and Hewitt et al. (1992) presented "MacGregor-type" stellate cell models in which IHT was treated as a digital entity; that is, it was "on" during an action potential (AP) but "off" at other times. In general, the preceding HH-like and MacGregor-like stellate models were successful in that they replicated many of the response characteristics of stellate cells in vitro and in vivo. For example, during a depolarizing current pulse, the models exhibited repetitive firing (i.e. a Type I current-clamp response) and when stimulated with auditory-nerve-like synaptic input, the models exhibit a "chopping" response in their poststimulus time histograms (PSTHs), reflecting their regular discharge. These models were also successful in replicating responses to more complex stimuli (Arle and Kim 1991; Hewitt et al. 1992; Wang and Sachs 1995).
Models of VCN bushy cells have also been described. Rothman et al. (1993), for example, presented a model based on the voltage-clamp study of Manis and Marx (1991). Because only the low-threshold current (ILT) had been fully characterized at the time, models of INa and IHT were based on those of Frankenhaeuser and Huxley (1964). Perney and Kaczmarek (1997) also presented a model a VCN bushy cells. In this case, ILT was based on the data of Manis and Marx (1991) and IHT was based on a study of a Shaw voltage-activated K+ channel (KCNC1) expressed in NIH 3T3 fibroblasts (Kanemasa et al. 1995) because KCNC1 is thought to be a component of the delayed rectifier in VCN neurons (Perney and Kaczmarek 1997). Like the stellate cell models, these bushy cell models successfully replicated many of the response characteristics of bushy cells in vitro and in vivo, including a phasic discharge of APs in response to a depolarizing current pulse (i.e. a Type II current-clamp response) and the ability to follow a train of auditory-nerve-like synaptic inputs at relatively high frequencies.
Although these models have generally been successful in replicating the response characteristics of stellate and bushy cells, they were developed on the basis of incomplete information regarding the K+ channels. Figure 1 compares the voltage dependence and kinetics of some of the K+ currents used in previous models. In this figure, normalized activation functions, whole cell currents, and time constants of the model "high-threshold" currents are plotted on the left (n
, IHT, and 
n), and those of the model "low-threshold" currents are plotted on the right (w, ILT, and w). The thin lines correspond to the models, and the bold lines pertain to the mean experimental data presented in our previous studies (Rothman and Manis 2003a,b). Not included in this figure are the MacGregor-type models because these models have unrealistic K+ currents in that they turn on and off instantaneously. Comparison between the model and experimental data reveals several discrepancies. First, the activation functions of the model high-threshold currents (n) all show significant deviation from the experimental data in that they have shallow voltage dependencies and activate over different voltage ranges. When these activation functions are converted into whole cell currents (IHT), the discrepancies become even more pronounced (note y axis log scaling). All of the descriptions show significant activation for potentials just above rest: their activation thresholds (Vth) fall between -52 and -62 mV. According to our experimental data (Rothman and Manis 2003a), such currents would be classified not as high threshold but low threshold. Indeed, one IHT model with Vth near -60 mV (Perney and Kaczmarek 1997) used a large value for its total conductance (50 µS; Fig. 1, trace 2), and consequently behaves entirely like ILT. For the bushy cell models, this means activation of IHT overlaps that of ILT and therefore contributes to the proposed effects of ILT. For the stellate cell models, this means their intrinsic membrane properties (i.e. input resistance and membrane time constant) are more similar to those of bushy cells than stellate cells.
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The last two discrepancies in Fig. 1 pertain to the time constants of IHT (n) and ILT (w). For n, discrepancies occur at potentials less than -60 mV. For w, discrepancies occur at all potentials. Although the discrepancies in time constants probably do not lead to dramatic differences in membrane properties at suprathreshold potentials (i.e. during an action potential), they could have subtle effects on membrane properties at subthreshold potentials, such as refractoriness.
Hence, given the many discrepancies between the previous models and our new experimental data, we thought it important and necessary to theoretically reexamine the roles K+ currents play in regulating the electrical activity of VCN neurons.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: MODEL EQUATIONSACKNOWLEDGMENTSREFERENCES |
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The VCN model described in this paper consists of a single electrical compartment with a membrane capacitance (Cm) connected in parallel with a fast-inactivating A-type K+ current (IA), a fast-activating slow-inactivating low-threshold K+ current (ILT), a high-threshold K+ current (IHT), a fast-inactivating TTX-sensitive Na+ current (INa), a hyperpolarization-activated cation current (Ih), a leakage current (Ilk), an excitatory synaptic current (IE), and an external electrode current source (Iext). For such an electrical circuit, the membrane potential V is described by the following first-order differential equation
 | (1) |
), and are collectively given in the APPENDIX. Because INa and Ih were not studied in our voltage-clamp experiments, their models were derived from other studies, as described in the following text. Equations for Ilk and IE are also described in the following text. Except for the current-clamp simulations, Iext = 0. For all simulations, Cm = 12 pF, the average value computed from our population of isolated VCN cells (Rothman and Manis 2003a). Given a typical neuronal specific membrane capacitance of 0.9 µF/cm2 (Gentet et al. 2000), the diameter of the soma model comes to 
21 µm, a value also in agreement with our isolated VCN cells (Rothman and Manis 2003a). Model sodium current
Due to the fast kinetics and large magnitude of INa in isolated VCN cells, we did not characterize this current, but instead looked to other voltage-clamp studies in mammalian neurons to derive its model. In particular, we used data from native TTX-sensitive Na+ currents at 22°C: (Belluzzi et al. 1985; Costa 1996; Ogata and Tatebayashi 1993; Parri and Crunelli 1998; Sah et al. 1988; Schild and Kunze 1997). Together, these studies provided sufficient data to compute steady-state activation/inactivation functions in the range -100 to +40 mV (m
and h), as well as activation/inactivation time constants (m and h; see APPENDIX). Based on the studies of Costa (1996) and Belluzzi et al. (1985), model INa was given the following instantaneous current-voltage (I-V) relation
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LTNa is the peak Na+ conductance, and VNa the reversal potential of INa (+55 mV). Comparison of the model current traces as well as steady-state I-V relations with the experimental data shows a close agreement. The activation threshold of model INa (Vth 
-49 mV) is also in agreement with the activation thresholds of the Na+ current measured from our population of isolated VCN neurons (data not shown). Model hyperpolarization-activated cation current
Results from our previous study indicate the majority of VCN neurons possess a hyperpolarization-activated inward current (Rothman and Manis 2003a). Because this inward current activated with slow kinetics, we assumed it was similar to the mixed-cation current Ih rather than the near-instantaneously activating inward current IKIR (Travagli and Gillis 1994). Evidence from experimental studies in the VCN supports this choice. First, the inward current measured in VCN neurons is resistant to blockade by barium (Bal and Oertel 2000; Schwarz and Puil 1997), suggesting it is a form of Ih rather than IKIR (Travagli and Gillis 1994). Second, the current reverses positive to the resting membrane potential (Bal and Oertel 2000; Rusznak et al. 1996), consistent with a mixed-cation selective conductance rather than a K+-selective conductance.
Like model INa, model Ih was derived from other voltage-clamp studies. Three of these studies pertained to auditory neurons (Banks et al. 1993; Fu et al. 1997; Rusznak et al. 1996), and two pertained to nonauditory neurons (Huguenard and McCormick 1992; Travagli and Gillis 1994). Based on these five studies, model Ih was given the following instantaneous I-V relation
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h is the maximum steady-state conductance, Vh the reversal potential of Ih (-43 mV), and r the time and voltage-dependent activation variable with steady-state value r and time constant r (see APPENDIX). Here, r was predominantly based on the Huguenard and McCormick study (1992).
Subsequent to the specification of our model Ih, Bal and Oertel (2000) published a characterization of Ih in VCN octopus cells. The properties of Ih in octopus cells differs from our model values in that the activation curve sits 10 mV more depolarized, the rates of activation and deactivation are faster, and h is larger than that used in this study. The faster rates probably reflect the higher experimental temperature in the Bal and Oertel study (33°C) in comparison to the Huguenard and McCormick study (23°C), and the large h is probably an unusual characteristic of octopus cells. We in fact explore the consequences of changing h in this study as well as shifting the activation curve of Ih to more depolarized potentials. We also decrease the rate constant r by a Q10 factor of 3 when investigating the model at higher temperatures.
Model leakage current
The leakage current was modeled as follows
 | (4) |
LK is the maximum steady-state conductance (2 nS), and Vlk the reversal potential of Ilk (-65 mV). Although it is common practice with HH-like models to adjust the resting membrane potential Vrest by adjusting Vlk, we fixed Vlk at -65 mV. Fixing Vlk allows Vrest to vary in response to changes in other currents, such as ILT and Ih. As will be shown in RESULTS, both ILT and Ih can play an important role in setting Vrest. Model excitatory post-synaptic current
Excitatory post-synaptic currents (EPSCs) from auditory nerve (AN) fibers were modeled by the following equation
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-wave of the form
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E determines the peak conductance and E the time to peak. For most of the simulations in this study, E = 0.4 ms, resulting in an EPSC half width 1 ms, a 10–90% rise time 0.2 ms, and a decay time 0.6 ms at 22°C. These values are comparable to those of fast non-NMDA receptor-mediated EPSCs recorded from auditory-pathway neurons at 22°C (Barnes-Davies and Forsythe 1995; Isaacson and Walmsley 1995; Zhang and Trussell 1994). Example waveforms of gE can be found in Fig. 6. To simulate the effects of converging AN input, the model includes up to 50 independent IE's in parallel.
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Model parameters and properties
Table 1 lists the parameter settings for the five different model configurations investigated in this study: the Type I-c, Type I-t, Type I-II, Type II-I, and Type II model. These canonical configurations were derived from the four cell types previously described (Rothman and Manis 2003a), where the Type I-c cells were those cells that, under voltage clamp, displayed only IHT, the Type I-t cells displayed both IHT and IA, and the Type II cells displayed both IHT and ILT. The two intermediate model types (Type I-II and Type II-I) refer to the intermediate experimental Type I-i cell. As in the experimental data, both intermediate types fall between Type I and Type II, due to the presence of small ILT; however, the Type I-II model is closer to being Type I, and the Type II-I model is closer to being Type II. Listed below each model type in Table 1 are the conductance values used in its simulation. Values for A, LT, and HT were based on experimental data (Rothman and Manis 2003a). For simplicity and ease of comparison between model responses, single values of Na and lk were chosen: 1,000 and 2 nS, respectively. The value of Na was chosen to give large, reliable APs in all five model types, and the value of LK was chosen to give the Type I-c model a realistic input resistance (see following text). The value of h, on the other hand, was not kept the same for all model simulations but instead was adjusted to keep Vrest near -64 mV. Note that we used Ih as a means of setting Vrest instead of Ilk since modulation of Ih could be a mechanism for setting Vrest, especially in Type II cells. For simulations where LT > 0 (i.e. the Type II, Type I-II and Type II-I simulations), h was LT. For the Type I-c and Type I-t model, h was set to a small value of 0.5 nS because larger values produced hyperex-citability. The value of h in the Type II model (20 nS) is similar to values reported in medial nucleus of the trapezoid body (MNTB) neurons (Banks et al. 1993). MNTB neurons share many physiological and anatomical features with VCN bushy cells (i.e. Type II cells), including the presence of strong inward rectification at hyperpolarized potentials. However, this value of h is lower than that estimated for VCN octopus cells (Bal and Oertel 2000).
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Table 1 also lists seven parameters computed from each model type that are consistent with the experimental data. The parameters are as follows.
-64 mV, a value consistent with VCN neurons (Oertel 1983). For the Type I-c and Type I-t models, Vrest is primarily determined by Ilk because this current is the largest at rest. For the Type I-II, Type II-I, and Type II models, however, Vrest is primarily determined by Ih and ILT because these currents are largest at rest. In fact, removal of Ilk from these models produces little change in Vrest. Hence it is possible to model the latter three models without Ilk. 1/LK. The other three models (Type I-II, Type II-I, and Type II) show decreasing values of Rrest, consistent with a parallel increase in h and LT. That Rrest is larger in the Type I-c and Type I-t models than in the Type II model is consistent with previous experimental findings that Rrest is significantly larger in VCN Type I cells [447 ± 265 (SD) M
] than in Type II cells (225 ± 160 M) (Manis and Marx 1991). rest, computed by multiplying Rrest by Cm. Again, rest is larger in the Type I-c and Type I-t models than in the Type II model, consistent with previous experimental findings that rest is significantly larger in Type I cells (6.5 ± 7 ms) than in Type II cells (1.69 ± 0.63 ms) (Manis and Marx 1991). Na = 0 and lk = 0 in these voltage-clamp simulations. Comparison of the model values with the experimental data (see Fig. 4) again shows the model closely agrees with the experimental data. E
, denotes synaptic efficacy or strength of an individual AN synapse. Specifically, E is defined as the smallest one-tenth that of g value of E (Eq. 6) necessary for a single AN synapse to generate an AP. A comparison across model types in Table 1 shows that E is larger in the Type II model than in the Type I-c and Type I-t models. This disparity of E values is due to the magnitude difference of ILT: ILT, being a K+ current with significant activation near rest, acts to oppose synaptic depolarization.
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The fact that a given value of E may be suprathreshold in the Type I-c model but subthreshold in the Type II model poses a problem when trying to compare model responses to what one would like to be the "same" AN input. To resolve this problem, we normalized values of E with respect E, such that for all subthreshold inputs, E = 0.5E, and for all suprathreshold inputs, E = 3E. In this way, AN synapses share the same synaptic efficacy rather than the same peak conductance.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: MODEL EQUATIONSACKNOWLEDGMENTSREFERENCES |
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Model current-clamp responses
MODEL CURRENT-CLAMP RESPONSES RESEMBLE THOSE OF REAL VCN NEURONS. Because VCN neurons are typically classified by their response to current injection as either Type I or Type II (Manis and Marx 1991; Wu and Oertel 1984), it was of particular interest to investigate the behavior of the VCN model under current-clamp conditions. Hence, each model listed in Table 1 was investigated with respect to hyperpolarizing and depolarizing current steps, similar in magnitude to those used in vitro. Results are summarized in Fig. 2, where model responses to a depolarizing (+) and hyperpolarizing (-) current pulse are plotted concurrently. In A and B, the response of the Type I-c and Type I-t model to small current injection is clearly Type I, responding to a small depolarizing current pulse (+50 pA) with a train of regularly spaced APs and responding to a small hyperpolarizing current pulse (-50 pA) with an exponential decay of the membrane potential. These models are nearly indistinguishable except that the Type I-t model shows a higher discharge rate than the Type I-c model during the same current step. VCN neurons with such a "Type I" response have been morphologically identified as stellate cells (Wu and Oertel 1984).
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In Fig. 2C, the response of the Type II model to large current injection is clearly Type II, responding to a large depolarizing current pulse (+300 pA) with a single AP followed by a steady depolarization and responding to a large hyperpolarizing current pulse (-300 pA) with a non-exponential decay of the membrane potential, followed by a "sag" back to Vrest (i.e. inward rectification). At the termination of the hyperpolarizing pulse, the Type II model displays an anodal-break spike, yet another classic sign of the Type II response. VCN neurons with such a Type II response have been morphologically identified as bushy cells (Wu and Oertel 1984).
In Fig. 2D, the response of the Type I-II model is intermediate in that it displays one or two APs in response to a small depolarizing current pulse (+100 pA) but a regular discharge of APs in response to a larger depolarizing current pulse (+150 pA). Similarly, the Type II-I model displays one or two APs in response to a small depolarizing current pulse. However, in response to a larger depolarizing current pulse, the Type II-I model displays only three or four APs rather than a regular train of APs (see Fig. 4). VCN neurons with such intermediate current-clamp responses have in fact been observed in several studies (Francis and Manis 2000; Manis and Marx 1991; Oertel 1991; Schwarz and Puil 1997).
ILT IS RESPONSIBLE FOR THE CLASSIC TYPE II CURRENT-CLAMP RESPONSE. The Type I-c and Type II models differ only in the magnitudes of ILT and Ih (see Table 1). To determine the relative roles these currents play in producing the Type II current-clamp response, the Type II model was simulated with either ILT = 0 or Ih = 0. When Ih = 0, the model no longer displayed inward rectification during a hyperpolarizing current pulse but still displayed the characteristic Type II response of a single AP at the onset of a depolarizing current pulse (Fig. 3B). When ILT = 0, on the other hand, the model displayed a regular discharge of APs in response to a depolarizing current pulse (Fig. 3D), similar to the canonical Type I current-clamp response in Fig. 2A. Hence, ILT is responsible for the phasic discharge pattern of the Type II model. These results coincide with those of Brew and Forsythe (1995) and Rathouz and Trussell (1998), who found that auditory neurons that normally display a single AP in response to a depolarizing current pulse (rat MNTB neurons and avian n. magnocellularis neurons, respectively) displayed a regular discharge of APs in response to a depolarizing current pulse after blocking ILT with dendrotoxin.
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Figure 3 reveals three other notable results. First, when IHT = 0 (C), the only change in the model is a slight change in AP shape, which is not evident in this figure. Second, Vrest shifts negative when Ih = 0 (B), and shifts positive when ILT = 0 (D). The shifts arise because Vrest is not constrained but varies according to the sum of the membrane currents (these shifts are explained further in the following text). It could be argued that the shifts in Vrest are the source of change in the model's current-clamp response; however, when Vrest is held constant across conditions via a steady holding current, the same conclusions about ILT are drawn (not shown). Finally, the model does not show an anodal-break spike when Ih = 0, suggesting Ih plays an influential role in generating anodal-break spikes. The influence of Ih is even more apparent when ILT = 0, in which case the model now responds with a train of APs at the end of a hyperpolarizing current pulse, lasting 400 ms (D). When the depolarizing effects of Ih are counteracted with a holding current of -50 pA, this train of APs disappears (not shown).
VARIATION OF ILT LEADS TO A VARIATION OF CURRENT-CLAMP RESPONSES. The preceding results demonstrate ILT is the main current responsible for producing the Type II current-clamp response in the Type II model. However, results from the other simulations (Type I-II and Type II-I) with lower amounts of ILT demonstrate that the simple presence of ILT is not enough to create a Type II current-clamp response, but rather a certain level of ILT is necessary. This is demonstrated in Fig. 4, which shows the model's current-clamp response as a function of LT. As this figure shows, changing LT over the physiologically measured range 0–600 nS significantly affects the model's current-clamp response. When LT < 20 nS, the model fires regularly, regardless of the injected current. When LT > 35 nS, the model fires one or two action potentials, regardless of the injected current. When LT is the range 20–35 nS, however, the model shows intermediate behavior that is level dependent: single APs are produced in response to small depolarizing currents, but two or more APs in response to larger depolarizing currents.
Because it was also of interest to relate these findings to the results presented in our previous study (Rothman and Manis 2003a), Vth and S-50/-70 values were computed from the model's I-V relation. As Fig. 4 shows, a variation of LT produces a range of Vth and S-50/-70 values (
): from LT 0 (Vth = -38.2 mV, S-50/-70 = 0.3 nS), to LT gLT = gLT = 600 nS (Vth = -63 mV, S-50/-70 = 147.7 nS). Remarkably, the trajectory of model simulations as LT increases follows the same trajectory of the experimental data (
) (from Rothman and Manis 2003a, Fig. 7). These results lend strong support to the hypothesis that a gradient of ILT is responsible for the large dispersion of Vth and S-50/-70 values observed experimentally. In addition, the same threshold and slope analyses was computed for simulations in which ILT = 0, and the magnitude of IHT varied (HT = 50–600 nS, where 600 nS is well above the highest value measured from the isolated VCN cells). Under these conditions, Vth only spanned from -44 to -32 mV, and S-50/-70 remained near zero. Hence, a variation of IHT cannot account for the dispersion of Vth and S-50/-70 values observed in the experimental data.
TYPE II MODEL EXHIBITS STRONG OUTWARD AND INWARD RECTIFICATION. I-V relations of the models listed in Table 1 were computed from their steady-state current-clamp responses, similar to those in Fig. 2 (INa = 0 to inhibit AP generation). As the results in Fig. 5 show, all model types exhibit outward rectification at V > Vrest, due to their outward currents. However, outward rectification in the Type I-c and Type I-t models occurs at higher potentials (V > -45 mV, where IHT begins to activate) in comparison to the Type II and intermediate models (V > Vrest, where ILT begins to activate). The difference in outward rectification is a clear demonstration of how increasing levels of ILT cause increasing amounts of outward rectification. Figure 5 also shows that, at V < Vrest, all model types exhibit inward rectification, due to Ih. Hence, increasing levels of Ih cause increasing amounts of inward rectification, as denoted in Fig. 5.
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The difference in slope between the Type I-c and Type II model I-V relations is dramatic: whereas the Type I-c I-V relation is steep from -100 to -50 mV, the Type II I-V relation is relatively flat. As just mentioned, these differences are due to ILT and Ih, which tend to flatten out, or rectify the model's I-V relation above and below Vrest. The input resistance at Vrest shows a sevenfold difference between models (Table 1, Rrest). The sevenfold difference in input resistance in turn reflects a sevenfold difference in the resting membrane time constant because both models share the same membrane capacitance (m = RrestCm at rest).
Model EPSPs
In this section, the effects of the different currents on EPSP integration are examined. For simplicity, only results from the Type II and Type I-c models are presented because these models represent the extreme cases of having and not having ILT. It should be noted, however, that as before, results of the Type I-t model are similar to those of the Type I-c model, and results from the intermediate models fall between those of the Type I-c and Type II models.
WHEREAS THE TYPE I-C MODEL ACTS AS AN EPSP INTEGRATOR, THE TYPE II MODEL ACTS AS AN EPSP COINCIDENCE DETECTOR. Figure 6A shows that the half-width of the EPSP elicited by a synaptic current with E = 1 nS is significantly briefer in the Type II model (1.6 ms) than in the Type I-c model (7.1 ms). Inspection of the EPSPs reveals faster EPSP rise and decay times in the Type II model. These differences in EPSP shape are due to a smaller m in the Type II model, as described in the preceding text.
One consequence of the long-duration EPSP in the Type I-c model and a very brief EPSP in the Type II model is that the Type I-c model operates as an EPSP integrator, whereas the Type II model operates as an EPSP coincidence detector. This is demonstrated in Fig. 6B, which shows model responses to pairs of subthreshold synaptic inputs of equal amplitude, separated in time. When the interval between the two EPSPs is short, both models show EPSP summation (EPSPs that rise above – – –). However, EPSP summation occurs over significantly longer intervals in the Type I-c model (40 ms) than in the Type II model (2 ms). Hence, the Type I-c model temporally integrates its synaptic input, whereas the Type II model acts as a synaptic coincidence detector. Interestingly, an afterhyperpolarization (AHP) follows each EPSP in the Type II model (Fig. 6A, ) due to activation of ILT. As Fig. 6B shows, inputs that arrive during this AHP are smaller than the primary EPSP (), thus enhancing coincidence detection in the Type II model.
Figure 6C shows the Type I-c model response to a train of subthreshold synaptic inputs arriving at 333 Hz (top). Here, consecutive EPSPs sum together to cause a depolarization that eventually leads to the generation of APs. When this simulation is carried out for a longer time, the model displays repetitive firing reminiscent of the current-clamp response in Fig. 2A. Moreover, repeating the same analysis at two different frequencies demonstrates that the rate of firing is directly proportional to the rate of incoming EPSPs (17 spikes/s at 250 Hz, 25 spikes/s at 333 Hz, 67 spikes/s at 1,000 Hz). Hence the Type I-c model is well suited to measure the average rate, or intensity, of its synaptic input. This contrasts to the Type II model, where subthreshold synaptic inputs do not sum together in time, and therefore do not generate output spikes (Fig. 6C, middle).
ILT AND IH HAVE DISTINCT EFFECTS ON THE TYPE II MODEL EPSP. Although both Ih and ILT act to reduce m near Vrest, their effect on the Type II model's EPSP is quite distinct. This is apparent when the effect of each conductance is studied independently; i.e., setting both Ih and ILT to zero in the Type II model and then adding them back one at a time. Such an analysis is shown in Fig. 7A, where the EPSP of the reduced Type II model is plotted on the left (-ILT - Ih). As Fig. 7A shows, adding Ih to the reduced model has two effects: a small decrease in EPSP width and a depolarization of Vrest (-ILT + Ih). In contrast, adding ILT to the reduced model produces a dramatic decrease in EPSP width and a hyperpolarization of Vrest (+ILT - Ih). Here the reductions in EPSP width are due to a decrease in the model's input resistance, which leads to a concomitant decrease in m. The shifts in Vrest are due to the fact that both Ih and ILT act to move Vrest toward their respective reversal potential (-43 and -70 mV). From this analysis, we see that if the goal is to have the briefest EPSP possible (that of approximating an ideal EPSP coincidence detector), then Ih alone is an inefficient means of doing so: not only is Ih's effect on the EPSP width small, but it depolarizes Vrest toward the threshold of AP generation, producing a hyperexcitable model (see Fig. 3D). Using ILT alone, on the other hand, is an efficient means of reducing the model's EPSP width without inducing hyperexcitability. And yet, the scenario with ILT alone is suboptimal because ILT hyperpolarizes Vrest toward VK, in which case <2% of ILT is activated. To produce the briefest EPSP possible, it is necessary to have a larger fraction of ILT activated because this would produce an even briefer EPSP. Hence with these observations in mind, one can see how a combination of Ih and ILT act to produce a very brief EPSP, more so than with each conductance alone: 1) ILT acts to decrease the EPSP width by reducing m, while 2) Ih counteracts the hyperpolarizing effects of ILT, thereby maintaining Vrest at potentials where a significant amount of ILT is activated. This is also demonstrated in Fig. 7A, where the addition of Ih to the reduced model with ILT alone (+ILT - Ih) depolarizes Vrest back to -64 mV, thereby raising the activation level of ILT to 6%, thereby reducing the EPSP width even further (+ILT + Ih). Interestingly, after ILT and Ih are reinstated to the Type II model, an AHP appears at the end of the EPSP (arrowhead). The AHP is a consequence of ILT equilibrating back to its steady-state value at rest, and this effect is more pronounced when Vrest is further away from VK (+ILT + Ih) than near VK (+ILT - Ih).
Another effect of ILT and Ih on the Type II model EPSP is a reduction in amplitude, as is clear in Fig. 7A (+ILT + Ih). This result too has implications for coincidence detection in that, with smaller EPSPs, it takes a larger number of coincident inputs to drive the membrane potential towards the threshold of AP generation, in which case the requirements for coincidence detection are even more stringent.
MODULATION OF IH MODULATES THE TYPE II MODEL EPSP WIDTH AND AMPLITUDE. It has previously been reported that the activation curve of Ih can shift as much as 15–20 mV positive in the presence of either norepinephrine (NE) or the membrane permeable analogue of cyclic-AMP, 8-Br-cAMP (Banks et al. 1993; Cuttle et al. 2001). This modulation of Ih could play an important role in increasing temporal acuity of VCN neurons by decreasing the width and amplitude of their EPSPs, thereby enhancing coincidence detection. We tested this hypothesis in the Type II model by shifting the steady-state activation curve of Ih (r) 15 mV positive. Results are shown in Fig. 7B, where model EPSPs are shown before and after the shift in r. Comparison of EPSPs shows both a reduction in width and amplitude. The explanation of these results follows the same reasoning as in Fig. 7A: the shift in activation of Ih caused Vrest to depolarize, increasing the activation level of ILT, thereby reducing the EPSP width and amplitude. The end result is a more ideal coincidence detector, as we demonstrate in Model responses to AN-like inputs.
Model action potentials
ILT AND IH REDUCE THE MODEL REFRACTORY PERIOD. Besides producing a brief EPSP, another consequence of a small m in the Type II model is a brief AHP following an AP, one that is significantly briefer than that of the Type I-c model (Fig. 8A). Fitting single exponential functions to the AHP time course reveals a twofold difference in time constants between the Type II and Type I-c model (6 and 13 ms).
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The shorter AHP in the Type II model coincides with a shorter refractory period (RP), as demonstrated in Fig. 8, B–D. In each panel of Fig. 8 are 30 Type II model responses to two suprathreshold synaptic inputs of equal amplitude (E = 3E) separated by various intervals. When both Ih and ILT are set to zero, the model displays a long RP (B, 9.6 ms), similar to the Type I-c model. When ILT is reinstated to the model, the RP is reduced (C, 8.2 ms) but not as significantly as when both ILT and Ih are reinstated together (D, 6.5 ms). Note although the shifts in Vrest are not obvious in this figure, the same voltage shifts occur as those in Fig. 7A.
Because the Type II model has a shorter RP than the Type I-c model, it follows a train of suprathreshold inputs better than the Type I-c model. This is demonstrated in Fig. 9, where the Type II model (B) follows a suprathreshold AN input at 140 Hz better than the Type I-c model (A). The ability to follow a synaptic input one-to-one can be quantified by an entrainment index, which is simply the ratio of output spikes to synaptic input events. Hence at 140 Hz, the entrainment index is 0.5 for the Type I-c model and 1.0 for the Type II model. Computing entrainment indexes over a range of frequencies (100–250 Hz) shows that the Type II model is consistently better than the Type I-c model at following a suprathreshold synaptic input one-to-one (Fig. 9C).
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IHT SERVES TO REPOLARIZE THE MEMBRANE DURING AN AP. In the Type I-c model, the only K+ current besides Ilk is IHT; hence, it is primarily responsible for repolarizing the membrane during an AP. As Fig. 10A shows, modulation of IHT produces a dramatic change in the rate of repolarization during the downstroke of the AP. When IHT was set to very small values (HT < 15 nS), the membrane failed to repolarize back to Vrest. In the Type I-t model, there are two K+ currents that contribute to the downstroke of the AP: IHT and IA. However, due to the inactivating behavior of IA, and its smaller magnitude, its influence on the rate of repolarization is considerably less than that of IHT (Fig. 10B). In the Type II model, there are also two K+ currents that contribute to the downstroke of the AP: IHT and ILT. In this case, both IHT and ILT repolarize the membrane equally well, in which case modulation of IHT does not produce a dramatic effect on the rate of repolarization (Fig. 10C, left) nor does modulation of ILT (right). If, on the other hand, IHT is removed from the Type II model, then modulation of ILT produces dramatic changes in the rate of repolarization of the AP (Fig. 10D).
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IA AND ILT CAN MODULATE THE RATE OF REPETITIVE FIRING. In Fig. 6C, subthreshold EPSPs sum together in time to produce a train of APs spaced at regular intervals. For the same simulation, we found that increasing levels of IA (A = 50–100 nS) decreased the rate of this repetitive firing by increasing E. This is because IA, being an outward current, acts to oppose excitatory depolarization (compare Type I-c and Type I-t E gA values in Table 1). When A > 100 nS, E was raised high enough to prevent AP generation. The same effects could be achieved with ILT, although the conductance levels necessary were significantly smaller (LT = 1–6 nS). Although similar effects could be achieved by increasing levels of IHT, the conductance levels necessary to produce the same effects were substantially larger (HT = 300–1,000 nS), due to IHT's higher activation voltage range. Hence, small amounts of ILT, as well as modest amounts of IA, were effective in modulating the rate of repetitive firing in the Type I models.
Model responses to auditory-nerve-like inputs
In this final section, the effects of applying simulated AN inputs to the VCN somatic model are discussed. As described in METHODS, AN inputs are applied by activating the excitatory post-synaptic current, IE. Up to this point, the activation time of IE has occurred at specified times. In this section, however, the activation time is controlled by a model that accurately describes the spiking discharge pattern of real AN fibers during presentation of pure-tone stimuli. A detailed description of the AN spike generator model has previously been given (Rothman et al. 1993). Further details of the model can be found in the legend of Fig. 11.
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), which display phaselocking. —, phaseloc