Using kinetic data from three different K+ currents in acutely isolated neurons, a single electrical compartment representing the soma of a ventral cochlear nucleus (VCN) neuron was created. The K+ currents include a fast transient current (IA), a slow-inactivating low-threshold current (ILT), and a noninactivating high-threshold current (IHT). The model also includes a fast-inactivating Na+ current, a hyperpolarization-activated cation current (Ih), and 1–50 auditory nerve synapses. With this model, the role IA, ILT, and IHT play in shaping the discharge patterns of VCN cells is explored. Simulation results indicate that IHT mainly functions to repolarize the membrane during an action potential, and IA functions to modulate the rate of repetitive firing. ILT is found to be responsible for the phasic discharge pattern observed in Type II cells (bushy cells). However, by adjusting the strength of ILT, both phasic and regular discharge patterns are observed, demonstrating that a critical level of ILT is necessary to produce the Type II response. Simulated Type II cells have a significantly faster membrane time constant in comparison to Type I cells (stellate cells) and are therefore better suited to preserve temporal information in their auditory nerve inputs by acting as precise coincidence detectors and having a short refractory period. Finally, we demonstrate that modulation of Ih, which changes the resting membrane potential, is a more effective means of modulating the activation level of ILT than simply modulating ILT itself. This result may explain why ILT and Ih are often coexpressed throughout the nervous system.
A major goal of studying neuronal mechanisms of information processing is to determine precisely how each mechanism contributes to the electrical activity of a neuron. Because voltage-gated ionic currents introduce strong nonlinearities into a cell's electrical behavior, computational models are often necessary to provide the appropriate predictive power. In our previous papers, we provided a detailed description of three K+ currents expressed across a population of ventral cochlear nucleus (VCN) neurons (Rothman and Manis 2003a,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 (nλ, IHT, and τn), and those of the model “low-threshold” currents are plotted on the right (wλ, ILT, and τw). 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 (nλ) 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.
The second discrepancy in Fig. 1 pertains to ILT: the magnitudes of the model currents are approximately 10 times smaller than indicated by the new experimental data. Thus the full extent of the ILT's ability to reduce the membrane time constant was not fully explored in the previous models.
The last two discrepancies in Fig. 1 pertain to the time constants of IHT (τn) and ILT (τw). For τn, discrepancies occur at potentials less than -60 mV. For τw, 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.
VCN somatic model
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 Equations for IA, ILT and IHT were derived from experimental data, as previously described (Rothman and Manis 2003b), 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 (τm and τh; 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 2 where ḡLTNa 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 3 (3) where ḡh 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 τr (see APPENDIX). Here, τr 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 ḡh 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 ḡh is probably an unusual characteristic of octopus cells. We in fact explore the consequences of changing ḡh in this study as well as shifting the activation curve of Ih to more depolarized potentials. We also decrease the rate constant τr by a Q10 factor of 3 when investigating the model at higher temperatures.
Model leakage current
The leakage current was modeled as follows 4 where ḡLK 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 5 where VE is the reversal potential of IE (0 mV), and gE the time-dependent conductance change in response to an excitatory synaptic input. For simplicity, gE was modeled as an α-wave of the form 6 where ḡE determines the peak conductance and τE the time to peak. For most of the simulations in this study, τE = 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.
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 ḡA, ḡLT, and ḡHT were based on experimental data (Rothman and Manis 2003a). For simplicity and ease of comparison between model responses, single values of ḡNa and ḡlk were chosen: 1,000 and 2 nS, respectively. The value of ḡNa was chosen to give large, reliable APs in all five model types, and the value of ḡLK was chosen to give the Type I-c model a realistic input resistance (see following text). The value of ḡh, 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 ḡLT > 0 (i.e. the Type II, Type I-II and Type II-I simulations), ḡh was ḡLT. For the Type I-c and Type I-t model, ḡh was set to a small value of 0.5 nS because larger values produced hyperex-citability. The value of ḡh 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 ḡh is lower than that estimated for VCN octopus cells (Bal and Oertel 2000).
Table 1 also lists seven parameters computed from each model type that are consistent with the experimental data. The parameters are as follows.
) The resting membrane potential, Vrest. For all models, Vrest ≅ -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.
) The resting membrane resistance, Rrest, computed by taking the reciprocal of the sum of all conductances at Vrest. As Table 1 shows, the Type I-t and Type I-c models show the largest Rrest, which equals ∼1/ḡLK. The other three models (Type I-II, Type II-I, and Type II) show decreasing values of Rrest, consistent with a parallel increase in ḡh and ḡLT. 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) MΩ] than in Type II cells (225 ± 160 MΩ) (Manis and Marx 1991).
) The resting membrane time constant, τrest, computed by multiplying Rrest by Cm. Again, τrest is larger in the Type I-c and Type I-t models than in the Type II model, consistent with previous experimental findings that τrest 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).
) Vth and S-50/-70 values, computed from model voltage-clamp simulations (i.e. steady-state I-V relations), where Vth is the voltage at which the whole cell steady-state current reaches a value of 0.1 nA, and S-50/-70 is the slope of a steady-state I-V relation from -50 to -70 mV. To simulate the same experimental conditions of our isolated VCN data, that is, to simulate the block of INa with TTX and to simulate off-line linear leak subtraction, ḡNa = 0 and ḡlk = 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.
) The last parameter, ḡEθ, denotes synaptic efficacy or strength of an individual AN synapse. Specifically, ḡEθ is defined as the smallest one-tenth that of g value of ḡE (Eq. 6) necessary for a single AN synapse to generate an AP. A comparison across model types in Table 1 shows that ḡEθ is larger in the Type II model than in the Type I-c and Type I-t models. This disparity of ḡEθ values is due to the magnitude difference of ILT: ILT, being a K+ current with significant activation near rest, acts to oppose synaptic depolarization.
Definition of sub- and suprathreshold synaptic inputs
The fact that a given value of ḡE 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 ḡE with respect ḡEθ, such that for all subthreshold inputs, ḡE = 0.5ḡEθ, and for all suprathreshold inputs, ḡE = 3ḡEθ. In this way, AN synapses share the same synaptic efficacy rather than the same peak conductance.
We present our results in four sections. In the first section, we examine current-clamp responses of each model type and compare them to responses of real VCN neurons. In the second section, we compare model EPSPs resulting from the same effective synaptic input and explore the effects of subthreshold currents, such as ILT and Ih, on the shape of these EPSPs. In the third section, we explore the effects of each K+ current on AP shape as well as the rate of repetitive firing in the Type I models. In the last section, we explore the responses to simulated trains of convergent AN inputs with respect to phase locking, PSTHs, and regularity analysis, again comparing results to those of real VCN neurons.
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).
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.
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 ḡLT. As this figure shows, changing ḡLT over the physiologically measured range 0–600 nS significantly affects the model's current-clamp response. When ḡLT < 20 nS, the model fires regularly, regardless of the injected current. When ḡLT > 35 nS, the model fires one or two action potentials, regardless of the injected current. When ḡLT 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 ḡLT produces a range of Vth and S-50/-70 values (▵): from ḡLT 0 (Vth = -38.2 mV, S-50/-70 = 0.3 nS), to ḡLT gLT = gLT = 600 nS (Vth = -63 mV, S-50/-70 = 147.7 nS). Remarkably, the trajectory of model simulations as ḡLT 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 (ḡHT = 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.
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 (τm = RrestCm at rest).
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 ḡE = 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 τm 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 τm 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 τm. 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 τm, 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 τm 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).
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 (ḡE = 3ḡEθ) 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).
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 (ḡHT < 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).
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 (ḡA = 50–100 nS) decreased the rate of this repetitive firing by increasing ḡEθ. This is because IA, being an outward current, acts to oppose excitatory depolarization (compare Type I-c and Type I-t ḡEθ gA values in Table 1). When ḡA > 100 nS, ḡEθ was raised high enough to prevent AP generation. The same effects could be achieved with ILT, although the conductance levels necessary were significantly smaller (ḡLT = 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 (ḡHT = 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.
TEMPERATURE SCALING. Up until now, the operating temperature of the VCN model has been 22°C, the temperature at which most of the in vitro data used in its construction was recorded. In this section, it was of interest to consider the behavior of the model at normal body temperature (38°C) because these results could be compared to in vivo studies. Hence, to correct for the difference in temperature, all model time constants were multiplied by 0.17 (i.e., a reciprocal Q10 factor of 3) and all peak conductance values (except for ḡE) were multiplied by 3.03 (i.e., a Q10 factor of 2). These Q10 values are approximations to those reported for Na+ currents (Belluzzi et al. 1985; Sah et al. 1988), K+ currents (Connor and Stevens 1971a; Frankenhaeuser and Huxley 1964; Hodgkin and Huxley 1952; Kros and Crawford 1990), and excitatory postsynaptic currents (EPSCs) (Zhang and Trussell 1994). The exact Q10 values for peak conductance values in VCN neurons are unknown and therefore are purely speculative at this time.
Inspection of the model current-clamp responses after the preceding temperature corrections shows that, at 38°C, all model types respond to depolarizing current injection as they do at 22°C, except for the intermediate Type II-I model, in which case only very large depolarizing current pulses elicit repetitive firing. Responses to EPSCs are also qualitatively the same as those described at 22°C; however, EPSCs decay at a faster rate (τE = 0.4 ms at 22°C and 0.07 ms at 38°C). Consequently, there is an increase in ḡEθ values (see Table 1).
CONVERGENCE OF AN FIBERS WITH LOW BF (PHASELOCKING). The AN inputs in this section were patterned after AN fibers with low best frequency (BF; Fig. 11A). Such fibers phaselock to sinusoidal stimuli ≤5 kHz (Johnson 1980; Joris et al. 1994). To quantify phaselocking, we use the synchronization index (S), which takes on values from zero (when all spikes occur randomly throughout the stimulus) to one (when all spikes are synchronized to the stimulus) (Goldberg and Brown 1969).
Figure 12A shows the results of simulations in which both the Type I-c and Type II model receive 50 subthreshold low-BF AN inputs. Because the AN inputs are subthreshold, summation of two or more EPSPs is required to produce a spike output. At f ≤ 500 Hz, both models show phaselocking exceeding that of their AN input (S > 0.9). Such “enhanced” phaselocking has been reported in VCN bushy cells (Joris et al. 1994) and is a consequence of converging AN inputs with the same BF. Although enhanced phaselocking can be achieved by a convergence of either subthreshold AN inputs or suprathreshold AN inputs, the highest synchronization with the lowest signal-to-noise ratio is achieved by converging subthreshold AN inputs (Rothman et al. 1993). The mechanism of enhanced phaselocking with converging subthreshold AN inputs is due to the necessity of coincidental inputs to drive a post-synaptic spike: during the peak phase of a sinusoidal stimulus, coincidence of inputs is high, and therefore the output discharge rate is high, whereas during the off-peak phase, coincidence of inputs is near zero, and the output discharge rate is near zero. The end result is higher synchronization in the output than in any one of the AN input.
The fact that the Type I-c model shows enhanced phaselocking at f ≤ 500 Hz means that, like the Type II model, it is capable of performing coincidence detection at low frequencies. This is because the average inter-arrival time of the EPSPs at f ≤ 500 Hz is sufficiently long (≥2 ms) to allow EPSPs to decay to low levels before the arrival of the next EPSP. For f > 500 Hz, however, the EPSP inter-arrival time is <2 ms, in which case EPSPs sum together to form a steady depolarization of the membrane, inducing repetitive firing (Fig. 6C). Hence, phaselocking in the Type I-c model rapidly degrades at f > 500 Hz (Fig. 12A, circles). The Type II model, in contrast, having such a brief EPSP, is capable of sustaining coincidence detection ≤2–3 kHz. For f > 1 kHz, synchronization indexes of the Type II model are comparable to that of its AN input, but not exactly the same (Fig. 12A, triangles).
Model simulations with <50 subthreshold low-BF AN inputs were also investigated (not shown). In all cases, synchronization indexes were less than that for 50 subthreshold inputs, demonstrating what has previously been reported: the greater the number of converging subthreshold AN inputs, the greater the degree of phaselocking (Rothman and Manis 1996; Rothman and Young 1996; Rothman et al. 1993).
Figure 12B shows the results of simulations in which both the Type I-c and Type II models receive one suprathreshold AN input. In this case, the suprathreshold AN input can produce a spike output whenever the model is not refractory. Hence, these simulations test how well the Type I-c and Type II model follow a suprathreshold AN input one-to-one, as in Fig. 9. Not surprisingly, the Type II model, having the shorter RP, is capable of following the suprathreshold AN input one-to-one better than the Type I-c model at all frequencies. Again, for f > 1 kHz, synchronization indexes of the Type II model are comparable to those of its AN input, but not exactly the same.
Model simulations with more than one suprathreshold AN input were also investigated (not shown). These simulations demonstrate that, for all model types, increasing the number of suprathreshold inputs degrades phase locking at f ≥ 500 Hz but enhances phase locking at f < 500 Hz. The enhancement of phaselocking at low frequencies with suprathreshold inputs is due to convergence of several AN inputs with similar BF, as noted previously (Rothman and Young 1996). However, the number of suprathreshold AN inputs necessary to achieve enhanced phase locking comparable to that seen in real VCN neurons is unusually (and perhaps unrealistically) high: ≥10. If one considers no more than five suprathreshold inputs, then the added benefit of having more than one suprathreshold input at low frequencies is small.
When the results in Fig. 12, A and B, are compared to those of real VCN neurons (C), one sees two important similarities. First, the degradation of phaselocking at f > 500 Hz in the Type I-c model with all subthreshold AN inputs parallels the degradation seen in VCN chopper units (Blackburn and Sachs 1989), which are stellate cells (i.e. Type I cells). However, phaselocking is noticeably better in the Type I-c model than in the VCN chopper units at all frequencies. The discrepancy could be due to a difference in membrane properties; however, the more likely explanation is a difference in placement of AN synapses: whereas the Type I-c model receives its AN inputs on its soma, the VCN chopper units are likely to receive their AN inputs on their dendrites (Cant 1981), in which case there would be significant dendritic filtering of the AN inputs and therefore a larger degradation in phaselocking (Banks and Sachs 1991; White et al. 1994).
The second similarity between the real and simulated data is that phaselocking of the Type II model (with both subthreshold and suprathreshold AN inputs) parallels that of VCN bushy cells (Blackburn and Sachs 1989; Joris et al. 1994). Specifically, at f < 1 kHz, both the Type II model and real bushy cells show phaselocking equal to and greater than AN fibers, and at f > 1 kHz, both the Type II model and real bushy cells show phaselocking slightly below that of AN fibers (Fig. 12C). This time there is little discrepancy between the Type II model and real bushy cells.
Simulations with phaselocked AN input were also computed for the Type I-t and intermediate models (not shown). Results of these simulations demonstrate that the Type I-t model phase-locks nearly the same as the Type I-c model and the intermediate models phaselock between that of the Type I-c and Type II models.
CONVERGENCE OF AN FIBERS WITH HIGH BF (PSTHS AND REGULARITY). Presented in this section are results from model simulations with non-phaselocked AN inputs (Fig. 11B; BF = 6 kHz). As in the previous section, the analysis was broken into two scenarios of converging AN input: sub- versus supra-threshold. Because neither the input nor output are phaselocked to a sinusoidal stimuli, we present results in the form of PSTHs, which plot the model's instantaneous discharge rate versus time, as well as regularity analysis, which plots the mean (μISI) and standard deviation (σISI) of the model's inter-spike interval (ISI) versus time (see Fig. 11B). A common measure of regularity is the coefficient of variation: CV = σISI/μISI. For a deadtime-modified Poisson process, σISI = μISI - ARP, where ARP is the absolute refractory period (Goldberg et al. 1964), in which case CV = 1 - ARP/μISI. Hence an increase in the ARP, without any other change in the underlying Poisson process, will cause an unexpected increase in CV. A better measure of regularity, therefore, is CV-prime (CV′), which corrects for refractoriness of the spike generator (Rothman et al. 1993): CV′ = σISI/(μISI - ARP). In this way, spike generators with noticeably different refractory characteristics can be compared directly. For the AN spike train in Fig. 11B, CV′ = 0.91 (ARP = 0.7 ms), a value comparable to that of real AN fibers (Rothman et al. 1993).
Figure 13 shows a summary of the results for the Type I-c (top) and Type II (bottom) model simulations. On the left appear the model PSTHs when all AN inputs are subthreshold (ḡE = 0.5ḡEθ), and on the right appear the model PSTHs when all AN inputs are suprathreshold (ḡE = 3ḡEθ). These model PSTHs are meant to be representative of a variety of scenarios, described as follows.
) The Type I-c model with a large number of subthreshold AN inputs exhibits a “chopping” response (e.g., 50 subthreshold inputs). In this case, inputs from many asynchronous subthreshold AN synapses sum together to produce a large steady depolarization of the model's membrane potential, resulting in a regular discharge of APs, similar to that in Fig. 6C. The regular discharge of APs results in a PSTH with successive peaks at the onset of the stimulus (i.e., chopping), where the model tends to fire consistently at the same time from trial to trial. The peaks gradually disappear as the model tends to fire less consistently from trial to trial due to the stochastic nature of the AN inputs. The regular discharge of APs also results in a very narrow interspike interval distribution, evident in the small values of σISI throughout the stimulus. Hence, CV′ is significantly smaller (0.45) in comparison to the AN inputs (0.91).
) The Type I-c model with a small number of subthreshold AN inputs exhibits an onset response (e.g., 10 subthreshold inputs). In this case, the number of converging inputs is not enough to induce a significant amount of repetitive firing during the 50-ms stimulus period. Nevertheless, there is a small secondary peak in the PSTH, and the discharge is regular (CV′ = 0.44).
) The Type II model with both a large and small number of subthreshold AN inputs exhibits an onset response (e.g., 10 and 50 subthreshold inputs). In this case, a fast τm and strong outward rectification prevent the asynchronous subthreshold AN inputs from summating, as demonstrated in Fig. 6C; hence, there is very little spike activity during the steady-state portion of the stimulus (<100 spikes/s), even with a large number of AN inputs. There is, however, a near-instantaneous change of rate at the onset of the stimulus due to a large number of coincidental inputs at this time. The variance of this initial onset response is similar to that of the Type I-c onset response. The Type II and Type I-c onset responses are distinguishable by their regularity: the Type II onset responses, for either 10 or 50 subthreshold AN inputs, are more irregular (CV′ = 0.58 and 0.69, respectively) than the Type I-c onset responses (CV′ = 0.44 and 0.45, respectively).
) The Type I-c model with one suprathreshold AN input exhibits a primarylike-with-notch (“Pri-notch”) response. In this case, a long RP prevents the model from following its suprathreshold input one-to-one. The result is not only a reduction in the steady-state discharge rate, but a notch in the PSTH following the initial onset response. The notch occurs because the suprathreshold AN input forces the model to discharge at the beginning of the stimulus for almost every trial, in which case the ensuing RP prevents the model from spiking again until ∼4 ms later. Interestingly, a comparison of CV values seems to suggest the model's output is regular in comparison to its input (CV = 0.51 and 0.80, respectively). However, a comparison of CV′ values shows the model's output is actually as irregular as its input (CV′ = 0.96 and 0.91, respectively), demonstrating how a simple change in refractoriness can make an irregular process appear regular when using CV as a measure of regularity.
) The Type II model with one suprathreshold AN input exhibits a “primarylike” response. In this case, a short RP allows the model to follow the suprathreshold input better than the Type I-c model. There is no notch following the initial onset response, and the steady-state discharge rate is closer to that of the AN input. Again, due to the effects of refractoriness, the output appears more regular than the AN input when comparing CV values (CV = 0.63 and 0.80, respectively) but just as irregular when comparing CV′ values (CV′ = 0.96 and 0.91, respectively).
) The Type I-c model with several suprathreshold AN inputs exhibits a chopping response (e.g., 3 suprathreshold inputs). This scenario is similar to the one with converging subthreshold AN inputs except there is less temporal precision in the output spike train. In comparison to the model's response to converging subthreshold inputs, for example, the model's response to converging suprathreshold inputs shows a less precisely timed onset response, fewer peaks following the onset response, and a larger σISI.
) The Type II model with several suprathreshold AN inputs exhibits a Pri-notch response (e.g., 3 suprathreshold inputs). In this case, convergence of more than one suprathreshold input forces the model to fire at the onset of the stimulus more precisely than if there was just one input. Because the model is refractory for ∼1.7 ms after the initial onset spike, a notch appears in the PSTH. A comparison of CV′ values shows that as the number of suprathreshold inputs increases, the Type II model becomes more regular (for 3 suprathreshold inputs, CV′ = 0.59, for 5 suprathreshold inputs, CV′ = 0.40). Here, the increase in regularity is due to the fact that, with several suprathreshold AN inputs, the model tends to fire the moment it is no longer refractory. The end result is a narrow interspike interval distribution, or small σISI. Therefore, increasing the number of suprathreshold AN inputs decreases CV′, as reported by Rothman et al. (1993).
The same simulations in Fig. 13 were also computed for the Type I-t model and the two intermediate models (not shown). Results from these simulations show that with respect to PSTHs and regularity, the Type I-t model is nearly indistinguishable from the Type I-c model, and, the intermediate models fall between the Type I-c and Type II models, where the Type I-II model appears more like the Type I-c model, and the Type II-I model appears more like the Type II model. However, with respect to classification based on PSTH appearance, most of the intermediate responses have Pri-notch appearances.
MODULATION OF IH ENHANCES TEMPORAL ACUITY IN THE TYPE II MODEL. As demonstrated in Fig. 7B, a shift in activation of Ih causes a reduction in EPSP width and amplitude of the Type II model EPSP. It was hypothesized that this could enhance coincidence detection. To test this hypothesis further, simulations of the Type II model with converging subthreshold high-BF AN inputs (Fig. 11B) were compared before and after a positive shift in activation of Ih. Because the AN inputs were subthreshold in these simulations, the Type II model was acting as an EPSP coincidence detector. Results are shown in Fig. 14, where model PSTHs before and after the 15-mV shift in Ih (+NE) are plotted. Here, a comparison of PSTHs reveals three changes due to the shift in Ih: a more precisely timed onset response, a lower sustained discharge rate, and a lower spontaneous discharge rate. Because the probability of coincidence is high at the onset of the stimulus, but low everywhere else (especially before and after the stimulus), these results are expected for a mechanism that enhances coincidence detection.
In this paper we present a robust model of VCN neurons based on our previous experimental findings (Rothman and Manis 2003a,b). This model replicates many of the complex behaviors associated with VCN neurons, including the Type I current-clamp response of stellate cells and the Type II current-clamp response of bushy cells. The model also replicates many of the PSTH, regularity and phaselocking responses associated with these neurons. Although previous models have achieved similar success in replicating the behavior of VCN neurons (see INTRODUCTION), the current model presented here is considered more accurate because it was derived from a complete characterization of the K+ currents, rather than ad hoc assumptions. The largest difference between the models is in the voltage dependence of the high-threshold K+ current (IHT) and the magnitude of the low-threshold current (ILT). In most previous models, these two currents were confounded, such that IHT behaved more like ILT, limiting the interpretation of the relative roles of the currents, especially with respect to VCN bushy cells. In the future, it will be important to characterize the behavior of INa and Ih in VCN neurons, since our characterizations of these two currents were derived from other neurons.
Functional significance of IA
Since the discovery of the fast transient K+ current (IA) in crab (Connor and Stevens 1971b) and molluscan neurons (Hagiwara et al. 1961), IA has been localized to a variety of neurons that display repetitive firing (Rudy 1988). Similarly, we find IA only in our population of Type I cells, i.e., the stellate cells, which are thought to encode their output in repetitive firing. Hence, IA may play a role in modulating the rate of repetitive firing in VCN stellate cells. Our modeling simulations indicate IA could in fact play such a role. In these simulations, increasing levels of IA counteracted the depolarizing effects of excitatory synaptic input, thereby increasing the threshold of AP generation. The end result was a decrease in discharge rate. When the magnitude of IA was large enough, repetitive firing was abolished altogether.
These conclusions are somewhat different from those of IA in the dorsal cochlear nucleus (Kanold and Manis 1999, 2001). However, neurons with IA in the dorsal cochlear nucleus show a more complex discharge pattern than VCN Type I cells, suggesting IA plays different roles in the two sets of cells. IA in the dorsal cochlear nucleus in fact shows considerable difference in activation/inactivation kinetics in comparison to IA in VCN neurons, again suggesting two distinct fast-inactivating K+ currents.
In future modeling studies it will be important to also simulate the nature of inhibitory inputs to Type I cells because hyperpolarization of their membrane will produce a dramatic effect on the magnitude of IA due to the de-inactivation that occurs in the voltage range above and below their resting membrane potential.
Functional significance of IHT
Results from our modeling simulations indicate IHT is primarily responsible for repolarizing the membrane during the downstroke of an AP. The one exception is when IHT and ILT are “expressed” together, in which case both IHT and ILT act to repolarize the membrane. In fact, ILT repolarizes the membrane nearly as well as IHT. Why would a cell, then, express both IHT and ILT, if ILT alone is sufficient to repolarize the membrane? One possibility is that IHT functions as the steadfast means of repolarizing the membrane during periods of modulation of ILT. As Fig. 10D demonstrates, in the absence of IHT, modulation of ILT produces a dramatic effect on the rate of repolarization during an AP. This is not true when IHT is present (Fig. 10C). Why a cell might need to modulate ILT is discussed in the following text.
It was previously suggested that IHT in bushy cells contributes to the precise conveyance of rapid temporal information (Perney and Kaczmarek 1997; Wang et al. 1998). This hypothesis stemmed from modeling results that showed IHT significantly reduced the duration of “large” EPSPs and IHT allowed a bushy cell model to follow a high-frequency train of synaptic inputs (100–600 Hz), which was otherwise not possible when IHT was absent. However, the simulations that demonstrate IHT's ability to improve phaselocking at high frequencies are problematic. In these simulations, the maximum conductance of IHT was set to an unusually large value (50 μS), in which case IHT activated at potentials near -60 mV (see Fig. 1, IHT, trace 2). Hence, IHT in these simulations can no longer be considered high threshold but rather low threshold. In addition, the large EPSPs that showed significant reduction in duration with increasing levels of IHT were suprathreshold EPSPs with peak amplitude near 0 mV. Such large EPSPs would normally elicit APs and would therefore not be directly observed in real bushy cells. These large EPSPs did not elicit APs in the model because the Na+ current was removed from the simulations. The reduction in width of these very large EPSPs confirms the finding that, at suprathreshold potentials (V > -50 mV), IHT contributes to repolarization of the membrane, as it does during the course of an AP. These results are however consistent with the idea that ILT makes the major contribution to the precise conveyance of rapid temporal information.
Functional significance of ILT
The modeling results in this paper suggest several functions of ILT. The most noteworthy of these functions is in reducing the membrane time constant (τm) at subthreshold potentials. By far, of all the currents investigated in this study, ILT was most adept at reducing τm in the range -70 to -50 mV. Perhaps the best illustration of ILT's ability in reducing τm is in Fig. 7A, which shows the model's response to a subthreshold synaptic input with and without ILT. Here, the ability of ILT to reduce τm is clearly reflected in the difference in decay rate of the two EPSPs. These results suggest a neuron acting as an EPSP coincidence detector would be well served to express ILT because ILT greatly reduces the width of subthreshold EPSPs, thereby shortening the window over which EPSPs sum together. The next best illustration of ILT's ability in reducing τm at subthreshold potentials is given in Fig. 8, B and C. Here, the ability of ILT to reduce τm is reflected in the reduction of the refractory period after the addition of ILT to the model. Hence a neuron that aspires to follow its synaptic input at very fast rates would also be well served to express ILT.
There are two additional roles for ILT. First, ILT could modulate the rate of repetitive firing, similar to the role proposed for IA. Our modeling results show that small amounts of ILT produce a dramatic change in the rate of repetitive firing, and these effects are qualitatively indistinguishable from the changes produced by larger values of IA. Neurons that encode their output in the form of repetitive firing might be well served to express small amounts of ILT. One potential advantage of utilizing ILT is its relatively weak inactivation voltage dependence, which will limit the kind of behavior that can be produced by rapidly inactivating IA currents (Kanold and Manis 1999, 2001). Second, ILT could potentially change suprathreshold events into subthreshold events. For example, neurons that express ILT in the initial segment of their axons or proximal dendrites might have much more limited back-propagation of APs into the dendritic tree.
Modulation of ILT
All of the preceding functions of ILT could easily be regulated by modulating the activation level of ILT. An upregulation of ILT, for example, would cause a further reduction in τm, thereby reducing the width and amplitude of EPSPs. For a neuron acting as an EPSP coincidence detector, the reduction in width and amplitude of EPSPs would lead to an enhancement of temporal acuity; that is, an increase in precision of spike occurrences during the peak phase of a stimulus, and a decrease in spike occurrences during the off-peak phase, including periods of spontaneous activity (Fig. 14). A reduction in τm would also lead to a reduction in the refractory period following an AP, thereby allowing a neuron to fire at a higher discharge rate. Hence an upregulation of ILT would boost a neuron's ability to preserve and even enhance the temporal encoding of its synaptic inputs.
A simple means of upregulating the activation level of ILT would be to simply increase ILT, by either increasing its peak conductance or removing its inactivation. However, increasing ILT leads to a concomitant hyperpolarization of the resting membrane potential (Vrest) toward the K+ reversal potential because the increase in ILT, an outward current, is not opposed by an increase in an inward current (Vrest is the potential where the sum total of outward currents is counterbalanced by the sum total of inward currents). In other words, Vrest moves to more hyperpolarized potentials to deactivate ILT. The end result is the same steady-state activation level of ILT, now at a lower Vrest. As investigated in Fig. 7, a possible solution to this problem rests with the hyperpolarization-activated cation current (Ih). Because Ih is an inward current below -40 mV, it serves to counterbalance the hyperpolarizing force of ILT. To increase the activation level of ILT, then, a neuron might increase Ih because an increase in Ih leads to a concomitant depolarization of Vrest, which in turn leads to an increase in activation of ILT. The end result is a larger activation level of ILT, and a slightly higher Vrest. Or, a neuron might increase both ILT and Ih simultaneously in order to increase the activation level of ILT without significantly changing Vrest.
The hypothesis that Ih is used as a means of modulating ILT by changing Vrest was proposed by Banks et al. (1993). In their study, Banks et al. found that Ih in MNTB neurons is modulated by norepinephrine (NE) as well as a membrane permeable analogue of cyclic-AMP (8-Br-cAMP). If Ih in VCN neurons is similar to Ih in MNTB neurons, then it might be possible that NE and/or cyclic-AMP act to modulate Ih in VCN neurons. If ILT is also modulated by cAMP, the effects could be synergistic with modulation if Ih. There is some evidence in expression systems that KCNA1 (a likely component of ILT) (see Rothman and Manis 2003b) can be regulated by phosphorylation by cyclic-AMP-dependent kinases (Ivanina et al. 1994; Levin et al. 1995; Matthias et al. 2002), with a resulting increase in current through the channels at a given voltage. Whether this occurs in native channels remains to be determined.
Interestingly, an in vivo study of bat VCN neurons found that iontophoretic application of NE during acoustic stimulation enhanced the onset response of these neurons as well as decreased the sustained and spontaneous discharge rates (Kossl and Vater 1989). The cells under investigation were located in the anterior region of the VCN and often exhibited “primarylike” responses to tone bursts, suggesting these cells were bushy cells or Type II cells. Hence, it is possible that NE was acting on Ih (and possibly ILT) in these VCN neurons, as it did in MNTB neurons.
Implications for bushy cells
An interesting distinction between bushy cells in the VCN is the size and number of their AN inputs. Spherical bushy cells, for example, receive a small number of endbulbs of Held (1 or 2), which presumably form secure (i.e. suprathreshold) synapses. In the most rostral region of the VCN, spherical bushy cells are contacted by a single endbulb of Held (Ryugo and Sento 1991). With such an exclusive suprathreshold input, these cells are poised to act as relay cells from the cochlear nucleus to the superior olivary complex. The only condition preventing a spherical bushy cell from acting as a relay cell would be a long post-synaptic refractory period. Spherical bushy cells, therefore, appear to express high levels of ILT to help reduce the refractory period following an AP. In this way, spherical bushy cells are able to follow their suprathreshold AN input one-to-one better than if they did not express ILT (see Fig. 9).
The expression of large ILT in spherical bushy cells comes with a cost, however: a reduction in synaptic efficacy. This is evident in Fig. 7, which shows the effect of ILT is not only to decrease the width of an EPSP but its height as well. One solution to this problem is to increase the synaptic conductance. Results from this study indicate that, for every additional 50 nS of ḡLT added to an excitable membrane, the peak synaptic conductance has to be increased by ∼75% to keep synaptic efficacy constant (T = 22°C). This might explain why spherical bushy cells receive such unusually large synapses from AN fibers, the endbulbs of Held, with many active zones.
In contrast to spherical bushy cells, globular bushy cells receive a large number of small AN inputs. The difference in AN input configuration between spherical and globular bushy cells clearly suggests a difference in function. Indeed, comparison of the temporal response patterns between spherical and globular bushy cells demonstrates that globular bushy cells have a more precisely timed onset response, and an enhanced ability to phaselock to a sinusoidal stimulus (Blackburn and Sachs 1989; Joris et al. 1994). How globular bushy cells achieve this “enhancement” of timing can be explained by a coincidence detection model, whereby coincidence of two or more excitatory inputs is necessary to generate an output spike (Joris et al. 1994; Rothman et al. 1993). Not surprisingly, the best coincidence detectors are those with a large number of inputs, each of which produces a small, brief EPSP. Globular bushy cells, therefore, appear to express high levels of ILT, not only to reduce the refractory period following an AP, but to reduce the width of subthreshold EPSPs. In this way, globular bushy cells are able to act as coincidence detectors far better than if they did not express ILT.
APPENDIX: MODEL EQUATIONS
The model currents presented below have voltage and time dependencies similar to those of the original Hodgkin and Huxley model (1952). In these equations, currents are governed by an activation/inactivation variable x whose rate of change is defined by the following first-order differential equation A1 where τx is the time constant of x, x∞ is the steady-state value of x (i.e. the value of x when t >> τx), and x itself represents the activation/inactivation variables a, b, c, w, z, n, p, m, h, and r in the following text. Although the formalism of the preceding equation is different from the original HH formalism in which activation/inactivation variables are expressed in terms of “open” and “close” rate constants α and β, they are nevertheless mathematically equivalent when x∞ = α/(α + β) and τx = 1/(α + β). Reversal potentials are: VK = -70 mV, VNa = +55 mV, Vh = -43 mV, and Vlk = -65 mV.
Fast transient K+ current A2 A3 A4 A5 A6 A7 A8
Low-threshold K+ current A9 A10 A11 A12 A13
High-threshold K+ current A14 A15 A16 A17 A18
Fast Na+ current A19 A20 A21 A22 A23
Hyperpolarization-activated cation current A24 A25 A26
Leak current A27
This work was supported by National Institute of Deafness and Other Communication Disorders Grants P60DC-00979 (Research and Training Center in Hearing and Balance), Subproject 2, and R01 DC-04551 to P. B. Manis. J. S. Rothman was also partially supported by Training Grant T32DC-00023.
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- Copyright © 2003 by the American Physiological Society