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Kinetic Analyses of Three Distinct Potassium Conductances in Ventral Cochlear Nucleus Neurons

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, The University of North Carolina, Chapel Hill, North Carolina 27599

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
Neurons in the ventral cochlear nucleus (VCN) express three distinct K⁺ currents that differ in their voltage and time dependence, and in their inactivation behavior. In the present study, we quantitatively analyze the voltage-dependent kinetics of these three currents to gain further insight into how they regulate the discharge patterns of VCN neurons and to provide supporting data for the identification of their channel components. We find the transient A-type K⁺ current (I_A) exhibits fourth-order activation kinetics (a⁴), and inactivates with one or two time constants. A second inactivation rate (leading to an a⁴bc kinetic description) is required to explain its recovery from inactivation. The dendrotoxin-sensitive low-threshold K⁺ current (I_LT) also activates with fourth-order kinetics (w⁴) but shows slower, incomplete inactivation. The high-threshold K⁺ current (I_HT) appears to consist of two kinetically distinct components (n² + p). The first component activates 10 mV positive to the second and has second-order kinetics. The second component activates with first-order kinetics. These two components also contribute to two kinetically distinct currents upon deactivation. The kinetic behavior of I_HT was indistinguishable amongst cell types, suggesting the current is mediated by the same K⁺ channels amongst VCN neurons. Together these results provide a basis for more realistic modeling of VCN neurons, and provide clues regarding the molecular basis of the three K⁺ currents.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
Previously we have shown that isolated VCN neurons possess one or more of the following three distinct macroscopic K⁺ currents (Rothman and Manis 2003a): a rapidly inactivating A-type current (I_A), a rapidly activating, slowly inactivating low-threshold current (I_LT), and a non-inactivating high-threshold current (I_HT). Although these currents have been described before (Manis and Marx 1991; Manis et al. 1996), their detailed kinetic behavior has not been elucidated. Because the kinetic behavior of K⁺ currents differs amongst the various known channel types (Chandy and Gutman 1994; Coetzee et al. 1999; Rudy 1988), kinetic analysis can provide insight into the identity of each participating channel type. In addition, kinetic analysis can provide critical information necessary to understand how and which channels regulate the subthreshold integration of synaptic inputs as well as the generation of complex action potential patterns. In this paper, we describe the kinetics of I_A, I_LT, and I_HT in VCN neurons. We use these data to construct mathematical models of each current, which are compared directly to the experimental currents. The kinetic descriptions are used collectively in the following paper to simulate the electrical behavior of VCN neurons (Rothman and Manis 2003b).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
Cell isolation and voltage-clamp procedures

Procedures to isolate VCN neurons are as described in our previous paper (Rothman and Manis 2003a). For the data presented in this paper, all whole cell voltage-clamp recordings were made with an Axopatch 200 amplifier at 22°C. The extracellular solution [(in mM) 130 NaCl, 5 KCl, 2.5 CaCl₂, 1.3 MgCl₂, 30 glucose, and 10 HEPES] contained TTX and Cd²⁺ to block sodium, calcium, and calcium-dependent currents. Electrodes were filled with a K-gluconate solution [(in mM) 130 K-gluconate, 4 NaCl, 1 EGTA, 5 sucrose, 10 HEPES, and 4 Mg₂ATP], having resistances in the range 3–10 M. Only cells with stable 75–95% online compensation were included in the analyses. Under these conditions, the voltage-step rise time _s was in the range 8–40 µs.

Kinetic analysis

Our kinetic description of a membrane current follows the classical formalism of Hodgkin and Huxley (1952), where the relationship between the channel conductance and measured current is defined as

(1)

In this equation, g_max is the peak conductance, a the activation state variable, the order of activation, b the inactivation state variable, V_r the current's reversal potential, and V the membrane potential. The rate of change of states a and b are governed by the following first-order differential equation

(2)

where is the state time constant, and x its steady-state value (i.e., the value of x when t >> _x). Although this equation is different from the original HH formalism, in which x is expressed in terms of "opening" and "closing" rate constants and , it is nevertheless mathematically equivalent when x = /( + ) and _x = 1/( + ). Because the solution to Eq. 2 for a voltage step at t = 0 is

(3)

where x₀ is the value of x at t = 0, Eq. 1 can be rewritten as

(4)

In this study, we describe the steady-state values of a and b by a Boltzmann function of the form

(5)

where V_0.5 is the half-activation voltage where y = 0.5, and k is the slope factor that determines the steepness of the Boltzmann function. To simplify fitting the Boltzmann relationships to our current data points, we used a modified Boltzmann function in which conductance has been translated into current

(6)

In this case, fits proceed against current measurements rather than calculated conductance, thereby eliminating large variations in estimated conductance that occur near V_r. This was important in the present study because I_LT activates close to the K⁺ reversal potential.

Time constants as a function of voltage were fit to a bell-shaped function of the form

(7)

where SF is a scale factor, and M is a constant to set a minimum value. This is similar to the HH formalism when = C exp((V + 60)/V) and = C exp(-(V + 60)/V).

Statistics

Statistical significance was assessed using a two-sided t-test for unpaired samples at the significant level P indicated. Significant differences are denoted with asterisks as follows: (0.01 < P <= 0.05, *) (0.001 < P <= 0.01, **) (P <= 0.001, ***). All results are reported as means ± SE. Error bars in the various figures also represent SE.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION ACKNOWLEDGMENTS REFERENCES

 
Kinetic analysis of I_A

In this section, we present a quantitative description of the rapidly inactivating A-type current, I_A. As stated in our previous paper, I_A was found only in a subpopulation of VCN Type I cells, the Type I-t cells (Rothman and Manis 2003a).

ISOLATION OF I_A. Figure 1 shows the voltage-clamp protocol used to isolate I_A. In this protocol, alternating preconditioning voltage steps (V_pp) were used to either remove inactivation of I_A (Fig. 1A; V_pp = -110 mV) or inactivate I_A (Fig. 1B; V_pp = -40 mV). Depolarizing voltage steps (V_cmd) applied after V_pp then evoked currents with and without I_A, respectively. Point-by-point subtraction of alternating current traces yielded I_A in isolation (Fig. 1C). Because the more positive V_pp often activated small amounts of I_HT and/or I_LT in our population of Type I-t cells (Fig. 1B, arrowhead), a 2- or 3-ms hyperpolarizing step to -110 mV (V_int) was added after every V_pp to deactivate most of I_HT and/or I_LT before V_cmd. The addition of V_int also generated identical capacitative electrode currents at the beginning of V_cmd, which could then be eliminated by subtraction (Fig. 1C). Although V_int removed 2% of I_A, it did not affect its subsequent kinetic analyses.

View larger version (20K): [in this window] [in a new window]   FIG. 1. Isolation of IA. A: outward currents elicited by the voltage-clamp protocol at top, where an 82-ms conditioning prepulse to -110 mV (Vpp; of which only the last 7 ms is shown) preceded a 100-ms depolarizing command step to potentials between -110 and 0 mV (Vcmd; of which only the 1st 50 ms is shown). During Vpp, inactivation of IA was removed. The fast transient current before each command step is the electrode capacity current (0.5 ms). B: same as A, except Vpp consisted of an 80-ms step to -42 mV, followed by a 2-ms step to -110 mV (Vint). During Vpp, IA was inactivated; also, a small sustained outward current was activated (). During Vint, the small sustained outward current was deactivated, while inactivation of IA remained relatively unchanged. Note the absence of IA during Vcmd. C: isolated IA obtained by subtracting B from A. Here, current traces during 100-ms Vcmd are shown in their entirety. Scale bars for A–C.

ACTIVATION AND INACTIVATION OF I_A. To estimate the time course of activation and inactivation of I_A, isolated current traces of I_A, as obtained in Fig. 1, were fit to the following equation

(8)

where I_max = g_max(a)(V - V_r). This equation was derived from Eq. 4 assuming a₀ = 0 and b₀ = 1 (i.e., all A channels are closed and all inactivation is removed at the start of V_cmd) and b = 0 (true for V_cmd > -50 mV; see Fig. 3C). Examples of fits to I_A are shown in Fig. 2A, with time represented on a log scale. Because a number of current traces showed both fast and slow inactivation components, the inactivation variable b was modified to include both a fast and slow time constant (_bf and _bs), in which the fraction attributed by each component was set by f. In our initial analysis, the order of activation , which measures the degree of sigmoidal activation, was allowed to vary from trace to trace; it was found, however, that a value near 4 usually produced the best fits. Hence, was set to 4 for the entire analysis. Results from nine Type I-t cells are shown in Fig. 2B, where each variable in Eq. 8 is plotted versus V_cmd. As this figure shows, I_A is strongly voltage dependent with respect to both activation (I_max and _a) and inactivation (_bf). However, the slow inactivation time constant (_bs), with values in the range 18–300 ms, is only sometimes voltage dependent. In all cases, the fast component of inactivation was significantly larger than the slow component (f > 0.7), and in 70% of the cells it was the only component.

View larger version (21K): [in this window] [in a new window]   FIG. 3. Steady-state inactivation of IA. A: outward currents elicited by the voltage-clamp protocol at top, where a 100-ms depolarizing voltage step to -14 mV (Vcmd; of which only the 1st 50 ms is shown) followed an 80-ms conditioning prepulse to potentials between -110 and -23 mV (Vpp; of which only the last 7 ms is shown) and a 2 ms step to -110 mV (Vint). During Vpp, various levels of IA were inactivated; also, a sustained outward current was activated at V > -50 mV (). During Vint, the sustained outward current was partially deactivated, while inactivation of IA remained relatively unchanged. B: same as A, except Vpp was held at -110 mV. This sequence, being interleaved with that in A, measured the maximum IA available at the beginning of each sequence run. C: isolated IA obtained by subtracting B from A. Current traces during the 100-ms command step are shown in their entirety. IA < 0 not shown. Note, after subtraction, the largest current trace corresponded to Vpp = -110 mV and the smallest to Vpp = -23 mV. Inset: steady-state inactivation of IA. To obtain these data, peak currents in C were computed as a function of voltage. The resulting I-V relation was then fit to Eq. 6 (Imax = 2.0 nA, V0.5 = -66.8 mV, k = -7.5 mV) where Imax = gmax(V - Vr). Using Imax, both the I-V relation () and Boltzmann function (bold line) were normalized and subtracted from 1 to show inactivation in the conventional manner. Thin lines are Boltzmann functions derived from 9 other Type I-t cells. Scale bars for A–C.

View larger version (27K): [in this window] [in a new window]   FIG. 2. Characterization of IA. A: IA from Fig. 1C (—) plotted on a log time axis to allow better visualization of activation. —, fits to Eq. 8 ( = 4). B: fit results from the cell in A () and 8 other Type I-t cells (). x axis denotes Vcmd. Top left: Imax, maximum activation current. —, the mean modified Boltzmann function (Eq. 6; gmax = 61.5 nS, V0.5 = -30.8 mV, k = 6.1 mV). Bottom left: a, activation time constant. —, the model time constant (Eq. 7; C = 7 ms-1, V = 14 mV, C = 29 ms-1, V = 24 mV, SF = 100, M = 0.1 ms). Top right: f, proportion attributed to the fast and slow inactivation components, where f = 1 denotes all fast and no slow. Hence, most of inactivation is due to the fast component. Note y axis spans 0.5–1.0. Bottom right: bf and bs, fast (, ) and slow (,) inactivation time constants.

To compute the steady-state activation of I_A, modified Boltzmann functions (Eq. 6) were independently fit to the I_max-V relations in Fig. 2B and averaged together. Mean values for g_max, V_0.5, and k were: 61.5 ± 7.2 nS, -30.8 ± 1.2 mV, and 6.1 ± 0.6 mV, respectively (n = 6).

Steady-state inactivation of I_A was next investigated with the prepulse protocol in Fig. 3. This protocol is similar to the one in Fig. 1, except V_pp was varied while V_cmd was held constant. Again, V_int was used to deactivate any sustained outward current that might have activated during V_pp and to provide identical capacitative transients at the beginning of V_cmd. Although not apparent in this figure, sequence B, which consists of the same repeated voltage step, was interleaved with sequence A, in which V_pp was changed. The interleaved protocol was used to detect time-dependent rundown of I_A. Subtraction of traces in B from those in A resulted in isolated I_A in C. From C, an estimate of steady-state inactivation was obtained by computing peak current as a function of V_pp (inset) and fitting the resulting I-V relations to Eq. 6 (see legend for details). Repeating the same analysis for nine other Type I-t cells (inset, thin lines) yielded the following mean values for V_0.5 and k: -66.1 ± 1.6 and -6.9 ± 0.4 mV.

Recovery from inactivation of I_A was studied with the twin-pulse protocol shown in Fig. 4A, where two voltage steps to the same potential (V_S1 and V_S3) were separated by a hyperpolarizing voltage step (V_S2) of variable length (T_S2). As indicated by the current traces in Fig. 4A, recovery from inactivation was related to T_S2 in an exponential-like manner. In Fig. 4B, current traces evoked by V_S3 are plotted concurrently after subtraction of I_HT (see legend for details). From these current traces, recovery functions were obtained by computing peak current as a function of T_S2 (inset). Close inspection of these recovery functions shows a significant voltage-dependent delay in removal of inactivation: for V_S2 = -100 mV, the delay is 3 ms; for V_S2 = -70 mV, the delay is 8 ms. Furthermore, after the delay, the functions do not follow a single-exponential time course, but a bi-exponential one. Although the recovery functions could be satisfactorily fit to a sum of exponentials, a product of exponentials was adopted instead, because it could be readily modeled with a two-state function in the form bc (see Fig. 5B)

(9)

Fits to the recovery functions in Fig. 4B are plotted as dashed lines (inset), demonstrating the adequacy of Eq. 9. In Fig. 4C, time constants _b and _c are plotted versus V_S2 (squares and triangles; n = 8), in which case both time constants show strong voltage dependence.

View larger version (24K): [in this window] [in a new window]   FIG. 4. Removal of inactivation of IA. A: outward currents elicited by the double-pulse protocol at top. None of the 11 current traces (i1–i11) have been leak subtracted. Only the last voltage trace, v11, is displayed. TS2 ranged from 2 (i1) to 200 (i11) ms. VS1 and VS3 lasted 100 ms. B: isolated IA during VS3, achieved by 2 subtractions. First, trace i', equal to i11 during VS1 and VS2, was subtracted from i1 through i11 (during VS3, i' was set equal to the mean value of i11 during the last 5 ms of VS2). This subtraction eliminated time-dependent changes in the membrane current due to fast deactivating tails at the end of VS1, and, if present, slow activation of Ih during VS2. Second, after i1–i11 were realigned to the beginning of VS3, trace i1, which consisted almost exclusively of IHT, was subtracted from i1 through i11. Inset: peak current vs. TS2 for 4 different VS2 (circles; data normalized to steady-state values). Dashed lines, fits to Eq. 9: b = 5.0 ms, c = 18.3 ms (VS2 = -100 mV); b = 7.0 ms, c = 22.7 ms (VS2 = -90 mV); b = 13.0 ms, c = 29.9 ms (VS2 = -80 mV); b = 14.6 ms, c = 46.3 ms (VS2 = -70 mV). C: b (squares) and c (triangles) as computed in B (n = 8). Circles, bf in Fig. 2B. Bold lines, model b (Eq. 7; C = 14 ms-1, V = 27 mV, C = 29 ms-1, V = 24 mV, SF = 1,000, M = 1 ms), and model c (Eq. 7; C = 0 ms-1, C = 0.7 ms-1, V = 17 mV, SF = 90, M = 10 ms).

View larger version (30K): [in this window] [in a new window]   FIG. 5. Model IA. A: model IA in response to a prepulse protocol similar to the one in Fig. 1A, where Vpp = -110 mV (100 ms) and Vcmd = -42, -32, -22, -14, and -6 mV (100 ms). Inset: comparison of model IA in A (—) to experimental IA in Fig. 2A (—) plotted on log time axis. B: model recovery functions (—) compared to experimental ones in Fig. 4B (—). As in Fig. 4, VS2 = -100, -90, -80, and -70 mV.

NUMERICAL RECONSTRUCTION OF I_A. From the above analysis, the following kinetic model of I_A was developed

(10)

In this equation, _A is the maximum conductance, which we leave as a free parameter because the magnitude of I_A varied from cell to cell. V_K is the reversal potential of I_A. Because V_K was not experimentally determined for I_A, it was set it to -70 mV, the experimentally determined V_K of I_LT and I_HT (see Figs. 7C and 10B). Parameter a⁴ is the activation variable whose steady-state value (a)⁴ is defined by Eq. 5 (V_0.5 = -31 mV, k = 6 mV; from Fig. 2B). Parameters b and c are the fast and slow inactivation variables whose steady-state values, b and c, are defined such that their product equals the mean inactivation Boltzmann function of I_A; specifically, b = c = y^1/2, where y is defined by Eq. 5 (V_0.5 = -66 mV, k = -7 mV; from Fig. 3). Time constants _b and _c were derived from the data in Fig. 4C and are shown as bold lines in that figure. _b describes the fast component of removal of inactivation at V < -50 mV (squares), and the fast component of development of inactivation at V > -50 mV (circles). _c describes the slow component of removal of inactivation at V < -50 mV (triangles). For V > -50 mV, _c is set to a large saturating value of 100 ms because it showed no signs of diminishing in the real data. _a was derived from the data in Fig. 2B. At V < -50 mV, the behavior of _a is unknown because deactivation of I_A was not investigated. We therefore assumed _a behaved qualitatively similar to _b at V < -50 mV, as it did at V > -50 mV (Fig. 2B). Hence, model _a = _bf/10 for all V.

View larger version (32K): [in this window] [in a new window]   FIG. 7. Activation and deactivation of ILT. A: ILT (—) elicited by the voltage-clamp protocol at top. —, fits to Eq. 11 ( = 4), where w0 was fixed so that (w0) = 0.09 for all traces (Eq. 5; V0.5 = -48 mV, k = 6, and V = -62 mV). B: w versus Vcmd (, n = 33).|, separates activation and deactivation time constants. —, a fit to Eq. 7 (C = 6 ms-1, V = 6 mV, C = 16 ms-1, V = 45 mV, SF = 100, M = 1.5 ms). C: Imax vs. Vcmd (n = 33). Vr computed where I = 0.| and ¦, mean and SE of Vr (-70 ± 0.8 mV).

View larger version (25K): [in this window] [in a new window]   FIG. 10. Deactivation of IHT. A: tail currents of IHT (—) elicited by the voltage-clamp protocol at top. —, a simultaneous fit to Eq. 16, where n and p were shared between traces and In and Ip were allowed to vary from trace to trace. Inset: results of the simultaneous fit. — through In and Ip data are fits to Eq. 6 for In (Imax = 3.7 nA, V0.5 = -16.8 mV, k = 5.2 mV) and Ip (Imax = 0.7 nA, V0.5 = -19.3 mV, k = 5.6 mV), where Imax = gmax (V - Vr). B: tail currents of IHT (—) elicited by the voltage-clamp protocol at top. —, fits to Eq. 16; no parameters were shared. Fit results are shown in insets B1 and B2. Ip was multiplied by 10 for comparison to In. Note both In and Ip reverse near -72 mV and show a linear voltage dependence. Current and time scales in A apply to B.

Model current traces of I_A are shown in Fig. 5A, computed with the same voltage protocol in Fig. 1A. For comparison, traces of model I_A are plotted against those of experimental I_A (inset). Because the model parameters are derived from the average of several Type I-t cells, the traces do not overlap exactly. However, the comparison shows that the time course of activation/inactivation of model I_A is similar to that of experimental I_A.

Despite the use of two inactivation variables b and c, inactivation of model I_A proceeds along an apparent single-exponential time course as if f = 1. However, the absence of a measurable slow component of inactivation (c) is expected: for V > -50 mV, the effective inactivation time constant is _bc/(_b + _c), which reduces to _b when _c >> _b. Experimentally, inactivation of I_A proceeded along a single-exponential time course in 70% of the cells, and when a bi-exponential time course was detected, the slow component was significantly smaller than the fast component (see Fig. 2B, parameter f).

The effects of parameter c are however apparent during recovery from inactivation as shown in Fig. 5B, where four different recovery functions of model I_A are plotted concurrently. For comparison, traces of model I_A (—) are plotted against those of experimental I_A (– – –; from Fig. 4B). Again, because the model parameters are derived from the average of several Type I-t cells, the traces do not overlap exactly. However, the comparison shows that the model recovery functions display an onset delay (sigmoidal kinetics) similar to the experimental recovery functions. These results demonstrate that a simple two-state model of inactivation, bc, can adequately describe the delay observed in the recovery from inactivation.

Kinetic analysis of I_LT

In this section we present a quantitative description of the rapidly activating, slowly inactivating low-threshold current, I_LT. The analyses of I_LT has been restricted to VCN Type II cells; these cells show large, unambiguous signs of I_LT (Manis and Marx 1991; Rothman and Manis 2003a).

I_LT OBEYS FOURTH-ORDER ACTIVATION KINETICS. To obtain an accurate estimate of the kinetics of I_LT, it was first necessary to determine its order of activation, . This was accomplished with the voltage-clamp protocol in Fig. 6A, where a command step to -50 mV (V_cmd), which activates I_LT but not I_HT, followed a 100-ms prepulse (V_pp) to various potentials below -50 mV. The response of the Type II cell at the bottom of Fig. 6A shows that, when V_pp > -80 mV, I_LT appears to activate with first-order kinetics. However, when V_pp < -80 mV, I_LT shows sigmoidal activation kinetics, indicating I_LT is mediated by a channel with multiple closed states.

View larger version (34K): [in this window] [in a new window]   FIG. 6. Activation of ILT. A: ILT (—) elicited by the prepulse protocol at top, showing delay in activation for Vpp < -80 mV. —, simultaneous fits to Eq. 11, where and w were shared between traces, and w0 was left as a free parameter, except for the 1 trace that corresponded to a -110-mV prepulse step, in which case w0 = 0 (i.e., all channels assumed closed at -110 mV). To reduce the number of free parameters in the fitting procedure, w was fixed in such a way that (w)4 = 0.5 for all traces; here, a value of 0.5 was determined from the average Boltzmann function (Eq. 5; V = -48 mV) of the DTX-sensitive low-threshold current in Type II cells: V0.5 = -48 mV, k = 6 (Rothman and Manis 2003a; Table 2). To get an accurate estimate of the initial time course of activation, the curve-fit analysis was restricted to t < 10 ms. Inset: w0 vs. Vpp, showing the variation in delay can be accounted for by w0. , points to a change in steady-state current levels due to a change in inactivation set by Vpp. B: development of delay () and recovery from inactivation (). Voltage-clamp protocol is shown at top on a reduced time scale. From -62 mV, the membrane was stepped to -110 mV for variable time T, and then to -52 mV to elicit ILT. Current traces (—) were shifted in time so that t = 0 corresponded to the beginning of the voltage step to -52 mV (¦). —, simultaneous fits to Eq. 11, as in A, except = 4, and w was fixed in such a way that (w)4 = 0.3 for all traces (Eq. 5; V0.5 = -48 mV, k = 6 and V = -52 mV). Inset B1: w0 vs. T, showing development of delay proceeds along a single-exponential time course. Inset B2: Imax vs. T, showing recovery from inactivation also proceeds along a single-exponential time course.

To determine the value of , current traces in Fig. 6A were simultaneously fit to the following equation

(11)

where I_max = g_maxz₀(V - V_r). This equation was derived from Eq. 4 using w and z to denote activation and inactivation. Because inactivation of I_LT is slow (_z > 150 ms; see results below), and the window of analysis comparatively brief (20 ms), the exponential decay of inactivation was ignored during the fit, in which case z was set to z₀ for all time. During the fitting procedure, _w and w were shared between traces because theoretically they should be the same for the same V_cmd. To determine if a single value of could account for the observed kinetic behavior of I_LT, was also shared between traces. The two remaining parameters, I_max and w₀, were allowed to vary from trace to trace to account for changes in the level of inactivation (z₀) and the level of activation at the start of V_cmd (w₀; Fig. 6A, inset). Fits thus obtained are shown in Fig. 6A (—). Clearly, current traces were satisfactorily fit with a single value of (4.1). Repeating the same analyses for 10 other Type II cells yielded the following estimate of : 4.2 ± 0.1 (n = 11). Hence, I_LT obeys fourth-order activation kinetics similar to the n⁴-kinetics of the delayed-rectifier current first described by Hodgkin and Huxley (1952).

The finding that I_LT obeys fourth-order kinetics differs from a previous finding that it obeys first-order kinetics (Manis and Marx 1991). The difference in findings is simply due to the use of prepulses in this study that deactivate I_LT to various degrees before stepping to V_cmd. Previously the kinetics of I_LT were analyzed from a holding potential near -60 mV, in which case I_LT was 10% activated (w₀ = 0.1), and therefore appeared to obey first-order kinetics when stepped positive (see Fig. 6A, 3rd trace from top).

DELAY OF ACTIVATION SHOWS EXPONENTIAL TIME COURSE. Results from the previous section demonstrate I_LT shows a delay in activation when stepped from potentials below -80 mV, similar to the delay observed in other delayed-rectifier currents (Hodgkin and Huxley 1952; Schoppa and Sigworth 1998; Young and Moore 1981). Because it has previously been shown that this delaying effect develops with an exponential time course (Begenisich 1979), it was of interest to determine if this was also true for I_LT.

Figure 6B shows the prepulse protocol used to study the evolution of delay of I_LT. In this protocol, a command step to -50 mV was preceded by a prepulse to -110 mV of variable length T, resulting in the 10 current traces shown as noisy dashed line. The delay of I_LT is pointed out () and should not to be confused with the removal of inactivation (). For small T, the delay was minimal, and I_LT activated with an exponential-like time course (top). For long T, the delay was greater, and I_LT activated with a sigmoidal time course (bottom).

To quantify the time evolution of delay, current traces of I_LT were simultaneously fit to Eq. 11, with parameters w, and _w shared between traces, and = 4 (see legend for details). Fits to the data in Fig. 6B are shown as black lines, again demonstrating I_LT is satisfactorily fit to Eq. 11 when = 4. In Fig. 6B1, the value of w at the end of the prepulse, w₀, is plotted versus the prepulse length T (). The solid line is a single-exponential fit to the data ( = 1.4 ms), demonstrating the time evolution of the delay follows an exponential time course. Repeating the same analysis for another Type II cell similarly yielded = 1.2 ms. Thus the delay in activation of I_LT can be accounted for by a simple w⁴-kinetic model. In this model, the current amplitude and time course of delay is determined by the initial open probability of the channel (w₀), which in turn is determined by both the level (Fig. 6A) and length (Fig. 6B) of the conditioning prepulse.

ACTIVATION KINETICS OF I_LT. Activation and deactivation kinetics of I_LT were computed by fitting Eq. 11 ( = 4) to whole cell current traces elicited by depolarizing and hyperpolarizing steps from -60 mV (Fig. 7A) or -70 mV. Again, by limiting the analysis to V < -40 mV, contributions from I_HT were minimized. For a number of cells, a hyperpolarization-activated inward current, I_h, began to activate at V < -80 mV; in these cases, fits were restricted to the first 10–20 ms, thereby minimizing contamination from I_h (which has a very slow activation time course on the order of seconds). Results are plotted in Fig. 7B, where _w is plotted versus V_cmd for 33 Type II cells (). Together, activation and deactivation _w describe a bell-shaped curve that peaks near -60 mV at 6 ms. However, the curve is not exactly bell-shaped in that activation _w shows a steeper voltage dependence than deactivation _w (2-fold). The difference in steepness arises from the fact that activation is described by a (1 - w)⁴ kinetic process, whereas deactivation is described by a single-exponential kinetic process (see Eq. 11). The curved solid line in Fig. 7B is a fit to Eq. 7 (see legend for values). Because the kinetic analysis of I_LT was limited to V < -40 mV, _w was given a lower limiting value of 1.5 ms, corresponding to the lowest value at -40 mV. However, it very well may be that _w falls below 1.5 ms for V > -40 mV.

INACTIVATION OF I_LT. Of the 49 cells classified as Type II in our study of VCN neurons, 41 showed visual signs of slow inactivation in their outward current traces at V > -50 mV (for example, see Rothman and Manis 2003a; Fig. 2C). For the majority of these cells, development of inactivation was too slow to allow an accurate estimate of its time course during the 100-ms window of observation (_z > 150 ms; n = 28). For the remaining cells, development of inactivation was fast enough to allow single-exponential fits to its time course; in these cases, _z had values 30–150 ms from -50 to 0 mV, with largest values near -50 mV (n = 13; not shown).

Our finding that the activation level of I_LT is sensitive to a preconditioning voltage step (Fig. 6, A and B; ) also indicates I_LT undergoes inactivation. Indeed, 77% of the Type II cells investigated in this study showed sensitivity to a preconditioning voltage step. However, just as _z showed significant variability, as described in the preceding text, the voltage dependence of inactivation of I_LT (z) showed significant variability. In nine Type II cells, z behaved in a Boltzmann-like manner (Eq. 6) with a visible lower limit near 0.5 for V > -60 mV (V_h = -70.9 ± 2.2 mV, k = -10.0 ± 0.7 mV; data not shown). In 11 Type II cells, z behaved in a more shallow Boltzmann-like manner that only appeared to have a lower limit near 0.5 (V_h = -50.9 ± 3.6 mV, k = -15.7 ± 2.1 mV; data not shown). In 10 Type II cells, z behaved in a linear-like manner and therefore did not fit well to a Boltzmann function. In no instance was inactivation of I_LT ever complete as it was for I_A. What the variability in inactivation of I_LT was due to is not clear.

For seven Type II cells, removal of inactivation was investigated with the twin-pulse protocol in Fig. 6B. Of these seven cells, four showed steady-state amplitude changes in response to a change in the interpulse time T. For the cell in Fig. 6B, removal of inactivation is denoted (). For small T, little inactivation was removed, and steady-state I_LT (I_max) remained at low levels (bottom). For large T, a significant amount of inactivation was removed, and I_max rose to higher levels (top). In Fig. 6B2, I_max from Eq. 11 is plotted versus T. Here, changes in I_max reflect changes in the initial level of inactivation, z₀, because g_max, V and V_r are constants. Clearly, I_max, and therefore z₀, varies with an exponential time course. The solid line is a single-exponential function with _z = 54 ms.

K⁺ SELECTIVITY OF I_LT. Given Eq. 11, V_r of I_LT could be estimated from the instantaneous I-V relationships in Fig. 7C where I = 0. Of the 33 Type II cells in this figure, 30 had sufficient data to compute V_r by linear interpolation (i.e., data points fell above and below the 0-current axis). Computed in this way, V_r was estimated at -69.9 ± 0.8 mV. The fact that this estimate is positive to the theoretical V_r (-80 mV, assuming perfect K⁺ selectivity) indicates either this estimate of V_r is imprecise, the ion channels that carry I_LT are not perfectly selective for K⁺, or intracellular and/or extracellular K⁺ concentrations are not as predicted (local accumulations of K⁺, for example, could shift V_r positive). Similar observations of a more positive V_r have been noted elsewhere (Bal and Oertel 2001; Manis and Marx 1991; Rathouz and Trussell 1998). Nevertheless, theoretical and empirical V_r suggest I_LT is predominantly carried by K⁺.

NUMERICAL RECONSTRUCTION OF I_LT. From the preceding analyses, the following kinetic model of I_LT was developed

(12)

In this equation, _LT is the maximum conductance, which we leave as a free parameter because the magnitude of I_LT tended to vary from cell to cell. Parameter V_K is the reversal potential of I_LT, set to -70 mV (Fig. 7C). Parameter w⁴ is the activation variable whose steady-state value (w)⁴ is described by the normalized Boltzmann function of the DTX-sensitive low-threshold current in Type II cells (Rothman and Manis, 2003a; Table 2) (V_0.5 = -48 mV, k = 6 mV), and whose time constant, _w, is described by the bell-shaped curve in Fig. 7B. Parameter z is the inactivation variable whose steady-state value z is described by the average of those inactivation functions that fit well to a Boltzmann function with steady-state offset () in the form

(13)

where = 0.5. The model inactivation time constant _z is plotted in Fig. 8B (inset), and was derived from the following assumptions based on experimental observations: _z is bell-shaped; _z 50 ms at -110 mV (Fig. 6B2); _z 100–300 ms at V > -50 mV; and _z > 300 ms near rest.

View larger version (32K): [in this window] [in a new window]   FIG. 8. Model ILT. A: model ILT in response to the voltage-clamp protocol at top with inactivation turned off ( = 1). Arrows denote classic signs of ILT. Inset: activating tail currents during the step to -50 mV at higher magnification that can be directly compared to those in Fig. 6A. B: same as A, with inactivation turned on ( = 0.5, LT = 272 nS). Slow inactivation is now apparent in the outward current traces during command step (arrow 3) as well as in the tail currents (arrow 4). Inset: inactivation time constant z (Eq. 7; C = 1 ms-1, V = 20 mV, C = 1 ms-1, V = 8 mV, SF = 1,000, M = 50 ms). C: development of inactivation and removal of inactivation as computed in Fig. 6B.

Figure 8A shows model I_LT first without inactivation ( = 1.0 in Eq. 13; _LT = 170 nS). Here, the model shows the signature features of I_LT (arrows) as described previously (Rothman and Manis, 2003a): a small, steady outward current at a -60 mV holding potential (1), a small deactivating inward current in response to voltage steps below V_K (2), and rapid activation (5). Furthermore, model I_LT shows sigmoidal activation kinetics (inset) comparable to that of the experimental data in Fig. 6A.

Figure 8B shows model I_LT now with inactivation ( = 0.5; _LT = 272 nS so that I_LT at -60 mV is the same as that in Fig. 8A). The same features of I_LT are visible (arrows 1, 2, and 5), but now slow inactivation is apparent in the current traces during V_cmd > -50 mV (arrow 3), as well as in the tail currents (arrow 5). Such signs of inactivation of I_LT was observed in the majority of Type II cells.

Figure 8C shows model I_LT ( = 0.5) when studied with the same twin-pulse protocol in Fig. 6B. The time constants for the time evolution of the delay are _d = 1.1 ms at -110 mV and for the removal of inactivation are _z = 53 ms at -110 mV. Again, the model accurately replicates the experimental data.

Kinetic analysis of I_HT

In this section, we present a quantitative description of the non-inactivating high-threshold current I_HT. As stated in our previous paper (Rothman and Manis 2003a), all VCN cell types possess I_HT; however, only the Type I-c cells appear to have I_HT as its sole outward current at V > -80 mV. Hence the analysis of I_HT in Type I-c cells is considered the prototype. Direct comparisons are then made to the DTX-insensitive current (I_DTX^-) in Type I and Type II cells, and to I_HT in Type I-t cells. These comparisons show that I_HT is kinetically indistinguishable amongst VCN neurons.

ACTIVATION OF I_HT. The time course of activation of I_HT was slightly more complex than expected for a single conductance. We found I_HT was better described by the sum of two components rather than a single component. This is demonstrated in Fig. 9A, where I_HT is better fit to a two-component model (n² + p) than to a one-component model (n²). Similarly for deactivation, I_HT was better fit to the sum of two exponentials rather than a single exponential (Fig. 10, A and B). These results agree with the previous finding of two components in the tail currents of I_HT (Manis and Marx 1991).

View larger version (29K): [in this window] [in a new window]   FIG. 9. Activation of IHT. A: fits to activating IHT in a Type I-c cell (—) elicited by the voltage-clamp protocol at top. —, fits to Eq. 15, where = 2 (n2 + p model). —, fits to Eq. 15, where = 2 and Ip = 0 (n2 model). B: difference curves between a single current trace of IHT (largest current trace in A) and its respective fit to various n + p models (Eq. 15). The n2 + p model fit activating IHT with the least amount of error. C: results of the n2 + p curve-fit analyses in A. — through In and Ip data, fits to Eq. 6 for In (gmax = 71.3 nS, V0.5 = -12.5 mV, k = 5.9 mV) and Ip (gmax = 26.8 nS, V0.5 = -25.7 mV, k = 6.7 mV).

After investigating several two-component models of I_HT, we found a n² + p model was better than a n + p model, and no worse than a n³ + p model (Fig. 9B). Although it is true a product of variables, such as np, can adequately describe the activation of I_HT (not shown), such a model was inadequate in describing the deactivation of I_HT because the tail currents of a np model decay with a single-exponential time constant = _np/(_p + _n), which reduces to _n/ when _p >> _n. Another description might be a n + np model, used previously to describe the activation of KCNC1 currents in NIH-3T3 cells (Kanemasa et al. 1995), and activation of a high-threshold DTX-insensitive current in principal cells of the MNTB (Wang, LY et al. 1998). However, deactivating tail currents in this model also decay as a single exponential ( _n/ when _p >> _n).

According to Eq. 4, activation and deactivation of a n + p model is described by the following equation

(14)

Here, n and p denote two distinct activation processes. Note that inactivation is not included in this equation because it was not apparent during the 100-ms command steps used in this study. When currents are activated from V < -45 mV, such as those in Fig. 9A, n₀ and p₀ can be assumed to be zero (i.e., all channels are closed), and Eq. 14 reduces to

(15)

where I_n = g_n(n)(V - V_n) and I_p = g_pp(V - V_p). On the other hand, when activated currents are deactivated by stepping to V < -45 mV, such as those in Fig. 10A, n and p can be assumed to be zero, and Eq. 14 reduces to

(16)

where I_n = g_n(n₀)(V - V_n) and I_p = g_pp₀(V - V_p). Results of fitting Eq. 15 ( = 2) to the activation of I_HT in Fig. 9A are shown in Fig. 9C. As Fig. 9C shows, _n < _p for all V and I_n > I_p for V > -20 mV. Furthermore, I_n appears to activate positive to I_p. This is apparent when V_0.5 values of I_n and I_p are compared, computed from Eq. 6: -13 versus -26 mV, respectively. Fitting Eq. 16 ( = 2) to the deactivation of I_HT in Fig. 10, A and B, gave similar results: _n < _p; I_n > I_p; and I_n activates positive to I_p. A summary of activation and deactivation analyses for a total of 16 Type I-c cells is shown in Fig. 11, where _n is plotted in A, _p in B, and n² and p in C. For V > -45 mV, data are from activation analysis (Fig. 9), and for V < -45 mV, data are from deactivation analysis (Fig. 10). The n² and p data in C were computed from I_n and I_p values, according to Eqs. 15 and 16, where n denotes n or n₀, and p denotes p or p₀. The bold lines in A and B are fits to Eq. 7, and the bold lines in C are average Boltzmann functions computed from the activation analysis (—) and the deactivation analysis (– – –). As A and B show, both _n and _p are described by bell-shaped curves that peak near -50 mV (peak values 4 and 17 ms, respectively). For all V, mean values of _p are 5 times greater those of _n. As C shows, n² activates 10 mV positive to p for both activation and deactivation (V_0.5 -15 and -25 mV, respectively). Hence, time constants and activation functions of the fast and slow component of I_HT are well separated. These results are similar to those of Manis and Marx (1991) who reported _p = 5.7_n at -60 mV, and V_0.5 = -13 and -24 mV for n and p, respectively (their n is equivalent to the n² reported here; however, their _n was multiplied by 2 to account for the difference in ).

View larger version (30K): [in this window] [in a new window]   FIG. 11. Summary of activation and deactivation of IHT. Activation (V > -45 mV) and deactivation (V < -45 mV) analysis of 16 Type I-c cells. For all cells, Vth > -48 mV, indicating the absence of ILT. A: fast time constant n and its fit to Eq. 7 (—; C = 11 ms-1, V = 24 mV, C = 21 ms-1, V = 23 mV, SF = 100, M = 0.7 ms). B: slow time constant p and its fit to Eq. 7 (—; C = 4 ms-1, V = 32 mV, C = 5 ms-1, V = 22 mV, SF = 100, M = 5 ms). —, the same line in A times 5. C: activation variable n2 () and p () computed from In and Ip data (see Fig. 9C; n denotes n, and p denotes p). —, average Boltzmann functions (Eq. 5) for n2[V0.5 = -14.2 ± 1.2 mV, k = 5.2 ± 1.0 mV (n = 5)] and p [V0.5 = -21.6 ± 1.0 mV, k = 5.8 ± 0.6 mV (n = 7)]. —, from the inset. Inset: activation variables n2 and p computed from tail currents (see Fig. 10A; n denotes n0, and p denotes p0). Data were normalized to Imax values. —, average Boltzmann functions (Eq. 5) for n2 [V0.5 = -16.2 ± 1.8 mV, k = 5.0 ± 0.2 mV (n = 6)] and p [V0.5 = -24.7 ± 4.4 mV, k = 5.8 ± 0.4 mV (n = 5)]. No significant differences were found between the inset tail current data and the activation data in C.

I_HT IS THE SAME AMONGST ALL VCN CELL TYPES. Using the same n² + p kinetic model to analyze the dendrotoxin-insensitive high-threshold current (I_DTX^-) described in our previous paper (Rothman and Manis 2003a), we found few statistical differences between I_DTX^- and I_HT in Fig. 11 (I_DTX^- data from 9 Type II cells and 4 Type I-i cells). Only for a few values of _n near -55 mV were there statistical differences with P < 0.01 (data not shown). Hence, this comparison supports the conclusion that I_HT and I_DTX^- constitute the same current.

Finally, we compared the n² + p deactivation analysis of I_HT in Type I-c cells to that of I_HT in Type I-t cells. Although Type I-t cells possess both I_A and I_HT, deactivating tail currents of I_HT in isolation could be obtained by stepping the membrane to V > -45 mV for 100–150 ms, at which time most of I_A is inactivated (see Fig. 1C) and then stepping to V < -45 mV. Results of this comparison shows that deactivation of I_HT in Type I-c and Type I-t cells is kinetically indistinguishable (data not shown) and therefore probably constitute the same current.

Although it was possible to obtain activating currents of I_HT in isolation in Type I-t cells with a prepulse protocol (V_pp > -45 mV), the resulting current traces had to be fit to Eq. 14 rather than Eq. 15; in this case, there were too many free parameters to obtain a unique solution. Whether activation of I_HT in Type I-t cells is best described by a n² + p model is therefore not known. However, visual inspection of I_HT in Type I-t cells shows clear signs of both a fast and slow component upon activation, indicating two components would be necessary to describe its time course.

K⁺ SELECTIVITY OF I_HT. The reversal potential of I_HT was estimated from tail currents evoked by hyperpolarizing steps from potentials above -45 mV (Fig. 10B). Because I_HT constituted a fast and slow component, denoted I_n and I_p, two reversal potentials were computed: V_n and V_p. Specifically, V_n was estimated at the point where I_n = 0, and V_p where I_p = 0 (Eq. 16; Fig. 10B1). Computed in this way, V_n and V_p were estimated at -68.1 ± 2.1 and -67.4 ± 1.1 mV, respectively (n = 5 Type I-c cells). The same analysis of I_HT in Type I-t cells resulted in similar estimates: -68.3 ± 1.0 and -63.6 ± 2.1 mV (n = 14 Type I-t cells). As did the same analysis for I_DTX^-: -70.6 ± 1.6 and -68.8 ± 3.8 mV (n = 5 Type II cells and 4 Type I-i cells; extracellular solution contained 10 or 100 nM DTX). Hence, both V_n and V_p are estimated near -70 mV, suggesting I_n and I_p are carried by K⁺.

Interestingly, while the above estimates of V_n and V_p are consistently near -70 mV for each cell type, they are not exactly the same. The small differences in average values are due to small differences in V_n and V_p on a cell-by-cell basis. This was sometimes apparent in the tail currents of I_HT, in which case the fast component reversed sign before the slow component, or visa versa (not shown). This also suggests I_n and I_p are independent K⁺ currents.

NUMERICAL RECONSTRUCTION OF I_HT. From the preceding analyses, the following kinetic model of I_HT was developed

(17)

In this equation, _HT is the maximum conductance, which we leave as a free parameter since the magnitude of I_HT tended to vary from cell to cell. Parameter V_K is the reversal potential of I_HT, set to -70 mV. Parameter n² is the fast activation variable with steady-state value n² described by the Boltzmann function in Fig. 11C, and time constant _n described by the bell-shaped curve in Fig. 11A. Parameter p is the slow activation variable with steady-state value p described by the Boltzmann function in Fig. 11C and time constant _p described by the bell-shaped curve in Fig. 11B. The fractional amplitude factor is set to 0.85, a value derived from the activation and deactivation analyses described in the preceding text.

Model current traces of I_HT are shown in Fig. 12A (_HT = 150 nS). The voltage-clamp protocol is similar to that in Figs. 9A and 10A and therefore can be directly compared to experimental I_HT. Fig. 12, B and C, shows the model's fast (I_n) and slow (I_p) components in isolation. Hence, a simple comparison between model I_HT and experimental I_HT clearly shows the sum of I_n and I_p describes I_HT better than I_n in isolation. A model in which I_p is simply scaled would also provide a poor description of I_HT because the tail currents would decay at a much slower rate than the experimental tail currents of Type I and II cells.

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