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Volume 16, Number 13, Issue of July 1, 1996 pp. 4089-4101
Copyright ©1996 Society for Neuroscience

The Role of K+ Currents in Frequency-Dependent Spike Broadening in Aplysia R20 Neurons: A Dynamic-Clamp Analysis

Minghong Ma1, 3, 4 and John Koester1, 2, 4

1 Center for Neurobiology and Behavior, 2 Department of Psychiatry, and 3 Department of Physiology and Cellular Biophysics, College of Physicians and Surgeons, Columbia University, and 4 The New York State Psychiatric Institute, New York, New York 10032

ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
FOOTNOTES
REFERENCES

ABSTRACT

The R20 neurons of Aplysia exhibit frequency-dependent spike broadening. Previously, we had used two-electrode voltage clamp to examine the mechanisms of this spike broadening (Ma and Koester, 1995). We identified three K+ currents that mediate action-potential repolarization: a transient A-type K+ current (IAdepol), a delayed rectifier current (IK-V), and a Ca2+-sensitive K+ current (IK-Ca). A major constraint in that study was the lack of completely selective blockers for IAdepol and IK-V, resulting in an inability to assess directly the effects of their activation and inactivation on spike broadening. In the present study, the dynamic-clamp technique, which employs computer simulation to inject biologically realistic currents into a cell under current-clamp conditions (Sharp et al., 1993a,b), was used either to block IAdepol or IK-V or to modify their inactivation properties.

The data in this paper, together with earlier results, lead to the following hypothesis for the mechanism of spike broadening in the R20 cells. As the spike train progresses, the primary responsibility for spike repolarization gradually shifts from IAdepol to IK-V to IK-Ca. This sequence can be explained on the basis of the relative rates of activation and inactivation of each current with respect to the constantly changing spike durations, the cumulative inactivation of IAdepol and IK-V, and the progressive potentiation of IK-Ca. Positive feedback interactions between spike broadening and inactivation contribute to the cumulative inactivation of both IAdepol and IK-V. The data also illustrate that when two or more currents have similar driving forces and partially overlapping activation characteristics, selectively blocking one current under current-clamp conditions can lead to a significant underestimate of its normal physiological importance.

Key words: spike broadening; dynamic clamp; K+ current; IAdepol; inactivation; Aplysia; R20

INTRODUCTION

Frequency-dependent spike broadening, an endogenously generated increase in spike duration that increases as a function of firing rate, has been shown in neurons from a variety of species to be correlated with enhanced transmitter release (Gillette et al., 1980; Coates and Bulloch, 1985) and to result primarily from inactivation of K+ currents (Aldrich et al., 1979a,b; Jackson et al., 1991; Bielefeldt et al., 1992; Crest and Gola, 1993; Quattrocki et al., 1994). Previously we had described the facilitatory synaptic effects and the mechanisms of frequency-dependent spike broadening in the two electrically coupled peptidergic R20 neurons of Aplysia (Ma and Koester, 1995). In addition, to analyze the mechanisms underlying spike broadening, several voltage-activated currents were isolated from the R20 cells by conventional voltage-clamp methods. These included a Na+ current (INa), a multi-component Ca2+ current (ICa), and three K+ currents---a high-threshold transient A-type current (IAdepol), a delayed rectifier current (IK-V), and a two component Ca2+-sensitive K+ current (IK-Ca). By using a tape-recorded train of broadening spikes as the command signal for the voltage clamp, we found that IAdepol, which is the largest outward current during a single nonbroadened spike, undergoes cumulative inactivation during such a train; IK-V exhibits bimodal changes, increasing during the early part of a train and decreasing during the latter part; IK-Ca facilitates throughout the train; and although the amplitude of ICa decreases during the train, its time integral increases. We hypothesized that the inactivation of IAdepol is the motive force driving spike broadening in the R20 cells and that most changes in the other currents occur secondary to increasing spike width (Ma and Koester, 1995). We further postulated that IK-V affects primarily the kinetics of broadening, whereas IK-Ca affects both the rate and extent of broadening. The goals of this study were to test directly the hypothesized roles of these three K+ currents in spike broadening in the R20 cells and to determine how their relative rates of activation and inactivation affect the voltage-mediated interactions between them.

Four complementary methods have been developed to study the role of voltage-gated membrane conductances in shaping the electrical activity patterns of a neuron. One approach, to block individual currents pharmacologically (Byrne et al., 1979; Schwindt, 1992; Tierney and Harris-Warrick, 1992), often is limited by a lack of specific blockers. For example, in the R20 cells, no known blockers are completely selective for IAdepol or IK-V. A second approach is to simulate the electrical activity of the cell by using mathematical descriptions of all of its voltage-gated membrane conductances. Then, individual conductances can be modified during the simulation (Byrne, 1980a,b; Golowasch et al., 1992; De Schutter and Bower, 1994). A drawback to this method is the possible difficulty in obtaining accurate descriptions of several voltage-sensitive conductances. A third approach is to express artificially the channel in a neuron in which it does not normally appear (Kaang et al., 1992) and then to observe the resultant changes in excitability properties. This approach is quite demanding technically. Moreover, as our results confirm, the functional role of a voltage-gated conductance is highly dependent on the other types of voltage-gated conductances in the neuron, because of interactions via changes in membrane potential (McCormick and Huguenard, 1992; Ducreux and Puizillout, 1995). A fourth method is the dynamic-clamp technique.

The dynamic clamp uses computer simulation to introduce, eliminate, or modify voltage-gated conductances in biological neurons (Sharp et al., 1993a,b). The resultant changes in excitability then provide a direct readout of the role of the simulated conductance. This method has the advantage over the standard simulation method (above) that only the specific current being simulated needs to be accurately modeled. The remaining currents are calculated and generated automatically by the membrane. The kinetics, voltage sensitivity, or ion selectivity of the conductance can be modified, and the resultant effects on excitability can be determined.

In this study we used the dynamic clamp to block IAdepol and/or IK-V or to modify their inactivation in the R20 neurons. The resultant changes in spike broadening can be explained by considering the kinetics of IAdepol, IK-V, and IK-Ca and the complex set of interactions among them that are mediated via changes in action-potential shape.

MATERIALS AND METHODS

Preparations. Aplysia californica (100-250 gm) were supplied by Marinus (Long Beach, CA). Each animal was anesthetized by the injection of isotonic MgCl2, and the abdominal ganglion was dissected out and desheathed after being bathed in artificial sea water (ASW) containing 0.5% glutaraldehyde for 45 sec to reduce spontaneous contractions of the sheath muscle. The R20 cells were identified as described previously (Alevizos et al., 1989). The main axon branches of the R20 cells were truncated by cutting short the branchial nerve and the commissure between the two hemiganglia to improve the space clamp of the R20 somata. Then the ganglion was allowed to recover for at least 1 hr before each experiment, which was performed at 15 ± 1°C.

Pharmacology. The following compounds were used: tetrodotoxin (TTX) (Calbiochem, San Diego, CA); Tris (pH 7.6), CdCl2, 4-aminopyridine (4-AP), and tetraethylammonium (TEA) (Sigma Chemical, St. Louis, MO). Each pharmacological blocker was applied by injecting a 100-fold concentrated bolus into the recording chamber directly by pipette. The composition of ASW was as follows (in mM): 460 NaCl, 10 KCl, 11 CaCl2, 55 MgCl2, 2.5 NaHCO3, and 10 Tris.

The various ionic currents isolated as difference currents were INa (blocked by 60 µM TTX), total ICa (blocked by 2 mM CdCl2 after INa and all K+ currents had been blocked), IAdepol (blocked by 1 mM 4-AP after IK-V and IK-Ca had been blocked), IK-V (blocked by 40 mM TEA after IK-Ca had been blocked), and IK-Ca (blocked by 3-4 mM TEA). See Ma and Koester (1995) for a discussion of the selectivity of these drugs on the R20 cells.

Voltage clamp. Standard two-electrode voltage-clamp and current-clamp methods were implemented using an Axon Instruments Axoclamp 2A (Foster City, CA), as described previously (Ma and Koester, 1995). In voltage-clamp mode, rectangular voltage steps combined with pharmacological blockers were used to identify IAdepol and IK-V conductances in the R20 cells and to determine the parameters for simulating them with the dynamic clamp.

In some experiments, tape-recorded action-potential trains were played back as commands for the voltage clamp in order to determine directly how individual currents change during action-potential trains (for details, see Ma and Koester, 1995). In short, a train of gradually broadening action potentials evoked by injecting brief current pulses into the soma of an R20 cell was tape-recorded under current clamp in normal ASW. Then, with the same cell voltage-clamped, the action-potential train was played back as the command signal for the voltage clamp. When the command was repeated before (control) and after adding a specific channel blocker to the bath, the difference currents obtained by subtracting the currents with blocker from the control currents gave the waveforms of the specific currents that contributed to action-potential generation in current-clamp mode. The action-potential trains were 7 Hz for ~10 sec in all experiments. Every other spike and its corresponding current waveform are plotted in each figure where data from spike trains are presented. Duration of an action potential was defined as the time from the peak to the midpoint of its falling phase. All group data are expressed as mean ± SEM.

Empirical determination of IAdepol and IK-V. Both IAdepol and IK-V were simulated by the dynamic clamp using standard Hodgkin-Huxley-type equations (Hodgkin and Huxley, 1952) of the form:
I<SUB>x</SUB> = G<SUB>max</SUB> m<SUP>4</SUP>h(V<SUB>m</SUB> − E<SUB>rev</SUB>) (1)
m(t + &Dgr;t) = m<SUB>∞</SUB> − (m<SUB>∞</SUB> −m(t))exp(−&Dgr;t/&tgr;<SUB>m</SUB>) (2)
h(t + &Dgr;t) = h<SUB>∞</SUB> − (h<SUB>∞</SUB> − h(t))exp(−&Dgr;t/&tgr;<SUB>h</SUB>), (3)
in which Gmax is the conductance when all channels of type x are open, Erev is the reversal potential for Ix, minfinity and hinfinity are the steady-state values of the activation and inactivation variables at a given potential, Delta t is the time step used in the integration, and t is the time at the end of the last integration step. The time constants tau m and tau h and steady-state values minfinity and hinfinity at different voltage levels were obtained by conventional voltage-clamp protocols. In short, Ix (obtained as a pharmacological difference current) was activated by a series of depolarizing steps, the rising phases of the current were fit with fourth-power exponential functions to calculate the activation time constant (tau m), and the decay phases were fit with single exponential functions to give the inactivation time constants (tau h). (Deviations of IAdepol or IK-V from typical Hodgkin-Huxley kinetics and the resultant approximations introduced into our model are described in Results.) For voltage steps that were too negative to activate Ix, the deactivation time constants (tau m) were determined by tail-current analysis. This protocol also gave the reversal potential for Ix and revealed a linear relationship between the instantaneous tail currents and voltage in the range from -50 to +50 mV for both IAdepol and IK-V. The time constants for recovery from inactivation (tau h) were determined by a two-pulse protocol (see inset in Fig. 3A). The maximum activation of the conductance underlying Ix (Gmax) and the voltage dependence of the activation variable (minfinity ) were examined by plotting peak conductance (Gp) versus membrane potential. Gp was calculated as: Gp = Ipeak/(Vm - Erev), in which Ipeak was measured by extrapolating the exponential decay of Ix to t = 0, the start of the pulse. The resulting conductances (Gp) and the voltages (Vm) were fit to a Boltzmann function of the form: Gp = Gmax/ 1 + exp((Vm - V1/2)/k), to estimate the maximum conductance, in which V1/2 is the potential when Gp becomes half of Gmax and k is the slope factor of the curve. The value of minfinity at each membrane potential was calculated from equation 1 by setting h to its value at t = 0 and substituting Ipeak for Ix. The voltage dependence of steady-state inactivation of Ix (hinfinity ) was estimated by a two-pulse protocol: a 100 msec test pulse to +30 mV was preceded by 2 sec pulses to a prepulse potential, which on different trials ranged from -80 to 0 mV. The value of Gp during each test pulse, normalized to its value after a -80 mV-conditioning pulse, was plotted against the corresponding prepulse potential and fitted to a Boltzmann function, which described the curve of hinfinity versus voltage.

Fig. 3. A mathematical model, which accurately simulates IAdepol, illustrates the role of cumulative inactivation in the evolution of IAdepol traces during a spike train. A, The recovery from inactivation of recorded IAdepol has two time constants. A two-pulse protocol (inset) was used to determine the recovery from inactivation caused by the first 100 msec pulse from -50 to +30 mV. The peak current during the second pulse, normalized to that during the first one (filled circles), was plotted against the time interval between the two pulses. The curve (smooth line) was fit by a double exponential function; the major component had a tau h value of 0.9 sec, whereas tau h for the minor component was 18.5 sec. To simulate IAdepol, the recovery from inactivation was approximated by using just the faster time constant, which was 1.1 sec at -50 mV, averaged from four experiments. The accuracy of this approximation was tested by using the same double-pulse protocol (inset) to drive the dynamic-clamp circuit in open-loop mode. The peak current calculated during the second pulse, normalized to that during the first pulse (open circle), was plotted against the time interval between two pulses, which was fit by a single exponential curve (dotted line). B, The model simulates IAdepol during rectangular depolarizing steps. B1, Empirically measured IAdepol in response to 200 msec depolarizing steps in 10 mV increments from -40 to +30 mV from a holding potential of -50 mV. B2, IAdepol was simulated by driving the dynamic-clamp circuit in open-loop mode with the same set of depolarizing steps as those used in B1. C, The simulated IAdepol traces during action-potential trains were quite similar to those recorded as 4-AP difference current. C1, An action-potential train evoked by injecting brief, depolarizing current pulses into the soma was recorded for use as a command signal. The holding potential was -40 mV. C2, IAdepol was recorded as a 4-AP difference current under voltage clamp while playing back the spike train recorded in C1. C3, C4, IAdepol was simulated by playing back the action-potential train from C1 to the dynamic-clamp circuit in open-loop mode with two different versions of the model. C3, IAdepol waveforms that were simulated with the unmodified version of the model matched the empirically determined values in C2. C4, When noninactivating IAdepol was simulated by fixing h = 1, the same command led to greatly enhanced values of IAdepol. C5, Changes in peak values of recorded and simulated IAdepol from C2-C4 are plotted against spike number. IAdepol was measured as a 1 mM 4-AP difference current after 60 µM TTX, 2 mM CdCl2, and 40 mM TEA had been added to block INa, IK-Ca, and IK-V. The value of Gmax used in the simulations C3, C4 was 1900 nS. The first and last traces in C are indicated by the white and black arrows, respectively.
[View Larger Version of this Image (27K GIF file)]

The rate factors of alpha m (activation), beta m (deactivation), alpha h (recovery from inactivation), and beta h (inactivation) at each value of membrane potential were calculated from time constants of activation and inactivation and steady-state value of m and h, according to: alpha m = minfinity /tau m, beta m = (1 - minfinity )/tau m, alpha h = hinfinity /tau h, and beta h = (1 - hinfinity )/tau h. Then alpha m, beta m, alpha h, and beta h were fit by appropriate versions of the expression (a + bV)/c + exp((d + V)/f), with a, b, c, d, and f as adjustable parameters that were used to represent the different rate factors in the dynamic-clamp program. The dynamic-clamp software computed the current flow through each type of channel by integrating equations 2 and 3 and plugging the outputs into equation 1.

Dynamic clamp protocols. The setup for dynamic clamping is illustrated in Figure 1 (Sharp et al., 1993a,b). Dclamp 2.0 simulation software (Dyna-Quest Technologies, Sudbury, MA) running on a Tangent 486, 33 MHz computer connected to the preparation via a TL-1 DMA interface (Axon Instruments) was used to simulate either IAdepol or IK-V. The sampling rate for simulating a single conductance by dynamic clamp was 5 kHz. Parameters for the Hodgkin-Huxley-type equations describing the time and voltage dependence of the IAdepol and IK-V conductances, which had been determined empirically in other R20 cells, were entered into the dynamic-clamp program. Any one of three kinds of membrane potential signal (Vm) could be used as input to the dynamic-clamp program: an (off-line) tape-recorded action-potential train (V1), a rectangular voltage step from a waveform generator (V2), or membrane potential recorded from the neuron on-line (V3). On the basis of the Vm signal and equations 1-3 (above) describing the desired conductance, the current i (IAdepol or IK-V) flowing through the conductance was computed by the dynamic clamp and expressed as a voltage signal (Vi) proportional to i. In open-loop mode (position 1 or 2), the dynamic clamp was used simply to calculate off-line the current flowing in response to a series of voltage steps or during a spike train. For these simulations, Vi was tape recorded for later playback and analysis. In closed-loop (on-line) mode (position 3), when the dynamic clamp was used to inject current (i) into the cell, Vi was used to control the injected current by driving the current-generating amplifier of the Axoclamp, and Vi, a signal proportional to i, was tape recorded for later playback and analysis. All voltage and current data were recorded on a modified VCR (PCM Data Recorder; A. R. Vetter, Rebersburg, PA) (sampling rate 22 kHz per channel) and on a Gould Brush 2400 chart recorder. The data played back from the tape recorder were analyzed by pClamp 6.0 software (Axon Instruments), which sampled the analog output of the tape recorder at 10 kHz per channel. All the current traces were filtered at 500 Hz by an RC software filter in pClamp 6.0. Output from the pClamp software was printed on a laser printer.

Fig. 1. The dynamic clamp was used in open-loop mode to simulate current waveforms off-line in response to predetermined voltage commands consisting of either (1) a train of recorded spike waveforms, (2) rectangular voltage steps, or (3) in closed-loop mode to inject, block, or modify currents on-line.
[View Larger Version of this Image (29K GIF file)]

To block endogenous IAdepol (or IK-V) in the R20 cells, the dynamic clamp computed IAdepol (or IK-V) with reversed polarity and injected it into the cell. To block the inactivation of IAdepol (or IK-V), two currents were injected simultaneously: one was reversed IAdepol (or IK-V) to block the endogenous IAdepol (or IK-V) through the membrane, and the other was IAdepol (or IK-V) with normal polarity but with the h inactivation variable fixed at a value of 1.0.

The kinetic parameters for the dynamic-clamp equations describing the voltage- and time-dependent behavior of IK-V and IAdepol were determined from group data obtained in separate experiments, but Gmax for each of the two currents varied significantly between cells. Therefore, we did not use Gmax values averaged from several cells to calculate Gmax for a given experiment in which the K+ current of an R20 cell was modified on-line by the dynamic clamp. Rather, in each such experiment, several values of Gmax were used for a given protocol. These steps bracketed the average value from group data for each current (1700 nS for IAdepol, 2100 nS for IK-V) in steps of ± 10%. Then, at the end of the experiment, we measured Gmax for each current directly. Gmax was determined by measuring 40 mM TEA difference currents for IK-V and 1 mM 4-AP difference currents for IAdepol for large depolarizing steps that maximally activated each conductance. In this procedure, INa, ICa, and IK-Ca were preblocked by 60 µM TTX and 2 mM CdCl2 to improve space clamp and to eliminate the nonspecific effects of TEA and 4-AP on IK-Ca (see below). The data for the dynamic-clamp run in which Gmax most closely matched the Gmax value measured at the end of the experiment were chosen for further analysis. The one exception to this protocol was an experiment in which IK-V, IAdepol, and IK-Ca were blocked at the start of the experiment with 50 mM TEA and 10 mM 4-AP (see Fig. 10). In this experiment, Gmax for IAdepol was adjusted such that the first spike duration matched that of the control. Then Gmax for IK-V was adjusted such that broadening with IAdepol and IK-V added back roughly matched the broadening pattern seen when IK-Ca was blocked pharmacologically (see Fig. 14 in Ma and Koester, 1995).

Fig. 10. Addition of either IAdepol or IK-V with the dynamic clamp was sufficient to cause spike broadening in a cell in which K+ currents had been preblocked pharmacologically. An action-potential train was evoked: A, In normal ASW. B, After 50 mM TEA and 10 mM 4-AP had been added to block IK-V, IK-Ca, and IAdepol. C, With both IAdepol and IK-V added back. D, With IAdepol added back. E, With IK-V added back. F, With IK-V, modified to express only the slow, nonstate-dependent inactivation, added back. Resting potential was -47 mV. Gmax for IAdepol, 1500 nS; Gmax for IK-V, 1500 nS (n = 6).
[View Larger Version of this Image (31K GIF file)]

In experiments in which conductances were added to a cell, the results were relatively insensitive to the value of Gmax chosen, with values that varied by as much as ±50% in some cases providing qualitatively similar results. Blocking a conductance with the dynamic clamp was much more sensitive to the value of Gmax chosen, however, as inadvertent overblock of either of the K+ conductances resulted in a negative-slope conductance that destabilized the cell (Sharp et al., 1993b).

RESULTS

Voltage-clamp data revealed changing current waveforms correlated with spike broadening

R20 cells undergo an unusually high degree of frequency-dependent spike broadening. When fired in a train of 60 spikes at fixed frequencies ranging from 1 to 10 Hz, the duration of the action potentials in R20 neurons increases in a range from twofold, at 1 Hz, to a maximum broadening of 5- to 10-fold in the range of 4-8 Hz (Ma and Koester, 1995). In studying the mechanisms of spike broadening in the R20 cells, we previously had combined conventional voltage-clamp techniques with various channel blockers to isolate five major ionic currents. Figure 2 illustrates the results obtained by using a tape-recorded train of broadening action potentials as the command signal for the voltage-clamp circuit for generating various pharmacological difference currents, as described in Materials and Methods (see also Ma and Koester, 1995). INa inactivated only slightly during a spike train and did not contribute to spike broadening directly. The peaks of ICa showed modest inactivation, but the duration of Ca2+ influx increased throughout the train, causing the time integrals of ICa to facilitate by about 2.5- to 4.0-fold. The prolonged ICa maintained the shoulders of the broadened spikes. IAdepol, which was the largest outward current during a single nonbroadened spike, underwent steadily increasing inactivation during such a train. This cumulative inactivation was postulated to be the critical factor leading to spike broadening. IK-V exhibited bimodal changes that were proposed to influence the kinetics of spike broadening in two ways: IK-V increased during the early part of a train, slowing spike broadening, and decreased during the latter part, thereby accelerating broadening. The time integrals of IK-Ca facilitated throughout the train, opposing spike broadening. This increased IK-Ca, along with the gradual inactivation of ICa, limited the maximal extent of broadening.
Fig. 2. Characteristic patterns of changes in individual currents accompany frequency-dependent spike broadening. A 7 Hz, 9.3 sec action-potential train (left, first row) was evoked by injecting brief, depolarizing current pulses into a cell with a resting potential of -48 mV. The five major ionic currents mediating the spike train were isolated by playing back the spike train as the command to the conventional voltage clamp. Between each repetition of the train, the following sequence of blocking drugs was added, one at a time, to the bath: 60 µM TTX, 3 mM TEA, 40 mM TEA, 1 mM 4-AP, or 2 mM CdCl2. The resulting difference currents measured before and after adding each compound revealed INa, IK-Ca, IK-V, IAdepol, and ICa, respectively. The five major currents during the first spike, the 27th spike (at which point IK-V reached its peak), the 43rd spike (at which point the maximum broadening was reached), and the last spike (65th) are shown. The normalized values of spike duration and of the time integrals of each current are plotted against spike number in the right-hand column.
[View Larger Version of this Image (17K GIF file)]

Although these experiments allowed us to determine how various currents change during a spike train, we were not able to use conventional methods to block all of the currents individually and determine the resulting effects on spike shape. Although selective blockers exist for IK-Ca (BAPTA or EGTA injection or low-dose external TEA), the other outward currents (IAdepol and IK-V) could not be blocked in isolation because of the side effects of the available blockers. However, both IAdepol and IK-V could be blocked specifically by a certain drug if the other currents affected by that compound had been preblocked by another compound. We used the following protocol to measure the voltage sensitivity and kinetics of each current. IAdepol was measured as a 1 mM 4-AP difference current. Because 1 mM 4-AP also can increase a slow, outward current in the R20 cells, which appears to be IK-Ca (Ma and Koester, 1995), we preblocked IK-Ca and IK-V by adding 2 mM CdCl2 (to block ICa) and 40 mM TEA before measuring the baseline current. Likewise, IK-V was isolated as a 40 mM TEA difference current. Because 40 mM TEA also blocks IK-Ca, we again had to preblock IK-Ca by adding 2 mM CdCl2. Descriptions of the reversal potentials, kinetics, and voltage sensitivities of the IAdepol and IK-V conductances measured in this way were programmed into the dynamic-clamp software, which was used in later experiments to add, subtract, or modify the IAdepol and IK-V conductances in living R20 cells.

The dynamic clamp can simulate IAdepol

The first step in implementing the dynamic-clamp protocol is to obtain voltage-clamp data that can be used to create a model to simulate the current in question. IAdepol could be modeled accurately by the classical Hodgkin-Huxley-type equations (see Materials and Methods) with two minor deviations. First, the time course of inactivation varied somewhat across cells: in most cells, it was well fit by a single exponential, whereas, in others, a relatively minor component of the current decayed at a somewhat slower exponential rate. Because the dynamic-clamp software we used cannot simulate a current with more than one time constant of inactivation, the rate of decay of IAdepol at each voltage step was approximated by a single exponential, with time constant tau h, over the voltage range from -20 to +50 mV. Second, time course of recovery from inactivation at -50 mV was described best by a double exponential function (Fig. 3A; Furukawa et al., 1992); the major component of recovery had a time constant of 1.1 ± 0.2 sec, although there was a much smaller component that had a time constant of 19.5 ± 2.8 sec (n = 4). Again, only the faster time constant at each voltage was used to approximate the kinetics of inactivation of IAdepol (Fig. 3A). The data from conventional voltage-clamp experiments gave the following results for IAdepol: Erev = -73.0 ± 1.7 mV (n = 5); Gmax = 1700 ± 250 nS (n = 8); alpha m = 300/(0.9 + exp((-6 + Vm)/(-15))); beta m = 300/ (3 + exp ((50 + Vm)/12)); alpha h = 1.8/exp((62 + Vm)/20); and beta h = 8.5/(0.43 + exp((20 + Vm)/(-5))); V1/2 = -11.5 and -34.0 mV for m4 and h, respectively.

The accuracy of the mathematical model of IAdepol was tested in two ways. First, it was used to simulate IAdepol waveforms in response to 200 msec depolarizing steps from -40 to +30 mV from a holding potential of -50 mV. The results were quite similar to the IAdepol waveforms recorded as 1 mM 4-AP difference currents (Fig. 3B1,B2). Second, the values of IAdepol during an action-potential train were simulated. The simulated currents were similar, but not identical, to those isolated pharmacologically. Although the calculated IAdepol underwent cumulative inactivation similar to that shown by 1 mM 4-AP difference currents measured during a train of broadening spikes (Fig. 3C1), the final extent of inactivation of simulated IAdepol at the end of the train was a bit less than that for the empirically measured IAdepol (Fig. 3C2,C3). The reason was presumed to be that IAdepol was simulated with only the fast-time constant for recovery from inactivation instead of two time constants, a fast one as well as a slow one (Fig. 3A). The lack of a major discrepancy between recorded and simulated currents suggests that the fast recovery process accounts for the bulk of the cumulative inactivation during a train, whereas the slow component plays a relatively minor role.

The importance of inactivation in shaping IAdepol waveforms during a spike train was determined by blocking the inactivation of simulated IAdepol currents while using the recorded spike train from Fig. 3C1 as the input to the dynamic clamp. Blocking inactivation increases the peak of simulated IAdepol during the first spike in the train by only 1.3-fold ± 0.1 (n = 4) (compare first traces in Fig. 3C3 and C4), but as the train progresses and spike durations increase, the differences between the currents with or without inactivation increase dramatically. With inactivation intact, the gradual accumulation of IAdepol channels in the inactivated state causes a progressive decrease in peak IAdepol (Fig. 3C3). With inactivation blocked, peak IAdepol increases during the train as the voltage command signals (spike waveforms) get longer and longer (Fig. 3C4). The maximum potentiation of IAdepol caused by removing its inactivation reaches 27.0-fold (± 3.9; n = 4) at the end of the train (Fig. 3C5). This build-up of amplitude potentiation caused by removing inactivation occurs because the first spikes in the train are too brief to allow the IAdepol activation variable to approach its steady-state value (minfinity ) at the peak of the spike. This potentiation of modified (noninactivating) IAdepol demonstrates that the degree of progressive inactivation that normally occurs for native IAdepol during a train is even greater than one would conclude by simply observing the decline in IAdepol peaks (Fig. 3C5).

The dynamic clamp can simulate IK-V

The empirical determination of parameters describing the kinetics and voltage sensitivity of the IK-V channels was based on protocols similar to those described for IAdepol, with one important exception. Brief depolarizing steps were found to be more effective in inactivating IK-V than was a long step of duration equivalent to that of the sum of the short steps (Fig. 4A1,A2). A similar phenomenon has been described previously for delayed rectifier currents in neurons of various species (Aldrich et al., 1979a,b; Marom and Levitan, 1994; Quattrocki et al., 1994; Baukrowitz and Yellen, 1995). The excess inactivation that results when a long depolarization is broken up into an equivalent train of short pulses is sometimes called ``cumulative inactivation'' (Aldrich, 1981). It is attributed to a state-dependent phenomenon, based on the coupling of activation and inactivation variables. (We use the term ``cumulative inactivation'' in a more generic sense to refer to any build-up in inactivation that occurs when repeated voltage pulses are applied, without implying a particular type of kinetic scheme.)
Fig. 4. A mathematical model, which accurately simulates IK-V, illustrates the role of cumulative inactivation in the evolution of IK-V traces during a spike train. A, IK-V exhibits state-dependent inactivation, which causes it to inactivate more rapidly in response to several brief pulses than to one long pulse of equivalent duration. Depolarizing steps were from a holding potential of -50 mV to +20 mV. A1, IK-V inactivated relatively slowly during a 2 sec depolarizing step. A2, IK-V during four 500 msec pulses repeated at 1 Hz decayed to a lower final level than in A1. The dotted line in A1 marks the corresponding level of IK-V at the end of four brief voltage steps in A2. B, Recorded and simulated IK-V were similar during 50 msec repetitive depolarizing pulses (-10 to +50 mV) at 7 Hz from a holding potential of -50 mV. B1, IK-V was measured empirically as a TEA difference current using the voltage-step protocol shown in the inset. B2, IK-V was simulated by driving the dynamic-clamp circuit in open-loop mode. C, The simulated waveforms of IK-V during action-potential trains were similar to those recorded as TEA difference currents. C1, An action-potential train evoked by injecting brief depolarizing current pulses was recorded for use as a command signal. The holding potential was -50 mV. C2, IK-V was recorded as a TEA difference current under voltage clamp while playing back the spike train recorded in C1. C3-C5, IK-V was simulated by playing back the action-potential train from C1 to the dynamic-clamp circuit in open-loop mode with three different versions of the model. C3, Simulated IK-V had normal inactivation. C4, Noninactivating IK-V was simulated by fixing h = 1. C5, The state-dependent nature of IK-V inactivation was ignored; tau h was estimated from the decay during long (2 sec) voltage steps (A1) rather than from high-frequency trains of steps as in B. C6, Peak values of recorded and simulated IK-V from C3-C5 are plotted against spike number. IK-V was measured as a 40 mM TEA difference current after 60 mM TTX and 2 mM CdCl2 had been added to block INa, ICa, and IK-Ca. The value of Gmax used for the simulations in C was 1850 nS. The first and last traces in C are indicated by the white and black arrows, respectively.
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State-dependent coupling of activation and inactivation variables cannot be modeled by the classical Hodgkin-Huxley model encoded in the Dclamp 2.0 software. As a result, inactivation rates determined during a long depolarizing step cannot simulate accurately the kinetics of inactivation during a train of relatively brief action potentials. Instead, we used inactivation rates measured during a train of brief voltage steps (Fig. 4B1), which occurred at the same rate as the spikes in a train, to provide a Hodgkin-Huxley-type approximation of the state-dependent inactivation of IK-V. In other words, the inactivation time constants of IK-V (tau h) were not estimated from the decay of exponential curves during long pulses but were approximated by fitting the decay of the peak currents during a 7 Hz train of repeated 50 msec pulses (-10 to 50 mV) from a holding potential of -50 mV (inset of Fig. 4B). With this estimate of IK-V inactivation rates, the dynamic clamp could simulate rather accurately the empirically determined changes in IK-V during frequency-dependent spike broadening. This approach required sacrificing the accuracy of IK-V simulations during long voltage-clamp steps (>100 msec), which are not physiologically relevant in this case. With this modification in protocol, the kinetics and voltage sensitivity of the IK-V currents were fit by the equations described in Materials and Methods: Erev = -62.0 ± 3.1 mV (n = 3); Gmax = 2100 ± 370 nS (n = 9); alpha m = (53 + 0.22 Vm)/(0.65 + exp((-5 + Vm)/(-13))); beta m = (3.4 - 0.06Vm)/exp ((-10 + Vm)/65); alpha h = 1/exp((143 + Vm)/30); and beta h = 1.7/(0.83 + exp((7.4 + Vm)/(-6.7))); V1/2 = 0.1 and -33.0 mV for m4 and h, respectively.

The accuracy of the mathematical model of IK-V was checked in two ways. First, the simulated waveforms of IK-V in response to four 7 Hz, 50-msec-depolarizing repetitive steps (-10 to +50 mV) from a holding potential of -50 mV were found to be quite similar to the values of IK-V measured empirically as 40 mM TEA difference currents (Fig. 4B). Second, the simulated IK-V waveforms during an action-potential train matched quite closely the waveforms measured empirically (Fig. 4C1-C3). The importance of using inactivation kinetics determined from high-frequency pulses (Fig. 4B1) was tested by trying an alternative kinetic model that used values of tau h for IK-V that were determined by measuring the decay rate during a series of long (2 sec) voltage steps. When control action-potential waveforms were used to drive simulations based on such a conventional, nonstate-dependent model of IK-V inactivation, the computed IK-V waveforms deviated considerably from the empirically measured difference currents. The simulated currents showed much greater facilitation during the train, as well as a much lower level of inactivation at the end of the train (compare Fig. 4C3 and C5).

The role of inactivation in shaping IK-V waveforms during a spike train was determined by modifying the inactivation of simulated IK-V currents while using the recorded spike train from Figure 4C1 as the input to the dynamic clamp. Blocking inactivation has no significant effect on the simulated IK-V during the first spike in the train (compare Fig. 4C3 to C4), but as the train progresses, the differences between the currents with or without inactivation rapidly increase. In both cases, as the spikes begin to broaden during the train, peak IK-V increases as the spike waveforms get longer and longer, allowing the channels to approach more closely full activation. When inactivation is intact, however, this potentiating trend is overwhelmed rapidly by the gradual cumulative inactivation of IK-V channels, which causes a progressive decrease in peak IK-V, beginning about a third of the way into the train (Fig. 4C3). In contrast, with inactivation blocked, peak IK-V increases throughout the train as the spike waveforms get longer and longer (Fig. 4C4). Only near the end of the train, as spike amplitude and duration decrease slightly because of potentiation of IK-Ca and inactivation of ICa (Ma and Koester, 1995), does IK-V begin to decrease slightly. The early onset of differences in IK-V patterns under the two conditions indicates that inactivation of IK-V plays an important role in shaping IK-V behavior even at the beginning of the control train, when IK-V is increasing from spike to spike. It also demonstrates that the degree of progressive inactivation of normally inactivating IK-V during a train is even greater than one would conclude simply by observing the decline in control IK-V peaks (Fig. 4C6). Blocking inactivation causes a much greater potentiation of IK-V than of IAdepol in this protocol, because the rapid activation kinetics of IAdepol allow it to approach much more closely full activation during the first spike of the train than does IK-V, which activates relatively slowly. Potentiation measured as the ratio of the peak of the largest current trace to the peak of the first trace in the train was 10.6-fold ± 1.7 (n = 4) for IK-V but only 1.8 ± 0.4 (n = 4) for IAdepol (compare Figs. 3C5, 4C6).

Both IAdepol and IK-V control spike broadening

The dynamic clamp was used to study the effects of selectively blocking IAdepol or IK-V on spike broadening and to study the effects of blocking the inactivation of either current both on spike broadening and on the dynamics of the other current. Pharmacological methods are not adequate for such manipulations, because the known blockers of the two types of ion channel are nonselective (Hermann and Gorman, 1981a,b; Ma and Koester, 1995), and there are no known pharmacological blockers of inactivation of these channels. Blocking activation or inactivation of IAdepol modulates spike broadening Previously we had examined the roles of the three voltage-activated K+ currents in spike repolarization (Ma and Koester, 1995). Our data showed that blocking both IK-V and IK-Ca with TEA so that IAdepol was the only voltage-activated outward current remaining had no significant effect on the width of a single spike. When 1 mM 4-AP was added subsequently, it blocked IAdepol selectively, because the previous addition of TEA had eliminated nonselective potentiation of IK-Ca from 4-AP. Under these conditions, pharmacological block of IAdepol broadened a single spike by ~10-fold (Ma and Koester, 1995). These results suggested that, of the three outward currents, IAdepol is the only one to affect significantly the repolarization rate for a single spike. Moreover, in the same study, cumulative inactivation of IAdepol was found to correlate well with spike broadening during a high-frequency train (see also Figs. 2,3). One might expect, therefore, that selective block of IAdepol would cause the first spike of a train to broaden to near its maximal value, but further broadening during the train would be negligible. In the present study, however, applying 1 mM 4-AP alone to block IAdepol made a single spike only 2.2-fold (±0.2) wider and 9.6% (±1.3%) larger (n = 8), and major spike broadening still occurred later in the train (data not shown). Given that 4-AP also can facilitate IK-Ca (Hermann and Gorman, 1981a), the relatively minor effects of blocking IAdepol on spike width could result from a countervailing enhancement of IK-Ca. In fact, late in a train when IK-Ca reaches its peak value (Fig. 2 in Ma and Koester, 1995), broadening was more pronounced in controls than in cells treated with 4-AP (data not shown). These results suggest that 4-AP may indeed have potentiated IK-Ca. To eliminate this possibility, we repeated this experiment by using the dynamic clamp, rather than 4-AP, to block IAdepol. We found that selectively blocking IAdepol with the dynamic clamp had results on a single spike similar to those produced by 4-AP---~two-fold broadening (Fig. 5A1,A2, 6A). Unlike 4-AP block, it showed no tendency to reduce broadening late in the train. We therefore restricted our analyses of the effect of blocking IAdepol to experiments in which the dynamic clamp was used to cancel it out.
Fig. 5. Blocking IAdepol activation (or inactivation) with the dynamic clamp accelerated (or eliminated) spike broadening, which in turn modified the dynamics of IK-V activation and inactivation. A, Action-potential trains were evoked under control conditions (A1), after blocking IAdepol (A2), or after blocking inactivation of IAdepol (A3). The dynamic clamp was used in closed-loop mode either to simply cancel out IAdepol (A2) or to replace it with a noninactivating version of IAdepol (A3). B, Simulated IK-V changed during a spike train recorded while IAdepol (or its inactivation) was blocked. B1, Simulated IK-V during a control spike train showed bimodal changes. B2, Simulated IK-V during an action-potential train recorded while IAdepol was blocked underwent cumulative inactivation from the increased initial value. B3, Simulated IK-V during a spike train recorded while inactivation of IAdepol was blocked underwent only modest cumulative inactivation. A4, B4, Changes in spike width (A1-A3) and peak values of simulated IK-V (B1-B3) caused by modifying IAdepol are plotted against spike number. Resting potential was -50 mV. Gmax used by the dynamic clamp for blocking or modifying IAdepol was 1650 nS. Gmax used for simulating IK-V in B1-B3 was 2050 nS.
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Given our earlier results on the importance of IAdepol, why did blocking IAdepol have such a small effect on initial spike width and subsequent broadening during a train (Figs. 5A, 6A)? One possibility is that IK-V and IK-Ca, which normally activate relatively slowly, had sufficient time to activate more fully as a result of the twofold spike broadening caused by blocking IAdepol. As a result of such increased activation, IK-V and IK-Ca could become the dominant outward currents underlying spike repolarization. In that case, cumulative inactivation of IK-V and a parallel decrease of IK-Ca could mediate the subsequent spike broadening. (Although the time integrals of IK-Ca increased overall during a spike train (Fig. 2), the peaks of IK-Ca did show a transient dip during the initial part of the train (Figs. 6B1,B3; Fig. 9 in Ma and Koester, 1995), perhaps because of the inactivation of Ca2+ channels nearest the IK-Ca channels.) This hypothesis for the enhanced role of IK-V and IK-Ca was tested by using two sets of action-potential trains, which had been recorded either under control conditions or while IAdepol was blocked by the dynamic clamp. For IK-V, these recorded spike trains were used as commands to the dynamic clamp, which simulated the IK-V waveforms off-line in response to the two different spike trains (Fig. 5). For IK-Ca, the same recorded spike trains were used as commands to the voltage clamp, and 3 mM TEA was applied to isolate IK-Ca during the two different spike trains (Fig. 6). As predicted, (1) the simulated IK-V (Fig. 5B) and the measured IK-Ca (Fig. 6B) activated much more completely during the first spike compared with the first spike when IAdepol was normal; (2) IK-V underwent cumulative inactivation during spike trains recorded when IAdepol was blocked (Fig. 5B2,B4); and (3) peaks of IK-Ca decreased during the early part of the spike train, as though the initial transient dip were enhanced (Fig. 6B2,B3).

Fig. 6. Blocking IAdepol activation with the dynamic clamp accelerated spike broadening, which, in turn, enhanced the activation of IK-Ca and modified its kinetics. A, The spike trains recorded either under control conditions (A1) or with IAdepol activation blocked (A2) by the dynamic clamp were used as commands to the voltage clamp for isolating IK-Ca pharmacologically (B1, B2). B, IK-Ca was measured as 3 mM TEA difference currents under control condition (B1) and after blocking IAdepol (B2). A3, B3, The durations of the spikes of the two trains (A1, A2) and the corresponding peaks of IK-Ca currents (B1, B2) are plotted against spike number. Resting potential was -50 mV, and Gmax used for the IAdepol simulations (A2) was 1350 nS.
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Fig. 9. Using the dynamic clamp to block activation or inactivation of IAdepol or of IK-V caused changes in frequency-dependent spike broadening that were measured in the same cell. A, Action-potential trains generated with K+ conductances in various different states. A1, Control; A2, with IAdepol blocked; A3, with IK-V blocked; A4, with IAdepol and IK-V blocked; A5, with inactivation of IAdepol blocked; A6, with inactivation of IK-V blocked. B, Durations of spikes in A1-A6 are plotted against spike number. Resting potential was -50 mV. Currents were blocked or modified by using Gmax for IAdepol = 1750 nS and Gmax for IK-V = 1650 nS.
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We had postulated that in control conditions inactivation of IAdepol is critical for spike broadening, because IAdepol is normally the largest outward current during a single spike and because its cumulative inactivation mirrors the increase in spike duration (see also Fig. 2; Ma and Koester, 1995). This hypothesis was supported by an experiment in which the inactivation of IAdepol was blocked by the dynamic clamp. Under these conditions spike broadening was negligible (n = 14). As a result of spike width staying constant during such a train, the simulated peak values of IK-V did not facilitate. Instead, they exhibited only steadily increasing inactivation during a train of nonbroadened spikes (Fig. 5A3,B3,B4; see also below). Although the model used to simulate IK-V can only approximate its complex inactivation kinetics, similar results were obtained when IK-V was measured directly as a TEA difference current in a similar protocol (Fig. 12 in Ma and Koester, 1995).

An important positive feedback interaction between spike broadening and IK-V inactivation was revealed by examining the simulated IK-V currents under two different conditions. When IAdepol inactivation was blocked, thereby preventing spike broadening, the simulated IK-V currents reached a maximal cumulative inactivation of only ~50% (Fig. 5B3). When broadening was allowed to occur, IK-V inactivated to ~12% of its maximum value earlier in the train (Fig. 5B2).

Blocking activation or inactivation of IK-V modulates spike broadening IK-V normally starts out small for the first spike in the train and then increases, before eventually decreasing toward its initial value late in the train (see also Fig. 2; Ma and Koester, 1995). Thus, we predicted that blocking IK-V with the dynamic clamp would have relatively little effect on the first or last spikes in the train but would increase the rate of broadening during the early and middle parts of the train, when IK-V normally increases in amplitude. As expected, blocking IK-V had negligible effects on the durations of the first and last spikes in the train, but it accelerated the process of spike broadening dramatically (Figs. 7A1,A2). Just as blocking IAdepol could affect the behavior of IK-V via its effects on spike shape, blocking IK-V likewise influenced IAdepol waveforms. The increase in average spike duration during a train of spikes generated with IK-V blocked resulted in cumulative inactivation of IAdepol that was faster than that in control conditions (Fig. 7B1,B2,B4).
Fig. 7. Blocking IK-V activation (or inactivation) with the dynamic clamp accelerated (or reduced) spike broadening, which, in turn, modified the rate and extent of IAdepol inactivation. A, Action-potential trains were evoked under control conditions (A1), after blocking IK-V (A2), or after blocking inactivation of IK-V (A3). The dynamic clamp was used in closed-loop mode either to simply cancel out IK-V (A2) or to replace it with a noninactivating version of IK-V (A3). B, Compared with control (B1), simulated IAdepol underwent faster inactivation during a spike train with IK-V blocked (B2) or slower and less complete inactivation with inactivation of IK-V blocked (B3). A4, B4, Changes in spike width (A1-A3) and peak values of IAdepol (B1-B3) caused by modifying IK-V are plotted against spike number. Resting potential was -50 mV. Gmax used by the dynamic clamp for blocking or modifying IK-V was 1700 nS. Gmax of IAdepol used for simulating IAdepol in B1-B3 was 1950 nS.
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Next we investigated whether blocking inactivation of IK-V would affect spike broadening during a train. The amplitude of IK-V during a single spike is relatively small compared with IAdepol (Fig. 2), and blocking it completely has no effect on the duration of a single spike (Fig. 7; Fig. 13 in Ma and Koester, 1995). One might therefore predict that eliminating its inactivation would have a negligible effect on spike broadening. On the contrary, blocking inactivation of IK-V with the dynamic clamp reduced maximal spike broadening during a train by ~75% (Fig. 7A1,A3,A4). This ability of blocking IK-V inactivation to limit strongly the spike broadening can be explained as follows. Under normal conditions, IK-V becomes a major component of repolarizing current in the early-to-middle portion of the train, because cumulative inactivation of IAdepol causes longer spike durations, which in turn generate enhanced activation of IK-V. Eventually, IK-V amplitudes begin to decline later in the train, as cumulative IK-V inactivation outstrips the increase in IK-V activation (Figs. 2, 5B; Fig. 8 in Ma and Koester, 1995). As a result of the gradual rise and fall of IK-V during the train, blocking its inactivation can add significantly to the repolarizing drive of the cell, thereby limiting the broadening process.

Fig. 8. Replacing the rapid, state-dependent inactivation of IK-V by a slower, Hodgkin-Huxley-type version reduced both the rate and final extent of spike broadening. A, An action-potential train was evoked by injecting brief depolarizing current pulses under control conditions. B, An action-potential train was evoked after eliminating the state-dependent inactivation of IK-V, which was achieved by injecting two currents into the cell simultaneously: IK-V with normal state-dependent inactivation but with reversed sign to block the existing IK-V of the cell and IK-V with only slow inactivation and normal polarity to replace the endogenous current. C, The durations of spikes for these two trains are plotted against the number of spike. Resting potential was -48 mV. Gmax used by the dynamic clamp for blocking or modifying IK-V was 2100 nS.
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Simulation of IAdepol waveforms with the dynamic clamp driven by the spike trains recorded with IK-V in various different functional states revealed an important positive feedback interaction between inactivation of IAdepol and spike broadening. There was a positive correlation between mean-spike duration during a train and rate of inactivation of simulated IAdepol under three conditions: (1) inactivation of IK-V blocked, (2) normal IK-V and (3) activation of IK-V blocked (Fig. 7A4,B4). Because IAdepol-gating properties were not modified in this series of experiments, the three different patterns of its inactivation must be secondary to changes in action-potential waveforms caused by changing IK-V. These data indicate that normally, as inactivation of IAdepol during the course of a train contributes to spike broadening, the broadening in turn allows additional inactivation of IAdepol to occur, thereby further enhancing broadening. Any other process that contributes to broadening, such as IK-V inactivation, would be amplified by this positive feedback effect.

How would the balance between increasing activation and inactivation of IK-V during a train be shifted if IK-V did not undergo state dependent inactivation? This question was examined by comparing spike broadening under two conditions: (1) with native IK-V inactivation kinetics, or (2) with the IK-V conductance replaced by a modified version in which IK-V inactivation parameters are fit by the current decay during a single, long voltage step, rather than by the state-dependent inactivation kinetics approximated from peak values of IK-V measured during a train of brief steps (Fig. 4A,B). The nonstate-dependent value of tau h is slower than the state-dependent version, and substituting it in the dynamic-clamp model of IK-V limits cumulative inactivation of IK-V and reduces by 22% (±1.4%; n = 4) the amount of spike broadening that occurs (Figs. 4C5, 8). Thus the enhanced inactivation resulting when depolarization is broken up into short pulses potentiates the rate and extent of broadening.

IAdepol and IK-V contribute differently to spike broadening

To compare the relative effects of IAdepol and IK-V on spike broadening directly, we used the dynamic clamp to block or to modify the two currents in the same cell. As described above, blocking IAdepol caused a doubling in width of the first spike in the train (Fig. 9A1,A2,B). In contrast, blocking IK-V had no effect on the width of the first spike but caused an acceleration of broadening in the first half of the train (Fig. 9A1,A3,B). When both IAdepol and IK-V were blocked simultaneously, the effects on spike width were variable. In most cells (9 of 14), the first spike in the train was not as wide as the maximally broadened spikes in a normal train (Fig. 9A4,B). As the train progressed, the spikes broadened significantly, presumably because of the transient decay of IK-Ca (Fig. 6B3). Spike width then decayed as IK-Ca facilitated, and ICa underwent progressive inactivation (compare Figs. 2 and 6). In other cells (5 of 14), the first spike in the train was broader than the widest spike in a normal train. In these cells, with both IAdepoland IK-V blocked, IK-Ca cells have a limited ability to take over as the major repolarizing influence early in the train.

The relative effects of blocking inactivation of either IK-V or IAdepol also were examined in the same cell (n = 14). As expected from results described above (Figs. 5, 7), blocking inactivation of IAdepol completely blocked broadening, whereas blocking inactivation of IK-V blocked most, but not all, of the broadening (Fig. 9A5,A6). These results further support the hypothesis that cumulative inactivation of IAdepol is essential for initiating spike broadening, and cumulative inactivation of IK-V is essential to allow broadening to reach its full extent.

Simulated IAdepol and IK-V are sufficient to cause spike broadening

To test further our conclusions about the relative roles of IAdepol and IK-V in broadening, we blocked the entire complement of depolarization-activated K+ currents pharmacologically and then added back IAdepol and/or IK-V with the dynamic clamp to see how they affected broadening. After first recording a control spike train, we blocked IAdepol, IK-V, and IK-Ca by addition of 50 mM TEA and 10 mM 4-AP. With the outward currents blocked in this way, the first spike in the train was broadened maximally, and no significant change in spike duration occurred throughout the train (Fig. 10A,B). While maintaining the pharmacological block, IAdepol, IK-V, or the two together were added back to the cell. Unlike the experiments described above, we did not measure the values of Gmax for IAdepol and IK-V in this set of experiments. Instead, the average Gmax of both currents determined in earlier experiments was used as a starting point for the dynamic-clamp program (1700 nS for IAdepol and 2100 nS for IK-V). Then several different runs were made in which these initial values were bracketed by values of Gmax that varied in steps of ±20%. We found that a wide range of Gmax values for the two currents gave qualitatively similar results. The maximum value used was twice as large as the minimum value for both currents, i.e., Gmax of IAdepol ranged from 1300 to 2600 nS and Gmax of IK-V ranged from 1500 to 3000 nS.

Adding back both IAdepol and IK-V resulted in a spike train similar to that recorded under control conditions. The main difference was that, with IK-Ca still blocked by TEA, broadening at the end of the train, when IK-Ca normally has its maximal effect, was slightly greater than in the control (compare Fig. 10A and C). The incomplete inactivation of simulated IAdepol at the end of a train (Fig. 3C3) seems to work against this extra broadening, as indicated by the fact that the protocol in Fig. 10C enhances broadening less than does blocking IK-Ca by BAPTA injection (Fig. 14A1,A2 in Ma and Koester, 1995). Adding back IAdepol alone resulted in a spike train in which the first spike had a normal duration, and broadening occurred much more rapidly than in the control condition (Fig. 10A,D), because the normally potentiating IK-V (Fig. 4C2) as well as IK-Ca still were blocked pharmacologically. Restoring IK-V alone resulted in a train in which the width of the first spike was threefold longer than the first control spike (Fig. 10A,E). This enhanced duration, with IK-Ca and IAdepol blocked, is larger than that seen by blocking only IAdepol (Figs. 5A, 6A), confirming the hypothesis that IK-Ca can play a significant role in repolarization of a single spike if IAdepol is blocked (Fig. 6). If the IK-V that was added back had the slow time constant of inactivation measured from long pulses, rather than the state-dependent inactivation tau h approximated by the brief pulse-train protocol, broadening was reduced (Fig. 10F). This difference between the results achieved with the Hodgkin-Huxley versus the state-dependent model of inactivation (see also Fig. 8) indicates that the relatively rapid state-dependent IK-V inactivation mechanism plays an important role in allowing inactivation of this current to build up during a spike train. Overall, the results obtained by adding back IAdepol or IK-V with the dynamic clamp were consistent with those obtained with the complementary approach of blocking the currents with the dynamic clamp.

DISCUSSION

Application of the dynamic-clamp method to the R20 neurons

There were several technical limitations in this study. The space clamp was imperfect, as significant neuritic stumps remained connected to the soma. Using different concentrations of the same drug (TEA) to block either IK-Ca or IK-V quite likely resulted in a minor contamination of the difference currents obtained. The software used could not simulate two time constants for inactivation or recovery from inactivation (for IAdepol) or state-dependent inactivation (for IK-V), so these processes had to be approximated. The value of Gmax used by the program had to be estimated for each individual cell from measurements made at the end of the experiment. Despite these limitations, the results obtained were highly consistent from preparation to preparation, and there was a high degree of internal consistency when results were compared from current-clamp, voltage-clamp, and dynamic-clamp protocols, in both this and the earlier study (Ma and Koester, 1995). All of these approaches provided data that fit a unified interpretation of spike broadening summarized in Figure 11.
Fig. 11. Summary of the predominant mechanisms underlying spike broadening in the R20 neurons, determined in this and the previous study (Ma and Koester, 1995). Some interactions that play a minor role in spike broadening are not included in this diagram. For example, spike broadening also facilitates the activation of IAdepol (Fig. 3C4). This effect is overwhelmed by the progressive increase in inactivation, however. Broadening also enhances cumulative inactivation of ICa (Ma and Koester, 1995). This effect has two opposing effects on spike broadening: by reducing inward current it limits broadening, and by reducing the build-up of IK-Ca, it enhances broadening.
[View Larger Version of this Image (25K GIF file)]

Relative roles of IAdepol, IK-V, and IK-Ca in spike repolarization

Several lines of evidence indicate that IAdepol is the major current responsible for repolarization of a single spike in the R20 cells. During the first spike in a train, IAdepol is the largest and most rapidly activating of the three K+ currents (Fig. 2). Moreover, with IK-V and IK-Ca blocked, IAdepol is sufficient to cause normal repolarization of a single spike, but blocking all three K+ currents caused ~10- to 15-fold spike broadening (Fig. 10A,B; Fig. 13 in Ma and Koester, 1995). Likewise, if IK-V, IK-Ca, and IAdepol all are blocked pharmacologically, adding back IAdepol with the dynamic clamp restores normal spike width (Fig. 10A,B,D). These results suggest that IAdepol is the major current responsible for spike repolarization. However, blocking IAdepol alone causes only a twofold increase in spike duration (Figs. 5A, 6A). The cause of this surprisingly small effect of blocking IAdepol on spike width was revealed by simulating IK-V and measuring IK-Ca during the spike trains recorded with IAdepol blocked. These data showed that IK-V amplitude is increased two- to threefold in response to twofold spike broadening, thereby allowing it to become a major repolarizing influence (Fig. 5B2,B4). This enhanced activation occurs because IK-V activation is slow compared with normal spike duration (compare Fig. 4B and C). IK-Ca also activates relatively slowly (Fig. 5 in Ma and Koester, 1995). It is potentiated 2.5- to fourfold by twofold spike broadening and likewise contributes to limiting spike broadening when IAdepol is blocked (Fig. 6). These results illustrate how, in a system with several voltage-gated conductances, the effect produced by blocking a single current (e.g., IAdepol) can underestimate significantly the physiological effect of that current (cf. McCormick and Huguenard, 1992). In this case, blockade of IAdepol is compensated largely by the emergence of two latent currents, IK-V and IK-Ca.

Cumulative inactivation of IAdepol, potentiated by a positive feedback interaction with spike broadening, is essential for initiation of broadening

Several lines of evidence confirm that IAdepol inactivation is essential for spike broadening. (1) It is the largest outward current during the first spike in the train, but as the train progresses, it inactivates to 0-10% of its initial value (Figs. 2, 3C2, 7B1). (2) Changes in the other voltage-activated currents during the train are insufficient to explain broadening either qualitatively or quantitatively (Fig. 2; Ma and Koester, 1995). (3) Keeping spike width constant reduces only partially the progressive decrease in IAdepol amplitudes during a train, indicating that a significant portion of its progressive inactivation is not secondary to broadening but, rather, acts as a primary cause of broadening (Fig. 12D in Ma and Koester, 1995). (4) When the other voltage-gated outward currents (IK-V and IK-Ca) are blocked pharmacologically, either endogenous IAdepol or simulated IAdepol that is generated by the dynamic clamp can mediate frequency-dependent broadening (Fig. 10D; Fig. 14A3 in Ma and Koester, 1995). (5) Blocking inactivation of IAdepol completely eliminates frequency-dependent broadening (Figs. 5A3, 9A5).

Unlike other neurons in which the mechanisms of frequency-dependent spike broadening have been examined in detail (Aldrich et al., 1979a,b; Quattrocki et al., 1994), the progressive inactivation of the critical current in the R20 cells, IAdepol, is not attributable primarily to state-dependent, cumulative inactivation. It has been proposed that the AKv1.1a gene product forms the ion channels for native IAdepol measured in Aplysia neurons (Pfaffinger et al., 1991). Furukawa (1995) described pronounced state-dependent, cumulative inactivation of the AKv1.1a channels expressed in frog oocytes. When brief depolarizing pulses from a holding potential of -50 mV were repeated at 0.1 Hz, the evoked AKv1.1.a current pulses declined during the train in a manner that could be fit best by a kinetic model with state-dependent inactivation (Furukawa, 1995). We observed no such build-up of inactivation of IAdepol in the R20 cells using the identical protocol. Higher frequency (7 Hz) trains of depolarizing voltage steps, designed to mimic the spike trains used in this study, did produce modest cumulative inactivation of IAdepol from step to step in the R20 cells. However, this progressive inactivation could be simulated by the conventional Hodgkin-Huxley-type kinetic scheme described above without invoking state-dependent processes (data not shown). This difference in inactivation properties may indicate that the AKv1.1a gene product is not a component of the IAdepol channels. More likely, given the other strong similarities between native IAdepol and IAKv1.1a (Pfaffinger et al., 1991; Kaang et al., 1992), they differ in their inactivation properties either because of differences in ion composition or post-translational modification between oocytes and Aplysia neurons or because the AKv1.1a channels lack unidentified beta -subunits or heterologous alpha -subunits that may be present in native IAdepol channels (Sheng et al., 1993; Rettig et al., 1994).

The mechanism by which IAdepol inactivation builds up during a train can be explained by the relation of IAdepol channel kinetics to spike duration and interspike interval. Because of its rapid inactivation rate, significant, though not complete, inactivation of IAdepol can occur during a single spike (Fig. 3C5). The time constant of recovery of IAdepol from inactivation is on the order of 1 sec at -50 mV. Thus, for a 7 Hz train, a significant amount of inactivation that occurs during each spike will persist into the onset of the succeeding spike and beyond. Over the course of a 10 sec high-frequency train, one would expect the balance between inactivation and recovery from inactivation to reach an equilibrium within a few seconds if spike duration were constant (Fig. 12 in Ma and Koester, 1995); but empirically IAdepol is found to continue to inactivate throughout the entire train if normal spike broadening is allowed to occur (Figs. 2, 3C). This extended inactivation occurs because spike broadening caused by persistent IAdepol inactivation from previous spikes causes IAdepol in succeeding spikes to undergo more complete inactivation, as h approaches more fully its steady-state value, hinfinity . That is, there is positive feedback between inactivation and broadening. This coupling is illustrated by the positive correlation between spike width and the rate and extent of IAdepol inactivation when spike width is manipulated artificially (Fig. 7; compare Figs. 7C and 12D in Ma and Koester, 1995). Broadening can also increase activation, which would provide a negative feedback force tending to resist broadening; because IAdepol activates quickly with respect to spike rise time, increasing spike duration has only a relatively modest tendency to cause IAdepol activation to facilitate. Enhanced inactivation is the predominant effect (Fig. 3C5).

Cumulative state-dependent inactivation of IK-V, potentiated by a positive feedback interaction with spike broadening, is essential for maximum broadening

During a spike train, there is a constantly varying interplay between the changes in activation and inactivation of IK-V. Early in the train there is a large scope for facilitation of IK-V in response to the spike broadening that is caused by IAdepol inactivation, because IK-V activates so slowly relative to spike duration. As a result, IK-V increases during the first part of the train (Figs. 2, 4C), and blocking IK-V speeds up broadening (Figs. 7A, 9B; Fig. 14 in Ma and Koester, 1995). As spike width begins to approach its limit, IK-V approaches full activation, so the dominant effect of broadening on IK-V becomes to potentiate inactivation, which is a much slower process than activation. The importance of inactivation dynamics throughout the train is illustrated by the fact that cumulative inactivation contributes to limiting the amplitude of IK-V even as the IK-V peaks are increasing from spike to spike early in the train (Fig. 4C6). Thus, blocking inactivation of IK-V greatly reduces spike broadening (Figs. 7A4, 9A6).

As in the case of IAdepol, there is a positive feedback interaction between spike broadening and inactivation of IK-V. For simulated and recorded IK-V, only a moderate cumulative inactivation occurs when spike broadening is prevented during a train. This inactivation is enhanced greatly when broadening is allowed to occur (Fig. 5A4,B4) (see Fig. 12 in Ma and Koester, 1995).

The state-dependent nature of IK-V inactivation, which causes it to be inactivated more effectively by brief pulses, is important in determining the role of IK-V in spike broadening. When native IK-V currents are substituted by IK-V currents with slower, nonstate-dependent inactivation kinetics, the rate of broadening is slower and less complete than the rate that occurs normally (Fig. 8). Moreover, after blocking all outward currents pharmacologically, adding back such a slowly inactivating current with the dynamic clamp is insufficient to restore significant frequency-dependent broadening (Fig. 10F). In contrast, the model that approximated state-dependent inactivation kinetics generated robust frequency-dependent broadening (Fig. 10E).

Summary and conclusions

The major interactions between currents and membrane potential that determine the dynamics of frequency-dependent spike broadening are summarized in Figure 11. During the first action potential in a train, IAdepol activates relatively rapidly and therefore dominates the repolarization process. However, the inactivation that occurs during each spike accumulates from spike to spike, causing progressive broadening. In addition, because inactivation of IAdepol is relatively slow with respect to spike duration, the broadening has a positive feedback effect on IAdepol inactivation. IK-V and IK-Ca are latent currents that have no significant effect on repolarization during the first spike in the train. The broadening also allows IK-V and IK-Ca, which activate relatively slowly, to turn on more fully during successive spikes in the train, thereby playing a greater role in repolarization. The broader spikes also prolong ICa. This added Ca2+ influx, on the one hand, supports the shoulders of the broadened spikes but, on the other hand, limits spike broadening by elevating cytoplasmic Ca2+, which potentiates IK-Ca. As the train progresses, the enhanced IK-V activation caused by broadening gradually is overwhelmed by progressive build-up of IK-V inactivation, which is amplified by a positive feedback relation between broadening and inactivation. As a result of these complex dynamics, control of repolarization gradually devolves from IAdepol to IK-V to IK-Ca during the course of a high-frequency spike train.

FOOTNOTES

Received March 1, 1996; revised April 16, 1996; accepted April 18, 1996.

  

This work was supported by National Institutes of Health Grant NS14385.

We thank Drs. I. Kupfermann, A. MacDermott, and S. Siegelbaum for comments on this manuscript.

Correspondence should be addressed to Dr. John Koester, Center for Neurobiology and Behavior, New York State Psychiatric Institute, 722 West 168th Street, New York, NY 10032.


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