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

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). I_Nainactivated only slightly during a spike train and did not contribute to spike broadening directly. The peaks of I_Cashowed modest inactivation, but the duration of Ca²⁺ influx increased throughout the train, causing the time integrals ofI_Ca to facilitate by about 2.5- to 4.0-fold. The prolonged I_Ca maintained the shoulders of the broadened spikes. I_Adepol, 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. I_K-V exhibited bimodal changes that were proposed to influence the kinetics of spike broadening in two ways: I_K-V increased during the early part of a train, slowing spike broadening, and decreased during the latter part, thereby accelerating broadening. The time integrals ofI_K-Ca facilitated throughout the train, opposing spike broadening. This increased I_K-Ca, along with the gradual inactivation of I_Ca, limited the maximal extent of broadening.

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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 mmCdCl₂. The resulting difference currents measured before and after adding each compound revealed I_Na,I_K-Ca, I_K-V,I_Adepol, and I_Ca, respectively. The five major currents during the first spike, the 27th spike (at which point I_K-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.

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 forI_K-Ca (BAPTA or EGTA injection or low-dose external TEA), the other outward currents (I_Adepol and I_K-V) could not be blocked in isolation because of the side effects of the available blockers. However, both I_Adepol andI_K-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.I_Adepol 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 beI_K-Ca (Ma and Koester, 1995), we preblockedI_K-Ca and I_K-V by adding 2 mm CdCl₂ (to blockI_Ca) and 40 mm TEA before measuring the baseline current. Likewise, I_K-V was isolated as a 40 mm TEA difference current. Because 40 mm TEA also blocks I_K-Ca, we again had to preblock I_K-Ca by adding 2 mmCdCl₂. Descriptions of the reversal potentials, kinetics, and voltage sensitivities of the I_Adepol andI_K-V conductances measured in this way were programmed into the dynamic-clamp software, which was used in later experiments to add, subtract, or modify theI_Adepol and I_K-Vconductances in living R20 cells.

The dynamic clamp can simulate I_Adepol

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. I_Adepol 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 I_Adepol at each voltage step was approximated by a single exponential, with time constant τ_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 ofI_Adepol (Fig. 3A). The data from conventional voltage-clamp experiments gave the following results for I_Adepol:E_rev = −73.0 ± 1.7 mV (n = 5);G_max = 1700 ± 250 nS (n = 8); α_m = 300/(0.9 + exp((−6 + V_m)/(−15))); β_m = 300/ (3 + exp ((50 + V_m)/12)); α_h = 1.8/exp((62 + V_m)/20); and β_h = 8.5/(0.43 + exp((20 + V_m)/(−5)));V_1/2 = −11.5 and −34.0 mV form⁴ and h, respectively.

The accuracy of the mathematical model ofI_Adepol was tested in two ways. First, it was used to simulate I_Adepol 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 theI_Adepol waveforms recorded as 1 mm4-AP difference currents (Fig. 3B1,B2). Second, the values of I_Adepol during an action-potential train were simulated. The simulated currents were similar, but not identical, to those isolated pharmacologically. Although the calculatedI_Adepol 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 simulatedI_Adepol at the end of the train was a bit less than that for the empirically measured I_Adepol(Fig. 3C2,C3). The reason was presumed to be thatI_Adepol 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 shapingI_Adepol waveforms during a spike train was determined by blocking the inactivation of simulatedI_Adepol currents while using the recorded spike train from Fig. 3C1 as the input to the dynamic clamp. Blocking inactivation increases the peak of simulatedI_Adepol 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 I_Adepolchannels in the inactivated state causes a progressive decrease in peakI_Adepol (Fig. 3C3). With inactivation blocked, peak I_Adepol increases during the train as the voltage command signals (spike waveforms) get longer and longer (Fig. 3C4). The maximum potentiation ofI_Adepol 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 I_Adepol activation variable to approach its steady-state value (m_∞) at the peak of the spike. This potentiation of modified (noninactivating)I_Adepol demonstrates that the degree of progressive inactivation that normally occurs for nativeI_Adepol during a train is even greater than one would conclude by simply observing the decline inI_Adepol peaks (Fig. 3C5).

The dynamic clamp can simulate I_K-V

The empirical determination of parameters describing the kinetics and voltage sensitivity of the I_K-V channels was based on protocols similar to those described forI_Adepol, with one important exception. Brief depolarizing steps were found to be more effective in inactivatingI_K-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.)

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Fig. 4.

A mathematical model, which accurately simulatesI_K-V, illustrates the role of cumulative inactivation in the evolution of I_K-V traces during a spike train. A, I_K-Vexhibits 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,I_K-V inactivated relatively slowly during a 2 sec depolarizing step. A2, I_K-Vduring four 500 msec pulses repeated at 1 Hz decayed to a lower final level than in A1. The dotted line inA1 marks the corresponding level ofI_K-V at the end of four brief voltage steps inA2.B, Recorded and simulatedI_K-V were similar during 50 msec repetitive depolarizing pulses (−10 to +50 mV) at 7 Hz from a holding potential of −50 mV. B1, I_K-V was measured empirically as a TEA difference current using the voltage-step protocol shown in the inset. B2, I_K-V was simulated by driving the dynamic-clamp circuit in open-loop mode.C, The simulated waveforms of I_K-Vduring 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,I_K-V was recorded as a TEA difference current under voltage clamp while playing back the spike train recorded inC1.C3–C5, I_K-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 I_K-V had normal inactivation. C4, NoninactivatingI_K-V was simulated by fixing h = 1. C5, The state-dependent nature ofI_K-V inactivation was ignored; τ_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 I_K-V fromC3–C5 are plotted against spike number.I_K-V was measured as a 40 mm TEA difference current after 60 mm TTX and 2 mmCdCl₂ had been added to block I_Na,I_Ca, and I_K-Ca. The value of G_max used for the simulations in Cwas 1850 nS. The first and last traces in C are indicated by the white and black arrows, respectively.

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 I_K-V. In other words, the inactivation time constants ofI_K-V (τ_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 I_K-V inactivation rates, the dynamic clamp could simulate rather accurately the empirically determined changes inI_K-V during frequency-dependent spike broadening. This approach required sacrificing the accuracy ofI_K-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 I_K-V currents were fit by the equations described in Materials and Methods: E_rev = −62.0 ± 3.1 mV (n = 3); G_max = 2100 ± 370 nS (n = 9); α_m = (53 + 0.22 V_m)/(0.65 + exp((−5 + V_m)/(−13))); β_m = (3.4 − 0.06V_m)/exp ((−10 + V_m)/65); α_h = 1/exp((143 + V_m)/30); and β_h = 1.7/(0.83 + exp((7.4 + V_m)/(−6.7))); V_1/2 = 0.1 and −33.0 mV for m⁴ and h, respectively.

The accuracy of the mathematical model of I_K-Vwas checked in two ways. First, the simulated waveforms ofI_K-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 ofI_K-V measured empirically as 40 mmTEA difference currents (Fig. 4B). Second, the simulatedI_K-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 τ_h for I_K-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 I_K-V inactivation, the computedI_K-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 I_K-Vwaveforms during a spike train was determined by modifying the inactivation of simulated I_K-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 I_K-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 I_K-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 I_K-V channels, which causes a progressive decrease in peak I_K-V, beginning about a third of the way into the train (Fig.4C3). In contrast, with inactivation blocked, peakI_K-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 I_K-Ca and inactivation of I_Ca (Ma and Koester, 1995), doesI_K-V begin to decrease slightly. The early onset of differences in I_K-V patterns under the two conditions indicates that inactivation of I_K-Vplays an important role in shaping I_K-V behavior even at the beginning of the control train, whenI_K-V is increasing from spike to spike. It also demonstrates that the degree of progressive inactivation of normally inactivating I_K-V during a train is even greater than one would conclude simply by observing the decline in controlI_K-V peaks (Fig. 4C6). Blocking inactivation causes a much greater potentiation ofI_K-V than of I_Adepol in this protocol, because the rapid activation kinetics ofI_Adepol allow it to approach much more closely full activation during the first spike of the train than doesI_K-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 I_K-V but only 1.8 ± 0.4 (n = 4) for I_Adepol(compare Figs. 3C5, 4C6).

Both I_Adepol andI_K-V control spike broadening

The dynamic clamp was used to study the effects of selectively blocking I_Adepol or I_K-Von 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 I_Adepol 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 bothI_K-V and I_K-Ca with TEA so that I_Adepol 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 I_Adepol selectively, because the previous addition of TEA had eliminated nonselective potentiation ofI_K-Ca from 4-AP. Under these conditions, pharmacological block of I_Adepol broadened a single spike by ∼10-fold (Ma and Koester, 1995). These results suggested that, of the three outward currents,I_Adepol is the only one to affect significantly the repolarization rate for a single spike. Moreover, in the same study, cumulative inactivation of I_Adepol 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 I_Adepol 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 blockI_Adepol 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 facilitateI_K-Ca (Hermann and Gorman, 1981a), the relatively minor effects of blocking I_Adepol on spike width could result from a countervailing enhancement ofI_K-Ca. In fact, late in a train whenI_K-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 I_K-Ca. To eliminate this possibility, we repeated this experiment by using the dynamic clamp, rather than 4-AP, to blockI_Adepol. We found that selectively blockingI_Adepol 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 blockingI_Adepol to experiments in which the dynamic clamp was used to cancel it out.

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Fig. 5.

Blocking I_Adepol activation (or inactivation) with the dynamic clamp accelerated (or eliminated) spike broadening, which in turn modified the dynamics ofI_K-V activation and inactivation. A, Action-potential trains were evoked under control conditions (A1), after blocking I_Adepol(A2), or after blocking inactivation ofI_Adepol (A3). The dynamic clamp was used in closed-loop mode either to simply cancel outI_Adepol (A2) or to replace it with a noninactivating version of I_Adepol(A3). B, Simulated I_K-Vchanged during a spike train recorded whileI_Adepol (or its inactivation) was blocked.B1, Simulated I_K-V during a control spike train showed bimodal changes. B2, SimulatedI_K-V during an action-potential train recorded while I_Adepol was blocked underwent cumulative inactivation from the increased initial value. B3, SimulatedI_K-V during a spike train recorded while inactivation of I_Adepol was blocked underwent only modest cumulative inactivation. A4, B4, Changes in spike width (A1–A3) and peak values of simulatedI_K-V (B1–B3) caused by modifyingI_Adepol are plotted against spike number. Resting potential was −50 mV. G_max used by the dynamic clamp for blocking or modifying I_Adepolwas 1650 nS. G_max used for simulatingI_K-V in B1–B3 was 2050 nS.

Given our earlier results on the importance ofI_Adepol, why did blockingI_Adepol have such a small effect on initial spike width and subsequent broadening during a train (Figs.5A, 6A)? One possibility is thatI_K-V and I_K-Ca, which normally activate relatively slowly, had sufficient time to activate more fully as a result of the twofold spike broadening caused by blocking I_Adepol. As a result of such increased activation, I_K-V andI_K-Ca could become the dominant outward currents underlying spike repolarization. In that case, cumulative inactivation of I_K-V and a parallel decrease ofI_K-Ca could mediate the subsequent spike broadening. (Although the time integrals ofI_K-Ca increased overall during a spike train (Fig. 2), the peaks of I_K-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 theI_K-Ca channels.) This hypothesis for the enhanced role of I_K-V andI_K-Ca was tested by using two sets of action-potential trains, which had been recorded either under control conditions or while I_Adepol was blocked by the dynamic clamp. For I_K-V, these recorded spike trains were used as commands to the dynamic clamp, which simulated theI_K-V waveforms off-line in response to the two different spike trains (Fig. 5). For I_K-Ca, the same recorded spike trains were used as commands to the voltage clamp, and 3 mm TEA was applied to isolateI_K-Ca during the two different spike trains (Fig. 6). As predicted, (1) the simulated I_K-V(Fig. 5B) and the measured I_K-Ca(Fig. 6B) activated much more completely during the first spike compared with the first spike when I_Adepolwas normal; (2) I_K-V underwent cumulative inactivation during spike trains recorded whenI_Adepol was blocked (Fig. 5B2,B4); and (3) peaks of I_K-Ca decreased during the early part of the spike train, as though the initial transient dip were enhanced (Fig. 6B2,B3).

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Fig. 6.

Blocking I_Adepol activation with the dynamic clamp accelerated spike broadening, which, in turn, enhanced the activation of I_K-Ca and modified its kinetics. A, The spike trains recorded either under control conditions (A1) or withI_Adepol activation blocked (A2) by the dynamic clamp were used as commands to the voltage clamp for isolating I_K-Ca pharmacologically (B1, B2). B, I_K-Ca was measured as 3 mm TEA difference currents under control condition (B1) and after blocking I_Adepol(B2). A3, B3, The durations of the spikes of the two trains (A1, A2) and the corresponding peaks ofI_K-Ca currents (B1, B2) are plotted against spike number. Resting potential was −50 mV, andG_max used for the I_Adepolsimulations (A2) was 1350 nS.

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Fig. 9.

Using the dynamic clamp to block activation or inactivation of I_Adepol or ofI_K-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, withI_Adepol blocked; A3, withI_K-V blocked; A4, withI_Adepol and I_K-V blocked;A5, with inactivation of I_Adepolblocked; A6, with inactivation ofI_K-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 usingG_max for I_Adepol = 1750 nS and G_max for I_K-V = 1650 nS.

We had postulated that in control conditions inactivation ofI_Adepol is critical for spike broadening, because I_Adepol 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 I_Adepol 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 ofI_K-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 I_K-V can only approximate its complex inactivation kinetics, similar results were obtained whenI_K-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 andI_K-V inactivation was revealed by examining the simulated I_K-V currents under two different conditions. When I_Adepol inactivation was blocked, thereby preventing spike broadening, the simulatedI_K-V currents reached a maximal cumulative inactivation of only ∼50% (Fig. 5B3). When broadening was allowed to occur, I_K-V inactivated to ∼12% of its maximum value earlier in the train (Fig. 5B2).

Blocking activation or inactivation of I_K-Vmodulates spike broadening

I_K-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 blockingI_K-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 I_K-V normally increases in amplitude. As expected, blocking I_K-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 blockingI_Adepol could affect the behavior ofI_K-V via its effects on spike shape, blockingI_K-V likewise influencedI_Adepol waveforms. The increase in average spike duration during a train of spikes generated withI_K-V blocked resulted in cumulative inactivation of I_Adepol that was faster than that in control conditions (Fig. 7B1,B2,B4).

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Fig. 7.

Blocking I_K-V activation (or inactivation) with the dynamic clamp accelerated (or reduced) spike broadening, which, in turn, modified the rate and extent ofI_Adepol inactivation. A, Action-potential trains were evoked under control conditions (A1), after blocking I_K-V(A2), or after blocking inactivation ofI_K-V (A3). The dynamic clamp was used in closed-loop mode either to simply cancel outI_K-V (A2) or to replace it with a noninactivating version of I_K-V (A3).B, Compared with control (B1), simulatedI_Adepol underwent faster inactivation during a spike train with I_K-V blocked (B2) or slower and less complete inactivation with inactivation ofI_K-V blocked (B3). A4, B4, Changes in spike width (A1–A3) and peak values ofI_Adepol (B1–B3) caused by modifyingI_K-V are plotted against spike number. Resting potential was −50 mV. G_max used by the dynamic clamp for blocking or modifying I_K-V was 1700 nS. G_max of I_Adepol used for simulating I_Adepol in B1–B3 was 1950 nS.

Next we investigated whether blocking inactivation ofI_K-V would affect spike broadening during a train. The amplitude of I_K-V during a single spike is relatively small compared with I_Adepol(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 I_K-V with the dynamic clamp reduced maximal spike broadening during a train by ∼75% (Fig.7A1,A3,A4). This ability of blockingI_K-V inactivation to limit strongly the spike broadening can be explained as follows. Under normal conditions,I_K-V becomes a major component of repolarizing current in the early-to-middle portion of the train, because cumulative inactivation of I_Adepol causes longer spike durations, which in turn generate enhanced activation ofI_K-V. Eventually, I_K-Vamplitudes begin to decline later in the train, as cumulativeI_K-V inactivation outstrips the increase inI_K-V activation (Figs. 2, 5B; Fig.8 in Ma and Koester, 1995). As a result of the gradual rise and fall of I_K-V during the train, blocking its inactivation can add significantly to the repolarizing drive of the cell, thereby limiting the broadening process.

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Fig. 8.

Replacing the rapid, state-dependent inactivation of I_K-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 I_K-V, which was achieved by injecting two currents into the cell simultaneously:I_K-V with normal state-dependent inactivation but with reversed sign to block the existingI_K-V of the cell and I_K-Vwith 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. G_max used by the dynamic clamp for blocking or modifying I_K-V was 2100 nS.

Simulation of I_Adepol waveforms with the dynamic clamp driven by the spike trains recorded withI_K-V in various different functional states revealed an important positive feedback interaction between inactivation of I_Adepol and spike broadening. There was a positive correlation between mean-spike duration during a train and rate of inactivation of simulatedI_Adepol under three conditions: (1) inactivation of I_K-V blocked, (2) normalI_K-V and (3) activation ofI_K-V blocked (Fig. 7A4,B4). BecauseI_Adepol-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 I_K-V. These data indicate that normally, as inactivation of I_Adepol during the course of a train contributes to spike broadening, the broadening in turn allows additional inactivation ofI_Adepol to occur, thereby further enhancing broadening. Any other process that contributes to broadening, such asI_K-V inactivation, would be amplified by this positive feedback effect.

How would the balance between increasing activation and inactivation ofI_K-V during a train be shifted ifI_K-V did not undergo state dependent inactivation? This question was examined by comparing spike broadening under two conditions: (1) with native I_K-Vinactivation kinetics, or (2) with the I_K-Vconductance replaced by a modified version in whichI_K-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 ofI_K-V measured during a train of brief steps (Fig. 4A,B). The nonstate-dependent value of τ_h is slower than the state-dependent version, and substituting it in the dynamic-clamp model ofI_K-V limits cumulative inactivation ofI_K-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.

I_Adepol and I_K-Vcontribute differently to spike broadening

To compare the relative effects of I_Adepoland I_K-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 I_Adepolcaused a doubling in width of the first spike in the train (Fig.9A1,A2,B). In contrast, blockingI_K-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 bothI_Adepol and I_K-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 ofI_K-Ca (Fig. 6B3). Spike width then decayed as I_K-Ca facilitated, andI_Ca 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 I_Adepoland I_K-Vblocked, I_K-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 eitherI_K-V or I_Adepol also were examined in the same cell (n = 14). As expected from results described above (Figs. 5, 7), blocking inactivation ofI_Adepol completely blocked broadening, whereas blocking inactivation of I_K-V blocked most, but not all, of the broadening (Fig. 9A5,A6). These results further support the hypothesis that cumulative inactivation ofI_Adepol is essential for initiating spike broadening, and cumulative inactivation of I_K-Vis essential to allow broadening to reach its full extent.

Simulated I_Adepol andI_K-V are sufficient to cause spike broadening

To test further our conclusions about the relative roles ofI_Adepol and I_K-V in broadening, we blocked the entire complement of depolarization-activated K⁺ currents pharmacologically and then added back I_Adepol and/orI_K-V with the dynamic clamp to see how they affected broadening. After first recording a control spike train, we blocked I_Adepol, I_K-V, and I_K-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, I_Adepol,I_K-V, or the two together were added back to the cell. Unlike the experiments described above, we did not measure the values of G_max forI_Adepol and I_K-V in this set of experiments. Instead, the average G_max of both currents determined in earlier experiments was used as a starting point for the dynamic-clamp program (1700 nS forI_Adepol and 2100 nS forI_K-V). Then several different runs were made in which these initial values were bracketed by values ofG_max that varied in steps of ±20%. We found that a wide range of G_max 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.,G_max of I_Adepol ranged from 1300 to 2600 nS and G_max ofI_K-V ranged from 1500 to 3000 nS.

Adding back both I_Adepol andI_K-V resulted in a spike train similar to that recorded under control conditions. The main difference was that, withI_K-Ca still blocked by TEA, broadening at the end of the train, when I_K-Ca normally has its maximal effect, was slightly greater than in the control (compare Fig.10A and C). The incomplete inactivation of simulated I_Adepol 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 I_K-Ca by BAPTA injection (Fig. 14A1,A2 in Ma and Koester, 1995). Adding back I_Adepol 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 potentiatingI_K-V (Fig. 4C2) as well asI_K-Ca still were blocked pharmacologically. Restoring I_K-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, withI_K-Ca and I_Adepolblocked, is larger than that seen by blocking onlyI_Adepol (Figs. 5A, 6A), confirming the hypothesis that I_K-Ca can play a significant role in repolarization of a single spike ifI_Adepol is blocked (Fig. 6). If theI_K-V that was added back had the slow time constant of inactivation measured from long pulses, rather than the state-dependent inactivation τ_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-dependentI_K-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 backI_Adepol or I_K-V with the dynamic clamp were consistent with those obtained with the complementary approach of blocking the currents with the dynamic clamp.

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 I_K-Ca orI_K-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 I_Adepol) or state-dependent inactivation (for I_K-V), so these processes had to be approximated. The value of G_max 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.

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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 ofI_Adepol (Fig. 3C4). This effect is overwhelmed by the progressive increase in inactivation, however. Broadening also enhances cumulative inactivation ofI_Ca (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 ofI_K-Ca, it enhances broadening.

Relative roles of I_Adepol,I_K-V, and I_K-Ca in spike repolarization

Several lines of evidence indicate thatI_Adepol is the major current responsible for repolarization of a single spike in the R20 cells. During the first spike in a train, I_Adepol is the largest and most rapidly activating of the three K⁺ currents (Fig. 2). Moreover, with I_K-V andI_K-Ca blocked, I_Adepol 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 I_K-V,I_K-Ca, and I_Adepol all are blocked pharmacologically, adding backI_Adepol with the dynamic clamp restores normal spike width (Fig. 10A,B,D). These results suggest thatI_Adepol is the major current responsible for spike repolarization. However, blocking I_Adepolalone causes only a twofold increase in spike duration (Figs.5A, 6A). The cause of this surprisingly small effect of blocking I_Adepol on spike width was revealed by simulating I_K-V and measuringI_K-Ca during the spike trains recorded withI_Adepol blocked. These data showed thatI_K-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 I_K-V activation is slow compared with normal spike duration (compare Fig. 4Band C). I_K-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 I_Adepol 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., I_Adepol) can underestimate significantly the physiological effect of that current (cf. McCormick and Huguenard, 1992). In this case, blockade ofI_Adepol is compensated largely by the emergence of two latent currents, I_K-V andI_K-Ca.

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

Several lines of evidence confirm thatI_Adepol 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 inI_Adepol 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 (I_K-V andI_K-Ca) are blocked pharmacologically, either endogenous I_Adepol or simulatedI_Adepol that is generated by the dynamic clamp can mediate frequency-dependent broadening (Fig. 10D; Fig. 14A3 in Ma and Koester, 1995). (5) Blocking inactivation ofI_Adepol 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, I_Adepol, 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 I_Adepol measured inAplysia 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 I_Adepol 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 ofI_Adepol 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 I_Adepolchannels. More likely, given the other strong similarities between native I_Adepol andI_AKv1.1a (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 β-subunits or heterologous α-subunits that may be present in native I_Adepol channels (Sheng et al., 1993; Rettig et al., 1994).

The mechanism by which I_Adepolinactivation builds up during a train can be explained by the relation of I_Adepol channel kinetics to spike duration and interspike interval. Because of its rapid inactivation rate, significant, though not complete, inactivation ofI_Adepol can occur during a single spike (Fig.3C5). The time constant of recovery ofI_Adepol 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 I_Adepol 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 persistentI_Adepol inactivation from previous spikes causesI_Adepol in succeeding spikes to undergo more complete inactivation, as h approaches more fully its steady-state value, h_∞. 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 I_Adepol 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; becauseI_Adepol activates quickly with respect to spike rise time, increasing spike duration has only a relatively modest tendency to cause I_Adepol activation to facilitate. Enhanced inactivation is the predominant effect (Fig.3C5).

Cumulative state-dependent inactivation ofI_K-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 ofI_K-V. Early in the train there is a large scope for facilitation of I_K-V in response to the spike broadening that is caused by I_Adepolinactivation, because I_K-V activates so slowly relative to spike duration. As a result, I_K-Vincreases during the first part of the train (Figs. 2, 4C), and blocking I_K-V speeds up broadening (Figs.7A, 9B; Fig. 14 in Ma and Koester, 1995). As spike width begins to approach its limit, I_K-Vapproaches full activation, so the dominant effect of broadening onI_K-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 ofI_K-V even as the I_K-Vpeaks are increasing from spike to spike early in the train (Fig.4C6). Thus, blocking inactivation ofI_K-V greatly reduces spike broadening (Figs.7A4, 9A6).

As in the case of I_Adepol, there is a positive feedback interaction between spike broadening and inactivation ofI_K-V. For simulated and recorded I_K-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 I_K-V inactivation, which causes it to be inactivated more effectively by brief pulses, is important in determining the role of I_K-V in spike broadening. When native I_K-V currents are substituted by I_K-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,I_Adepol 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 I_Adepol is relatively slow with respect to spike duration, the broadening has a positive feedback effect on I_Adepol inactivation.I_K-V and I_K-Ca are latent currents that have no significant effect on repolarization during the first spike in the train. The broadening also allowsI_K-V and I_K-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 I_Ca. 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 potentiatesI_K-Ca. As the train progresses, the enhancedI_K-V activation caused by broadening gradually is overwhelmed by progressive build-up of I_K-Vinactivation, which is amplified by a positive feedback relation between broadening and inactivation. As a result of these complex dynamics, control of repolarization gradually devolves fromI_Adepol to I_K-V toI_K-Ca during the course of a high-frequency spike train.