A Novel Method for the Description of Voltage-Gated Ionic Currents Based on Action Potential Clamp Results—Application to Hippocampal Mossy Fiber Boutons

Introduction

The action potential clamp technique (AP-clamp) is a paradigm in which an AP recorded from a neuron in current clamp is applied to that same cell in voltage clamp mode before and after the addition of a specific ion channel blocker to the external medium (Llinás et al., 1982; Bean, 2007). In this way the role of that current during an AP can be determined. The AP can also be applied computationally to a mathematical model of that current constructed from voltage step results in order to provide an additional test of the model. This approach was recently applied to I_Naand I_Ca in suprachiasmatic nucleus neurons (Jackson et al., 2004; Clay, 2015). In this report the method is applied to the AP-clamp recordings of Alle et al. (2009) of I_Na and I_K in rat hippocampal mossy fiber boutons (MFBs) at physiological temperatures (T = 36−37°C). Those results demonstrate a significant separation in time during an AP of I_Na and I_K, an important result for efficient neuronal signaling (Crotty et al., 2006; Alle et al., 2009; Sengupta et al., 2010). The method was implemented using digitized results from MFBs (personal communication, Dr. H. Alle). A digitized representation of an AP from an MFB was applied computationally to the models of I_K and I_Na in MFBs of Engel and Jonas (2005) for model testing. The I_K AP-clamp analysis revealed a significant discrepancy between theory and experiment, which can be resolved using the Goldman-Hodgkin-Katz (GHK) equation for the fully activated current-voltage relation for I_K (Clay, 2009). The revised I_K model provides a significant improvement in the description of this component compared with a model in which a linear dependence of I_K on (V-E_K) was used. Computational analysis of the I_Na AP-clamp result (Alle et al., 2009) using the Engel and Jonas (2005) I_Na model was in agreement with experiment.

Materials and Methods

A data set corresponding to an AP from MFBs was applied computationally to the models of I_K and I_Na of Engel and Jonas (2005) which are given by I_K = g_Kn⁴(V-E_K) and I_Na = g_Nam³h(V-E_Na), respectively, similar to the original models of I_K and I_Na in squid giant axons of Hodgkin and Huxley (1952), with g_K and g_Naconstants, E_K and E_Na the K⁺, and Na⁺ reversal potentials and

$\begin{array}{rc}\text{d}x∕\text{d}t=-({\alpha }_x+{\beta }_x)x+{\alpha }_x& \left(1\right)\hfill \end{array}$

with x = n, m, or h, and time t in msec. The rate parameters in Equation (1) are given by Engel and Jonas (2005).

$\begin{array}{l}{\alpha }_n=-0.01(V+55)∕\left\{\text{exp}\left[-(V+55)∕10\right]-1\right\};\\ {\beta }_{\text{n}}=0.125\text{ exp}\left[-(V+65)∕80\right]& \left(2\right)\hfill \end{array}$

$\begin{array}{l}{\alpha }_m=-93.8(V-105)∕\left\{\text{exp}[-(V-105)∕17.7)-1\right\};\\ {\beta }_m=0.17\text{ exp}(-V∕23.3)\\ {\alpha }_h=0.00035\text{ exp}(-V∕18.7);\\ {\beta }_h=6.6∕\left\{\text{exp}\left[-(V+17.7)∕13.3\right]+1\right\}.& \left(3\right)\end{array}$

The expressions for α_n and β_n (Equation 2) were taken from the experimental procedures of Engel and Jonas (2005). The expressions for α_m, β_m, α_h, and β_h (Equation 3) were taken from Supplementary Table 1 of their paper. All αs and βs in Equations (2) and (3) are in units of inverse milliseconds. The model of I_K was based on the voltage step recordings of this component in MFBs by Geiger and Jonas (2000) obtained at T = 34°C. It was extrapolated to T = 37°C using a Q₁₀ of 2.2 (Fohlmeister, 2015). That is, α_n and β_n (Equation 2) were each multiplied by 1.27. The reversal potential for K⁺ used in the analysis was E_K = −110 mV. [E_K = kT/q ln(${\text{K}}_{\text{o}}^{+}$/${\text{K}}_{\text{i}}^{+}$) where k is the Boltzmann constant, T is the absolute temperature, q is the unit electron charge (kT/q = 26.7 mV for T = 37°C), ${\text{K}}_{\text{o}}^{+}$=2.5 mM and ${\text{K}}_{\text{i}}^{+}$ = 155 mM (Alle et al., 2009)]. The recordings of I_Na of Engel and Jonas (2005) were obtained at T = 23°C. Their model of I_Na was extrapolated to T = 37°by multiplying each of the αs and βs in Equation (3) by a factor of 2.8. The reversal potential for Na⁺ was E_Na = 62 mV (Alle et al., 2009).

The V_i vs. t_i data set (i = 1,2,3…  ; Supplementary Table 1) of the MFB AP from Figure 1B of Alle et al. (2009) is represented in the top panel of Figure 1 with lines connecting adjacent points. It was applied to Equation (1) with x = n, m, or h using NDSolve in Mathematica (Wolfram Research, Inc., Champaign, IL). At t₁ = 0, V₁ = −80 mV. The initial value of n, n₁, was given by α_n/(α_n+β_n) with V = −80 mV and α_nand β_n given by Equation (2), i.e., n₁ = 0.1288. The following point of the AP data set is t₂ = 0.039 ms, V₂ = −77.7 mV. The corresponding value of n, n₂, was determined from Equation (1) and NDSolve using V(t) = V₁+(V₂−V₁)(t−t₁)/(t₂−t₁) for t₁< t< t₂. The result was n₂ = 0.1289. More significant changes in n occur later as the membrane potential is depolarized throughout the AP. A similar analysis was applied to x = m and h.

FIGURE 1

Figure 1. Top panel: This AP corresponds to the AP in the top panel of Figure 1B from Alle et al. (2009). A digitized version of this waveform is given in Supplementary Table 1. The points in that data set were connected by lines to given the result shown. Bottom panel: The data points with error bars (n = 9; ±SEM) correspond to AP-clamp measurements of I_K in MFBs (Alle et al., 2009). The mean values of these results are also given in Supplementary Table 1. The theoretical results (models 1 and 2) are as described in the text.

Results

K⁺ Current

A digital representation of the AP-clamp recordings of I_K from Alle et al. (2009) is given in the bottom panel of Figure 1 of this report (data points with the error bars representing ±SEM, n = 9; Supplementary Table 1). These results are the differences obtained by application of the AP in Figure 1 to MFBs in voltage-clamp before and after addition to the bath of 1 mM 4-aminopyridine (4-AP), which was sufficient to completely block I_K elicited by an AP (Alle et al., 2009, 2011). They were scaled to match the I_K result in the bottom panel of Figure 1B of Alle et al. (2009). Also shown in Figure 1 is the prediction of the Engel and Jonas (2005) I_K model (dashed line; model 1) starting from the maximum level of I_K close to the peak of the AP. The rising phase of the model is not shown. These results (also given in Supplementary Table 1) correspond to I_K,_i = g_Kn_i⁴(V_i-E_K), i = 1.2.3… , with n_i determined as described above (Section Materials and Methods), E_K = −110 mV, and g_K = 36 mS/cm². This model—model 1, the Engel and Jonas (2005) I_K model—does not provide a good description of the I_K AP-clamp result. The difficulty most likely concerns the fully activated current-voltage relation for I_K. This result for squid axons is well described by the GHK dependence on (V-E_K) rather than by (V-E_K) itself (Clay et al., 2008), a relationship given by I_K(n = 1) = aGHK[(V-E_K)] where a is a constant, and GHK[(V-E_K)] = (qV/kT) {exp[q(V-E_K)/kT]-1}/[exp(qV/kT)-1]. In their original analysis of squid axon currents Hodgkin and Huxley (1952) obtained the I_K activation curve, an important result for models of I_K, by normalizing a family of I_K records with (V-E_K), a linear dependence on driving force. Normalization by GHK[(V-E_K)] should be used instead. This analysis is illustrated for I_K from MFBs in Figure 2 using the results of Geiger and Jonas (2000). Their I_Kactivation curve is shown in Figure 2 (open circles) along with a description of this result by $n_{\infty }^4$(V) with n_∞(V) = α_n/(α_n+β_n) and α_n and β_n as given above (Equation 2). Their result for V = 50 mV (Figure 1C of Geiger and Jonas, 2000) was not included. The I_K component in squid axons is partially blocked by ${\text{Na}}_i^{+}$ in a voltage-dependent manner for strong depolarizations such as V ≥ 50 mV (Bezanilla and Armstrong, 1972; French and Wells, 1977). Geiger and Jonas (2000) used an intracellular solution containing 21 mM Na⁺. A partial block of I_K in MFBs at 50 mV by this level of ${\text{Na}}_i^{+}$ cannot be ruled out and so this point was excluded from the analysis. The remaining results from V = −70 to +30 mV were multiplied by (V-E_K) to remove the linear normalization they used to obtain their result. The next step was normalization with GHK[(V-E_K)] as described in Clay (2009)—closed circles in Figure 2. The result is an I_K activation curve that is significantly steeper than the one obtained using normalization by (V-E_K). A single modification in the Engel and Jonas (2005) model is sufficient to describe these results, namely a change in β_n from 0.125 exp[−(V+65)/80] ms⁻¹ to 0.125 exp[−(V+65)/20] ms⁻¹, the curve labeled “$n_{\infty }^4$ revised” in Figure 2. The same modification in the original Hodgkin and Huxley (1952) model, namely replacing “80” in the exponential term of β_n to “20,” is sufficient to describe the I_K activation curve in squid axons obtained using GHK normalization of a family of I_K records (Clay et al., 2008). The AP-clamp result for this version of the Engel and Jonas (2005) I_K model—model 2—is given by an_i⁴ GHK[(V_i-E_K)], continuous line in Figure 1, with a = 1.3 mA/cm² and n_i determined from Equation (1) using the modified version of β_n. This result provides a significant improvement in the description of the falling phase of the experimental record compared with model 1. The rising phase of both models underestimates the delay in the rise of I_K during an AP, a result that is similar to the Cole and Moore (1960) effect for voltage steps in squid axons (Discussion).

FIGURE 2

Figure 2. Open channel probability, p_o, for I_K in MFB's as a function of V. The open circles were taken from Figure 5C from Geiger and Jonas (2000). The curve describing those results corresponds to $n_{\infty }^4$(V) with n_∞(V) = α_n∕(α_n+β_n) and α_n = −0.01(V+55)/{exp[−(V+55)/10]−1} msec⁻¹, β_n = 0.125 exp[−(V+65)/80] msec⁻¹ (Engel and Jonas, 2005). These results were obtained from normalization of a family of I_K records with (V-E_K) and E_K = −85 mV (Geiger and Jonas, 2000). They were multiplied by (V+85) and renormalized using GHK[(V−E_K)] as described in the text with kT/q=26.5 mV (T = 34°C) and E_K = −104 mV from K_o = 2.5 mM and K_i = 125 mM (Geiger and Jonas, 2000). The renormalized results are represented by the filled circles. The theoretical curve describing those results is given by $n_{\infty }^4$(V) with α_n= −0.01(V+55)/{exp[−(V+55)/10]−1} msec⁻¹ and β_n = 0.125 exp[−(V+65)/20] msec ⁻¹.

The revised I_K result in Figure 1—model 2—is further illustrated by the current-voltage trajectory for I_K during the AP in Figure 1 (Figure 3—dashed line). The I_K gate—the n variable—is maximally activated during the AP to a level of 0.283, which occurs near the latter part of the repolarization phase. The GHK current voltage relation with n = 0.283, also shown in Figure 3, is tangent to the trajectory at this point (arrow labeled a). The trajectory lies close to the GHK relation a considerable distance on either side of a, indicating that the time course of I_K in model 2 during repolarization is largely determined by the GHK equation.

FIGURE 3

Figure 3. Current-voltage trajectory (dashed line) of I_K—model 2—for the AP in Figure 1. The arrows indicate the direction of time. Also shown is the GHK current-voltage relation, an⁴ GHK(V-E_K) with a = 1.3 mA/cm², n = 0.283, and GHK(V-E_K) as given in the text. The I_K trajectory during the AP is tangent to the GHK relation at point a.

Na⁺ Current

Activation curve results for I_Na from MFBs—both experimental and theoretical—taken from Engel and Jonas (2005) are shown in Figure 4 along with the revised I_K results described above. The I_K and I_Na activation curves overlap almost completely (Figure 4) an observation that may be consistent with known structural similarities of voltage-gated Na⁺ and K⁺ channels (MacKinnon, 1991, 1995; Jan and Jan, 1997; Hanlon and Wallace, 2002).

FIGURE 4

Figure 4. Open channel probability, p_o, for I_K and I_Na for MFBs as a function of V, both experimental and theoretical. The experimental results for I_K were taken from Figure 5C of Geiger and Jonas (2000) as modified in Figure 2. The solid line labeled “n^4∞(V) EJ revised” corresponds to [α_n/(α_n+β_n)]⁴ with α_n = −0.01(V+55)/{exp[−(V+55)∕10]−1} msec⁻¹ and β_n = 0.125 exp[−(V+65)∕20] msec ⁻¹. The experimental results for I_Na were taken from Figure 2B of Engel and Jonas (2005). The dashed curve labeled “m^3∞(V) EJ” corresponds to [α_m/(α_m+β_m)]³ with α_m = −93.8(V−105)/{exp[−(V−105)∕17.7]−1} msec⁻¹ and β_m = 0.17 exp(−V∕23.3) msec⁻¹.

Pooled results of AP-clamp recordings of I_Na from Alle et al. (2009) are illustrated in Figure 5 (data points with error bars representing ±SEM, n = 9; Supplementary Table 1). They were scaled to match the I_Na result in the bottom panel of Figure 2B of Alle et al. (2009). The results are the differences obtained by applying the AP shown in the top panel of Figure 5 to MFBs before and after the addition of 1 μM tetrodotoxin (TTX) to the bathing medium. Also shown in Figure 5 is the prediction of the Engel and Jonas (2005) model described above, I_Na,_i = g_Nam_i³h_i(V_i-E_Na), with i = 1.2,3…, g_Na = 110 mS/cm², m_i and h_i determined as described in Section Materials and Methods and E_Na = 62 mV (Alle et al., 2009). The model provides a good description of the experimental results. The arrow in Figure 5 highlights a slight secondary increase of I_Na during repolarization in the Engel and Jonas (2005) I_Na model attributable to an overlap of activation and inactivation. A similar result is not apparent in the experimental recordings.

FIGURE 5

Figure 5. Top panel: Same AP as in the top panel of Figure 1. Bottom panel: The data points with error bars (n = 8; ±SEM) correspond to AP-clamp measurements of I_Na in MFBs (Alle et al., 2009). The mean values of those results are also given in Supplementary Table 1. The theoretical curve is as described in the text.

The mean of the pooled results for I_K and –I_Na from Figures 1, 5, respectively, are shown in Figure 6 scaled as described in Alle et al. (2009) along with the predictions of the I_Na model and I_K model 2 described above. The arrow in Figure 6 highlights a slight overlap of I_Na with I_Kduring the AP, an energetically inefficient result (Alle et al., 2009).

FIGURE 6

Figure 6. Pooled results for –I_Na and I_K from MFBs. The mean values of these measurements are shown here along with the theoretical curves for –I_Na and I_K are the same as described in Figures 1, 5, respectively. The arrow highlights a slight overlap in time of the two results (Alle et al., 2009).

Discussion

This report provides further illustration of a method recently described for the analysis of ionic currents recorded with the AP-clamp technique (Clay, 2015). The work also provides an example of the utility of the GHK equation for the analysis of I_K from a mammalian neuronal preparation. Traditionally, those results have been described by I_K = g_K(V-E_K) with g_K a constant (Hodgkin and Huxley, 1952). This expression implies, by definition, that the slope conductance for I_K at a given potential below E_K is the same as the slope conductance positive to E_K. This result is theoretically impossible when E_K ≠ 0 because I_K is proportional to ${\text{K}}_o^{+}$ with V well below E_K and I_K is proportional to ${\text{K}}_i^{+}$ when V is well above E_K. The fully activated current-voltage relation for I_K outwardly rectifies, a result that is well described by the GHK equation (Clay, 2009). A similar result applies for I_Na with a caveat. The fully activated current-voltage relation for I_Na in squid axons in Ca²⁺-free seawater is consistent with the GHK equation (Vandenberg and Bezanilla, 1991; their Figure 3). It inwardly rectifies since ${\text{Na}}_o^{+}$ is much greater than ${\text{Na}}_i^{+}$. Calcium ions in normal seawater partially block I_Na in a voltage-dependent manner with the blockade increasing as the membrane potential is hyperpolarized relative to E_Na. This effect counterbalances the inward rectification of I_Na in the absence of divalent cations so that I_Na is approximately proportional to (V-E_Na) for physiological conditions over the range of potentials spanned by an AP (Vandenberg and Bezanilla, 1991). A similar mechanism may apply to I_Na from other preparations (Worley et al., 1986; Green et al., 1987).

One consequence of the original prediction by Hodgkin and Huxley (1952) that I_K = g_K(V-E_K) is that activation curves for voltage gated K⁺ channels have typically been determined by normalizing a family of I_K records using (V-E_K). An activation curve with a shallow voltage dependence is obtained (open circles, Figure 2). In contrast, normalization of those results by GHK[(V-E_K)] yields an activation curve having a steepness similar to that noted by Sigworth (2003) for voltage-gated K⁺ channels. Moreover, the revised K⁺ channel activation curve is similar to the Na⁺ channel activation curve (Figure 4). High sensitivity of these channels to voltage is important because cellular voltage changes are small (Sigworth, 2003).

Models 1 and 2 for I_K both fail to account for the delay in the rising phase of this component during an AP (Figures 1, 6), a result that is similar to the Cole and Moore (1960) effect in squid axons. Specifically, the delay in the rising phase of I_K following a voltage clamp step from relatively negative holding potentials is greater than the prediction of the Hodgkin and Huxley (1952) n⁴ model (Cole and Moore, 1960). This result is significant in squid axons even for moderately hyperpolarized holding potentials such as −75 mV (Figure 5; Clay et al., 2008). The discrepancy between theory and experiment reported here for the rising phase of I_K elicited during AP-clamp from a holding potential of −80 mV in MFBs is a corollary of the Cole and Moore (1960) effect.

The I_K component underlying repolarization in rat hippocampal MFBs is the result of the entry of K⁺ through a mixture of channels, Kv1, Kv3, and BK (Alle et al., 2011). BK channels appear not to be significant for basal APs, i.e., APs recorded under normal physiological conditions (Alle et al., 2011). The model of I_K in MFB's by Engel and Jonas (2005) is based, implicitly, on the assumption of a homogeneous population of K⁺ channels. They noted that their model provided “a relatively accurate description of the voltage-dependence of activation of K⁺ channels in MFBs.” This view is not necessarily at odds with the results of Alle et al. (2011) especially with regard to Kv, channels that are activated rapidly. The kinetics of Kv1 and Kv3 may well be described by the same, or similar, Hodgkin and Huxley (1952) type model. In any event the falling phase of I_K obtained in AP-clamp from MFBs is consistent with a homogeneous population of K⁺ channels with their fully-activated current-voltage relation described by GHK(V-E_K).

The emphasis in this report is on a method, for analyzing ionic currents in neurons with an application to MFBs. The method is general. It can be applied to ionic currents in any excitable cell for which a specific blocker is available, such as TTX for I_Na. The method requires a digitized representation of an experimentally recorded AP as well as a model of the ionic current in question obtained from voltage clamp step results such as the Hodgkin and Huxley (1952) m³h model for I_Na. The analysis given above for I_Na in MFBs is largely confirmatory of the m³h model as given by Engel and Jonas (2005). The analysis for I_K in MFBs reveals two discrepancies between experiment and the Hodgkin and Huxley (1952) model of I_K, one concerning the rising phase of I_K during an AP similar to the Cole and Moore (1960) effect and a discrepancy in the falling phase that can be accounted for by changing the fully-activated current-voltage for I_K from a linear dependence upon the K⁺ driving force to the GHK dependence on (V-E_K). The method provides a complementary test of models constructed from voltage step results. An AP-clamp rapidly scans the range of membrane potentials corresponding to this waveform. In this way the GHK dependence of I_K on (V-E_K) can be elucidated for the physiological range of membrane potentials more readily than is possible with voltage steps.

The original work of Hodgkin and Huxley (1952) continues to influence the design and analysis of experimentals in membrane neuroscience. The method described in this report provides a variation of their approach that can yield additional insight to the relationship between membrane excitability and the ionic currents that underlie excitability.

Conflict of Interest Statement

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

This research was supported by the Intramural Research Program of the National Institute of Neurological Disorders and Stroke, National Institutes of Health, Bethesda, Maryland USA. The author gratefully acknowledges Dr. Henrik Alle for providing a table of digitized results from Alle et al. (2009).

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/article/10.3389/fncel.2015.00514

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