Novel description of ionic currents recorded with the action potential clamp technique: application to excitatory currents in suprachiasmatic nucleus neurons

Abstract

The traditional method of recording ionic currents in neurons has been with voltage-clamp steps. Other waveforms such as action potentials (APs) can be used. The AP clamp method reveals contributions of ionic currents that underlie excitability during an AP (Bean BP. Nat Rev Neurosci 8: 451–465, 2007). A novel usage of the method is described in this report. An experimental recording of an AP from the literature is digitized and applied computationally to models of ionic currents. These results are compared with experimental AP-clamp recordings for model verification or, if need be, alterations to the model. The method is applied to the tetrodotoxin-sensitive sodium ion current, I_Na, and the calcium ion current, I_Ca, from suprachiasmatic nucleus (SCN) neurons (Jackson AC, Yao GL, Bean BP. J Neurosci 24: 7985–7998, 2004). The latter group reported voltage-step and AP-clamp results for both components. A model of I_Na is constructed from their voltage-step results. The AP clamp computational methodology applied to that model compares favorably with experiment, other than a modest discrepancy close to the peak of the AP that has not yet been resolved. A model of I_Ca was constructed from both voltage-step and AP-clamp results of this component. The model employs the Goldman-Hodgkin-Katz equation for the current-voltage relation rather than the traditional linear dependence of this aspect of the model on the Ca²⁺ driving force. The long-term goal of this work is a mathematical model of the SCN AP. The method is general. It can be applied to any excitable cell.

in their original work, Hodgkin and Huxley (1952) used rectangular voltage-clamp steps to construct models of the ionic currents underlying excitability in squid giant axons. The advantage of this approach is that it allows the investigator to control the primary independent variable of an excitable membrane, the membrane potential V, and to make predetermined changes in that variable to activate the primary dependent variable, the net membrane ionic current. Voltage steps also short-circuit the membrane capacitance current, at least ideally. Waveforms other than steps, such as action potentials (APs), can also be used, the so-called AP clamp technique (Bean 2007). In this approach a previously recorded AP is applied to the cell with the amplifier in voltage-clamp mode. The contributions of specific ionic currents during the AP can be determined with the use of ion channel blockers. In contrast, the dynamic clamp technique is an indirect approach in which firing of a cell in current-clamp mode is altered by changes of particular currents with simulations in real time (Sharp et al. 1993).

The primary purpose of this report is a demonstration of a novel application of the AP clamp technique. It is used here as an aid in the construction of models of ion channel conductances. Specifically, AP-clamp results provide a test of models constructed from voltage-step results. Moreover, the two sets of results together provide a richer base from which to construct a model ab initio than rectangular step results alone. The method is illustrated with results from suprachiasmatic nucleus (SCN) neurons (Jackson et al. 2004). This group recorded the tetrodotoxin (TTX)-sensitive Na⁺ current (I_Na) and the Ca²⁺ current (I_Ca), using voltage steps and the AP clamp for both components. A significant feature of the I_Ca analysis is the use of the Goldman-Hodgkin-Katz (GHK) equation for the fully activated current-voltage relation of this component (Clay 2009; Goldman 1943; Hodgkin and Katz 1949). The descriptions of I_Na and I_Ca given below may ultimately be incorporated in a mathematical model of the AP in the SCN, the long-term goal of this work.

As noted above, the AP clamp computational method can be used to test models of ion current components including models in the literature. This feature of the analysis was applied to models of I_Na and I_Ca in SCN neurons (Diekman et al. 2013; Sim and Forger 2007). The I_Ca model of Diekman et al. (2013) utilized I_Ca = g_Ca(V − E_Ca) as the fully activated current-voltage relation for I_Ca, where g_Ca is the conductance of this component and E_Ca is the Ca²⁺ reversal potential. This approach has been traditional for models of ion channel currents beginning with Hodgkin and Huxley (1952). The I_Ca model in this report is based on the GHK equation. The comparison of these models given below illustrates the utility of the GHK equation in the analysis of I_Ca.

METHODS

This work requires an experimental recording of an AP from an SCN neuron such as the result in the inset of Fig. 5B of Jackson et al. (2004). SCN neurons fire APs spontaneously, especially during the daytime. The waveform used was taken from a train of spontaneously occurring APs. That waveform was digitized (Fig. 1). The same result is shown with the points connected by straight lines (Fig. 1). This V_i vs. t_i data set (t is time in milliseconds, V is membrane potential in millivolts) was applied computationally to models determined from voltage steps. The procedure is illustrated by consideration of a generic ion current component, I_x. We assume that voltage-step analysis indicates that I_x = g_xq²(t)(V − E_x), where g_x and E_x are the conductance and the reversal potential of this component, respectively. The gating variable q is given by

$$\text{d}q\left(t\right)\text{/d}t=-\left[{\alpha }_q\left(V\right)+{\beta }_q\left(V\right)\right]q\left(t\right)+{\alpha }_q\left(V\right)$$ (1)

Fig. 5.

AP-clamp analysis of the Sim and Forger (2007) model of I_Na (SF). The arrows labeled a and b indicate discrepancies between their model and the experimental recording as described in the text.

Fig. 1.

Example of the digitization procedure used for action potentials (APs) taken from Jackson et al. (2004). The AP in the inset of their Fig. 5B was digitized as indicated. That membrane potential (V_i) vs. time (t_i) data set is also shown shifted downward 50 mV with the points connected by solid lines. This neuron was spontaneously firing at a rate of 11 Hz. The AP shown is a signal-averaged result from a 5 s recording—55 APs. Those waveforms were aligned by the peaks of the APs and averaged (Jackson et al. 2004) to give the result shown.

where α_q and β_q are voltage-dependent functions based on chemical reaction rate theory similar to the αs and βs used by Hodgkin and Huxley (1952) for their models of I_Na and I_K in squid axons. Note that q is raised to the second power in Eq. 1, which is used to describe a slight delay in the onset of I_x immediately following a voltage step, assuming such a result occurs. Hodgkin and Huxley (1952) used this approach in their models of I_K and I_Na. They observed that both components exhibited a sigmoidal time-dependent activation following voltage steps—a delay—that they modeled by raising the activation gating variables n (I_K) and m(I_Na) to a power −n⁴ for I_K and m³ for I_Na. We assume that raising q to the second power is sufficient for this result for I_x. We further assume that the parameters in the expressions for α_q and β_q (Eq. 1) were adjusted so that the model provided a good fit by eye to a family of I_x records, a procedure similar, again, to the approach of Hodgkin and Huxley (1952) in their analysis of I_Na and I_K. The V_i vs. t_i data set in Fig. 1 can be applied to the I_x model in AP clamp, computationally, with software routines such as the ones in Mathematica (Wolfram Research, Champaign, IL). The start point of the AP in Fig. 1 is t₀ = 0, V₀ = −56.7 mV. If V₀ is below the activation range of I_x, the start value of the gating variable q is given by its steady-state value q = q₀ = α_q(V₀)/[α_q(V₀)+ β_q(V₀)]. The next iterative value of q, q₁, is determined from Eq. 1 with NDSolve (Mathematica), using V(t) = V₀ + (V₁ − V₀)(t − t₀)/(t₁ − t₀) for t₀ <t < t₁ (V₁ = −56.1 mV; t₁ = 0.44 ms). This procedure is continued throughout the V_i vs. t_i data set of the AP waveform. The resulting digitized values of I_x can be determined according to I_x,i = g_xq_i²(V_i − E_x). This result is compared with an experimental AP-clamp recording of I_x obtained by applying the waveform in Fig. 1 to the cell before and after the addition to the bathing medium of a blocker of I_x, assuming such a blocker exists. The difference between the two results gives the time course and amplitudes of I_x during the AP. If the comparison between theory and experiment for I_x is satisfactory, AP-clamp analysis would be merely confirmatory of the model obtained from rectangular steps. On the other hand, if significant differences are found, an attempt would be made to modify the model by changing α_q and β_q and perhaps also the gating scheme from q² to, for example, q³, so as to provide a good description of both AP-clamp and voltage-step results.

The other part of the I_x model, the fully activated current-voltage relation, g_x(V − E_x), may also require revision. Many ion channel current components have a nonlinear dependence on driving force, which in some instances, I_Ca for example, is well described by the GHK equation (Clay 2009; McCormick and Huguenard 1992). The GHK equation is used in the I_Ca results given below.

RESULTS

I_Na component: voltage-step results.

The first part of the analysis is the development of a model based on voltage-step recordings, in this case the I_Na results in Fig. 8, B and C, of Jackson et al. (2004). A model widely used throughout neuroscience including the results of Engel and Jonas (2005) on I_Na in hippocampal mossy fiber boutons is that of Hodgkin and Huxley (1952), I_Na = g_Nam³h(V − E_Na), where m refers to channel activation, h refers to inactivation, and E_Na refers to the reversal potential for Na⁺ (E_Na = 45 mV). The m and h variables are determined by

$$\begin{array}{c}\text{d}m\text{/d}t=-\left[{\alpha }_m\left(V\right)+{\beta }_m\left(V\right)\right]m+{\alpha }_m\left(V\right);\\ \text{d}h\text{/d}t=-\left[{\alpha }_h\left(V\right)+{\beta }_h\left(V\right)\right]h+{\alpha }_h\left(V\right)\end{array}$$ (2)

Fig. 8.

Time-dependent profile of Ca_s during the AP at top as determined from Eq. 5.

The αs and βs in Eq. 2 for squid axons do not provide a good description of the I_Na rectangular step results of Jackson et al. (2004). The start point here is the equations of Engel and Jonas (2005):

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

The αs and βs are in units of inverse milliseconds. The above expressions were modified to give a qualitative description, by eye (Fig. 2A), for the I_Na results in Fig. 8B of Jackson et al. (2004) as well as a description of steady-state inactivation, i.e., the h_∞(V) curve (Huang 1993) and the m³_∞(V) curve (Fig. 8C of Jackson et al. 2004), with h_∞(V) = α_h(V)/[α_h(V) + β_h(V)] and m_∞(V) = α_m(V)/[α_m(V) + β_m(V)]. The h_∞(V) and m_∞³(V) curves are illustrated in Fig. 2B. The results in Fig. 2A correspond to I_Na = 120 m³(t)h(t)(V − 45), with I_Na in picoamperes and voltage steps V = −53, −48, −43, −38, −33, −28, −18, and −8 mV; holding potential = −78 mV. The αs and βs obtained from this analysis, modified from Eq. 3, were

$$\begin{array}{c}{\alpha }_m\left(V\right)=-14\left(V-88\right)/\left\{\exp \left[-\left(V-88\right)/17.7\right]-1\right\};{\beta }_m\left(V\right)=0.026\exp \left[-\left(V+12\right)/8\right];\\ {\alpha }_h\left(V\right)=1.2\times {10}^{-4}\exp \left[-\left(V+7\right)/8\right];{\beta }_h\left(V\right)=2.3/\left\{\exp \left[-\left(V+24.7\right)/13.3\right]+1\right\}\end{array}$$ (4)

Fig. 2.

A: sodium ion current (I_Na) records from Eqs. 2 and 4 for V = −53, −48, −43, −38, −33, −28, −18, and −8 mV; holding potential = −78 mV. B: inactivation, h_∞ = α_h(V)/[α_h(V) + β_h(V)], and activation, m³_∞(V), with m_∞(V) = α_m(V)/[α_m(V) + β_m(V)]. Data for the inactivation curve are from Fig. 6A of Huang (1993). The activation results are from Fig. 8C of Jackson et al. (2004). The αs and βs are given by Eq. 4.

The expressions in Eq. 4 were used for the voltage-step results (Fig. 2) and the AP-clamp results that follow.

I_Na component: AP-clamp results.

Jackson et al. (2004) applied the AP in Fig. 1 (shown again in Fig. 3, top) to an SCN neuron in voltage clamp before and after the addition of TTX to the bathing medium. The difference current is shown in Fig. 3, bottom (solid line). The AP-clamp computational result (dashed line in Fig. 3, bottom) obtained with the procedure described above (methods) provides a good description of experiment, other than a slight discrepancy near the peak of the AP (arrow in Fig. 3, bottom). This result was obtained without modifications in the model constructed from voltage-step results (Eqs. 2 and 4).

Fig. 3.

AP-clamp analysis of the I_Na model (Eqs. 2 and 4). The AP at top is the same as in Fig. 1. The experimental record at bottom is a digitized version of the I_Na record taken from the middle panel of Fig. 5B of Jackson et al. (2004). The experimental and theoretical I_Na results are shown superimposed below the AP. The arrow indicates a discrepancy between theory and experiment described in the text.

The AP-clamp result described above illustrates the behavior of inactivation, h(t), and activation, m³(t), during repolarization of the AP (Fig. 4A). Inactivation rapidly goes to completion (h = 0) close to the peak of the AP [Fig. 4A, middle, h(t) at bottom left]. Before this occurs, h and m³ (which is effectively 1 at this point of the AP) have an overlap as indicated by the small shaded area (Fig. 4A, middle) resulting in I_Na in Fig. 4A, bottom. If h(t) went to 0 even more rapidly during this phase of the AP, the discrepancy between theory and experiment indicated above (Fig. 3) would be reduced, or perhaps eliminated. Attempts to accomplish this result without modifying other aspects of the results were unsuccessful. Activation, m³(t), rapidly changes from 1 to 0 during the midportion of repolarization (Fig. 4A) just before inactivation begins to recover from 0. Consequently, no additional overlap of h(t) and m³(t) occurs throughout the remainder of the AP (Fig. 4A).

Fig. 4.

A: analysis of m³(t) and h(t) during repolarization of the SCN AP. Inactivation is almost—but not quite—complete near the initial part of this phase of the AP, resulting in an overlap of m³(t) and h(t) as indicated by the small shaded area, middle. The I_Na component corresponding to this overlap is shown at bottom. B: similar analysis of the Hodgkin and Huxley (1952) model, which is given by CdV/dt = −120m(t)³h(t)(V − 55) − 36n(t)⁴(V + 72) − 0.3(V + 49), with C = 1 μF/cm², α_m(V) = −0.1(V + 35)/{exp[−0.1(V + 35)]−1}, β_m(V) = 4exp[−(V + 60)/18], α_h(V) = 0.07exp[−(V + 60)/20], β_h(V) = 1/{exp[−0.1(V + 30)] + 1}, α_n(V) = −0.01(V + 50)/{exp[−0.1(V + 50)] − 1}, and β_n(V) = 0.125exp[−(V + 60)/20]. An AP was elicited from the model by a 1-ms-duration 10 μA/cm² amplitude pulse (not shown). Results at middle and bottom are described in the text.

The above results are compared with a similar analysis of the Hodgkin and Huxley (1952) model with their original expressions for the αs and βs for I_Na (Fig. 4B legend). Inactivation in their model does not go to completion during the AP. Moreover, m³(t) does not return to 0 as early in the AP as is the case in the SCN I_Na analysis. For both reasons, h(t) and m³(t) overlap considerably during repolarization, which leads to a significant I_Na especially since the driving force for Na⁺ increases as the membrane potential travels down the repolarization phase of the AP. Indeed, peak I_Na during the AP is twice as large as the peak I_Na during the upstroke phase (Fig. 4B, bottom). The analysis in Fig. 4B appears to be valid for APs in squid giant axons (Clay 2013). As a number of groups have noted, squid giant axon APs are not energetically efficient since I_K must overcome I_Na during the AP to return the membrane potential to rest (Alle et al. 2009; Carter and Bean 2009; Crotty et al. 2006; Sengupta et al. 2010). This inefficiency does not appear to have adverse consequences for squid since the giant axon triggers the rapid escape response of the animal, a relatively infrequent occurrence that is triggered by one or at most two or three APs (Otis and Gilly 1990). APs occur much more frequently in mammalian neurons, a result that requires significant separation in time of the I_Na and I_K components during an AP for energetically efficient signaling. This result is accomplished when the I_Na component flows primarily during the upstroke phase of the AP, as has been shown for hippocampal mossy fiber boutons (Alle et al. 2009), cortical pyramidal neurons (Carter and Bean 2009), and SCN neurons (Jackson et al. 2004).

Alternative model for I_Na in SCN neurons.

Sim and Forger (2007) have described a model of the AP from SCN neurons. The equations for I_Na in their work were based on the voltage-step results of Jackson et al. (2004), as was the case for the model of I_Na described above. They used I_Na = 229 m³(t)h(t)(V − 45), with dm/dt = (m_∞ − m)/τ_m, dh/dt = (h_∞ − h)/τ_h, and m_∞ = {1 + exp[−(V + 35.2)/7.9]}⁻¹, τ_m = exp[−(V + 286)/160], h_∞ = {1 + exp[(V + 62)/5.5]}⁻¹, and τ_h = 0.51 + exp[−(V + 26.6)/7.1]. AP-clamp analysis was applied to these equations (Fig. 5). A discrepancy between theory and experiment is apparent during the latter part of the declining phase of I_Na (arrow b in Fig. 5), similar to the results described above (Fig. 3), suggesting that this portion of the AP-clamp recording is a challenge for models of I_Na. More significantly, the Sim and Forger (2007) model does not provide a successful description for I_Na just prior to and during the initial upstroke phase of the AP (arrow a in Fig. 5), in contrast to the model of I_Na in this report (Fig. 3). The I_Na component just prior to upstroke of the AP is significant for near-threshold excitability.

I_Ca component.

Jackson et al. (2004) reported voltage-step and AP-clamp recordings of I_Ca from SCN neurons by applying steps and APs in control followed by application of the same waveforms in voltage clamp after replacement of Ca²⁺ in the bathing medium by Mg²⁺, a nonpermeant divalent cation. They also reported measurements of I_Ca before and after the addition to the bath of nimodipine, which blocks a portion of I_Ca in SCN neurons (Pennartz et al. 2002). The model of I_Ca contains two intracellular compartments for Ca²⁺ (Diekman et al. 2013; Yamada et al. 1998). One compartment corresponds to a thin spherical shell 0.1 μm in width near the membrane surface (McCormick and Huguenard 1992). The Ca²⁺ concentration in this compartment is denoted by Ca_s. The other compartment corresponds to the cytosol. The Ca_s parameter is given by

$${\text{dCa}}_{\text{s}}/\text{d}t=-K_1{I}_{\text{Ca}}-K_2{\text{Ca}}_{\text{s}}+c_{\text{s}}$$ (5)

where K₁ = 3 × 10⁻⁵ M/nC, K₂ = 0.04 ms⁻¹ (Purvis and Butera 2005), and c_s = 2 nM/ms, so that Ca_s = 50 nM when I_Ca = 0 (McCormick and Huguenard 1992).

As noted above, the fully activated current-voltage relation for I_Ca is given by the GHK equation (McCormick and Huguenard 1992),

$$I_{\text{Ca}}=-{\text{z}}^2{\text{AP}}_{\text{Ca}}\left(F^{2}V/\text{RT}\right)\left[{\text{Ca}}_{\text{i}}{{\displaystyle }}^{2+}\exp \left(\text{z}FV/\text{RT}\right)-{\text{Ca}}_o{{\displaystyle }}^{2+}\right]/\left[\exp \left(\text{z}FV/\text{RT}\right)-1\right]$$ (6)

where z is the ionic valence for Ca²⁺ (z = 2), P_Ca is the membrane's permeability to Ca²⁺, A is the membrane surface area, F is the Faraday constant, T is absolute temperature (RT/F = 25 mV at room temperature), and Ca_o²⁺ is the extracellular Ca²⁺ concentration (Ca_o²⁺ = 1.2 mM) (Jackson et al. 2004). In this analysis Ca_i²⁺ is represented by Ca_s, as noted above. Since Ca_i²⁺ is significantly less than Ca_o²⁺, Eq. 6 is considerably simplified using Ca_i²⁺ = 0 so that I_Ca = −aGHK(V), where a is a constant in picoamperes and GHK(V) = (zFV/RT)/[exp(zFV/RT) − 1] = (V/12.5)/[exp(V/12.5) − 1]. Note that GHK[V = 0] = 1. As shown below, the assumption of Ca_i²⁺ = 0 for the GHK equation does not significantly alter the results (discussion).

Voltage-step recordings of the nimodipine-blocked or nimodipine-sensitive I_Ca component—bottom panel of Fig. 9A of Jackson et al. (2004)—appear to be consistent with an activation gating model consisting of a single gate (no delay in onset of the current following a step). In contrast, the recordings of the nimodipine-sensitive component—middle panel of Fig. 9A of Jackson et al. (2004)—do exhibit a slight delay. Those results are modeled by r₁²(t) with dr₁/dt = −[α_r1(V) + β_r1(V)]r₁ + α_r1(V), where α_r1(V) and β_r1(V) are voltage-dependent functions similar to those used for I_Na. Since the kinetics of the nimodipine-blocked I_Ca component do not exhibit a delay, they are modeled by r₂(t) with dr₂/dt = −[α_r2(V) + β_r2(V)]r₂ + α_r2(V).

Fig. 9.

A: role of Goldman-Hodgkin-Katz (GHK) in the I_Ca AP-clamp result for the model described in this report. The dashed line represents I_Ca vs. voltage for the simulation in Fig. 7A, with the direction of time indicated by the arrows. The curve labeled GHK I_Ca represents Eq. 6 for Ca_i²⁺ = 0 and 18.5 μM. The trajectory is tangential to the GHK current-voltage relation at point a. I_Ca is maximal at point b. B: similar analysis of the Diekman et al. (2013) I_Ca model. In this model I_Ca = g_Ca(V − E_Ca). The trajectory is tangent to this line at a, where the r₁ and r₂ channels are maximally activated. The maximum current of the trajectory is indicated by b. This result is also shown in the inset (bottom right) by the dashed line along with the experimental AP-clamp recording of I_Ca.

Inactivation of I_Ca in the model was ascribed to a calcium-dependent process (Fox et al. 2002; Kay 1991; Luo and Rudy 1994; Standen and Stanfield 1982). This parameter, f(t), is given by

$$\text{d}f\left(t\right)/\text{d}t=\left[f_1\left({\text{Ca}}_{\text{i}}{{\displaystyle }}^{2+}\right)-f\left(t\right)\right]/{\tau }_{f\text{Ca}}$$ (7)

with f₁(Ca²⁺) = 1/[1 + (Ca_i²⁺/K_d)³] and K_d = 0.18 μM; τ_fCa = 30 ms (Fox et al. 2002). The full model for I_Ca is

$$I_{\text{Ca}}\left(V,t\right)=-\text{GHK}\left(V\right)f\left(t\right)\left[a_1{r}_1{{\displaystyle }}^2\left(V,t\right)+a_2{r}_2\left(V,t\right)\right]$$ (8)

with a₁ = 305 pA and a₂ = 31 pA.

As noted above, a significant result for developing models of voltage-gated ion channel conductances is the activation curve of channel gating. Hodgkin and Huxley (1952) obtained this result for I_Na in squid giant axons by dividing peak I_Na during a voltage step by the driving force (V − E_Na). Jackson et al. (2004) used a similar approach to obtain the I_Na activation curve for SCN neurons (Fig. 2B). This method is appropriate for I_Na since the fully activated current-voltage relation of this component is directly related to the driving force. In contrast, I_Ca has a nonlinear dependence on the driving force for Ca²⁺, a result well described by the GHK equation. Consequently, peak I_Ca results during voltage steps should be divided by GHK(V) rather than by (V − E_Ca). The latter approach with E_Ca = 57 mV was used by Jackson et al. (2004) to obtain the data points in their Fig. 9B. Those results were multiplied by (V − 57) and then divided by GHK(V) = (V/12.5)/[exp(V/12.5) − 1] to give the activation curves shown here in Fig. 6, bottom. These results are described by r_1∞²(V) =[α_r1/(α_r1 + β_r1)]² for the nimodipine-insensitive I_Ca component and r_2∞(V) = α_r2/(α_r2+ β_r2) for the nimodipine-sensitive component with

$$\begin{array}{c}{\alpha }_{r1}\left(V\right)=-0.028\left(V+11\right)/\left\{\exp \left[-0.1\left(V+11\right)\right]-1\right\};{\beta }_{r1}\left(V\right)=0.352\exp \left[-0.015\left(V+12\right)\right];\\ {\alpha }_{r2}\left(V\right)=-0.154\left(V+27\right)/\left\{\exp \left[-0.1\left(V+27\right)\right]-1\right\};{\beta }_{r2}\left(V\right)=1.92\exp \left[-0.1\left(V+37\right)\right]\end{array}$$ (9)

Fig. 6.

Top left: calcium ion current (I_Ca) voltage-step results from the I_Ca model (Eq. 8) with V = −48, −38, −28, −18, −8, and +2 mV for the nimodipine-insensitive component (a₂ = 0 in Eq. 8) and α_r1(V) and β_r1(V) as given in Eq. 9. Top right: I_Ca voltage-step results for the nimodipine-sensitive component (a₁ = 0 in Eq. 8) with V = −58, −53, −48, −40, −30, −20, −10, and 0 mV and α_r2(V) and β_r2(V) as given in Eq. 9. Bottom: activation curves for the I_Ca components, r_1∞²(V) and r_2∞(V) with r_i,∞(V) = α_ri(V)/[α_ri(V) + β_ri(V)] and α_ri(V) and β_ri(V) given by Eq. 9 for i = 1 and 2. The data points were taken from Fig. 9A, top, of Jackson et al. (2004) and modified as described in the text.

The predictions of Eq. 8 for voltage steps with the αs and βs in Eq. 9 are shown in Fig. 6, top.

I_Ca: AP-clamp results.

An I_Ca result from the AP-clamp recordings of Jackson et al. (2004)—their Fig. 12—is shown in Fig. 7A, bottom (solid line) in response to the AP in Fig. 7A, top. The computational result (dashed line in Fig. 7A, bottom) was obtained in a manner similar to the I_Na analysis given above. Specifically, the r₁, r₂, and f variables were iterated throughout the AP (methods). The model provides a favorable description of experiment. It mimics the inflection in I_Ca that occurs during the upstroke phase of the AP, although the current level at which the inflection occurs does not match experiment. A significant feature of the model is the Ca²⁺ concentration adjacent to the internal surface of the membrane—Ca_s (Eq. 5). This result for the simulation in Fig. 7A is illustrated in Fig. 8. The maximum value of Ca_s is 18.5 μM, which occurs near the latter part of the repolarization phase of the AP. These results for Ca_s are significantly higher than the 50–100 nM level observed for Ca²⁺ in the cytosol of spontaneously firing SCN neurons with calcium-sensitive dyes (Diekman et al. 2013; Irwin and Allen 2007). A similar distribution of Ca_i²⁺ occurs at nerve terminals immediately following an AP. The AP triggers entry of Ca²⁺ into the terminal via voltage-gated Ca²⁺ channels. The resulting Ca²⁺ concentration in the immediate vicinity of synaptic vesicle release sites may briefly rise to levels as high as 20 μM (Augustine et al. 2003; Neher 1998). The increases in Ca_s during and after an AP in the simulation of Fig. 8 do not have a similarly brief duration. However, the effect of Ca_s on Ca²⁺-dependent K⁺ current, in particular the large-conductance Ca²⁺-dependent K⁺ channel known as BK, is limited to the duration of the AP (Jackson et al. 2004, their Fig. 12). The membrane potential following an AP is in the −80 to −60 mV range, which lies below the activation range of BK channels even with Ca_s as high as 10–20 μM (Cui et al. 1997; Shelley et al. 2013). In the model Ca_s relaxes back to baseline well before the subsequent AP in a spontaneously firing SCN neuron.

Fig. 7.

A: AP-clamp analysis of I_Ca for the model described in this report (dashed line) superimposed on the experimental record. B: similar analysis of the Diekman et al. (2013) I_Ca model as described in the text.

The GHK aspect of the I_Ca analysis is illustrated in Fig. 9A. The GHK current-voltage relation (Eq. 6) is shown (Fig. 9A), scaled as indicated below, along with the current-voltage trajectory of total I_Ca of the model during the AP in Fig. 7A (dashed line in Fig. 9A). The arrows on the trajectory indicate the direction of time (Fig. 9A). The GHK result with Ca_i²⁺ = 0 is shown along with the corresponding result for Ca_i²⁺ = Ca_s = 18.5 μM, the highest value attained by Ca_s during an AP (Eq. 5 and Fig. 8). The current-voltage relations for Ca_i²⁺ = 0 and 18.5 μM differ only slightly for V > 0 and, for practical purposes, not at all for V < 0. Consequently, Ca_s = 0 was used, since this assumption simplifies the analysis as indicated above. During the upstroke of an AP and immediately thereafter the gates of both I_Ca components are strongly activated. Maximal activation of the gates of the nimodipine-insensitive I_Ca (r₁ = 0.8; r₁² = 0.64) occurs at the point labeled a in Fig. 9A. The gates of the nimodipine-sensitive component are fully activated (r₂ = 1) throughout the top portion of the AP because of the position of the activation curve of this component on the voltage axis (Fig. 6). The inactivation variable, f, is ∼0.7 at this point. These values of r₁, r₂, and f (r₁ = 0.8, r₂ = 1, f = 0.7) were used in Eq. 8 to determine the GHK current-voltage relation shown in Fig. 9A. Subsequent to point a, r₂ and f change relatively little, whereas r₁ begins to deactivate, i.e., closure of those gates thereby causing the trajectory to move away from the GHK curve. Before the r₁ (and r₂) gates close completely, I_Ca significantly increases to point b in Fig. 9A in a relatively brief period of time because of rectification of the GHK current-voltage relation.

Alternative model for I_Ca in SCN neurons.

Diekman et al. (2013) described an I_Ca model for SCN neurons in which I_Ca = g_Ca(V − E_Ca) for the fully activated current-voltage relation, i.e., the traditional approach. The I_Ca components are given by I₁ = g₁r₁f₁(V − E_Ca) and I₂ = g₂r₂f₂(V − E_Ca) with g₁ = 20 nS, g₂ = 6 nS, E_Ca = 54 mV, dr_i/dt = (r_i∞ − r_i)/τ_i, i = 1,2, with r_1∞ = {1 + exp[−(V + 21.6)/6.7]}⁻¹, r_2∞ = {1 + exp[−(V + 36)/5.1]}⁻¹, τ₁ = τ₂ = 3.1 ms, df₁/dt = (f_1∞ − f₁)/τ_f1, with f_1∞ = {1 + exp[(V + 260)/65]}⁻¹, τ_f1 = exp[−(V − 444)/220], and f₂ = 39 nM/(655 nM + Ca_s), where Ca_s is given by Eq. 5 with K₁ = 1.65 × 10⁻⁴ M/nC, K₂ = 10 ms⁻¹, and c_s = 540 nM/ms. The AP-clamp computational methodology was applied to this model. The result (Fig. 7B) does not provide an accurate description of the I_Ca AP-clamp recording. One problem with this model concerns the kinetics, τ₁ = τ₂ = 3.1 ms, i.e., a lack of voltage dependence and the same kinetics for both I_Ca components. The voltage-step recordings of Jackson et al. (2004) demonstrate that the kinetics of the nimodipine-sensitive component are considerably faster than the kinetics of the nimodipine-insensitive component and that both are voltage dependent. One approach to this problem is to graft the kinetics from the I_Ca model described in this report onto the Diekman et al. (2013) model while retaining I_Ca = g_Ca(V − E_Ca) as the fully activated current-voltage relation for the model. That is, τ₁ = τ₂ = 3.1 ms is replaced by τ₁ =(α_r1 + β_r1)⁻¹ and τ₂ = (α_r2 + β_r2)⁻¹ with the αs and βs given by Eq. 9. The AP-clamp analysis of this model (Fig. 9B, inset bottom right) provides an improved description of experiment compared with Fig. 7B, although the peak current of the simulation is considerably less than the peak current of the experimental recording. The I_Ca current-voltage trajectory of the model (dashed line in Fig. 9B) is tangent to I_Ca = g_Ca(V − E_Ca) at a point where activation of the r₁ and r₂ channel gates is maximal during the simulation (point a in Fig. 9B). The maximum current of the simulation (point b in Fig. 9B) is 20% larger than the current at point a. In contrast, the current at point b in the GHK-based model is 65% larger than the current at point a (Fig. 9A). The GHK-based model has a more robust mechanism for an increase in I_Ca during repolarization than either version of the alternative model, a result due to rectification of the GHK current-voltage relation, which is consistent with experiment (Fig. 7A).

Subthreshold I_Na and I_Ca.

Jackson et al. (2004) recorded I_Na and I_Ca with the AP clamp during the interval between spikes in spontaneously firing SCN neurons (their Fig. 5, A and C). In addition, they also recorded the persistent, subthreshold TTX-sensitive Na⁺ current (reproduced in Fig. 10) sometimes referred to as I_NaP (Crill 1996), using a slow voltage ramp (20 mV/s). The record in the presence of TTX (Fig. 10) was attributed to the leak current, I_L = 0.07(V + 10) pA, plus the steady-state delayed-rectifier K⁺ current, I_K,DR, in SCN neurons (Bouskila and Dudek 1995) as indicated by the line through this recording. The model of the latter (Fig. 10 legend) was taken from Clay (2009). The I_NaP component is given by (Clay 2003; McCormack and Huguenard 1992)

$$I_{\text{NaP}}=2.8\left(V/25\right)\left\{\exp \left[\left(V-45\right)/25\right]-1\right\}/\left\{\left[\exp \left(V/25\right)-1\right]\left[1+\exp \left(-\left\{V+62\right\}/3.5\right)\right]\right\}$$ (10)

Fig. 10.

Background currents in SCN neurons from Fig. 13 of Jackson et al. (2004) reproduced with permission from the Journal of Neuroscience. The noisy traces represent current measurements before and after the addition of 300 nM TTX to the bathing medium. These records are in response to voltage ramps (20 mV/s) applied from −98 to +12 mV. A portion of those recordings is shown here. The solid curves correspond to I_L + I_K,DR +I_NaP in control and I_L + I_K,DR with TTX, where I_L = 0.07(V + 10), I_K,DR = 16 n_∞⁴(V)(V + 96) with n_∞(V) = α_n/(α_n + β_n) and α_n = −0.01(V + 27)/{exp[−0.08(V + 27)] − 1} and β_n = 0.125exp[−(V + 37)/30] based on recordings of I_K in SCN neurons (Bouskila and Dudek 1995; Clay 2009). The I_NaP component is given by Eq. 10.

This result, together with I_L and I_K,DR, is represented by the line describing the control result (Fig. 10).

The voltage waveform during the interspike interval of Fig. 5A of Jackson et al. (2004) was digitized as indicated in Fig. 11, top. The corresponding AP-clamp recordings of I_Na and I_Ca are reproduced in Fig. 11, bottom. Also shown are results for I_NaP from Eq. 10 (filled circles). This component provides a good description of I_Na during the latter part of the interspike interval. The nimodipine-sensitive I_Ca component is sufficient for the I_Ca results in Fig. 11 (open circles). The nimodipine-insensitive I_Ca component is activated at potentials that are considerably more positive (Fig. 6) than those used in Fig. 11.

Fig. 11.

I_Na (red) and I_Ca (blue) during an interspike interval of a spontaneously firing SCN neuron (Fig. 5C of Jackson et al. 2004 reproduced with permission from the Journal of Neuroscience). The voltage waveform at top is a digitized version of the voltage tracing in Fig 5A of Jackson et al. (2004). The recordings at bottom were obtained in AP-clamp mode. The filled circles (I_NaP) were determined from Eq. 10. The open circles represent the nimodipine-sensitive I_Ca component as described in the text.

DISCUSSION

This report describes a novel method for constructing models of ionic currents in excitable preparations in which AP-clamp recordings are used in conjunction with voltage-step results. The traditional approach relies on voltage-step results alone (Hodgkin and Huxley 1952). Models of the voltage-gated ion current components underlying excitability, I_Na and I_K in the case of squid axons, are fitted to voltage-step recordings. The equations for I_Na and I_K obtained from this analysis are then used to simulate an AP. A comparison of simulated and experimental APs provides a test of the model. An intermediate step in this process is proposed here in which the model of each individual ionic component obtained from voltage-step analysis, I_Na for example, is further tested by an experimental recording of an AP that is applied computationally to the model and compared to a recording of I_Na obtained in AP clamp for model verification, or if need be, alterations in the model. In the I_Na results above, the model developed from voltage steps was not altered for the AP-clamp result (Fig. 3). That analysis reveals a discrepancy between experiment and theory that has not yet been resolved. Voltage-step and AP-clamp results were used together for construction of the I_Ca model.

I_Na component.

The model of Engel and Jonas (2005) was a significant part of this analysis. Initial attempts to simulate I_Na during an AP from an SCN neuron based on modifications of the Hodgkin and Huxley (1952) αs and βs for I_Na led to a substantial I_Na during repolarization (simulations not shown), a result that is appropriate for squid axons but not for SCN neurons. The AP model of Engel and Jonas (2005) predicts a separation of the I_Na and I_K components on the time axis (Clay 2013), consistent with experiment for hippocampal neurons as well as SCN neurons (Alle et al. 2009; Jackson et al. 2004), which makes their model of I_Na gating a more appropriate starting point for building a model of I_Na gating for SCN neurons than the Hodgkin and Huxley (1952) model. The αs and βs for the SCN I_Na (Eq. 4) correspond to results obtained at room temperature (Jackson et al. 2004). They are ∼10 times smaller than the αs and βs for hippocampal mossy fiber I_Na (Eq. 3), results also obtained at room temperature (Engel and Jonas 2005). This comparison is consistent with considerably faster I_Na gating for the latter preparation compared with the SCN. Both results have been described with the m³h kinetic scheme, a squid-based model (Hodgkin and Huxley 1952). Later work found their model to be incomplete for squid axons based primarily on gating currents (Vandenberg and Bezanilla 1991a and other studies cited therein). Vandenberg and Bezanilla (1991b) proposed an alternative model that describes most if not all I_Na results from squid axons. Their model is, unfortunately, cumbersome—not easy to use—and perhaps not applicable to mammalian preparations. The original m³h model is relatively simple, and it does describe whole cell currents from squid axons and mammalian preparations (Clay 2013). Therefore, its continued use appears to be appropriate provided the relevant αs and βs are also used.

Subthreshold I_Na.

A time-independent TTX-sensitive Na⁺ current activated close to, or slightly below, AP threshold and having relatively small amplitudes—I_NaP—has been reported in many mammalian neuron preparations (Bean 2007). The molecular basis of I_NaP has not yet been resolved. Some groups have suggested that it is attributable to a set of channels that are distinct from the traditional I_Na channel (Crill 1996). Other investigators have suggested that it is, in fact, attributable to I_Na (Taddese and Bean 2002). The former approach was used for the results in Figs. 10 and 11 since it is simpler, computationally, than requiring I_NaP to be incorporated in the m³h kinetic scheme.

I_Ca component.

The voltage-step recordings of I_Ca of Jackson et al. (2004) in the absence of similar results with nimodipine (their Fig. 9A) do not clearly indicate the presence of two kinetically distinct I_Ca components. In contrast, AP-clamp recordings of I_Ca (their Fig. 12) are suggestive of this result, in particular the inflection on the rising phase of I_Ca during the initial phase of AP repolarization. The nimodipine-sensitive component is activated at relatively negative potentials with sufficiently rapid kinetics (Fig. 6) so that it is nearly in step with the membrane potential during the upstroke phase of the AP. The nimodipine-insensitive component is activated at a slower rate and at potentials that are depolarized relative to the nimodipine-sensitive component, thereby accounting for the inflection in I_Ca in the AP-clamp recording. These results provide further evidence for the utility of the AP clamp methodology.

In contrast to I_Na, the I_Ca component clearly does contribute during repolarization of the SCN AP. It may help “shape” the AP. In particular, the “surge” in I_Ca during repolarization that in the I_Ca model is attributable to GHK rectification (Fig. 9A) may explain the relatively long half-width of the SCN AP, ∼2.5 ms (Jackson et al. 2004). Moreover, the influx of Ca²⁺ associated with I_Ca influences the Ca²⁺-dependent current, I_K,Ca, in particular K⁺ current associated with BK channels (Jackson et al. 2004). A transient rise of Ca_i²⁺ in the vicinity of these channels somewhere in the 10–20 μM range is believed necessary to shift the voltage-dependent BK activation curve to within the range of potentials spanned by an AP (Berkenfeld et al. 2006; Fakler and Adelman 2008). The simulations in Fig. 7A and Fig. 8 predict that Ca_i²⁺ briefly reaches the 18.5 μM level during the latter part of the AP, a result that serendipitously falls within the 10–20 μM range. Jackson et al. (2004) reported a robust I_K,Ca component obtained with the AP clamp that they attributed to BK. They did not report voltage-step results for I_K,Ca, which would be necessary for the analysis described here.

In addition to being permeable to calcium ions, Ca²⁺ channels are known to have a small permeability to intracellular K⁺ (Hille 2001), which influences the reversal potential for the channel but has relatively little effect on Ca²⁺ currents for V < 25 mV, the range of potentials investigated in this report.

Other current components in SCN neurons.

Analysis of non-Ca²⁺-activated K⁺ currents in SCN neurons with the AP clamp technique does not appear straightforward. Jackson et al. (2004) found that addition of 10 mM TEA⁺ to the bath completely removed net outward current during an AP. In contrast, 30 mM TEA⁺ does not completely block the delayed rectifier K⁺ current, I_K,DR, in voltage-step analysis (Bouskila and Dudek 1995). Perhaps modeling of both sets of results may lead to a resolution of this apparent paradox. SCN neurons also have the transient, rapidly inactivating K⁺ current, I_A (Bouskila and Dudek 1995; Itri et al. 2010) that is completely blocked by 5 mM 4-aminopyridine (4-AP; Huang et al. 1993). Moreover, 4-AP modifies spiking behavior of SCN neurons and the shape of the SCN AP (Itri et al. 2005). This result suggests that AP-clamp analysis of SCN neurons before and after bath application of 4-AP would be of interest.

SCN and circadian rhythms: relationship to ionic currents.

Circadian rhythms in mammals are coordinated by the hypothalamic SCN (Brancaccio et al. 2013). Spontaneously occurring APs in the SCN exhibit diurnal patterning. During the day SCN neurons are more active than at night, having firing frequencies of 8–10 Hz. At night, activity is suppressed to <2 Hz, on average, with many neurons in the silent state (Green and Gillette 1982; Groos and Hendriks 1982; Inouye and Kawamura 1979; Shibata et al. 1982; Yamazaki et al. 1998). Potential roles of I_Ca, I_A, and I_K,Ca have been emphasized in the regulation of day-night differences of SCN firing rate (Kent and Meredith 2008). Pennartz et al. (2002) found that Ca²⁺-free bathing media reversibly suppressed firing of APs in SCN neurons during the day. Moreover, firing of APs during the day was blocked by the addition of 2 μM nimodipine to the bathing medium. At night I_Ca was reduced ∼50%, which is consistent with a reduction in firing rate at night (Pennartz et al. 2002). Nimodipine-sensitive I_Ca is activated close to AP threshold with relatively fast kinetics, so that blockade of this component is consistent with block of firing during the day even though this portion of I_Ca is relatively small (Jackson et al. 2004; Pennartz et al. 2002). Itri et al. (2005) found a day-night difference in the amplitude of I_A in SCN neurons, with a reduction of this component by ∼50% at night relative to the day. Moreover, 4-AP, a blocker of I_A, reduces firing rate during the day by ∼50% (Itri et al. 2005). Finally, Meredith et al. (2006) found that daily expression of I_K,Ca, BK channels, is controlled by the intrinsic circadian clock. Specifically, BK channel-null mice have increased spontaneous firing rates selectively at night and weak circadian amplitudes in multiple behaviors timed by the SCN. A number of reports including Jackson et al. (2004) have provided indirect evidence for the presence of BK channels and other I_K,Ca channel subtypes in SCN neurons from electrophysiological recordings (Cloes and Sather 2003; Pitts et al. 2006; Teshima et al. 2003). A complete description of this component will also require voltage-step results. That work necessitates the use of the excised patch-clamp technique in the inside-out configuration so that Ca_i²⁺ and membrane potential V can both be controlled during the experiments. This approach has been used for BK currents from mslo channels heterologously expressed in Xenopus oocytes (Cui et al. 1997) and BK splice variants heterologously expressed in HEK cells (Shelley et al. 2013). Similar results from SCN neurons at various points in the day-night cycle will provide information on diurnal changes in I_K,Ca in the SCN.

The results of Jackson et al. (2004), the focus of this work, were taken from acutely dissociated single SCN neurons. This approach appears necessary to ensure isopotentiality during voltage-step and AP-clamp recordings especially for I_Na. These cells are physiologically relevant since they exhibit electrical properties similar to those described in SCN brain slices (Pennartz et al. 2002). Nevertheless, contributions to excitability of ion channels in dendritic spines and axons, anatomical features that are not clearly present in dissociated cells, cannot be excluded. Furthermore, SCN neurons are not homogeneous. The preparation used by Jackson et al. (2004) was enriched with neurons stained for arginine vasopressin (AVP). These neurons exhibited rhythmic subthreshold oscillations in the presence of TTX but were still likely a heterogeneous population reflected by a dramatically different ratio of I_Na and I_Ca between SCN neurons that were firing at the same frequency (Fig. 6, Jackson et al. 2004).

Dynamic clamp: comparison with AP and voltage clamp step techniques.

The AP clamp technique is similar to the dynamic clamp method, an approach originally used in cardiac electrophysiology that also has been used in neuroscience and in other fields (Goaillard and Marder 2006; Wilders 2006). As the name implies, dynamic clamping involves real-time injection of current into the preparation under investigation during an ongoing experiment, a procedure not required with AP clamp. The AP clamp method does require the preparation under study to be sufficiently stable to permit several steps: 1) recording of an AP, 2) application of the AP waveform to the preparation in voltage-clamp mode in control conditions, 3) voltage-step recordings in control with a particular ionic conductance in mind, I_Na, for example, and 4) application of the AP waveform and voltage steps in voltage clamp after the addition of TTX to the bath. The results are the differences in membrane currents in control and test conditions both for the AP and for voltage steps. This report describes an extension of the method in which an experimentally recorded AP is digitized and the resulting V_i vs. t_i data set applied to mathematical models of the relevant ionic conductance, such as I_Na. The result is compared with an experimental AP-clamp recording of that component either for model validation or for modification of model parameters as noted above. A review of the literature did not yield any other reports in which this procedure has been used. A related approach concerning models of I_Na has been reported for raphe pacemaker neurons (Milescu et al. 2008). Moreover, waveforms other than rectangular steps have been used previously. For example, Fohlmeister and Adelman (1985) used sinusoids in voltage clamp to measure I_Na gating currents in squid giant axons, and they analyzed their results with the Hodgkin and Huxley (1952) I_Na model. An AP waveform is, perhaps, of greater interest compared with sine waves or rectangular steps since it has direct physiological relevance.

Summary.

A novel extension of the AP clamp technique is described involving models of the ionic conductances in the cell from which the AP was recorded. The method is general. It has been applied to SCN neurons with a goal of describing a complete mathematical model of the AP in these cells. The equations for I_Na and I_Ca given above form one part of that model, a model that may have broad applicability not only for the SCN but also for other mammalian preparations.

GRANTS

This research was supported by the Intramural Research Program of the National Institute of Neurological Disorders and Stroke, National Institutes of Health, Bethesda, MD.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

Author contributions: J.R.C. conception and design of research; J.R.C. analyzed data; J.R.C. interpreted results of experiments; J.R.C. prepared figures; J.R.C. drafted manuscript; J.R.C. edited and revised manuscript; J.R.C. approved final version of manuscript.