Sodium Currents Activate without a Hodgkin and Huxley-Type Delay in Central Mammalian Neurons

Currents recorded at 12°C had properties of typical fast sodium currents

Electrically compact neurons were obtained by acute dissociation of rat prefrontal cortex. Cells with pyramidal shape and a short (<20 μm) stump of the apical dendrite were selected for the recordings. In these neurons, the TTX-sensitive currents recorded at 12°C activated very rapidly (<2 ms) and then decayed with time constants ranging from ∼1 to ∼35 ms (Fig. 1A). Hence, the recorded current was similar to the fast TTX-sensitive sodium current described in prefrontal cortex pyramidal neurons (Maurice et al., 2001).

Because several types of neurons are found in the layer V of the rat frontal cortex (Kawaguchi, 1993), we used single-cell RT-PCR to characterize the cell type in a subset of the recorded neurons (n = 5). In these neurons, cell identification was performed by testing the presence of the CaMKII and GAD₆₇ mRNAs. As shown in Figure 1A (inset), all neurons were GAD₆₇ negative but positive for CaMKII, demonstrating that all of these neurons were glutamatergic. All five neurons satisfied the below described requirements for the voltage-clamp speed and were used for the sodium current analysis.

Voltage-clamp quality was sufficient to determine the activation time course of sodium currents

Because the goal of these experiments was to establish the activation kinetics of sodium currents, it was critical to verify the quality of voltage clamp in our experimental conditions. To this aim, tail currents representing the deactivation kinetics were recorded (Fig. 1B,C). At constant voltage, the time course of sodium channel closure can be usually fit with a monoexponential function (Hodgkin and Huxley, 1952). Because the time constant of the monoexponential function is voltage sensitive, any loss of voltage control will result in a non-monoexponential decay of the current. In addition, suboptimal voltage control will translate into unstable driving force for sodium ions, thus affecting the amplitude of the sodium current and additionally deteriorating the monoexponential time course of the current decay. The current time course matched very closely the monoexponential fit function after ∼120 μs from the start of the voltage step (Fig. 1D). Hence, there is a reasonable voltage control of the current after ∼120 μs from the start of the voltage command for voltage steps of <30 mV.

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Figure 1.

The speed of voltage clamp in acutely dissociated prefrontal cortex neurons is sufficient to investigate activation kinetics of sodium current at 12°C. A, An example of current traces evoked by the voltage-clamp protocol shown above the traces. The inset shows a gel obtained after the RT-PCR analysis of the recorded neuron, and its current traces are shown in A-D and F. CaMKII mRNA but not GAD₆₇ mRNA was detected in this neuron. B, The voltageclamp protocol shown above the traces was used to study the deactivation kinetics of sodium currents (tail currents). The dark thick trace corresponds to the -56 mV voltage step, the dark thin trace corresponds to the -46 mV voltage step, and the light thick trace corresponds to the -36 mV voltage step. C, The same traces of B are shown on an expanded time scale. Note the increase in the peak amplitude of the tail current at more hyperpolarized membrane potentials. The first ∼100 μs after the start of the test voltage step were blanked for clarity. D, A semilogarithmic plot of the fast component of the traces shown in C demonstrates the monoexponential decay of the tail currents starting ∼100 μs after the start of the voltage step. The slow component of the traces was subtracted as shown in Figure 2G. Dashed lines represent the monoexponential fits. E, Total amplitude of the tail current measured at the start of the test step (120-220 μs after the start of the test pulse) plotted versus the voltage of the test step. The thin line represents the Goldman-Hodgkin-Katz equation fit of the data points. The arrow points to the extrapolated reversal potential of the tail currents (-30.5 mV). F, The reversal potential of approximately -30 mV was obtained by direct measurement. All traces are from the same neuron and were obtained by subtracting the currents obtained in the presence of 0.5 μm TTX from the control traces. All traces were digitally filtered for clarity (2 kHz for A, B, F; 10 kHz for C, D). Traces in A-D are the averages of four sweeps.

An additional control for the speed and quality of the voltage clamp can be made by plotting the peak amplitude of the tail current against the voltage of the command step. In this case, the fraction of open channels at the beginning of the voltage command is the same for all test voltages, and the peak current amplitude must be proportional to the driving force for sodium ions. Figure 1, E and F, shows that the sodium ion reversal potential estimated from the tail currents was similar to the reversal potential obtained from the peak current measurements in the same cell. The value of the measured reversal potential for the sodium current was close to the calculated reversal potential of -22 mV for the solutions used during these recordings (10 mm Na⁺ in the external solution and 25 mm of Na⁺ in the pipette solution; see Materials and Methods).

Similar analysis indicated that an acceptable voltage-clamp control was achieved after ≤150 μs in nine cells recorded at 12-14°C (including five cells used for RT-PCR) and seven cells recorded at room temperature (22-24°C). Only these cells were further analyzed.

The time resolution of the currents (∼150 μs) was sufficient to detect the delay of currents activation when the activation time constants are >0.5 ms (for details, see Appendix).

Sodium currents activate without the mⁿ type delay at 12°C in prefrontal cortex neurons

For traces obtained at 12°C, a preliminary examination of the sodium current activation revealed no delay at less than -40 mV (Fig. 2A). However, a more detailed examination of the time course of the sodium current activation can be performed by the mathematical elimination of inactivation (Keynes and Rojas, 1976). In this procedure, the current trace is treated as a multiple of two components: the raising phase or “activation” component 1 - exp(-t/τ_act)ⁿ, where n = 1-3, and the decaying phase or “inactivation” component exp(-t/τ_inact). By dividing the current trace by the decaying component exp(-t/τ_inact), the raising phase or activation component 1 - exp(-t/τ_act) is revealed as a non-decaying trace (Fig. 2D, inset) (for details, see Appendix). Hence, following this procedure, the time course of current raising phase should be determined solely by the function of exp(-t/τ_act).

It should be noted that, for such analysis, we assume that macroscopic inactivation time constants correspond to the microscopic inactivation time constants and that inactivation and activation processes are independent. Single-channel recordings have shown that macroscopic inactivation rates may not correspond to the microscopic inactivation rates (Aldrich et al., 1983); since then, the sodium channel inactivation process is usually assumed to be either voltage independent (Kuo and Bean, 1994) or little voltage dependent (Vandenberg and Bezanilla, 1991). The relevance of these models to the data presented here is discussed in the section of the model selection for computer simulations. Nevertheless, computer simulations show that, in most of the cases, the mathematical procedure used in this study “to remove” inactivation is suitable to reveal the kinetics of the step(s) controlling activation process. For instance, in the case of the sequential model “closed state” ↔ “open state” ↔ “inactivated state” in which the first transition is slow whereas the second is fast (Hille, 2001), this procedure will identify the second transition time constant, the one that determines the activation kinetics in this model. Moreover, for voltage steps to less than -40 mV, the correction is minimal and unlikely to introduce any artifacts.

Nevertheless, we realize that it is a purely mathematical procedure, and caution should be used while applying the procedure. Hence, we used a double-pulse protocol to detect possible presence of a rapid inactivation that may interfere with the activation process (Hille, 2001). Figure 2C shows that there was a good match between the current decay after a step to -46 mV and the estimated inactivation time course measured with a protocol shown in Figure 2B. Importantly, there is no additional rapid decay phase during the first 2 ms after the start of the test command. In both cases, the time course could be fit with a monoexponential function, and the obtained time constants differed by <30%. This difference may be attributable to the relatively brief duration of the test command (two to three times that of the inactivation time constant).

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Figure 2.

Sodium current activation can be fit with a monoexponential function at 12°C. A, An example of the sodium current activation recorded from an acutely dissociated prefrontal cortex neuron. Currents were evoked by the voltage-clamp protocol shown above the traces. B, Voltage clamp protocol used to test the presence of a fast component of inactivation. Currents were evoked by the protocol shown above the traces; the inactivation time course at -46 mV was estimated by measuring the amplitude of the peak current at -36 mV. C, There was a clear overlap between the inactivation time course obtained by the protocol shown in B and the decay of sodium current evoked with a voltage step to -46 mV. No fast inactivation was detected during the first 2-5 ms. D, The inactivation of the current trace shown in C was eliminated mathematically by dividing the current trace by the following function: 28.8 pA + 134.5 pA × exp(-t/16.1 ms). The inset shows that this procedure converted the decaying current to a nondecaying current. The main panel shows the same converted trace on an expanded time scale and after the subtraction of the trace from 1. The light line represents the monoexponential fit of the activation. Note the absence of any detectable delay. For details of all procedures, see Appendix. E, A semilogarithmic plot of the trace from D. The monoexponential fit is shown as a thick dashed line. Two smooth lines correspond to the quadrate (thin line) and the cube (thick line) of an exponential function. F, Tail current evoked by the voltage-clamp protocol shown above the traces. G, A semilogarithmic plot of the tail current shown in F reveals the presence of two distinct components. The slowly decaying component was fit with a monoexponential function with a time constant of 18 ms, which is close to the inactivation rate of the current at -46 mV (C). This slow component was subtracted from the original trace, and the semilogarithmic plot of the fast component is shown in H. H, The monoexponential fit for the fast component of the tail current (dark line) is shown as a light dashed line. Note that the obtained rate of tail current decay is similar to the activation rate at this voltage (0.80 vs 0.85 ms; E). For comparison, the activation current trace from E is shown as a thin light line.

The elimination of inactivation was performed by dividing the current trace by the monoexponential fit function (see Appendix). The result is a nondecaying trace (Fig. 2D, inset). The absence of any significant deviation of the obtained current amplitude from 1 (except during the activation process for the first 2-3 ms) confirms that the original current trace decayed as a monoexponential function. The same observation was made for all nine neurons that were analyzed using this method for voltages between -56 and -40 mV. At membrane potentials above -40 mV, a second slower component of inactivation could be detected. However, this second component comprised <30% of the total current amplitude and was very slow (inactivation time constants >20 ms at 12-14°C). Therefore, during the analysis of the sodium current activation time course, the slower inactivation component was included into the non-inactivating term of the exponential function used to eliminate inactivation (for details, see Appendix).

Figure 2D shows that the current activation time course could be fit with a monoexponential function. Therefore, the current trace obtained in the semilogarithmic plot follows a straight line (Fig. 2E). This linear portion of the trace was used to determine the activation time constant τ_m. For comparison, the lines corresponding to the square and the cube of an exponential function with the same time constant τ_m have been added to this plot (Fig. 2E) (for details, see Appendix). It is obvious that the activation time course is very different from the square or the cube of an exponential function.

The same clear difference between the m² and the m³ function and the actual current traces was observed in all nine neurons that satisfied the above described criteria for the quality of the voltage clamp at 12-14°C. In Figure 2E, the duration of the delay can be measured as the time interval between the start of the voltage command and the time when the extrapolated monoexponential fit function is equal to 1 (Fig. 2E) (for details, see Appendix). For current traces obtained at the test voltage of -46 mV (holding potential of -66 or -76 mV), the average delay was 140 μs (0.13 ± 0.02 × τ_m, ranging between 0.02 × τ_m and 0.19 × τ_m, where τ_m is the time constant of the monoexponential fit; n = 9). Similar observations were made for test voltages between -56 and -36 mV. In this range of voltages and for holding potentials of either -76 or -66 mV, the delay never exceeded 270 μs and was always <0.3 × τ_m. Because the estimated membrane charging time is <100-150 μs (Fig. 1), such brief delays may be caused at least in part by membrane charging. However, these data do not exclude the existence of a brief delay (see below) similar to what reported for the Kv3 potassium currents (Lien and Jonas, 2003).

For holding potentials of -76 or -66 mV, it is possible that the delay in the activation of the sodium current is reduced because, at such depolarized potentials, a large fraction of the gates is already in the open position and therefore the sodium current can be generated much faster. However, for the test voltage steps to -36 mV, the delay was increased by <100 μs when a 500 ms conditioning step to -116 mV instead of the step to -66 mV preceded the test pulse (n = 3) (for the protocol, see Fig. 4C). Such a small increase in the delay may be attributable to the longer time required to charge the membrane after a large voltage step. In any case, for the test voltage steps to -36 mV, the delay after the -116 mV conditioning step never exceed 0.4 × τ_m (n = 3) and was significantly shorter than the m² function delay of the 0.69 × τ_m (for the formula, see Appendix).

Most of the sodium current kinetic properties are consistent with a single transition model

The monoexponential activation time course suggests that a single transition controls the kinetics of sodium current activation. If this is the case, for a test step to a given voltage, activation and deactivation time constant of the current should be the same. To test this hypothesis, we analyzed the tail currents obtained with a two-step protocol in the same cell (Fig. 2F, G).

Voltage steps to -46 mV that were preceded by a conditioning step to -36 mV evoked currents with a two-component decay (Fig. 2F, G). We assumed that the slowly decaying component represents the inactivation of the activated current, whereas the rapidly decaying component represents the deactivation process (Oxford, 1981) (for an additional analysis, see below). As shown in Figure 2H, the time constant of the monoexponential fit of the rapidly decaying component was similar to the current activation time constant (Fig. 2E), and, consequently, the traces overlap after normalization.

The close match of activation and deactivation time constants was evident for all of the voltages investigated (Fig. 3A). Because of technical difficulties, only a limited range between -56 and -31 mV could be tested. The same close match of activation and deactivation time constants could be detected also at room temperature (Fig. 3B). The voltage dependence of the time constants remained essentially unaltered at both temperatures: the fits used for the time constants obtained at 12-14°C could be used for the time constants measured at 22-24°C after division of the fit function by 2.8, a value close to the expected Q₁₀ value (Hille, 2001). These results suggest that no unusual changes in the kinetics of sodium channels occur between 12-14°C and 22-24°C. At room temperature, a direct test of the presence of the delay was less accurate because the first 100-150 μs of the recordings after the voltage step were unreliable, and this time is comparable with the time range obtained for the activation time constants (150-400 μs) (Fig. 3B) (but see below, section about nucleated patch recordings at room temperature in hippocampal CA1 pyramidal neurons and dentate gyrus granule cells). Nevertheless, at room temperature, in prefrontal cortex pyramidal neurons, there was no indication that sodium currents activate after the mⁿ (n ≥ 2) type delay.

In a two-state model, the voltage dependence of the current amplitude is usually tied to the voltage dependence of transition rates (Hille, 2001). In prefrontal cortex pyramidal neurons, the activation and deactivation rates increased e-fold for each ∼12 mV (this value tended to increase at membrane potentials below -60 mV and above -10 mV). This voltage dependence of the activation/deactivation rates predicts a value of 6 mV for the slope factor of the activation voltage dependence, which matches the observed slope value (Fig. 3C). For this graph, it was assumed that current decay represents inactivation process, and the total current amplitude was obtained by extrapolation of the decay of the current to the start of the voltage command (for details, see Materials and Methods).

However, it is possible that the actual microscopic time constants do not correspond to the macroscopic time constants (Aldrich et al., 1983; Hille, 2001). For sodium channels, it has been proposed that inactivation process is voltage independent and the apparent voltage dependence of inactivation rate constants is attributable to coupling with activation process (Aldrich et al., 1983; Kuo and Bean, 1994). The following experiments were aimed to test whether a model with a single open state and voltage-independent inactivation can account for the sodium currents behavior in prefrontal cortex pyramidal neurons.

First, we investigated the voltage dependence of the sodium current at below -35 mV. To this end, in the same neuron, we measured the peak current amplitude (Fig. 1A protocol) and divided it by the instantaneous tail current amplitude (Fig. 1B protocol) at the same voltage. If our assumption that the activation process is much faster than the inactivation process is correct, most of the channels are already activated at the peak of the current. Hence, we should obtain relative permeability numbers that should match closely the data obtained in Figure 3C. At membrane voltages above -51 mV, the obtained data (Fig. 3D, filled circles) showed similar voltage dependence to the graph in Figure 3C. Interestingly, at membrane voltages less than -51 mV, the peak current amplitude seemed to exhibit stronger voltage dependence than at more depolarized membrane potentials.

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Figure 3.

Summary of the properties of sodium current activation in prefrontal cortex pyramidal neurons. A, Time constants of activation (crosses) and deactivation (circles) at 12°C plotted as a function of the test voltage. All time constants were measured as shown in Figure 2. The thin line represents the fit function of 1/(11/(0.4 + exp(-(V + 6)/12)) + 0.035/(0.0015 + exp((V + 6)/12))), where V is the membrane voltage in millivolts. This function was used for computer simulations (for details, see Materials and Methods). B, Time constants of activation (circles) and deactivation (crosses) at 23°C are plotted as a function of the test voltage. The thin line represents the fit obtained by dividing the fit function for the 12°C by 2.8. C, Peak permeability is plotted as a function of voltage. Permeability was obtained from the corrected peak current amplitudes using the Goldman-Hodgkin-Katzequation. The correction of peak current amplitude was performed by extrapolating the exponential part of the current decay to the start of the test voltage command. Thin line represents a Boltzmann function fit of the form 1/(1 + exp[(V_h - V)/V_c], where V is the membrane potential, V_h is the half activation voltage, and V_c is the slope factor. D, Slowly decaying component of tail current (Fig. 2G, open circles) and peak current amplitude (Fig. 1 A, filled circles) are plotted as a function of the test voltage. All data are presented as a relative permeability that was calculated by dividing current amplitude by the instantaneous total tail current amplitude (Fig. 1 B protocol) at the same voltage. Thereafter, the attained values were normalized to the relative permeability calculated for -36 mV. The amplitude of the slowly decaying component was obtained as shown in Figure 2, G and H. The straight line represents the monoexponential fit to the relative permeability values of the slow tail current component. Peak currents were measured during voltage-clamp protocol shown in Figure 1 A. E, Deactivation time constants saturate at approximately -36 mV. The fast component of tail currents was obtained after peeling of the slow component as it is shown in Figure 2 E-H. The voltage-clamp protocol was the same as in Figures 1 B and 2G, except that the conditioning step was to -26 mV. To facilitate deactivation rate comparison, tail current amplitudes were set to 1 at the beginning of the test voltage command (100 μs after the end of the conditioning step). The first 100 μs after the conditioning step to -26 mV are blanked for clarity. F, There is an increase in the sodium current activation rates below -41 mV. Currents were evoked by the voltage-clamp protocol shown at the top. To facilitate activation rate comparison, the amplitude of all currents was set to -1 at the peak of the current.

In addition, we measured the amplitude of the slow tail current component normalized to the instantaneous tail current amplitude at the same voltages. If the fast tail current component represents deactivation process and the slow component represents inactivation process, the normalized slow current amplitude should have voltage dependence similar to the voltage dependence of the peak current. The obtained data (Fig. 3D, open circles) show that both datasets have similar voltage dependence, although some differences could be detected. Notably, the data points corresponding to the slow tail current component were somehow lower than the data points corresponding to the peak current measurements. However, similar behavior could be observed in the computer model described below.

It is important to notice that voltage dependence of both macroscopic activation and deactivation rates had clear peaks (Fig. 3A). Although it was difficult to show a peak of deactivation rates in a single cell (Fig. 3E), activation rates were clearly the slowest at approximately -40 mV and started to accelerated at more hyperpolarized membrane potentials as well as at more depolarized potentials (Fig. 3F). All of these observations will be discussed in the next section while justifying the proposed sodium channel model.

The action potential shape in prefrontal cortex neurons can be faithfully reproduced with computer simulations

The activation kinetics of sodium currents affects the shape of action potential (Moore and Cox, 1976; Sangrey et al., 2004). Therefore, the validity of the model in which a single transition determines the sodium current activation kinetics can be tested by comparing the shape of action potentials recorded in the prefrontal cortex neurons with the shape of the computer-generated action potentials. To this aim, we developed a computer model that reproduces the obtained current traces during voltage-clamp experiments. Before we discuss the model selection, we present here the data on sodium current inactivation that are necessary for a complete model (Fig. 4).

As shown in Figure 2, C and D, at membrane voltages below -40 mV, the current decay could be fit by a monoexponential function with a non-inactivating component, whereas at potentials above -40 mV, an additional slower component was detected. This component, however, accounted for <30% of the total current. Similarly, the recovery from inactivation could be usually fit with a monoexponential function. Because we were interested only in the shape of the action potential, we made no attempt to include such a slow current component into the model. Figure 4B shows the summary of the inactivation and recovery from inactivation time constants corresponding to the fast component of each process (Fig. 4A,B). These time constants peaked at approximately -65 to -70 mV, which was close to the half-inactivation voltage (Fig. 4C,D). These results indicate that a single transition between two states is sufficient to describe the inactivation/deinactivation process.

Based on these results, a six-state model was constructed (Fig. 5A). Because a complete sodium channel model was not our goal, we selected the simplest model that could reproduce all key observations reported here. First, at the same voltage, macroscopic activation and deactivation rates were similar (Figs. 2H, 3A). Second, there was a clear peak of activation rate voltage dependence (Fig. 3A,F) and good evidence that deactivation rates had a similar peak (Fig. 3A, E). Third, the total area under the current normalized to the driving force for the sodium ions decreases for voltages more than -20 mV (data not shown) that contradicts the predictions of a tight coupling between activation and inactivation (Bean, 1981). Moreover, computer simulations (data not shown) suggest that one open-state and multiple closed-state models with voltage independent inactivation (Kuo and Bean, 1994) are unlikely to explain why, for membrane potential change from -41 to -51 mV, the macroscopic activation time constant decreased (Fig. 3F), whereas the macroscopic inactivation time constants increased (Fig. 4B). More sophisticated schemes with several open states (Vandenberg and Bezanilla, 1991) or a modified and more complex version of the Kuo and Bean model (Kuo and Bean, 1994) could explain the observed current behavior; however, we opted for a simpler scheme in which the parameters were more easily related to the observed data.

In our model, the inactivation and activation processes are independent, as it was proposed in the original Hodgkin and Huxley model. We did not use another alternative model that is the one with slow activation and fast voltage-dependent inactivation rates (Hille, 2001). Such a model can well reproduce most of the data presented here (Figs. 1, 2, 3). However, we did not see why it should be preferred over an easily interpretable model presented here.

In our model, the activation process was simulated by a two-step process. There were several reasons for the addition of a second closed state. First, there was the ∼5 mV difference between the half-activation voltage and the voltage at which activation/deactivation time constants reached their peak values (Fig. 3, compare A, C). Second, at membrane voltages below -60 mV, >12 mV corresponded to the e-fold change in the deactivation time constant (Fig. 3A). Meanwhile, at membrane potentials below -50 mV, the voltage dependence of the current amplitude became steeper (Fig. 3D). This steeper voltage dependence was consistent with the low action potential threshold (Fig. 5D). The addition of a fast voltage-dependent transition between the two closed states was sufficient to account for both observations, although it could not explain all complex voltage dependence of the sodium currents in prefrontal cortex pyramidal neurons (for comparison, see below).

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Figure 4.

Inactivation properties of sodium currents in prefrontal cortex pyramidal neurons. A, Current traces evoked by the voltage protocol used to determine the recovery from inactivation time constants. The protocol is shown above the traces. B, Time constants of inactivation and recovery from inactivation plotted as a function of the amplitude of the conditioning voltage step. Inactivation rates were measured by fitting the current decay with the monoexponential or biexponential function. The biexponential fit was required only for voltages above -40 mV. In these cases, the dominant fast component was used for the plot. The thin line for the data points at 12°C represents the fit function of 1/(2/(2 + exp(-(V + 6)/12)) + 0.00005 × exp(-(V + 6)/13)), where V is the membrane potential in millivolts. The fit function for the data points at 23°C was obtained by dividing the fit function at 12°C by 2.4. The parameters of the fit function were used for the computer simulations (for details, see Materials and Methods). C, Currents evoked by the voltage-clamp protocol shown above the traces. The conditioning step duration was 500 ms. D, Amplitude of the current at -36 mV plotted as a function of the conditioning step voltage. The thin line represents a Boltzmann function fit.

Finally, there was short delay (∼100 μs) of the sodium current activation; imperfect voltage-clamp might account for this delay, but it is also possible that it was not an experimental artifact (see below) (Lien and Jonas, 2003). Therefore, the peak value of the time constants corresponding to the transition between two closed states was set to be ∼100 μs.

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Figure 5.

A two-step activation model accurately reproduces the sodium current behavior and the shape of action potentials in prefrontal cortex pyramidal neurons. A, States and transitions scheme used for the computer simulations of the sodium current. C1 and C2 represent the closed states; I1-I3 represent the inactivated states; O represents the open state. For functions determining the individual rates, see Materials and Methods. B, Simulated currents evoked by a voltage-clamp protocol shown above the traces. The inset shows the start of the activation of the simulated (dark lines) and recorded (light lines) currents for two voltages (-46 and -41 mV). The dashed line represents the simulated current response to the -41 mV voltage step obtained with the m³ model of Mainen et al. (1995). The peak current amplitude was normalized to the peak current amplitude of the model shown in A. All traces correspond to the 12°C. C, Simulated currents were evoked by the voltage-clamp protocol shown above the traces. D, In the model, activation (gray circles) and deactivation (gray triangles) rates are similar at the same voltage. Crosses correspond to experimental data from Figure 3A. Line represents deactivation rates of the m³ model (⅓ of the time constant corresponding to the transition from the closed to the open state in the HH model). E, Activation voltage dependence obtained for simulated currents. Dark thin line represents a Boltzmann function fit. Light thick line corresponds to the experimental data shown in Figure 3C. F, In the model, the voltage dependence of the tail current slow component amplitude (open circles) and the peak current amplitude (filled dark circles) are similar. Experimental data of peak current amplitude are shown as gray filled circles. All data are presented as relative permeability obtained by dividing current amplitudes by the instantaneous total tail current amplitude at the start of the test pulse at the same voltage. Thereafter, the attained values were normalized to the relative permeability value obtained at -26 mV. G, H, The simulated with the m¹ sodium channel model action potential (light thick line) is very similar in shape to the action potential recorded experimentally (thin dark line). In G, sodium channels density is the same in soma and axons. In H, sodium channels density is approximately eight times higher in axons compared with soma. For current densities used during simulations, see Materials and Methods. For comparison, an action potential simulated with the m³ model is shown (dotted line). Note a smoother transition to spike in this model. I, J, The membrane potential depolarization speed is reproduced well in the model with the higher sodium channel density in axons. In I, sodium channel density is the same in soma and axons. In J, sodium channel density is approximately eight times higher in axons compared with soma. Trace notation is the same as in G and H. Note clearly asymmetrical time course of the depolarization speed in the recorded spike and in the models with higher axonal channels density.

Our model reproduces well both the voltage dependence and kinetics of the sodium currents recorded from prefrontal cortex pyramidal neurons (Fig. 5B, C). The inset of Figure 5B demonstrates that the time course of sodium current activation matched well the time course of the simulated currents. In contrast, at -41 mV, the amplitude of the current simulated with the m³ kinetics model [HH model (Mainen et al., 1995)] was approximately threefold smaller during the first 0.5 ms after the start of the voltage step (Fig. 5B, inset), despite the fact that there is only ∼ 30% difference between the time constant of the m³ model and the time constant of the model presented here.

For a single trace, it is possible to obtain a much better fit with the HH model; however, in this model, there is a problem of fitting currents at all voltages because voltage dependence of activation and deactivation rates is tied to the voltage dependence of the amplitude of the current. According to the m³ model, the voltage dependence of the amplitude of the current will be 3 × 2 times steeper than the voltage dependence of activation and deactivation rates because of the presence of the m³ variable. Because the observed voltage dependence of activation and deactivation rates is approximately two times less steep than the observed voltage dependence of the amplitude of the current (Fig. 3A, C), it is impossible to obtain a satisfactory match of the data reported here with the m³ kinetics HH model. In addition, the m³ kinetics HH model cannot explain the similarity of activation and deactivation rates at the same voltage. For all of these reasons, we used the m³ kinetics model that was used by several groups without any change in the parameters of the models.

As shown in Figure 5C, all essential inactivation features are reproduced by the model. Although the simulated currents do not contain the more slowly decaying component detected in the recorded currents at potentials above -41 mV (compare Figs. 1A, 5B), this slower component could be added by extending our model, as it has been done for the classical Hodgkin and Huxley model.

One of the key observations reported here is the similarity of activation and deactivation rates at the same voltage over relatively wide range of voltages (from -46 to -31 mV) (Fig. 3A). Hence, it is crucial that our proposed model can reproduce this phenomenon as shown in Figure 5D. Meanwhile, in the model, the same middle point of the voltage dependence of the current amplitude was attained, although there was a large difference in the slope values between the model and the experimental data (3.7 vs 6.4 mV) (Fig. 5E). In our model, the increased shift of the half-activation voltage resulted in the decrease of the slope factor. We selected to match the halfactivation voltage while ignoring the slope factor of the Boltzmann function fit. Such choice was dictated by the observation that, for membrane voltages below -40 mV, there was quite good correspondence between this model and the experimental data (Fig. 5F). In any case, these discrepancies between the model and the experimental data indicate that our model is an oversimplification of the real channel behavior that is not entirely unexpected.

Finally, we tested the models ability to reproduce the shape of action potential. Because the actual sodium channel density in axons is still a matter of debate (Mainen et al., 1995; Safronov et al., 2000), we tested two channel distributions during simulations. First, we assumed that the sodium channel density is the same in the axon and the soma. Under these conditions, a fairly good reproduction of spike shape could be obtained with our model, assuming a sodium channel density within the range of the experimentally observed values. For instance, the observed average sodium channel permeability was 2.5 × 10^-4 cm/s (ranging from 1.8 × 10^-4 to 3.9 × 10^-4), whereas for this simulation, a 3.2 × 10^-4 value was used (Fig. 5G,I). The main difference between the recorded and the simulated voltage traces is a smoother onset of the spike in the trace of the model (Fig. 5G, inset). In the case of the m³ HH model, the simulated spike onset was even slower, although the sodium channel permeability was similar (Fig. 5G) (for the parameters of the model, see Materials and Methods). The steep rising phase of the action potential is attained in the model with approximately eight times higher sodium channel density in the axonal compartment compared with the soma (Fig. 5H, light thick trace). Interestingly, the simulations incorporating the m³ sodium channel model showed little improvement in the steepness of the rising phase (Fig. 5H, dotted trace). This result suggests that our model is more sensitive to local changes in the channels density, probably because of the much faster activation kinetics.

Detailed information on the action potential shape can be obtained by examining the time course of the membrane potential depolarization speed. As shown in Figure 5I, in the case of the same sodium channel density in the soma and the axon, simulations with our model overestimated the maximal depolarization speed, whereas simulations with the m³ sodium channel model underestimated the maximal depolarization speed (Fig. 5F). In addition, both simulations failed to reproduce the asymmetrical profile of the experimentally observed time course of the depolarization speed: the rise to the maximum was slow, whereas the subsequent drop was much faster. However, in the case of high sodium channel density in the axonal compartment, simulations with our model were able to closely reproduce both the time course and the amplitude of the depolarization speed (Fig. 5F, thick light trace), whereas simulation with the m³ sodium channel model underestimated approximately two times the peak depolarization speeds. In addition, the initial rising phase of the depolarization speed was very slow compared with the experimental data.

Hence, the presented model is sufficient to reproduce accurately most of the properties of sodium currents and the shape of the action potentials recorded in prefrontal cortex pyramidal neurons. In contrast, the shape of action potentials generated by the model incorporating sodium currents with the m³ kinetics differed from the shape of the experimentally recorded action potential (Fig. 5G-J). In particular, the time course of depolarization speed could be matched only with simulations incorporating our model. We conclude that the presented model can reproduce the observed shape of action potential more faithfully than the traditional Hodgkin and Huxley model.

Sodium current activates with minimal delay in hippocampal CA1 pyramidal neurons and dentate gyrus granule cells

The described lack of the Hodgkin and Huxley-type delay in the sodium current activation in the prefrontal cortex pyramidal neurons is at odds with the current prevailing view that most, if not all, fast sodium currents activate after the Hodgkin and Huxley-type delay. Therefore, to test whether the sodium current activation time course in prefrontal cortex pyramidal neurons was unique to this cell type or common to other mammalian brain neurons, we recorded sodium currents from two additional types of CNS neurons. We selected hippocampal CA1 pyramidal neurons and dentate gyrus granule cells because they are two well studied and distinct neuron types.

These experiments were performed using the nucleated patch configuration, which provides, probably, the best voltage-clamp conditions (Martina and Jonas, 1997).

In addition, the use of a different experimental approach gave us the opportunity to test whether the observed properties of sodium current activation in the prefrontal cortex neurons were associated with the acute dissociation procedure, which involves protease treatment, the age of animals (>4 weeks for acutely dissociated neurons vs 16-18 d for nucleated patch recordings), the type of used voltage compensation (hypercharge in acutely dissociated cells vs traditional exponential charging), or the specific solution composition used to record sodium currents (for details, see Materials and Methods). Finally, the -124 mV prepulse was used during nucleated patch recordings to boost the recorded current amplitude. At -120 mV, we expected all sodium channels to be closed, preventing any possible complications associated with the relatively depolarized holding potential used in the previous experiments.

All experiments on nucleated patches were performed at room temperature (21-23°C).

Despite the above-mentioned differences in the experimental details, the properties of the sodium current activation in the hippocampal CA1 pyramidal neurons and dentate gyrus granule cells were qualitatively the same as in the prefrontal cortex pyramidal neurons. Figure 6A shows a typical set of traces obtained from an hippocampal CA1 neuron. Looking at the expanded time scale traces shown in the Figure 6B, at first glance, one might have the impression that sodium currents activate after a relatively long delay. However, because of the extreme speed of the sodium current activation, first we need to establish experimentally how quickly the voltage clamp was established after the voltage step.

Most commercial amplifiers, including the Axopatch 200B used for these experiments, in addition to the devices that speed up membrane charging at the beginning of the voltage command (“active compensation” according to the Axopatch 200B manual), have a circuitry that subtracts from the recorded trace the membrane charging transient current (“passive compensation”). Hence, the absence of any charging current in the traces does not mean that the membrane charging process has been completed and the applied voltage is sensed by the membrane. Because we applied test voltages of -64 to -44 mV from the holding potential of -124 mV, a 20% reduction in the voltage command would result in a voltage difference of ∼10 mV. Figure 6A shows that, for the command step from -124 to -44 mV, no current would be detected if the membrane sensed only 80% of the 80 mV voltage difference, which translates into a membrane potential of -44 mV - 0.2 × 80 mV = -60 mV. Therefore, we need to know the time required to achieve ∼80% of the voltage command step amplitude on the cell membrane. This time can be estimated from the tail currents by identifying the time point at which the current trace deviates >20% from the monoexponential function.

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Figure 6.

Sodium currents activated with minimal delay in hippocampal CA1 pyramidal neurons (A-E) and the dentate gyrus granule cells (F-H) at 22°C. A, B, Sodium current activation recorded in a nucleated patch obtained from a hippocampal CA1 pyramidal neuron. Currents were evoked by the voltage-clamp protocol shown above the traces. In B, traces are shown on expanded time scale. An arrow indicates an estimated time point when the membrane charging was completed by ∼80% as estimated from tail current traces shown in D. C, D, Tail currents evoked by the protocol shown above the traces. In D, traces are shown in the expanded semilogarithmic plot. The light arrow indicates the onset of the voltage step command, and the dark arrow indicates the time point when current traces differed <20% from the monoexponential fit of the linear part of the trace in the semilogarithmic plot. E, A semilogarithmic plot of the modified current trace that was obtained from A following the procedure shown in Figure 2. This procedure included two steps of modifications. First, inactivation of the current was eliminated mathematically by dividing the current trace by the following function: 17.8 pA + 227.4 pA × exp(-t/2.22 ms). Second, the final trace was obtained after the subtraction of the obtained trace from 1. For details of all procedures, see Appendix. The thick light line represents the monoexponential fit. The two smooth lines correspond to the quadrate (thin line) and the cube (thick line) of an exponential function. The light arrow indicates the time point when the membrane charging was completed by ∼80% as estimated from tail current traces. E, Sodium current activation recorded in a nucleated patch obtained from a dentate gyrus granule cell. F, Tail currents plotted in the semilogarithmic plot were evoked by the protocol shown above the traces. The light arrow indicates the onset of the voltage step command, and the dark arrow indicates the time point when current traces differed <20% from the monoexponential fit of the linear part of the trace in the semilogarithmic plot. H, A semilogarithmic plot of the modified current trace that was obtained from F following the procedure shown in Figure 2. All notations are the same as in E.

The amplitude of the voltage steps used to evoke tail currents was similar to the amplitude of the voltage steps used for the delay measurements (Fig. 6A, B). Figure 6D shows tail current traces on an expanded semilogarithmic scale. It is clear that >30 μs was required to charge the membrane, because no tail currents were detectable during the first 30 μs. The arrow indicates the time point corresponding to the average membrane charging duration that was sufficient to ensure that the deviation of the current trace from the monoexponential time course was <20%. This time point was ∼35 μs after the onset of the voltage step command. In this recording, the estimated series resistance was 10 MΩ, and the patch capacitance that corresponds to the patch area of ∼70 μm² was 1 pF. Hence, for uncompensated series resistance, the calculated time constant would be ∼10 μs. This estimate shows that other factors, such as digital filter time constant (∼10 μs of the 20 kHz low-pass filter), command lag (∼5 μs), compensation circuitry lag (∼15 μs), and pipette stray capacitance, have contributed to a relatively slow onset of the tail current.

The arrow in the Figure 6B indicates the time point 35 μs after the start of the voltage step. Now the detected delay is approximately two times shorter than one would measure without the correction derived from the tail current analysis. Following the procedure described in Figure 2, a direct comparison of the observed delay and the expected delay can be performed (Fig. 6E). The traces corresponding to the m² and m³ functions were shifted to the right by 35 μs. There are no obvious differences between Figures 2E and 6E, except for much faster activation time constant at 22°C. In hippocampal CA1 pyramidal neurons, for voltage steps to -44 mV, the average delay was 71 ± 13 μsor (0.16 ± 0.04) × τ_m (n = 4). Similarly, in dentate gyrus granule cells, for voltage steps to -44 mV, the observed average delay was 50 ± 5 μs or (0.18 ± 0.02) × τ_m (n = 4) (Fig. 6F-H).

Interestingly, the average delay changed little with voltage. In hippocampal CA1 pyramidal neurons, the delay was 71 μsat -44 mV and 69 μs at -29 mV (n = 4). Meanwhile, in the same neurons, the activation time constant differed ∼ 1.7 times for these two membrane potentials (396 μsat -44 mV and 237 μsat -29 mV; n = 4). This relative insensitivity to the membrane voltage in this range of membrane potentials is inconsistent with the Hodgkin and Huxley model, which predicts a linear relationship between the delay and the activation time constant. Nevertheless, there was a clear decrease in the delay for voltage steps to less than -45 mV. For voltage steps to -49 mV, the delay duration observed in the currents from the hippocampal CA1 pyramidal neurons was 41 μs (n = 3). Although the small current amplitude complicated the analysis at these voltages, the observed trend is consistent with the proposed model of sodium current activation.