Mathematical analysis of depolarization block mediated by slow inactivation of fast sodium channels in midbrain dopamine neurons

Abstract

Dopamine neurons in freely moving rats often fire behaviorally relevant high-frequency bursts, but depolarization block limits the maximum steady firing rate of dopamine neurons in vitro to ∼10 Hz. Using a reduced model that faithfully reproduces the sodium current measured in these neurons, we show that adding an additional slow component of sodium channel inactivation, recently observed in these neurons, qualitatively changes in two different ways how the model enters into depolarization block. First, the slow time course of inactivation allows multiple spikes to be elicited during a strong depolarization prior to entry into depolarization block. Second, depolarization block occurs near or below the spike threshold, which ranges from −45 to −30 mV in vitro, because the additional slow component of inactivation negates the sodium window current. In the absence of the additional slow component of inactivation, this window current produces an N-shaped steady-state current-voltage (I-V) curve that prevents depolarization block in the experimentally observed voltage range near −40 mV. The time constant of recovery from slow inactivation during the interspike interval limits the maximum steady firing rate observed prior to entry into depolarization block. These qualitative features of the entry into depolarization block can be reversed experimentally by replacing the native sodium conductance with a virtual conductance lacking the slow component of inactivation. We show that the activation of NMDA and AMPA receptors can affect bursting and depolarization block in different ways, depending upon their relative contributions to depolarization versus to the total linear/nonlinear conductance.

the mechanisms by which dopamine neurons enter into depolarization block are potentially of interest for three reasons. First, antipsychotics used to treat schizophrenia have been hypothesized to exert their therapeutic effects by inducing depolarization block in mesolimbic dopamine neurons (Grace and Bunney 1986). Second, drugs that induce depolarization block in nigrostriatal neurons cause extrapyramidal side effects. Finally, the tendency of these neurons to go into depolarization block affects their ability to generate the high firing rates that are achieved during behaviorally relevant bursts in vivo (Hyland et al. 2002).

Depolarization block limits the maximum firing rate of dopamine neurons in vitro (Richards et al. 1997). Depolarizing current steps elicit steady firing rates up to 10 Hz, and additional depolarization causes a cessation of spiking, although transients as fast as 30 Hz have been evoked by current steps (Blythe et al. 2009). Our previous modeling work (Kuznetsova et al. 2010) and experimental studies (Deister et al. 2009) suggested that the failure to recover from sodium channel inactivation between spikes was responsible for entry into depolarization block and predicted that increasing the availability of sodium channels would delay entry into depolarization block. Subsequently, we showed that decreasing the sodium conductance pharmacologically causes dopamine neurons to go into depolarization block with lower maximal frequencies at lower values of applied current, whereas augmenting this conductance with the dynamic clamp has the opposite effect (Tucker et al. 2012). Understanding the precise mechanism of depolarization block in these neurons may lead to improved therapeutics.

Despite this firing rate limitation, dopamine neurons in freely moving rats can fire at high rates in phasic bursts (Hyland et al. 2002) signaling behaviorally relevant information about rewards (Schultz 1998, 2002). Also, various manipulations in vitro (Deister et al. 2009; Ji et al. 2012; Ping and Shepard 1996; Yu et al. 2014) can evoke a train of action potentials of variable frequency prior to entering depolarization block. Previous models (Drion et al. 2011; Kuznetsova et al. 2010; Oster and Gutkin 2011) do not capture critical aspects of the firing rate limitation and how it is circumvented. Therefore we examined depolarization block in a simple model that faithfully reproduces the sodium current measured in these neurons (Seutin and Engel 2010) and also in a model with an additional slow component of sodium channel inactivation, recently observed in these neurons (Ding et al. 2011). We discovered that these models entered into depolarization block in two different ways and experimentally demonstrated that a dopamine neuron can enter into depolarization block either way, depending upon the experimental conditions. Furthermore, previous studies (Blythe et al. 2007; Deister et al. 2009) reported conflicting results with respect to the contributions of AMPA and NMDA receptors to burst firing, and we attempted to clarify these roles using modeling.

MATERIALS AND METHODS

Model description.

Our goal was to construct a model of a dopamine neuron that was as simple as possible in order to capture the essential features of entry into depolarization block. Hence only three currents were included in a single compartment: a fast Na current [I_Na = g_Nam³hh_s(v − E_Na)], a delayed rectifier K current [I_K = g_Kn³(v − E_K)], and a leak current [I_leak = g_leak(v − E_leak)], where g_x and E_x are the maximal conductance and reversal potential for current x. The gating variables are m (activation of I_Na), h (fast inactivation of I_Na), h_s (slow inactivation of I_Na), and n (activation of I_K). The equations governing these gating variables are of the form dz/dt = −[z − z_inf(v)]/τ_z(v), where z_inf(v) and τ_z(v) are the voltage-dependent steady-state and time constants of the gating variable z (i.e., m, h, h_s, and n) at membrane potential v. The voltage dependence of z_inf(v) is described by a Boltzmann function of the form z_inf(v) = 1/{1 + exp[−(v − z_half)/z_slope]} (Hodgkin and Huxley 1952). Values for z_half and z_slope are given in Table 1, along with the expressions for the time constants as a function of membrane potential. The maximal conductances are (in mS/cm²) g_Na = 8, g_K = 0.6, and g_leak = 0.013, and the reversal potentials (in mV) are E_Na = 60, E_K = −85, and E_leak = −60. Square conductance pulses of AMPA and NMDA were injected in some simulations, with the following equations (Jahr and Stevens 1990) for the currents:

$$I_{\text{AMPA}}=g_{\text{AMPA}}\left(v-E_{\text{AMPA}}\right)$$

$$I_{\text{NMDA}}=g_{\text{NMDA}}\left(v-E_{\text{NMDA}}\right)/\left[1+\left(\left[\text{Mg}\right]/3.57\right)\exp \left(0.062v\right)\right]$$

with [Mg] = 1.4 mM.

Table 1.

Time and voltage dependence of model gating variables

z	zhalf, mV	zslope, mV	Time Constant τz, ms
m	−30.0907	9.7264	0.01 + 1/(a + b),
			a = −(15.6504 + 0.4043v)/[exp(−19.565 − 0.50542v) − 1]
			b = 3.0212 exp(−7.4630 × 10−3v)
h	−54.0289	−10.7665	0.4 + 1/(a + b),
			a = 5.0754 × 10−4 exp(−6.3213 × 10−2v)
			b = 9.7529 exp(0.13442v)
hs	−54.8	−1.57	20 + 160/{1 + exp[(v + 47.2)/1]}
n	−25	12	1 + 19exp(−{ln[1 + 0.05(v + 40)]/0.05}2/300)

See text for definitions.

To investigate the mechanism of depolarization block in dopamine neurons given the critical postulated role of the sodium channel current in depolarization block, it was imperative that we honor the known data regarding the sodium channel current in these neurons. The dynamics of the activation (m) and fast inactivation (h) of the Na current were taken directly from Ji et al. (2012) and Tucker et al. (2012), because this description faithfully reproduces the peak currents observed during the single-step voltage-clamp experiments described by Seutin and Engel (2010) and shown in Fig. 1A. During steps to depolarized membrane potentials (> −30 mV), we hypothesize that a second slow component of sodium channel inactivation is present but cannot be observed because it is occluded by the faster entry into the fast inactivated state, and therefore inactivation has a monoexponential onset. During recovery from inactivation, however, a biexponential recovery is evident, which indicates that inactivation has a fast and a slow component (Ding et al. 2011; Seutin and Engel 2010). Although variable fractions of the channels recorded in nucleated patches in these studies were reported to exhibit slow inactivation, in our model all channels exhibit slow inactivation.

Fig. 1.

Model development and reduction. A: adding a second, slow component of inactivation (solid lines) does not compromise the fit of the original sodium current description (dashed lines) to previously published data (gray dots) from the literature (Seutin and Engel 2010) of the peak voltage-clamp currents obtained as described in the text. For the model, the results are given for a 1-cm² patch of membrane with g_Na = 5.92 nS. B: slow inactivation of the sodium current summates when repeated 3-ms depolarizing pulses to 0 mV are applied to the model at 100-ms intervals from a holding potential of −70 mV. To match Ding et al. (2011), g_Na was set to 9.12 nS. C: the time course of the activation variable n (gray curves) during the simulation shown in D1 was fit to an instantaneous polynomial function [f(h), black] of the inactivation variable h. D1: pacemaking in full 5-dimensional (5D) model at applied current (I_app) = 0 μA/cm². D2: pacemaking in reduced 3-dimensional (3D) model at I_app = 0 μA/cm².

To test the possible contribution of the second, slow component of inactivation (h_s), we used a simple, switchlike description: this component inactivates completely above about −54.8 mV and recovers completely below that level, with a time constant of 20 ms above about −47.2 and 200 ms below. The relatively hyperpolarized half-inactivation point and the relatively steep voltage dependence of the steady-state value of h_s are consistent with Fernandez and White (2010) and Migliore et al. (1999). We confirmed that the addition of slow inactivation did not compromise the fit to the experimental data by simulating the voltage-clamp protocols described in Seutin and Engel (2010). The voltage-clamp data were obtained with nucleated patches from dopamine neurons in the substantia nigra and previously published by others. After a 50-ms prepulse to −120 mV, injected currents were recorded during 30-ms steps to more depolarized potentials. In Fig. 1A, the peak currents observed in the simulations of the sodium channel kinetics with and without h_s are compared to the peaks from the previously published experimental data (from Fig. 3E of Seutin and Engel 2010) to show that there is generally good agreement in both cases. Furthermore, this description allows inactivation to accumulate during a multiple-pulse protocol (Fig. 1B), consistent with Fig. 8D of Ding et al. (2011).

Fig. 3.

Analysis of entry into depolarization in 3D model. A1: the model transitions from pacemaking at I_app = 0 μA/cm² to depolarized silence at I_app = 0.16 μA/cm². The applied current step is shown at bottom. Inset at top right shows the instantaneous frequency (filled circles) for each interspike interval after the current step and prior to entry into depolarization block. A2: the time course of slow inactivation h_s (dashed curves) is compared to that of the total inactivation h_total (solid curves), the product of h and h_s. B: nullcline analysis. 2D phase portraits are given by treating the slow variable h_s as a constant. B1: nullcline portrait at I_app = 0 μA/cm² and h_s set to its approximate average value of 0.6. The membrane potential nullcline (dashed curve) and the sodium channel inactivation nullcline (solid curve) intersect at an unstable fixed point (open circle). B2: nullcline portrait at I_app = 0.16 μA/cm² and h_s set to its approximate initial value of 0.6 immediately after the step change in applied current. B3: nullcline portrait at I_app = 0.16 μA/cm² and h_s set to its approximate average value of 0.2 toward the end of the spike train. B4: nullcline portrait at I_app = 0.16 μA/cm² and h_s set to its value of 0.05 near the steady state during depolarization block. The membrane potential nullcline (dashed curve) and the sodium channel inactivation nullcline (solid curve) intersect at a stable fixed point (filled circle). C: bifurcation analysis with h_s as the bifurcation parameter. The trajectory (thin black curve) shows the leftward movement in the state space toward the fixed point after the current step in A1 destabilizes pacemaking. The slow time course of h_s leads to multiple spirals corresponding to action potential. D: speeding up slow inactivation by a factor of 2 leads to faster convergence toward the fixed point and fewer spikes.

Model reduction.

The simplified model described above consists of five state variables: v, m, h, h_s, and n. The dimensionality of the model was reduced with the separation of timescales method to facilitate the bifurcation analysis. The activation of the sodium current is very fast compared with the time course of the other state variables; therefore it is set to its steady-state value in the reduced models. The time constants for the activation of I_K (n) and for the fast inactivation of I_Na (h) have comparable timescales; hence the former was dynamically yoked to the latter (Rinzel 1985) with the following polynomial: n = f(h) = a₀ + a₁h + a₂h² + a₃h³, with a₀ = 0.8158, a₁ = −3.8768, a₂ = 6.8838, and a₃ = −4.2079 and the caveat that n is not allowed to drop below 0 or increase beyond 1. The coefficients for polynomial was obtained by a least-squares fit: The free (n) and yoked [f(h)] variables are compared in Fig. 1C. Pacemaking in the full model with five state variables (Fig. 1D1) and in the reduced model with only three state variables (Fig. 1D2) are quite similar, and all subsequent simulations and analyses employ this reduction. We then compared two versions of the reduced, minimal model of dopamine neurons: a three-dimensional (3D) version with v, h, and h_s and a two-dimensional (2D) version with only v and h, with h_s set to 1 to eliminate slow inactivation. All simulations, bifurcation diagrams, and nullclines were conducted with XPPAUT (Ermentrout 2002).

Slice electrophysiology.

All experiments were conducted according to University of Pittsburgh Institutional Animal Care and Use Committee-approved protocols. The midbrain slice preparation was performed as previously described (Tucker et al. 2012). In brief, postnatal day 14–21 male Sprague-Dawley rats (Hilltop Labs, Scottdale, PA) were anesthetized with isoflurane followed by decapitation. The brain was then quickly removed and placed into ice-cold 95% O₂ and 5% CO₂-saturated, sucrose-modified artificial cerebrospinal fluid (s-ACSF) containing the following (in mM): 87 NaCl, 75 sucrose, 2.5 KCl, 25 NaHCO₃, 1.25 NaH₂PO₄, 0.5 CaCl₂, 7 MgSO₄, 25 glucose, 0.15 ascorbic acid, and 1 kynurenic acid, pH 7.4. Coronal midbrain slices (250 μm) were cut with a Vibratome 3000 (Vibratome, St. Louis, MO) in ice-cold 95% O₂ and 5% CO₂-saturated s-ACSF, followed by incubation in room temperature s-ACSF in an interface chamber for at least 1 h. The slices were then held until use in an interface chamber at room temperature in normal ACSF containing the following (in mM): 124 NaCl, 4 KCl, 25.7 NaHCO₃, 1.25 NaH₂PO₄, 2.45 CaCl₂, 1.2 MgSO₄, 11 glucose, and 0.15 ascorbic acid, pH 7.4. All salts were purchased from either Sigma-Aldrich or Thermo Fisher Scientific.

Patch electrodes were fabricated from Corning 7056 Patch Glass (Warner Instruments, Hamden, CT) and coated near the tip with beeswax to reduce the pipette capacitance. The pipette solution contained the following (in mM): 120 potassium gluconate, 20 KCl, 10 HEPES hemisodium salt, 2 MgCl₂, 0.1 EGTA, and 1.2 ATP disodium salt, pH 7.3. Carbogenated normal ACSF was used for bath perfusion of midbrain slices at a rate of 2 ml/min. Substantia nigra pars compacta dopamine neurons were identified by location and electrophysiological characteristics as described previously (Grace and Bunney 1983; Lacey et al. 1989; Richards et al. 1997).

Whole cell recordings in the current- and dynamic-clamp configurations were performed as previously described (Tucker et al. 2012). Briefly, the current- and dynamic-clamp recording system included an A-M Systems Patch-Clamp Amplifier model 2400 (National Instruments, Austin, TX), a National Instruments PXI-1002 CPU for real-time loop computation, and an IBM MT-M 8310-47U CPU with Pentium 4 processor used for user-G-clamp interface. LabVIEW-RT 8.1 (National Instruments) was used to run the G-clamp 2.02 software (http://hornlab.neurobio.pitt.edu). Current- and dynamic-clamp measurements were performed with a feedback loop rate of 20 kHz, which has been previously shown to be sufficient for fast Na_V conductances (Bettencourt et al. 2008; Kullmann et al. 2004).

In the current-clamp recordings, substantia nigra DA neurons were held at −60 mV with bias current to prevent spontaneous pacing in between 2-s 25-pA current steps from −100 pA to 200 pA with an 8-s interpulse interval. With this paradigm, the characteristic sag in the membrane potential in response to hyperpolarization, slow pacemaker-like activity with small depolarizations, and depolarization block with larger depolarizations was monitored. Native Na_V channels were then blocked by 10-min incubation with 1 μM tetrodotoxin (TTX) (Alomone Labs, Jerusalem, Israel) and replaced with virtual Na_V channels without the slow inactivation component by using the dynamic clamp. The equations used in G-clamp to describe the virtual Na_V channel current are listed in Table 1 and are the same as those described for the 2D model except that m was allowed to vary dynamically to more closely approximate the native current.

RESULTS

As stated in the introduction, previous models of dopamine neurons, including our own, do not capture the manner in which real dopamine neurons enter depolarization block. After a current step is applied to a real neuron with the minimum amplitude required to cause cessation of firing via depolarization block, several spikes are emitted and then spiking fails abruptly but the membrane remains relatively hyperpolarized (Richards et al. 1997). In previous models (Kuznetsova et al. 2010), spiking ceases as action potentials devolve into small-amplitude oscillations centered at a depolarized potential, and then the membrane potential hangs up at a relatively depolarized level; the failure mode of the model is not consistent with the experimental data. We hypothesize that the mechanisms underlying depolarization block in response to strong depolarizing current in vitro are relevant to the therapeutic efficacy of antipsychotic drugs as well as to the gating of high-frequency bursts observed in vivo (Grace and Bunney 1986); therefore, we closely examined the mathematical bifurcation structure leading to different types of depolarization block failure in both the 2D and 3D models described in materials and methods.

Phase portrait analysis of depolarization block in 2D model.

Figure 2A, left, shows regular pacemaking activity of the 2D model with no applied current. Figure 2A, right, shows the application of a current step that causes spike failure shortly after step onset, at a depolarization level corresponding to the middle of an action potential. This stimulus is near the minimum amount of current required to induce depolarization block, which in this model cannot be observed below about −19 mV. When the applied current is quite close to this minimum level, a small-amplitude oscillation (“ringing”) can be observed after spike failure. Figure 2B analyzes the entry into depolarization block with a phase portrait analysis (Ermentrout and Terman 2010) in terms of the only two state variables in the model, v and h. First, we plot the system nullclines for h and v. The nullcline for a given variable is comprised of the pairs of values for which the derivative of this variable is zero. Therefore the membrane potential (or v) nullcline contains the pairs (v, h) for which the net ionic membrane current is zero, and the h nullcline is the steady-state inactivation curve for this variable. The salient feature of the v nullcline is that the positive feedback due to the activation of the sodium current causes the v nullcline to have three distinct branches: a left branch on which the sodium channels are not activated, a middle branch on which they are partially activated, and a right branch on which they are essentially fully activated (or at least the increase in activation is offset by the decrease in driving force). At any intersection of the nullclines, all temporal derivatives are zero; therefore each intersection is a fixed point of the system. If this point is stable, it sets the resting membrane potential. If we assume that h changes slowly with respect to membrane potential, we can perform a fast-slow analysis (Izhikevich 2007) to determine the stability of the fixed points. Above the v nullcline membrane potential increases because of depolarizing net ionic membrane current, which implies rightward motion of trajectories under the fast-slow assumption, whereas below it membrane potential decreases because of hyperpolarizing net ionic membrane current, which implies leftward motion. Under the fast-slow assumption, any fixed point on the left or right branch is stable but one on the middle branch is unstable, resulting in a pacemaking oscillation instead of quiescence. This is the case for the phase plane portrait in Fig. 2B1 corresponding to the control pacemaking oscillation. As the stimulus current is increased, the fixed point modes toward the right branch of the membrane potential nullcline. If h were to remain slow compared with v, the fixed point would lose stability via a Hopf bifurcation when it lands on the right branch. The fast-slow analysis explained above is an approximation; it is possible for fixed points on the middle branch to gain stability near the turns (Herrera-Valdez et al. 2013). In this particular example, the analysis based on a separation of timescales breaks down because the voltage-dependent time constant for h becomes quite fast at depolarized potentials, so the Hopf bifurcation occurs slightly before the right branch is reached. Nonetheless, the rightward motion of the fixed point is the underlying mechanism that stabilizes the fixed point. This rightward motion results because increasing the applied current flattens the cubic nullcline. We introduce the fast-slow analysis simply to provide an intuitive understanding of the conventional explanation of depolarization block as “excitation block” (Izhikevich 2007), in order to show that the mechanism for depolarization block explained in the next section is novel in contrast.

Fig. 2.

Analysis of entry into depolarization block in 2-dimensional (2D) model. A: the model transitions from pacemaking at I_app = 0 μA/cm² to depolarized silence at I_app = 3.5 μA/cm². The applied current step is shown at bottom. B: nullcline analysis. B1: nullcline portrait at I_app = 0 μA/cm². The membrane potential nullcline (dashed curve) and the sodium channel inactivation nullcline (solid curve) intersect at an unstable fixed point (open circle). B2: nullcline portrait at I_app = 3.5 μA/cm². The membrane potential nullcline (dashed curve) and the sodium channel inactivation nullcline (solid curve) intersect at a stable fixed point (filled circle).

Phase portrait analysis of depolarization block in 3D model.

The only difference between the 2D and 3D models is the addition of a second, slower component of inactivation of the sodium channel (Colbert et al. 1997) represented by the state variable h_s. This variable recovers from inactivation at the hyperpolarized potentials between action potentials more slowly than it enters the inactivated state at depolarized potentials during an action potential (Fig. 1B), providing a mechanism for spike frequency adaptation. The 3D model emits a number of spikes after the minimum depolarizing current step is applied to induce depolarization block (Fig. 3A1). The product h h_s (which is h_total) determines the availability of the sodium current prior to action potential initiation, and Fig. 3A2 shows that prior to the current step h_s recovers to the same level after each spike, which is sufficient to permit the initiation of the next spike. In contrast, after the current step is applied, the faster frequency does not allow sufficient time between spikes for h_s to recover to its previous level after each spike, resulting in depolarization block. We hypothesize that since dopamine neurons enter depolarization block in a similar manner, a similar mechanism underlies entry into depolarization block in dopamine neurons, and it is predicated on the existence of a second, slower component of sodium channel inactivation.

The slow entry into depolarization block (lasting 19 spikes, see Fig. 3A1) requires the presence of a slow variable with a timescale in the tens to hundreds of milliseconds. The slow time course of this variable allows us to apply the phase portrait method used in the 2D model by treating h_s as a parameter (see Baer et al. 1989 for limitations of this approach). In other words, we take a stroboscopic snapshot of the system at each decreasing value of h_s averaged over a cycle and examine the phase portrait at that value of h_s. Figure 3B shows a series of these phase portraits starting from the basal condition (Fig. 3B1) and continuing through the train of action potentials at the onset of the current stimulus (Fig. 3, B2 and B3) until a quiescent state is reached (Fig. 3B4). The value of h_s is the same in Fig. 3, B1 and B2, only the value of I_app was changed, whereas Fig. 3, B3 and B4, show the effect of the decrease in h_s; the arrows indicate the point in time at which the nullcline snapshots were taken. For fixed values of h_s (and I_app), the voltage nullcline tells you how much fast inactivation h is required at each value of the membrane potential in order to balance the net current, with 1 indicating no inactivation and 0 meaning full inactivation. After the current step is applied, the average value of h_s over a cycle begins to decrease. The reduction in h_s results in less availability of sodium channels to be recruited by depolarization, which causes the voltage nullcline to shift toward larger values of h to compensate for the decrease in h_s, moving the fixed point from the unstable middle branch toward the stable left branch. We used XPPAUT to trace the bifurcation with h_s considered as a constant parameter. The bifurcation diagram in Fig. 3C was generated at the higher level of applied current shown in Fig. 3A and shows the stable fixed points corresponding to quiescent resting membrane potentials and the maxima and minima of the membrane potential oscillation in the pacemaker regime surrounding the unstable branch of fixed points. Also shown is a projection of the trajectory in the v-h_s plane of the simulation in Fig. 3A1 after the current step is applied. The trajectory spirals to the left as time proceeds in a forward direction and becomes quiescent after passing through a supercritical Hopf bifurcation.

The other critical factor in producing multiple spikes during the transition to depolarization block cannot be determined from the bifurcation diagram, because it is determined by the transient dynamics rather than the bifurcation diagram, indicating the asymptotic solutions of the differential equations as time goes to infinity. Note that the frequency corresponding to the first interspike interval after the current step in Fig. 3A1 is 9.4 Hz and the final one is 7.4 Hz, so slow inactivation of the sodium channel provides a mechanism for spike frequency adaptation. This is not the case for the “ringing” observed in the 2D case in Fig. 2A. The ringing consists of damped oscillations rather than full spikes, and the frequency of these oscillations increases during the approach to depolarization block, from 60 Hz for the first to 130 Hz for the last. The role of the slow variable in eliciting multiple spikes prior to entry into depolarization block is illustrated in Fig. 3D. Halving the time constant τ_hs at all potentials reduces the number of spikes before depolarization block from 19 in Fig. 3A to 4 in Fig. 3D, showing that the slow timescale of this parameter controls the number of action potentials emitted prior to block.

Two distinct modes of entry into depolarization block.

Figure 4 contrasts the two modes of entry into depolarization block using nullclines (Fig. 4A), bifurcation diagrams (Fig. 4B), and current-voltage (I-V) curves (Fig. 4C). Figure 4A1 treats I_stim as the bifurcation parameter and summarizes the analysis presented in Fig. 2B that increasing I_stim moves the fixed point toward the right stable branch of the voltage nullcline. On the other hand, Fig. 4A2 treats h_s as the bifurcation parameter and summarizes the analysis presented in Fig. 3B that decreasing h_s moves the fixed point toward the left stable branch of the voltage nullcline. Thus we would expect the latter to go into depolarization block at more hyperpolarized potentials and the former to fail at the more depolarized potentials traversed by an action potential.

Fig. 4.

Distinct dynamic changes underlying 2 modes of entry into depolarization block. A: the dependence of the voltage nullcline (red) on the bifurcation parameter, while the h nullcline (green) remains constant, leads to stabilization of the fixed point for both the 2D and 3D models. A1: right shift. Nullclines for the 2D model as I_app is increased from 0 to 3.5 μA/cm² in the direction of the arrow. A2: left shift. Nullclines for the 3D model as h_s is decreased from 0.6 to 0.2 to 0.05 in the direction of the arrow. B: bifurcation diagrams as I_app is varied. Green lines indicate the maxima and minima of the pacemaking oscillation. B1: 2D model. The critical feature is that to the left of the pacemaking region; there are 3 branches of fixed points, of which only the lower is stable (red curve). B2: bifurcation diagram for 3D model as I_app is varied. The critical difference is that there is a single branch of fixed points. C: current-voltage (I-V) curves. C1: nonmonotonic I-V curve for 2D model. C2: monotonic I-V curve for 3D model. I_stim, stimulus current.

To further differentiate these mechanisms, we constructed bifurcation diagrams, this time using the external applied stimulus current I_stim as the bifurcation parameter for both models. Red lines in Fig. 4B indicate stable fixed points whereas black dots indicate unstable ones, and in the pacemaking regime the minima and maxima of the oscillation in membrane potential are indicated by green line(s); solution branches of fixed points (red lines plus black dots) are differentiated by a change in the sign of the slope, and the stability is not necessarily consistent across a branch. The critical difference between the two types of entry into depolarization block is that the fixed points (Fig. 4B1) for the left shift mechanism illustrated for the 2D model in Fig. 2B have three branches (positive slope, negative slope, then positive again). In contrast, there a single solution branch (Fig. 4B2), with a monotonic positive slope, for the right shift mechanism illustrated for the 3D model in Fig. 3B. The reason the three branches are important is that the gap between the membrane potential at the rightmost stable value on the bottom branch and the leftmost stable value on the top branch produces a “forbidden zone” in which there are no stable resting potentials. Consequently, depolarization block can only be observed at potentials that are quite depolarized compared with the hyperpolarized silent state. The forbidden zone can be quite small for the case with a single solution branch, depending upon the slope in the unstable region. Spiking is initiated in the 2D model by a saddle node bifurcation and in the 3D model by a supercritical Hopf with a very steep dependence of the amplitude of the oscillation on applied current. In both cases, spiking terminates at a saddle node of periodics (SNP), and there is a region of bistability (Dovzhenok and Kuznetsov 2012) between depolarization block (rightmost red branch of stable fixed points) and pacemaking (green branches indicate maxima and minima during spiking). The bifurcation diagrams can be presented in a form more familiar to electrophysiologists, the I-V curve. A nonmonotonic I-V curve as illustrated in Fig. 4C1 is required for a saddle node bifurcation to initiate spiking (Izhikevich 2007), whereas a monotonic curve as in Fig. 4C2 is often associated with a Hopf bifurcation for spike initiation, although other bifurcation structures are possible (Herrera-Valdez et al. 2013). The bifurcation structure we predict for dopaminergic neurons implies that the I-V curve measured with long-duration (on the order of seconds) voltage-clamp steps to each potential or with a very slow ramp current should produce a monotonic I-V curve, but fast ramps that do not allow sufficient slow inactivation of the sodium channel current should produce a nonmonotonic I-V curve.

Experimental support for role of slow sodium channel inactivation.

To test our hypothesis that the slow component of sodium channel inactivation is critical for the mechanism of entry into depolarization block in real dopamine neurons, we utilized the dynamic clamp (Sharp et al. 1993a, 1993b) technique, which gives us the opportunity to manipulate dynamic ion channel variables in real time. With current-clamp slice electrophysiology in the whole cell configuration, substantia nigra dopamine neurons were held silent at −60 mV with a small bias current followed by a 2-s, 75- to 100-pA step to restore regular pacinglike behavior (Fig. 5A1) or a 200- to 300-pA step to induce depolarization block (Fig. 5A2). This method of entry into depolarization block appears to be analogous to the mechanism observed in the 3D model and illustrated in Fig. 3 and Fig. 4A2. Native fast voltage-gated sodium channels were then blocked by applying 1 μM TTX and replaced, using the dynamic-clamp technique, with a virtual sodium conductance characterized by fast activation and only the faster component of inactivation.

Fig. 5.

Experimental demonstration of the need for the slow component of sodium channel inactivation. The slow component of sodium channel was selectively removed by applying tetrodotoxin (TTX) to remove the native voltage-dependent sodium channel (Na_V) and replacing it with a virtual Na_V that did not contain slow inactivation. A: native Na_V. A1: control before TTX application. A2: depolarization block induced in the presence of native Na_V occurs near the spike threshold, which ranges from −45 to −30 mV in vitro (Grace and Onn 1989), with multiple spikes following the application of the current step. B: virtual Na_V with no slow inactivation. B1: pacemaking restored in the presence of 1 μM TTX by injecting virtual Na_V with the dynamic clamp. This virtual conductance did not have a slow component of inactivation. B2: depolarization block induced in the presence of native Na_V occurs mid-action potential after just a couple of spikes with weak afterhyperpolarizations. The maximum Na_V conductance was 400 nS.

If the native current indeed contains a very slow inactivation component, then this dynamic-clamp experiment provides a means to selectively “remove” the slow component in order to assess its contribution. Interestingly, the virtual conductance restored the neuron's ability to pace (Fig. 5B1). However, there was little accommodation, quantified as the ratio of the height of the last action potential to the height of the first action potential, compared with that observed with the native current (Fig. 5A1; 1.02 ± 0.01 vs. 0.87 ± 0.03, respectively, n = 4). Thus we find it likely that this component of the sodium current is responsible for both the decrease in spike height and the decrease in frequency due to progressive decreases in sodium channel availability. The same level of applied current that induced depolarization block with the native sodium current present also induced depolarization block in the presence of the virtual current (Fig. 5B2). However, latency to depolarization block entry was shorter (85.6 ± 10 ms vs. 540.1 ± 168 ms, n = 4), action potential height accommodation, a ratio of the height of the last action potential to that of the first, was larger (0.38 ± 0.02 mV vs. 0.78 ± 0.06 mV, n = 4), and depolarization block potential was more depolarized (−16.6 ± 1.66 mV vs. −35.6 ± 1.47 mV, n = 4) with the non-slowly inactivating conductance than the native. This method of entry into depolarization block is more analogous to the mechanism exhibited by the 2D model illustrated in Fig. 2 and Fig. 4A1. Therefore the slow component of inactivation seems critical in allowing a train of normal spikes to be produced after a large depolarization but prior to entering depolarization block. Another striking difference is that during entry into depolarization block the instantaneous frequency decreases from 20.5 ± 3.8 Hz for the first interspike interval to 11.5 ± 2.3 Hz for the last (n = 4) under control conditions but increases from 56 ± 9.6 Hz to 73.3 ± 14.6 Hz (n = 4) with the virtual conductance lacking slow inactivation. These results parallel those observed in the model, suggesting that slow inactivation mediates spike frequency adaptation.

The simple switchlike representation used here for slow sodium channel inactivation was chosen only to illustrate the essence of the mechanism in a reduced one-compartment model with no spatial extent and without a full complement of the currents expressed in dopamine neurons. It is far simpler to selectively remove the slow component than to replace it, because we have available to us a complete experimental characterization of the fast components of the sodium current but not of the slow inactivation. Thus the scope of this study is limited to removing and not replacing the slow component of inactivation.

Relative roles of NMDA and AMPA receptors in bursting and depolarization block.

A more physiologically relevant method of depolarization of dopamine neurons is the activation of glutamate receptors. There is a clear connection between high-frequency bursts in dopamine neurons and reward-mediated learning (Schultz 1998, 2002). Furthermore, genetic knockout of the NMDA subtype of glutamate receptors has been shown to selectively attenuate burst firing in dopamine neurons and to selectively impair cue-dependent learning (Zweifel et al. 2009). One previous study (Blythe et al. 2007) showed that both AMPA and NMDA receptor activation contributed to bursting evoked by synaptic stimulation or pressure-pulse application of glutamate, in contrast to in vivo studies that found that only NMDA receptors were implicated in burst firing (Chergui et al. 1993; Overton and Clark 1992; Tong et al. 1996). Another previous study (Deister et al. 2009) used the dynamic clamp to demonstrate that a square pulse of virtual AMPA or NMDA conductance of equal amplitude drives dopamine neurons to depolarization block, but only NMDA elicited a train of action potentials prior to depolarization block. They attributed this result to the nonlinearity of the NMDA conductance. We examined how the activation of AMPA versus NMDA subtypes of glutamate receptors interacts with the slow inactivation of the sodium channel in our model, and what this implies for their effects on bursting and entry into depolarization block.

Activation of AMPA and NMDA receptors depolarizes the membrane and increases the membrane conductance either linearly or nonlinearly. To separate the effects of increased depolarization from increased conductance, we normalized their effects in two ways. First (Fig. 6A), we examined the minimum amplitude of a square pulse of NMDA conductance, AMPA conductance, or applied current that is sufficient to induce depolarization block in the 3D model. The activation of AMPA receptors produced depolarization block at −50 mV compared with −43 mV for NMDA receptors and −48 mV for depolarization block due to a square pulse of current. It is clear that all three manipulations can induce fast frequency spiking that terminates in depolarization block, analogous to a burst and consistent with a contribution of AMPA receptor activation to bursting (Blythe et al. 2007; Canavier and Landry 2006). However, activation of NMDA receptors allows repetitive spiking to persist at more depolarized potentials, whereas activation of AMPA receptors causes depolarization block to be entered at more hyperpolarized potentials.

Fig. 6.

Activation of NMDA but not AMPA receptors can delay entry into depolarization block. A: square pulses of 6-s duration and the minimal amplitude of conductance or current required to induce depolarization block in the 3D model are turned on at 2 s and off at 8 s. A1: NMDA conductance pulse (60 nS/cm²). A2: AMPA conductance pulse (2.3 nS/cm²). A3: applied current pulse (160 nA/cm²). B: comparison of square pulses that induce the same level of depolarization. B1: NMDA conductance pulse (60 nS/cm²) (same as A1). B2: AMPA conductance pulse (0.7 nS/cm²) speeds entry into depolarization block. B3: applied current pulse (320 nA/cm²).

Next we compared the same three manipulations at a constant final level of depolarization (Fig. 6B) in order to eliminate the effect of depolarization and focus on the effect of the linearity versus nonlinearity of the activated conductance. At the same level of depolarization, activation of NMDA receptors delayed entry into depolarization block, allowing many more spikes to be emitted than allowed by depolarization alone. This is because the nonlinearity of the NMDA conductance allows it to contribute to the upstroke of the spike, thereby compensating in part for the progressive reduction in the availability of fast sodium current due to slow inactivation. In contrast, at the same level of depolarization, activation of AMPA receptors actually decreased the number of spikes emitted prior to depolarization block from five to four, consistent with experimental observations that AMPA receptor antagonists can increase the number of spikes in a burst evoked by glutamate iontophoresis (Deister et al. 2009). Therefore AMPA contributes to bursting by virtue of additional depolarization, but its linear conductance opposes bursts and favors depolarization block. On the other hand, NMDA not only enhances bursting by virtue of the depolarization that it contributes but also staves off depolarization block by virtue of the positive feedback evoked by its nonlinearity.

DISCUSSION

The mechanism for bursting in midbrain dopamine neurons is of great interest because of its behavioral significance (Schultz 1998). Three burst mechanisms (Morikawa and Paladini 2011) have been proposed: depolarization resulting from plateau potentials (Johnson and Wu 2004; Ping and Shepard 1996), depolarization resulting from large excitatory postsynaptic potentials (EPSPs) (Blythe et al. 2007, 2009), or interactions between intrinsic and synaptic properties, for example, an oscillatory synaptic input that removes sodium channel inactivation with appropriately timed inhibition (Deister et al. 2009). We concur with the view that the first two mechanisms by themselves are not sufficient and that synaptic events must interact with the intrinsic properties of dopamine neurons. The slow, subthreshold oscillation in intracellular free Ca²⁺ (Wilson and Callaway 2000) has been proposed (Kuznetsov et al. 2006) and rejected (Deister et al. 2009) as the intrinsic mechanism underlying the limitation of the firing rate. We demonstrate that the second, slow component of sodium channel inactivation is instead the relevant, rate-limiting intrinsic mechanism in juvenile rats in vitro. Although dopamine neurons do not typically burst in vitro, this limitation on the firing rate is likely also relevant in vivo. This result is consistent with and provides a partial mechanistic explanation for previous theories that depolarization block limits the firing rate of dopamine neurons (Grace et al. 1997). Previously, we have shown that the density (Kuznetsova et al. 2010) and distribution (Tucker et al. 2012) of sodium channels are important for setting the lower limit of the firing rate. Here we show that omitting slow inactivation raises the upper limit on the firing rate.

Significance of depolarization block for antipsychotic drugs.

Significant therapeutic benefits are achieved within the first days of antipsychotic administration (Kapur and Seeman 2001), although several weeks are required for maximal efficacy. Acute treatment of normal rats with antipsychotic drugs increases rather than decreases population activity by blocking inhibitory D2 autoreceptors (Grace and Bunney 1986). Three weeks of chronic treatment of normal rats is required before population activity decreases as a result of driving hyperexcited neurons into depolarization block (Grace and Bunney 1986), which does not support a primary role for depolarization block in the therapeutic benefits of antipsychotics. However, one hypothesis (Lodge and Grace 2011) suggests that psychosis results from hyperactivity in hippocampal subfields that excite the nucleus accumbens, which inhibits the ventral pallidum, and thereby disinhibits mesolimbic dopamine neurons. Accordingly, rats in a developmental model of schizophrenia had constitutively high levels of mesolimbic dopamine neuron population activity and exhibited depolarization block when antipsychotic drugs were administered acutely (Valenti et al. 2011), which does support a role for depolarization block in the therapeutic benefits of antipsychotics.

The availability of sodium channels largely determines whether a neuron will continue spiking versus entering depolarization block; thus the effect of antipsychotic drugs may be exerted at least in part through their effects on sodium channel availability. Currents that contribute to the afterhyperpolarization following each spike can increase sodium channel availability because an enhanced afterhyperpolarization more effectively removes sodium channel inactivation. The ERG potassium channel, for example, slowly activates but rapidly inactivates during an action potential, and inactivated ERG channels must pass through the open state before they can close at hyperpolarized potentials. Therefore, the presence of unblocked ERG channels enhances the removal of sodium channel inactivation, and we have shown (Ji et al. 2012) that blocking these channels accelerates entry into depolarization block. This is particularly relevant because both typical and atypical antipsychotics block the ERG channel as a side effect (Shepard et al. 2007).

Diversity of dopamine neurons.

Dopamine neurons have been traditionally regarded as relatively homogeneous; however, an unexpected degree of diversity (Roeper 2013) has recently been identified among dopaminergic subpopulations. In particular, the 10-Hz frequency limitation applies only to dopaminergic neurons projecting to the striatum and the lateral shell of the nucleus accumbens; the remaining mesolimbic and mesocortical neurons can sustain firing at up to 20 Hz (Lammel et al. 2008). All dopamine neurons go into depolarization block, but the fast-firing neurons have a pronounced reduction in action potential amplitude during entry into depolarization block that is not observed in the slower-firing neurons modeled in this study. This suggests that the amplitude of the SNP indicated in Fig. 4B is decreased in the fast-firing neurons compared with the slower ones of this study.

Significance for reward signaling.

Midbrain dopamine neurons receive several types of afferent excitatory inputs; for example, the pedunculopontine nucleus provides both cholinergic and glutamatergic afferents to dopamine neurons (Lokwan et al. 1999; Pan and Hyland 2005). Only activation of NMDA receptors, by virtue of their nonlinearity, tends to overcome the intrinsic limitation imposed on dopamine neuron firing rates by recovery from slow sodium channel inactivation. When AMPA receptor activation is required for (Blythe et al. 2007) or facilitates (Canavier and Landry 2006) bursting, based on the results in Fig. 6, the AMPA receptors are probably just providing sufficient depolarization to effectively engage nonlinear conductances such as the fast sodium channel and the NMDA receptors.

Depolarization block in other brain areas.

Depolarization block may also participate in function and dysfunction in other brain areas such as the hippocampus. Recently, Bianchi et al. (2012) suggested that depolarization block can be induced in CA1 pyramidal neurons with a random background activity from only a few hundred synapses, and they further suggested that depolarization block may be a rate-limiting mechanism protecting CA1 pyramidal neurons from excessive spiking. Depolarization block in hippocampal interneurons has been observed during in vitro seizurelike events (Ziburkus et al. 2006), in particular within a network oscillation in which interneurons alternated activities between firing in bursts and exhibiting depolarization block. While the interneurons were spiking, the pyramidal neurons were hyperpolarized. Conversely, while the interneurons were silent because of depolarization block, the pyramidal neurons became disinhibited and generated spikes. A related form of network bursting has been observed between two excitatory neurons alternating between spiking and depolarization block (Sieling et al. 2009). Preventing depolarization block in interneurons was suggested as a therapeutic target for seizure control (Ziburkus et al. 2006). Therefore the significance of understanding the mechanisms governing depolarization block is not confined to midbrain dopamine neurons.

I-V curves determined under voltage clamp as a diagnostic tool.

Figure 4C makes a critical experimental prediction. Because of our presumed bifurcation structure, we predict that the steady-state I-V curve for dopamine neurons is monotonic as in Fig. 4C2 rather than nonmonotonic as in Fig. 4C1. On the other hand, if a fast ramp is utilized to obtain the I-V curve and the speed is too fast to engage slow sodium channel inactivation, then a nonmonotonic I-V curve resembling Fig. 4C1 could be obtained. In a current-clamp mode, both models predict hysteresis in that less current is required to maintain depolarization block than is required to induce it (Dovzhenok and Kuznetsov 2012). Our simple model has focused on the mechanism of depolarization block and ignores the subthreshold calcium oscillations that can be observed in substantia nigra pars compacta cells (Wilson and Callaway 2000). We expect the results presented here will still apply, because blocking either of the two main currents underlying this oscillation, L-type calcium channels (Blythe et al. 2009) or SK potassium channels (Deister et al. 2009; Shepard and Bunney 1991), does not change the mode of entry into depolarization block.

Figure 9 of Bianchi et al. (2012) shows a bifurcation diagram for their model of a hippocampal CA1 pyramidal neuron that is similar to the bifurcation diagram in our Fig. 4B2. They also show, in Fig. 8 of their paper, a bifurcation diagram that is similar to our Fig. 4B1, except that the upper branch remains stable at membrane potentials to the left of the knee that terminates the lower branch. The result is that the model neuron cannot pace, because there is at least one stable fixed point at every value of the membrane potential. Although Bianchi et al. (2012) do not explicitly state this, their bifurcation diagram captures the essence of Hodgkin's (1948) type 3 excitability in which step current changes induce a train of spikes but can never induce sustained pacemaking. Therefore the I-V curve determined under voltage clamp may be a helpful diagnostic tool for characterizing the bifurcation structure of neural oscillators, although other techniques are required to complement the I-V curves (Herrera-Valdez et al. 2013).

Conclusion.

We find that the second, slow component of recovery of the sodium channels from inactivation contributes to the experimentally observed limitations on the firing rate of dopamine neurons (Richards et al. 1997) in vitro by causing depolarization block. We show that there are two dynamically different ways to enter depolarization block and show that we can switch the mode of entry in real dopamine neurons. We explain contradictory reports on the effects of activation of AMPA and NMDA receptors on burst firing by considering the effects of increased depolarization and increased conductance separately.

GRANTS

This work was funded by National Institutes of Health Grants R01 NS-061097 to C. C. Canavier and F32 NS-078994 to K. R. Tucker and utilized resources provided by the computational core of P30 GM-103340.

DISCLOSURES

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

AUTHOR CONTRIBUTIONS

Author contributions: K.Q., N.Y., and K.R.T. performed experiments; K.Q. prepared figures; K.Q. drafted manuscript; N.Y., K.R.T., and C.C.C. edited and revised manuscript; N.Y., K.R.T., E.S.L., and C.C.C. approved final version of manuscript; K.R.T. analyzed data; E.S.L. and C.C.C. conception and design of research; E.S.L. and C.C.C. interpreted results of experiments.