Fig 1.
A. Markov model with 5 states for NaV 1.2. Conductance is proportional to the open O1 state. C1 and C2 are closed states, I1 is the fast inactivated state and I2 is the long-term inactivated state. Voltage dependent rate functions for the transition rates that label each arrow are given in Table 1. The maximal I1-I2 transition rate was set to 26.7 s-1, but the voltage clamp protocol in the next panel is not sensitive to long-term inactivation due to the short duration test pulses and the lack of direct transition for O1 to I2 state. B and C. Simulated normalized model voltage clamp currents (top) using the protocols (bottom) of [15]. B. Activation. C. Fast inactivation. D. Simulated “steady state” dashed model activation curves compared to solid curves summarizing experimental data from [15]. All parameters are set to atypical values in Table 1.
Fig 2.
Atypical maximum frequency is limited by recovery from fast inactivation, but some long-term inactivation is needed for hysteresis.
A. No long-term inactivation. A1. Weak triangular ramp (50 pA peak relative to baseline) in atypical model with no long-term inactivation (see panel A3) does not induce DP block. Time course of occupancy in the fast inactivated state, blue trace, bottom. A2. Strong triangular current ramp (600 pA) in atypical model with no long-term inactivation induces DP block with little hysteresis, such that spiking is reinstated on the down branch (membrane potential, upper trace). The fraction of channels in the fast inactivated state saturates (blue trace at bottom). A3. For I1-I2 = 0, the long-term inactivated state is effectively removed from the Markov model. B. Long-term inactivation added. The maximal I1-I2 transition rate is 26.7 s-1 as in Fig 1. B1. Response of atypical model (voltage trace, top) with long-term inactivation to a weak triangular current ramp (50 pA). B2. A stronger triangular ramp current (100 pA) induces depolarization block at a lower current amplitude than in A2. Bottom traces in each panel: time course of occupancy in I1 (blue) and I2 (green) states and their sum (red trace). B3. The green arrow in the Markov model shows which parameter was changed between A and B. Ramp currents are calibrated such that the initial hyperpolarizing step is identical for all cells (-25 pA) but the peak current relative to the hyperpolarized baseline is variable. Parameters are the ‘atypical’ values given in Table 1.
Fig 3.
Known differences between atypical and conventional populations do not fully account for differences in peak firing rate or failure mode.
A. Effects of varying maximal HCN conductance on A1, the minimum depolarizing current ramp amplitude required to induce depolarization block and A2, the peak firing rate (red) and number of evoked spikes (blue) on entry into depolarization block at the ramp amplitude from the previous panel. Dashed lines indicate values used for control and representative value in F-I. A1. B. Same as A except varying the Kv4 inactivation time constant. C. Same as A except varying the maximal SK conductance. D Same as A except varying the delayed rectifier and the sodium current by the same scale factor. E. Atypical control as in Fig 2B2 except in response to 75 pA ramp. Dashed line indicates time of first spike for control. Scale bar: 20 mV and 1 s. F. Increased H conductance. G. Faster Kv4.3 inactivation time constant. H. Elevated SK conductance in response to 110 pA ramp. I. Effect of increasing the primary spiking conductances (voltage gated sodium and the delayed rectifier) by an identical factor of 2 in response to 250 pA ramp. F. All parameters are set to atypical values in Table 1 except the parameter that is varied. The spike scale factor proportionally scales both gK,dr and gNa,V.
Table 1.
Differences between conventional and atypical exemplar models.
The differences in parameters are based on the literature and explained in the text for Fig 3, except the difference in the maximal transition rate from the fast to the long-term inactivated state, CI1I2, which is based on Fig 5.
Fig 4.
Additional Long-term Inactivation Converts Atypical to Conventional Failure.
A. Atypical Model. During ramp stimulation, each spike produces a small uptick in occupancy in the long-term inactivated state I2 (green curves) A1. A 40 pA ramp does not induce depolarization block in the atypical model. A2. An 80 pA ramp results in depolarization block in the atypical model. B. Atypical Model with increased occupancy in the slowly inactivated state. B1. Increasing the maximum rate of the I1 to I2 transition from 26.7 s-1 to 80 ms-1 leads to failure at substantially lower frequencies for 40 pA ramp current levels. Large upticks in the level of slow inactivation (green trace) are observed for each spike. B2. A larger ramp with peak amplitude of 80 pA, beyond that required to induce depolarization block. C. Bifurcation diagrams with respect to applied current. C1. Atypical model. Oscillations terminate via supercritical Hopf. C2. Elevated cI1-I2. Oscillations terminate via saddle node of periodics (SNP) following subcritical Hopf where the unstable limit cycle (LC) originates. D, Effect of I1-I2 transition rate on entry in depolarization block. D1. Dependence of the minimum ramp amplitude required to induce depolarization block on the I1-I2 transition rate (red curve) and the frequency upon entry into depolarization block (blue curve). For the atypical value of 26.7 s-1 for the maximum I1-I2 transition rate, an 80 pA peak ramp amplitude is sufficient to induce depolarization block with a maximum frequency of about 23 Hz (vertical dashed line labeled A2). Increasing the I1-I2 maximum transition rate to 80 s-1 drops the required peak ramp amplitude to 50 pA and the maximum frequency to about 13 Hz (vertical dashed line labeled B1). D2. The larger the peak ramp current tolerated prior to entry into depolarization block, the larger the maximum frequency observed during the ramp. All parameters are set to atypical values in Table 1 except CI1-I2, which is variable.
Fig 5.
Calibration of Long-term Sodium Channel Inactivation of Conventional Model Dopamine Cells Compared to Atypical Model.
Fit of long-term inactivation to data. The protocol of [15] consisting of five brief voltage clamp steps was used to calibrate the conventional model (black dots) and compare to the data taken from [15] (blue diamonds). The red squares are the prediction for the atypical cells. The maximum rate of the I1 to I2 transition was increased to 100 s-1 for conventional from 26.7 s-1 for atypical.
Fig 6.
Calibration of Electrophysiological Profile of Atypical and Conventional Model Dopamine Cells Compared to Experimental Data from the Literature.
A. Atypical Model. A1. Spontaneous pacemaking at 5 Hz. A2. Single action potential from A1 with a peak of 11 mV, width of 5 ms at -30 mV, and minimum AHP depth of -51 mV. A3. Phase plot of dV/dt vs V. A4. Response to a 2 s, 25 pA hyperpolarization. B. Conventional model. B1. Spontaneous pacemaking at 2 Hz. B2. Single action potential from B1 with a width of 3 ms at -30 mV, peak amplitude of 28 mV and minimum AHP of -64 mV. B3. Phase plot of dV/dt vs V. B4. Response to a 2 s, 75 pA hyperpolarization. C1-C4. Same protocol as A1-A4 except recorded from an identified mesocortical VTA atypical dopamine neuron. D1-D4. Same protocol as B1-B4 except recorded from an identified mesostriatal SN conventional dopamine neuron. E- H, Responses from a hyperpolarized, silent state to depolarizing current ramps. E. Atypical model. E1. Response to 50 pA triangular ramp. E2. Response to 100 pA ramp. F1-F2. same protocol as E1-E2 as except recorded from an identified mesolimbic medial shell projecting atypical dopamine neuron. G. Conventional model. G1. Response to a 50 pA triangular ramp. G2. Response to 100 pA triangular ramp. H1-H2. same protocol as G1-G2 except recorded from an identified mesostriatal projecting conventional dopamine neuron. C, D, F and H were modified with permission from Figs 4A–4D and 5B and 5C in the complete Elsevier source [10].
Fig 7.
Explanation of Depolarization Block Mechanisms Using Square Pulses. A.
Atypical model A1. Voltage traces (top) of a spontaneously pacing model neuron in response to a 2 sec, 75 pA square current pulse, with an additional 200 ms 50 pA step (bottom) applied after the model enters depolarization block. Depolarization block is entered via a gradual decrease in spike amplitude, with a maximum firing rate of 28 Hz. An additional current step during depolarization block does not evoke additional spikes. A2. Time course of available sodium channels (purple) (C1+C2+O1). Time course of occupancy in I1 (blue), I2 (green), and their sum (red). Inset: Low available pool prevents large oscillations following entry into depolarization block. A3. dV/dt for voltage trace in A1. Oscillations rapidly fall below the spike threshold (5 V/s), and do not pass above that point in response to added current. B. Conventional cell model. As in the previous figure, the conventional model has twice the surface area as the atypical. Voltage traces (top) of a spontaneously pacing model neuron in response to a 75 pA step current with an additional 200 ms, 50 pA step following entry into depolarization block. Depolarization block occurs abruptly at 10 Hz with a large amplitude spike, but an additional spike can be evoked with additional current. B2. Time course of available sodium channels (C1+C2+O1) (purple). Occupancy in I1 (blue), I2 (green), and their sum (red) for voltage trace in B1. Inset: relative to B2, larger fractions of sodium channel remain available for spikes evoked by additional current. B3. dV/dt for voltage trace in B1. The action potential evoked by additional current has peak slope well above spike threshold.
Fig 8.
Response of midbrain dopamine cells with identified projection targets to additional depolarization after inducing depolarization block with a current step.
A1, B1. Double labeling of cells with TH (tyrosine hydroxylase) to confirm dopaminergic phenotype and retrobeads (RB) to confirm projection target is medial shell (A1) or lateral shell (B2). A2, B2. Rebound response of identified cells to 2s of hyperpolarizing current. Cell from A1 does not show significant sag potential and has a 1.9 s rebound delay, confirming atypical phenotype (A2). Cell from B1 has a post sag potential of +19 mV and a rebound delay of 145 ms, consistent with conventional phenotype (B2). A3. Action potential waveform during pacing of identified medial shell projecting cell from A1. Action potential threshold is -30 mV and width at threshold is 5.2 ms. A4. Phase plot (slope vs voltage) of action potential in A3. Peak slopes are +30 and -18 V/s. B3. Action potential waveform during pacing of identified lateral shell projecting cell from B1. Action potential threshold is -27 mV and width at threshold is 3.7 ms. B4. Phase plot of action potential in B3. Peak slopes are +70 and -33 V/s. A5. Voltage trace of entry into depolarization block in response to 2s 125 pA pulse with an additional 200 ms pulse of 75 pA after 1.5 s for medial projecting cell identified in A1. Spike amplitude decays continuously into near threshold oscillation with a peak frequency (avg of first 3 ISI) is 40.5 Hz and a terminal frequency (average of final 3 ISI) of 20.7 Hz. Additional current pulse evokes a 20 mV amplitude oscillation. A6. Plot of dV/dt for voltage trace in A5. Slope declines continuously to below 5 V/s spike threshold. Peak slope of evoked oscillation is 2.6 V/s. B5. Voltage vs time in response to 2s 300 pA pulse with additional 200 ms pulse of 75 pA after 1.5 s in lateral projecting cell identified in B1. Spike amplitude remains consistent with a peak frequency of 15.7 hz and terminal frequency of 8.7 Hz. Additional current pulse evokes a 44 mV amplitude spike. B6. Plot of dV/dt for voltage trace in B5. Slope declines from spike to spike with final spike having a peak amplitude of 25 V/s. Evoked spike has a peak slope of 18 V/s, well above spike threshold of 5 V/s.
Fig 9.
Statistical Summary of Experimental Results.
A. Pacemaker frequency. B. Spike threshold as defined by 5 mV/ms rise rate during pacing. C. Spike width at threshold from B. D. Peak rate of depolarization during pacing (V/s) E. Peak rate of repolarization during pacing. F. Minimum AHP (mV) during pacing. G. Sag potential following hyperpolarization to -80 mV. H. Rebound delay (s) following hyperpolarization from G. I. Peak frequency (Hz) at first observed entry into depolarization block (25 pA intervals). J. Amplitude of evoked response to 75 pA, 200 ms pulse at 1.5 seconds into depolarizing pulse from I. K. Peak dV/dt for evoked response to 200 ms pulse from J. L. All lateral projecting, but only some medial projecting cells clear 5 mV/ms spike threshold. N = 13 for medial shell projecting (red), N = 7 for lateral shell projecting (blue).
Table 2.
Comparison with previous study.
Fig 10.
Single Compartment Dopamine Neuron Model.
A. Circuit diagram of one compartment model with leak, sodium (NaV 1.2), delayed rectifier potassium, A-type potassium (Kv4.3), Calcium activated potassium (SK), L and N type calcium channels, and an HCN channel. B. Calcium balance. Calcium is dynamically buffered using the mechanisms from [62]. The volume was restricted to that of a single submembrane shell (0.5 μm thick) to approximate a dendritic calcium response.
Table 3.
Parameters for Markov sodium channel transition rates.
RO1-I1 is the sum of the two Boltzmann functions respectively parameterized by the first and 2nd values. The maximal rate for the I1-I2 transition varies between populations (see Table 1).