Effects of Delayed Sodium Channel Kinetics

The Wang-Buzsaki model makes an idealization that the activation of sodium channels is instantaneous. In reality the activation time is finite, although the detailed properties of Na⁺channel gating and activation kinetics are still to be uncovered for cortical neurons. The activation time constant estimated from in vitro recordings of Na⁺currents in cortical neurons is around200 µs at23°C, and is expected to be even faster at the physiological temperature (Baranauskas &

Martina, 2006; Engel & Jonas, 2005).

To study the impact of delayed channel kinetics we introduced the voltage- and time- dependent activation of sodium channels into our baseline model. Following Hodgkin and Huxley’s formalism, the Na⁺current is controlled by two kinetic processes, activation and inactivation:

IN a = ¯gN amⁿh(V −EN a), (11.17) where m is the probability that any one of the n independent gating events for activation has occurred, and the exponent n indicates the number of subunit gates.

A recent study by Baranauskas & Martina (2006) suggested that the activa-tion time course was better fit by a mono-exponential funcactiva-tion rather than them³ kinetics originally proposed in the Hodgkin-Huxley model. For a systematic com-parison of the different candidate models, we therefore constructed two types of channel kinetics, the WBm¹ model and theWBm³ model. To facilitate compari-son, in both models we use the same activation curve as shown in Figure 11.11A and the same voltage dependent activation time constant:

τ_m(V) =ϕ/(α_m(V) +β_m(V)), (11.18) whereϕis a scaling factor to set the maximum activation time constantτ_m⁰ around 200 µs (Figure 11.11B, solid line). In both models, if the maximum activation time constant was larger than 200 µs, slow activation of Na⁺channels coupled with faster inactivation resulting in bursting of APs (data not shown).

In the WBm¹ model the activation variable m approaches its steady state value m³_∞ mono-exponentially:

gN a = ¯gN amh, (11.19)

τm(V)dm

dt =m³_∞(V)−m. (11.20)

And the WBm³ model follows the original HH model formulation:

gN a = ¯gN am³h, (11.21) τm(V)dm

dt =m∞(V)−m. (11.22)

To depict the different channel kinetics in the WB,WBm¹andWBm³ models, we applied a voltage step from -65 mV to -40 mV and plotted the channel opening probabilities and the corresponding Na⁺currents in Figure 11.11. As expected from the channel kinetics, the WB model with instantaneous activation exhibited a step increase of the Na⁺current and then decayed exponentially due to the channel inactivation. In both WBm¹ and WBm³ models Na⁺currents increased as a continuous functions at time initially rising linearly or parabolically (∽t³), respectively.

Figure 11.12A plots spike patterns of these models. Different activation ki-netics did not affect the rheobase current to generate action potentials. Given the same level of injected currents, the first AP fired simultaneously in the three models. However, the subsequent APs showed deviation in their time course:

APs in theWBm¹ model (green trace) preceded those in theWBm³ model, while the WB model exhibited the longest inter-spike intervals. Thus fast activation and inactivation would space the subsequent spikes more sparsely. The phase plots in Figure 11.12B verify that the onset dynamics agrees with the channel kinetics, i.e., fast kinetics resulted in fast onset dynamics. To obtain the same

0 0.5 1

Figure 11.11: Na⁺Currents in Wang-Buzsaki model with delayed channel kinetics. (A)Activation curve (solid line) and inactivation curve (dashed line).

(B)Activation time constant (solid line) and inactivation time constant (dashed line). (C) The channel open probabilities change with a step of the command voltage from -65 to -40 mV at t = 0. (D) The Na⁺currents under the same voltage clamp protocol. red: WB model; green: WBm¹ model; blue: WBm³ model.

0 0.2 0.4 0.6 0.8 1

Figure 11.12: Single compartment WB Models with m¹ and m³ kinetics.

Red trace: WB model; Green trace: WBm¹ model; Blue trace: WBm³ model.

(A) The spike patterns generated by the same constant current input: Ie = 10 nA/mm², ¯gNa = 600 pS/µm². (B)Phase plots explicit onset dynamics of the three models. The maximum Na⁺channel conductances were adjusted such as to obtain similar levels of peak dV /dt values. WB model: ¯gNa = 300 pS/µm²; WBm¹ model: ¯gNa = 600pS/µm²; WBm³ model: ¯gNa= 750 pS/µm².

level of peak V˙, a higher maximum Na⁺channel conductance was required in compensate of the slower activation kinetics.

Then we applied the different kinetic models to the multi-compartment sim-ulations of the geometric neuron. The AP onset rapidness was quantified for different combinations of the three parameters L, α, and¯gN a using the same pro-cedure as described before for the WB model. The simulation results are shown in Figure 11.13. The parameter dependencies of the AP onset dynamics were similar as that in the WB model except that the peak onset rapidness was dra-matically reduced from 70 ms^-1 (WB model) to 20 ms^-1 (WBm¹ model) and 10 ms^-1 (WBm³ model).

Another Hodgkin-Huxley type m³ model widely used in cortical neuron simu-lations is the Mainen-Sejnowski model (Mainen et al., 1995). More details about its channel kinetics can be found in the appendix to Chapter 9. Figure 11.14 summarizes the onset rapidness of somatic APs obtained by applying Mainen-Sejnowski channel kinetics in the geometric model. The effect induced by propa-gation was qualitatively the same but quantitatively much smaller in this model, with the maximum value of onset rapidness below 10ms^-1 in all simulations.

0 500 1000 1500 2000

500 1000 1500 2000 2500 3000 0

0 500 1000 1500 2000

0

500 1000 1500 2000 2500 3000 0 onset rapidness of somatic AP depends on (A1, B1) the distance from ini-tiation site, (A2, B2) the soma size factor and (A3, B3) the maximum Na⁺conductance.

0 1000 2000

Figure 11.14: Somatic AP onset in the Mainen-Sejnowski modelThe onset rapidness of the somatic AP depends on (A) the distance from initiation site, (B)the soma size factor, and(C)the maximum Na⁺conductance. (D1-D3)The soma size effects vary at different distance ranges.

11.8 Summary and Discussion

In this chapter we presented a comprehensive study of the somatic action poten-tial waveforms using an idealized geometric model with axonal AP initiation and back propagation.

First, the waveforms of the somatic APs were quantified by the measure of onset rapidness. Its parameter dependence was characterized with respect to the distance from the AP initiation site to the soma, the soma size, and the overall Na⁺channel density.

In our simulations the waveforms of APs exhibited a transition from an ini-tially slow onset to a rapid onset with increasing propagation distance. If the spikes were triggered within 50 µm away from the soma, the onset dynamics of somatic APs was as slow as that at the initiation site. At a large propagation dis-tance, the traveling APs reached an invariant waveform. For these traveling APs, we derived the onset speed as a function of the propagation velocity (Eq. 11.16).

The impact of soma geometry was found to depend on the propagation dis-tance. If the propagation distance was small (L ≪ λ), expansion of the soma area introduced a larger current load, so that the AP onset was slowed down;

If the propagation distance was large (L > λ), the traveling wave had reached an invariant waveform such that increasing the soma size had hardly any im-pact on the onset dynamics of the somatic AP; At an intermediate distance in the transition regime (L≈λ) with a high Na⁺channel density, the local current source outweighed the current load, thus a larger somatic membrane area actually speeded up the AP onset in the middle point of the soma.

Na⁺currents provide the major current source for the generation of APs. A higher Na⁺channel density was reflected by increase of the peak rate of risedV /dt.

The impact on the AP onset dynamics differed at different propagation distance.

Close to the initiation site, the onset dynamics remained slow and was rather insensitive to an increase in Na⁺channel density. At an intermediate propagation distance, increasing Na⁺channel density dramatically increased the onset speed;

At a large distance, increasing Na⁺channel density speeded up the propagation velocity, thus the onset speed was increased correspondingly.

Second, we investigated the relative impacts of the lateral and local currents.

Our results suggested that APs propagating over a distance on the order of one electrotonic length established a strong potential field in space, generating a large lateral current. The contributions of lateral currents and local Na⁺currents to the somatic AP waveforms were revealed by the double peaks on the traces of dV /dtand d²V /dt². Two components in the first and second derivatives could be detected only if the soma size, the distance, and the channel density were within a certain range. Particularly for a larger lateral current to preced a smaller local Na⁺current, this could happen only within a small range of the propagation distance between 30 and 60 µm. The corresponding somatic APs exhibit slow onset regardless of the double peaks inV¨. Thus biphasic APs, even with a salient

peak induced by invasion of axonal APs, do not necessarily result in fast onset.

Finally, different models of Hodgkin-Huxley type show qualitatively the same parameter dependence of the AP onset dynamics. The changes induced by prop-agation were more prominent with faster Na⁺channel activation kinetics, i.e., the onset dynamics of somatic APs was fastest in the WB model, slower in theWBm¹ and WBm³ model, and even slower in the Mainen-Sejnowski models.

Dynamics of AP Initiation in the