Numerical Model of Ectopically Active Axon

Although it seems almost intuitive that a rapidly activating, persistent Na+ current and kinetically slow K+ channels may conspire together to generate a slow rhythmic discharge, a numerical simulation can help to more objectively rule-in persistent Na+ current as a candidate for the bio physical basis of ectopic activity. A numerical model of a space-clamped, lumped node and demyelinated internode from a large axon, was written in Visual Basic (Microsoft). The simulations contained the same channel types as those included in the axon model of Bostock et al. (1991), with voltage-dependent gating parameters taken from Schwarz et al. (1995), but omitted inward rectification. They also included a persistent Na+ current that activated 20 mV more negative than transient Na+ current (c.f., Bostock and Rothwell, 1997), where the value of persistent Na+ conductance was 1.5% that of the transient Na+ conductance. The model output could be made to mimic the low-frequency, regular discharge recorded in demyelinated axons (although the balance of Na+ and slow K+ channels was critical).

Although real discharge cannot be generated by a depolarization elicited by raised external K+ alone, the discharge frequency in the model was increased by raising external K+, because GKs becomes a less efficient check on the excitatory influence of persistent Na+ current. This effect is similar to the effect of blocking GKs (Baker and Bostock, 1992). Using the kinetic gating parameters published by Schwarz et al. (1995) for 20°C and running the model at a higher temperature, it did not seem possible to get 10 Hz activity at 37°C, whereas at 20 or 30°C, it was possible (at the higher temperature, a higher frequency discharge occurred with sufficient Na+ current). Increasing the temperature tends to stabilize the membrane potential, preventing the lowest frequency ectopic activity through an action on the kinetics of GKs. Thus, while this model clearly demonstrates the probable role of persistent Na+ current in generating ectopic activity, it is too simple. Two shortcomings were a poor reproduction of ectopic activity at 37°C (probably related to the kinetics of GKs), and the lack of a Na+/K+ ATPase, that would normally tend to reduce the frequency of discharge and contribute to burst discharges interspersed with quiet periods.

Typical output of the model can be seen in Fig. 5A and B, along with the effects of very slightly raising external [K+]. Action potentials were generated at 10.5 Hz, a slow and similar frequency to that of real ectopic activity. As in real activity, this was close to the lowest frequency attainable. The pacemaker potential was caused by the low-threshold persistent Na+ current, whose presence meant that the membrane current was inward (and never zero) after the repolarization phase of an action potential (Fig. 5D).

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