Sodium channel slow inactivation normalizes firing in axons with uneven conductance distributions

The Na⁺ channels that are important for action potentials show rapid inactivation, a state in which they do not conduct, although the membrane potential remains depolarized.^1,2 Rapid inactivation is a determinant of millisecond-scale phenomena, such as spike shape and refractory period. Na⁺ channels also inactivate orders of magnitude more slowly, and this slow inactivation has impacts on excitability over much longer timescales than those of a single spike or a single inter-spike interval.^{3,4,5,6,7,8,9,10} Here, we focus on the contribution of slow inactivation to the resilience of axonal excitability^11,12 when ion channels are unevenly distributed along the axon. We study models in which the voltage-gated Na⁺ and K⁺ channels are unevenly distributed along axons with different variances, capturing the heterogeneity that biological axons display.^13,14 In the absence of slow inactivation, many conductance distributions result in spontaneous tonic activity. Faithful axonal propagation is achieved with the introduction of Na⁺ channel slow inactivation. This "normalization" effect depends on relations between the kinetics of slow inactivation and the firing frequency. Consequently, neurons with characteristically different firing frequencies will need to implement different sets of channel properties to achieve resilience. The results of this study demonstrate the importance of the intrinsic biophysical properties of ion channels in normalizing axonal function.