Optimization of battery strengths in the Hodgkin–Huxley model

We study the effects of the ionic reversal potentials for sodium (ENa) and potassium (EK) on the dynamics and energetics of the action potential in the Hodgkin–Huxley model of the squid giant axon, finding that larger action potentials, whose size is controlled primarily by the difference between ENa and EK, can be evoked more rapidly, travel at a faster speed, and consume more metabolic energy than smaller ones. We then systematically investigate whether the biological values of the reversal potentials are optimal for any of a large class of objective functions combining common functional properties of the axon. There appear to be no unconstrained local maxima or minima in these functions over either the two-dimensional {ENa,EK} parameter space or, more significantly, over higher-dimensional parameter spaces which also include the ionic conductances and the axon diameter. This implies that any optimization of these functions in the Hodgkin–Huxley model must necessarily involve some kind of constraint among the parameters. We identify and discuss some possible constrained optimizations: if ENa is fixed, then the biological value of EK is optimal for action potential velocity and for the metabolic energy divided by the maximum firing frequency.