Dynamic compensation mechanism gives rise to period and duty-cycle level sets in oscillatory neuronal models

Rhythmic oscillation in neurons can be characterized by various attributes, such as the oscillation period and duty cycle. The values of these features depend on the amplitudes of the participating ionic currents, which can be characterized by their maximum conductance values. Recent experimental and theoretical work has shown that the values of these attributes can be maintained constant for different combinations of two or more ionic currents of varying conductances, defining what is known as level sets in conductance space. In two-dimensional conductance spaces, a level set is a curve, often a line, along which a particular oscillation attribute value is conserved. In this work, we use modeling, dynamical systems tools (phase-space analysis), and numerical simulations to investigate the possible dynamic mechanisms responsible for the generation of period and duty-cycle levels sets in simplified (linearized and FitzHugh-Nagumo) and conductance-based (Morris-Lecar) models of neuronal oscillations. A simplistic hypothesis would be that the tonic balance between ionic currents with the same or opposite effective signs is sufficient to create level sets. According to this hypothesis, the dynamics of each ionic current during a given cycle are well captured by some constant quantity (e.g., maximal conductances), and the phase-plane diagrams are identical or are almost identical (e.g., cubic-like nullclines with the same maxima and minima) for different combinations of these maximal conductances. In contrast, we show that these mechanisms are dynamic and involve the complex interaction between the nonlinear voltage dependencies and the effective time scales at which the ionic current's dynamical variables operate.