Subthreshold amplitude and phase resonance in models of quadratic type: Nonlinear effects generated by the interplay of resonant and amplifying currents

Abstract

We investigate the biophysical and dynamic mechanisms of generation of subthreshold amplitude and phase resonance in response to sinusoidal input currents in two-dimensional models of quadratic type. These models feature a parabolic voltage nullcline and a linear nullcline for the recovery gating variable, capturing the interplay of the so-called resonant currents (e.g., hyperpolarization-activated mixed-cation inward and slow potassium) and amplifying currents (e.g., persistent sodium) in biophysically realistic parameter regimes. These currents underlie the generation of resonance in medial entorhinal cortex layer II stellate cells and CA1 pyramidal cells. We show that quadratic models exhibit nonlinear amplifications of the voltage response to sinusoidal inputs in the resonant frequency band. These are expressed as an increase in the impedance profile as the input amplitude increases. They are stronger for values positive than negative to resting potential and are accompanied by a shift in the phase profile, a decrease in the resonant and phase-resonant frequencies, and an increase in the sharpness of the voltage response. These effects are more prominent for smaller values of 𝜖 (larger levels of the time scale separation between the voltage and the resonant gating variable) and for values of the resting potential closer to threshold for spike generation. All other parameter fixed, as 𝜖 increases the voltage response becomes “more linear”; i.e., the nonlinearities are present, but “ignored”. In addition, the nonlinear effects are strongly modulated by the curvature of the parabolic voltage nullcline (partially reflecting the effects of the amplifying current) and the slope of the resonant current activation curve. Following the effects of changes in the biophysical conductances of realistic conductance-based models through the parameters of the quadratic model, we characterize the qualitatively different effects that resonant and amplifying currents have on the nonlinear properties of the voltage response. We identify different classes of resonant currents, represented by h- and slow potassium, according to whether they enhance (h-) or attenuate (slow potassium) the nonlinear effects. Finally, we use dynamical systems tools to investigate the dynamic mechanisms of generation of resonance and phase-resonance. We show that the nonlinear effects on the voltage response (e.g., amplification of the voltage response in the resonant frequency band and shifts in the resonant and phase-resonant frequencies) result from the ability of limit cycle trajectories to follow the unstable (right) branch of the voltage nullcline for a significant amount of time. This is a canard-related mechanism that has been shown to underlie the generation of intrinsic subthreshold oscillations in quadratic type models such as medial entorhinal cortex stellate cells. Overall, our results highlight the complexity of the voltage response to oscillatory inputs in nonlinear models and the roles that resonant and amplifying currents have in shaping these responses.

Appendix: Model functions

Below we present the activation/inactivation curves x _∞ (V) and voltage-dependent time scales τ _x (V) used in the kinetic (3) for the gating variables x corresponding to the various models presented in Section 2.1.

1.1 A.1 HNap model 1

$$\begin{array}{ll} p_{\infty}(V) = 1 / (1 + e^{-(V+38)/6.5})\\ \tau_{p}(V) = 0.15\\ r_{f,\infty}(V) = 1 / (1+e^{(V+79.2)/9.78}),\\ \tau_{r_{f}}(V) = 0.51 / (e^{(V-1.7)/10}+e^{-(V+340)/52})+1.\\ \end{array} $$

1.2 A.2 KsNap model

$$\begin{array}{ll} p_{\infty}(V) = 1 / (1 + e^{-(V+38)/6.5})\\ \tau_{p}(V) = 0.15\\ q_{\infty}(V) = 1 / (1 + e^{-(V+35)/6.5}),\\ \tau_{q}(V) = 90. \end{array} $$