Interplay of activation kinetics and the derivative conductance determines resonance properties of neurons

In a neuron with hyperpolarization activated current ($I_h$), the correct input frequency leads to an enhancement of the output response. This behavior is known as resonance and is well described by the neuronal impedance. In a simple neuron model we derive equations for the neuron's resonance and we link its frequency and existence with the biophysical properties of $I_h$. For a small voltage change, the component of the ratio of current change to voltage change ($dI/dV$) due to the voltage-dependent conductance change ($dg/dV$) is known as derivative conductance ($G_h^{\text{Der}}$). We show that both $G_h^{\text{Der}}$ and the current activation kinetics (characterized by the activation time constant ${\tau }_h$) are mainly responsible for controlling the frequency and existence of resonance. The increment of both factors ($G_h^{\text{Der}}$ and ${\tau }_h$) greatly contributes to the appearance of resonance. We also demonstrate that resonance is voltage dependent due to the voltage dependence of $G_h^{\text{Der}}$. Our results have important implications and can be used to predict and explain resonance properties of neurons with the $I_h$ current.

Figure 1

Slope conductance of $I_h$ and its properties. (a) Voltage dependence of $I_h$ slope conductance ($G_h$), chord conductance $g_h$ and the derivative conductance $G_h^{\text{Der}}$. (b) Voltage dependence of the steady state activation variable $A_h^{\infty }$. (c) Voltage dependence of $d{A}_h^{\infty }/dV$.

Figure 2

Impedance dependence of $V$ and ${\tau }_h$. (a–e) Impedance profiles for the passive case (only leak, $Z_L$) and for the case of leak plus $I_h$ ($Z_{L+h}$) for different ${\tau }_h$ values (10, 100, and 1000 ms) and for different membrane potentials from $V=-60$ to $V=-140$ mV, which is the full range of $I_h$ activation. Notice that resonance peaks are not present for all potentials. (f) $I_h$ derivative conductance ($G_h^{\text{Der}}$). The arrows indicate the membrane potential used in the impedance profiles in (a–e). (g) Regions where there is resonance. In blue (i) there is no resonance; in (ii) green and above there is resonance for ${\overline{g}}_h=10$ nS; in light red (iii) and above, there is resonance for ${\overline{g}}_h=5$ nS; and in dark red (iv) there is resonance for ${\overline{g}}_h=1$ nS.

Figure 3

Evolution of trajectories for different membrane voltages. (a) Voltage dependence of the steady-state activation variable. Horizontal arrows indicate which region is studied on the plots to the right. (b–j) phase plots of activation variable vs membrane potential of the trajectories for different membrane potentials: (b–d) $V=-120$ mV, (e–g) $V=-80$ mV, (h–j) $V=-40$ mV and for different stimulation frequencies ($\omega$): (b, e, h) low frequency; (e, f, i) resonance frequency ${\omega }_{\text{res}}$; and (d, g, j) high frequency. The green (light gray) line represents the slope of $d{A}_h^{\infty }/dV\propto G_h^{\text{Der}}$. In all panels ${\tau }_h=50\mathrm{ms}$

Figure 4

Evolution of trajectories for different ${\tau }_h$. Phase plots of activation variable versus membrane potential of the trajectories for different activation time constant: (a–c) ${\tau }_h=50$, and (d–f) ${\tau }_h=5\mathrm{ms}$. There are different stimulation frequencies ($\omega$): (a, d) low frequency, (b, e) resonance frequency ${\omega }_{\text{res}}$, and (c, f) high frequency. The membrane potential was fixed at $V=-90$ mV. Green (light gray) line slope represents the $d{A}_h^{\infty }/dV\propto G_h^{\text{Der}}$.

Figure 5

Resonance frequency dependence with ${\tau }_h$ and $V$. (a–c) Resonance frequency (${\omega }_{\text{res}}$) for different values of ${\tau }_h$ and $V$. (d–f) Resonance frequency (${\omega }_{\text{res}}$) for different values of ${\tau }_h$ and $g_L$. (a, d) ${\overline{g}}_h=1$ nS, (b, e) ${\overline{g}}_h=5$ nS, and (c, f) ${\overline{g}}_h=10$ nS. In all cases $V=-85$ mV.

Figure 6

Impedance profiles and their dependency with $V$ and ${\tau }_h$. (a) Impedance profiles for different membrane potentials, and (b) for different ${\tau }_h$. (c, d) Variation of $A_h$. In (a, c) ${\tau }_h=50\mathrm{ms}$. In (b, d) we fixed $V=-90\mathrm{mV}$ and varied ${\tau }_h$ (5, 50, and 100 ms).

Figure 7

Schematic diagram explaining the mechanism by which the interaction between frequency and activation variable time evolution determines the amount of the change of the activation variable $\mathrm{\Delta }{A}_h$. The maximum amplitude of the change of the activation variable is determined by $d{A}_h^{\infty }/dV$, while the time evolution is determined by ${\tau }_h$. (a) For the same value of $d{A}_h^{\infty }/dV$, $\mathrm{\Delta }{A}_h$ is bigger for the faster ${\tau }_h$ values. (b) For the same ${\tau }_h$, $\mathrm{\Delta }{A}_h$ is bigger for the bigger $d{A}_h^{\infty }/dV$ values.

Figure 8

Impedance at vanishing frequency is voltage-dependent. Impedance magnitude when $\omega =0\mathrm{Hz}$.

Figure 9

Impedance profiles. Impedance magnitude for $V=-80$ mV. The blue (solid) curve is the case with only a leak current. Orange (dotted and dashed), green (dashed), and red (solid with stars) curves correspond to the cases with leak current plus $I_h$ with ${\tau }_h=10,100$, and $1000\mathrm{ms}$, respectively. The green (dashed) and red (solid with stars) curves cross at $\approx 25\mathrm{Hz}$, the orange (dotted and dashed) and blue (solid) curves cross at $\approx 85\mathrm{Hz}$, and the red (solid with stars) and blue (solid) curves never cross.