Frequency preference in two-dimensional neural models: a linear analysis of the interaction between resonant and amplifying currents

Abstract

Many neuron types exhibit preferred frequency responses in their voltage amplitude (resonance) or phase shift to subthreshold oscillatory currents, but the effect of biophysical parameters on these properties is not well understood. We propose a general framework to analyze the role of different ionic currents and their interactions in shaping the properties of impedance amplitude and phase in linearized biophysical models and demonstrate this approach in a two-dimensional linear model with two effective conductances g _L and g ₁. We compute the key attributes of impedance and phase (resonance frequency and amplitude, zero-phase frequency, selectivity, etc.) in the g _L  − g ₁ parameter space. Using these attribute diagrams we identify two basic mechanisms for the generation of resonance: an increase in the resonance amplitude as g ₁ increases while the overall impedance is decreased, and an increase in the maximal impedance, without any change in the input resistance, as the ionic current time constant increases. We use the attribute diagrams to analyze resonance and phase of the linearization of two biophysical models that include resonant (I _h or slow potassium) and amplifying currents (persistent sodium). In the absence of amplifying currents, the two models behave similarly as the conductances of the resonant currents is increased whereas, with the amplifying current present, the two models have qualitatively opposite responses. This work provides a general method for decoding the effect of biophysical parameters on linear membrane resonance and phase by tracking trajectories, parametrized by the relevant biophysical parameter, in pre-constructed attribute diagrams.

Appendices

Appendix A. Two-dimensional linear systems: eigenvalues, natural frequency and impedance

Consider the following two-dimensional linear system

$$ \left\{\begin{array}{l}X\prime = aX+ bY+{A}_{i n}{e}^{i\omega t}\hfill \\ {}Y\prime = cX+ dY\hfill \end{array}\right. $$

(17)

a, b, c and d are constant, ω > 0 and A _in  ≥ 0.

1.1 Intrinsic oscillations and natural frequency

The Jacobian of the corresponding homogeneous system (A _in  = 0) is given by

$$ J=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ {}\hfill c\hfill & \hfill d\hfill \end{array}\right). $$

The roots of the characteristic polynomial are given by

$$ {r}_{1,2}=\frac{\left(a+d\right)\pm \sqrt{{\left(a-d\right)}^2+4 bc}}{2}. $$

(18)

From Eq. (18), the homogeneous system displays stable oscillatory solutions (focus points) for values of the parameters satisfying

$$ \begin{array}{r}\hfill {\left(a-d\right)}^2+4 bc<0\\ {}\hfill a+d\le 0.\end{array} $$

The natural frequency is given by

$$ \mu =\frac{\sqrt{4 bc+{\left(a-d\right)}^2}}{2}. $$

(19)

1.2 Impedance function

The particular solutions to system (17) have the form

$$ \begin{array}{c}\hfill {X}_p(t)={A}_{out}{e}^{i\omega t}\hfill \\ {}\hfill {Y}_p(t)={B}_{out}{e}^{i\omega t}\hfill \end{array} $$

Substituting into system (17) and rearranging terms we obtain

$$ \left(\begin{array}{cc}\hfill \left( i\omega -a\right)\hfill & \hfill -b\hfill \\ {}\hfill -c\hfill & \hfill \left( i\omega -d\right)\hfill \end{array}\right)\left(\begin{array}{c}\hfill {A}_{out}\hfill \\ {}\hfill {B}_{out}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {A}_{in}\hfill \\ {}\hfill 0\hfill \end{array}\right). $$

(20)

By solving the algebraic system (20) we obtain

$$ \begin{array}{l}Z\left(\omega \right)=\frac{A_{out}}{A_{in}}=\frac{-d+ i\omega}{\left( ad- bc-{\omega}^2\right)- i\omega \left(a+d\right)},\hfill \\ {}{\left|Z\left(\omega \right)\right|}^2=\frac{A_{out}^2}{A_{in}^2}=\frac{d^2+{\omega}^2}{{\left( ad- bc-{\omega}^2\right)}^2+{\left(a+d\right)}^2{\omega}^2}.\hfill \end{array} $$

(21)

The impedance Z peaks at the resonant frequency

$$ {\omega}_{res}=\sqrt{-{d}^2+\sqrt{b^2{c}^2-2 abcd-2{d}^2 bc}}. $$

(22)

The impedance phase is given by

$$ \tan \left(\phi \right)=\frac{\left( ad- bc-{\omega}^2\right)\omega -\left(a+d\right)\omega d}{\left( ad- bc-{\omega}^2\right)d+\left(a+d\right){\omega}^2}=-\frac{ bc+{d}^2+{\omega}^2}{\left( ad- bc\right)d+a{\omega}^2}\omega . $$

(23)

Appendix B. The 2D biophysical models

The voltage-dependent activation/inactivation curves x _∞(V) and voltage-dependent time scales τ _x (V) used in the kinetic Eq. (3) for the gating variables x are frequently expressed in terms of

$$ \begin{array}{c}\hfill {x}_{\infty }(V)=\frac{\alpha_x(V)}{\alpha_x(V)+{\beta}_x(V)}\hfill \\ {}\hfill {\tau}_x(V)=\frac{1}{\alpha_x(V)+{\beta}_x(V)}.\hfill \end{array} $$

(24)

Below we present the functions and parameters corresponding to the various models we use in this paper.

2.1 I _h  + I _p model

This model has been modified from (Acker et al. 2003). It has a persistent sodium current and a two-component (fast and slow) h-current given by I _p  = G _p p(V − E _Na ) = G _p p _∞(V)(V − E _Na ) and I _h =G _h r _∞(V)(V--E _h ) respectively with E _h  = −20 mV, E _Na  = 55 mV. The voltage-dependent activation/inactivation and time constants are given by

$$ \begin{array}{c}\hfill {p}_{\infty }(V)=1/\left(1+{e}^{-\left(V+38\right)/6.5}\right)\hfill \\ {}\hfill {r}_{f,}{}_{\infty }(V)=1/\left(1+{e}^{\left(V+79.2\right)/9.78}\right)\hfill \\ {}\hfill {\tau}_{r_f}(V)=0.51/\left({e}^{\left(V-1.7\right)/10}+{e}^{-\left(V+340\right)/52}\right)+1.\hfill \end{array} $$

2.2 I _Ks  + I _p model

This model has been proposed in (Acker et al. 2003). It has a persistent sodium current and a slow potassium (M-type) current given by I _p  = G _p p (V − E _Na ) = G _p p _∞(V)(V − E _Na ) and I _Ks  = g _q q (V − E _K ) with E _Na  = 55 mV and E _K  = 90 mV. The voltage-dependent activation/inactivation and time constants are given by

$$ \begin{array}{l}{p}_{\infty }(V)=1/\left(1+{e}^{-\left(V+38\right)/6.5}\right)\hfill \\ {}{q}_{\infty }(V)=1/\left(1+{e}^{\left(V+35\right)/6.5}\right)\hfill \\ {}{q}_{\tau }(V)=90\hfill \end{array} $$