Resonance modulation, annihilation and generation of anti-resonance and anti-phasonance in 3D neuronal systems: interplay of resonant and amplifying currents with slow dynamics

Abstract

Subthreshold (membrane potential) resonance and phasonance (preferred amplitude and zero-phase responses to oscillatory inputs) in single neurons arise from the interaction between positive and negative feedback effects provided by relatively fast amplifying currents and slower resonant currents. In 2D neuronal systems, amplifying currents are required to be slave to voltage (instantaneously fast) for these phenomena to occur. In higher dimensional systems, additional currents operating at various effective time scales may modulate and annihilate existing resonances and generate antiresonance (minimum amplitude response) and antiphasonance (zero-phase response with phase monotonic properties opposite to phasonance). We use mathematical modeling, numerical simulations and dynamical systems tools to investigate the mechanisms underlying these phenomena in 3D linear models, which are obtained as the linearization of biophysical (conductance-based) models. We characterize the parameter regimes for which the system exhibits the various types of behavior mentioned above in the rather general case in which the underlying 2D system exhibits resonance. We consider two cases: (i) the interplay of two resonant gating variables, and (ii) the interplay of one resonant and one amplifying gating variables. Increasing levels of an amplifying current cause (i) a response amplification if the amplifying current is faster than the resonant current, (ii) resonance and phasonance attenuation and annihilation if the amplifying and resonant currents have identical dynamics, and (iii) antiresonance and antiphasonance if the amplifying current is slower than the resonant current. We investigate the underlying mechanisms by extending the envelope-plane diagram approach developed in previous work (for 2D systems) to three dimensions to include the additional gating variable, and constructing the corresponding envelope curves in these envelope-space diagrams. We find that antiresonance and antiphasonance emerge as the result of an asymptotic boundary layer problem in the frequency domain created by the different balances between the intrinsic time constants of the cell and the input frequency f as it changes. For large enough values of f the envelope curves are quasi-2D and the impedance profile decreases with the input frequency. In contrast, for f ≪ 1 the dynamics are quasi-1D and the impedance profile increases above the limiting value in the other regime. Antiresonance is created because the continuity of the solution requires the impedance profile to connect the portions belonging to the two regimes. If in doing so the phase profile crosses the zero value, then antiphasonance is also generated.

Appendices

Appendix

A Impedance and phase profiles for 2D and 3D linear systems: Analytic expressions

In order to analytically compute he impedance and phase profiles for 3D linear generic systems we use

$$ \left\{\begin{array}{ll} x^{\prime} = a\, x + b\, y + c\, z + A_{in}\, e^{i\, \omega\, t},\\ y^{\prime} = \alpha\, x + p\, y \\ z^{\prime} = \beta\, x + q\, z, \end{array} \right. $$

(44)

where a, b, c, α, β, p and q are constant, ω > 0 and A _{i n} ≥ 0.

The characteristic polynomial for the corresponding homogeneous system (A _{i n} = 0) is given by

$$\begin{array}{@{}rcl@{}} r^3 &-& (a + p + q)\, r^2 + (a\, p + a\, q + p\, q - c\, \beta - b\, \alpha )\, r\\ &&+ b\, \alpha\, q + c\, \beta\, p - a\, p\, q = 0. \end{array} $$

(45)

The particular solution to system (44) has the form

$$\begin{array}{@{}rcl@{}} x_p(t) = A_{out} e^{i \omega t}, y_p(t) \,=\, B_{out} e^{i \omega t} \text{and} z_p(t) = C_{out} e^{i\, \omega t},\\ \end{array} $$

(46)

Substituting (46) into system (44), rearranging terms, and solving the resulting algebraic system one obtains

$$ Z(\omega) = \frac{A_{out}}{A_{in}} = \frac{P_r(\omega) + i\, P_i(\omega)}{Q_r(\omega) + i\, Q_i(\omega)} $$

(47)

where

$$ P_{r}(\omega) = p\, q - \omega^2, $$

(48)

$$ P_i(\omega) = -(p + q)\, \omega, $$

(49)

$$ Q_r(\omega) = (a + p + q)\, \omega^2 - a\, p\, q + b\, \alpha\, q + c\, \beta\, p, $$

(50)

and

$$ Q_i(\omega) = (a\, p + a\, q + p\, q - b\, \alpha - c\, \beta - \omega^2)\, \omega. $$

(51)

From Eq. (47)

$$ |Z|^2(\omega) := \frac{A_{out}^2}{A_{in}^2} = \frac{P_r^2(\omega) + P_i^2(\omega)}{Q_r^2(\omega) + Q_i^2(\omega)} $$

(52)

and

$$ \phi = \tan^{-1} \frac{P_r(\omega)\, Q_i(\omega)-P_i(\omega)\, Q_r(\omega)}{P_r(\omega)\, Q_r(\omega) + P_i(\omega)\, Q_i(\omega)}. $$

(53)

For a 2D linear system (c = q = 0), the characteristic polynomial for the corresponding homogeneous system (A _{i n} = 0) is given by

$$ r^2 - (a + p)\, r + (a\, p - b\, \alpha )\, = 0. $$

(54)

The roots of the characteristic polynomial are given by

$$ r_{1,2} = \frac{(a + p) \pm \sqrt{(a - p)^2 + 4\, b\, \alpha}}{2}. $$

(55)

From Eq. (55), the homogeneous (unforced) system displays oscillatory solutions with a natural frequency f _{n a t} (Hz) given by

$$ f_{nat} = \mu\, \frac{1000}{2\, \pi}, \ \ \ \ \ \ \ \ \ \ \ \ \mu = \sqrt{-4 b \alpha -(a-p)^2}, $$

(56)

provided 4 b α + (a − p)² < 0.

The impedance amplitude and phase are given, respectively, by

$$ |Z(\omega) |^2 := \frac{A_{out}^2}{A_{in}^2} = \frac{p^2 + \omega^2}{[\, a\, p - b\, \alpha - \omega^2\, ]^2 + (a+p)^2\, \omega^2}, $$

(57)

and

$$ \phi(\omega) = \tan^{-1} \frac{(a\, p - b\, \alpha - \omega^2)\ \omega - (a+p)\, \omega\, p}{(a\, p - b\, \alpha - \omega^2)\, p + (a + p)\ \omega^2}. $$

(58)