The shaping of intrinsic membrane potential oscillations: positive/negative feedback, ionic resonance/amplification, nonlinearities and time scales

Abstract

The generation of intrinsic subthreshold (membrane potential) oscillations (STOs) in neuronal models requires the interaction between two processes: a relatively fast positive feedback that favors changes in voltage and a slower negative feedback that opposes these changes. These are provided by the so-called resonant and amplifying gating variables associated to the participating ionic currents. We investigate both the biophysical and dynamic mechanisms of generation of STOs and how their attributes (frequency and amplitude) depend on the model parameters for biophysical (conductance-based) models having qualitatively different types of resonant currents (activating and inactivating) and an amplifying current. Combinations of the same types of ionic currents (same models) in different parameter regimes give rise to different types of nonlinearities in the voltage equation: quasi-linear, parabolic-like and cubic-like. On the other hand, combinations of different types of ionic currents (different models) may give rise to the same type of nonlinearities. We examine how the attributes of the resulting STOs depend on the combined effect of these resonant and amplifying ionic processes, operating at different effective time scales, and the various types of nonlinearities. We find that, while some STO properties and attribute dependencies on the model parameters are determined by the specific combinations of ionic currents (biophysical properties), and are different for models with different such combinations, others are determined by the type of nonlinearities and are common for models with different types of ionic currents. Our results highlight the richness of STO behavior in single cells as the result of the various ways in which resonant and amplifying currents interact and affect the generation and termination of STOs as control parameters change. We make predictions that can be tested experimentally and are expected to contribute to the understanding of how rhythmic activity in neuronal networks emerge from the interplay of the intrinsic properties of the participating neurons and the network connectivity.

Appendix A: Alternative linear model formulations

1.1 A.1 Alternative dimensional formulation

The 3D system (4)–(6) can be viewed as a 2D system (4)–(5) coupled to the third variable ( w ₂) whose dynamics is given by Eq. (6). This effective coupling depends not only on the linearized conductance g ₂, but also on the time constant $\bar {\tau }_{2}$. To capture this, it is useful to express system (4)–(6) in an alternative way by defining

$$ w = w_1 + \frac{g_2}{g_1}\, w_2. $$

(13)

By substituting into (4)-(5) and rearranging terms we obtain

$$ \frac{dv}{dt} = \frac{g_L}{C}\, \left[\, -v - \gamma\, w \right], $$

(14)

$$ \frac{d w}{dt} = \frac{1}{\bar{\tau}_1}\, [\, \alpha\, v - w - \beta\, w_2\, ], $$

(15)

$$ \frac{d w_2}{dt} = \frac{1}{\bar{\tau}_2}\, [\, v - w_2\, ]. \hspace{1cm} $$

(16)

where

$$\gamma = \frac{g_1}{g_L}, \ \ \ \ \ \ \ \ \ \ \alpha = 1+\frac{\bar{\tau}_1}{\bar{\tau}_2}\, \frac{g_2}{g_1} \ \ \ \ \ \ \ \ \ \ \text{and} $$

$$ \beta = \frac{g_2}{g_1}\, \left( \frac{\bar{\tau}_1}{\bar{\tau}_2}-1\right) = \alpha -1 - \frac{g_2}{g_1}. $$

(17)

Note that α, β and γ are dimensionless but system (14)–(16) is not. For g ₂=0 ( α=1 and β=0), the 3D system reduces to the 2D system studied in Rotstein and Nadim (2014). The strength of the coupling between w ₂ and this system depends on both g ₂ and τ ₂. Changes in g ₂ and $\bar {\tau }_{2}$ cause changes in both α and β. In particular, if g ₂>0 ( g ₂<0), corresponding to a resonant (amplifying) gating variable x ₂, then α>1 ( α<1) and β>0 ( β<0).

1.2 A.2 Nondimensionalized linearized system

System (4)–(6) can be rescaled in order to reduce the number of parameters that effectively govern its dynamics (with no loss of information) and to uncover the effective strength of the terms in Eq. (4). We follow Richardson et al. (2003) and define the following dimensionless time and parameters

$$ \hat{t} = \frac{t}{\bar{\tau}_1} \ \ \ \ \ \ \gamma_L = \frac{g_L \bar{\tau}_1}{C} \ \ \ \ \ \ \gamma_1 = \frac{g_1 \bar{\tau}_1}{C} \ \ \ \ \ \ \gamma_2 = \frac{g_2 \bar{\tau}_1}{C} \ \ \ \ \ \ \kappa=\frac{\bar{\tau}_1}{\bar{\tau}_2}. $$

(18)

Substituting into (4)–(5) we obtain

$$ \frac{dv}{d \hat{t}} = -\gamma_L v - \gamma_1 w_1 - \gamma_2 w_2, $$

(19)

$$ \frac{d w_1}{d \hat{t}} = v - w_1, \hspace{2.7cm} $$

(20)

$$ \frac{d w_2}{d \hat{t}} = \kappa\, [ v - w_2 ]. \hspace{2.3cm} $$

(21)

Note that since v, w ₁ and w ₂ have the same units (of voltage) it is not necessary to rescale them (equivalently to rescale them by 1 mV).

1.3 Appendix B: Characterisic polynomial for 2D and 3D linearized systems

The characteristic polynomial for system (4)–(6) is given by

$$\begin{array}{@{}rcl@{}} C\, \bar{\tau}_1\, \bar{\tau}_2\, r^3 &+& [\, g_L\, \bar{\tau}_1\, \bar{\tau}_2 + C\, ( \bar{\tau}_1 + \bar{\tau}_2)\, ]\, r^2\\ & +& [\, g_L\, (\bar{\tau}_1 + \bar{\tau}_2) + C + g_1\, \bar{\tau}_2 + g_2\, \bar{\tau}_1\, ]\, r \\ & +&+ [\, g_L + g_1 + g_2\, ] = 0 \end{array} $$

(22)

For the 2D linear system (4)–(5) with g ₂=0, the characteristic polynomial reduces to

$$ C\, \bar{\tau}_1\, r^2 + [\, g_L\, \bar{\tau}_1+ C\, ]\, r + [\, g_L + g_1\, ] = 0. $$

(23)

The roots of Eq. (23) are given by

$$ r_{1,2} = \frac{-(\, C + g_L\, \bar{\tau}_1\, ) \pm \sqrt{ (\, C - g_L\, \bar{\tau}_1\, )^2 - 4\, C\, g_1\, \bar{\tau}_1\, }}{2\, C\, \bar{\tau}_1} $$

(24)

and

$$ \eta = \frac{g_L}{2\, C} + \frac{1}{2\, \bar{\tau}_1}. $$

(25)