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.
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Acknowledgments
The authors thank Diana Martinez and David Fox for their comments on this manuscript.
Grants
Supported by NSF DMS1313861 (HGR) and NIH MH060605, NS083319 (FN).
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The authors declare that they have no conflict of interest.
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Appendices
Appendix A. Two-dimensional linear systems: eigenvalues, natural frequency and impedance
Consider the following two-dimensional linear system
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
The roots of the characteristic polynomial are given by
From Eq. (18), the homogeneous system displays stable oscillatory solutions (focus points) for values of the parameters satisfying
The natural frequency is given by
1.2 Impedance function
The particular solutions to system (17) have the form
Substituting into system (17) and rearranging terms we obtain
By solving the algebraic system (20) we obtain
The impedance Z peaks at the resonant frequency
The impedance phase is given by
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
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
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
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Rotstein, H.G., Nadim, F. Frequency preference in two-dimensional neural models: a linear analysis of the interaction between resonant and amplifying currents. J Comput Neurosci 37, 9–28 (2014). https://doi.org/10.1007/s10827-013-0483-3
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DOI: https://doi.org/10.1007/s10827-013-0483-3