Frequency-domain analysis of membrane polarization in two-compartment model neurons with weak alternating electric fields

Abstract

Transcranial alternating current stimulation (tACS) is widely used in studying brain functions and the treatment of neuropsychiatric diseases in a frequency-specific manner. However, how tACS works on neuronal activity has been poorly understood. In this paper, we use linear system analysis to investigate how weak alternating electric fields (EFs) affect the membrane polarization of neurons in the frequency domain. Two biophysically realistic conductance-based two-compartment models of cortical pyramidal neurons are developed to simulate subthreshold membrane polarization with weak alternating EFs. We linearize the original nonlinear models at the stable equilibrium points and further simplify them to the two- or three-dimensional linear systems. Thus, we calculate the transfer functions of the low-dimensional linear models to model neuronal polarization patterns. Based on the transfer functions, we compute the amplitude- and phase-frequency characteristics to describe the relationship between weak EFs and membrane polarization. We also computed the parameters (gain, zeros, and poles) and structures (the number of zeros and poles) of transfer functions to reveal how neuronal intrinsic properties affect the parameters and structure of transfer functions and thus the frequency-dependent membrane polarization with alternating EFs. We find that the amplitude and phase of membrane polarization both strongly depended on EF frequency, and these frequency responses are modulated by the intrinsic properties of neurons. The compartment geometry, internal coupling conductance, and ionic currents (except I_h) affect the frequency-dependent polarization by mainly changing the gain and pole of transfer functions. Larger gain contributes to larger amplitude-frequency characteristics. The closer the pole is to the imaginary axis, the lower phase-frequency characteristics. However, I_h changes the structure of transfer function in the dendrite by introducing a new pair of zero-pole points, which decrease the amplitude at low frequencies and thus lead to a visible resonance. These results highlight the effects of passive properties and active ion currents on subthreshold membrane polarization with alternating EFs in the frequency domain, which provide an explainable connection of how intrinsic properties of neurons modulate the neuronal input-output functions with weak EF stimulation.

Keywords: Frequency characteristic; Transfer function; Two-compartment neuron model; Weak alternating EFs.

Fig. 1

Schematics of two-compartment a Model I and b Model II in the presence of weak alternating EFs V_DS. The dendritic compartment is passive in Model I and incorporates four active currents in Model II

Fig. 2

a Membrane voltage responses in somatic and dendritic compartments. Weak alternating EF intensity is 2 mV. b Phase lag between membrane polarization in somatic/dendritic compartment and weak EFs. c The maximum membrane voltage deflection and d the phase difference in the soma (top) and dendrite (bottom) induced by weak alternating EFs

Fig. 3

Amplitude-frequency and phase-frequency characteristics in the a somatic and b dendritic compartments in the Model I, L-Model I, and LR-Model I

Fig. 4

a The amplitude- and b phase-frequency characteristics of the soma with different p (color coded). c Variation of the polarization amplitude and phase of the soma with respect to p, filed frequency is 50 Hz. d The amplitude- and e phase-frequency characteristics of the dendrite. f Variation of the polarization amplitude and phase of the dendrite with respect to p, filed frequency is 50 Hz. The solid lines correspond to the analytical results of LR-Model I while the dashed lines correspond to the numerical integrations of nonlinear Model I. The 50 Hz in c and f are marked in a-b and d–e using the grey dotted lines

Fig. 5

a The parameters (gain, zeros and poles) of somatic transfer function change with p. b The asymptotes of somatic log amplitude-frequency characteristics with different p (0.05, 0.25, 0.5, 0.75, and 0.95 along the direction of the arrow). c The phase-frequency characteristics of different substructures of $G_S(s,p)$

Fig. 6

a The parameters of dendritic transfer function change with $p$. b The asymptotes of dendritic log amplitude-frequency characteristics with different p (0.05, 0.25, 0.5, 0.75 and 0.95 along the direction of the arrow). c The log amplitude-frequency characteristics of different substructures (color coded) of $G_D(s)$ with $p=0.5$. d The derivative of $A_D(\omega )$ with respect to ω with $p=0.5$. e The phase-frequency characteristics of different substructures of $G_D(s,p)$ with different p

Fig. 7

a The amplitude- and b phase-frequency characteristics of the soma with different g_c (color coded). c Variation of the polarization amplitude and phase of the soma with respect to g_c, filed frequency is 50 Hz. d The amplitude- and e phase-frequency characteristics of the dendrite. f Variation of the polarization amplitude and phase of the dendrite with respect to g_c, filed frequency is 50 Hz. The solid lines correspond to the analytical results of LR-Model I while the dashed lines correspond to the numerical integrations of nonlinear Model I. The 50 Hz in c and f are marked in a–b and d–e using the grey dotted lines

Fig. 8

a The parameters (gain, zero and poles) of somatic transfer function changing with g_c. b The asymptotes of somatic log amplitude-frequency characteristics with different g_c (0.01, 0.12, 0.3, 0.5 and 1.0 along the arrow). c The phase-frequency characteristics of different substructures of $G_S(s,g_c)$ with different g_c

Fig. 9

a The parameters of dendritic transfer function changing with g_c. b The asymptotes of dendritic log amplitude-frequency characteristics with different g_c (0.01, 0.12, 0.3, 0.5 and 1.0 along the arrow). c The phase-frequency characteristics of different substructures of $G_D(s,g_c)$ with different g_c

Fig. 10

a $A_S(\omega )$ and b ${\phi }_S(\omega )$ with different ionic currents (color coded) of the soma. c The parameters of $G_S(s)$ change with g. d The asymptotes of somatic log amplitude-frequency characteristics with different somatic ion currents. The corresponding values of g are − 0.0158 (I_Na(S)), 0.0012 (I_K(S)) and − 3.2449e⁻⁴ (I_Nap(S)), respectively

Fig. 11

a $A_D(\omega )$ and b ${\phi }_D(\omega )$ with different ionic currents (color coded) of the dendrite. c The parameters of $G_D(s)$ change with $g$. d The asymptotes of dendritic log amplitude-frequency characteristics. The corresponding values of g are − 2.1975e⁻⁴ (I_Na(D)), 1.7385e⁻⁴ (I_K(D)), − 3.2324e⁻⁴ (I_Nap(D)) and 5.0e⁻³ (I_h), respectively

Fig. 12

a The dendritic amplitude-frequency characteristics of different models. b The log amplitude-frequency characteristics of $G_D(s)$ with LR-Model II and LR-Model II $-I_h$. c The polarization phase versus ω of different substructures of $G_D(s)$