A linearized modeling framework for the frequency selectivity in neurons postsynaptic to vibration receptors

Abstract

Vibration is an indispensable part of the tactile perception, which is encoded to oscillatory synaptic currents by receptors and transferred to neurons in the brain. The A2 and B1 neurons in the drosophila brain postsynaptic to the vibration receptors exhibit selective preferences for oscillatory synaptic currents with different frequencies, which is caused by the specific voltage-gated Na⁺ and K⁺ currents that both oppose the variations in membrane potential. To understand the peculiar role of the Na⁺ and K⁺ currents in shaping the filtering property of A2 and B1 neurons, we develop a linearized modeling framework that allows to systematically change the activation properties of these ionic channels. A data-driven conductance-based biophysical model is used to reproduce the frequency filtering of oscillatory synaptic inputs. Then, this data-driven model is linearized at the resting potential and its frequency response is calculated based on the transfer function, which is described by the magnitude–frequency curve. When we regulate the activation properties of the Na⁺ and K⁺ channels by changing the biophysical parameters, the dominant pole of the transfer function is found to be highly correlated with the fluctuation of the active current, which represents the strength of suppression of slow voltage variation. Meanwhile, the dominant pole also shapes the magnitude–frequency curve and further qualitatively determines the filtering property of the model. The transfer function provides a parsimonious description of how the biophysical parameters in Na⁺ and K⁺ channels change the inhibition of slow variations in membrane potential by Na⁺ and K⁺ currents, and further illustrates the relationship between the filtering properties and the activation properties of Na⁺ and K⁺ channels. This computational framework with the data-driven conductance-based biophysical model and its linearized model contributes to understanding the transmission and filtering of vibration stimulus in the tactile system.

Introduction

Peripheral cells in the auditory (Rudnicki et al. 2015), vestibular (Highstein et al. 2005) and somatosensory (Witham and Baker 2012) encode the displacement of the body and receptors (Suver et al. 2019). In the tactile sensory system, the vibration, a common displacement occurred on the surface of skin (Shao et al. 2016; Ding and Bhushan 2016), is also encoded to behavior features of mechanical stimulus, which is distributed over a wide range of frequency (Felicetti et al. 2023). The selective preference for oscillatory synaptic inputs with particular frequencies (Gescheider et al. 2009) is widely observed in receptors and peripheral cells, such as the A2 and B1 neurons postsynaptic to the Johnston’s organ in drosophila brains (Azevedo and Wilson 2017) and vertebrate hair cells (Fuchs and Evans 1990; Goodman and Art 1996). It is essential to examine such frequency selectivity of vibration stimulus for understanding the integration of encoded synaptic currents from receptors in the tactile sensory system. This contributes to further exploring the role of tactile vibration integration in the related behaviors, including the detection of touch (Sandykbayeva et al. 2022) and the enhancements in emotions (Sharp et al. 2019; Raheel et al. 2020).

In the previous study, the A2 and B1 neurons exhibit diverse frequency preferences to vibrations (Azevedo and Wilson 2017). In contrast to the uniform filtering of the A2 neurons to vibrations, B1 neurons present as diverse bandpass filters to vibrations. To investigate the special intrinsic properties that cause B1 neurons to exhibit bandpass filtering to vibrations, Azevedo and Wilson (2017) eliminate the synaptic inputs and focus on the intrinsic currents of A2 and B1 neurons (Azevedo and Wilson 2017; Hudspeth and Lewis 1988; Combe et al. 2018; Roberts et al. 1990; Fettiplace and Hackney 2006). When the voltage-gated Na⁺ and K⁺ currents are blocked by tetrodotoxin (TTX) and tetraethylammonium chloride (TEA), the steady-state current of A2 neurons is not affected and the current of B1 neurons tend to be suppressed. Then, all B1 neurons exhibit the similar filtering properties as A2 neurons. In this way, the voltage-gated Na⁺ and K⁺ currents are discovered to suppress low-frequency synaptic inputs by clamping the membrane potential at different values, which is different from the role of Na⁺ and K⁺ currents oscillating below the threshold voltage in other cells (Wilbers et al. 2022; Kamiya 2022). Since the B1 neuron rarely generates the action potential, only the subthreshold oscillatory responses of the A2 and B1 neuron are concerned in this study. The unusual suppression of slow variation in voltage by Na⁺ and K⁺ currents makes the B1 neurons prefer the oscillatory synaptic current with intermediate frequency, combined with the passive membrane property that inhibits the high-frequency synaptic inputs (Azevedo and Wilson 2017). Indeed, the filtering properties of A2 and B1 are determined by the interaction of the passive membrane property and the active currents, which depends on the activation properties of Na⁺ and K⁺ channels. Thus, it is important to interpret the relationship between the cellular frequency selectivity and the specific activation properties of Na⁺ and K⁺ channels for understanding the encoding and transmission of tactile vibration.

An effective method used to explain the link between the activation properties of ionic channels and the subthreshold oscillations in the membrane potential is the linearization of the model. The linearization of models provides an accurate but parsimonious description of the subthreshold oscillatory dynamics (Huang et al. 2023; Rotstein 2013, 2015; Rotstein and Nadim 2014). The frequency response of the biophysical model can be characterized sufficiently by the transfer function calculated from the linearized model, which allows to relate the responses in frequency domain to the causes. In the previous studies, the linearized model provides a clear explanation for how the neural properties modulate the field-induced frequency-dependent membrane polarization (Huang et al. 2023). In this study, the membrane potential of A2 and B1 neurons always oscillate around the resting potential below the threshold voltage. Thus, we linearize the biophysical model at its stable equilibrium to capture the frequency property of subthreshold oscillations in the membrane potential (Huang et al. 2023; Rotstein 2013, 2015; Rotstein and Nadim 2014). Although the absence of the nonlinearities in the linearized model leads to a decrease in voltage response, the preferred frequency of oscillatory synaptic currents that makes the neuron generate the optimal voltage response in magnitude is not affected by this predictive deviation (Rotstein 2014). Instead of applying stimulus with various frequencies to the biophysical model and measuring the amplitude of the voltage response, the magnitude of the voltage response could be directly derived from the transfer function of the linearized model. Further, the filtering properties could be easily observed from the shape of magnitude–frequency curves. The linearized model provides a compact way to relate the biophysical parameters to the magnitude of subthreshold voltage oscillations (Moran et al. 2007).

In this study, we develop a computational framework to understand how the Na⁺ and K⁺ currents shape the frequency selectivity in A2 and B1 neurons by a data-driven conductance-based biophysical model and its linearized model. We use the data-driven conductance-based biophysical model based on in-vivo experimental recording to reproduce the filtering of oscillatory synaptic currents in A2 and B1 neurons (Azevedo and Wilson 2017; Rotstein and Nadim 2019), which allows to systematically examine the role of biophysical parameters of Na⁺ and K⁺ channels in shaping the fluctuation of Na⁺ and K⁺ currents below the threshold voltage for action potential generation. The sinusoidal alternating current is injected into the conductance-based biophysical model to simulate synaptic currents from antennal vibration receptors, forcing the membrane potential to oscillate around the resting potential. Then, we linearize the model at its resting potential and further derive the transfer function to predict the frequency response of models. We find that the fluctuation of the Na⁺ and K⁺ currents based on different biophysical parameters change the filtering properties of A2 and B1 neurons by shifting the dominant pole of the transfer function. Our framework explains how the peculiar biophysical properties of Na⁺ and K⁺ currents modulate the interaction of the active current and passive membrane properties in A2 and B1 neurons, which leads to diverse frequency selectivity in these neurons.

Methods and models

We use the conductance-based biophysical model to simulate the frequency filtering of oscillatory synaptic inputs in the A2 and B1 cells postsynaptic to the Johnston’s organ in drosophila brains. The biophysical parameters of the Na⁺ and K⁺ channels in the model are respectively fit to their steady-state conductance recorded in vivo. To explore the effects of biophysical parameters on the filtering function, we linearize the data-driven A2 and B1 models at their resting potential, and use the transfer function of the resulting linearized model to examine the frequency filtering in each cell. The zeros and poles of the transfer function shape the magnitude–frequency curve of the model neuron (Beauchene et al. 2021), which contribute to understanding the relation between biophysical parameters and frequency filtering properties.

Conductance-based A2 and B1 models

We use a conductance-based model to reproduce the dynamics of membrane potential V in A2 and B1 neurons, which is modified from the model proposed by Prescott et al. (2008), Rinzel and Ermentrout (1989), Morris and Lecar (1981), Schreiber et al. (2004). The model consists of the Na⁺, K⁺ (active currents) and leakage (passive current) currents. The system is described as

$$C_M\frac{\mathit{dV}}{\mathit{dt}}=-I_{\mathit{Na}}-I_K-I_L+I_{\mathit{stim}}(t)$$ 1

$$I_L=g_L(V-E_l)$$ 2

$$I_{\mathit{Na}}={\overline{g}}_{\mathit{Na}}m(V-E_{\mathit{Na}})$$ 3

$$I_K={\overline{g}}_{K}w(V-E_K)$$ 4

where C_M is the membrane capacitance, V is the membrane potential, t is time, I_Na and I_K are Na⁺ and K⁺ currents, I_L is the leakage current, I_stim is the stimulus current injected to the model, g_L is the leakage conductance, ${\overline{g}}_{\mathit{Na}}$ is the maximum Na⁺ conductance and ${\overline{g}}_K$ is the maximum K⁺ conductance. Note that a gating variable m is introduced to Na⁺ channel to capture the subthreshold dynamics of I_Na at different stimulus frequencies. The gating variables m and w are functions of V, which are given by

$$\frac{\mathit{dm}}{\mathit{dt}}=\frac{m_{\infty }(V)-m}{{\tau }_m}$$ 5

$$\frac{\mathit{dw}}{\mathit{dt}}=\frac{w_{\infty }(V)-w}{{\tau }_w}$$ 6

$$m_{\infty }(V)=-tanh\left(\frac{V-{\beta }_m}{{\gamma }_m}\right)$$ 7

$$w_{\infty }(V)=1+tanh\left(\frac{V-{\beta }_w}{{\gamma }_w}\right)$$ 8

where m_∞ and w_∞ are the steady-state values, τ_m and τ_w are the time constants.

To simulate different frequency filtering in A2 and B1 cells, the biophysical parameters of Na⁺ and K⁺ conductances are fit to the experimental data. The steady-state I_Na, I_K and I_L that flow through A2 and B1 cells are measured by voltage-clamp recordings when V is stepped to − 20, − 10, − 5, − 2.5, 2.5, 5, 10 and 15 mV, which are all near the resting potential V_rest. The steady-state conductance of these currents is then derived and interpolated with the step of 0.1 mV by a spline function. The functions of m_∞ and w_∞ are modified as Eq. (7) and (8) for realization of the conductance-voltage properties recorded in experiments in A2 and B1 models. We use the least square method (Ermentrout 1994) to fit the six parameters of Na⁺ and K⁺ channels, including ${\overline{g}}_{\mathit{Na}}$, β_m, γ_m, ${\overline{g}}_K$, β_w and γ_w, which are set to 1 initially. Since only the subthreshold response near V_rest is considered in this study, the steady-state conductance is clipped over the range of V ∈ [-15, 20] mV. Other parameters used in A2 and B1 models follow Azevedo and Wilson (2017), which are shown in Table 1.

Table 1.

Parameters used in A2 and B1 models

Parameters	A2 model	B1 model
${\overline{g}}_{\mathit{Na}}$(mS/cm2)	0.0248	0.3522
βm (mV)	34.6186	15.2255
γm (mV)	149.9946	27.8269
${\overline{g}}_K$(mS/cm2)	0.0083	2.1365
βw (mV)	− 12.2349	14.8127
γw (mV)	7.0652	14.0203
CM (μF/cm2)	1.6	1.7
Vrest (mV)	0	0
gL (mS/cm2)	1	1.92
El (mV)	0	0
ENa (mV)	162	162
EK (mV)	− 51	− 51
τm (ms)	10	6
τw (ms)	10	10

Linearization of A2 and B1 models

To avoid the derivation of the frequency responses from the outputs of A2 and B1 model in time domain, which consumes a lot of time to calculate the voltage amplitude as functions of the oscillation frequency, the explicit transfer function is used to predict the steady-state responses in frequency domain. Thus, the A2 and B1 models are linearized at their stable fixed-points with V below the threshold for action potential generation (Rotstein and Nadim 2014; Richardson et al. 2003). As only the subthreshold responses around V_rest are considered in this study, the linearization is confined to a holding voltage V* = V_rest. The linearized A2 (L-A2) and linearized B1 (L-B1) models are defined as

$$C_M\frac{\mathit{dV}}{\mathit{dt}}=-g_{M}V-g_m{x}_m-g_w{x}_w+I_{\mathit{stim}}$$ 9

$${\tau }_m\frac{d{x}_m}{\mathit{dt}}=V-x_m$$ 10

$${\tau }_w\frac{d{x}_w}{\mathit{dt}}=V-x_w$$ 11

Here variable x_m and x_w have units of mV. g_M is the slope of the instantaneous I-V curve at V*, which is given by

$$g_M=g_L+{\overline{g}}_{\mathit{Na}}{m}_{\infty }^{\ast }+{\overline{g}}_K{w}_{\infty }^{\ast }$$ 12

Here, the superscript asterisk * is used to denote the value of each variable at V = V*. The parameters g_m and g_w are used to respectively quantify the strength of the effects of x_m and x_w on V, which are described as

$$g_m={\overline{g}}_{\mathit{Na}}(V^{\ast }-E_{\mathit{Na}}){\left(\frac{d{m}_{\infty }}{\mathit{dV}}\right)}^{\ast }$$ 13

$$g_w={\overline{g}}_K(V^{\ast }-E_K){\left(\frac{d{w}_{\infty }}{\mathit{dV}}\right)}^{\ast }$$ 14

$$\frac{d{m}_{\infty }}{\mathit{dV}}=-\frac{1}{{\gamma }_m}\left[1,-,{tanh}^2,\left(\frac{V-{\beta }_m}{{\gamma }_m}\right)\right]$$ 15

$$\frac{d{w}_{\infty }}{\mathit{dV}}=\frac{1}{{\gamma }_w}\left[1,-,{tanh}^2,\left(\frac{V-{\beta }_w}{{\gamma }_w}\right)\right]$$ 16

The linearized model is then transformed into the transfer function form for the calculation of frequency responses (David and Friston 2003; David et al. 2004). The three-dimensional linear time-invariant system is reorganized in the generic form as

$$\begin{array}{c}\dot{X}=AX+Bu\\ Y=CX+Du\\ \end{array}$$ 17

where $X=\left[\begin{array}{c}{x}_m\\ x_w\\ V\end{array}\right]$ is a three-dimensional state vector, the output of the system Y is V and the input of the system u is I_stim. The state matrix A, input matrix B, output matrix C and direct transmission matrix D are

$$A=\left[\begin{array}{ccc}-1/{\tau }_m& 0& 1/{\tau }_m\\ 0& -1/{\tau }_w& 1/{\tau }_w\\ -g_m/C_M& -g_w/C_M& -g_M/C_M\end{array}\right],B=\left[\begin{array}{c}0\\ 0\\ 1/C_M\end{array}\right],C=\left[\begin{array}{ccc}0& 0& 1\end{array}\right],D=0$$ 18

The transfer function of the model is calculated as the following pole-zero form

$$G(s)=C{(sI-A)}^{-1}B+D=\frac{k{\prod }_{i=1}^z(s-z_i)}{{\prod }_{j=1}^p(s-p_j)}$$ 19

where s is the Laplace operator, I is the unit matrix, z_i is the ith zero, p_j is the jth pole, n_z is the number of zeros, n_p is the number of poles and k is the gain of transfer function. In the stable linear system, the magnitude–frequency response is determined by cutting along the jω axis at α = 0 of s = α + jω (Prsa et al. 2019), which is

$$A_m(\omega )=\left|G(j\omega )\right|=\left|\frac{k{\prod }_{i=1}^z(j\omega -z_i)}{{\prod }_{l=1}^p(j\omega -p_l)}\right|=\frac{\left|k\right|{\prod }_{i=1}^z\sqrt{{\omega }^2+z_i^2}}{{\prod }_{l=1}^p\sqrt{{\omega }^2+p_l^2}}$$ 20

where j is the imaginary unit, ω = 2πf / 1000 is the frequency in rad/ms and A_m is the magnitude in dB. The magnitude–frequency curves of the transfer functions of L-A2 and L-B1 models in the s-phase provide the frequency characteristics intuitively in the frequency domain.

The magnitude–frequency curve can be described by the inflection points of its asymptote, which presents as the distance from zeros and poles to the imaginary axis. The slope of the asymptote increases 20 dB/dec when a first-order differential element occurs. A first-order inertial element leads to a decrease of 20 dB/dec in the slope. Thus, the zeros and poles of the linear system qualitatively describe the frequency corresponding to the peak of the curve, which exhibits the filtering characteristic of models. All models are simulated in MATLAB 2022A based on the forward Euler method with a time step of 0.01 ms.

Results

Validation of A2 and B1 models

To validate the data-driven biophysical models, we examine the voltage response in A2 and B1 model cells to the stimulus with different frequencies. A series of sinusoidal alternating stimulus I_stim are injected to the models to simulate synaptic currents from antennal vibration receptors. The A_stim is set to 0 μA/cm² at first 100 ms to lead the models to reach a stable state. Then, A_stim is set to 10 μA/cm² for 2 s, which is used to simulate the synaptic currents evoked by step displacements of the antenna. The strength of oscillatory stimulus equals to the amplitude of the synaptic current recorded in all A2 and B1 cells, which is not able to cause an action potential generation. The frequency f_s includes 25 Hz, 50 Hz, 75 Hz, 100 Hz, 125 Hz, 150 Hz, 175 Hz and 200 Hz.

In the Fig. 1a, the steady-state Na⁺ and K⁺ conductances in each data-driven model exhibit similar values to the experimental recordings over a range of V (Azevedo and Wilson 2017). The Na⁺ and K⁺ conductances in A2 model are both much lower than those in the B1 model (Fig. 1a), and thus the steady-state I_Na and I_K in A2 model are several times smaller than B1 model (Fig. 1b). Specifically, the steady-state I_Na of A2 model is less than 1.1 μA/cm², and the steady-state I_K is also at a low level. In this case, the combination of I_Na and I_K in A2 model oscillates less than 0.5 μA/cm² over the range of stimulation frequencies, which is much lower than the capacitive current $I_C=C_{M}dV/dt$ (Fig. 2a). Since I_C strongly opposes the high-frequency voltage oscillations, the amplitude of V decays with the increase of f_s (Fig. 2b). As a result, the A2 model exhibits the low-pass filter to the oscillatory stimulus. All these simulations are in accordance with the experimental recordings in A2 cells (Azevedo and Wilson 2017).

Fig. 1.

a Steady-state Na⁺ and K⁺ conductances as a function of voltage V in data-driven models and voltage clamp recordings. The steady-state conductance of Na⁺ and K⁺ channels is computed by ${\overline{g}}_{\mathit{Na}}{m}_{\infty }$ and ${\overline{g}}_K{w}_{\infty }$. The experiment recordings are described by Azevedo and Wilson (2017). b Steady-state I_Na and I_K as a function of voltage V in A2 and B1 models

Fig. 2.

a Sample responses of membrane currents and voltage evoked by sinusoidal alternating currents stimulus I_stim. The stimulus frequency f_s is 25, 50, …, and 200 Hz, and the duration is 2 s. We extend the time axis and show I_Na, I_K, I_C and V in the last cycle of 2 s. The current direction flowing out of the cell is defined as positive. b Voltage amplitude as a function of f_s in A2 and B1 models

In the B1 model, the steady-state I_Na and I_K change dramatically with V (Fig. 1b). When B1 model is depolarized, the steady-state I_Na decreases and I_K increases, which inhibits the depolarization of V. When the B1 model is hyperpolarized, I_Na becomes stronger and I_K decreases, which pushes the V back toward V_rest. The amplitudes of I_Na and I_K reach about 30 μA/cm², and the combination of these two currents oscillates between ± 10 μA/cm² (Fig. 2a). In this situation, I_Na and I_K work together to oppose the variations in membrane potential. As f_s increases to a high level, especially when f_s is higher than 100 Hz, the fluctuations of I_Na and I_K in the Fig. 2a become small. The combination of I_Na and I_K is less than I_C, which indicates that I_C dominates the membrane dynamics in response to high-frequency stimulus. When the stimulus is at an intermediate f_s (from about 100 Hz to 140 Hz), the voltage oscillation is not strongly opposed by I_Na, I_K or I_C, and thus the stimulus is favored by the B1 model. As a result, the amplitude of V first increases and then decreases with the frequency f_s, which reaches a maximum at f_s = 127 Hz (Fig. 2b). That is, the B1 model is approximate to a band-pass filter. All these simulations are in accordance with the experimental recordings in B1 cells (Azevedo and Wilson 2017).

Filtering properties in linearized A2 and B1 models

Having reproduced the filtering properties of A2 and B1 cells in data-driven biophysical models, our next step is to linearize each model at the resting potential to generate L-A2 and L-B1 models. We perform the same simulations in each linearized model to examine its frequency filtering. The amplitude of the stimulus I_stim applied to linearized models is also A_stim = 10 μA/cm².

The maximum V of L-A2 model is about 1 mV smaller than the A2 model over a range of the stimulus frequency f_s (Fig. 3a, left). Even so, the voltage amplitude of L-A2 model decreases as f_s increases. That is, the L-A2 model acts as a low-pass filter, which is similar to the original A2 model. We also examine the current fluctuation in L-A2 model to determine the effects of ionic currents on filtering properties. The total active current I_active in the linearized model consists of the linearized terms corresponding to Na⁺ and K⁺ currents, which is described as $I_{\mathit{active}}=g_m{x}_m+g_w{x}_w$. The capacitive current I_C interacts with I_active to modulate dynamics of voltage V at different frequency f_s. The fluctuation of I_active is much smaller than that of I_C (Fig. 3b, left), and I_C dominates the low-pass filtering of L-A2 model. The response properties of these currents in L-A2 model match those in A2 model at different stimulus frequencies f_s. These simulations indicate that the L-A2 model captures the frequency filtering in nonlinear A2 model.

Fig. 3.

a Voltage amplitude as a function of f_s in L-A2 and L-B1 models. b Fluctuations of I_active and I_C as a function of f_s in each model. The current fluctuation is computed by the difference between the maximum and minimum of the current

The response in L-B1 model is highly consistent with B1 model over the range of stimulus frequency f_s (Fig. 3a, right), which exhibits the band-pass filtering to the oscillatory stimulus. Their voltage amplitudes both reach the peak at f_s = 127 Hz. Different from the I_active in L-A2 model, the fluctuation of I_active in L-B1 model is much higher than that of I_C when the stimulus frequency is f_s < 100 Hz (Fig. 3b). In this situation, I_active is the main source of voltage oscillation. When f_s is higher than 100 Hz, the fluctuation of I_active is lower than that of I_C, and thus I_active is not able to strongly oppose the voltage oscillation. This result is consistent with the decay of Na⁺ and K⁺ currents with frequency f_s in B1 model. Thus, the L-B1 model also capture the filtering properties in nonlinear B1 model.

The transfer function of linearized models is used to generate the magnitude–frequency curve of the bode diagram, which describes the frequency responses of the L-A2 and L-B1 models. The state-space representation of L-A2 model used for the transfer function is given by

$$A_{A2}=\left[\begin{array}{ccc}-0.1& 0& 0.1\\ 0& -0.1& 0.1\\ -0.0159& -0.0052& -0.6386\end{array}\right],B_{A2}=\left[\begin{array}{c}0\\ 0\\ 0.625\end{array}\right],C_{A2}=\left[\begin{array}{ccc}0& 0& 1\end{array}\right],D_{A2}=0$$ 21

In the same way, the state-space representation of L-B1 model is

$$A_{B1}=\left[\begin{array}{ccc}-0.1& 0& 0.1\\ 0& -0.1667& 0.1667\\ -0.9192& -1.8711& -1.5196\end{array}\right],B_{B1}=\left[\begin{array}{c}0\\ 0\\ 0.5882\end{array}\right],C_{B1}=\left[\begin{array}{ccc}0& 0& 1\end{array}\right],D_{B1}=0$$ 22

Correspondingly, the transfer functions of L-A2 and L-B1 models are

$$G_{A2}(s)=\frac{0.625{(s+0.1)}^2}{(s+0.6347)(s+0.1039)(s+0.1)}$$ 23

$$G_{B1}(s)=\frac{0.5882(s+0.1)(s+0.1667)}{(s+0.1128)(s+0.5853)(s+1.0882)}$$ 24

In the generation of frequency response, the frequency f is set to a wide range of 20–400 Hz, which is the range of the oscillatory synaptic current produced by vibration receptors. Accordingly, the ω is set to 0.1275–2.5133 rad/ms. The shape of magnitude–frequency curve of L-A2 model is similar to the function of voltage amplitude-stimulus frequency, which decreases monotonically (Fig. 4a). The curve of L-B1 model reaches a peak at the frequency f = 121 Hz and then decreases, which matches the band-pass filtering in B1 model (Fig. 4a). These results show that the shape of magnitude–frequency curve accurately but parsimonious describes the filtering properties of the linearized model.

Fig. 4.

a The magnitude–frequency curves of the L-A2 and L-B1 models. The abscissa is the frequency of the stimulus and the ordinate is the magnitude of the response of the system. b Zeros and poles in the complex plane of L-A2 and L-B1 models

To understand the mechanistic relationship between magnitude–frequency curve and frequency filtering, we show the zeros and poles of each transfer function in a complex plane. The curve is considered to start at $20lg(\left|k{\prod }_{i=1}^z(-z_i)/{\prod }_{l=1}^p(-p_l)\right|)$ dB. When a zero occurs, the slope of the asymptote increases by 20 dB/dec. In contrast, the slope decreases by 20 dB/dec when a pole appears. There are two zeros and three poles in the complex plane of the L-A2 and L-B1 models. In the L-A2 model, two poles (-0.1 and − 0.1039) occur in a region very closed to two zeros (-0.1 and − 0.1), which counteracts the effects of these zero-poles on the magnitude–frequency curve (Fig. 4b, left). The last pole (-0.6347) far from the origin leads the curve to decay with frequency. Thus, the magnitude–frequency curve of L-A2 model keeps decaying with frequency f ignoring two zeros and two poles. In the L-B1 model, there is only one pole (-0.1128) approximately coincide with a zero (-0.1). The curve is raised by the zero (-0.1667) at the beginning of frequency f and then reduced by two poles (-0.5853 and − 1.0882) (Fig. 4b, right). As a result, the curve is convex and reaches a peak near the preferred frequency of L-B1 model. The relationship between the shape of magnitude–frequency curve and the distribution of zero-pole suggests that the filtering of oscillatory stimulus depends on the zeros and poles of the transfer function.

Effects of biophysical parameters on filtering properties

Here, we change the biophysical parameters in L-A2 and L-B1 models to examine the distributions of zero-poles derived from the transfer functions, which contribute to understanding how filtering properties are inferred from the zero-poles in a complex plane. The ranges of six biophysical parameters are set as ${\overline{g}}_{\mathit{Na}}\in [0.0248,0.3522]$ mS/cm², β_m ∈ [15.2255, 34.6186] mV, γ_m ∈ [27.8269, 149.9946] mV, ${\overline{g}}_K\in [0.0083,2.1365]$ mS/cm², β_w ∈ [-12.2349, 14.8127] mV and γ_w ∈ [7.0652, 14.0203] mV, which are the values in A2 and B1 models.

We first examine the effects of varying ${\overline{g}}_K$ or ${\overline{g}}_{\mathit{Na}}$ on the filtering properties of L-A2 model. When ${\overline{g}}_K$ increases from 0.0083 to 2.1365 mS/cm², two zeros and the green pole of the transfer function are fixed in the complex plane. In contrast, the red pole moves from (-0.6348, 0) to (-3.1807, 0), and the blue pole moves from (-0.1038, 0) to (-0.1374, 0), which are away from the origin (Fig. 5a, left). The blue pole determines the inflection point, where the magnitude–frequency curve begins to decay. That is, it is a dominant pole. With 0.0083 < ${\overline{g}}_K$ ≤ 0.0935 mS/cm², the dominant pole only moves a distance of 0.0066, and the membrane of L-A2 model still acts as low-pass filter (Fig. 5b, c, left). With ${\overline{g}}_K$ > 0.0935 mS/cm², the shift distance of the blue dominant pole becomes so large that the frequency corresponding to the peak of the magnitude–frequency curve starts to increase from 21 Hz. The second blue pole from the right corresponding to ${\overline{g}}_K$ = 0.2212 mS/cm² moves a distance of 0.0133 (Fig. 5a, left). In this case, the L-A2 model is switched from the low-pass filtering to band-pass filtering of oscillatory stimulus. In fact, the switch of filtering property is due to that the active K⁺ current increases with ${\overline{g}}_K$. This further increases the total active current I_active, which is the main source of the suppression of the variation in membrane voltage. We use the fluctuation of I_active to describe the strength of this suppression, which is the difference between the maximum and minimum of the steady-state active current over the range of V ∈ [-20, 15] mV. The fluctuation of I_active increases from 0.6501 to 36.8419 μA/cm² as ${\overline{g}}_K$ increases (Fig. 5d, left). With ${\overline{g}}_K$ > 0.0935 mS/cm², the I_active becomes strong enough to oppose the variation of voltage V at low stimulus frequencies (< 100 Hz), which transforms the model from low-pass filter to band-pass filter. As ${\overline{g}}_{\mathit{Na}}$ increases from 0.0248 to 0.3522 mS/cm², the blue dominant pole moves from (-0.1038, 0) to (-0.1424, 0) (Fig. 5a, right). With ${\overline{g}}_{\mathit{Na}}$ = 0.123 mS/cm², the blue pole moves from − 0.1038 to − 0.1155 (Fig. 5a, right), which corresponds to an increase in the sensitive frequency of the model from 21 to 40 Hz (Fig. 5b, c, right). It shows that the L-A2 model exhibits band-pass filtering of oscillatory stimulus with ${\overline{g}}_{\mathit{Na}}$ > 0.123 mS/cm². In fact, increasing ${\overline{g}}_{\mathit{Na}}$ increases Na⁺ current, and thus the fluctuation of I_active increases from 0.6501 to 6.709 μA/cm² (Fig. 5d, right). With ${\overline{g}}_{\mathit{Na}}$ > 0.123 mS/cm², the fluctuation of I_active is strong enough to oppose slow voltage oscillation, which transforms the model to a band-pass filter. These results indicate that the position of the dominant pole is highly correlated with the strength of the suppression of slow voltage variation by I_active, which is an effective description of frequency filtering of the A2 model as ${\overline{g}}_K$ or ${\overline{g}}_{\mathit{Na}}$ is varied.

Fig. 5.

Frequency responses of L-A2 model over the range of ${\overline{g}}_K\in [0.0083,2.1365]$ mS/cm² and ${\overline{g}}_{\mathit{Na}}\in [0.0248,0.3522]$ mS/cm². a Distributions of zeros and poles. The red arrow denotes the direction of red pole’s movement with the increase of ${\overline{g}}_K$ or ${\overline{g}}_{\mathit{Na}}$. The blue arrow indicates the direction of dominant pole’s movement as ${\overline{g}}_K$ or ${\overline{g}}_{\mathit{Na}}$ increases. b Left: The magnitude of the response of L-A2 model as a function of ${\overline{g}}_K$ and the frequency of sinusoidal stimulus. Right: The response magnitude of L-A2 model as a function of ${\overline{g}}_{\mathit{Na}}$ and stimulus frequency. c Left: Five magnitude–frequency curves of L-A2 model with ${\overline{g}}_K$ = 0.0083, 0.5404, 1.0724, 1.6045 mS/cm². Right: The magnitude–frequency curves of L-A2 model with ${\overline{g}}_{\mathit{Na}}$ = 0.0248, 0.1067, 0.1885, 0.2704 and 0.3522 mS/cm². The membrane potential of L-A2 model reaches the maximum value when the oscillatory stimulus is at the frequency corresponding to the red star, which denotes the peak of the curve. d Fluctuation of I_active as a function of frequency f in L-A2 model

We also examine the effects of varying ${\overline{g}}_K$ or ${\overline{g}}_{\mathit{Na}}$ on the filtering properties of L-B1 model. As ${\overline{g}}_K$ increases from 0.0083 to 2.1365 mS/cm², the blue dominant pole moves from (-0.1925, 0) to (-0.5578, 0) (Fig. 6a, left). Different from L-A2 model, the dominant pole starts at a position far from two zeros (-0.1, 0) and (-0.1667, 0). Thus, the magnitude–frequency curve all increases and then decreases with stimulus frequency (Fig. 6b, c, left). That is, L-B1 model keeps band-pass filtering of oscillatory stimulus. Meanwhile, the sensitive frequency moves to a high level as ${\overline{g}}_K$ increases. Increasing ${\overline{g}}_K$ enlarges the K⁺ current, and thus the fluctuation of I_active increases from 54.3522 to 158.6206 μA/cm². Such strong I_active is always able to inhibit the slow variations in voltage V (Fig. 6d, left), which results in a band-pass filtering of oscillatory inputs. The similar results occur when ${\overline{g}}_{\mathit{Na}}$ is varied. As ${\overline{g}}_{\mathit{Na}}$ increases, the dominant pole moves a short distance of 0.0658 away from the origin (Fig. 6a, right), and the peak of the magnitude–frequency curve is shifted right by 10 Hz (Fig. 6b, c, right). Here, the fluctuation of I_active increases from 108.4756 to 158.6206 μA/cm², which is always strong enough to oppose slow membrane voltage oscillations and leads to band-filtering of oscillatory inputs (Fig. 6d, right). These results indicate that the position of the dominant pole in the complex plane could predict the effects of varying ${\overline{g}}_K$ and ${\overline{g}}_{\mathit{Na}}$ on the filtering properties of L-B1 model as well as their relation with the underlying active current I_active.

Fig. 6.

Frequency responses of L-B1 model over the range of ${\overline{g}}_K\in [0.0083,2.1365]$ mS/cm² and ${\overline{g}}_{\mathit{Na}}\in [0.0248,0.3522]$ mS/cm². a Distributions of zeros and poles. The direction of the arrow reveals the direction of the point’s movement with the increase of ${\overline{g}}_K$ and ${\overline{g}}_{\mathit{Na}}$. b Left: The magnitude of the response of L-B1 model as a function of ${\overline{g}}_K$ and the frequency of sinusoidal stimulus. Right: The response magnitude of L-B1 model as a function of ${\overline{g}}_{\mathit{Na}}$ and stimulus frequency. c Left: Five magnitude–frequency curves of L-B1 model with ${\overline{g}}_K$ = 0.0083, 0.5404, 1.0724, 1.6045 and 2.1365 mS/cm². Right: The magnitude–frequency curves of L-B1 model with ${\overline{g}}_{\mathit{Na}}$ = 0.0248, 0.1067, 0.1885, 0.2704 and 0.3522 mS/cm². d Fluctuation of I_active as a function of frequency f in L-B1 model

Subsequently, we vary β_w, γ_w, β_m or γ_m in linearized models, and repeat above simulations to examine how varying each parameter affects the filtering property of the model. As β_w, γ_w, β_m and γ_m increase respectively, the dominant pole of L-A2 model moves a distance shorter than 0.0068, which maintains the shapes of magnitude–frequency curves (Fig. 7). Meanwhile, the dominant pole is kept close to the zero of (-0.1, 0). Thus, the magnitude–frequency curves all decrease monotonically with these parameters and the sensitive frequency is limited to 21 Hz (Fig. 7). That is, the L-A2 model keeps low-pass filtering of oscillatory stimulus. Here, the fluctuation of I_active is always less than 1.7129 μA/cm², which is not able to inhibit the slow variations in voltage V and leads to the low-pass filtering of oscillatory inputs. While increasing β_w, γ_w, β_m or γ_m, the dominant pole of L-B1 model is always far from two zeros of (-0.1, 0) and (-0.1667, 0) (Fig. 8). The magnitude–frequency curves both increases and then decreases with stimulus frequency, which suggests that the L-B1 model acts as a band-pass filter. Meanwhile, the fluctuation of I_active is kept larger than 80 μA/cm², which is strong enough to oppose slow variations in voltage. These results show that a large fluctuation of I_active is reflected by the long distance between the dominant pole and the zeros, which makes the peak of the magnitude–frequency curve occur at a frequency higher than 21 Hz (Fig. 8). Therefore, the position of the dominant pole could directly assess the effects of varying β_w, γ_w, β_m or γ_m on the fluctuation of Na⁺ and K⁺ currents, which is a compact description of filtering properties caused by the suppression of slow voltage variations.

Fig. 7.

a The effects of varying β_w on the L-A2 model. When β_w increases from − 12.2349 to 14.8127 mV, there is only a minimal change in the position of the dominant pole, the shape of magnitude–frequency curves and the fluctuation of I_active. b The effects of increasing γ_w from 7.0625 to 14.0203 mV on the L-A2 model. c Variations of L-A2 model with β_m ∈ [15.2255, 34.6186] mV. d Variations of L-A2 model with γ_m increasing from 27.8269 to 149.9946 mV. The left panel shows the distribution of zero-poles of the L-A2 model. The middle panel contains the magnitude–frequency curves, which are the directly description of frequency responses of the L-A2 model. The fluctuation of I_active is shown in the right panel

Fig. 8.

a The effects of varying β_w on the L-B1 model. When β_w increases from − 12.2349 to 14.8127 mV, there is only a minimal change in the position of the dominant pole, the shape of magnitude–frequency curves and the fluctuation of I_active. b The effects of increasing γ_w from 7.0625 to 14.0203 mV on the L-B1 model. c Variations of L-B1 model with β_m ∈ [15.2255, 34.6186] mV. d Variations of L-B1 model with γ_m increasing from 27.8269 to 149.9946 mV. The left panel shows the distribution of zero-poles of the L-B1 model. The direction of the arrow reveals the direction of the movement of the dominant pole. The middle panel contains the magnitude–frequency curves, which are the directly description of frequency responses of the L-B1 model. The fluctuation of I_active is shown in the right panel

Conclusion

In this study, we use a data-driven conductance-based biophysical model to reproduce the different frequency preferences of A2 and B1 neurons for oscillatory synaptic currents. To understand the unusual role of Na⁺ and K⁺ currents in shaping the filtering property of the A2 and B1 cells, we linearize the conductance-based biophysical model at the resting potential. The linearized model provides an accurate and compact prediction of the frequency response. Although the absence of nonlinear terms in the linearized model leads to a lower voltage response, it could qualitatively describe the shape of the magnitude–frequency response curve. We use the linearized model to explore the relationship between the Na⁺ and K⁺ current fluctuations and the biophysical parameters. Then the current fluctuation is linked to the position of the dominant pole calculated from the transfer function of the linearized model with different biophysical parameters, which is used to characterize the filtering properties. Based on this linearized modeling framework, we systematically explain how the biophysical parameters regulate the frequency selectivity of the A2 and B1 neurons. The biophysical parameters modulate the suppression of slow voltage variation by changing the current fluctuation, and further regulate the balance between the active ionic currents and the passive membrane property that inhibits the fast voltage variation. Thus, A2 and B1 neurons exhibits different filtering properties for oscillatory synaptic currents due to the shift of this equilibrium, which is shown as the movement of the dominant pole of the linearized model.

The frequency selectivity for vibrations is widely observed in the cells or tissues of the tactile pathway, such as the Pacinian corpuscles, the primary afferent fibers and the primary somatosensory cortex (Arabzadeh et al. 2005). Although the selective preferences for vibrations with different frequencies are found to result from the special ionic currents (Azevedo and Wilson 2017; Fuchs and Evans 1990; Goodman and Art 1996; Rotstein and Nadim 2019) and the cyclical entrainment (Arabzadeh et al. 2005; Prsa et al. 2019), the relationship between the cellular frequency property and the activation properties of these ionic channels is not clear. In this study, we develop a linearized modeling framework to understand how the activation properties of the Na⁺ and K⁺ channels change the filtering properties of A2 and B1 neurons by regulating the suppression of slow voltage variation. The data-driven modeling allows to systematically change the biophysical parameters of the conductance-based biophysical model to control the activation of Na⁺ and K⁺ channels. Further, the response in the time domain of the data-driven model with different biophysical parameters is transformed to the response in the frequency domain by the transfer function of its linearized model, which provides a natural and direct prediction of the magnitude of steady-state voltage oscillation. Based on the transfer function, the magnitude–frequency response could be derived directly from the biophysical parameters of the linearized model, which allows to assess the role of biophysical parameters in shaping the filtering property of the A2 and B1 neurons. Such linearized models are widely used in the studies that focus on the subthreshold frequency responses, i.e., the description of the relationship between spectral response and biophysical parameters (Moran et al. 2007), and the effects of subthreshold resonance on the dynamics of the firing rate (Richardson et al. 2003). Combined with the data-driven modeling method, the linearized modeling framework for A2 and B1 neurons stimulated by subthreshold oscillatory synaptic currents provides an accurate and compact description of the relationship between frequency selectivity of oscillatory stimulus and activation properties of Na⁺ and K⁺ channels.

In the data-driven model, the Na⁺ current decays with the increase of membrane potential, and the K⁺ current is enhanced with depolarization. These ionic currents, collectively named the active currents, both inhibit oscillations in voltage. However, as the frequency of stimulus increases, the active current is not able to reach the steady state with such high frequency, which leads to an attenuation of suppression of slow voltage variation. Therefore, the passive membrane properties begin to dominate the suppression of high-frequency stimulus (Baruah et al. 2019). The interaction between the active currents and the passive membrane properties determines the different filtering properties of the linearized model, which is consistent with the specialized properties of membranes and ion channels of A2 and B1 neurons (Azevedo and Wilson 2017). Further, the filtering properties of the linearized model are intuitionally inferred by the magnitude–frequency curve, as the frequency corresponding to the peak of the curve indicates the sensitive frequency of the linearized model. The position of the dominant pole of the linearized model shapes the magnitude–frequency curve and then qualitatively predicts the filtering properties. Meanwhile, the position of the dominant pole depends on the biophysical parameters of the linearized model, which determine the activations of Na⁺ and K⁺ currents. When the fluctuation of the active current is enlarged, the suppression of slow voltage variation by the active current is enhanced, and the equilibrium between the active current and the passive membrane properties is moved to a higher frequency. Correspondingly, the peak of the magnitude–frequency curve of the linearized model also moves to a higher frequency, which is reflected by the dominant pole moving away from the origin. This intuitional understanding of the relationship between the activation properties of ionic channels and the filtering properties provides greater insight into cellular frequency selectivity in the tactile sensory system than a purely phenomenological description of current variations, which provides a detailed and reliable explanation for how vibration signals are encoded and transmitted in tactile pathways.

Funding

This work was supported by the National Natural Science Foundation of China [Grant No. 62071324], and by the National Natural Science Foundation of China [Grant No. 62171311].

Declarations

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.