Recordings were made in postnatal day (P) 19–25 (average: 21.5) B57BL6/6N mice, Mongolian gerbils (VNLL: P24–27, average: P25.5; MSO: P15/16 average 15.5 and >P60) and Brown Norway rats (P18–P27 average: 23.4) of either sex. Mice were purchased from Charles River or taken from our own breeding colonies; gerbils were obtained from own breeding colonies and rats were ordered from Charles River (Sulzfeld, Germany).

All experiments complied with institutional guidelines, and national and regional laws.

Animals were decapitated in deep isoflurane anesthesia. Brains were quickly removed in preparation solution containing (in mM) D-Saccharose 120, NaCl 25, NaHCO3 25, NaH2PO4 1.25, KCl 2.5, D-Glucose 25, L-Ascorbic acid 0.4, Myo-Inositol 3, Na-pyruvate 2, MgCl2 3, CaCl2 0.1 at a pH 7.4 and was oxygenated with 95% O2 and 5% CO2. After brains were trimmed 120 μm (MSO) or 180–200 μm (LSO and VNLL) thick sections were cut using a vibratome (Leica VT1200, Leica Microsystems GmbH, Germany). Slices were incubated for about 45 min at 34–35°C in extracellular recording solution containing (in mM) NaCl 125, NaHCO3 25, NaH2PO4 1.25, KCL 2.5, D-Glucose 25, L-Ascorbic acid 0.4, Myo-Inositol 3, Na-pyruvate 2, MgCl2 1, CaCl2 2 oxygenated with 95% O2 and 5% CO2.

Slices were transferred into the recording chamber integrated into upright BX50 or 51 WI Olympus microscopes and continuously perfused with recording solution. Data were acquired at near physiological temperatures of 34-36°C. Electrophysiological recordings were carried out with an EPC 10/2 amplifier (HEKA, Lambrecht/Pfalz, Germany). Electrode resistances were constrained between 3 and 5.5 MΩs. Stimulus generation and presentation was controlled by the PatchMaster software. Visual identification of brain structures and individual cells was carried out by CCD-cameras (TILL-Imago VGA, Retiga 2000DC) controlled by TILLvisION imaging system (FEI Munich GmbH, Munich, Germany). Constant background conductances were injected with an analog conductance amplifier (SM-1 Cambridge Conductance, Royston UK). Throughout these conductance clamp experiments, reversal potential of the leak was set to resting potential of a given cell. Recordings were performed in whole-cell configuration using an intracellular solution containing (in mM) K-gluconate 145, KCl 4.5, HEPES 15, Mg-ATP 2, K-ATP 2, Na2-GTP 0.3, Na2-phosphocreatine 7, K-EGTA 0.5, Alexa 488/594 0.05. Alexa labeling was used to control for cell localization within appropriate brain structure. Data were acquired with 20 kHz for ZAP stimulation (see ZAP Stimuli) and hyperpolarizing current steps. Only for neurons of >P60 MSO the sample frequency for hyperpolarizing steps was increased to 100 kHz. All data was high-pass filtered by 3 Hz. Access resistance was compensated in voltage clamp mode before switching into current clamp, where bridge balance was set to 100%. Data was not corrected for the liquid junction potential of about 15 mV.

Each cell was challenged with a long step current injection at various hyper and depolarizing intensities to test for the characteristic sub- and supra-threshold voltage behavior. Only neurons that generated action potentials toward depolarizing current injections were taken for further analysis. Also in each cell a −5 pA current injection between 10 and 500 ms length and 80–400 repetitions was applied. The voltage response to this hyperpolarization was used to determine membrane decay time constant (τ_p) by exponential fits. Using Ohm's law the input resistance at the beginning of the hyperpolarization (R_p) and during steady state (R_s) was calculated. From the derived values the cell's effective capacitance C was calculated according to τ_p = R_pC.

Following Puil et al. (1986), the neuronal sub-threshold resonance behavior was examined by applying a sinusoidal current I(t) with exponentially increasing frequency f(t) (ZAP-current).

The phase function is chosen such that the frequency at time 0 equals the starting frequency f(0) = f_s and after a time D the frequency stops at an end frequency f(D) = f_e. In between the frequency increases exponentially in time. The function

fulfills these three requirements. For LSO neurons in mice the frequency increased from 2.5 to 280 Hz within 96 s and in rats from 2.5 to 280 Hz, 2.5 to 300 Hz, or 2.5 to 330 Hz always within 96 s. For gerbil VNLL neurons the ZAP frequency increased from 1.8 to 260 Hz within 99 s. For P15 MSO neurons the stimulation increased from 2.5 to 330 Hz within 96 s. For MSO neurons of P60 or older animals the ZAP stimulation increased from 4 to 700 or 10 to 850 Hz over 99 s. The duration of ZAP currents was chosen long enough to approximate quasi-stationary membrane oscillations on the logarithmic frequency scale.

The depolarizing resonance frequency f_r was calculated from the electrophysiological recordings by measuring the time t^* when the maximal depolarization occurred and converted it into the frequency f(t^*) of the stimulus at this given time point.

ZAP stimuli were applied for varying current amplitudes and at varying holding currents. To control for low-frequency build up of the membrane voltage during ZAP stimuli, in most cells we also applied stimuli with time-reversed frequency profile (see Figure 1D).

We assume that for small input currents I(t), the voltage follows a 2-d linear dynamics (Rotstein, 2014)

where υ denotes the voltage deflection from rest, and u is a relaxation variable that describes the linearized net effect of the subthreshold channels. Membrane capacitance C and the onset membrane resistance R_p comprise two physiologically interpretable parameters. Conversely, α, β, and γ are as yet unspecified parameters. All three parameters will be positive since clamping the voltage υ to a depolarized value should lead to an increase (γ > 0) in conductance up to the positive steady state value u = (γ/β)υ (hence β > 0). Clamping the outward conductance u to a positive value, however, should lead to a decrease in voltage υ = −α R_p u and thus α > 0.

In the Fourier domain, the dynamics transforms to a set of linear algebraic equations linking voltage and relaxation variable by

and, hence,

Solving for υ gives υ = Z(ω)I, with the impedance

The steady state input resistance R_s is obtained as

At the resonance frequency ω_r, the modulus of the impedance is maximized,

From ddω|Z(ω)|=0, we obtain this resonance frequency as

where τ_p = R_pC denotes the time constant of the onset conductance and τ_s = R_sC denotes the membrane time constant resulting from the (steady state) input resistance. Both τ_p and τ_s are experimentally accessible parameters (Figure 5) and thus the resonance frequency depends on only one free parameter β, the rate of change of the relaxation variable. Note that ω_r is the radial resonance frequency, i.e., the stimulus frequency at which the model displays the resonance equals f_r = ω_r/(2π).

All analysis of electrophysiological data was performed in IGOR Pro (Version 6.37, Wavemetrics) and Microsoft Excel 2010. Model fitting was performed by custom-made MATLAB code. Statistical analysis of the Pearson correlation coefficient was performed by the MATLAB function corrcoef in Figure 4B and by GraphPad Prism 7 in Figure 2.

We probed subthreshold membrane potential resonances in auditory brainstem neurons of the medial and lateral superior olive (MSO and LSO) and the ventral nucleus of the lateral lemniscus (VNLL). To efficiently determine resonance frequencies, we applied sinusoidal current injections with exponential increasing frequency, also called ZAP stimuli (Puil et al., 1986) (Figure 1A). Using ZAP current stimuli of 96–99 second duration, neurons in all tested nuclei showed sub-threshold resonance behavior. We generally observe resonances of the hyperpolarizing troughs and the depolarizing peaks of the voltage response, where the resonance frequency of the troughs is usually lower than that of the peaks (Figure 1B). The resonance of hyperpolarizing troughs was mediated by HCN channels, which we could show using ZD7288 that selectively blocked the hyperpolarizing resonance in all five tested >P60 MSO neurons. Compensation of the holding current during ZD7288 application recovered the depolarizing membrane resonance. In one case a nearly full restoration of the depolarizing membrane resonance was achieved by adding an additional leak conductance (Figure 1C).

For larger amplitudes of the ZAP stimulus, the depolarizing resonance often facilitated the generation of action potential locked to the stimulus peaks (Figure 1B). Such a frequency dependence of supra-threshold action potential firing was observed in all neuron types. In the following we therefore focussed on the resonance frequency f_r of the depolarizing peaks.

To validate our experimental approach, we also applied temporally reversed ZAP inputs (Figure 1D), and found good agreement between the resonance frequencies obtained from both stimulus directions (Figure 1E).

In the initial data the depolarizing resonance frequency f_r appeared to scatter over a large range for individual neuron types (Figure 1E). In previous papers (Rotstein, 2015; Mikiel-Hunter et al., 2016), it was shown that f_r increases with the average membrane depolarization indicating a non-linear voltage-dependent gating of the subthreshold conductances. Thus, we expected, that our observed scatter of f_r can be explained by such factors as well. We therefore derived resonance frequencies for varying amplitudes I(f) of the ZAP stimulus (Figures 2A,B; 14 neurons, 91 trials of P15 MSO; 12 neurons, 82 trials of >P60 MSO) and varying magnitudes I₀ of an additionally applied constant current (Figures 2C,D; 10 neurons, 41 trials of P15 MSO; 10 neurons, 32 trials >P60 MSO). As previously described (Mikiel-Hunter et al., 2016), both an increase in amplitude and an increase in I₀ yielded generally larger resonance frequencies (Figures 2B,D). The correlation between resonance frequency f_r and amplitude was significant in all groups (Pearson's r = 0.71, p < 0.0001 for P15, n = 91 stimuli, r = 0.60, p < 0.0001 for P > 60, n = 82; r = 0.80, p < 0.0001 for both age groups, n = 173). The correlation between resonance frequency and holding current was only significant in P > 60 animals and the combined data, but not for P15 animals (Pearson's r = 0.06, p = 0.35 for P15, n = 41 stimuli, r = 0.72, p < 0.0001 for P > 60, n = 32; r = 0.60, p < 0.0001 for both age groups, n = 73), potentially indicating heterogeneous maturation states, in which some cells may still have very low densities of active channels. The holding current is thus not fully predictive for the resonance frequency.

We then reasoned that if the scatter in f_r is due to the differential opening of subthreshold conductances, we should observe a stronger correlation by adding conductance artificially. To test this prediction we first used conductance clamp to simulate the change in input resistance and find that this manipulation indeed shifts the resonance frequency as well (Figure 3A). We assessed the effect of artificial conductance by the instantaneous input resistance RZAP=V^/I(f), in which the voltage amplitude V^ is averaged over the first three cycles to obtain a robust estimate. When the constant background leak was increased R_ZAP decreased and the resonance frequency f_r increased. Conversely, when decreasing the leak, R_ZAP increased and the resonance frequency decreased (Figures 3B,C).

The definition of the instantaneous input resistance R_ZAP allowed us to compare resonance frequencies for all three stimulus paradigms shown in Figures 2, 3 and revealed a very consistent and strong correlation. We found a general monotonic decrease for MSO Neurons of both age groups (Figure 3D), exhibiting less variable resonance frequencies than in the original analyses from Figures 2B,D. Also for mouse and rat LSO and gerbil VNLL neurons (Figure 3E) we find a similiar relation, however, with VNLL data scattered more broadly along the f_r axes. Thus, our data indicates that the instantaneous input resistance R_ZAP is a major determinant governing the subthreshold membrane resonance frequency.

To test, whether the resonance amplitude is comparable for all probed conditions, we computed Q factors defined as the ratio of the depolarizing amplitude at the resonance frequency over the depolarizing amplitude at frequency 0 (Figure 3F). Q factors generally decreased with the instantaneous input resistance R_ZAP (Figure 3G), however, their distributions were relatively narrow and largely independent of the neuronal population except for only six P > 60 MSO neurons with exceptionally high values (above 3.5). For the majority of about 90% of the stimulations, the Q factor was between 1.05 and 2.5. The Q factor was above 2.5 or below 1.05 in only about 5% of the stimulations each.

On a single cell level the relation between R_ZAP and f_r can be fitted by a power law fr=a(RZAP)-b (Figure 4A). This dependence allowed us to identify a stimulus independent resonance frequency f₀ by extrapolating f_r to the steady state input resistance R_s that is defined as the impedance value at frequency 0, R_s = Z(0). The steady state resistance R_s is typically close to the cell specific maximum of R_ZAP since it is defined for small current stimuli with only few additional channels opening. Our measurements thus not only allow us to define stimulus independent resonance frequency f₀ but also relate it to an experimentally accessible, passive membrane property R_s (Figure 5A).

Resonance frequencies decrease with R_ZAP on a single cell level, but can we also find input resistance dependent effects across cells? We therefore correlated f₀ with R_s of all cells (Figure 4B) and found a strong negative correlation (r = −0.85, p = 1.8 · 10⁻¹⁵, n = 52) across all experimental groups. Thus, high resonance frequencies tend to require leaky membranes, whereas low resonance frequencies are generated by non-leaky membranes. We excluded 10 out of 61 cells from this and further analyses because we considered the estimated resonance frequency f₀ as too remote from the resonance frequencies f_r obtained for these cells from individual recordings [when min(f_r) − f₀ exceeded 33.3% of the mean f_r].

The strong negative correlation between resonance frequency f₀ and input resistance R_s suggests a general underlying biophysical principle. We therefore analyzed a linear dynamical model (Rotstein, 2014, 2015) to explain this general correlation (see Methods). The model consists of two coupled linear differential Equations (1) that describe the temporal evolution of small voltage deflections from rest. The model's resonance frequency from Equation (3) only depends on three biophysically defined parameters: (i) the onset membrane time constant τ_p = CR_p, in which R_p denotes the peak input resistance derived from the voltage maximum upon a brief low-amplitude current stimulus (Figure 5A), (ii) the steady state membrane time constant τ_s = CR_s, and (iii) the decay rate β of the subthreshold conductances. Whereas τ_s and τ_p are directly measurable, the last parameter β remains as a fit parameter.

To fit our model, we determined τ_p (Figure 5B), and the input resistances R_s and R_p (Figure 5C) for each cell and derived its effective capacitance and steady state time constant via C = τ_p/R_p and τ_s = R_sC, respectively. As expected we found characteristic differences between the three different groups of cells; see Table 1.

When correlating R_s and R_p we found a strong linear correlation (Figure 5C). This is not surprising since both parameters result from the same subthreshold channels and only reflect different dynamical states. Less expectedly, the correlation seems to be largely independent of the group of cells, suggesting that the subthreshold dynamics underlying the membrane resonance are similar across all groups. We therefore generally eliminated R_p from the model by fitting a power law (Figure 6A)

The resonance frequency from Equation (3) thus solely depends on the steady state time constant τ_s.

We next analyzed the dependence of resonance frequencies f₀ on the steady state membrane time constant τ_s. As expected f₀ generally decreased with τ_s. Moreover, the different cell groups showed similar dependence (Figure 6B). We thus asked, whether we could fit the model for all groups with only one parameter β. Comparing the least squares fit with the data showed indeed good agreement, for the optimal rate constant β = 333.7 Hz. We also fitted the five groups of cells independently and asked whether the variability in β would be consistent with the null hypothesis that they all arise from the same β. We therefore used 10, 000 random permutations of group labels to construct a null distribution of β (Figure 6C) and found that the mouse LSO and gerbil VNLL cells are outside the 5% significance range, whereas gerbil MSO cells and rat LSO cells could arise from the same β. We thus conclude that the different cell groups may follow the same general law, however, the mechanisms may have different molecular contributions (i.e., different heterotetrameres) and hence result in different values β for the effective channel kinetics.

To compare the quality of the fit between the different cell groups we rescaled the resonance frequencies and time constants by the group-specific fits of β (Figure 6D). We observed that only the data from juvenile (P15) MSO neurons (blue) is not completely fitted by the model. Shortly after hearing onset (P14), however, we expect a large heterogeneity in developmental states and thus a single fit parameter β might not be appropriate at this age and the scatter might simply reflect the heterogeneity in channel composition.

A further conclusion from rescaling (Figure 6D) is that we can determine the critical value τ_c for the steady state membrane time constant above which no membrane resonance is possible. This value was determined numerically as τ_c = 0.42/β. Accordingly, for neurons (like in the MSO) where the resonance is supposed to be mostly supported by Kv1 channels with kinetic constants of β⁻¹ ≈ 2 ms resonances require membrane time constants below τ_c ≈ 840μs. However, if the resonance is due to channels with slower kinetics up to a range of few hundreds of milliseconds (e.g., HCN channels) the critical value increases to about τ_c ⪆ 100 ms. As a consequence, high resonance frequencies require fast channels and leaky membranes (low τ_s), whereas low resonance frequencies (e.g., supporting theta oscillations) can be achieved with relatively moderate time constants and slow channels.

Finally, we computed Q-factors from the theoretical model (Figure 6E) and compared them with the experimentally obtained values (Figure 6F). The general decrease with τ_s was similar for both data and theory. However, the theoretical Q-values were only about 2/3 as large as the experimental ones indicating that our measured data also includes amplifications by non-linear channel dynamics.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.