Distinct mechanisms underlie electrical coupling resonance and its interaction with membrane potential resonance

Abstract

Neurons in oscillatory networks often exhibit membrane potential resonance, a peak impedance at a non-zero input frequency. In electrically coupled oscillatory networks, the coupling coefficient (the ratio of post- and prejunctional voltage responses) could also show resonance. Such coupling resonance may emerge from the interaction between the coupling current and resonance properties of the coupled neurons, but this relationship has not been clearly described. Additionally, it is unknown if the gap-junction mediated electrical coupling conductance may have frequency dependence. We examined these questions by recording a pair of electrically coupled neurons in the oscillatory pyloric network of the crab Cancer borealis. We performed dual current- and voltage-clamp recordings and quantified the frequency preference of the coupled neurons, the coupling coefficient, the electrical conductance, and the postjunctional neuronal response. We found that all components exhibit frequency selectivity, but with distinct preferred frequencies. Mathematical and computational analysis showed that membrane potential resonance of the postjunctional neuron was sufficient to give rise to resonance properties of the coupling coefficient, but not the coupling conductance. A distinct coupling conductance resonance frequency therefore emerges either from other circuit components or from the gating properties of the gap junctions. Finally, to explore the functional effect of the resonance of the coupling conductance, we examined its role in synchronizing neuronal the activities of electrically coupled bursting model neurons. Together, our findings elucidate factors that produce electrical coupling resonance and the function of this resonance in oscillatory networks.

Keywords: central pattern generator; gap junctions; oscillation; resonance; stomatogastric.

FIGURE 1

The two PD neurons produce synchronized slow wave bursting due to their strong electrical coupling. (A) Somatic recording of the two PD neurons shows that they produce bursting oscillations that are synchronized in their slow-wave activity. (B) Measurement of coupling coefficient between the two PD neurons. The prejunctional PD₁ neuron is voltage clamped with steps ranging from −80 to −40 mV from a holding potential of −60 mV. The postjunctional PD₂ neuron membrane potential is recorded in current clamp. The coupling coefficient CC is measured as the slope of the linear fit to the values of V _post plotted vs. V _pre . Each data point is the mean value of voltage during the step, as seen in the grey point, corresponding to the lowest steps (arrows). (C) Measurement of coupling conductance between the two PD neurons. The prejunctional PD₁ neuron is voltage clamped as in panel B, while the postjunctional PD₂ neuron is voltage clamped at a steady holding potential of −60 mV (not shown). The coupling conductance G _c is measured as the slope of the linear fit to the values of I _post plotted vs. V _pre . Each data point is the mean value the step, as seen in the grey point, corresponding to the lowest V _pre and highest I _post steps (arrows).

FIGURE 2

The coupling coefficient (CC) between the two PD neurons shows resonance. A ZAP current, sweeping a frequency range of 0.1–4 Hz, was applied to one PD neuron to simultaneously measure the voltage changes in both PD neurons. (Ai) Both neurons showed a peak amplitude response at an intermediate frequency (marked by arrowheads). Schematic shows the two coupled neurons monitored in current clamp. (Aii) The prejunctional impedance (Z _pre ) and CC of the data shown in Ai. A 6th order polynomial fit (smooth curves) to the raw data was used to measure the peak amplitude and resonance frequency (circled). (B) Z _pre and CC have distinct resonances. Averaged frequency profiles of CC and Z _pre are shown, both normalized to their amplitude at 0.1 Hz. CC had a smaller resonance frequency than Z _pre (p < 0.001) and higher resonance power (p = 0.037). N = 19, paired Student’s t-test. (C, D) The resonance frequency of CC was correlated with the resonance frequency of both Z _pre and Z _post (C), while its maximum amplitude was only correlated with that of Z _post (D).

FIGURE 3

The coupling conductance shows a frequency-dependent resonance which is distinct from the resonance of the coupled PD neurons. (Ai) The two PD neurons were voltage clamped, the prejunctional neuron with a ZAP waveform, sweeping a frequency range of 0.1–4 Hz and a voltage range of −60 to −30 mV, while the postjunctional neuron was held at constant holding potential of −60 mV (not shown), and the current flow in both neurons was measured. I _pre showed a minimum value at an intermediate frequency, reflecting the intrinsic resonance of the prejunctional neuron (magenta arrowhead), while I _post showed a peak at a distinct frequency (blue/bronze arrowheads). Schematic represents the two coupled neurons in voltage clamp. (Aii) The prejunctional impedance (Z _pre ) and G _c measured from the data shown in Ai. A 6th order polynomial fit (smooth curves) to the raw data was used to measure the peak amplitude and resonance frequency (circled). The peak of G _c corresponds to the bronze color arrowhead in (Ai). (Bi) The frequency profile of G _c across experiments shows a peak below 1 Hz. (Bii) Z _pre and G _c have distinct resonances. Averaged frequency profiles of G _c and Z _pre are shown, both normalized to their amplitude at 0.1 Hz. G _c had a smaller resonance frequency than Z _pre (p < 0.001) but comparable resonance power Z _PD (p = 0.525). N = 20, paired Student’s t-test. (C, D) Neither the resonance frequency (C), nor the resonance amplitude (D) of G _c was correlated with that of Z _pre or Z _post .

FIGURE 4

(A) The right panel schematically shows the protocols of this figure, in which we examine the current-clamp responses of two coupled identical biophysical model neurons (A), two coupled linear resonators with distinct resonance frequencies measured analytically (B) and two coupled biophysical model neurons with distinct resonance frequencies (C). (Ai) Two identical biophysical model neurons (parameters in Table 2) were coupled (G _c = 20 nS) and a ZAP current sweeping frequencies of 0.1–4 Hz was injected in both neurons to measure the changes in their membrane potentials. Arrows in Ai show the peak (resonance) values in membrane potential amplitudes. (Aii) Membrane impedance amplitudes of the pre- and postjunctional model neurons (Z _pre and Z _post , respectively) of panel (Ai) shown in raw form (dots) and with a non-linear curve fit (solid curves). The impedance profile of the isolated identical pre- and post-junctional cells (1 and 2) are also shown in gray. The coupling coefficient (CC) shows a resonance frequency at a value close to those of Z _pre and Z _post . The peak (resonance) frequencies are shown as open circles. (Bi) Simulation of a ZAP current injected in one of two coupled linear resonators, shown for comparison with the biophysical simulations. (Bii) Analytical calculations (see Supplementary Appendix SA1) show that coupling two linear resonators with distinct resonance frequencies (shown in Bi) brings the resonance peaks (open circles) closer to each other. Z _pre and Z _post show the impedance amplitude profiles of the coupled neurons whose isolated impedance profiles are shown in Z ₁ and Z ₂. In contrast, resonance frequency of CC does not fall between those of Z ₁ and Z ₂. (Biii) Bottom panel shows the resonance frequencies (f values of open circles in Bi) as a function of increasing coupling conductance γ _c (=G _c /C; C is the membrane capacitance). Top panel shows the resonance amplitudes (Z values of open circles in Bii). (C) Panels (Ci–Ciii) show simulations of two coupled biophysical neurons [as in (A)], confirming the findings of the analytical model (panel B). Panel descriptions are the same as in (B). Cell 1 was made to have a different resonance frequency by adjusting the parameters as indicated in Table 2. Cell 2 is identical to that of panel (A). (D) The linear model is used to compare the effect of G _c resonance on the coupling coefficient CC. The left panel shows the two cases compared. In one (scale factor of 1), G _c is kept constant whereas in the other G _c is scaled by an inverted U function, mimicking the resonance measured experimentally in Figure 3 B. The right panel compares Z _pre , Z _post and CC for the two cases, using the linear models of panel (B). Note the amplification of CC and the shift in its resonance frequency when G _c shows resonance.

FIGURE 5

Coupling to a third resonant neuron can produce resonance in the coupling current between two voltage-clamped neurons. (A) The coupling current between two identical model neurons with resonant properties was measured in voltage clamp [schematic in (Ai)]. The prejunctional neuron was voltage clamped with a ZAP waveform spanning from 0.1 Hz to 4 Hz and voltage range of −60 to −45 mV. The postjunctional neuron was voltage clamped at a holding potential of −60 mV. The postjunctional current amplitude showed no frequency dependence (Aii). As a function of input frequency, the prejunctional impedance shows resonance, but the post junctional current remains constant. For comparison, Z _pre and I _post are normalized to their value at 0.1 Hz. (B) The same protocol as A, but the two neurons are both coupled to a third (identical) neuron which is not voltage clamped [schematic in (Bi)]. The addition of the third cell leads to a frequency-dependent response in the voltage of the third neuron (Bii) and in resonance in the postjunctional current (Biii). For comparison, Z _pre and I _post are normalized to their value at 0.1 Hz.

FIGURE 6

The effect of space clamp error on the measurement of G _c . Two ball-and-stick models were examined, one with a standard cylindrical neurite of constant diameter (10 µm), the other with a tapering diameter (20–0.5 µm). Neurite lengths were set to 1,000 µm and the position of the electrical coupling was shifted from the beginning to the end of the neurite, as shown schematically in the top panels. Both somas were voltage clamped and step voltage was applied to one cell. G _c was calculated from the current measured in the second soma. The bottom panel shows the effect of the electrical coupling position on the measurement G _c .

FIGURE 7

Resonance in the coupling conductance influences the level of synchrony between two model bursting neurons. (A) The level of synchrony between two model bursting neurons, coupled with a resonant G _c (schematic), depends on the network oscillation frequency. The three columns show superimposed phase-locked oscillations of two model bursting neurons at three frequencies. The second row is a zoom in to a single burst. The third row shows lowpass filtered traces (slow), highlighting the level of asynchrony of the burst slow waves. The bottom row shows the high pass filtered traces (fast = full - slow), highlighting the lack of synchrony of spiking activity. Gray boxes correspond to frequencies and G _c values as shown in panel (B) (B) Coupling conductance is modeled to show resonance at f = 0.75 Hz. The level of synchrony between the two coupled neurons, measured as a coefficient of determination R ² of their voltage waveforms depends on the network frequency. Changing the network frequency increases synchrony of the slow and full waveforms, but not the fast spiking activity. (C) R ² increases with the coupling conductance.