A balance of outward and linear inward ionic currents is required for generation of slow-wave oscillations

Abstract

Regenerative inward currents help produce slow oscillations through a negative-slope conductance region of their current-voltage relationship that is well approximated by a linear negative conductance. We used dynamic-clamp injections of a linear current with such conductance, I_NL, to explore why some neurons can generate intrinsic slow oscillations whereas others cannot. We addressed this question in synaptically isolated neurons of the crab Cancer borealis after blocking action potentials. The pyloric network consists of a distinct pacemaker and follower neurons, all of which express the same complement of ionic currents. When the pyloric dilator (PD) neuron, a member of the pacemaker group, was injected with I_NL with dynamic clamp, it consistently produced slow oscillations. In contrast, all follower neurons failed to oscillate with I_NL To understand these distinct behaviors, we compared outward current levels of PD with those of follower lateral pyloric (LP) and ventral pyloric (VD) neurons. We found that LP and VD neurons had significantly larger high-threshold potassium currents (I_HTK) than PD and LP had lower-transient potassium current (I_A). Reducing I_HTK pharmacologically enabled both LP and VD neurons to produce I_NL-induced oscillations, whereas modifying I_A levels did not affect I_NL-induced oscillations. Using phase-plane and bifurcation analysis of a simplified model cell, we demonstrate that large levels of I_HTK can block I_NL-induced oscillatory activity whereas generation of oscillations is almost independent of I_A levels. These results demonstrate the general importance of a balance between inward pacemaking currents and high-threshold K⁺ current levels in determining slow oscillatory activity.NEW & NOTEWORTHY Pacemaker neuron-generated rhythmic activity requires the activation of at least one inward and one outward current. We have previously shown that the inward current can be a linear current (with negative conductance). Using this simple mechanism, here we demonstrate that the inward current conductance must be in relative balance with the outward current conductances to generate oscillatory activity. Surprisingly, an excess of outward conductances completely precludes the possibility of achieving such a balance.

Keywords: compensation; ionic currents; model; phase space; rhythmic activity.

Fig. 1.

Effect of I_NL injection in PD neurons. A: I-V curves of I_MI (gray), I_NL (red), and the truncated version of I_NL (I_NL-cut, blue). Note that both I_NL and I_NL-cut are good approximations to the negatively sloped portion of I_MI, where $I_{\mathrm{M}\mathrm{I}}=g_{\mathrm{M}\mathrm{I}}{m}_{\infty }(v)[v-E_{\mathrm{M}\mathrm{I}}],\hspace{0.17em}{m}_{\infty }(v)=\frac{1}{1+e^{-[v+10]/6}}$. B: effect of dynamic-clamp injection of I_NL (left) and I_NL-cut (right) on the activity of PD neurons. Both traces are from the same neuron. C: injection of I_NL (left) and comparison with injection of constant current of the same time-averaged amplitude as I_NL (I_DC, right). Different preparation than in B. Bottom traces in B and C show the current injected by the dynamic clamp (I_dclamp).

Fig. 2.

Effect of I_NL parameters on PD neuron oscillations. Identified PD neurons (n = 7) were injected with dynamic-clamp I_NL, and parameters g_NL and E_rev were modified over a broad range of values. A: the presence or absence of oscillations (no. of oscillatory preparations out of 7) graphed as a function of the change in E_rev relative to each cell’s resting voltage (ΔE_NL = 0 = V_rest – 2 mV) and the value of the negative conductance injected (g_NL). Red symbols are the experimental data. The smooth surface is a Gaussian surface fit to the experimental data. Adjusted R² = 0.9185. B: mean period vs. g_NL. C: mean amplitude vs. g_NL. D: mean period vs. E_NL. E: mean amplitude vs. E_NL. All data shown in A–E are from the same set of cells. Error bars are SE.

Fig. 3.

Dynamic-clamp I_NL injection in pyloric network neurons cannot induce oscillatory activity in follower cells. Identified neurons with characteristic activity response to I_NL injection (g_NL = 0.16 μS, E_NL = V_rest – 2 mV) of the cells listed. PD neurons responded with oscillatory activity in 100% of the preparations tested. Numbers next to the cell type name indicate how many of the total cells tested showed any oscillatory activity. All cells were tested with the same combinations of g_NL and ΔE_NL as the PD neurons shown in Fig. 2A. Top trace in each panel is the membrane potential; bottom trace is the dynamic-clamp current injected.

Fig. 4.

Potassium current levels in 2 key pyloric network neurons. A: sample traces of I_HTK (left) and I_A (right). Vertical arrows point to the times at which I_HTK peak, I_HTK SS, and I_A are measured. B: I-V curves of the peak of the high-threshold current I_HTK. The curves are significantly different (2-way ANOVA, P < 0.029). C: steady-state I-V curves of I_HTK. The curves are significantly different (2-way ANOVA, P < 0.001). Asterisks indicate the source of the difference from 2-way ANOVA analysis and Tukey pairwise post hoc comparisons: *P < 0.05, **P < 0.01, *** P < 0.001. D: I-V curves of the peak of the transient K⁺ current I_A. The curves are significantly different (2-way ANOVA, P < 0.001). Data in black for PD neurons and in gray for LP neurons. E: activation curves for I_HTK peak and the shown parameters are calculated as in D. F: mean of the voltage-dependent activation curves of I_A derived from the I-V curves of each cell used in A. The I-V curves were fit with the equation $I=g_{\max }\frac{1}{1+e^{[V_{1/2}-v]/V_{\text{slope}}}}[v-E]$. E was fixed at −80 mV, and the other 3 parameters were determined from the fit (V_1/2 and V_slope in mV, g_max in μS). P values are for Student’s t-test analysis for each parameter between the 2 cells. All error bars are SE.

Fig. 5.

High I_HTK levels are responsible for the inability of LP neurons to generate oscillatory activity. A: I-V curves of the peak I_HTK in LP neurons under control (0 mM TEA) and 8 mM TEA. A significant decrease is observed overall (2-way RM ANOVA, n = 3, P = 0.008), with asterisks indicating the source of the difference from Tukey pairwise comparisons: ***P < 0.001. B: dose-dependent sensitivity of peak I_HTK to TEA. I_HTK peak was normalized to the current measured in control (0 mM TEA). Mean ± SE is plotted. The overall effect was statistically significant (1-way RM ANOVA, n = 3, P = 0.006). Dashed trace is an exponential fit with a saturation level at ~20%, and an ID₅₀ = 1.6 mM. C: effect of dynamic-clamp injection of I_NL (g_NL = −0.16 μS; E_NL = −52 mV) in control conditions. Top: membrane potential. Bottom: current injected by the dynamic-clamp circuit. D: effect of dynamic-clamp injection of I_NL (same as in C) but in the presence of 8 mM TEA.

Fig. 6.

I_A does not appear to be involved in the ability of PD neurons to generate oscillatory activity. A: I-V curves of the peak I_A in PD neurons under control (0 mM 4-AP) and 1 mM 4-AP. A significant decrease was observed overall (2-way RM-ANOVA, n = 3, P = 0.012), with asterisks indicating the source of the difference from Tukey pairwise comparisons: **P < 0.01, ***P < 0.001. B: effect of dynamic-clamp injection of I_NL in the absence of 4-AP (g_NL = −0.16 μS; E_NL = −72 mV). Top: membrane potential. Bottom: dynamic-clamp current. C: effect of dynamic-clamp injection of I_NL in the presence of 1 mM 4-AP (g_NL = −0.16 μS; E_NL = −72 mV).

Fig. 7.

High I_HTK levels are responsible for the inability of VD neurons to generate oscillatory activity. A: I-V curves show that the peak I_HTK in VD neurons is almost twice as large as in PD neurons (2-way ANOVA, P = 0.001) over most of its activation range. B: I-V curves of I_A show no significant difference between PD and VD neurons (2-way ANOVA, P = 0.787). C: I-V curves of the peak I_HTK in VD neurons under control (0 mM TEA) and 8 mM TEA. A significant decrease is observed overall (2-way RM ANOVA, n = 4, P = 0.003), with asterisks indicating the source of the difference from Tukey pairwise comparisons: *P < 0.05, **P < 0.01, ***P < 0.001. D: sensitivity of I_HTK peak of the VD neuron to TEA. I_HTK peak was normalized to the current measured in control (0 mM TEA). Mean ± SE is plotted. The overall reduction of I_HTK peak by TEA was significant (1-way RM ANOVA, n = 4, P = 0.001). E and F: effect of dynamic-clamp injection of I_NL (g_NL = −0.08 μS; E_NL = −70 mV) in normal saline (E) and in the presence of 8 mM TEA (F). Top: membrane potential. Bottom: current injected by the dynamic-clamp circuit.

Fig. 8.

Effect of I_HTK on the existence of oscillations. A: A1 shows the v-w phase plane for low values of g_HTK in which an oscillatory solution exists (shown as dark closed path). For larger values of g_HTK the oscillatory solution fails to exist, as shown in A2, because the stronger I_HTK stabilizes the fixed point at the intersection of the 2 nullclines. B: bifurcation diagram showing how the existence and amplitude of oscillatory solutions depend on g_HTK. Gray curve indicates the voltage limits of oscillatory solutions, whereas the black line indicates a fixed point. Solid and dashed lines indicate stable and unstable fixed points, respectively. Arrows from A1 and A2 show the values of g_HTK used to produce those simulations. Oscillatory solutions are lost through a Hopf bifurcation as g_HTK increases. C: bifurcation diagrams showing behavior of solutions at fixed values of g_HTK while g_NL is varied. The range of g_NL values for which oscillations exist decreases as g_HTK increases (compare C1 and C2). If g_HTK is too large (C3), no amount of g_NL can restore oscillations. D1 and D2: voltage traces corresponding to the dashed lines in C1 and C2, respectively. E: behavior of the model in the g_HTK-g_NL parameter space. Oscillations are limited to parameter values in the gray region.

Fig. 9.

Effect of I_A on the existence of oscillations. A1: projection onto the v-w phase plane for g_A = 0.033 µS. No qualitative difference is found from the g_A = 0 µS case (shown in Fig. 8A1). The third variable, h (not shown), also oscillates but remains small. The v-nullcline shown is for the largest value of h along the periodic solution. The intersection of the projection of the v and w-nullclines is not a fixed point. A2: projection onto the v-w phase plane at g_A = 0.195 µS for which bistability exists between the periodic solution and a stable fixed point. The lower of the two v-nullclines occurs for h = 0.45 at the value corresponding to a stable fixed point (open circle). The upper of the 2 v-nullclines occurs at the largest h value along the periodic solution. B: bifurcation diagram showing how the bistability of solutions depends on g_A. Gray curves indicate the voltage limits of oscillatory solutions, whereas the black line indicates a fixed point. Solid and dashed lines, respectively, indicate stable and unstable. Arrows from A1 and A2 show the values of g_A used to produce those simulations. The unstable fixed point (dashed black curve) undergoes a subcritical Hopf bifurcation as g_A increases, resulting in a stable fixed point and an unstable branch of periodic solutions.

Fig. 10.

Effect of I_NL parameters on oscillation period and amplitude. A and C are model analogs of Fig. 2, B and D, showing how period depends on g_NL and E_NL, respectively. The simulations qualitatively match the behavior of the experimental system. B1 and D1 plot the bifurcation diagrams obtained by respectively varying g_NL and E_NL. B2 and D2 are model analogs of Fig. 2, C and E, showing how oscillation amplitude depends on g_NL and E_NL, respectively. Amplitude is calculated as a difference between the minimum (bottom gray) and maximum (top gray) voltage for each parameter value in B1 and D1. E_NL = −65 mV in A and B; g_NL = −0.0195 µS in C and D; and g_A = 0 µS in all panels.