Cooperation of intrinsic bursting and calcium oscillations underlying activity patterns of model pre-Bötzinger complex neurons

Abstract

Activity of neurons in the pre-Bötzinger complex (pre-BötC) within the mammalian brainstem drives the inspiratory phase of the respiratory rhythm. Experimental results have suggested that multiple bursting mechanisms based on a calcium-activated nonspecific cationic (CAN) current, a persistent sodium (NaP) current, and calcium dynamics may be incorporated within the pre-BötC. Previous modeling works have incorporated representations of some or all of these mechanisms. In this study, we consider a single-compartment model of a pre-BötC inspiratory neuron that encompasses particular aspects of all of these features. We present a novel mathematical analysis of the interaction of the corresponding rhythmic mechanisms arising in the model, including square-wave bursting and autonomous calcium oscillations, which requires treatment of a system of differential equations incorporating three slow variables.

Appendix: Return maps

We first computed the durations of the active phase and silent phase by treating hp as a parameter. With initial condition hp ₀, we integrated the averaged equations of hp (Eqs. (11) and (12)) over the silent phase duration and the active phase duration sequentially. The value of hp at the end of active phase, F (hp ₀) in Eq. (17), is the map that we used.

For quick reference, we first rewrite the governing equation of hp,

$$ \frac{d \,hp}{dt} = \frac{hp_{\infty}(V) - hp}{\tau_{h} (V)} = \frac{hp_{\infty}(V)}{\tau_{h} (V)} - \frac{1}{\tau_{h} (V)} hp. $$

(10)

We regard hp as a parameter to compute the duration of the silent phase, say SP(hp), and the active phase, say AP(hp), of the bursting solution for each fixed hp between 0.2 and 0.8. With hp fixed, the activity of the solution of the full system is purely driven by the dendritic Ca ^2 + dynamics and the projection of the solution onto ($g_{\rm CAN _{Tot}}$, hp)-space is a horizontal line segment. Now we average the right-hand side of Eq. (10) over time duration AP(hp). The averaged equation over the active phase is given as

$$ \begin{array}{rll} \frac{d \,hp^{A}_{av}}{dt}&=& \left [ \frac{1}{T^A_F-T^A_I}\int_{T^A_I}^{T^A_F} \frac{hp_{\infty}(V)}{\tau_{h}(V)} dt \right ]\\ && - \left [\frac{1}{T^A_F-T^A_I}\int_{T^A_I}^{T^A_F} \frac{1}{\tau_{h}(V)} dt \right ] hp^A_{av} \end{array} $$

(11)

where $T^A_I (T^A_F)$ denotes the initial (final) time of the active phase AP(hp). Since τ _h (V) is sensitive to the range of voltage V within silent phase, we also average the right-hand side of Eq. (10) over time duration SP(hp) to get

$$ \begin{array}{rll} \frac{d \,hp^{S}_{av}}{dt}&=& \left [ \frac{1}{T^S_F-T^S_I}\int_{T^S_I}^{T^S_F} \frac{hp_{\infty}(V)}{\tau_{h}(V)} dt \right ]\\ &&- \left [\frac{1}{T^S_F-T^S_I}\int_{T^S_I}^{T^S_F} \frac{1}{\tau_{h}(V)} dt \right ] hp^S_{av} \end{array} $$

(12)

where $T^S_I (T^S_F)$ denotes the initial (final) time of the silent phase SP(hp). The averaged hp variable over the active phase, $hp^A_{av}$, has a tendency to approach the fixed point of Eq. (11), call it FP _A (hp).

$${\rm FP}_A(hp) = \int_{T^A_I}^{T^A_F} \frac{hp_{\infty}(V)}{\tau_{h}(V)} dt / \int_{T^A_I}^{T^A_F} \frac{1}{\tau_{h}(V)} dt $$

(13)

Similarly, $hp^S_{av}$ tends to approach FP _S (hp), which is given as

$${\rm FP}_S(hp) = \int_{T^S_I}^{T^S_F} \frac{hp_{\infty}(V)}{\tau_{h}(V)} dt / \int_{T^S_I}^{T^S_F} \frac{1}{\tau_{h}(V)} dt $$

(14)

We computed FP _A (hp) and FP _S (hp) over hp values from 0.2 to 0.8 with stepsize 0.05 and [IP₃] values from 0.95 to 1.4 with stepsize 0.01. The results are shown in Fig. 13(a), where the solid curves correspond to the mean values of FP _A (lower curve) and FP _S (upper curve) averaged over the [IP₃] range and the standard deviation from this mean over the range of [IP₃] is indicated with error bars. From the small size of the error bars, we see that the mean values of FP _A and FP _S show only slight dependence on [IP₃] values. This observation is consistent with our earlier claim that the hp values at transitions between phases are relatively independent of [IP₃].

Fig. 13

Return maps to estimate the minimal values of hp over bursting solutions. (a) Fixed points of averaged Eqs. (11) and (12), FP _A (hp) and FP _S (hp), are computed over hp values from 0.2 to 0.8 and [IP₃] from 0.95 to 1.4. Upper (lower, resp.) solid curve with error bars denote the mean value of FP _S (hp) (FP _A (hp), resp.) averaged over the [IP₃] range and standard deviation from this mean. (b) Examples of the return map F(hp) when IP₃ = 1 (black, upper) and 1.3 (black, lower), relative to the identity line (gray). (c) Collection of stable fixed points of the return map (Eq. (17)) (gray) with the results from numerical simulation (black)

Now, suppose that the silent phase begins at t = 0 with initial condition hp ₀. Let F _S (hp ₀) be the value of hp at the end of the silent phase predicted by integrating the averaged equation of hp for the duration of silent phase, SP(hp ₀). Then we have

$$ \begin{array}{rll} {\rm F}_S(hp_0) &=& {\rm FP}_S(hp_0) \\&&+\, (hp_0 - {\rm FP}_S(hp_0))\exp(-k_S{\rm SP}(hp_0)) \end{array} $$

(15)

where

$$ k_S=\frac{1}{T^S_F-T^S_I}\int_{T^S_I}^{T^S_F} \frac{1}{\tau_{h}(V)} dt. $$

(16)

Now, we integrate Eq. (11) again for the duration of active phase, AP(hp ₀), with initial condition F _S (hp ₀). Let F (hp ₀) be the resulting value of hp, which is the desired map and given as

$$ \begin{array}{rll} {\rm F}(hp_0)&=& {\rm FP}_A(hp_0) + ({\rm F}_S(hp_0) \\ &&-\, {\rm FP}_A(hp_0))\exp(-k_A{\rm AP}(hp_0)) \end{array} $$

(17)

where

$$ k_A=\frac{1}{T^A_F-T^A_I}\int_{T^A_I}^{T^A_F} \frac{1}{\tau_{h}(V)} dt $$

(18)

Two examples of maps with a line of identity (gray diagonal line) are shown in Fig. 13(b) when [IP₃] = 1 (upper) and 1.3 (lower). In each map, there is a stable fixed point. Figure 13(c) shows the collection of these stable fixed points (gray), which matches well with the numerical results (black). Thus, Eq. (17) captures the decrease in the hp value where the balance of silent and active phase durations occurs, specifically in the hp value at the end of the active phase, as [IP₃] increases.