Dynamics and control of mixed bursting in nonlinear pre-Bötzinger complex systems

Abstract

The respiratory rhythm is initiated and controlled by the respiratory center. The pre-Bötzinger complex is a key part of the origin of respiratory rhythms in mammals. The regulatory mechanism of the respiratory rhythm has been a prominent topic for researchers. By monitoring the changes in oxygen and carbon dioxide concentrations in the blood, it has been found that peripheral chemoreceptors can significantly influence the respiratory rhythm. In this study, using a closed-loop respiratory control model based on pre-Bötzinger complex neurons, by incorporating the motor pool, lung volume, lung oxygen, blood oxygen, and chemoreceptors, we investigate how the response of the system changes after the same hypoxic perturbation when the initial state of the neuron system is different types of mixed bursting patterns. The results show that the system can recover when the initial state consists of one or two somatic bursters and one dendritic burster. However, the system cannot recover when the initial state consists of three or four somatic bursters and one dendritic burster. Due to the complexity of calcium currents, we simulate the complex calcium current with a square wave current to achieve consistent hypoxic responses in the system. Subsequently, using bifurcation analysis and control theory, we investigate the dynamic mechanisms underlying the changes in the system after hypoxic perturbation. According to the bifurcation analysis, we show that different initial states lead to changes of the bifurcation structure of the system, which result in different responses after hypoxic perturbation. This study contributes to a better understanding of the interaction between the respiratory center and peripheral chemical sensory feedback and their influence on the respiratory rhythm.

Appendices

Appendix A

The specific expressions of functions in models (1a)-(1j):

The equilibrium function ${n}_{\infty }\left(V\right)$, ${h}_{\infty }\left(V\right)$, ${m}_{\infty }\left(V\right)$, ${p}_{\infty }\left(V\right)$ has the following form:

$x_\infty (V) = \frac{1}{{1 + \exp [{{(V - \theta_x )} / {\sigma_x }}]}},x \in \left\{ {n,h,m,p} \right\}$.

Time scale function: $\tau_x (V) = \frac{{\overline{\tau }_x }}{{\cosh [{{(V - \theta_x )} / {2\sigma_x }}]}},x \in \left\{ {n,h} \right\}$.

Other function expressions:

$f\left( {\text{[Ca]}} \right) = {1 / {\left( {1 + \left( {{{K_{{\text{CAN}}} } / {[{\text{Ca}}]}}} \right)^{n_{{\text{CAN}}} } } \right)}}$, $\left[ {{\text{Ca}}} \right]_{{\text{ER}}} = \frac{{\left[ {{\text{Ca}}} \right]_{{\text{Tot}}} - \left[ {{\text{Ca}}} \right]}}{\sigma }$,$\begin{aligned}J_{{\text{ER}}_{{\text{IN}}} } = \left( {L_{{\text{IP}}_{3} } + P_{{\text{IP}}_{3} } \left[ {\frac{{\left[ {{\text{IP}}_{3} } \right]\left[ {{\text{Ca}}} \right]l}}{{\left( {\left[ {{\text{IP}}_{3} } \right] + K_I } \right)\left( {\left[ {{\text{Ca}}} \right] + K_a } \right)}}} \right]} \right) \times \left( {\left[ {{\text{Ca}}} \right]_{{\text{ER}}} - \left[ {{\text{Ca}}} \right]} \right)\end {aligned}$,$J_{{\text{ER}}_{{\text{OUT}}} } = V_{{\text{SERCA}}} \frac{{\left[ {{\text{Ca}}} \right]^2 }}{{K_{{\text{SERCA}}}^2 + \left[ {{\text{Ca}}} \right]^2 }}$, $[T] = \frac{{T_{\max } }}{{1 + \exp [{{ - (V - V_T )} / {K_P }}]}}$,${\text{Sa}}_{{\text{O}}_{2} } = \frac{{{\text{Pa}}_{{\text{O}}_{2} }^c }}{{{\text{Pa}}_{{\text{O}}_{2} }^c + K^c }}$,

$J_{{\text{LB}}} = \left( {\frac{{{\text{PA}}_{{\text{O}}_{2} } - {\text{Pa}}_{{\text{O}}_{2} } }}{{\tau_{{\text{LB}}} }}} \right)\left( {\frac{{{\text{vol}}_{\text{L}} }}{{{\text{RT}}}}} \right)$, $J_{{\text{BT}}} = M\zeta (\beta_{{\text{O}}_{2} } {\text{Pa}}_{{\text{O}}_{2} } + \eta {\text{Sa}}_{{\text{O}}_{2} } )$,

$\frac{{\partial {\text{Sa}}_{{\text{O}}_{2} } }}{{\partial {\text{Pa}}_{{\text{O}}_{2} } }} = c{\text{Pa}}_{{\text{O}}_{2} }^{c - 1} \left[ {\frac{1}{{{\text{Pa}}_{{\text{O}}_{2} }^c + K^c }} - \frac{{{\text{Pa}}_{{\text{O}}_{2} }^c }}{{({\text{Pa}}_{{\text{O}}_{2} }^c + K^c )^2 }}} \right]$,

$\zeta = {\text{vol}}_{\text{B}} \times \left( {\frac{{{\text{mol}}_{\,} {\text{O}}_{2} }}{{22400_{\,} {\text{ml}}_{\,} {\text{O}}_2 }}} \right)$, $\eta = {\text{[Hb]}} \times \left( {\frac{{1.36_{\,} {\text{ml}}_{\,} {\text{O}}_{2} }}{{g_{\,} {\text{Hb}}}}} \right)$.

See Tables

Table 3 Parameter list

3,

Table 4 Parameter values for each Case

4.

Appendix B

The model of the square wave current fitting calcium current is as follows (2a-2h):

$$ \frac{{{\text{d}}V}}{{{\text{d}}t}} = {{\left( { - I_{\text{K}} - I_{{\text{NaP}}} - I_{{\text{Na}}} - I_{\text{L}} - I_{{\text{tonic}}} + I_{{\text{ex}}} } \right)} / C}, $$

(2a)

$$ \frac{{{\text{d}}n}}{{{\text{d}}t}} = {{\left( {n_\infty (V) - n} \right)} / {\tau_n (V)}}, $$

(2b)

$$ \frac{{{\text{d}}h}}{dt} = {{\left( {h_\infty (V) - h} \right)} / {\tau_h (V)}}, $$

(2c)

$$ \frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = r_a [T](1 - \alpha ) - r_d \alpha , $$

(2d)

$$ \frac{{\text{d}}}{{{\text{d}}t}}({\text{vol}}_{\text{L}} ) = E_1 \alpha - E_2 ({\text{vol}}_{\text{L}} - {\text{vol}}_{0} ), $$

(2e)

$$ \frac{{\text{d}}}{{{\text{d}}t}}({\text{PA}}_{{\text{O}}_{2} } ) = \frac{{{\text{P}}_{{\text{ext}}} {\text{O}}_{2} - {\text{PA}}_{{\text{O}}_{2} } }}{{{\text{vol}}_{\text{L}} }}\left[ {\frac{{\text{d}}}{{{\text{d}}t}}({\text{vol}}_{\text{L}} )} \right]_+ - \frac{{{\text{PA}}_{{\text{O}}_{2} } - {\text{Pa}}_{{\text{O}}_{2} } }}{{\tau_{{\text{LB}}} }}, $$

(2f)

$$ \begin{gathered} \frac{{\text{d}}}{{{\text{d}}t}}({\text{Pa}}_{{\text{O}}_{2} } ) = \frac{{J_{{\text{LB}}} - J_{{\text{BT}}} }}{{\zeta \left( {\beta_{{\text{O}}_{2} } + \eta \frac{{\partial {\text{Sa}}_{{\text{O}}_{2} } }}{{\partial {\text{Pa}}_{{\text{O}}_{2} } }}} \right)}}, \hfill \\ \hfill \\ \end{gathered} $$

(2g)

$$ g_{{\text{tonic}}} = \phi \left[ {1 - \tanh \left( {\frac{{{\text{Pa}}_{{\text{O}}_{2} } - \theta_g }}{\sigma_g }} \right)} \right]. $$

(2h)

The parameters are consistent with Appendix A.