The logic behind neural control of breathing pattern

Abstract

The respiratory rhythm generator is spectacular in its ability to support a wide range of activities and adapt to changing environmental conditions, yet its operating mechanisms remain elusive. We show how selective control of inspiration and expiration times can be achieved in a new representation of the neural system (called a Boolean network). The new framework enables us to predict the behavior of neural networks based on properties of neurons, not their values. Hence, it reveals the logic behind the neural mechanisms that control the breathing pattern. Our network mimics many features seen in the respiratory network such as the transition from a 3-phase to 2-phase to 1-phase rhythm, providing novel insights and new testable predictions.

Figure 1

Examples of Boolean networks and their characteristic response to an input signal C₁. An action potential (spike) is represented by “1” and the time that passes between action potentials is signified by a sequence of zeros. Panel (A) shows a network where the memory (represented by $S_1,S_2,\mathrm{···},S_k$) is preserved after an action potential has been generated in $X_1$. Panel (B) shows a network with self excitation (depicted by $I_1$) where some of the memory is erased after a spike has been generated (the nodes $S_1,S_2,\mathrm{···},S_m$ convey the memory that remains). Panel (C) shows the response of Network A and Panel (D) shows the response of Network B to changes in the period of $C_1$. In both networks when the period of $C_1$, $p$, is low (the spiking frequency is high), $X_1$ exhibits tonic spiking with period 1 (i.e. $X_1=111\mathrm{···}$). When $p=k$/$N+1$ in Network A, $p=m+2$ in Network B, bursting appears (for example, $X_1=11100001110000\mathrm{···}$). When $p\ge k$/$(N-1)$ in Network A, $p\ge k$ in Network B, $X_1$ exhibits silence (i.e. $X_1=000\mathrm{···}$). The number of consecutive “1” within a burst is reduced in Network A as the period of $C_1$ increases but stays constant in Network B (the only exceptions are when $p=m+1$ and $p=k-1$ where we get different kinds of bursting, see Theorem 5). This characteristic response does not depend on the actual values of $k$ (memory size), $m$ (size of memory that was not erased) and $N$ (threshold of $X_1$ in Network A).

Figure 2

Schematic description of a larger network that provides better control of expiration and inspiration time. The Net A and Net B sub-networks are shown in Fig. 1, Panels (A and B) respectively. Here we only show the output and the first node to which the control input connects ($S_1^i$, where $i$ is the sub-network number, this is equivalent to $S_1$ in Fig. 1, see Figs S9, S10 and S11 for more details). This structure can be related to the schematic representation of the respiratory neural network hypothesized in Smith et al.. Sub-network $X_2$, is deliberately missing from our diagram. Unlike Smith et al., we found that this sub-network is not essential for generating and controlling the bursting signal.

Figure 3

Transition from 3- to 2- to 1-phase pattern in the larger network (Fig. 2). In the 3-phase pattern, $X_1$ is active during phase “I” (inspiration), $X_4$ is active during phase “E₁” (first phase of expiration) and $X_3$ is active during phase “E₂” (second phase of expiration). In the 2-phase pattern, $X_1$ is active during inspiration, $X_3$ is active during expiration and $X_4$ is inactive. In the 1-phase pattern only $X_1$ is bursting. We used the following parameters to generate this figure. For $X_1$ and $X_3$, $k=400$, $N=2,$ $m=100$. For $X_4$, $k=800$, $N=3$. The 3-phase pattern is shown here when the period of $C_1=5$, the period of $C_3=110$ and the period of $C_4=32$. For the depicted 2-phase pattern, the period of $C_4$ is increased to $1000$. The 1-phase pattern is shown here when the period of $C_4=1000$ (same as for the 2-phase pattern), the period of $C_3=500$ and the period of $C_1=110$.

Figure 4

Controlling inspiration and expiration times within the 3-phase pattern. The period of breathing and expiration time can be increased (keeping inspiration time constant) by increasing the period of $C_3$ (RTN, Panel (B)). Expiration time can be decreased and inspiration time increase (keeping the period of breathing constant) by increasing the period of $C_4$ (Pons, Panel (C)). The inverse effect (increasing expiration time and decreasing inspiration time while keeping the period of breathing constant) can be achieved by increasing the period of $C_1$ (NTS, Panel (A)). This figure also shows that there is some variability in the timing of the bursting signals and that this variability increases when the periods of $C_1$ and $C_4$ increase. We used the following parameters to generate the figure. For $X_1$ and $X_3$, $k=400$, $N=2,$ $m=100$. For $X_4$, $k=800$, $N=3$. When it is not varied the period of $C_1=5$, the period of $C_3=110$ and the period of $C_4=32$.

Figure 5

Controlling inspiration and expiration times within the 2-phase pattern. Increasing the period of $C_1$ (NTS, Panel (A)) results in increasing expiration time (number of “0”) and decreasing inspiration time (number of “1”s) while keeping the period of breathing constant on average. Increasing the period of $C_3$ (RTN, Panel (B)) increases the inspiration time (number of “1”s). The period of bursting is not shown in Panel (B) - its value increases as the period of $C_3$ increases with the same increasing tendency as the inspiration time. The parameters used to generate this figure are as follows: for $X_1$ and $X_3,k=400,N=2,m=100$; for $X_4,k=800,N=3$, and the period of $C_4=1000$. In panel (A) the period of $C_3=110$. In panel (B) the period of $C_1=5$.

Figure 6

Other types of breathing patterns predicted by the model. Panel (A) shows periodic breathing - a dynamic change in inspiration time (marked by I) caused by an increase in the periods of $C_1$, $C_3$ and $C_4$ (decrease in the spiking frequency of NTS, RTN and the Pons respectively). As a result, $X_3$ (Aug-E) is silent and $X_4$ (Post-I) is bursting. $X_1$ (Pre-I) would have been bursting had it been in isolation. In Panel (B), the periodic breathing is suppressed by increasing the spiking frequency of NTS (decreasing the period of $C_1$). This leads to another way by which a 2-phase pattern can be achieved. We used the following parameters to generate this figure. For $X_1$ and $X_3$, $k=400$, $N=2$, $m=100$. For $X_4$, $k=800$, $N=3$. The period of $C_3=500$ and the period of $C_4=350$. The period of $C_1=110$ in Panel (A) and the period of $C_1=50$ in Panel (B).

Figure 7

Mechanism of the 3-phase pattern generation. $X_3$ is in a bursting state due to the period of $C_3$. While it is active both $X_1$ and $X_4$ are inactive. When $X_3$ turns itself down (due to its bursting state), both $X_1$ and $X_4$ can be activated by their control signals $C_1$ and $C_4$ respectively. Due to the lower threshold of $X_1$ and the higher frequency of its control signal, $X_1$ is activated first. When $X_4$ is activated, it terminates $X_1$. When $X_3$ becomes active again (due to its bursting state), $X_4$ is terminated.

Figure 8

Creating a periodic signal: (a) $C_1=(\overline{10})$ (period 2); (b) $C_1=(\overline{100})$ (period 3); (c) $C_1=(\overline{1000})$ (period 4).