The brain produces movement and also controls homeostasis of the internal physiological state of the brain and body. These two facets of brain function converge in the neural system that controls breathing. Breathing is a primal homeostatic neural process, regulating brain and body oxygen and carbon dioxide levels; rhythmic movements, generated continuously throughout life by the nervous system, are responsible for the process. The automatic and simple act of breathing hides the complex brain processes involved. Indeed, neurophysiologists have been conducting intensive investigations of the underlying mechanisms for more than a century. Mathematicians and other brain modelers have gotten into the act and are making profound contributions. For experimentalists, a major focus has been to discover, at the level of nerve cells and circuits, how the breathing rhythm in mammals is generated. For modelers, the problem, like problems elsewhere in the brain, entails analyses of how oscillations and patterns of neural activity evolve in nonlinear dynamical systems. The experimental breakthrough occurred in the early 1990s, when, after decades of searching, the rhythm generator was found. Experiments on rodent nervous systems showed that the oscillator—at least the excitatory components that activate inspiration—resides in a brainstem structure, named the pre-Bötzinger complex.
Thank you for reading this post, don’t forget to subscribe!Astonishingly, researchers were able to isolate a slice of tissue containing the pre-Bötzinger complex from the rodent brainstem and keep it alive in a dish in vitro!—an experimentalist’s dream come true with immediate prospects for discovery. Indeed, it was soon established that the in vitro pre-Bötzinger complex had autorhythmic properties: Some type of neural pacemaker was generating inspiration [1,4].
Establishing the biophysical mechanisms, however, proved to be a much more technically formidable challenge. Fortunately, the problem immediately captured the attention of dynamical systems modelers fascinated by neural oscillators. In a classic example of a multidisciplinary team, Robert Butera (an engineer), John Rinzel (a mathematician), and I (an experimental neuroscientist) joined forces to formulate hypothetical biophysical models for the pre-Bötzinger complex pacemaker cells [1,2]—at the time with limited biophysical data. The models aimed to explain how individual cells and the pre-Bötzinger complex network produce bursts of activity. These bursts alternate with silent periods. They occur over a few seconds. These bursts alternate with silent periods and happen over a few seconds. The models are simple electrodynamic representations of neurons. They include ionic mechanisms that create electrical current across the neuronal membrane. Kinetic mechanisms control these currents. This approach is common among biophysicists who aim to capture realistic mechanisms. Our group’s work is a case study in the application of approaches from nonlinear dynamics to deal with systems of differential equations describing a complex biological process. The state variables were dimensionally reduced by a fast–slow variable decomposition analysis—a framework developed earlier by Rinzel and colleagues—that dissects the dynamics into a fast subsystem coupled to a slowly varying subsystem.
The advanced system operates on two distinctive timescales: a fast system that models the membrane voltage and neuronal firing during bursts of activity on a millisecond scale, and a slow system that models the underlying slowly varying ionic currents on a second scale. This dual system was used to mimic the rhythmic bursts seen in the pre-Bötzinger complex cells. A new groundbreaking hypothesis emerged from the models: rhythm generation might centrally depend on a special type of ionic current, specifically a persistent sodium current. This sodium current initiates cellular bursting, slowly deactivates during ongoing activity, and then gradually reactivates. This oscillation was depicted by the slow subsystem of the model. Moreover, the model suggested an ongoing leak of potassium ions through a separate membrane current, which dynamically countered the deactivating sodium current. It also proposed a mechanism to regulate the oscillation frequency. These hypothesized biophysical mechanisms are now open to empirical testing. Mathematical techniques and modeling shed light on how groups of pacemaker cells achieve dynamic synchronization within an excitatory network, critical for the distributed operation of the pre-Bötzinger complex across both sides of the brainstem, ensuring symmetrical inspiratory movements.
Other mathematicians—David Terman, Jon Rubin, and colleagues—joined the modeling effort [3,6], and several remarkable network properties were deduced. The same cellular burst-generating mechanism involving persistent sodium current was found to promote network-wide synchronization. Moreover, new oscillatory dynamics were found for networks. The network oscillations were more robust, in the sense that they could occur over a wider dynamic range than those produced by single cells. This robustness arose from the heterogeneity of cellular properties, which was incorporated within model networks to account for known variability of bursting behavior among real cells. And, remarkably, heterogeneous networks were shown to have several dynamically different rhythm-generating states (see discussions in [3]). These surprising insights could not have been derived from experiments alone.
The work provided neurophysiologists with tantalizing and satisfying results: An oscillator as vital as the inspiratory rhythm generator must be robust; the existence of multiple dynamical mechanisms for generating rhythms implied that the pre-Bötzinger complex is a functionally flexible neural machine. Stunning experimental confirmation of some of the model predictions has followed. The persistent sodium current was searched for, found, and confirmed to underlie pre-Bötzinger complex pacemaking. A remarkable finding is that oxygen levels may directly regulate this mechanism, which becomes critical when the brain is severely deprived of oxygen (see [5]). The molecular identity of the postulated potassium leak channel was revealed recently, and this channel was shown to have precisely the biophysical properties predicted.
Multiple mechanisms for rhythm generation, reflecting different dynamical states of the pre-Bötzinger complex, have also been identified. This is a triumphant example of a story A New Mathematics-Inspired Understanding of Breathing and the Brain that began with models, motivated by rudimentary phenomenological observations, and went on to detailed experimental confirmation. Many opportunities for modeling remain. The pre-Bötzinger complex is an oscillatory “kernel,” embedded in a much more spatially distributed system that generates both inspiratory and expiratory activity. Populations of cells active during expiration inhibit the pre-Bötzinger complex and introduce novel dynamics. There is actually a hierarchy of interacting network components that can transform the rhythm-generation process between pacemaker-driven and inhibitory network-based mechanisms. Again, these observations indicate remarkable functional flexibility, beyond the rhythmogenic capabilities of the pre-Bötzinger complex.
References
[1] R.J. Butera, J. Rinzel, and J.C. Smith, Models of respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons, J. Neurophysiol., 82 (1999), 382–397.
[2] R.J. Butera, J. Rinzel, and J.C. Smith, Models of respiratory rhythm generation in the pre-Bötzinger complex. II. Populations of coupled pacemaker neurons, J. Neurophysiol., 82 (1999), 398–415.
[3] R.J. Butera, J. Rubin, D. Terman, and J. Smith, Oscillatory bursting mechanisms in respiratory pacemaker neurons and networks, in Bursting: The Genesis of Rhythm in the Nervous System, S. Coombes and P.C. Bressloff, eds., World Scientific Press, London, 2005, 303–347.
[4] N. Koshiya and J.C. Smith, Neuronal pacemaker for breathing visualized in vitro, Nature, 400 (1999), 360–363.
[5] J.F.R. Paton, A.P.L. Abdala, H. Koizumi, J.C. Smith, and W.M. St-John, Respiratory rhythm generation during gasping depends on persistent sodium current, Nat. Neurosci., 9 (2006), 311–313.
[6] J. Rubin and D. Terman, Synchronized bursts and loss of synchrony among heterogeneous conditional oscillators, SIAM J. Appl. Dyn. Syst., 1 (2002), 146–174. Jeffrey Smith is a researcher in the Cellular and Systems Neurobiology Section, National Institute of Neurological Disorders and Stroke, National Institutes of Health, in Bethesda, Mayland.
The pre-Bötzinger complex: Unraveling the mysteries of respiratory rhythm with advanced modeling and mathematical techniques.
An innovative dual-timescale system has been developed to model the complex behavior of the pre-Bötzinger complex cells, offering groundbreaking insights into the generation of respiratory rhythms. By operating on both millisecond and second scales, the model reveals the crucial role of ionic currents, specifically a persistent sodium current, in initiating and regulating cellular bursting. This sodium current’s slow deactivation and reactivation give rise to oscillations, counterbalanced by a leak of potassium ions.
Furthermore, the model proposes a mechanism for controlling oscillation frequency and sheds light on dynamic synchronization within excitatory networks. These findings have significant implications for understanding the symmetrical inspiratory movements ensured by the distributed operation of the pre-Bötzinger complex across the brainstem. With these hypotheses now open to empirical testing, the study showcases the power of mathematical modeling in unraveling the mysteries of respiratory control.
Key Takeaways:
- Dual-Timescale Modeling: The advanced system models membrane voltage and neuronal firing on a millisecond scale and slow ionic currents on a second scale, capturing the complex behavior of the pre-Bötzinger complex cells.
- Ionic Current Hypothesis: A persistent sodium current is hypothesized to initiate cellular bursting, with its slow deactivation and reactivation giving rise to oscillations, counterbalanced by potassium ion leakage.
- Frequency Regulation: The model proposes a mechanism for controlling oscillation frequency, offering insights into respiratory rhythm regulation.
- Dynamic Synchronization: Mathematical techniques reveal how pacemaker cells synchronize within excitatory networks, ensuring the symmetrical operation of the pre-Bötzinger complex across the brainstem for coordinated inspiratory movements.
- Empirical Testing: The study generates testable hypotheses, paving the way for further experimental research to validate the modeled mechanisms.
The pre-Bötzinger complex, with its intricate cellular dynamics, continues to yield its secrets through the innovative use of mathematical modeling and simulation techniques, advancing our understanding of respiratory control and offering potential therapeutic insights for respiratory disorders.