Interacting oscillations in neural control of breathing: modeling and qualitative analysis

Exploring the role of the Kölliker-Fuse nucleus in breathing variability by mathematical modelling

The Kölliker-Fuse nucleus (KF), which is part of the parabrachial complex, participates in the generation of eupnoea under resting conditions and the control of active abdominal expiration when increased ventilation is required. Moreover, dysfunctions in KF neuronal activity are believed to play a role in the emergence of respiratory abnormalities seen in Rett syndrome (RTT), a progressive neurodevelopmental disorder associated with an irregular breathing pattern and frequent apnoeas. Relatively little is known, however, about the intrinsic dynamics of neurons within the KF and how their synaptic connections affect breathing pattern control and contribute to breathing irregularities. In this study, we use a reduced computational model to consider several dynamical regimes of KF activity paired with different input sources to determine which combinations are compatible with known experimental observations. We further build on these findings to identify possible interactions between the KF and other components of the respiratory neural circuitry. Specifically, we present two models that both simulate eupnoeic as well as RTT-like breathing phenotypes. Using nullcline analysis, we identify the types of inhibitory inputs to the KF leading to RTT-like respiratory patterns and suggest possible KF local circuit organizations. When the identified properties are present, the two models also exhibit quantal acceleration of late-expiratory activity, a hallmark of active expiration featuring forced exhalation, with increasing inhibition to KF, as reported experimentally. Hence, these models instantiate plausible hypotheses about possible KF dynamics and forms of local network interactions, thus providing a general framework as well as specific predictions for future experimental testing. KEY POINTS: The Kölliker-Fuse nucleus (KF), a part of the parabrachial complex, is involved in regulating normal breathing and controlling active abdominal expiration during increased ventilation. Dysfunction in KF neuronal activity is thought to contribute to respiratory abnormalities seen in Rett syndrome (RTT). This study utilizes computational modelling to explore different dynamical regimes of KF activity and their compatibility with experimental observations. By analysing different model configurations, the study identifies inhibitory inputs to the KF that lead to RTT-like respiratory patterns and proposes potential KF local circuit organizations. Two models are presented that simulate both normal breathing and RTT-like breathing patterns. These models provide testable hypotheses and specific predictions for future experimental investigations, offering a general framework for understanding KF dynamics and potential network interactions.

Perspectives on adaptive dynamical systems

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems, such as the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges and give perspectives on future research directions, looking to inspire interdisciplinary approaches.

The role of Kölliker-Fuse nucleus in breathing variability

The Kölliker-Fuse nucleus (KF), which is part of the parabrachial complex, participates in the generation of eupnea under resting conditions and the control of active abdominal expiration when increased ventilation is required. Moreover, dysfunctions in KF neuronal activity are believed to play a role in the emergence of respiratory abnormalities seen in Rett syndrome (RTT), a progressive neurodevelopmental disorder associated with an irregular breathing pattern and frequent apneas. Relatively little is known, however, about the intrinsic dynamics of neurons within the KF and how their synaptic connections affect breathing pattern control and contribute to breathing irregularities. In this study, we use a reduced computational model to consider several dynamical regimes of KF activity paired with different input sources to determine which combinations are compatible with known experimental observations. We further build on these findings to identify possible interactions between the KF and other components of the respiratory neural circuitry. Specifically, we present two models that both simulate eupneic as well as RTT-like breathing phenotypes. Using nullcline analysis, we identify the types of inhibitory inputs to the KF leading to RTT-like respiratory patterns and suggest possible KF local circuit organizations. When the identified properties are present, the two models also exhibit quantal acceleration of late-expiratory activity, a hallmark of active expiration featuring forced exhalation, with increasing inhibition to KF, as reported experimentally. Hence, these models instantiate plausible hypotheses about possible KF dynamics and forms of local network interactions, thus providing a general framework as well as specific predictions for future experimental testing.

Key points

The Kölliker-Fuse nucleus (KF), a part of the parabrachial complex, is involved in regulating normal breathing and controlling active abdominal expiration during increased ventilation. Dysfunction in KF neuronal activity is thought to contribute to respiratory abnormalities seen in Rett syndrome (RTT). This study utilizes computational modeling to explore different dynamical regimes of KF activity and their compatibility with experimental observations. By analyzing different model configurations, the study identifies inhibitory inputs to the KF that lead to RTT-like respiratory patterns and proposes potential KF local circuit organizations. Two models are presented that simulate both normal breathing and RTT-like breathing patterns. These models provide plausible hypotheses and specific predictions for future experimental investigations, offering a general framework for understanding KF dynamics and potential network interactions.

Population spiking and bursting in next-generation neural masses with spike-frequency adaptation

Spike-frequency adaptation (SFA) is a fundamental neuronal mechanism taking into account the fatigue due to spike emissions and the consequent reduction of the firing activity. We have studied the effect of this adaptation mechanism on the macroscopic dynamics of excitatory and inhibitory networks of quadratic integrate-and-fire (QIF) neurons coupled via exponentially decaying post-synaptic potentials. In particular, we have studied the population activities by employing an exact mean-field reduction, which gives rise to next-generation neural mass models. This low-dimensional reduction allows for the derivation of bifurcation diagrams and the identification of the possible macroscopic regimes emerging both in a single and in two identically coupled neural masses. In single populations SFA favors the emergence of population bursts in excitatory networks, while it hinders tonic population spiking for inhibitory ones. The symmetric coupling of two neural masses, in absence of adaptation, leads to the emergence of macroscopic solutions with broken symmetry, namely, chimera-like solutions in the inhibitory case and antiphase population spikes in the excitatory one. The addition of SFA leads to new collective dynamical regimes exhibiting cross-frequency coupling (CFC) among the fast synaptic timescale and the slow adaptation one, ranging from antiphase slow-fast nested oscillations to symmetric and asymmetric bursting phenomena. The analysis of these CFC rhythms in the θ-γ range has revealed that a reduction of SFA leads to an increase of the θ frequency joined to a decrease of the γ one. This is analogous to what has been reported experimentally for the hippocampus and the olfactory cortex of rodents under cholinergic modulation, which is known to reduce SFA.

Advancing respiratory-cardiovascular physiology with the working heart-brainstem preparation over 25 years

Twenty‐five years ago, a new physiological preparation called the working heart–brainstem preparation (WHBP) was introduced with the claim it would provide a new platform allowing studies not possible before in cardiovascular, neuroendocrine, autonomic and respiratory research. Herein, we review some of the progress made with the WHBP, some advantages and disadvantages along with potential future applications, and provide photographs and technical drawings of all the customised equipment used for the preparation. Using mice or rats, the WHBP is an in situ experimental model that is perfused via an extracorporeal circuit benefitting from unprecedented surgical access, mechanical stability of the brain for whole cell recording and an uncompromised use of pharmacological agents akin to in vitro approaches. The preparation has revealed novel mechanistic insights into, for example, the generation of distinct respiratory rhythms, the neurogenesis of sympathetic activity, coupling between respiration and the heart and circulation, hypothalamic and spinal control mechanisms, and peripheral and central chemoreceptor mechanisms. Insights have been gleaned into diseases such as hypertension, heart failure and sleep apnoea. Findings from the in situ preparation have been ratified in conscious in vivo animals and when tested have translated to humans. We conclude by discussing potential future applications of the WHBP including two‐photon imaging of peripheral and central nervous systems and adoption of pharmacogenetic tools that will improve our understanding of physiological mechanisms and reveal novel mechanisms that may guide new treatment strategies for cardiorespiratory diseases.

A homeostasis criterion for limit cycle systems based on infinitesimal shape response curves

Homeostasis occurs in a control system when a quantity remains approximately constant as a parameter, representing an external perturbation, varies over some range. Golubitsky and Stewart (J Math Biol 74(1-2):387-407, 2017) developed a notion of infinitesimal homeostasis for equilibrium systems using singularity theory. Rhythmic physiological systems (breathing, locomotion, feeding) maintain homeostasis through control of large-amplitude limit cycles rather than equilibrium points. Here we take an initial step to study (infinitesimal) homeostasis for limit-cycle systems in terms of the average of a quantity taken around the limit cycle. We apply the "infinitesimal shape response curve" (iSRC) introduced by Wang et al. (SIAM J Appl Dyn Syst 82(7):1-43, 2021) to study infinitesimal homeostasis for limit-cycle systems in terms of the mean value of a quantity of interest, averaged around the limit cycle. Using the iSRC, which captures the linearized shape displacement of an oscillator upon a static perturbation, we provide a formula for the derivative of the averaged quantity with respect to the control parameter. Our expression allows one to identify homeostasis points for limit cycle systems in the averaging sense. We demonstrate in the Hodgkin-Huxley model and in a metabolic regulatory network model that the iSRC-based method provides an accurate representation of the sensitivity of averaged quantities.

Respiratory rhythm and pattern generation: Brainstem cellular and circuit mechanisms

Breathing movements in mammals are driven by rhythmic neural activity automatically generated within spatially and functionally organized brainstem neural circuits comprising the respiratory central pattern generator (CPG). This chapter reviews up-to-date experimental information and theoretical studies of the cellular and circuit mechanisms of respiratory rhythm and pattern generation operating within critical components of this CPG in the lower brainstem. Over the past several decades, there have been substantial advances in delineating the spatial architecture of essential medullary regions and their regional cellular and circuit properties required to understand rhythm and pattern generation mechanisms. A fundamental concept is that the circuits in these regions have rhythm-generating capabilities at multiple cellular and circuit organization levels. The regional cellular properties, circuit organization, and control mechanisms allow flexible expression of neural activity patterns for a repertoire of respiratory behaviors under various physiologic conditions that are dictated by requirements for homeostatic regulation and behavioral integration. Many mechanistic insights have been provided by computational modeling studies driven by experimental results and have advanced understanding in the field. These conceptual and theoretical developments are discussed.

Heartbeats entrain breathing via baroreceptor-mediated modulation of expiratory activity

NEW FINDINGS

  Cardio-ventilatory coupling refers to the onset of inspiration occurring at a preferential latency following the last heartbeat (HB) in expiration. According to the cardiac-trigger hypothesis, the pulse pressure initiates an inspiration via baroreceptor activation. However, the central neural substrate mediating this coupling remains undefined. Using a combination of animal data, human data and mathematical modelling, this study tests the hypothesis that the HB, by way of pulsatile baroreflex activation, controls the initiation of inspiration that occurs through a rapid neural activation loop from the carotid baroreceptors to Bötzinger complex expiratory neurons.

ABSTRACT

Cardio-ventilatory coupling refers to a heartbeat (HB) occurring at a preferred latency prior to the next breath. We hypothesized that the pressure pulse generated by a HB activates baroreceptors that modulate brainstem expiratory neuronal activity and delay the initiation of inspiration. In supine male subjects, we recorded ventilation, electrocardiogram and blood pressure during 20-min epochs of baseline, slow-deep breathing and recovery. In in situ rodent preparations, we recorded brainstem activity in response to pulses of perfusion pressure. We applied a well-established respiratory network model to interpret these data. In humans, the latency between a HB and onset of inspiration was consistent across different breathing patterns. In in situ preparations, a transient pressure pulse during expiration activated a subpopulation of expiratory neurons normally active during post-inspiration, thus delaying the next inspiration. In the model, baroreceptor input to post-inspiratory neurons accounted for the effect. These studies are consistent with baroreflex activation modulating respiration through a pauci-synaptic circuit from baroreceptors to onset of inspiration.

On the Organization of the Locomotor CPG: Insights From Split-Belt Locomotion and Mathematical Modeling

Rhythmic limb movements during locomotion are controlled by central pattern generator (CPG) circuits located in the spinal cord. It is considered that these circuits are composed of individual rhythm generators (RGs) for each limb interacting with each other through multiple commissural and long propriospinal circuits. The organization and operation of each RG are not fully understood, and different competing theories exist about interactions between its flexor and extensor components, as well as about left-right commissural interactions between the RGs. The central idea of circuit organization proposed in this study is that with an increase of excitatory input to each RG (or an increase in locomotor speed) the rhythmogenic mechanism of the RGs changes from "flexor-driven" rhythmicity to a "classical half-center" mechanism. We test this hypothesis using our experimental data on changes in duration of stance and swing phases in the intact and spinal cats walking on the ground or tied-belt treadmills (symmetric conditions) or split-belt treadmills with different left and right belt speeds (asymmetric conditions). We compare these experimental data with the results of mathematical modeling, in which simulated CPG circuits operate in similar symmetric and asymmetric conditions with matching or differing control drives to the left and right RGs. The obtained results support the proposed concept of state-dependent changes in RG operation and specific commissural interactions between the RGs. The performed simulations and mathematical analysis of model operation under different conditions provide new insights into CPG network organization and limb coordination during locomotion.

Inhibitory control of active expiration by the Bötzinger complex in rats

KEY POINTS

Contraction of abdominal muscles at the end of expiration during metabolic challenges (such as hypercapnia and hypoxia) improves pulmonary ventilation. The emergence of this active expiratory pattern requires the recruitment of the expiratory oscillator located on the ventral surface of the medulla oblongata. Here we show that an inhibitory circuitry located in the Bötzinger complex is an important source of inhibitory drive to the expiratory oscillator. This circuitry, mediated by GABAergic and glycinergic synapses, provides expiratory inhibition that restrains the expiratory oscillator under resting condition and regulates the formation of abdominal expiratory activity during active expiration. By combining experimental and modelling approaches, we propose the organization and connections within the respiratory network that control the changes in the breathing pattern associated with elevated metabolic demand.

ABSTRACT

The expiratory neurons of the Bötzinger complex (BötC) provide inhibitory inputs to the respiratory network, which, during eupnoea, are critically important for respiratory phase transition and duration control. Here, we investigated how the BötC neurons interact with the expiratory oscillator located in the parafacial respiratory group (pFRG) and control the abdominal activity during active expiration. Using the decerebrated, arterially perfused in situ preparations of juvenile rats, we recorded the activity of expiratory neurons and performed pharmacological manipulations of the BötC and pFRG during hypercapnia or after the exposure to short-term sustained hypoxia - conditions that generate active expiration. The experimental data were integrated in a mathematical model to gain new insights into the inhibitory connectome within the respiratory central pattern generator. Our results indicate that the BötC neurons may establish mutual connections with the pFRG, providing expiratory inhibition during the first stage of expiration and receiving excitatory inputs during late expiration. Moreover, we found that application of GABAergic and glycinergic antagonists in the BötC caused opposing effects on abdominal expiratory activity, suggesting complex inhibitory circuitry within the BötC. Using mathematical modelling, we propose that the BötC network organization and its interactions with the pFRG restrain abdominal activity under resting conditions and contribute to abdominal expiratory pattern formation during active expiration observed during hypercapnia or after the exposure to short-term sustained hypoxia.

Spinal V3 Interneurons and Left-Right Coordination in Mammalian Locomotion

Commissural interneurons (CINs) mediate interactions between rhythm-generating locomotor circuits located on each side of the spinal cord and are necessary for left-right limb coordination during locomotion. While glutamatergic V3 CINs have been implicated in left-right coordination, their functional connectivity remains elusive. Here, we addressed this issue by combining experimental and modeling approaches. We employed Sim1^Cre/+; Ai32 mice, in which light-activated Channelrhodopsin-2 was selectively expressed in V3 interneurons. Fictive locomotor activity was evoked by NMDA and 5-HT in the isolated neonatal lumbar spinal cord. Flexor and extensor activities were recorded from left and right L2 and L5 ventral roots, respectively. Bilateral photoactivation of V3 interneurons increased the duration of extensor bursts resulting in a slowed down on-going rhythm. At high light intensities, extensor activity could become sustained. When light stimulation was shifted toward one side of the cord, the duration of extensor bursts still increased on both sides, but these changes were more pronounced on the contralateral side than on the ipsilateral side. Additional bursts appeared on the ipsilateral side not seen on the contralateral side. Further increase of the stimulation could suppress the contralateral oscillations by switching to a sustained extensor activity, while the ipsilateral rhythmic activity remained. To delineate the function of V3 interneurons and their connectivity, we developed a computational model of the spinal circuits consisting of two (left and right) rhythm generators (RGs) interacting via V0_V, V0_D, and V3 CINs. Both types of V0 CINs provided mutual inhibition between the left and right flexor RG centers and promoted left-right alternation. V3 CINs mediated mutual excitation between the left and right extensor RG centers. These interactions allowed the model to reproduce our current experimental data, while being consistent with previous data concerning the role of V0_V and V0_D CINs in securing left–right alternation and the changes in left–right coordination following their selective removal. We suggest that V3 CINs provide mutual excitation between the spinal neurons involved in the control of left and right extensor activity, which may promote left-right synchronization during locomotion.

Robustness of respiratory rhythm generation across dynamic regimes

A central issue in the study of the neural generation of respiratory rhythms is the role of the intrinsic pacemaking capabilities that some respiratory neurons exhibit. The debate on this issue has occurred in parallel to investigations of interactions among respiratory network neurons and how these contribute to respiratory behavior. In this computational study, we demonstrate how these two issues are inextricably linked. We use simulations and dynamical systems analysis to show that once a conditional respiratory pacemaker, which can be tuned across oscillatory and non-oscillatory dynamic regimes in isolation, is embedded into a respiratory network, its dynamics become masked: the network exhibits similar dynamical properties regardless of the conditional pacemaker node's tuning, and that node's outputs are dominated by network influences. Furthermore, the outputs of the respiratory central pattern generator as a whole are invariant to these changes of dynamical properties, which ensures flexible and robust performance over a wide dynamic range.

Neural network model of an amphibian ventilatory central pattern generator

The neuronal multiunit model presented here is a formal model of the central pattern generator (CPG) of the amphibian ventilatory neural network, inspired by experimental data from Pelophylax ridibundus. The kernel of the CPG consists of three pacemakers and two follower neurons (buccal and lung respectively). This kernel is connected to a chain of excitatory and inhibitory neurons organized in loops. Simulations are performed with Izhikevich-type neurons. When driven by the buccal follower, the excitatory neurons transmit and reorganize the follower activity pattern along the chain, and when driven by the lung follower, the excitatory and inhibitory neurons of the chain fire in synchrony. The additive effects of synaptic inputs from the pacemakers on the buccal follower account for (1) the low frequency buccal rhythm, (2) the intra-burst high frequency oscillations, and (3) the episodic lung activity. Chemosensitivity to acidosis is implemented by an increase in the firing frequency of one of the pacemakers. This frequency increase leads to both a decrease in the buccal burst frequency and an increase in the lung episode frequency. The rhythmogenic properties of the model are robust against synaptic noise and pacemaker jitter. To validate the rhythm and pattern genesis of this formal CPG, neurograms were built from simulated motoneuron activity, and compared with experimental neurograms. The basic principles of our model account for several experimental observations, and we suggest that these principles may be generic for amphibian ventilation.

Reduced computational modelling of Kölliker-Fuse contributions to breathing patterns in Rett syndrome

KEY POINTS

Reduced computational models are used to test effects of loss of inhibition to the Kölliker-Fuse nucleus (KFn). Three reduced computational models that simulate eupnoeic and vagotomized respiratory rhythms are considered. All models exhibit the emergence of respiratory perturbations associated with Rett syndrome as inhibition to the KFn is diminished. Simulations suggest that application of 5-HT_1A agonists can mitigate the respiratory pathology. The three models can be distinguished and tested based on their predictions about connections and dynamics within the respiratory circuit and about effects of perturbations on certain respiratory neuron populations.

ABSTRACT

Rett syndrome (RTT) is a developmental disorder that can lead to respiratory disturbances featuring prolonged apnoeas of variable durations. Determining the mechanisms underlying these effects at the level of respiratory neural circuits would have significant implications for treatment efforts and would also enhance our understanding of respiratory rhythm generation and control. While experimental studies have suggested possible factors contributing to the respiratory patterns of RTT, we take a novel computational approach to the investigation of RTT, which allows for direct manipulation of selected system parameters and testing of specific hypotheses. Specifically, we present three reduced computational models, developed using an established framework, all of which successfully simulate respiratory outputs across eupnoeic and vagotomized conditions. All three models show that loss of inhibition to the Kölliker-Fuse nucleus reproduces the key respiratory alterations associated with RTT and, as suggested experimentally, that effects of 5-HT_1A agonists on the respiratory neural circuit suffice to alleviate this respiratory pathology. Each of the models makes distinct predictions regarding the neuronal populations and interactions underlying these effects, suggesting natural directions for future experimental testing.

Interaction between the retrotrapezoid nucleus and the parafacial respiratory group to regulate active expiration and sympathetic activity in rats

The retrotrapezoid nucleus (RTN) contains chemosensitive cells that distribute CO₂-dependent excitatory drive to the respiratory network. This drive facilitates the function of the respiratory central pattern generator (rCPG) and increases sympathetic activity. It is also evidenced that during hypercapnia, the late-expiratory (late-E) oscillator in the parafacial respiratory group (pFRG) is activated and determines the emergence of active expiration. However, it remains unclear the microcircuitry responsible for the distribution of the excitatory signals to the pFRG and the rCPG in conditions of high CO₂. Herein, we hypothesized that excitatory inputs from chemosensitive neurons in the RTN are necessary for the activation of late-E neurons in the pFRG. Using the decerebrated in situ rat preparation, we found that lesions of neurokinin-1 receptor-expressing neurons in the RTN region with substance P-saporin conjugate suppressed the late-E activity in abdominal nerves (AbNs) and sympathetic nerves (SNs) and attenuated the increase in phrenic nerve (PN) activity induced by hypercapnia. On the other hand, kynurenic acid (100 mM) injections in the pFRG eliminated the late-E activity in AbN and thoracic SN but did not modify PN response during hypercapnia. Iontophoretic injections of retrograde tracer into the pFRG of adult rats revealed labeled phox2b-expressing neurons within the RTN. Our findings are supported by mathematical modeling of chemosensitive and late-E populations within the RTN and pFRG regions as two separate but interacting populations in a way that the activation of the pFRG late-E neurons during hypercapnia require glutamatergic inputs from the RTN neurons that intrinsically detect changes in CO₂/pH.

The parafacial respiratory group and the control of active expiration

Breathing at rest is typically characterized by three phases: active inspiration, post-inspiration (or stage 1 expiration), and passive expiration (or stage 2 expiration). Breathing during periods of increased respiratory demand, on the other hand, engages active expiration through recruitment of abdominal muscles in order to increase ventilation. It is currently hypothesized that different phases of the respiratory rhythm are driven by three coupled oscillators: the preBötzinger Complex, driving inspiration, the parafacial respiratory group (pFRG), driving active expiration and the post-inspiratory Complex, driving post-inspiration. In this paper we review advances in the understanding of the pFRG and its role in the generation of active expiration across different developmental stages and vigilance states. Recent experiments suggest that the abdominal recruitment varies across development depending on the vigilance state, possibly following the maturation of the network responsible for the generation of active expiration and neuromodulatory systems that influence its activity. The activity of the pFRG is tonically inhibited by GABAergic inputs and strongly recruited by cholinergic systems. However, the sources of these modulatory inputs and the physiological conditions under which these mechanisms are used to recruit active expiration and increase ventilation need further investigation. Some evidence suggests that active expiration during hypercapnia is evoked through disinhibition, while during hypoxia it is elicited through activation of catecholaminergic C1 neurons. Finally, a discussion of experiments indicating that the pFRG is anatomically and functionally distinct from the adjacent and partially overlapping chemosensitive neurons of the retrotrapezoid nucleus is also presented.

Organization of the core respiratory network: Insights from optogenetic and modeling studies

The circuit organization within the mammalian brainstem respiratory network, specifically within and between the pre-Bötzinger (pre-BötC) and Bötzinger (BötC) complexes, and the roles of these circuits in respiratory pattern generation are continuously debated. We address these issues with a combination of optogenetic experiments and modeling studies. We used transgenic mice expressing channelrhodopsin-2 under the VGAT-promoter to investigate perturbations of respiratory circuit activity by site-specific photostimulation of inhibitory neurons within the pre-BötC or BötC. The stimulation effects were dependent on the intensity and phase of the photostimulation. Specifically: (1) Low intensity (≤ 1.0 mW) pulses delivered to the pre-BötC during inspiration did not terminate activity, whereas stronger stimulations (≥ 2.0 mW) terminated inspiration. (2) When the pre-BötC stimulation ended in or was applied during expiration, rebound activation of inspiration occurred after a fixed latency. (3) Relatively weak sustained stimulation (20 Hz, 0.5-2.0 mW) of pre-BötC inhibitory neurons increased respiratory frequency, while a further increase of stimulus intensity (> 3.0 mW) reduced frequency and finally (≥ 5.0 mW) terminated respiratory oscillations. (4) Single pulses (0.2-5.0 s) applied to the BötC inhibited rhythmic activity for the duration of the stimulation. (5) Sustained stimulation (20 Hz, 0.5-3.0 mW) of the BötC reduced respiratory frequency and finally led to apnea. We have revised our computational model of pre-BötC and BötC microcircuits by incorporating an additional population of post-inspiratory inhibitory neurons in the pre-BötC that interacts with other neurons in the network. This model was able to reproduce the above experimental findings as well as previously published results of optogenetic activation of pre-BötC or BötC neurons obtained by other laboratories. The proposed organization of pre-BötC and BötC circuits leads to testable predictions about their specific roles in respiratory pattern generation and provides important insights into key circuit interactions operating within brainstem respiratory networks.

Theoretical perspectives on central chemosensitivity: CO2/H+-sensitive neurons in the locus coeruleus

Central chemoreceptors are highly sensitive neurons that respond to changes in pH and CO₂ levels. An increase in CO₂/H⁺ typically reflects a rise in the firing rate of these neurons, which stimulates an increase in ventilation. Here, we present an ionic current model that reproduces the basic electrophysiological activity of individual CO₂/H⁺-sensitive neurons from the locus coeruleus (LC). We used this model to explore chemoreceptor discharge patterns in response to electrical and chemical stimuli. The modeled neurons showed both stimulus-evoked activity and spontaneous activity under physiological parameters. Neuronal responses to electrical and chemical stimulation showed specific firing patterns of spike frequency adaptation, postinhibitory rebound, and post-stimulation recovery. Conversely, the response to chemical stimulation alone (based on physiological CO₂/H⁺ changes), in the absence of external depolarizing stimulation, showed no signs of postinhibitory rebound or post-stimulation recovery, and no depolarizing sag. A sensitivity analysis for the firing-rate response to the different stimuli revealed that the contribution of an applied stimulus current exceeded that of the chemical signals. The firing-rate response increased indefinitely with injected depolarizing current, but reached saturation with chemical stimuli. Our computational model reproduced the regular pacemaker-like spiking pattern, action potential shape, and most of the membrane properties that characterize CO₂/H⁺-sensitive neurons from the locus coeruleus. This validates the model and highlights its potential as a tool for studying the cellular mechanisms underlying the altered central chemosensitivity present in a variety of disorders such as sudden infant death syndrome, depression, and anxiety. In addition, the model results suggest that small external electrical signals play a greater role in determining the chemosensitive response to changes in CO₂/H⁺ than previously thought. This highlights the importance of considering electrical synaptic transmission in studies of intrinsic chemosensitivity.

The sensory mechanism by which changes in CO₂ and H⁺ levels are detected in the brain is known as central chemoreception. Altered chemoreception is common to a wide variety of clinical conditions, including sleep apnea, sudden infant death syndrome, hyperventilation, depression, anxiety and asthma. In addition, CO₂/H⁺-sensitive neurons are present in some regions of the brain that have been identified as drug targets for the treatment of anxiety and panic disorders. We are interested in understanding the cellular mechanisms that determine and modulate the behavior of these neurons. We previously investigated possible mechanisms underlying their behavior in rats to elucidate whether they respond to changes in intracellular or extracellular pH, CO₂, or a combination of these stimuli. To study the roles that signals and ion channel targets play in individual neurons we develop mathematical models that simulate their electrochemical behavior and their responses to hypercapnic and/or acidotic stimuli. Nowadays, we are focused on using computational tools to explore the firing pattern of such neurons in response to chemical (CO₂/H⁺) and electrical (synaptic) stimulation. Our results reveal significant effects of electrical stimulation on the responses of brainstem neurons and highlight the importance of considering synaptic transmission in experimental studies of chemosensitivity.

The Kölliker-Fuse nucleus orchestrates the timing of expiratory abdominal nerve bursting

Coordination of respiratory pump and valve muscle activity is essential for normal breathing. A hallmark respiratory response to hypercapnia and hypoxia is the emergence of active exhalation, characterized by abdominal muscle pumping during the late one-third of expiration (late-E phase). Late-E abdominal activity during hypercapnia has been attributed to the activation of expiratory neurons located within the parafacial respiratory group (pFRG). However, the mechanisms that control emergence of active exhalation, and its silencing in restful breathing, are not completely understood. We hypothesized that inputs from the Kölliker-Fuse nucleus (KF) control the emergence of late-E activity during hypercapnia. Previously, we reported that reversible inhibition of the KF reduced postinspiratory (post-I) motor output to laryngeal adductor muscles and brought forward the onset of hypercapnia-induced late-E abdominal activity. Here we explored the contribution of the KF for late-E abdominal recruitment during hypercapnia by pharmacologically disinhibiting the KF in in situ decerebrate arterially perfused rat preparations. These data were combined with previous results and incorporated into a computational model of the respiratory central pattern generator. Disinhibition of the KF through local parenchymal microinjections of gabazine (GABA_A receptor antagonist) prolonged vagal post-I activity and inhibited late-E abdominal output during hypercapnia. In silico, we reproduced this behavior and predicted a mechanism in which the KF provides excitatory drive to post-I inhibitory neurons, which in turn inhibit late-E neurons of the pFRG. Although the exact mechanism proposed by the model requires testing, our data confirm that the KF modulates the formation of late-E abdominal activity during hypercapnia. NEW & NOTEWORTHY The pons is essential for the formation of the three-phase respiratory pattern, controlling the inspiratory-expiratory phase transition. We provide functional evidence of a novel role for the Kölliker-Fuse nucleus (KF) controlling the emergence of abdominal expiratory bursts during active expiration. A computational model of the respiratory central pattern generator predicts a possible mechanism by which the KF interacts indirectly with the parafacial respiratory group and exerts an inhibitory effect on the expiratory conditional oscillator.

Different roles for inhibition in the rhythm-generating respiratory network

Unraveling the interplay of excitation and inhibition within rhythm-generating networks remains a fundamental issue in neuroscience. We use a biophysical model to investigate the different roles of local and long-range inhibition in the respiratory network, a key component of which is the pre-Bötzinger complex inspiratory microcircuit. Increasing inhibition within the microcircuit results in a limited number of out-of-phase neurons before rhythmicity and synchrony degenerate. Thus unstructured local inhibition is destabilizing and cannot support the generation of more than one rhythm. A two-phase rhythm requires restructuring the network into two microcircuits coupled by long-range inhibition in the manner of a half-center. In this context, inhibition leads to greater stability of the two out-of-phase rhythms. We support our computational results with in vitro recordings from mouse pre-Bötzinger complex. Partial excitation block leads to increased rhythmic variability, but this recovers after blockade of inhibition. Our results support the idea that local inhibition in the pre-Bötzinger complex is present to allow for descending control of synchrony or robustness to adverse conditions like hypoxia. We conclude that the balance of inhibition and excitation determines the stability of rhythmogenesis, but with opposite roles within and between areas. These different inhibitory roles may apply to a variety of rhythmic behaviors that emerge in widespread pattern-generating circuits of the nervous system.NEW & NOTEWORTHY The roles of inhibition within the pre-Bötzinger complex (preBötC) are a matter of debate. Using a combination of modeling and experiment, we demonstrate that inhibition affects synchrony, period variability, and overall frequency of the preBötC and coupled rhythmogenic networks. This work expands our understanding of ubiquitous motor and cognitive oscillatory networks.

Computational study on neuronal activities arising in the pre-Bötzinger complex

Experimental investigations have shown that the pre-Bötzinger complex (pre-BötC) within the mammalian brainstem generates the inspiratory phase of respiratory rhythm. Based on a single-compartment model of a pre-BötC inspiratory neuron, we, in this paper, use semi-analytical, numerical as well as fast-slow dynamical methods to investigate the effects of sodium conductance ([Formula: see text]) and potassium conductance ([Formula: see text]) on the firing activities of pre-BötC and try to reveal the dynamical mechanisms behind them. We show how [Formula: see text] and [Formula: see text] affect the bifurcations of the fast-subsystem and how the the firing patterns of pre-BötC transit according to the bifurcations.

The Kölliker-Fuse nucleus acts as a timekeeper for late-expiratory abdominal activity

While the transition from the inspiratory to the post-inspiratory (post-I) phase is dependent on the pons, little attention has been paid to understanding the role of the pontine respiratory nuclei, specifically the Kölliker-Fuse nucleus (KF), in transitioning from post-I to the late expiratory (late-E) activity seen with elevated respiratory drive. To elucidate this, we used the in situ working heart-brainstem preparation of juvenile male Holtzman rats and recorded from the vagus (cVN), phrenic (PN) and abdominal nerves (AbN) during baseline conditions and during chemoreflex activation [with potassium cyanide (KCN; n=13) or hypercapnia (8% CO2; n=10)] to recruit active expiration. Chemoreflex activation with KCN increased PN frequency and cVN post-I and AbN activities. The inhibition of KF with isoguvacine microinjections (10mM) attenuated the typical increase in PN frequency and cVN post-I activity, and amplified the AbN response. During hypercapnia, AbN late-E activity emerged in association with a significant reduction in expiratory time. KF inhibition during hypercapnia significantly decreased PN frequency and reduced the duration and amplitude of post-I cVN activity, while the onset of the AbN late-E bursts occurred significantly earlier. Our data reveal a negative relationship between KF-induced post-I and AbN late-E activities, suggesting that the KF coordinates the transition between post-I to late-E activity during conditions of elevated respiratory drive.

Computational models of the neural control of breathing

The ongoing process of breathing underlies the gas exchange essential for mammalian life. Each respiratory cycle ensues from the activity of rhythmic neural circuits in the brainstem, shaped by various modulatory signals, including mechanoreceptor feedback sensitive to lung inflation and chemoreceptor feedback dependent on gas composition in blood and tissues. This paper reviews a variety of computational models designed to reproduce experimental findings related to the neural control of breathing and generate predictions for future experimental testing. The review starts from the description of the core respiratory network in the brainstem, representing the central pattern generator (CPG) responsible for producing rhythmic respiratory activity, and progresses to encompass additional complexities needed to simulate different metabolic challenges, closed-loop feedback control including the lungs, and interactions between the respiratory and autonomic nervous systems. The integrated models considered in this review share a common framework including a distributed CPG core network responsible for generating the baseline three-phase pattern of rhythmic neural activity underlying normal breathing. WIREs Syst Biol Med 2017, 9:e1371. doi: 10.1002/wsbm.1371 For further resources related to this article, please visit the WIREs website.

Coordinated Respiratory Motor Activity in Nerves Innervating the Upper Airway Muscles in Rats

Maintaining the patency of the upper airway during breathing is of vital importance. The activity of various muscles is related to the patency of the upper airway. In the present study, we examined the respiratory motor activity in the efferent nerves innervating the upper airway muscles to determine the movements of the upper airway during respiration under normocapnic conditions (pH = 7.4) and in hypercapnic acidosis (pH = 7.2). Experiments were performed on arterially perfused decerebrate rats aged between postnatal days 21–35. We recorded the efferent nerve activity in a branch of the cervical spinal nerve innervating the infrahyoid muscles (CN), the hypoglossal nerve (HGN), the external branch of the superior laryngeal nerve (SLN), and the recurrent laryngeal nerve (RLN) with the phrenic nerve (PN). Inspiratory nerve discharges were observed in all these nerves under normocapnic conditions. The onset of inspiratory discharges in the CN and HGN was slightly prior to those in the SLN and RLN. When the CO₂ concentration in the perfusate was increased from 5% to 8% to prepare for hypercapnic acidosis, the peak amplitudes of the inspiratory discharges in all the recorded nerves were increased. Moreover, hypercapnic acidosis induced pre-inspiratory discharges in the CN, HGN, SLN, and RLN. The onset of pre-inspiratory discharges in the CN, HGN, and SLN was prior to that of discharges in the RLN. These results suggest that the securing of the airway that occurs a certain time before dilation of the glottis may facilitate ventilation and improve hypercapnic acidosis.

Central control of interlimb coordination and speed-dependent gait expression in quadrupeds

KEY POINTS

Quadrupeds express different gaits depending on speed of locomotion. Central pattern generators (one per limb) within the spinal cord generate locomotor oscillations and control limb movements. Neural interactions between these generators define interlimb coordination and gait. We present a computational model of spinal circuits representing four rhythm generators with left-right excitatory and inhibitory commissural and fore-hind inhibitory interactions within the cord. Increasing brainstem drive to all rhythm generators and excitatory commissural interneurons induces an increasing frequency of locomotor oscillations accompanied by speed-dependent gait changes from walk to trot and to gallop and bound. The model closely reproduces and suggests explanations for multiple experimental data, including speed-dependent gait transitions in intact mice and changes in gait expression in mutants lacking certain types of commissural interneurons. The model suggests the possible circuit organization in the spinal cord and proposes predictions that can be tested experimentally.

ABSTRACT

As speed of locomotion is increasing, most quadrupeds, including mice, demonstrate sequential gait transitions from walk to trot and to gallop and bound. The neural mechanisms underlying these transitions are poorly understood. We propose that the speed-dependent expression of different gaits results from speed-dependent changes in the interactions between spinal circuits controlling different limbs and interlimb coordination. As a result, the expression of each gait depends on (1) left-right interactions within the spinal cord mediated by different commissural interneurons (CINs), (2) fore-hind interactions on each side of the spinal cord and (3) brainstem drives to rhythm-generating circuits and CIN pathways. We developed a computational model of spinal circuits consisting of four rhythm generators (RGs) with bilateral left-right interactions mediated by V0 CINs (V0_D and V0_V sub-types) providing left-right alternation, and conditional V3 CINs promoting left-right synchronization. Fore and hind RGs mutually inhibited each other. We demonstrate that linearly increasing excitatory drives to the RGs and V3 CINs can produce a progressive increase in the locomotor speed accompanied by sequential changes of gaits from walk to trot and to gallop and bound. The model closely reproduces and suggests explanations for the speed-dependent gait expression observed in vivo in intact mice and in mutants lacking V0_V or all V0 CINs. Specifically, trot is not expressed after removal of V0_V CINs, and only bound is expressed after removal of all V0 CINs. The model provides important insights into the organization of spinal circuits and neural control of locomotion.

Effects of ion channel noise on neural circuits: an application to the respiratory pattern generator to investigate breathing variability

Neural activity generally displays irregular firing patterns even in circuits with apparently regular outputs, such as motor pattern generators, in which the output frequency fluctuates randomly around a mean value. This "circuit noise" is inherited from the random firing of single neurons, which emerges from stochastic ion channel gating (channel noise), spontaneous neurotransmitter release, and its diffusion and binding to synaptic receptors. Here we demonstrate how to expand conductance-based network models that are originally deterministic to include realistic, physiological noise, focusing on stochastic ion channel gating. We illustrate this procedure with a well-established conductance-based model of the respiratory pattern generator, which allows us to investigate how channel noise affects neural dynamics at the circuit level and, in particular, to understand the relationship between the respiratory pattern and its breath-to-breath variability. We show that as the channel number increases, the duration of inspiration and expiration varies, and so does the coefficient of variation of the breath-to-breath interval, which attains a minimum when the mean duration of expiration slightly exceeds that of inspiration. For small channel numbers, the variability of the expiratory phase dominates over that of the inspiratory phase, and vice versa for large channel numbers. Among the four different cell types in the respiratory pattern generator, pacemaker cells exhibit the highest sensitivity to channel noise. The model shows that suppressing input from the pons leads to longer inspiratory phases, a reduction in breathing frequency, and larger breath-to-breath variability, whereas enhanced input from the raphe nucleus increases breathing frequency without changing its pattern.

NEW & NOTEWORTHY

A major source of noise in neuronal circuits is the "flickering" of ion currents passing through the neurons' membranes (channel noise), which cannot be suppressed experimentally. Computational simulations are therefore the best way to investigate the effects of this physiological noise by manipulating its level at will. We investigate the role of noise in the respiratory pattern generator and show that endogenous, breath-to-breath variability is tightly linked to the respiratory pattern.

Chemoreception and neuroplasticity in respiratory circuits

The respiratory central pattern generator must respond to chemosensory cues to maintain oxygen (O₂) and carbon dioxide (CO₂) homeostasis in the blood and tissues. To do this, sensorial cells located in the periphery and central nervous system monitor the arterial partial pressure of O₂ and CO₂ and initiate respiratory and autonomic reflex adjustments in conditions of hypoxia and hypercapnia. In conditions of chronic intermittent hypoxia (CIH), repeated peripheral chemoreceptor input mediated by the nucleus of the solitary tract induces plastic changes in respiratory circuits that alter baseline respiratory and sympathetic motor outputs and result in chemoreflex sensitization, active expiration, and arterial hypertension. Herein, we explored the hypothesis that the CIH-induced neuroplasticity primarily consists of increased excitability of pre-inspiratory/inspiratory neurons in the pre-Bötzinger complex. To evaluate this hypothesis and elucidate neural mechanisms for the emergence of active expiration and sympathetic overactivity in CIH-treated animals, we extended a previously developed computational model of the brainstem respiratory-sympathetic network to reproduce experimental data on peripheral and central chemoreflexes post-CIH. The model incorporated neuronal connections between the 2nd-order NTS neurons and peripheral chemoreceptors afferents, the respiratory pattern generator, and sympathetic neurons in the rostral ventrolateral medulla in order to capture key features of sympathetic and respiratory responses to peripheral chemoreflex stimulation. Our model identifies the potential neuronal groups recruited during peripheral chemoreflex stimulation that may be required for the development of inspiratory, expiratory and sympathetic reflex responses. Moreover, our model predicts that pre-inspiratory neurons in the pre-Bötzinger complex experience plasticity of channel expression due to excessive excitation during peripheral chemoreflex. Simulations also show that, due to positive interactions between pre-inspiratory neurons in the pre-Bötzinger complex and expiratory neurons in the retrotrapezoid nucleus, increased excitability of the former may lead to the emergence of the active expiratory pattern at normal CO₂ levels found after CIH exposure. We conclude that neuronal type specific neuroplasticity in the pre-Bötzinger complex induced by repetitive episodes of peripheral chemoreceptor activation by hypoxia may contribute to the development of sympathetic over-activity and hypertension.

Mixed-mode oscillations and population bursting in the pre-Bötzinger complex

This study focuses on computational and theoretical investigations of neuronal activity arising in the pre-Bötzinger complex (pre-BötC), a medullary region generating the inspiratory phase of breathing in mammals. A progressive increase of neuronal excitability in medullary slices containing the pre-BötC produces mixed-mode oscillations (MMOs) characterized by large amplitude population bursts alternating with a series of small amplitude bursts. Using two different computational models, we demonstrate that MMOs emerge within a heterogeneous excitatory neural network because of progressive neuronal recruitment and synchronization. The MMO pattern depends on the distributed neuronal excitability, the density and weights of network interconnections, and the cellular properties underlying endogenous bursting. Critically, the latter should provide a reduction of spiking frequency within neuronal bursts with increasing burst frequency and a dependence of the after-burst recovery period on burst amplitude. Our study highlights a novel mechanism by which heterogeneity naturally leads to complex dynamics in rhythmic neuronal populations.

DOI:http://dx.doi.org/10.7554/eLife.13403.001

Each breath we take removes carbon dioxide from the body and exchanges it for oxygen. A structure called the brainstem, which connects the brain with the spinal cord, generates the breathing rhythm and controls its rate. While this process normally occurs automatically, we can also control our breathing voluntarily, such as when singing or speaking.

Within the brainstem, a group of neurons in the area known as the pre-Bötzinger complex is responsible for ensuring that an animal breathes in at regular intervals. Recordings of the electrical activity from slices of brainstem show that pre-Bötzinger neurons display rhythmic activity with characteristic patterns called “mixed-mode oscillations”. These rhythms consist of bursts of strong activity (“large amplitude bursts”), essential for triggering regular breathing, separated by a series of bursts of weak activity (“small amplitude bursts”). However, it is not clear how mixed-mode oscillations arise.

Bacak, Kim et al. now provide insights into this process by developing two computational models of the pre-Bötzinger complex. The first model consists of a diverse population of 100 neurons joined by a relatively small number of weak connections to form a network. The second model is a simplified version of the first, consisting of just three neurons. By manipulating the properties of the simulated networks, and analysing the data mathematically, Bacak, Kim et al. identify the properties of the neurons that allow them to generate mixed-mode oscillations and thus rhythmic breathing.

The models suggest that mixed-mode oscillations result from the synchronization of many neurons with different levels of activity (excitability). Neurons with low excitability have low bursting frequencies, but generate strong activity and recruit other neurons, ultimately producing large amplitude bursts that cause breathing.

Many parts of the nervous system are also made up of networks of neurons with diverse excitability. A challenge for future studies is thus to investigate whether other networks of neurons similar to the pre-Bötzinger complex generate rhythms that control other repetitive actions, such as walking and chewing.

DOI:http://dx.doi.org/10.7554/eLife.13403.002

Mechanisms of left-right coordination in mammalian locomotor pattern generation circuits: a mathematical modeling view

The locomotor gait in limbed animals is defined by the left-right leg coordination and locomotor speed. Coordination between left and right neural activities in the spinal cord controlling left and right legs is provided by commissural interneurons (CINs). Several CIN types have been genetically identified, including the excitatory V3 and excitatory and inhibitory V0 types. Recent studies demonstrated that genetic elimination of all V0 CINs caused switching from a normal left-right alternating activity to a left-right synchronized “hopping” pattern. Furthermore, ablation of only the inhibitory V0 CINs (V0_D subtype) resulted in a lack of left-right alternation at low locomotor frequencies and retaining this alternation at high frequencies, whereas selective ablation of the excitatory V0 neurons (V0_V subtype) maintained the left–right alternation at low frequencies and switched to a hopping pattern at high frequencies. To analyze these findings, we developed a simplified mathematical model of neural circuits consisting of four pacemaker neurons representing left and right, flexor and extensor rhythm-generating centers interacting via commissural pathways representing V3, V0_D, and V0_V CINs. The locomotor frequency was controlled by a parameter defining the excitation of neurons and commissural pathways mimicking the effects of N-methyl-D-aspartate on locomotor frequency in isolated rodent spinal cord preparations. The model demonstrated a typical left-right alternating pattern under control conditions, switching to a hopping activity at any frequency after removing both V0 connections, a synchronized pattern at low frequencies with alternation at high frequencies after removing only V0_D connections, and an alternating pattern at low frequencies with hopping at high frequencies after removing only V0_V connections. We used bifurcation theory and fast-slow decomposition methods to analyze network behavior in the above regimes and transitions between them. The model reproduced, and suggested explanation for, a series of experimental phenomena and generated predictions available for experimental testing.

Movements of left and right limbs in mammals during locomotion are controlled by distinct rhythm-generating neuronal circuits in the spinal cord. Complex interactions between these circuits provide flexible coordination of limb movements in different gaits. It was shown that interactions between left and right spinal circuits are mediated by commissural interneurons. Genetic ablation of a particular type of these interneurons, called V0, leads to switching from a regular, left-right alternating “walking” activity to a left-right synchronous “hopping” pattern. Moreover, the V0 commissural interneurons have excitatory and inhibitory subtypes that appear to play different roles in the left-right coordination depending on locomotor speed. In this theoretical study, we build a simplified mathematical model of spinal circuits that describes left and right rhythm generators interacting bilaterally via several types of commissural connections. Using this model, we simulate different experimental manipulations, analyze the resultant alternating and synchronous regimes of activity, and propose explanations for the results of experimental studies. We show that although both excitatory and inhibitory V0 commissural pathways support left-right alternation, the resultant locomotor pattern and gait depend on the balance between different commissural interactions, which in turn may depend on the level of neuronal excitation and locomotor speed.

A closed-loop model of the respiratory system: focus on hypercapnia and active expiration

Breathing is a vital process providing the exchange of gases between the lungs and atmosphere. During quiet breathing, pumping air from the lungs is mostly performed by contraction of the diaphragm during inspiration, and muscle contraction during expiration does not play a significant role in ventilation. In contrast, during intense exercise or severe hypercapnia forced or active expiration occurs in which the abdominal “expiratory” muscles become actively involved in breathing. The mechanisms of this transition remain unknown. To study these mechanisms, we developed a computational model of the closed-loop respiratory system that describes the brainstem respiratory network controlling the pulmonary subsystem representing lung biomechanics and gas (O₂ and CO₂) exchange and transport. The lung subsystem provides two types of feedback to the neural subsystem: a mechanical one from pulmonary stretch receptors and a chemical one from central chemoreceptors. The neural component of the model simulates the respiratory network that includes several interacting respiratory neuron types within the Bötzinger and pre-Bötzinger complexes, as well as the retrotrapezoid nucleus/parafacial respiratory group (RTN/pFRG) representing the central chemoreception module targeted by chemical feedback. The RTN/pFRG compartment contains an independent neural generator that is activated at an increased CO₂ level and controls the abdominal motor output. The lung volume is controlled by two pumps, a major one driven by the diaphragm and an additional one activated by abdominal muscles and involved in active expiration. The model represents the first attempt to model the transition from quiet breathing to breathing with active expiration. The model suggests that the closed-loop respiratory control system switches to active expiration via a quantal acceleration of expiratory activity, when increases in breathing rate and phrenic amplitude no longer provide sufficient ventilation. The model can be used for simulation of closed-loop control of breathing under different conditions including respiratory disorders.

Respiratory rhythm generation in vivo

The cellular and circuit mechanisms generating the rhythm of breathing in mammals have been under intense investigation for decades. Here, we try to integrate the key discoveries into an updated description of the basic neural processes generating respiratory rhythm under in vivo conditions.

The cellular building blocks of breathing

Respiratory brainstem neurons fulfill critical roles in controlling breathing: they generate the activity patterns for breathing and contribute to various sensory responses including changes in O2 and CO2. These complex sensorimotor tasks depend on the dynamic interplay between numerous cellular building blocks that consist of voltage-, calcium-, and ATP-dependent ionic conductances, various ionotropic and metabotropic synaptic mechanisms, as well as neuromodulators acting on G-protein coupled receptors and second messenger systems. As described in this review, the sensorimotor responses of the respiratory network emerge through the state-dependent integration of all these building blocks. There is no known respiratory function that involves only a small number of intrinsic, synaptic, or modulatory properties. Because of the complex integration of numerous intrinsic, synaptic, and modulatory mechanisms, the respiratory network is capable of continuously adapting to changes in the external and internal environment, which makes breathing one of the most integrated behaviors. Not surprisingly, inspiration is critical not only in the control of ventilation, but also in the context of "inspiring behaviors" such as arousal of the mind and even creativity. Far-reaching implications apply also to the underlying network mechanisms, as lessons learned from the respiratory network apply to network functions in general.

Integrative approaches for modeling regulation and function of the respiratory system

Mathematical models have been central to understanding the interaction between neural control and breathing. Models of the entire respiratory system-which comprises the lungs and the neural circuitry that controls their ventilation-have been derived using simplifying assumptions to compartmentalize each component of the system and to define the interactions between components. These full system models often rely-through necessity-on empirically derived relationships or parameters, in addition to physiological values. In parallel with the development of whole respiratory system models are mathematical models that focus on furthering a detailed understanding of the neural control network, or of the several functions that contribute to gas exchange within the lung. These models are biophysically based, and rely on physiological parameters. They include single-unit models for a breathing lung or neural circuit, through to spatially distributed models of ventilation and perfusion, or multicircuit models for neural control. The challenge is to bring together these more recent advances in models of neural control with models of lung function, into a full simulation for the respiratory system that builds upon the more detailed models but remains computationally tractable. This requires first understanding the mathematical models that have been developed for the respiratory system at different levels, and which could be used to study how physiological levels of O2 and CO2 in the blood are maintained.

Spontaneous autoresuscitation in a model of respiratory control

We introduce a closed-loop model of respiratory control incorporating a conductance-based central pattern generator (CPG), low-pass filtering of CPG output by the respiratory musculature, gas exchange in the lung, metabolic oxygen demand, and chemosensation. The CPG incorporates Butera, Rinzel and Smith (BRS)'s (1999) conditional pacemaker model. BRS model cells can support quiescent, bursting, or beating activity depending on the level of excitatory drive; we identify these activity modes with apnea (cessation of breathing), eupnea (normal breathing), and tachypnea (excessively rapid breathing). We demonstrate the coexistence of two dynamically stable behaviors in the closed-loop model, corresponding respectively to eupnea and tachypnea. The latter state represents a novel failure mode within a respiratory control model. In addition, the closed-loop system exhibits a form of autoresuscitation: conductances intrinsic to the BRS model buffer the CPG against brief episodes of hypoxia, steering the system away from catastrophic collapse as can occur with tachypnea.

Brainstem respiratory networks: building blocks and microcircuits

Breathing movements in mammals are driven by rhythmic neural activity generated within spatially and functionally organized brainstem neural circuits comprising the respiratory central pattern generator (CPG). This rhythmic activity provides homeostatic regulation of gases in blood and tissues and integrates breathing with other motor acts. We review new insights into the spatial-functional organization of key neural microcircuits of this CPG from recent multidisciplinary experimental and computational studies. The emerging view is that the microcircuit organization within the CPG allows the generation of multiple rhythmic breathing patterns and adaptive switching between them, depending on physiological or pathophysiological conditions. These insights open the possibility for site- and mechanism-specific interventions to treat various disorders of the neural control of breathing.

Computational models and emergent properties of respiratory neural networks

Computational models of the neural control system for breathing in mammals provide a theoretical and computational framework bringing together experimental data obtained from different animal preparations under various experimental conditions. Many of these models were developed in parallel and iteratively with experimental studies and provided predictions guiding new experiments. This data-driven modeling approach has advanced our understanding of respiratory network architecture and neural mechanisms underlying generation of the respiratory rhythm and pattern, including their functional reorganization under different physiological conditions. Models reviewed here vary in neurobiological details and computational complexity and span multiple spatiotemporal scales of respiratory control mechanisms. Recent models describe interacting populations of respiratory neurons spatially distributed within the Bötzinger and pre-Bötzinger complexes and rostral ventrolateral medulla that contain core circuits of the respiratory central pattern generator (CPG). Network interactions within these circuits along with intrinsic rhythmogenic properties of neurons form a hierarchy of multiple rhythm generation mechanisms. The functional expression of these mechanisms is controlled by input drives from other brainstem components,including the retrotrapezoid nucleus and pons, which regulate the dynamic behavior of the core circuitry. The emerging view is that the brainstem respiratory network has rhythmogenic capabilities at multiple levels of circuit organization. This allows flexible, state-dependent expression of different neural pattern-generation mechanisms under various physiological conditions,enabling a wide repertoire of respiratory behaviors. Some models consider control of the respiratory CPG by pulmonary feedback and network reconfiguration during defensive behaviors such as cough. Future directions in modeling of the respiratory CPG are considered.

Explicit maps to predict activation order in multiphase rhythms of a coupled cell network

We present a novel extension of fast-slow analysis of clustered solutions to coupled networks of three cells, allowing for heterogeneity in the cells' intrinsic dynamics. In the model on which we focus, each cell is described by a pair of first-order differential equations, which are based on recent reduced neuronal network models for respiratory rhythmogenesis. Within each pair of equations, one dependent variable evolves on a fast time scale and one on a slow scale. The cells are coupled with inhibitory synapses that turn on and off on the fast time scale. In this context, we analyze solutions in which cells take turns activating, allowing any activation order, including multiple activations of two of the cells between successive activations of the third. Our analysis proceeds via the derivation of a set of explicit maps between the pairs of slow variables corresponding to the non-active cells on each cycle. We show how these maps can be used to determine the order in which cells will activate for a given initial condition and how evaluation of these maps on a few key curves in their domains can be used to constrain the possible activation orders that will be observed in network solutions. Moreover, under a small set of additional simplifying assumptions, we collapse the collection of maps into a single 2D map that can be computed explicitly. From this unified map, we analytically obtain boundary curves between all regions of initial conditions producing different activation patterns.

Small is beautiful: models of small neuronal networks

Modeling has contributed a great deal to our understanding of how individual neurons and neuronal networks function. In this review, we focus on models of the small neuronal networks of invertebrates, especially rhythmically active CPG networks. Models have elucidated many aspects of these networks, from identifying key interacting membrane properties to pointing out gaps in our understanding, for example missing neurons. Even the complex CPGs of vertebrates, such as those that underlie respiration, have been reduced to small network models to great effect. Modeling of these networks spans from simplified models, which are amenable to mathematical analyses, to very complicated biophysical models. Some researchers have now adopted a population approach, where they generate and analyze many related models that differ in a few to several judiciously chosen free parameters; often these parameters show variability across animals and thus justify the approach. Models of small neuronal networks will continue to expand and refine our understanding of how neuronal networks in all animals program motor output, process sensory information and learn.

Computational modelling of 5-HT receptor-mediated reorganization of the brainstem respiratory network

Brainstem respiratory neurons express the glycine α(3) receptor (Glyα(3) R), which is a target of modulation by several serotonin (5-HT) receptor agonists. Application of the 5-HT(1A) receptor (5-HT(1A) R) agonist 8-OH-DPAT was shown (i) to depress cellular cAMP, leading to dephosphorylation of Glyα(3) R and augmentation of postsynaptic inhibition of neurons expressing Glyα(3) R (Manzke et al., 2010) and (ii) to hyperpolarize respiratory neurons through 5-HT-activated potassium channels. These processes counteract opioid-induced depression and restore breathing from apnoeas often accompanying pharmacotherapy of pain. The effect is postulated to rely on the enhanced Glyα(3) R-mediated inhibition of inhibitory neurons causing disinhibition of their target neurons. To evaluate this proposal and investigate the neural mechanisms involved, an established computational model of the brainstem respiratory network (Smith et al., 2007), was extended by (i) incorporating distinct subpopulations of inhibitory neurons (glycinergic and GABAergic) and their synaptic interconnections within the Bötzinger and pre-Bötzinger complexes and (ii) assigning the 5-HT(1A) R-Glyα(3) R complex to some of these inhibitory neuron types in the network. The modified model was used to simulate the effects of 8-OH-DPAT on the respiratory pattern and was able to realistically reproduce a number of experimentally observed responses, including the shift in the onset of post-inspiratory activity to inspiration and conversion of the eupnoeic three-phase rhythmic pattern into a two-phase pattern lacking the post-inspiratory phase. The model shows how 5-HT(1A) R activation can produce a disinhibition of inspiratory neurons, leading to the recovery of respiratory rhythm from opioid-induced apnoeas.

Intermittent hypoxia-induced sensitization of central chemoreceptors contributes to sympathetic nerve activity during late expiration in rats

Hypertension elicited by chronic intermittent hypoxia (CIH) is associated with elevated activity of the thoracic sympathetic nerve (tSN) that exhibits an enhanced respiratory modulation reflecting a strengthened interaction between respiratory and sympathetic networks within the brain stem. Expiration is a passive process except for special metabolic conditions such as hypercapnia, when it becomes active through phasic excitation of abdominal motor nerves (AbN) in late expiration. An increase in CO(2) evokes late-expiratory (late-E) discharges phase-locked to phrenic bursts with the frequency increasing quantally as hypercapnia increases. In rats exposed to CIH, the late-E discharges synchronized in AbN and tSN emerge in normocapnia. To elucidate the possible neural mechanisms underlying these phenomena, we extended our computational model of the brain stem respiratory network by incorporating a population of presympathetic neurons in the rostral ventrolateral medulla that received inputs from the pons, medullary respiratory compartments, and retrotrapezoid nucleus/parafacial respiratory group (RTN/pFRG). Our simulations proposed that CIH conditioning increases the CO(2) sensitivity of RTN/pFRG neurons, causing a reduction in both the CO(2) threshold for emerging the late-E activity in AbN and tSN and the hypocapnic threshold for apnea. Using the in situ rat preparation, we have confirmed that CIH-conditioned rats under normal conditions exhibit synchronized late-E discharges in AbN and tSN similar to those observed in control rats during hypercapnia. Moreover, the hypocapnic threshold for apnea was significantly lowered in CIH-conditioned rats relative to that in control rats. We conclude that CIH may sensitize central chemoreception and that this significantly contributes to the neural impetus for generation of sympathetic activity and hypertension.