Robustness, flexibility, and sensitivity in a multifunctional motor control model

Variational analysis of sensory feedback mechanisms in powerstroke-recovery systems

Although the raison d'etre of the brain is the survival of the body, there are relatively few theoretical studies of closed-loop rhythmic motor control systems. In this paper we provide a unified framework, based on variational analysis, for investigating the dual goals of performance and robustness in powerstroke-recovery systems. To demonstrate our variational method, we augment two previously published closed-loop motor control models by equipping each model with a performance measure based on the rate of progress of the system relative to a spatially extended external substrate-such as a long strip of seaweed for a feeding task, or progress relative to the ground for a locomotor task. The sensitivity measure quantifies the ability of the system to maintain performance in response to external perturbations, such as an applied load. Motivated by a search for optimal design principles for feedback control achieving the complementary requirements of efficiency and robustness, we discuss the performance-sensitivity patterns of the systems featuring different sensory feedback architectures. In a paradigmatic half-center oscillator-motor system, we observe that the excitation-inhibition property of feedback mechanisms determines the sensitivity pattern while the activation-inactivation property determines the performance pattern. Moreover, we show that the nonlinearity of the sigmoid activation of feedback signals allows the existence of optimal combinations of performance and sensitivity. In a detailed hindlimb locomotor system, we find that a force-dependent feedback can simultaneously optimize both performance and robustness, while length-dependent feedback variations result in significant performance-versus-sensitivity tradeoffs. Thus, this work provides an analytical framework for studying feedback control of oscillations in nonlinear dynamical systems, leading to several insights that have the potential to inform the design of control or rehabilitation systems.

A computational neural model that incorporates both intrinsic dynamics and sensory feedback in the Aplysia feeding network

Studying the nervous system underlying animal motor control can shed light on how animals can adapt flexibly to a changing environment. We focus on the neural basis of feeding control in Aplysia californica. Using the Synthetic Nervous System framework, we developed a model of Aplysia feeding neural circuitry that balances neurophysiological plausibility and computational complexity. The circuitry includes neurons, synapses, and feedback pathways identified in existing literature. We organized the neurons into three layers and five subnetworks according to their functional roles. Simulation results demonstrate that the circuitry model can capture the intrinsic dynamics at neuronal and network levels. When combined with a simplified peripheral biomechanical model, it is sufficient to mediate three animal-like feeding behaviors (biting, swallowing, and rejection). The kinematic, dynamic, and neural responses of the model also share similar features with animal data. These results emphasize the functional roles of sensory feedback during feeding.

Network attractors and nonlinear dynamics of neural computation

The importance of understanding the nonlinear dynamics of neural systems, and the relation to cognitive systems more generally, has been recognised for a long time. Approaches that analyse neural systems in terms of attractors of autonomous networks can be successful in explaining system behaviours in the input-free case. Nonetheless, a computational system usually needs inputs from its environment to effectively solve problems, and this necessitates a non-autonomous framework where typically the effects of a changing environment can be studied. In this review, we highlight a variety of network attractors that can exist in autonomous systems and can be used to aid interpretation of the dynamics in the presence of inputs. Such network attractors (that consist of heteroclinic or excitable connections between invariant sets) lend themselves to modelling discrete-state computations with continuous inputs, and can sometimes be thought of as a hybrid model between classical discrete computation and continuous-time dynamical systems. Bibliographic info here.

Tutorial: using NEURON for neuromechanical simulations

The dynamical properties of the brain and the dynamics of the body strongly influence one another. Their interaction generates complex adaptive behavior. While a wide variety of simulation tools exist for neural dynamics or biomechanics separately, there are few options for integrated brain-body modeling. Here, we provide a tutorial to demonstrate how the widely-used NEURON simulation platform can support integrated neuromechanical modeling. As a first step toward incorporating biomechanics into a NEURON simulation, we provide a framework for integrating inputs from a "periphery" and outputs to that periphery. In other words, "body" dynamics are driven in part by "brain" variables, such as voltages or firing rates, and "brain" dynamics are influenced by "body" variables through sensory feedback. To couple the "brain" and "body" components, we use NEURON's pointer construct to share information between "brain" and "body" modules. This approach allows separate specification of brain and body dynamics and code reuse. Though simple in concept, the use of pointers can be challenging due to a complicated syntax and several different programming options. In this paper, we present five different computational models, with increasing levels of complexity, to demonstrate the concepts of code modularity using pointers and the integration of neural and biomechanical modeling within NEURON. The models include: (1) a neuromuscular model of calcium dynamics and muscle force, (2) a neuromechanical, closed-loop model of a half-center oscillator coupled to a rudimentary motor system, (3) a closed-loop model of neural control for respiration, (4) a pedagogical model of a non-smooth "brain/body" system, and (5) a closed-loop model of feeding behavior in the sea hare Aplysia californica that incorporates biologically-motivated non-smooth dynamics. This tutorial illustrates how NEURON can be integrated with a broad range of neuromechanical models.

Code available at

https://github.com/fietkiewicz/PointerBuilder.

Heteroclinic cycling and extinction in May-Leonard models with demographic stochasticity

May and Leonard (SIAM J Appl Math 29:243-253, 1975) introduced a three-species Lotka-Volterra type population model that exhibits heteroclinic cycling. Rather than producing a periodic limit cycle, the trajectory takes longer and longer to complete each "cycle", passing closer and closer to unstable fixed points in which one population dominates and the others approach zero. Aperiodic heteroclinic dynamics have subsequently been studied in ecological systems (side-blotched lizards; colicinogenic Escherichia coli), in the immune system, in neural information processing models ("winnerless competition"), and in models of neural central pattern generators. Yet as May and Leonard observed "Biologically, the behavior (produced by the model) is nonsense. Once it is conceded that the variables represent animals, and therefore cannot fall below unity, it is clear that the system will, after a few cycles, converge on some single population, extinguishing the other two." Here, we explore different ways of introducing discrete stochastic dynamics based on May and Leonard's ODE model, with application to ecological population dynamics, and to a neuromotor central pattern generator system. We study examples of several quantitatively distinct asymptotic behaviors, including total extinction of all species, extinction to a single species, and persistent cyclic dominance with finite mean cycle length.

Variational and phase response analysis for limit cycles with hard boundaries, with applications to neuromechanical control problems

Motor systems show an overall robustness, but because they are highly nonlinear, understanding how they achieve robustness is difficult. In many rhythmic systems, robustness against perturbations involves response of both the shape and the timing of the trajectory. This makes the study of robustness even more challenging. To understand how a motor system produces robust behaviors in a variable environment, we consider a neuromechanical model of motor patterns in the feeding apparatus of the marine mollusk Aplysia californica (Shaw et al. in J Comput Neurosci 38(1):25-51, 2015; Lyttle et al. in Biol Cybern 111(1):25-47, 2017). We established in (Wang et al. in SIAM J Appl Dyn Syst 20(2):701-744, 2021. https://doi.org/10.1137/20M1344974 ) the tools for studying combined shape and timing responses of limit cycle systems under sustained perturbations and here apply them to study robustness of the neuromechanical model against increased mechanical load during swallowing. Interestingly, we discover that nonlinear biomechanical properties confer resilience by immediately increasing resistance to applied loads. In contrast, the effect of changed sensory feedback signal is significantly delayed by the firing rates' hard boundary properties. Our analysis suggests that sensory feedback contributes to robustness in swallowing primarily by shifting the timing of neural activation involved in the power stroke of the motor cycle (retraction). This effect enables the system to generate stronger retractor muscle forces to compensate for the increased load, and hence achieve strong robustness. The approaches that we are applying to understanding a neuromechanical model in Aplysia, and the results that we have obtained, are likely to provide insights into the function of other motor systems that encounter changing mechanical loads and hard boundaries, both due to mechanical and neuronal firing properties.

New insights from small rhythmic circuits

Small rhythmic circuits, such as those found in invertebrates, have provided fundamental insights into how circuit dynamics depend on individual neuronal and synaptic properties. Degenerate circuits are those with different network parameters and similar behavior. New work on degenerate circuits and their modulation illustrates some of the rules that help maintain stable and robust circuit function despite environmental perturbations. Advances in neuropeptide isolation and identification provide enhanced understanding of the neuromodulation of circuits for behavior. The advent of molecular studies of mRNA expression provides new insight into animal-to-animal variability and the homeostatic regulation of excitability in neurons and networks.

Cooperative stability renders protein complex formation more robust and controllable

Protein complexes are the fundamental units of many biological functions. Despite their many advantages, one major adverse impact of protein complexes is accumulations of unassembled subunits that may disrupt other processes or exert cytotoxic effects. Synthesis of excess subunits can be inhibited via negative feedback control or they can be degraded more efficiently than assembled subunits, with this latter being termed cooperative stability. Whereas controlled synthesis of complex subunits has been investigated extensively, how cooperative stability acts in complex formation remains largely unexplored. To fill this knowledge gap, we have built quantitative models of heteromeric complexes with or without cooperative stability and compared their behaviours in the presence of synthesis rate variations. A system displaying cooperative stability is robust against synthesis rate variations as it retains high dimer/monomer ratios across a broad range of parameter configurations. Moreover, cooperative stability can alleviate the constraint of limited supply of a given subunit and makes complex abundance more responsive to unilateral upregulation of another subunit. We also conducted an in silico experiment to comprehensively characterize and compare four types of circuits that incorporate combinations of negative feedback control and cooperative stability in terms of eight systems characteristics pertaining to optimality, robustness and controllability. Intriguingly, though individual circuits prevailed for distinct characteristics, the system with cooperative stability alone achieved the most balanced performance across all characteristics. Our study provides theoretical justification for the contribution of cooperative stability to natural biological systems and represents a guideline for designing synthetic complex formation systems with desirable characteristics.

Sensory feedback expands dynamic complexity and aids in robustness against noise

It has been hypothesized that sensory feedback is a critical component in determining the functionality of a central pattern generator. To test this, Yu and Thomas's recent work Yu and Thomas (Biol Cybern 115(2):135-160, 2021) built a model of a half-center oscillator coupled to a simple muscular model with sensory feedback. They showed that sensory feedback increases robustness against external noise, while simultaneously expanding the potential repertoire of functions the half-center oscillator can perform. However, they show that this comes at the cost of robustness against internal noise.

Quantitative comparison of the mean-return-time phase and the stochastic asymptotic phase for noisy oscillators

Seminal work by A. Winfree and J. Guckenheimer showed that a deterministic phase variable can be defined either in terms of Poincaré sections or in terms of the asymptotic (long-time) behaviour of trajectories approaching a stable limit cycle. However, this equivalence between the deterministic notions of phase is broken in the presence of noise. Different notions of phase reduction for a stochastic oscillator can be defined either in terms of mean-return-time sections or as the argument of the slowest decaying complex eigenfunction of the Kolmogorov backwards operator. Although both notions of phase enjoy a solid theoretical foundation, their relationship remains unexplored. Here, we quantitatively compare both notions of stochastic phase. We derive an expression relating both notions of phase and use it to discuss differences (and similarities) between both definitions of stochastic phase for (i) a spiral sink motivated by stochastic models for electroencephalograms, (ii) noisy limit-cycle systems-neuroscience models, and (iii) a stochastic heteroclinic oscillator inspired by a simple motor-control system.

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.

Organelle calcium-derived voltage oscillations in pacemaker neurons drive the motor program for food-seeking behavior in Aplysia

The expression of motivated behaviors depends on both external and internally arising neural stimuli, yet the intrinsic releasing mechanisms for such variably occurring behaviors remain elusive. In isolated nervous system preparations of Aplysia, we have found that irregularly expressed cycles of motor output underlying food-seeking behavior arise from regular membrane potential oscillations of varying magnitude in an identified pair of interneurons (B63) in the bilateral buccal ganglia. This rhythmic signal, which is specific to the B63 cells, is generated by organelle-derived intracellular calcium fluxes that activate voltage-independent plasma membrane channels. The resulting voltage oscillation spreads throughout a subset of gap junction-coupled buccal network neurons and by triggering plateau potential-mediated bursts in B63, can initiate motor output driving food-seeking action. Thus, an atypical neuronal pacemaker mechanism, based on rhythmic intracellular calcium store release and intercellular propagation, can act as an autonomous intrinsic releaser for the occurrence of a motivated behavior.

Sharpened and Mechanically Durable Carbon Fiber Electrode Arrays for Neural Recording

Bioelectric medicine treatments target disorders of the nervous system unresponsive to pharmacological methods. While current stimulation paradigms effectively treat many disorders, the underlying mechanisms are relatively unknown, and current neuroscience recording electrodes are often limited in their specificity to gross averages across many neurons or axons. Here, we develop a novel, durable carbon fiber electrode array adaptable to many neural structures for precise neural recording. Carbon fibers ( [Formula: see text] diameter) were sharpened using a reproducible blowtorchmethod that uses the reflection of fibers against the surface of a water bath. The arrays were developed by partially embedding carbon fibers in medical-grade silicone to improve durability. We recorded acute spontaneous electrophysiology from the rat cervical vagus nerve (CVN), feline dorsal root ganglia (DRG), and rat brain. Blowtorching resulted in fibers of 72.3 ± 33.5-degree tip angle with [Formula: see text] exposed carbon. Observable neural clusters were recorded using sharpened carbon fiber electrodes fromrat CVN ( [Formula: see text]), feline DRG ( [Formula: see text]), and rat brain ( [Formula: see text]). Recordings from the feline DRG included physiologically relevant signals from increased bladder pressure and cutaneous brushing. These results suggest that this carbon fiber array is a uniquely durable and adaptable neural recordingdevice. In the future, this device may be useful as a bioelectric medicine tool for diagnosis and closed-loop neural control of therapeutic treatments and monitoring systems.

Dynamical consequences of sensory feedback in a half-center oscillator coupled to a simple motor system

We investigate a simple model for motor pattern generation that combines central pattern generator (CPG) dynamics with a sensory feedback (FB) mechanism. Our CPG comprises a half-center oscillator with conductance-based Morris-Lecar model neurons. Output from the CPG drives a push-pull motor system with biomechanics based on experimental data. A sensory feedback conductance from the muscles allows modulation of the CPG activity. We consider parameters under which the isolated CPG system has either "escape" or "release" dynamics, and we study both inhibitory and excitatory feedback conductances. We find that increasing the FB conductance relative to the CPG conductance makes the system more robust against external perturbations, but more susceptible to internal noise. Conversely, increasing the CPG conductance relative to the FB conductance has the opposite effects. We find that the "closed-loop" system, with sensory feedback in place, exhibits a richer repertoire of behaviors than the "open-loop" system, with motion determined entirely by the CPG dynamics. Moreover, we find that purely feedback-driven motor patterns, analogous to a chain reflex, occur only in the inhibition-mediated system. Finally, for pattern generation systems with inhibition-mediated sensory feedback, we find that the distinction between escape- and release-mediated CPG mechanisms is diminished in the presence of internal noise. Our observations support an anti-reductionist view of neuromotor physiology: Understanding mechanisms of robust motor control requires studying not only the central pattern generator circuit in isolation, but the intact closed-loop system as a whole.

Shape versus timing: linear responses of a limit cycle with hard boundaries under instantaneous and static perturbation

When dynamical systems that produce rhythmic behaviors operate within hard limits, they may exhibit limit cycles with sliding components, that is, closed isolated periodic orbits that make and break contact with a constraint surface. Examples include heel-ground interaction in locomotion, firing rate rectification in neural networks, and stick-slip oscillators. In many rhythmic systems, robustness against external perturbations involves response of both the shape and the timing of the limit cycle trajectory. The existing methods of infinitesimal phase response curve (iPRC) and variational analysis are well established for quantifying changes in timing and shape, respectively, for smooth systems. These tools have recently been extended to nonsmooth dynamics with transversal crossing boundaries. In this work, we further extend the iPRC method to nonsmooth systems with sliding components, which enables us to make predictions about the synchronization properties of weakly coupled stick-slip oscillators. We observe a new feature of the isochrons in a planar limit cycle with hard sliding boundaries: a nonsmooth kink in the asymptotic phase function, originating from the point at which the limit cycle smoothly departs the constraint surface, and propagating away from the hard boundary into the interior of the domain. Moreover, the classical variational analysis neglects timing information and is restricted to instantaneous perturbations. By defining the "infinitesimal shape response curve" (iSRC), we incorporate timing sensitivity of an oscillator to describe the shape response of this oscillator to parametric perturbations. In order to extract timing information, we also develop a "local timing response curve" (lTRC) that measures the timing sensitivity of a limit cycle within any given region. We demonstrate in a specific example that taking into account local timing sensitivity in a nonsmooth system greatly improves the accuracy of the iSRC over global timing analysis given by the iPRC.

Control for multifunctionality: bioinspired control based on feeding in Aplysia californica

Animals exhibit remarkable feats of behavioral flexibility and multifunctional control that remain challenging for robotic systems. The neural and morphological basis of multifunctionality in animals can provide a source of bioinspiration for robotic controllers. However, many existing approaches to modeling biological neural networks rely on computationally expensive models and tend to focus solely on the nervous system, often neglecting the biomechanics of the periphery. As a consequence, while these models are excellent tools for neuroscience, they fail to predict functional behavior in real time, which is a critical capability for robotic control. To meet the need for real-time multifunctional control, we have developed a hybrid Boolean model framework capable of modeling neural bursting activity and simple biomechanics at speeds faster than real time. Using this approach, we present a multifunctional model of Aplysia californica feeding that qualitatively reproduces three key feeding behaviors (biting, swallowing, and rejection), demonstrates behavioral switching in response to external sensory cues, and incorporates both known neural connectivity and a simple bioinspired mechanical model of the feeding apparatus. We demonstrate that the model can be used for formulating testable hypotheses and discuss the implications of this approach for robotic control and neuroscience.

Rapid Adaptation to Changing Mechanical Load by Ordered Recruitment of Identified Motor Neurons

As they interact with their environment and encounter challenges, animals adjust their behavior on a moment-to-moment basis to maintain task fitness. This dynamic process of adaptive motor control occurs in the nervous system, but an understanding of the biomechanics of the body is essential to properly interpret the behavioral outcomes. To study how animals respond to changing task conditions, we used a model system in which the functional roles of identified neurons and the relevant biomechanics are well understood and can be studied in intact behaving animals: feeding in the marine mollusc Aplysia We monitored the motor neuronal output of the feeding circuitry as intact animals fed on uniform food stimuli under unloaded and loaded conditions, and we measured the force of retraction during loaded swallows. We observed a previously undescribed pattern of force generation, which can be explained within the appropriate biomechanical context by the activity of just a few key, identified motor neurons. We show that, when encountering load, animals recruit identified retractor muscle motor neurons for longer and at higher frequency to increase retraction force duration. Our results identify a mode by which animals robustly adjust behavior to their environment, which is experimentally tractable to further mechanistic investigation.

Computational model of the distributed representation of operant reward memory: combinatoric engagement of intrinsic and synaptic plasticity mechanisms

Operant reward learning of feeding behavior in Aplysia increases the frequency and regularity of biting, as well as biases buccal motor patterns (BMPs) toward ingestion-like BMPs (iBMPs). The engram underlying this memory comprises cells that are part of a central pattern generating (CPG) circuit and includes increases in the intrinsic excitability of identified cells B30, B51, B63, and B65, and increases in B63-B30 and B63-B65 electrical synaptic coupling. To examine the ways in which sites of plasticity (individually and in combination) contribute to memory expression, a model of the CPG was developed. The model included conductance-based descriptions of cells CBI-2, B4, B8, B20, B30, B31, B34, B40, B51, B52, B63, B64, and B65, and their synaptic connections. The model generated patterned activity that resembled physiological BMPs, and implementation of the engram reproduced increases in frequency, regularity, and bias. Combined enhancement of B30, B63, and B65 excitabilities increased BMP frequency and regularity, but not bias toward iBMPs. Individually, B30 increased regularity and bias, B51 increased bias, B63 increased frequency, and B65 decreased all three BMP features. Combined synaptic plasticity contributed primarily to regularity, but also to frequency and bias. B63-B30 coupling contributed to regularity and bias, and B63-B65 coupling contributed to all BMP features. Each site of plasticity altered multiple BMP features simultaneously. Moreover, plasticity loci exhibited mutual dependence and synergism. These results indicate that the memory for operant reward learning emerged from the combinatoric engagement of multiple sites of plasticity.

Where Computation and Dynamics Meet: Heteroclinic Network-Based Controllers in Evolutionary Robotics

In the fields of artificial neural networks and robotics, complicated, often high-dimensional systems can be designed using evolutionary/other algorithms to successfully solve very complex tasks. However, dynamical analysis of the underlying controller can often be near impossible, due to the high dimension and nonlinearities in the system. In this paper, we propose a more restricted form of controller, such that the underlying dynamical systems are forced to contain a dynamical object called a heteroclinic network. Systems containing heteroclinic networks share some properties with finite-state machines (FSMs) but are not discrete: both space and time are still described with continuous variables. Thus, we suggest that the heteroclinic networks can provide a hybrid between continuous and discrete systems. We investigate this innovated architecture in a minimal categorical perception task. The similarity of the controller to an FSM allows us to describe some of the system's behaviors as transition between states. However, other, essential behavior involves subtle ongoing interaction between the controller and the environment that eludes description at this level.

Propensity for Bistability of Bursting and Silence in the Leech Heart Interneuron

The coexistence of neuronal activity regimes has been reported under normal and pathological conditions. Such multistability could enhance the flexibility of the nervous system and has many implications for motor control, memory, and decision making. Multistability is commonly promoted by neuromodulation targeting specific membrane ionic currents. Here, we investigated how modulation of different ionic currents could affect the neuronal propensity for bistability. We considered a leech heart interneuron model. It exhibits bistability of bursting and silence in a narrow range of the leak current parameters, conductance (g_leak) and reversal potential (E_leak). We assessed the propensity for bistability of the model by using bifurcation diagrams. On the diagram (g_leak, E_leak), we mapped bursting and silent regimes. For the canonical value of E_leak we determined the range of g_leak which supported the bistability. We use this range as an index of propensity for bistability. We investigated how this index was affected by alterations of ionic currents. We systematically changed their conductances, one at a time, and built corresponding bifurcation diagrams in parameter planes of the maximal conductance of a given current and the leak conductance. We found that conductance of only one current substantially affected the index of propensity; the increase of the maximal conductance of the hyperpolarization-activated cationic current increased the propensity index. The second conductance with the strongest effect was the conductance of the low-threshold fast Ca²⁺ current; its reduction increased the propensity index although the effect was about two times smaller in magnitude. Analyzing the model with both changes applied simultaneously, we found that the diagram (g_leak, E_leak) showed a progressively expanded area of bistability of bursting and silence.

Eupnea, tachypnea, and autoresuscitation in a closed-loop respiratory control model

How sensory information influences the dynamics of rhythm generation varies across systems, and general principles for understanding this aspect of motor control are lacking. Determining the origin of respiratory rhythm generation is challenging because the mechanisms in a central circuit considered in isolation may be different from those in the intact organism. We analyze a closed-loop respiratory control model incorporating a central pattern generator (CPG), the Butera-Rinzel-Smith (BRS) model, together with lung mechanics, oxygen handling, and chemosensory components. We show that 1) embedding the BRS model neuron in a control loop creates a bistable system; 2) although closed-loop and open-loop (isolated) CPG systems both support eupnea-like bursting activity, they do so via distinct mechanisms; 3) chemosensory feedback in the closed loop improves robustness to variable metabolic demand; 4) the BRS model conductances provide an autoresuscitation mechanism for recovery from transient interruption of chemosensory feedback; and 5) the in vitro brain stem CPG slice responds to hypoxia with transient bursting that is qualitatively similar to in silico autoresuscitation. Bistability of bursting and tonic spiking in the closed-loop system corresponds to coexistence of eupnea-like breathing, with normal minute ventilation and blood oxygen level and a tachypnea-like state, with pathologically reduced minute ventilation and critically low blood oxygen. Disruption of the normal breathing rhythm, through either imposition of hypoxia or interruption of chemosensory feedback, can push the system from the eupneic state into the tachypneic state. We use geometric singular perturbation theory to analyze the system dynamics at the boundary separating eupnea-like and tachypnea-like outcomes.NEW & NOTEWORTHY A common challenge facing rhythmic biological processes is the adaptive regulation of central pattern generator (CPG) activity in response to sensory feedback. We apply dynamical systems tools to understand several properties of a closed-loop respiratory control model, including the coexistence of normal and pathological breathing, robustness to changes in metabolic demand, spontaneous autoresuscitation in response to hypoxia, and the distinct mechanisms that underlie rhythmogenesis in the intact control circuit vs. the isolated, open-loop CPG.