An inspiration-off attractor supports the robust and flexible control of breathing

Abstract

Breathing is a fundamental motor rhythm necessary to sustain life. The rhythm and pattern of breathing arises from the coordination of a rostrocaudally extended ventral respiratory column (VRC) in the medulla. Although recurrent excitation is sufficient for rhythmic bursting in reduced in vitro preparations, comparatively less is understood about how inhibition and excitation interact with feedback circuits to generate the breathing rhythm in vivo. Here, we expand on the classical “inspiratory-off switch” model that inhibitory signaling from lung stretch afferent pathways plays a critical role in generating the breathing rhythm in vivo. We perform high-density extracellular recordings along the length of the VRC during optogenetic and physiological manipulations and apply computational analyses to characterize how these perturbations govern low-dimensional neural dynamics. We show that stimulation of GABAergic afferents of the nucleus of the solitary tract advance or delay breathing in a manner similar to, but quantitatively distinct from, inhibition local to the VRC. We use generative dynamical models to show that inhibitory activity controls respiratory activity through transient stabilization of an attractive region of the low-dimensional manifold corresponding to expiration. Together, these results propose an “inspiratory-off attractor” model that underlies network level control of breathing in vivo.

INTRODUCTION

Rhythmicity is ubiquitous in nervous systems and occurs at multiple mechanistic, temporal, and functional levels; from molecules to brain-wide networks, from milliseconds to months, and from stereotyped motor patterns to cognition (1–7). Consequently, the fundamental questions of how these rhythms are generated and maintained, and how they functionally contribute to an organism’s fitness remain a subject of intense interest. Of these rhythms, the respiratory rhythm is essential for sustaining physiological function and life (8, 9). This rhythm dynamically responds to homeostatic demands and threats (10–13), serves as a medium for communication (14, 15), and influences cognitive and emotional states (16–19).

The respiratory rhythm and pattern is generated in a rostrocaudally extended, bilaterally synchronized column in the ventrolateral medulla called the ventral respiratory column (VRC) (20–24). Central among the VRC structures is the pre-Bötzinger complex (preBötC), which is intrinsically rhythmic in slices (25–29) and exerts exquisite control over breathing rhythm and pattern in vivo (30–33). The VRC has strong intracolumnar and collateral projections and diffuse inputs and projections with myriad other medullary and extra medullary brain regions (34, 35).

Early models of respiratory rhythmogenesis proposed that tonically active medullary centers are transiently and periodically interrupted by an “inspiratory-off switch” that originates from pontine centers and mechanical afferent feedback (36, 37). In the absence of these sources of transient inhibition, prolonged inspiratory spasms, termed apneusis, have been observed (38, 39). However, both classical and contemporary work demonstrate that, despite these observations, medullary circuits alone [in vitro (26, 40, 41) and deafferented in vivo (42)] are capable of periodicity, albeit reduced in complexity from true respiratory patterns. Intrinsic cellular mechanisms and/or recurrent excitatory signaling are sufficient for generating reduced rhythms without a synaptically driven inspiratory off switch. Hence, scientists working primarily in reduced systems largely dismissed the necessity of transient synaptic inhibition or the significance of inspiratory-off switch models for explaining respiratory rhythmogenesis.

In the intact animal, however, recurrent excitatory, inhibitory, and neuromodulatory circuits composed of diverse cell classes with complex connectivity patterns across anatomically distributed brain regions (27, 34, 40, 41, 43–53) coalesce to generate robust yet flexible motor outputs. Thus, fundamental attributes of the respiratory rhythm are vastly different in vivo and in vitro, emphasizing the potential for fundamentally different effective mechanisms that maintain rhythmicity and potentially override underlying kernel rhythms.

Inhibitory circuits are instrumental for generating and maintaining rhythms in various motor behaviors (54, 55). In the respiratory network, inhibitory populations that relay mechanoreceptive feedback about lung stretch (56–58) provide fast feedback that can both depress and accelerate breathing rates (59, 60). Many aspects of how inhibitory signaling from afferents impinge upon the underlying rhythmic primitives to exert bidirectional respiratory control remain unknown (12, 61, 62). Recent advances in cell type–specific manipulations and large-scale recordings allow for unprecedented interrogation of rhythmogenic mechanisms in the integrated, intact brain-body system, affording an opportunity to reconsider our understanding of respiratory rhythmogenesis.

Recently, we performed comprehensive large scale recordings along the extent of the VRC in vivo using high-density extracellular recording arrays (Neuropixels) (63). We showed that the coordinated activity of hundreds of simultaneously recorded single neurons in the VRC can be well described by projecting that high-dimensional activity onto a low-dimensional neural manifold (64–66). The concerted, time-varying activity of the entire population is reduced to a rotational trajectory in this low-dimensional representation. Those trajectories obey limit-cycle attractor dynamics in which deviations off the eupneic rotation inevitably reconverge. Consistent with reduced computational models of the VRC, we observed that at the end of each inspiration, the neural state accelerated toward, and dwelled in, a consistent, inspiratory-off attractive region in the latent space (41, 63, 67).

Here, we hypothesize that, in vivo, this acceleration is triggered by timed, phasic inhibition and is instrumental in setting the respiratory rate in the intact animal. We further hypothesize that inhibitory populations in a restricted part of the nucleus of the solitary tract at the level of the area postrema [NTS—second order lung stretch relay neurons, i.e., pump cells (68, 69)] could trigger this acceleration effect, thus contributing to bidirectional modulation of breathing rate.

We test these hypotheses by performing high-density neural recordings with Neuropixels from the extent of the VRC during direct activation of either inhibitory signaling in the contralateral VRC (VRC^GABA), inhibitory NTS (NTS^GABA) populations, or the endogenous lung stretch feedback circuit via the physiological lung stretch defense reflex [Hering-Breuer reflex (HBR) (70)]. We further contrast the differential effects of inhibitory neurons with stimulations of excitatory Vglut2⁺ neurons in the VRC (VRC^VGLUT2).

We show that an explicit, inhibitory inspiration-off signal is sufficient to both advance and delay respiratory phase during eupneic breathing in vivo with intact peripheral nerves. Unexpectedly, activation of the NTS^GABA populations exhibits a subphase specificity that is distinct from activating VRC^GABA populations.

We then analyze the low-dimensional population dynamics and show that activation of VRC^GABA, NTS^GABA, or HBR cause latent trajectories to converge toward a previously identified inspiration-off attractive region. Conversely, VRC^VGLUT2 activation causes the population dynamics to propel away from the attractive region. Last, we use a generative dynamical model (71, 72) in which optogenetic stimulations are modeled as uniform forces (i.e., vector fields) perturbing the on-going eupneic limit cycle dynamics.

This model captures essential results from the physiological experiments and extrapolates to predict experimental data not present in model fitting as well as results presented elsewhere (24). Strikingly, VRC^GABA stimulations exhibit separable inspiration promoting and delaying components, emphasizing the role of inhibition for in vivo rhythmogenesis that may be unneeded or overwhelmed in vitro (51, 53, 73).

Excitation and inhibition emerge to be diametrically opposed in this dynamic representation, and transiently destabilize or stabilize, respectively, the inspiratory-off region where the latent trajectories slow or briefly stop. The resultant antagonistic control reveals a push-pull relationship between inhibitory tone and the excitatory drive to breathe that is reminiscent of control strategies in diverse motor systems (74, 75).

Together, these data reframe and reemphasize the classical inspiratory off-switch hypothesis (37, 60). We offer experimental and computational evidence for an “inspiratory-off attractor” model in which concerted VRC activity can be represented as a low-dimensional limit cycle. Inhibition originating from afferent inputs impinges on core rhythm and pattern generating circuitry where antagonistic action of excitation and inhibition dynamically control the stability of the inspiration-off region.

RESULTS

Activating inhibitory populations in the NTS drives phase-dependent changes in breathing distinct from activation of excitatory and inhibitory neurons in the VRC

To test the effects on the VRC neural dynamics of perturbing constituent components of the brainstem respiratory neural circuitry, we inserted high-density extracellular electrode arrays (Neuropixel 1.0) probes along the length of the left VRC (63) of urethane-anesthetized mice. We simultaneously optogenetically stimulated inhibitory (Vgat⁺) neurons or excitatory (Vglut2⁺) neurons of the contralateral (right) VRC (VRC^GABA and VRC^VGLUT2, respectively), or Vgat⁺ neurons of the NTS (putative pump cells, NTS^GABA). To target VRC^GABA neurons, we generated Vgat^Cre;Ai32 (n = 8) or Vglut2^Cre;Ai32 (n = 7) crosses and acutely inserted a 200-μ optical fiber over VRC (Fig. 1A and fig. S1). We intentionally stimulate only contralaterally to the recorded neurons to interrogate the intact, integrated network response to subpopulation activation rather than the direct impact of ChR2 depolarization in the recorded neurons. Strong collateral projections within the VRC caused strong, short latency changes in the recorded populations (21). To target NTS^GABA neurons, we injected Cre-dependent ChRmine bilaterally into the NTS of Vgat^Cre mice (n = 9). ChRmine-oScarlet was well expressed in the NTS (fig. S1D); some animals were only transfected unilaterally, but similar physiological responses were observed, thus bilateral/unilaterally transfected mice were pooled.

Fig. 1. Activation of ventral and dorsal medullary groups differentially affect breathing.

(A) Experimental schematic. Created using https://alleninstitute.github.io/CCF-MAP/descriptions/mouse_ccf.html and (119). (B) Example integrated diaphragm traces during optogenetic stimulation. Shaded region is laser on. Color indicates wavelength used (blue: 473 nm, red: 635 nm). (C) Mean (± SEM) respiratory rate during before and during optogenetic stimulation during activation of different populations. Dots are individual animals. Solid lines indicate bilateral transfection/stimulation. (Paired t test two-sided, VRC^VGLUT2 n = 5, P = 0.036; VRC^GABA n = 6, P = 0.004; NTS^GABA n = 7, P = 0.006; *P < 0.05, **P < 0.01). (D) Latency (mean ± SEM) from laser offset to onset of first poststimulation breath. (Kruskal-Wallis test H = 10.2, dof = 2, P = 0.006; *P < 0.05 Holm-Bonferroni corrected). Dots are individual animals. (E) Example integrated diaphragm traces after the termination of a 2 s optogenetic stimulation (shaded region) for each stimulated population (one animal shown per stimulated population).

In agreement with previous reports, activating VRC^VGLUT2 neurons with 2 s of constant laser illumination increased respiratory rate, while activating inhibitory neurons in the VRC or the NTS decreased or abolished breathing [(VRC^GABA or VRC^VGLUT2) 473 nm, 8 to 10 mW; NTS^GABA 635 nm, 15 to 20 mW] (Fig. 1, B and C) (12, 62). Activation of NTS^GABA slightly decreased, and activation of VRC^VGLUT2 increased, heart rate (fig. S2B). The latency to the first breath after cessation of stimulation was shorter when stimulating VRC^GABA populations compared to either VRC^VGLUT2 or NTS^GABA populations (Kruskal-Wallis test P < 0.006). (Fig. 1, D and E).

We next replicated the previously reported phase-specific effects of stimulating VRC inhibitory or excitatory populations (59) and tested whether stimulation of NTS^GABA populations matched direct stimulation of VRC^GABA neurons. As expected, activation of VRC^VGLUT2 neurons during inspiration does not change respiratory rate but increases diaphragm electromyography (EMG) amplitude, and stimulation during expiration increases respiratory rate (Fig. 2, A and D, and fig. S2A). Conversely, stimulation of VRC^GABA and NTS^GABA neurons during expiration slows respiratory rate, while stimulation during inspiration increases respiratory rate (Fig. 2, B to D). We observed a reduction in diaphragm amplitude during inspiratory stimulation of both populations, while we only observed a reduction in amplitude during expiratory stimulations during activation of NTS^GABA populations (fig. S2A).

Fig. 2. Stimulation of inhibitory NTS neurons exerts phase dependent control of respiratory rate.

(A to C) Example integrated diaphragm EMG during laser stimulation of medullary cell types triggered by inspiration (top) or expiration (bottom). Color indicates laser wavelength. (D) Respiratory rate compared to baseline during phase-triggered stimulation. (red: inspiration trigger, blue: expiration trigger). (Two-sided one-sample t test; VRC^VGLUT2 n = 5, P_insp = 0.48, P_exp = 0.016; VRC^GABA n = 6, P_insp = 0.017, P_exp = 3.7 × 10⁻⁴; NTS^GABA n = 7, P_insp = 0.018, P_exp = 0.017; *P < 0.05, **P < 0.01, ***P < 0.001). (E). Phase reset driven by 50-ms laser pulses. Color represents stimulated population. Horizontal dashed line represents no change in rhythm; diagonal dashed line indicates theoretical limit on phase advance (i.e., immediate activation of next breath). Vertical lines represent average phase at which diaphragm features occur. (F) Timing of the phase reset curve intercept with 1 (no change in respiratory rate) relative to the peak or end of the diaphragm EMG signal. Color represents stimulated population. (Two-sided one sample t test; VRC^GABA n = 6, P_peak = 0.01, P_end = 0.69; NTS^GABA n = 6, P_peak = 0.14, P_end = 0.003; *P < 0.05). Marker indicates unilateral (crosses) or bilateral (circles) transfection/stimulation. In all panels, bars/shaded regions are means ± SEM. n.s., not significant.

Last, we presented brief, 50-ms pulse laser stimulations (n = 75) at random times throughout the cycle to observe how momentary activation of these populations perturbs ongoing respiratory rhythms. We then compute reset curves (Fig. 2E). As previously reported (59), stimulation of VRC^VGLUT2 neurons advances respiratory phase (curve below 1), specifically when activated during expiratory time (i.e., after the end of the diaphragm EMG activity). Stimulation of VRC^GABA advances the respiratory phase if stimulating during inspiration and delays the respiratory phase if stimulating during expiration. Stimulation of the NTS^GABA population advances the respiratory phase specifically during the first half of the inspiration, i.e., when the diaphragm amplitude is increasing. Otherwise, activation of these populations delays phase. To quantify this difference between activating VRC^GABA and NTS^GABA populations, we compute the point at which each animal’s phase reset curve crosses one and find that the intercept is at the end of the diaphragm activity for VRC^GABA stimulations but at the peak of the diaphragm for NTS^GABA stimulation (Fig. 2, E and F).

Activation of NTS^GABA populations selectively inhibits inspiratory VRC neural activity

We next asked how activation of these respiratory circuit components drives changes in the activity of recorded VRC populations. These stimulations are targeted to either contralateral (VRC^GABA or VRC^VGLUT2) or input (NTS^GABA) populations to minimize trivially observing the direct depolarization of recorded neurons. We analyzed single units recorded from the VRC during the 2-s continuous laser stimulations described above. After spike sorting and quality control steps, we categorized single units by their phasic activation patterns as inspiratory, expiratory, or tonic (see Materials and Methods). Example raster plots showing a subsample of the recorded population before, during, and after optogenetic stimulation show diverse, stimulus-specific changes in single-unit activity (Fig. 3, A to C, and fig. S3). We compare the firing rate in baseline control conditions against stimulation to see how much each unit is activated/inhibited as a result of stimulating our targeted populations (Fig. 3, D to F). While there is diversity in firing rate changes in all respiratory populations as a response to all stimulus conditions, we find that activating VRC^GABA neurons on average drastically inhibits all neuron types and most neurons in the VRC (Fig. 3, D and E, top). Conversely, activation of VRC^VGLUT2 increases the firing rate of most neurons on average (Fig. 3, D and E, bottom). Activation of NTS^GABA populations on average does not change the firing rates of expiratory neurons (Fig. 3, D and E, middle, and F top) and only minorly decreases firing of tonic neurons (Fig. 3, D and E, middle, and F bottom). Strikingly, these data suggest a functional specificity of activity of NTS^GABA populations to selectively decrease inspiratory activity without altering other respiratory population dynamics.

Fig. 3. NTS^GABA populations selectively inhibit inspiratory neurons.

(A to C) Example breathing and activity of example single units of tonic, inspiratory, and expiratory (10 per category; each row is a unit) in the VRC during optogenetic stimulation. (D) Average firing rate (spikes/s) of each unit during baseline control versus during 2-s laser stimulations. Each dot is a unit. Note the symmetric logarithmic scale. Black dashed diagonal is unity, and gray dashed lines indicate 2× or 0.5× firing rate during laser stimulus. Larger dots with borders are medians. (E) Cumulative distributions of firing rate ratios (stim versus control). x axis is log scaled. All comparisons are P < 0.05 except expiratory units during NTS^GABA stimulation (Wilcoxon signed rank test, control versus stimulus average firing rate). Right shifted curves indicate increases in firing rates during stimulations. (F) The same curves as (E), but phasic unit types are plotted together across stimulated populations. Color indicates stimulated population.

Activating inhibitory circuits shortcuts rotational population dynamics toward inspiration off

Recently, we showed that rotational latent dynamics underlie VRC population dynamics (63). Further, we showed that these rotations targeted an attractive-like region (potential well) that represents the offset of inspiration. Here, we use neuronal subtype-specific stimulations to test the hypothesis that activation of inhibitory circuits triggers an acceleration of the latent state toward that inspiration-off target to organize the respiratory rhythm. In addition, we tested a complementary hypothesis that activation of VRC^VGLUT2 populations drives the latent state away from the inspiration-off target. We observe that activation of both VRC^GABA and NTS^GABA populations during inspiration drives the network preemptively toward a point in the latent state that is near (although not coincident with) the inspiration-off target. Activation during expiration clamps the network in that potential well (Fig. 4, A and B, middle and right, and movies S1 and S2). Note that the inspiration-off target is not a true fixed point as the principal components analysis (PCA) model does not consider dynamics (see below). Activating VRC^VGLUT2 populations does not alter the rotation but does increase trajectory speed (Fig. 4, A and B, left; fig. S4; and movie S3). NTS^GABA stimulation most dramatically reduces trajectory speed during preinspiration and inspiration (fig. S4, C and D), emphasizing the selective inhibition of expiratory dynamics. We next examine the trajectories during and after the randomly timed, 50-ms laser pulses. Agreeing with the phase-triggered stimulus results, activating both VRC^GABA and NTS^GABA populations drives the network toward the inspiration-off target, with some quantitative differences described below. Conversely, activating VRC^VGLUT2 populations drives the network along its current trajectory. We quantify the degree of convergence of the trajectories by measuring the dispersion of the two-d latent trajectories 25 ms after the offset of the stimulus pulse (i.e., to address the question how similar the network states are shortly after the stimulus; see Materials and Methods). Activating VRC^VGLUT2 populations does not reduce the dispersion compared to randomly selected states during baseline, while activating either inhibitory population does. Similarly, as a metric of neural state dissimilarity, we compute the Euclidean distance in the PCA space from the poststimulation state to the average latent state at the offset of inspiration to quantify the difference in neural state. We observe that VRC^VGLUT2 pushes the network away from the inspiration-off state, while both inhibitory populations push the network toward inspiration-off (Fig. 4E and fig. S5A). Last, we compute this distance as a function of poststimulus time (Fig. 4F) and of latent state dimensionality (fig. S5). VRC^GABA immediately converges to a point advanced of the inspiration-off target and subsequently diverges (i.e. moves away from the inspiration-off point) more rapidly than the NTS^GABA stimulations, suggesting a fundamental difference in how these populations affect VRC dynamics.

Fig. 4. Inhibitory stimulations drive VRC populations dynamics toward inspiration off.

(A) Latent dynamic trajectories during inspiration and (B) expiration triggered stimulations of medullary populations for three example recordings. Blue-white-red gradient shows normal latent trajectory (color mapped to respiratory phase); black lines indicated trajectory during unstimulated periods; blue/orange lines indicate trajectories during stimulated periods corresponding to laser wavelength used to stimulate respective opsins. One stimulation replicate is shown (inspiration triggered: 15 s, expiration triggered: 2 s). (C) Randomly timed, 50-ms pulse stimulations (75 pulses) of each medullary population. Red-white-blue gradient lines indicate normal trajectory, and blue/orange lines indicate trajectory during stimulation as in (A). Black lines indicated trajectory for 25 ms after end of stimulation. Gray dots indicate latent state 25 ms after end of stimulation. Large white dot is the average latent state at the offset of the diaphragm contraction. (D) Dispersion (i.e., trace of the covariance matrix) of 2D latent state a t = 25-ms poststimulation, compared to a sample of randomly chosen points in the baseline period. Each dot is a recording. (Paired t test two-sided, VRC^VGLUT2 n = 7, P = 0.24; VRC^GABA n = 8, **P = 0.006; NTS^GABA n = 7, ***P = 8.3 × 10⁻⁴). (E) Distance between average 2D latent state at inspiration offset compared to 25 ms poststimulation. Random shuffled control as in (D). (Paired t test two-sided, VRC^VGLUT2 n = 7, P = 0.014; VRC^GABA n = 8, P = 0.0014; NTS^GABA n = 7, P = 2.7 × 10⁻⁵). (F) Distance to the inspiration off average latent state as in (E), for all poststimulation time delays. Dashed line indicates random shuffled controls. Lines, shaded regions are means ± SEM.

Neural population dynamics during lung stretch reflex suggests inspiratory-expiratory competition mechanism

The HBR (57, 70) is characterized by rapid and robust lung stretch–induced apnea often caused by exogenous increases in positive airway pressure. The HBR critically protects against damage to the lung due to over inflation. To evoke the HBR during recordings of the VRC, we present positive airway pressure (15-cm H₂O) by closing an exhaust valve in the airway line, diverting presented airflow into a water column for 2 or 5 s (Fig. 5A). This causes a decrease or cessation of inspiration and concomitant changes in VRC neural activity (Fig. 5, B and C, and fig. S6C). Inspirations that break through the inflation stimulation are shorter than control (Wilcoxon signed-rank test P = 3.0 × 10⁻⁵; Fig. 5, D and E). We compare the breath cycle–averaged firing rates of respiratory neurons between control and Hering-Breuer stimulation. Inspiratory neurons expectedly have shorter durations of activations during stimulation (Fig. 5, F and G). The reduction in both phase-normalized inspiratory firing rates (Fig. 5H) and average firing rate (Fig. 5I) is likely due specifically to a reduction in the inspiration time. Computing only the firing rate during inspiration reveals an average increase in inspiration-specific firing rate (fig. S6, E and F). Consequently, inspiratory neurons fire more spikes per inspiration but over fewer inspirations with shorter inspiratory times. Some expiratory neurons lose the characteristic gradual decrement over the expiratory phase that leads toward inspiration (Fig. 5, F to H) and transition to quiescence more rapidly during the transition into inspiration (Fig. 5, G and H), signifying a more abrupt state transition. The firing rate of expiratory neurons does not change on average, matching what was seen in direct activation of NTS^GABA populations. There is no notable anatomical specificity to changes in neural activity (fig. S6, A to D), suggesting that the Hering-Breuer affects the entire VRC network.

Fig. 5. Activation of airway protective HBR selectively reduces firing of inspiratory neurons.

(A) Experimental schematic. Created in BioRender. Oliveira, L. M. (2026) https://biorender.com/sdk3qbd. (B) Example diaphragm EMG and selection of single units during lung inflation. (C) Respiratory rate during 2- and 5-s lung inflations. (n = 16). (D) Normalized, breath-triggered average diaphragm EMG for normal breaths (black) and “break through” breaths during lung inflation (magenta). (E) Average inspiration duration (Wilcoxon signed rank test, n = 16, P = 3.0 × 10⁻⁵). Color as in (D). (F) Breath-triggered average firing rates for example single units during baseline (black) and lung inflation (magenta). (G) As in (F) but averaged over neurons of a given respiratory category in example recording. (H) Phase-normalized, breath-triggered average firing rates across all units. Color as in (G). Shaded region is mean ± SEM. (I) Firing rate of each unit of a given category in control against lung inflation. Note symmetric logarithmic scale. Black dashed diagonal is unity, and gray dashed lines indicate 2× or 0.5× firing rate during lung inflation. Large dots are medians. (J) Cumulative distributions of firing rate ratios (stimulation versus control) (Wilcoxon signed rank test control versus during lung inflation insp: Wilcoxon = 137,470, ***P = 6.4 × 10⁻⁴⁸, n = 1,064; exp.: Wilcoxon = 79,908, P = 0.43, n = 576; tonic: Wilcoxon = 976,308, ***P = 1.0 × 10⁻¹², n = 2,177). (K) Average firing rates for given category of units in the 50 ms before the onset of inspiration (inset) during control and Hering-Breuer stimulation. Firing rates are a normalized percentage of mean firing rate in control period. (Two-sided paired t test t_insp = −5.1, ***P_insp = 4.9 × 10⁻⁷; t_exp = −4.9, ***P_exp = 1.0 × 10⁻⁶; t_tonic = 1.5, P_tonic = 0.14). (L) Breath-triggered ratio of expiratory/inspiratory unit firing rates (top) or vice versa (bottom) for all units pooled across recordings during control (black) and lung inflation (magenta). (M) Ratio of firing rates of expiratory neurons to inspiratory neurons 10 ms before and after onset of inspiration. Dots/thin lines are individual recordings, and thick banded lines are means ± SEM. Shaded regions are means ± 95% confidence interval (CI) unless otherwise noted.

To further characterize how lung inflation changes the neuronal network activity during the transition into inspiration, we compute the firing rate of each neuron type (inspiratory, expiratory, and tonic) in the 50 ms before the onset of inspiration. Both inspiratory and expiratory neurons show higher firing rates in this period during the lung inflation stimulation compared to control (Fig. 5K, two-sided paired t test P_insp = 4.9 × 10⁻⁷, P_exp = 1.0 × 10⁻⁶). Last, we compare the ratio of inspiratory to expiratory firing rates, pooled across all neurons and recordings. The inspiration on transition occurs when the inspiratory/expiratory firing rate ratio is one in both control and stimulated conditions (Fig. 5, K and M), irrespective of the observation that the absolute firing rates are higher.

Lung inflation–induced latent dynamics resemble direct stimulation of inhibitory populations

We next examined the low-dimensional latent state dynamics of the VRC population during HBR activation evoked by lung inflation stimuli. Presentation of positive pressure causes that latent state to arrest at a single point in the latent space (Fig. 6A and movie S4). This arrest is exemplified by a reduction in latent trajectory speed (Fig. 6B) that is not significantly different from the reduction in speed when we stimulated either the VRC^GABA or NTS^GABA populations directly (Fig. 6C, Tukey post hoc test P = 0.937, P = 0.951 respectively, also compare to Fig. 4B). Trajectory speed during inspiration onset is higher during Hering-Breuer break through breaths (fig. S6G). The lung inflation stimulation converges the latent state toward the inspiration-off attractive target (Fig. 6, E and F) in a similar manner observed when we optogenetically stimulated VRC^GABA or NTS^GABA populations. Last, we asked whether the single neuron activity observed during lung-inflation stimulations resembled the single-neuron activity observed during the 2-s optogenetic stimuli applied previously (Fig. 6G). We compute the difference and Pearson correlation of the mean square root–transformed firing rates between the two stimulation paradigms. The difference indicates whether the average firing rates are similar across conditions, and the Pearson correlation indicates whether individual units are modulated similarly across the different conditions. While VRC^VGLUT2 stimulation and VRC^GABA stimulation showed higher and lower average firing rates than the lung inflation stimulation, respectively for all respiratory neuron types, NTS^GABA stimulation matched the average firing rates well (Fig. 6H). The correlation of the inspiratory neurons during NTS^GABA stimulation was notably lower than expiratory or tonic populations, while that of the inspiratory neurons during VRC^GABA stimulation was high, suggesting incomplete mimicry of the physiological stimulus with this circuit specific activation despite a strong match in the physiology.

Fig. 6. Lung inflation drives latent dynamics toward inspiration off.

(A) Example diaphragm EMG (top), first three PCs (grayscale), and latent trajectory speed (blue-yellow, scale arbitrary, not shared with PCs) during a 2-s lung inflation. (B) Trajectory speed of leading 3D for all recordings during 2-s laser stimulations (color indicates targeted population), or 5-s lung inflation (magenta). Shaded regions are means ± SEM. (C). Percent change in trajectory speed during stimulation compared to prestimulation. One-way analysis of variance (ANOVA) (F = 25.5, DF = 3, DF = 40, P = 2.2 × 10⁻⁹; Tukey post hoc test ***P < 0.001). (D) Example latent trajectories during baseline (red-white-blue lines colored by phase) and during one 5-s lung inflation (magenta). Average latent state at offset of inspiration as a large white dot. (E) Distance between average 2D latent state at inspiration offset compared to shuffled points (random) or 1 s after lung inflation onset (HB). Dots and thin lines are recordings. Color indicates targeted population in other experiments. Linear mixed effects model DF1 = 2, DF2 = 19; F_population = 9.14, P = 1.6 × 10⁻³; F_HBstim = 56.33, ***P = 4.3 × 10⁻⁷; F_interaction = 1.4, P = 0.27. (F) As (E) but showing dispersion of 2D latent state at 1 s after lung inflation onset. Linear mixed effects model DF1 = 2, DF2 = 19; F_population = 2.14, P = 0.145; F_HBstim = 56.2, ***P = 4.3 × 10⁻⁷; F_interaction = 1.98, P = 0.15). (G) Firing rate of individual units during lung inflation (x axis) and during 2-s optogenetic hold stimulation (y axis), stratified by stimulated population and respiratory type. Note the square root scaled axis. Dashed diagonal is unity, gray lines indicate 2× or 0.5× firing rate. Large dots are medians. (H) Comparison of firing rates of subpopulations during different stimulus modalities. x axis shows difference of the means of square root firing rates between optogenetic and lung inflation stimuli. Zero indicates similar firing rates in both conditions; positive values indicate higher firing rates during opto-genetic stimulus. y axis and marker size show correlation (ρ) between the firing rates of the two modalities. Color shows targeted population; marker shape indicates respiratory category. a.u., arbitrary units.

Activation of brainstem subpopulations are transient forces applied to ongoing dynamic landscapes

We previously reported that explicitly incorporating dynamics into our latent state models reveals a limit cycle attractor underlying VRC population dynamics that is highly consistent across animals (63). This limit cycle was well recovered by fitting recurrent switching linear dynamical systems (rSLDS) models (71). In our rSLDS model, the low-dimensional state (two-dimensional here) follows two distinct linear dynamical systems that govern the dynamics in two distinct partitions of the latent state. Both the dynamics matrices and the latent state partitioning (i.e., transition from inspiratory state to expiratory state) are fit from neural activity alone. Although the respiratory rhythm is typically described as having three phases, these partitions do not imply that there are two discrete phases of the cycle but rather that the nonlinear dynamics can be approximated with these partitions. We previously found that a simpler two-state model is sufficient (63) and supported by previous reduced dynamical models of the respiratory brainstem (67). Analyses of the trajectories uncovered by the PCA model above suggest that inhibition pushes the network state to the inspiratory off attractor. We test that hypothesis explicitly by incorporating a dependence of the 2D latent state ${\mathit{x}}_{\mathrm{t}}$ on an external input ${\mathit{u}}_{\mathrm{t}}$ (see Materials and Methods), where ${\mathit{u}}_{\mathrm{t}}$ is the presence of the laser pulse. We fit the latent state to the baseline (no stimulus) period and periods of stimulation with short (10 and 50 ms) laser pulses. The rSLDS model learns latent dynamics matrices and a resultant latent state, from which we can predict the integrated diaphragmatic activity (Fig. 7, A and B). The model partitioned the state such that inspiration was governed by spirals (i.e., complex eigenvalues n = 16 of 16 recordings, 8 unstable and 8 stable), while expiration was governed by stable sinks (i.e., real eigenvalues n = 13 of 16) or saddles (n = 3 of 16) (fig. S7). If the expiratory fixed point was a sink, the fixed point was located within the inspiratory partition and the trajectory was attracted toward inspiration. Conversely, if the expiratory fixed point was a saddle, then it was located in the expiratory partition and the trajectory was repelled toward inspiration. Although the eigenvalue composition of the discrete phases differ across recordings, they all generate qualitatively similar limit cycles in which the trajectories oscillate between these two partitions. The model estimates the imposed effect of the optogenetic stimulation on the dynamics (Fig. 7, B and C, and movies S5 and S6). The learned stimulus effect manifests as two uniform vector fields separated by the same partitions learned by the dynamics (i.e., respiratory phase). By manipulating a scalar multiple on the learned stimulus, we can elicit qualitative changes in the dynamics such that the limit cycle contracts (Fig. 7B, Stim = 1) or disappears as the latent state can approach the previously inaccessible expiratory fixed point (Fig. 7B). When the fixed point is stable, a persistent apnea is observed; when it is a saddle and the stimulus amplitude is sufficiently large and applied for sufficiently long (e.g., Fig. 7B, right), the state trajectory diverges. Such instances were discarded from further analyses. To compare the relative strength of the stimulus in the two phases, we compute the norm of the stimulus vectors. The VRC^VGLUT2 stimulus has a stronger effect during expiration, VRC^GABA has a stronger effect during inspiration, and NTS^GABA has a similar strength during both states (Fig. 7D). We next compare the learned inspiratory stimulus direction with the slow eigenvector of the expiratory dynamics. The slow expiratory eigenvector resembles the direction through which the latent state evolves as it transitions into inspiration (Fig. 7C). The cosine similarity of the inspiratory stimulus field with the expiratory slow dynamics is large and positive for VRC^VGLUT2 stimulation, indicating that the stimulus pushes the network toward the inspiratory partition and away from the expiratory fixed point (Fig. 7E). The stimulus opposes (i.e., large and negative) the normal dynamics during VRC^GABA stimulation and is more diffuse compared to NTS^GABA stimulations. These results are consistent with the experimental observations in Fig. 4.

Fig. 7. Activation of subpopulations drive generalizable changes to latent dynamics.

(A) Example fitting of rSLDS models. Example rasters/integrated diaphragm as in Fig. 3A. rSLDS model is fit to spiking observations during baseline, 10-ms, and 50-ms laser pulses. Fitted dynamics are regressed against observed diaphragm to map simulated diaphragm from latent state. (B) Simulated latent dynamics from data in (A), varying simulated stimulus amplitude. Red and blue streamlines are underlying dynamical system. Black lines are example latent evolutions for 5 s. Red and blue markers are fixed points of inspiratory/expiratory partitions of latent state, respectively. Circle is stable node; triangle is saddle. Expiratory fixed point is out of plotting range for amplitude <1. (C) Red and blue arrows represent the learned effect of optogenetic stimulation on the latent dynamics, restricted to the appropriate partition. Light-blue arrows show the eigenvectors of the expiratory partition in (B) superimposed on the learned stimulus fields. (D) Stimulus norm (i.e., stimulus strength) for the inspiratory/expiratory partitions, stratified by stimulated populations. Each connected pair is one recording. (E) Cosine similarity of the slow expiratory eigenvector with the direction of the inspiratory stimulus field, only for recordings where the expiratory dynamics are nonrotational (i.e., real eigenvalues). Each dot is a recording. Shuffle compares learned stimulus with eigenvectors from other recordings. (F) Example generalization of learned stimulus effect to recapitulate phase-dependent effects on respiratory rate in a simulated NTS^GABA recording, simulated from learned dynamics. Gray trace is simulated diaphragm; shaded red regions are triggered stimulations. (G) Example of varying stimulus strength in a simulated NTS^GABA recording, gradually abolishing respiration. Brown and gray traces are simulated latent, and black trace is simulated diaphragm. Red line is applied stimulation. (H) Quantification of (G) with high-density parameter sweep. Dot size and color indicate amplitude of simulated diaphragm during inspirations. In all panels, bars are means ± 95% CI.

We next asked whether applying this learned stimulus field could generalize to other types of stimulus patterns that were not part of the fitting procedure. We applied phasic-simulated NTS^GABA stimulations and found that inspiratory-triggered stimulations increase simulated respiratory rate, and vice versa (Fig. 7F and fig. S7, C and D). We then simulated long stimulus holds for a range of amplitudes (Fig. 7, G and H) and found that both simulated respiratory rate and amplitude decrease gradually as a function of stimulus strength until it is abolished. This amplitude sweep was not collected here for direct comparison but matches previously published data (24). As stimulus strength increases, the expiratory partition fixed points generally cross boundaries into the opposite phase partition, depending on their stability (sink versus saddle) (fig. S7).

Dynamic motifs generalize across stimulus types to predict neural activity and physiological behavior

Strikingly, we observed that the learned stimulus fields are qualitatively similar within a given experimental stimulus (e.g., VRC^GABA stimulus fields oppose the slow expiratory dynamics; Fig. 7E). We next asked whether we could design stimulus fields that, when applied, evoked simulated network behavior matching the experimental optogenetic results. We fit new rSLDS models without observations of optogenetic stimulation and applied stimulus vector fields (Fig. 8A) defined by the direction of the slow expiratory eigenvector (i.e., direction of E to I slow transition) as suggested by Fig. 7E. We matched all three optogenetic stimulus paradigms. Example stimulus fields and resultant reset curves defined as in Fig. 2E are shown in Fig. 8A (top and bottom): designed fields inline with slow expiratory eigenvector only during expiration matched the VRC^VGLUT2 stimulations (Fig. 8A, left); fields opposed to the expiratory eigenvector only during inspiration matched VRC^GABA (Fig. 8A, middle); and fields opposed to the expiratory eigenvector over the whole space matched NTS^GABA (Fig. 8A, right) (see Materials and Methods). Thus, the effect of perturbing a subset of the underlying network can be parameterized with respect to the latent dynamics of the unperturbed system. We verify this result by matching many of the observed experimental phenomena despite not having fit to experimental data for any of these scenarios. We exemplify this in Fig. 8 with NTS^GABA-like stimulations. We simulate reset curves across all recordings with 50-ms stimulation durations for all designed fields (Fig. 8, A and B). VRC^VGLUT2-like stimulations advance phase, while VRC^GABA-like and NTS^GABA-like stimulations show phase-specific respiratory frequency modulation as in Fig. 2E. This phase-dependent modulation is further evidenced by performing inspiration or expiration triggered stimulations. Stimulating during inspiration with an NTS^GABA-like field caused respiratory frequency to increase, while stimulation during expiration caused respiratory frequency decrease or apnea (Fig. 8, C and D, and fig. S8, A and C; compare to Fig. 2D). We then apply a long, constant, NTS^GABA-like field at varying amplitudes. We observe a gradual decrease in respiratory frequency until complete apnea is achieved (Fig. 8, E and F). Similar, but less consistent, decrease in frequency is observed in the VRC^GABA-like amplitude sweeps (Fig. 8F). Applying the VRC^VGLUT2-like field slightly increased respiratory frequency with a notable observation that large amplitude stimulation of VRC^VGLUT2-like fields caused a reduction or cessation of respiratory rate. We expect that this was due to a limitation of the piecewise linear model; the state would become restricted to the inspiratory region of the latent space, which was often a stable spiral and would subsequently fail to reach the expiratory region. Last, we find that the rebound latency (i.e., time to first breath after stimulus offset) of the VRC^GABA-like stimulation was shorter than either VRC^VGLUT2-like or NTS^GABA-like (fig. S8B), directionally matching experimental observations (Fig. 1D). Together, these results present a generative, network-level control mechanism for respiratory neural dynamics by which subpopulations exert specific influences on the dynamic landscape to reshape respiratory rhythms.

Fig. 8. Experimental stimulation of specific subpopulations is recapitulated by parametric, unlearned perturbations of underlying dynamical systems.

Simulated stimulations are indicated by orange. (A) Latent dynamics are fit without inclusion of optogenetic stimulation. (Left) Dynamics for an example recording [red/blue streamlines, fixed points as in (B)] and (right) the three candidate stimulus fields (black arrows). Direction of stimulus field is always aligned to slow expiratory eigenvector; the same direction for VRC^VGLUT2-like, opposite direction for inhibitory-like (bottom). Simulated phase reset curves for application of these stimulus fields (50-ms stimulation). Reset curves as in Fig. 2E. Stimulus amplitude is set to 4. Each dot is a stimulus. Each recording has different underlying dynamics and thus differently oriented stimulus fields. (B) Average reset curves applying the three stimulus fields in (A) to all recordings. Shaded region is mean ± 95% CI. All simulated fields generalize to all recordings, regardless of animal genotype recorded; n = 19 per group. (C) Example simulated phasic stimulation using an NTS^GABA -like (i.e., uniform) stimulus field. (D) Change in simulated respiration rate for the three stimulus fields applied during inspiration or expiration (each marker is a recording; crosses are omitted from statistics as outliers; two outlier points of Δ Resp. Rate > 400% are omitted). Stimulus amplitude is set at 4. (One sample Wilcoxon signed rank test **P < 0.01, ***P < 0.001; see source data for P values). Bars are median ± 95% CI. (E) Example varying strength of NTS^GABA -like stimulus for held period. Brown and gray traces are simulated latent; black trace is simulated diaphragm. Top orange line is duration of applied stimulation. (F) Average simulated respiratory rate as a function of supplied stimulus amplitude for the three stimulus types [color as in (B), n = 19 per stimulus application]. Shaded regions are means ± SEM.

DISCUSSION

Overview

Here, we provide a data-driven, low-dimensional dynamical systems framework for flexible respiratory control in vivo informed by high-spatiotemporal resolution neural recordings and causal manipulations of identified respiratory control circuit subpopulations. These results further the inspiration-off switch mechanism as a central respiratory pacing mechanism in vivo as originally proposed by systems-level approaches. In these original investigations of the inspiratory-off switch mechanism (58, 60, 76, 77), the primary readout was gross physiological activity (i.e., EMG, airflow, nerve recordings, and few single units) that represents the integrated output of a complex, heterogeneous, and spatially distributed neuronal network. These classical approaches allowed limited insights into the underlying neural mechanisms, and much of the support for these models was primarily inferred from computational biophysical simulations (41, 78, 79). Here, we leverage comprehensive, high-resolution electrophysiological recordings with Neuropixels and the optogenetic manipulation of genetically defined neuronal subtypes to unveil the neuronal network dynamics underlying flexible respiratory control.

Our inspiratory-off attractor model provides further evidence for, and follows in spirit from, the classical inspiratory off-switch model in that timed phasic inhibition from mechanosensory feedback resets the respiratory cycle to set breathing frequencies. In our inspiratory-off attractor model, inspired air generates a lung stretch signal relayed by primary afferents to inhibitory neurons of the NTS. Inhibition in turn stabilizes the inspiratory off region. Upon ceasing inspiratory effort, the lung stretch signal diminishes, and inspiration promoting mechanisms destabilize this region to initiate the next breath. When lung stretch remains high (as it does during the HBR), the inspiratory-off region remains stable, slowing or preventing inspiration. The inspiratory-off attractor model is corroborated by prior experimental evidence that respiratory rates decrease, and muscle contraction increases, when vagal afferents are removed (38, 59). We augment the classical inspiration off switch model by introducing a generalization that predicts bidirectional respiratory control from inhibitory populations, accounts for manipulations of excitatory populations, and predicts the activity of the entirety of the VRC neurons. The dynamics approach used here is especially powerful as it decomposes inputs and manipulations of the VRC as additive linear components. Experiments that activate or inhibit target populations with higher specificity than that used here may be able to further decompose these components to explain more fine-grained control of the respiratory limit cycle.

The results presented here corroborate previous experimental and associated modeling studies of the dynamics of reduced VRC populations (24, 41, 67). These studies originate the concept of VRC state as evolving along a limit cycle in which slow evolution of the neural population between bursts is punctuated by fast transitions between expiratory and inspiratory states. Similar to our experimental optogenetic results and vector field perturbations, these previous modeling studies showed that the relative stability and timing of transitions between states, and thus respiratory frequency, was tuned by manipulating the balance of inhibitory and excitatory tone and intrinsic cellular parameters such as the slowly inactivating sodium current.

An important distinction here is the architecture of models. In the reduced models, anatomical compartments of inhibitory and excitatory neurons are reduced to single point sources, and the rhythmic behavior emerges from the defined architecture. Here, we modeled perturbations to directly observed experimental data, and the latent dynamical behavior is fit from the observed spiking data of heterogeneous populations embedded in a fully intact neurorespiratory network. A strength of the reduced models is that they allow for detailed analyses of dynamics, including bifurcation analyses, which are not directly accessible here. That the previous reduced models and our data-driven models converge on limit cycle dynamics tightly governed by inhibitory/excitatory balance bolsters the fundamental importance of these latent dynamical patterns at multiple levels of granularity, circuit intactness, and biological mechanism.

Notably, our model is limited in the ability to fully analyze the underlying nonlinear dynamics, and we are limited to analyses of the linear partitions separately. For example, the spiral dynamics in the inspiratory phase are not consistently stable, nor are the expiratory dynamics always stable sinks (despite qualitatively similar limit cycles), complicating a unified bifurcation analysis. In addition, recent work has shown that oscillatory activity governed by piecewise linear dynamics lacks independent phase modulation (80). Methods to infer the fully nonlinear dynamics could allow for deeper insights into the control and manipulation of these latent dynamics (81).

Although the simultaneous high-resolution recordings from large populations of distinct, genetically defined neuronal subtypes and their phase-specific selective manipulations constitute a fundamental departure from previous, traditional in vivo studies, the present study builds on important insights that were gained by many of these original studies. Here, we show that breathing phase can be both advanced and delayed as a result of well-timed activity of inhibitory neurons in the NTS. Our experiments build on prior investigations of inhibitory cells in this region of the NTS, “pump cells,” that have been shown to receive direct innervation from slowly adapting lung mechanoreceptors (68, 82, 83). However, direct bidirectional control via optogenetics has not been shown. Because respiratory rhythmic activity can be maintained in vitro without this off-switch mechanism, there has been considerable uncertainty surrounding the importance of the off-switch mechanism as proposed for the in vivo network. Some work speculated that the burst-termination as seen in vitro may reflect a “fail-safe” mode that could play a role in gasping but not eupneic breathing (84).

It is known that in the absence of vagal inputs, respiration slows and diaphragmatic amplitude increases dramatically (38). Normal rates and amplitudes can be recovered by direct stimulation of the remaining vagal nerves (76) or by direct stimulation of inhibitory neurons of the preBötC (59). Whether activation of the NTS^GABA populations in conditions of vagotomy are sufficient to recover qualitatively normal respiration is yet unknown but an important next step. We speculate that the diverse molecular and anatomical targets of the vagi into the NTS make this plausible but unlikely to be the sole circuit involved. For example, our study did not address the role of the Kolliker-Fuse as an important contributor to the inspiratory off-switch. There are likely multiple pathways that can evoke identical, subtly different, or entirely different mechanisms that result in a similar physiological output, emphasizing the need for detailed and comprehensive neural recordings. Our data here show similarity between physiological responses NTS^GABA stimulation and lung-inflation but subtly different effects on the VRC (namely, a subpopulation of inspiratory VRC neurons that are unaffected in lung inflation). In particular, we expect that emerging techniques to target subpopulations based on more granular cell markers or restricted projection/input patterns will provide critical insights into the mechanisms that govern the generation of breathing in the largely intact animal (85–87).

Subphase specificity of respiratory modulation by different sources of VRC inhibition

We find that activating NTS^GABA populations plays a significant role in inspiratory rhythmogenesis in vivo by specifically advancing the respiratory phase during the rising half of the inspiration while delaying phase in the second half of inspiration. In contrast, direct stimulation of VRC^GABA populations advances phase throughout inspiration. We interpret this as having two critical implications. First, this agrees with the accepted role of the NTS^GABA population as relaying lung volume for the purpose of pacing via feedback inhibition and volutrauma protection. NTS^GABA activity amplifies the reafferent representation of lung volume, and the inspiration is terminated once this lung state representation reaches a threshold. Amplification of this reafferent signal during early inspiration will preemptively trigger a reset, while amplification of that signal in the latter half of inspiration will relay a lung-state representation that is larger than normal (as in a sigh). If NTS^GABA populations are activated in the interbreath interval, this conveys that the lung is overfull (as in the HBR) and delays respiratory phase. Second, the fact that activating reafferent circuitry during the latter half of inspiration delays respiration implies a role for inhibitory signaling in the VRC as rhythm advancing separate from mechanosensory feedback. Taken with the observation that latency to next breath after stimulation (i.e., rebound latency) is shorter for direct VRC^GABA stimulation than for NTS^GABA, and that VRC^GABA stimulation converges to a point advanced of inspiration-off, we conclude that nonspecific stimulation of inhibitory signaling within the VRC concurrently prepares for an inspiration while also opposing its execution.

The dynamical systems modeling results add further evidence for this functional separability of inhibition. We showed that NTS^GABA-like effects could be recreated by applying a force opposite the direction of evolution during the interbreath interval (Fig. 8), while VRC^GABA-like effects were matched by applying this force exclusively during inspiration and not during the interbreath period. Consequently, during VRC^GABA-like stimulation, the model systems are held at the edge of the transition into inspiration. This implies that VRC^GABA stimulation, but not NTS^GABA stimulation, prepares the dynamics to transition into inspiration. Moreover, VRC^GABA stimulation concurrently reduced the activity of expiratory unit activity (Fig. 3F), while NTS^GABA neurons did not (Fig. 3F). These experiments cannot specify whether the reduced expiratory activity is restricted to inhibitory neurons. However, we expect that this is likely as we have previously shown that expiratory neurons are more likely inhibitory in vivo (63). These data are consistent with the idea that VRC^GABA stimulation disinhibits some inspiration promoting components of the respiratory network. The separability of these inspiratory phase-promoting and phase-advancing components of inhibitory signaling suggests that purely excitatory mechanisms of rhythm generation, while well established as being sufficient for in vitro rhythms (51, 53, 73), are likely less important during eupneic breathing in vivo. In vitro respiratory rhythms rely on recurrent excitability within the preBötC and are terminated by intrinsic cellular mechanisms and local interactions (49). Our results offer an integrated view of the rhythmogenic process in vivo, which relies on the dynamic, well-timed activation of inhibitory neurons to coordinate inspiratory motor commands with proprioceptive inputs derived from lung-afferents via the NTS.

Of note, although disruption of GABAergic and/or glycinergic neurotransmission has been shown to have varied impacts on respiratory or fictive respiratory outputs (88–90), the separability of these two systems in vivo remains to be shown conclusively. GABAergic and glycinergic presynaptic mRNA is predominately coexpressed in the medulla (91), and it is unclear whether these neurotransmitters can be released differentially under normal circumstances. With the genetic techniques used here, we do not expect a difference in targeting GABAergic or glycinergic populations with Cre (or similar) driver approaches as these techniques would target and depolarize the same medullary population. We are unaware whether there is structured differential expression of postsynaptic GABA/glycine receptors in discrete subpopulations of the medulla that might separate these signals. Further investigations will be important to unravel exactly what subcircuits and cell types might be involved in the phase-promoting aspects of VRC inhibition.

In addition, we found that for our generative dynamical model to consistently recreate respiratory rate increases during inspiratory stimuli for the NTS^GABA-like stimuli, we had to restrict stimulation to the rising phase of the diaphragm (fig. S8A). This explains the difference in these results and recent reports in which inspiration timed NTS^GABA stimulations resulted in respiratory slowing (62). In those results, the phasic stimulations were long enough to potentially continue during expiration.

Antagonistic control of inspiratory transitions by expiratory/inspiratory balance

The HBR has been described for over a century (70), and the responses of single neurons of various subregions of the VRC have been well characterized (56, 61, 69, 92–97). However, the coordinated changes in activity that arise in these distributed neural populations of the VRC during specific activation of transcriptionally defined respiratory populations has not been investigated. We observe that during lung inflation, expiratory neural activity remains high until a breath is triggered (Fig. 5, G and K), contrasting the observation that expiratory neural activity decays in advance of the increase in inspiratory populations (Fig. 5G). The lung inflation keeps expiratory tone high, which in turn putatively inhibits inspiratory tone. Simultaneously, inspiratory tone increases over time likely because of a combination of intrinsic excitability (98, 99) and tonic excitatory input (10, 41, 67, 100). Whether and when the inspiratory tone overcomes this balance will likely be specific to the sensory input and internal state of the animal. Lung inflation opposes inspiratory tone and recruits expiratory neurons, and the inspiratory drive to overcome that expiratory “barrier” is increased. This “stiffening” of the transition to inspiration is supported by the observation of a coordinated increase in both inspiratory and expiratory firing rates just before inspiration (Fig. 5K), with the absence of a change in the ratios of firing (Fig. 5M). Break through breaths during lung inflation differ from normal breaths, which supports the idea of stiffening; inspiratory neural activity is higher (fig. S6F), inspirations are shorter (Fig. 5E), and neural trajectories are faster (fig. S6G). In other systems [e.g., arm movement (75) and fish locomotion (74)], similar mechanisms of antagonistic control are common strategies for fine-tuned control. Balance in two opposing forces results in stability, while imbalance moves the state with more precision than without antagonistic control. Such control precision is both more likely and potentially more advantageous when multiple feedback loops are intact and used in vivo as compared to the relatively unconstrained rhythm in vitro.

It is important to note the (expected) lack of a perfect match between neural activity during HBR responses and stimulation of either NTS^GABA or VRC^GABA populations (Fig. 6H). The optogenetic stimulus cannot completely mimic the lung inflation stimulus at the detailed level of neural populations. First, optogenetic stimulations in VRC^GABA experiments is unilateral (and contralateral to the recorded populations), while lung inflation will activate bilateral circuits. Second, lung-inflation stimulation specifically activates an endogenous inflation protection circuit, which includes activation of rapidly adapting stretch receptors (101, 102) and the pons (77). Conversely, the optogenetic stimulations likely activate off-target circuits, which may be involved in other aspects of respiratory control, or other, nonrespiratory functions (103). The strong match in the breathing data for both stimulus types without the perfect match of the neural data is a warning that behavioral/gross physiological readouts are not complete, and limits how much certainty can be said without comprehensive examination of the neural activity.

Contrasting dynamic motifs to modulate neural rhythms

Our data show that stimulation of VRC^VGLUT2, VRC^GABA, or NTS^GABA populations can differentially increase respiratory rate if done with precise timing. By interrogating latent population dynamics, we uncover that these populations elicit respiratory rate increase with strikingly different neural mechanisms. The inspiration-off target is a critical feature in this dynamic landscape. Neural trajectories are attracted to, and dwell in, this region. Activation of VRC^VGLUT2 populations speeds up neural trajectories by pushing them away from the inspiratory off target (i.e., destabilizing); conversely, activation of either inhibitory population temporarily increases the attractiveness of that region (i.e., stabilizing). Learned VRC^VGLUT2 stimulus fields are weak in the inspiratory phase (Fig. 7D) and are aligned with the slow expiratory eigenvector (fig. S6B). Inspiratory VRC^VGLUT2 stimulation is predicted to weakly increase diaphragm amplitude and potentially prolong expiration, matching physiological results in Fig. 2 and fig. S2 and (59). Together, VRC^VGLUT2 populations increase respiratory rate by increasing trajectory speeds, most strongly in between inspirations, while inhibitory stimulations “short cut” the latent trajectory.

In the transgenic (VRC^GABA or VRC^VGLUT2) mice, we stimulate not only local VRC neurons, but also glutamatergic or GABAergic inputs, and therefore are activating broadband excitatory and inhibitory inputs to the region. In addition, the specificity of optogenetic activation of VRC populations is limited to placement of the optic fiber, which is unilateral and contralateral to the recording site. As a result, we stimulate asymmetrically a bilaterally constrained system with strong commissural connectivity and inherent physiological symmetry. Bilateral activation of VRC^GABA neurons drives more pronounced respiratory changes (both increases and decreases) (59), and we suspect that the effects on latent dynamics would be qualitatively similar (but with stronger effects) to those reported here. Despite this limitation, we observed unexpectedly low variability in the physiological responses despite variability in fiber placement, underscoring that the VRC is a spatially distributed network where functional specificity is more critically affected by nonspatial features. We show here that lung stretch inputs are sufficient to drive the network back to the inspiration off attractor and likely play a primary role in shaping the normal eupneic dynamics. However, further experiments are required to delineate circuit, region, or input cell type–specific roles in sculpting or modulating those dynamics. In particular, we hypothesize that glutamatergic inputs from NMB⁺ RTN neurons may play a primary role in destabilizing the inspiratory off attractor, particularly since inhibiting and removing these populations drastically reduces the regularity and rate of the inspiratory rhythm in vivo (11).

Notably, VRC^GABA stimulation more quickly and directly drove the dynamics to the inspiration off target than did NTS^GABA stimulations. This agrees with our targeting of the NTS^GABA neurons that are presumptive secondary afferents receiving input from the SARs and affect breathing potentially via a multisynaptic pathway (61, 104, 105). Recent compelling work has implicated local inhibition of NTS^Phox2B neurons (62), but they do not rule out the function of additional collateral projections. Other possible mechanisms for these slower perturbation responses may include slower intrinsic dynamics of NTS^GABA cells, smaller postsynaptic effects, or lower levels of transfection in viral approaches compared to transgenic.

The conceptual model for respiratory control as presented here provides a differentiated view of the rhythmogenic process in which inhibitory and excitatory mechanisms play opposing roles in the coordination of the respiratory cycle via dynamic control of the stability of an inspiratory off attractor. Further, this control is in large part dictated by mechanoreceptive inputs from lung afferents. However, our study also raises important questions that remain unanswered. First, how is the lung stretch set point determined? What exogenous or endogenous subcircuits within the medulla, pons, and other areas of the brain can manipulate that set point? What are the contributions of various inspiration promoting mechanisms to the destabilization of the inspiratory-off fixed point? Blood gas concentrations are tightly regulated and drive diverse homeostatic regulatory mechanisms that affect respiratory control. Chemosensitive signals from the carotid bodies (106, 107) and the RTN/pFRG (10, 11, 108) relay information about peripheral and central blood gas concentrations, respectively. These modulate the homeostatic drive to breathe on generally slower timescales than mechanoreceptive feedback. We speculate signals from these sources may be well modeled as destabilizing the inspiration off region in a way that interacts with the transient stabilization resulting from stretch feedback.

Notably, our experiments were performed under 100% inspired O₂, which elevates arterial CO₂. In these conditions, pulmonary stretch receptor activity and subsequent lung stretch reflex responses are blunted (109–112). Under normal arterial CO₂, the chemosensitive drive to breathe will be reduced and the strength of feedback from stretch receptors will be increased as compared to the experiments performed here. Consequently, our model would predict that the inspiratory-off region would stabilize, as excitatory tone decreases and transient inhibitory strength increases. How that effect might interact with additional variables (e.g., arousal and complex behavior in awake animals) is unclear. Last, how do other inspiration terminating mechanisms [e.g., those triggered by irritants or e.g. during speech/vocalization (12, 14, 113)] or slowing mechanisms (18) affect VRC latent dynamics?

Together, these data emphasize the role of inhibitory reafferent feedback in shaping the normal ongoing dynamics of the diverse respiratory motor control populations that govern breathing. Viewing the activity of neural populations in a low-dimensional dynamic space allows us to decompose the impact of manipulating cell type–specific subcircuits and inputs into the respiratory control circuitry. This latent dynamics approach has been used to great effect in other motor and behavioral settings (114, 115), offering an exciting and powerful way to dissect neural circuits in exquisite detail beyond observing motor output (e.g., breathing behavior). Our finding that we can both define and predict the effects of activating specific subpopulations as directional stimulus fields suggests one could combine arbitrary fields to disentangle the functional effects of multiple subpopulations that may be disjoint or overlapping. Further, we showed that we could match experimental observations with this approach by parameterizing the stimulus fields relative to the underlying dynamics without having fit to the optogenetic perturbations. It may be possible to uncover these stimulus fields in separate, sequential experiments and determine subpopulation interactions through this latent decomposition as opposed to performing all possible combinations of intersectional experiments. When combined with perturbation of intersectionally defined cell type– and projection-specific populations (44, 45, 89), we can begin to disambiguate the functional role of neural populations that may have overlapping or indistinguishable impact on respiratory output.

MATERIALS AND METHODS

Animals

All procedures were approved by the Seattle Children’s Research Institute Institutional Animal Care and Use Committee (IACUC00058). Male and female mice aged 9 to 24 weeks were used. For targeting of VRC^VGLUT2 and VRC^GABA populations, homozygous Vglut2^Cre (Jax #028863) and Vgat^Cre (Jax# 016962) were crossed with homozygous Ai32 mice (Jax# 012569) allowing Cre-dependent expression of a channelrhodopsin-2/enhance yellow fluorescent protein (EYFP) fusion [ChR2(H134R)/EYFP].

Surgery

To target NTS^GABA populations, we injected AAV8-nEF-Con/Foff 2.0-ChRmine-oScarlet (30 nl; 137161-AAV8) into the NTS at the level of the area postrema (7.56 mm caudal from the bregma, ±0.5 mm lateral from the midline, and 3.9 mm below the dura mater) of Vgat^Cre (Jax# 016962) mice. Anesthesia was induced with 3% isoflurane and maintained at 1.5%. Body temperature was maintained at 37°C with a recirculating heated waterbed (Adroit medical). Perioperative ketoprofen (5 mg/kg, sc) was administered. Scalp fur was shaved and the scalp was sterilized with isopropyl alcohol and betadine before incision. The skull was cleared and a midline craniotomy extending over the left and right NTS was drilled (± 0.7 mm from the midline). Virus was loaded into a glass pipette attached to a Nanoject 2 (Drummond Scientific), and the pipette was targeted to the left NTS. After cranial injection, the pipette remained in place for ∼5 min before removal of the pipette and targeting of the right NTS. Skin was then sutured over the craniotomies and mice were allowed to recover. Ketoprofen (5 mg/kg, sc.) was administered for two consecutive days after the surgery. Virus was allowed to transfect for 10 to 28 days before recordings.

In vivo Neuropixel recordings

Recording preparations are modified from (63); see that reference for further details. Anesthesia was induced at 3% isoflurane and urethane (1.5 g/kg) was administered intraperitoneally before removing animals from isoflurane. Animals were placed on a feedback controlled heating bed (FHC 40-90-8D) to maintain body temperature at 37°C throughout the experiment. Two stainless steel EMG wires (AM systems 0.005 inch diameter) were inserted into the right diaphragm, and two were inserted to record electrocardiogram (one lead over the heart and one in the left intraperitoneal space). EMG wires were secured in the skin with cyanoacrylate (Krazy Glue). EMG signal was amplified (10,000×), hardware filtered (100 to 5000 Hz, AM-Systems 1700), and digitized at 10 kHz (NI PXIe-8381). Diaphragm EMG was hardware rectified and integrated (custom) to be used for closed-loop online triggering of optical stimulation. (N.B. This rectified and integrated signal was not used in downstream analyses; see Preprocessing). Scalp fur was removed with depilatory cream (Nair). The mouse was then placed in a stereotaxic frame (Stoelting), affixed with earbars, and the skull was leveled. The scalp and skin overlying the rostral neck were removed, and neck muscles were dissected to allow visualization of the foramen magnum and the first cervical vertebra. A stainless steel headplate (custom) was cemented to the skull with ultraviolet cure cement (Pulpdent Embrace Wetbond). A small craniotomy was opened over the right cortex, and a bare silver ground wire 0.005in (AM systems) soldered to a gold pin (AM systems) was inserted overlying the dura and secured with cyanoacrylate (Krazy Glue) as a ground. In experiments in VRC^GABA or VRC^VGLUT2 stimulations, an ∼0.5-mm craniotomy was drilled over the right VRC (1.25 mm lateral and 6.84 mm caudal from bregma) and overlying dura was carefully removed. The animal was then transferred to a custom head-fixing apparatus. A 3D-printed nose cone was placed over the animal’s snout, and 100% O₂ was supplied. If labored breathing was observed, atropine (0.1 mg/kg) was administered to alleviate mucous buildup. The dura overlying the left caudal brainstem surface was cut, and the brainstem surface was kept hydrated with phosphate-buffered saline (PBS). For recordings in VRC^GABA or VRC^VGLUT2 experiments, the optical fiber [200-μm diameter, 0.22 numerical aperture (NA), ThorLabs] was calibrated and then targeted above the right VRC (1.25 mm lateral, 6.84 mm caudal from bregma, and 5.3 mm ventral from bregma). In NTS^GABA experiments with animals expressing virally mediated, red-shifted ChRmine opsins bilaterally in the NTS, the optic fiber (400-μm diameter, 0.22 NA, Doric) was placed on the surface of the skull overlying the NTS (midline, 7.2 mm caudal from bregma). Neuropixels 1.0 probes coated in DiI (Thermo Fisher Scientific, V22885) were then targeted to the left VRC (1.25 mm lateral, 4.8 mm ventral from the intersection of the parietal and occipital bones at midline [see (63) for details]. The probe was then inserted ∼3.5 mm rostrally from the caudal brainstem surface at 10 μm/s (Scientifica PatchStar). The probe was then retracted 100 μm to reduce tissue distortion and allowed to settle for 15 min before recording. Ten to fifteen minutes of baseline neural and respiratory activity were recorded before optical stimulation protocols.

Stimulation protocols

All experimental hardware was controlled with custom python and Arduino software (https://github.com/nbush257/pyExpControl). Red (635 nm) or blue (473 nm) lasers (Cobalt MLD 635/473) were driven with an analog voltage (0 to 1 V) calibrated to output either: 635-nm light 20 mW at the tip outside the skull (400-μm diameter fiber) or 473-nm light 8 to 10 mW at the tip ∼300 μm above the VRC (200-μm fiber). A 2-ms sigmoidal ramp was added to the start and end of each pulse to minimize light artifacts on neural recordings. The following stimulus paradigms were applied: (i) Hold stimulus: 5 to 15 repetitions of 2-s illumination interleaved with 20-s intertrial interval; (ii) pulse stimulus: 50 repetitions of 10 ms and 50 repetitions of 50 ms pulses with 3-s intertrial interval; (iii) 10 repetitions of 10-s long inspiratory-triggered stimulations; (iv) 10 repetitions of 2- to 4-s long expiratory triggered stimulations.

Phasic (i.e., inspiratory and expiratory) triggered stimulations were performed by applying a hysteresis trigger to the hardware integrated diaphragm. During inspiratory stimulation, a positive threshold crossing turned on the laser and a negative crossing of the signal at 90% of the rising threshold turned off the laser. Expiratory stimulations used the same logic and thresholds, only the laser control was inverted (i.e., laser off when integrated diaphragm was above threshold, laser on when integrated diaphragm was below 90% threshold). Before recordings, the experimenter set the threshold manually with the light path interrupted (to not alter respiratory behavior that could alter the threshold choice) such that the threshold was as low as possible while maintaining phasic activation and avoiding constant illumination. To perform positive pressure experiments triggering the HBR, nose cone gas outlet was connected to a T-junction where one line led into a 15-cm water column and the other to a constitutively open solenoid valve. To apply positive pressure, the solenoid valve was closed, forcing exhaust gas into the water column.

Histology

After recording, the animal was euthanized and perfused with PBS followed by 4% PFA, and the brain was removed. The brain was then fixed overnight in 4% PFA, in 15% sucrose for 24 hours and finally 30% sucrose. The brain was embedded in embedding medium (NEG-50) and frozen at −80°C. Then, 25-μm sagittal or coronal sections were collected and imaged (Olympus BX61VS) at 4× to 10× magnification. Recording location was identified by the DiI fluorescent track and optical fiber placement confirmed by tissue damage. To minimize potential bias in the quantification, all photomicrography and cell counting were performed by a single researcher blinded to the experimental conditions. Cell counts were obtained using ImageJ software (version 1.41; National Institutes of Health). For each mouse, one-in-two coronal series of 25-μm brain sections was analyzed, resulting in sections spaced 50 μm apart. First, section alignment across brains was performed relative to a reference section of the NTS, which is anatomically bordered dorsally by the area postrema, laterally by the spinal trigeminal nucleus, and ventrally by the dorsal motor nucleus of the vagus. For sagittal sections, all brain slices were analyzed to identify the Neuropixel probe trajectory and optic fiber placement. Sections were mounted onto slides in phosphate buffer solution in rostrocaudal sequential order. After drying, they were coverslipped with 4′,6-diamidino-2-phenylindole–Fluoromount (0100-20; SouthernBiotech) and sealed with nail polish. The neuroanatomical nomenclature used in the experiments followed the atlas of Paxinos and Franklin (2012).

Preprocessing

Preprocessing of physiological and neural data was performed with custom python software available at https://github.com/nbush257/cibrrig. After recording, raw diaphragm EMG was rectified and median filtered, and electrocardiography artifact was removed using custom removal methods. Breaths were then detected using scipy.find_peaks from the resulting time series. Neural data were preprocessed using spikeinterface (116). Briefly, neural traces were destriped: Traces were high-pass filtered (250 Hz) and phase shifted, and noise channels were removed and spatial high-pass filtered (local median subtraction), and optogenetic-induced artifacts were removed. Recordings were then motion corrected and spike sorted with Kilosort 4 (117). Only units that passed the following quality control metrics were marked as acceptable and included in downstream analyses (noise cutoff <0.1, sliding refractory period <0.1, median amplitude >40.0 μV, number of spikes >500, presence ratio > 0.8; for details see https://spikeinterface.readthedocs.io/en/0.102.3/modules/qualitymetrics.html). For ease of reuse, extracted data were saved according to International Brain Lab Open Neurophysiology Environment conventions (ONE https://github.com/int-brain-lab/ONE).

Physiology analysis

Respiratory rates were computed during the stimulation period and compared to the equivalent length control period immediately preceding each stimulation. Relative changes in respiratory rate were computed as $\%\mathrm{\Delta }\mathit{RR}=\frac{100}{n}{\displaystyle {\sum }_{i=1}^n}\frac{R{R}_i^{\text{stim}}}{R{R}_i^{\text{control}}}$ for all i stimulations; relative changes in diaphragm amplitude were computed similarly. Reset curves were created by computing the time from each 50-ms pulse stimulus to the previous t_pre and the next breath t_next onset, and the average respiratory period $\overline{T}$. Then, we compute normalized values: $\mathit{\text{stim phase}}=\frac{t_{\text{pre}}}{\overline{T}}$ and $\mathit{\text{cycle duration}}=\frac{t_{\text{pre}}+t_{\text{next}}}{\overline{T}}$. We quantify the intersection (reset_intercept) of the phase reset curve with the y = 1 line by first smoothing the reset curve with a second-order Savitsky-Golay filter with a window size of 21 and then finding the normalized stimulus phase at which that smooth reset curve crosses y = 1. We then subtract the normalized phase time to the diaphragm peak and the diaphragm offset (end). We compute respiratory phase as in (54) such that respiratory phase ϕ is defined on [−π, π]; ϕ = 0 is defined as the onset of an inspiration (i.e., diaphragm activation), and ϕ = π is the offset of an inspiration (diaphragm inactivation). We set the sample immediately following ϕ = π to −π. Last, we linearly interpolate between [−π, 0) as expiration and [0, π] as inspiration.

Single-unit analyses

We restrict all analyses to units that passed quality control (see above Preprocessing). The response of each neuron as a function of phase is computed as $\mathit{FR}\left(\mathrm{\varphi }\right)=\mathit{dt}\ast \frac{P(\mathrm{\varphi }\mid \mathit{\text{spike}}=1)}{P\left(\mathrm{\varphi }\right)}$ where dt is the time step of the ϕ signal (1 ms) and ϕ is binned at a resolution of $\frac{\mathrm{\pi }}{25} \text{rad}$. We define the respiratory modulation and preferred respiratory phase as the vector mean of the phasic histogram from (118) where the normalized length of the vector is the respiratory modulation ∈ [0, 1] and the angle is the preferred phase ∈ [−π, π]. Units with a respiratory modulation <0.2 were classified as tonic; all others as phasic. Phasic units were classified as inspiratory if the preferred phase was ≥0; otherwise, they are expiratory. To compare firing rates between the two stimulus conditions (Hering-Breuer and opto-hold), we first take the square root transform of the mean firing rate of each unit in the stimulus period to transform the firing rate distributions before computing the correlation ρ. We also compute the difference in the mean square root firing rates: $\frac{1}{n}{\sum }_i^n\sqrt{F{R}_i^{\text{opto}}}-\frac{1}{n}{\sum }_i^n\sqrt{F{R}_i^{\text{HB}}}$, where $F{R}_i^{\text{stim}}$ is the mean firing rate of unit i for that stimulus across repetitions, to estimate the difference in firing rates on average between the two stimulus conditions.

Principal components projections

We project the simultaneously recorded neural data into a low-dimensional projection as previously (54). Briefly, we first compute the binned spike counts for each unit in 5-ms bins in the last 5 min of the baseline recording. The binned spike counts were smoothed with a gaussian kernel (σ = 10 ms) and square root transformed before applying PCA. In downstream analyses, we truncate the projection to a specific number of leading dimensions. We compute instantaneous trajectory speed as the Euclidean distance between two subsequent points (dt = 5 ms) in that truncated projection in time. We then compute the poststimulus point as the point in the latent space at a given time after the offset of the stimulus, for each stimulus repetition i. To compute dispersion (i.e., how far apart are the poststimulus points), we create a matrix of all stimulus points (# projection dimensions X # stimulus repetitions), compute its covariance matrix Σ, and then compute Tr(Σ). We compute the distance to inspiration-off by first computing the mean point in the latent space where respiratory phase is immediately after the offset of inspiration $[-\mathrm{\pi }<\mathrm{\varphi }<-\mathrm{\pi }+\frac{\mathrm{\pi }}{25}]$ in the last 100-s window of the baseline recording (i.e., no stimulus). We compute the mean Euclidean distance to all poststimulus points as the mean distance to I-off. All computations are performed in the truncated low-dimensional space indicated. Random shuffled controls were computed by choosing 100 uniformly distributed random points from the last 100 s of the baseline window to be the stimulus offset times. Because there is a delay between closing of the valve and buildup of pressure in the lung, we consider the poststimulus time to be 2.5 s after the valve closing.

Dynamical modeling

To account for temporal structure when fitting a latent representation of the neural population activity, we fit rSLDS (71) as in (63). Briefly, the rSLDS model fits a low-dimensional continuous latent state x_t where ${\mathit{x}}_t\in R^d$. The rSLDS also partitions the space such that the state of the model can be in any of k discrete states z_k, and the discrete state z_t is a function of the continuous state x_t. The dynamics in each partition are governed by separate linear dynamical systems subject to external inputs u_t such that the continuous state follows

$${\mathit{x}}_t={\mathit{A}}_k{\mathit{x}}_{t-1}+{\mathit{V}}_k{\mathit{u}}_t+{\mathit{b}}_k+{\mathit{w}}_t$$

where the fitted parameters are A_k, the dynamics matrix; V_k, the control matrix applied to the input u_t; and b_k, the offset vector, all associated with discrete state k. w_t is a noise term that we set to zero during simulation due to the observed stereotypy of the neural activity. We choose k = 2, d = 2, dt = 10 ms as we previously report this simple system to be sufficient to model these populations (54). u_t is 1D and binary; it takes a value of 1 when the optogenetic laser is on; 0 otherwise. Thus, V_k is a d-dimensional vector. For analyses in Fig. 7 where we analyze the learned stimulus field V_k, we fit jointly $\{{\mathit{A}}_k,{\mathit{b}}_k,{\mathit{V}}_k\}$ during the last 5 min of the baseline period and the brief pulse stimulations only (50 repetitions each of 10 and 50 ms optogenetic stimuli). In analyses in Fig. 8 where we impose arbitrary V_k, we fit only $\{{\mathit{A}}_k,{\mathit{b}}_k\}$ during the last 5 min of the baseline period (i.e., fit without optogenetic stimulus). While u_t is {0, 1} during fitting, the term V_ku_t, the effective stimulus field during the k_th discrete state is uniform over the k_th partition and differs in magnitude and direction for each discrete state. We restrict all subsequent analyses only for recordings that exhibited limit cycle dynamics (n = 16 for joint fitting of all parameters and n = 18 for fitting baseline only). We showed previously that when limit cycle dynamics were observed, the two discrete states z_k corresponded to inspiration and expiration. We computed the stimulus strength for a given phase as the norm of the real components of each V_k. Next, we computed the eigenvectors of the dynamics matrices A_k and determined slow and fast eigenvectors for each discrete state by the magnitude of the eigenvalue. To determine how aligned the dynamics matrices A_k were with the stimulus vector V_k, we computed the dot product between the normed eigenvectors and the normed stimulus vectors for the expiratory partition only for recordings where the expiratory dynamics have real eigenvalues (n = 13 of 16 recordings).

After fitting $\{{\mathit{A}}_k,{\mathit{b}}_k\}$ and optionally V_k, we could simulate latent states $\hat{\mathit{x}}$ by initializing at $\hat{\mathit{x}}=(0,0)$ and following the latent state forward in time as it reached the limit cycle attractor according to the state update rule. Then, simulated integrated diaphragm traces were generated by first fitting a nonlinear support vector regression (SVR) with the fitted 2D latent state x as input and the observed integrated diaphragm trace as the target followed by applying that SVR to $\hat{\mathit{x}}$. We simulated optogenetic stimulation by multiplying V_k by a scalar multiplier that represented the stimulus amplitude. This scalar was chosen ad hoc depending on the experiment performed to manipulate the relative strength of the stimulus. We performed simulated phasic-triggered simulations by setting a threshold on the simulated integrated diaphragm above (or below), which we multiplied V_k by some constant. We further controlled whether phasic stimulation occurred during the first (i.e., rising) or latter (i.e., falling) part of the inspiration, given the subphase specificity of NTS^GABA effects on breathing.

We manually examined a variety of arbitrary stimulus fields V (e.g, uniform across all directions, piecewise uniform, sinks, and sources). We chose three simple stimulus fields that qualitatively matched the effects of the optogenetic stimulation observed in experiment: VRC^VGLUT2-like, VRC^GABA-like, and NTS^GABA-like. VRC^VGLUT2-like was uniform in the direction of the slow expiratory eigenvector (constructive) with a sigmoidal decay in amplitude at the boundary of the inspiratory/expiratory partition. VRC^GABA-like was the mirror of VRC^VGLUT2-like (opposed to the slow expiratory eigenvector). NTS^GABA-like was also opposed to the expiratory eigenvector but constant amplitude over the entire space. The choice of a sigmoidal decay and the parameter of the decay was chosen manually that qualitatively matched observed data. To make stimulus amplitudes approximately comparable across recordings, stimulus amplitudes were scaled by the mean speed of the latent trajectory. Note that the amplitude scaling in the different fitting procedures (with or without including stimulus observations) are not comparable because of the explicit fitting of V_k in the former case. In some analyses, outlier data points were omitted after visual inspection showed that, for some recordings, the applied stimulus showed unexpected divergence or convergence to fixed points. These points are shown in figures, but excluded from statistical analyses.

Statistical analyses

Statistical analyses were computed using custom python software using statsmodels and pingouin. Analysis software is available at https://github.com/nbush257/inspiration-off-attractor.

Supplementary Materials

The PDF file includes:

Figs. S1 to S8

Tables S1 and S2

Legends for movies S1 to S6

Other Supplementary Material for this manuscript includes the following:

Movies S1 to S6