Figures and data
Model dynamics.
(A) Phase portrait of the biophysical model, with voltage variable v and adaptation variable a. Nullclines are shown for various constant stimuli strengths. The firing threshold (dashed line) separates active and quiet states. The system exhibits relaxation oscillations, with slow movement along nullcline-v and fast movement across nullcline-a to switch between active and quiet states. At strongly table equilibrium (open circle) bifurcates into a stable equilibrium (filled circle) corresponding to tonic and silent modes, respectively. (B) Activity of the biophysical model across stimulus strengths, with corresponding phase portrait trajectories. As strength increases, the neuron transitions from the stable silent mode to the periodic bursting mode to the stable tonic mode. (C) Phase portrait of the simplified model, with analogous voltage and adaptation variables. Nullclines are piecewise-linear “S”-shaped functions. Trajectories 1) and quiet (v = −1) legs until reaching the active and quiet bounds to trigger a fast state switch. The corner ds are unstable in bursting mode and stable in tonic and silent modes. (D) Activity of the simplified model responding phase portrait trajectories. Burst shape is controlled through a firing function y(v, a, x) that input x into a firing rate, with a linear-sloped shape used here to approximate the biophysical model. The features of the activity and phase portrait of the biophysical model.
Model features.
The simplified model enables explicit control of the bursting waveform through its interpretable parameters. (A) Burst duration is controlled through the intrinsic active duration Tactive and quiet duration Tquiet (achieved at input strength I = 0) and the active scale 
Biophysical model parameters for single-unit experiments.
Simplified model parameters for single-unit experiments.
Response to constant stimuli.
Top: Durations of the active state (left, filled bars) and quiet state (right, unfilled bars) at various stimulus strengths, along with the resulting duty cycles. Bottom: Activity at representative stimulus strengths. In both the biophysical model (A) and simplified model (B), the quiet duration decreases faster than the active duration as stimulus strength increases, thus increasing the duty cycle. The models transition into the silent mode (quiet only) at strong negative stimuli and the tonic mode (active only) at strong positive input. Figure 3—figure supplement 1. Biophysical model response waveforms to constant stimuli. Figure 3—figure supplement 2. Simplified model response waveforms to constant stimuli. Figure 3—figure supplement 3. Simplified model effect of parameters on constant response.
Response to pulse stimuli.
Top: Phase response curve showing phase shifts induced by various stimulus strengths and phases (calculated from intrinsic cycle duration T and perturbed cycle duration T ′). The phase of the intrinsic active-to-quiet state switch is marked with a vertical line. Pulse durations are 100 ms. Bottom: Activity of representative responses to excitatory and inhibitory pulses (strengths are the strongest values shown; phases are marked by arrows in the phase response curve). In both the biophysical model (A) and simplified model (B), pulse stimuli can advance or delay the waveform, depending on the stimulus strength and phase relative to the intrinsic cycle. Figure 4—figure supplement 1. Biophysical model response waveforms to pulse stimuli. Figure 4—figure supplement 2. Simplified model response waveforms to pulse stimuli. Figure 4—figure supplement 3. Simplified model effect of parameters on pulse response.
Response to periodic stimuli.
Top: Arnold tongue plot showing entrainment level induced by various stimulus strengths and periods (Arnold and Avez, 1989). The entrainment level is quantified by phase coherence (calculated from the circular mean of the phases of waveform peaks relative to the stimulus). Pulse durations are 100 ms. Periods are normalized by the intrinsic cycle duration. Bottom-left: Activity of representative responses to pulse trains with shorter and longer periods (strengths and periods are marked by arrows in the Arnold tongue). Bottom-right: Phase of waveform peaks relative to the stimulus, with stabilization of the phase indicating entrainment. In both the biophysical model (A) and simplified model (B), activity can entrain to positive and negative periodic stimuli. Stronger stimuli more effectively produce entrainment, as do stimuli with periods close to the intrinsic cycle duration. Activity returns to the intrinsic cycle once stimuli are removed. Figure 5—figure supplement 1. Biophysical model response waveforms to periodic stimuli. Figure 5—figure supplement 2. Simplified model response waveforms to periodic stimuli. Figure 5—figure supplement 3. Simplified model effect of parameters on periodic response.
Crustacean pyloric circuit.
(A) Schematic of the circuit adapted from Alonso and Marder (2020). The 3 neurons are connected through inhibitory synapses (blue circles). (B) Activity of the circuit with intact synapses. A periodic, triphasic rhythm emerges in which neurons burst in sequence: PD → LP → PY. (C) Activity of the circuit with isolated neurons after ablating synapses. Only the PD neuron bursts intrinsically, indicating that the LP and PY neurons were activated by post-inhibitory rebound effects. (D) Activity of the circuit under temperature perturbation, which was modeled as scaling of time-based parameters. (E) Effects of temperature perturbations on each neuron’s frequency, duty cycle, and phase. The simplified model reproduces these phenomena from Alonso and Marder (2020), in which burst frequency increases with temperature, but the average duty cycle and phase relationships between neurons remain constant.
Simplified model parameters for circuit experiments.
Mammalian locomotor circuit.
(A) Schematic of the circuit adapted from Danner et al.(2017); Ausborn et al. (2019); Zhang et al. (2022). The 4 limbs are each controlled by reciprocally inhibiting flexor and extensor bursting half-centers, which are connected through excitatory synapses (red arrows) and inhibitory synapses (blue circles) to non-bursting spinal interneurons. Brainstem command (BC) provides top-down modulation to half-centers and interneurons. Limbs are named fore-left (FL), fore-right (FR), hind-left (HL), and hind-right (HR). (B) Activity of the circuit in response to increasing brainstem command (BC). Top: Flexor activity for each limb. Bottom: Extensor activity for each limb (rendered as a simulated footfall plot to facilitate gait analysis). As BC increases, the gait transitions from: stand → walk → trot → gallop → bound. (C) Locomotor frequency increases with increasing BC. (D) Locomotor phase durations decrease with increasing BC, with the extension duration decreasing sharply compared to the flexion duration. The simplified model reproduces these phenomena from Danner et al. (2017). Figure 7—figure supplement 1. Activity of neuron types at different locomotor frequencies.
Biophysical model response waveforms to constant stimuli.
Activity for all stimulus strengths shown in Figure 3, along with the corresponding phase portraits. For stimulus strengths above 0.04, the activity does not have a quiet phase as the neuron fires at low rates between bursts. For stimulus strengths above 0.08, the neuron enters a stable tonic firing mode. For stimulus strengths below –0.04, the neuron enters a stable silent mode. Intervals show the active and quiet durations (in milliseconds) and the duty cycle (as a percentage of the oscillation period).
Simplified model response waveforms to constant stimuli.
Activity for all stimulus strengths shown in Figure 3, along with the corresponding phase portraits. Across stimulus strengths, the activity and phase portraits of the simplified model closely approximate that of the biophysical model. The simplified model decouples the shape of active ad quiet legs from the activity waveform’s shape and amplitude through the firing function.
Simplified model effect of parameters on constant response.
(A) Baseline response with default parameters from Table 2, modified to use a rectangular burst shape (in order to remove possible confounding effects of the firing function on waveform measurements). (B) Increasing only the active duration scale makes the active duration 65% longer than the intrinsic active duration Tactive at maximum input strength, while the quiet duration response is unchanged. (C) Removing both active and quiet duration scales results in the intrinsic durations Tactive and Tquiet being expressed at all input strengths. (D) Shortening the adaptation time constant narrows the response profile. (E) Lengthening the adaptation time constant widens the response profile. (F) Lowering both active and quiet stable point thresholds lowers the input strengths that induce tonic and silent modes, respectively.
Biophysical model response waveforms to pulse stimuli.
Activity for stimuli across various phases shown in Figure 4, along with the corresponding phase portraits. (A) Excitatory pulses (strength 0.08, duration 100 ms) tend to lengthen the active phase and shorten the quiet phase. (B) Inhibitory pulses (strength –0.08, duration 100 ms) tend to shorten the active phase and lengthen the quiet phase.
Simplified model response waveforms to pulse stimuli.
Activity for stimuli across various phases shown in Figure 4, along with the corresponding phase portraits. (A,B) Across both excitatory and inhibitory pulses, the activity and phase portraits of the simplified model closely approximate that of the biophysical model.
Simplified model effect of parameters on pulse response.
(A) Baseline response with default parameters from Table 2, modified to use a rectangular burst shape, no adaptation time constant scales, and no switching delay. (B) Changing the active and quiet duration scales (for positive input) results in different active and quiet bounds, which can dramatically alter the phase response curve in addition to the waveform duty cycle. In this example, excitatory and inhibitory pulses in the active phase have flipped advancing or delaying effects. (C) Shortening the active duration scale (for negative input) enables a stronger advancing effect by shortening the active phase. (D) Lengthening the adaptation time constant results in active and quiet bounds that are closer together (to maintain the desired active and quiet durations), with active and quiet nullcline legs overlapping less across different input strengths, leading to more responsive state switches due to falling off the legs. (E) Changing the adaptation time constant scales results in phase-independent speed changes along the active and quiet nullcline legs (corresponding to vertical translation of the flat segments in the phase response curve), while maintaining state switches due to falling off the legs (corresponding to the diagonal segments in the phase response curve). (F) Increasing the quiet switching delay creates a delay preceding the state switch, which is input-dependent since the delay is a constant fraction of the quiet duration.
Biophysical model response waveforms to periodic stimuli.
Activity for stimuli across various strengths and periods shown in Figure 5, along with the corresponding phase plots. The entrainment response is closely related to the phase shift response from the pulse stimuli analysis. (A) Shorter-period pulse trains (normalized period 0.85, duration 100 ms) entrain leftward to stable points in the phase response curve (active start for excitatory pulses, quiet start for inhibitory pulses). (B) Longer-period pulse trains (normalized period 1.15, duration 100 ms) entrain rightward to stable points in the phase response curve (active end for excitatory pulses, quiet end for inhibitory pulses).
Simplified model response waveforms to periodic stimuli.
Activity for stimuli across various strengths and periods shown in Figure 5, along with the corresponding phase plots. (A,B) Across both shorter-period and longer-period pulse trains, the activity and phase plots of the simplified model closely approximate that of the biophysical model.
Simplified model effect of parameters on periodic response.
(A) Baseline response with default parameters from Table 2, modified as in the pulse stimuli analysis. (B) Changing the active and quiet duration scales (for positive input) results in different active and quiet bounds, which can dramatically alter the entrainment response. In this example, entrainment is difficult due to the lack of stable points in the phase response curve (positive-sloped zero crossings), except for a small region for positive input at the quiet-to-active switch that is utilized by shorter-period, excitatory stimuli. (C) Shortening the active duration scale (for negative input) increases entrainment for inhibitory stimuli due to the larger phase shift response for negative input. (D) Lengthening the adaptation time constant increases entrainment due to the larger phase shift response. (E) Changing the adaptation time constant scales increases entrainment due to the larger and more stable phase shift response. (F) Increasing the quiet switching delay increases entrainment for longer-period, inhibitory stimuli by shifting the stable point for negative input earlier and enabling a subsequent pulse to arrive in the stabilizing region of the phase response curve.
Activity of neuron types at different locomotor frequencies.
Limbs are plotted with hindlimbs on the left (red) and forelimbs on the right (blue), with limb colors and neuron names corresponding to Figure 7. Gaits are lateral-sequence walk (at BC = 0.2), trot (at BC = 0.6), and bound (at BC = 1). Flexor half-centers are nominally in silent mode and increase bursting frequency with BC. Extensor half-centers are nominally in tonic mode and increase burst frequency with BC. The V0d neurons promote limb alternation and dominate at walk (both cross and diagonal descending) and trot (cross only). The V3f neurons promote limb synchronization and dominate at bound (cross only). The V3e neurons promote limb synchronization and are important for establishing the correct phase relations for lateral-sequence walk by suppressing rebound (Danner et al., 2016). The V0v neurons and V3a promote limb synchronization and dominate at trot (diagonal descending and diagonal ascending, respectively). These activity profiles approximate those from Zhang et al. (2022), which modeled the circuit using populations of conductance-based spiking neurons.
