A single-population recurrent network generates rhythm, segment-to-segment propagation, and left-right alternation.

A: Schematic connectivity diagram. B: Weight matrix illustrating outgoing inhibitory projections from a mid-body unit. C: Time-dependence of network activity.

Fast and slow speed modules enable control of locomotion frequency.

A: Connectivity schematic showing how fast and slow units project to one another. B: Map of outgoing projections from a mid-body unit (red). C: Time-dependent activity of fast- and slow-module units given different levels of tonic drive to the two populations.

Speed-module recruitment enables frequency and amplitude control.

A: Average amplitudes of the slow units (left), fast units (center), and averaged over both fast and slow units (right) during locomotion with different levels of tonic drive to fast and slow units. B: Levels of tonic drive to the fast and slow populations determine locomotion frequency. C: Average amplitude and frequency along the path shown as a solid line in (A) and (B). D: Average amplitude and frequency along the path shown as a dash-dot line in (A) and (B). E: Frequency-dependent recruitment of fast and slow units as a function of locomotion frequency. F: Phase difference between left and right units within each segment (dotted line corresponds to half of a period). G: Phase difference between pairs of units on the same side in adjacent segments (dotted line corresponds to 1/N, where N = 30 is the number of segments). Error bars in all panels denote standard deviation across units.

A model with excitatory and inhibitory populations.

A: Schematic connectivity diagram for the eight-population model. B: Detailed connectivity matrices for an example unit from each population. C: Time-dependent activity traces at slow (left) and fast (right) locomotion frequencies (traces are slightly offset for clarity).

Single-cell properties and excitatory connectivity influence locomotor frequency in an individual speed module.

A, B: Dependence of locomotion frequency on the axonal delay per segment and membrane time constant of units ((A) shows a broad range of values; (B) shows an inset from (A)). Stars denote experimentally observed values for fast and slow excitatory V2a cells in zebrafish [37, 33] C: Dependence of locomotion frequency on the projection distances of excitatory units.

Frequency range depends on excitatory projection strength and modularity.

A: Dependence of the range of possible locomotion frequencies on the global strength of excitatory projections relative to that of inhibitory projections. B: Dependence of the frequency range on connectivity modularity, which quantifies the strength of inter-module (fast-to-slow and slow-to-fast) projections relative to intra-module (fast-to-fast and slow-to-slow) projections. (Missing intermediate points correspond to cases where coordinated locomotion does not appear.) C: Dependence of the frequency range on connectivity modularity of excitatory units, where inhibitory units have modularity set to zero. D: Dependence of the frequency range on connectivity modularity of inhibitory units, where excitatory units have modularity set to zero. (In (A), modularity is set to zero; in (B)-(D), strength of excitation is set to 0.4.)

Ablating populations affects locomotion frequency and amplitude.

A: Dependence of locomotion frequency (normalized to its unperturbed value) on ablation of each of the four interneuron populations during slow speed oscillations (dashed lines; fast drive = 1.0, slow drive = 1.0; frequency = 9.3 Hz) and fast speed oscillations (dotted lines; fast drive = 2.0, slow drive = 0.5; frequency = 34.0 Hz). Asterisks mark points where the model failed to produce a coherent oscillation (see final subsection in Methods). B: Dependence of amplitude (normalized to its unperturbed value and averaged across all populations) on ablation of each of the four interneuron populations.

Recruitment of fast module at high frequencies inhibits slow module.

Data from Figure 3B plotted along paths with constant drive to the slow population (left: small drive; right: large drive) as drive to the fast population is varied.

Speed-module recruitment enables coordinated locomotion with frequency and amplitude control in an eight-population model.

A: Amplitude of the fast population (left), the slow population (center), and averaged over the fast and slow populations (right). B: Levels of tonic drive to the fast and slow populations determine locomotion frequency. C: Average amplitude and frequency along the path shown as a solid line in (A) and (B). D: Average amplitude and frequency along the path shown as a dash-dot line in (A) and (B). E: Frequency-dependent recruitment of fast and slow units as a function of locomotion frequency. F: Phase difference between units within each segment (dotted line corresponds to half of a period). G: Phase difference between pairs of units on the same side in adjacent segments (dotted line corresponds to 1/N, where N = 30 is the number of segments). Error bars in all panels denote standard deviation across units.

Frequency determination from time series is performed through calculating the period from the autocorrelation spectrum.

A: An example of a high-frequency rate time series from a single unit in the 8-population model. B: The autocorrelation spectrum corresponding to the time series in (A). The global minimum and resultant period are marked in dashed and solid lines, respectively. C: An example of a low frequency time series from a single unit in the 8-population model. D: The autocorrelation spectrum for the time series in (C). The global minimum and resultant period are marked in dashed and solid lines, respectively. E: An example time series without a single dominant frequency in the 8-population model. The failure was brought about by increasing the global strength of excitatory connections to 0.5 and the modularity to 0.4. F: The autocorrelation spectrum of the the time series in (E). The global minimum and resultant period are marked in dashed and solid black lines, respectively. The first local minimum and resultant period are marked in dashed and solid red lines, respectively.