We simulated an increase in treadmill speed in the tied-belt condition by the progressive increase of parameters β_L=β_R. As stated above, we assumed that intact cats voluntarily adjust supraspinal drives to the left and right RGs to the treadmill speed to maintain a fixed position relative to the external space. Therefore, in the intact model, the parameters α_L=α_R that characterized supraspinal drives were adjusted to correspond to the parameters β_L=β_R characterizing treadmill speed. In intact (Figure 5A) and spinal-transected (Figure 5B) models, changes in cycle and phase durations were qualitatively similar to experimental data (compare with Figure 1A and B). Specifically, the oscillation period (cycle duration) decreases with an increasing ‘speed’ due to a shortening of the extensor phase with a relatively constant flexor phase duration, so that the duty factor approaches or reaches 0.5. Comparison of changes in cycle and phase durations of the intact and spinal-transected models demonstrates similar dynamics of changes (Figure 5C), matching corresponding experimental data (Figure 1C). It is important to note that in the intact model at ‘slow treadmill speeds’ (low β_L=β_R) and the corresponding weak supraspinal drives (α_L=α_R, approximately at or below 0.4), both RGs are unable to intrinsically generate rhythmic activity and operate in a state-machine regime where switching from the extensor to the flexor phase is mainly controlled by SF-1 and SF-2. At higher values of ‘treadmill speeds’ and corresponding supraspinal drives, locomotion begins to be mainly controlled by supraspinal drives to the RGs. In this case, the role of limb sensory feedback in the control of phase durations is reduced because of increased presynaptic inhibition by increasing supraspinal drives (see Figure 5C). In contrast to the intact model, the transected model has no supraspinal drives to the RGs and can only operate in a state-machine regime at all treadmill speeds. Thus, the transition from extension to flexion and the duration of the extensor phase are entirely controlled by sensory feedback.

For split-belt locomotion, we simulated the speed of the ‘slow’ RG with a fixed value of β_L = 0.4 and the ‘fast’ RG with β_R progressively increasing from 0.5 to 1.0. Again, based on our assumption above, the supraspinal drives to the left and right RGs in the intact model were adjusted to the corresponding speeds of the treadmill belts. We set a constant drive α_L = 0.4 to the left RG and progressively increased the drive α_R to the right RG from 0.5 to 1.0. We show that in the intact model (Figure 6A), despite differences in the ‘speeds’ of left and right RGs, the left and right oscillation periods are equal, which corresponds to experimental data (Figure 2A).

In the spinal-transected model (Figure 6B), changes in cycle and phase durations are similar to those in the intact model and correspond to experimental data (Figure 2B). The oscillation periods on the left and right sides are equal, and changes in cycle and phase durations on the left slow side do not change much with a progressive increase of β_R from 0.5 to 1.0. The flexor phase duration of the fast RG increases with increasing β_R, which compensates for the decrease in the extensor phase duration.

An increase of flexion phase duration on the fast side with an increase of speed on that side in the spinal-transected model is much steeper than in the intact model (compare Figure 6B with Figure 6A) which reproduces corresponding experimental data (see Figure 2B vs. Figure 2A). The mechanism of this increase in the spinal case (spinal-transected model) is the following. With the increase in speed, the SF-E1 starts accumulating a tonic component that continues acting during flexion. This accumulated tonic activity provides a direct excitation to the ipsilateral F half-center during flexion, which is compensated by inhibition from the contralateral SF-E1 that inhibits the ipsilateral F half-center during the flexor phase (via contralateral V3-E CIN and ipsilateral InE neuron, see Figure 4). In tied-belt locomotion (Figure 5), this contralateral inhibition acts in balance with the accumulated ipsilateral excitation from SF-E1 and limits excitation of each F half-center during flexion, keeping both RGs within the flexor-driven operating regime with relatively constant flexor phase durations (as in Figure 3D, middle part of the graph). In split-belt locomotion, this contralateral inhibition from the slow (left) side remains relatively weak (it is fixed at a value corresponding to the constant slow left-side speed and does not change with the increasing speed on the right fast side), whereas the accumulated excitation on the fast side continues increasing with the speed. Therefore, the F half-center on the fast side becomes overexcited and the RG on the fast side starts operating in the classical half-center regime (like in Figure 3D, right part of the graph) with an increasing flexor phase duration as the speed on the fast side increases (Figure 6B). A similar mechanism operates in the intact split-belt case (Figure 6A), but the resultant increase of the flexor phase duration on the fast side is much weaker due to presynaptic inhibition of left and right SF-E1 feedback.

We used our model to simulate the effects of removing limb sensory feedback. The results of these simulations allowed us to formulate modeling predictions that could be tested in future experiments, providing an additional validation to our model. We specifically focused on the simulation of separate removal of SF-E1 feedback, activated with limb extension and involved in the stance-to-swing transition, or SF-E2 feedback involved in the generation of extensor activity and weight support during stance, as well as the removal of both feedback types.

In the transected model, the removal of SF-E1 (with or without SF-E2) prevented locomotion (not shown). Without supraspinal drive and SF-E1, both RGs could not switch to the flexor phase and generate locomotor activity. Following removal of SF-E2 in the transected model, sufficient extensor activity (necessary for ‘weight support’) could not be developed, leading to an abnormal locomotion with extremely short extensor phase durations (and oscillation periods), with an inability to adjust to treadmill speed (Figure 7). In this case, locomotion could be recovered by adding an additional activation of extensor half-centers during stance phases (not shown).

The described above effects of removing limb sensory feedback on locomotion in the spinal-transected model were expected. However, what happens after the removal of limb sensory feedback in the intact model, in which supraspinal drives are present? Removing SF-E1 in the intact model produced interesting results. It considerably increased the duration of the extensor phase (Figure 8A). However, the model only demonstrated oscillatory activity starting at some moderate speed, when β_L=β_R ≥ 0.5. This required the corresponding supraspinal drives to the RGs to be sufficient (α_L=α_R ≥ 0.5) to operate in the flexor-driven regime and produce phase transitions without any contributions from SF-E1. In contrast, removing SF-E2 in the intact model reduced the activity of the extensor half-centers, shortening extensor phase durations but preserving oscillations in the full range of considered ‘speeds’ (Figure 8B). Therefore, the removal of SF-E2 produced an opposite effect on extensor phase durations compared with removing SF-E1. Finally, removing both feedback types in the intact model shifted the start of oscillatory activity to the left compared to removal of only SF-E1, allowing oscillations to start at β_L=β_R ≥ 0.35 (Figure 8C). This allows the prediction that below 0.35 m/s, cats with diminished limb sensory feedback can only perform locomotion with patterned step-by-step supraspinal signals to produce phase transitions. This can also explain why intact cats do not usually walk consistently on a treadmill with a speed at or below 0.3 m/s.

Despite decades of research, a commonly accepted definition of the spinal locomotor CPG has not been formulated, and different authors use this term in relation to different entities depending on the context. For instance, the term CPG has been used to designate spinal circuits controlling and coordinating rhythmic movements of all limbs, or controlling only rhythmic movements of a single limb, or even a single joint (Grillner, 1981; Orlovsky et al., 1999; McCrea and Rybak, 2008; Brown, 1911; Brown, 1914; Grillner and Zangger, 1975; Grillner and Zangger, 1979; Rossignol, 1996; Grillner, 2006). Here, we use the term CPG for the spinal circuitry controlling and coordinating all limbs, and the term RG (rhythm generator) for the relatively independent part of the CPG that controls rhythmic movements of a single limb. We consider the CPG as a group of RGs, each controlling a single limb and interacting with each other through commissural and/or propriospinal pathways or circuits (Frigon, 2017; Rybak et al., 2015; Shevtsova et al., 2015; Zhang et al., 2022; Shevtsova et al., 2022; Danner et al., 2016; Danner et al., 2017; Danner et al., 2019).

Regardless of the exact name (RG or unit burst generator), a common view is that the function of the RG is to generate locomotor-like rhythmic bursting consisting of two major phases, flexion and extension. In each phase, the controlled limb operates in the corresponding functional state, defining the contraction and relaxation of specific sets of muscles. Thus, the RG is not just an oscillator, but rather a system with a dual function: it functions as a state-machine (Dzeladini et al., 2014; Spaeth et al., 2020; Di Russo et al., 2023), defining the state and operation of the controlled biomechanical system in each phase independent of the exact transition mechanisms, and as an oscillator defining the mechanisms and timing of transitions between states. These transitions may be fully defined by internal properties of the RG and/or require a contribution from external inputs, such as supraspinal signals or somatosensory feedback, that control or adjust the timing of transitions.

Here, we described a relatively simple model of a half-center RG. We identified three operation regimes: state-machine, flexor-driven, and classical half-center. In an intact system, at slow speeds (≤0.35 m/s), the spinal network operates in a state regime and requires external inputs for phase transitions, which can come from limb sensory feedback and/or volitional inputs (e.g. from the motor cortex). At higher speeds and greater supraspinal drives, the spinal network switches to a flexor-driven regime before transitioning to a classical half-center regime at higher speeds/drives. Following spinal transection, the spinal network can only operate in the state regime and entirely depends on limb sensory feedback for phase transitions.

Our modeling results also resolve a potential contradiction between the classical half-center and flexor-driven concepts of spinal RG operation. According to the classical half-center concept, both flexor and extensor half-centers are necessary for generating rhythmic activity (McCrea and Rybak, 2008; Brown, 1911; Brown, 1914). An alternative, flexor-driven concept was proposed by Pearson and Duysens (‘swing generator model’), where the flexor half-center is intrinsically rhythmic, unlike the extensor half-center, which exhibits rhythmic activity due to rhythmic inhibition from the flexor half-center (Pearson and Duysens, 1976; Duysens et al., 2013). Here, we follow our previous modeling studies (Latash et al., 2020; Ausborn et al., 2018) and show that these concepts do not contradict each other, but rather relate to different regimes of RG operation. With an increase of excitatory drive to an RG, the rhythmogenic mechanism changes from a flexor-driven to a classical half-center mechanism (Figure 3). This transition explains the increase of the flexor phase duration of the fast hindlimb or RG during split-belt treadmill locomotion in intact and spinal cats (Figures 2 and 6).

Following Maurice Philippson’s view formulated from studies in spinal dogs (Philippson, 1905), Charles Sherrington proposed that locomotion in decerebrate and spinal cats was generated by chains of reflexes (Sherrington, 1910a; Sherrington, 1910b). Yet, Thomas Graham Brown clearly showed that an intrinsic spinal mechanism alone, without somatosensory feedback, could generate rhythmic alternating activity (Brown, 1911), which later became the CPG concept. Our results support both concepts, depending on locomotor speed and the state of the animal. Based on our simulations, we suggest that in spinal cats at any speed and in intact cats at low speeds, the spinal network operates in a state-machine regime and requires some sensory feedback to locomote, consistent with Philippson’s/Sherrington’s viewpoint. In contrast, at higher (moderate) speeds (≥0.4 m/s), when a supraspinal drive is sufficient, the spinal network can intrinsically generate the basic locomotor activity controlling locomotion, which supports Brown’s concept.

Our results also suggest an important difference in the control of slow exploratory and faster escape locomotion (Caggiano et al., 2018; Ferreira-Pinto et al., 2018; Kim et al., 2017; Branco and Redgrave, 2020). Based on our predictions, slow (conditionally exploratory) locomotion is not ‘automatic’ but requires volitional (e.g. cortical) signals to trigger step-by-step phase transitions because the spinal network operates in a state-machine regime. In contrast, locomotion at moderate to high speeds (conditionally escape locomotion) occurs automatically under the control of spinal rhythm-generating circuits receiving supraspinal drives that define locomotor speed, unless voluntary modifications or precise stepping are required to navigate complex terrain (Drew et al., 1996; Beloozerova et al., 2010).

We also used our model to simulate and predict the effects of removing limb sensory feedback on locomotion. As expected, the spinal-transected model failed to generate locomotion without SF-E1 or normal locomotion without SF-E2 because the spinal network operates in a state-machine regime (Figure 7). The essential role of somatosensory feedback after spinal cord injury is well-known (Frigon et al., 2021; Takeoka, 2020; Takeoka and Arber, 2019; Takeoka et al., 2014; Bouyer and Rossignol, 2003; Rossignol and Frigon, 2011). However, in the intact model, limb sensory feedback, particularly from hip flexor afferents (SF-E1), is also required for locomotion and phase transitions at slow speeds to compensate for low supraspinal drives (Figure 8).

Another important implication of our results relates to the recovery of walking in movement disorders, where the recovered pattern is generally very slow. For example, in people with spinal cord injury, the recovered walking pattern is generally less than 0.1 m/s and completely lacks automaticity (Wagner et al., 2018; Angeli et al., 2018; Gill et al., 2018). Based on our predictions, because the spinal locomotor network operates in a state-machine regime at these slow speeds, subjects need volition, additional external drive (e.g. epidural spinal cord stimulation) or to make use of limb sensory feedback by changing their posture to perform phase transitions. Another paper commenting on observations in Parkinsonian patients also proposed that different CPG control systems operate at slow and fast speeds (Duysens and Nonnekes, 2021).

The concept of a spinal locomotor CPG in humans is strongly supported by a variety of experimental and clinical evidence (Duysens and Van de Crommert, 1998; Danner et al., 2015; Minassian et al., 2017; Yang et al., 2004). In human walking, the basic output of the CPG, reflected in the EMG pattern of leg muscles, is maintained even at very slow speeds, albeit with differences in amplitudes and some bursting patterns from slow to fast walking speeds (den Otter et al., 2004; Hof et al., 2002). This is consistent with the idea that the same CPG provides the basic motor program from slow to fast speeds but that additional inputs or drives are required. The idea of a flexible arrangement of CPG circuits has also been proposed to explain the generation of locomotion and other rhythmic motor behaviors (e.g. scratching, paw shaking) in various species, such as cats, turtles, zebrafish, and rats (Frigon and Gossard, 2010; Berkowitz et al., 2010; Berkowitz and Hao, 2011; Juvin et al., 2007; Harris-Warrick, 2011; Parker et al., 2018).

In this study, we developed a simplified model of the neural control of cat hindlimb locomotion in tied- and split-belt conditions to simulate and compare the locomotion of intact and spinal cats. The major limitation of the present model is the lack of biomechanical elements simulating multi-joint limbs and muscles. Our study and analysis were specifically focused on the proposed organization and operation of spinal CPG circuits, including left-right interactions within these circuits, and their control by supraspinal drives and limb sensory feedback. Note that the term ‘supraspinal drive in our model is used to represent supraspinal inputs providing both electrical and neuromodulator effects on spinal neurons to increase their excitability, which disappears after spinal transection. The other limitation of the present model is that it does not consider the possibility that afferent feedback can provide some constant level of excitation to the RG circuits after spinal transection, which can partly compensate for the lack of supraspinal drive and hence affect (shift) the timing of transitions between the considered regimes. We will consider this issue in the future. Another important limitation is that we did not consider possible mechanisms of plasticity following spinal transection. We know that the spinal connectome and sensorimotor interactions change after spinal cord injury (Rossignol and Frigon, 2011; Frigon and Rossignol, 2006). The precise nature of these changes is not well understood. Nevertheless, we believe that changes (increase) in the gain of sensory feedback, which in the model results from eliminating presynaptic inhibition, in a real situation may include a considerable contribution from plastic changes during the recovery period. However, even with the account of possible plastic changes during the recovery period, the functional role of the increased gain of sensory feedback in spinal-transected animals remains the same independent of the exact mechanisms for this increase (release from presynaptic inhibition, plastic changes during recovery, currently unknown mechanisms, or any combination of the above).

Although the model was based on simplifications and some assumptions, it reproduced and provided explanations for experimental results, such as the main locomotor characteristics (cycle and phase durations) at different speeds and left-right speed differences during tied-belt and split-belt locomotion. We are currently applying our model to simulate cat locomotion following incomplete spinal cord injury (lateral hemisection) during tied-belt and split-belt locomotion based on our recent experimental data (Audet et al., 2023; Mari et al., 2024; Lecomte et al., 2022). Using our model, this will allow us to remove supraspinal drives on the hemisected side and determine how the spinal network controls locomotion. We also plan to incorporate a full biomechanical model of the limbs to investigate the neuromechanical control of quadrupedal locomotion and its recovery following incomplete spinal cord injury (Markin, 2016; Prilutsky, 2016).

For this modeling study, we used previously published data obtained from intact and spinal cats during tied-belt (equal left-right speeds) and split-belt (different left-right speeds) locomotion (Frigon et al., 2015; Frigon et al., 2017; Latash et al., 2020). No new animals were used here. In those studies, all procedures were approved by the Animal Care Committee of the Université de Sherbrooke (Protocol 442–18) in accordance with policies and directives of the Canadian Council on Animal Care. Intact cats performed quadrupedal locomotion whereas spinal cats performed hindlimb-only locomotion with the forelimbs on a stationary platform. Intact and spinal cats performed tied-belt locomotion from 0.4 to 1.0 m/s and from 0.1 to 1.0 m/s, respectively, with 0.1 m/s increments. As stated earlier, intact cats cannot perform consistent quadrupedal tied-belt locomotion at or below 0.3 m/s. For split-belt locomotion, the slow belt (left) was 0.4 m/s while the fast belt (right) stepped at speeds of 0.5–1.0 m/s in 0.1 m/s increments.

In our model, we considered spinal circuits as a network of interacting neural populations. Each population is described by an activity-based neuron model (and is sometimes called a neuron in the text), in which the dependent variable V represents an average population voltage and the output f(V) (0≤f(V)≤1) represents the average or integrated population activity at the corresponding average voltage (Latash et al., 2020; Danner et al., 2016; Danner et al., 2017; Ausborn et al., 2019; Rubin et al., 2009; Ermentrout, 1994). This description allows an explicit representation of ionic currents, specifically of the persistent (slowly inactivating) sodium current (Rubin et al., 2009), which was proposed to be responsible for generating intrinsic bursting activity in the spinal cord (Brocard et al., 2013; Rybak et al., 2006a; Rybak et al., 2006b; Tazerart et al., 2007; Tazerart et al., 2008; Brocard et al., 2010; McCrea and Rybak, 2007; Zhong et al., 2012). Assuming that neurons within each population switch between silence and active spiking in a generally synchronized way, the dynamics of the average voltages are represented within a conductance-based framework used for a single neuron description, but without fast membrane currents responsible for spiking activity.

The dynamics of V for flexor and extensor half-centers, considered as INaP -dependent conditional bursters, are described by the following differential equation:(1)C⋅dVdt=−INaP−IL−ISynE−ISynI.

For other (non-bursting) populations, V is described as:(2)C⋅dVdt=−IL−ISynE−ISynI.

The output function f(V) transvers V to the integrated population output and is defined as follows:(3)f(V)={0,ifV<Vthr;(V−Vthr)/(Vmax−Vthr),ifVthr≤V<Vmax;1,ifV≥Vmax.

In Equations 1 and 2, C is the membrane capacitance, INaP is the persistent sodium current, IL is the leakage current, ISynE and ISynI represent excitatory and inhibitory synaptic currents, respectively. The leakage current was described as:(4)IL=gL⋅(V−EL)

where gL is the leakage conductance and EL represents the leakage reversal potential. The persistent sodium current in the flexor and extensor half-centers is described as:(5)INaP=g¯NaP⋅m(V)⋅h⋅(V−ENa),

where g¯NaP is the maximal conductance and ENa is the sodium reversal potential. Its voltage-dependent activation, m(V), is instantaneous, and its steady state is described as follows:(6)m(V)=m∞(V)={1+exp[(V−V1/2,m)/km]}−1.

The slow INaP inactivation was modeled by the following differential equation:(7)τh(V)⋅dhdt=h∞(V)−h,

where h∞(V) represents inactivation steady state and τh(V) is the inactivation time constant with maximal value τmax.(8)h∞(V)={1+exp[(V−V1/2,h)/kh]}−1;(9)τh(V)=τmax/cosh[(V−V1/2,τ)/kτ].

In Equations 6, 8, and 9, V1/2 and k represent the half-voltage and slope of the corresponding variables (m, h, and τ). Excitatory and inhibitory synaptic currents (ISynE and ISynI) for population i were described by:(10)ISynE,i=gSynE⋅{∑j[S(wji)⋅f(Vj)]+Di+SFiE1+SFiE2}⋅(Vi−ESynE);(11)ISynI,i=gSynI⋅{∑j[S(wji)⋅f(Vj)]}⋅(Vi−ESynE),

where gSynE and gSynI are synaptic conductances and ESynE and ESynI are the reversal potentials of the excitatory and inhibitory synapses, respectively; wji is the synaptic weight from population j to population t (wji > 0 for excitatory connections and wji < 0 for inhibitory connections).(12)S(x)={x,ifx≥0,0,ifx<0.

The weights of connections wji between populations for the main model (Figure 4) are shown in Table 1. Di in Equation 10 for the main model (Figure 4), represents the total excitatory drive to population i:(13)Di=kiαipsi⋅αipsi+kiαcontra+kiγ⋅γ,

where αipsi and αcontra are the ipsi- and contralateral excitatory drives, respectively, which depending on the side, represent left αL and right αR drives, respectively; parameter γ represents the constant drive to the ipsilateral (left or right) extensor half-center; kiαipsi , kiαcontra and kiγ define the weights of these drives to population i.

SFiE1 and SFiE2 in Equation 10 for the main model (Figure 4), define the effect of ipsilateral sensory feedback E1 (or SF-E1, see Figure 4) and E2 (or SF-E2), respectively, to population i. The gain of each feedback to population i (kiE1 and kiE2) is suppressed (reduced) by the ipsilateral drive αipsi (αL or αR depending on the side):(14)SFiE1=kiE1⋅E1/(1+kPSiE1⋅αipsi);(15)SFiE2=kiE2⋅E2/(1+kPSiE2⋅αipsi),

where kPSiE1 and kPSiE2 are the weights of presynaptic inhibition by αipsi of feedback inputs E1 and E2, respectively, to population i (see Figure 4A).

E1 feedback (SF-E1) represents an increase in the activity of length-dependent hip flexor afferents during limb extension. In our model, it is described as:(16)τE1⋅dE1dt=[kE1⋅βipsi1.25⋅(t−t0)]⋅sign[f(VE)−thr]−E1,whereτE1={0,ifdE1dt≥0,τ1,ifdE1dt<0,sign[y]={1,ify>0,0,ify≤0,

where kE1 is the parameter of E1 feedback; βipsi is the parameter characterizing the speed of the ipsilateral treadmill belt (left βL or right βR), f(VE) is the output of the ipsilateral E half-center, thr is the threshold, t0 is the starting time of the ipsilateral extensor burst and t is time (both in ms).

E2 feedback (SF-E2) that represents excitatory feedback from load-dependent afferents of limb extensor muscles to the extensor half-center during extension, described as follows:(17)τE2⋅dE2dt=[E2o−kE2⋅βipsi1.25.(t−t0)]⋅sign[f(VE)−thr]−E2,whereτE2={τ2, if dE2dt≥0,0,ifdE2dt<0,

where E2o and kE2 are the parameters of SF-E2.

Variables αL and αR characterize left and right supraspinal drives to RGs and define the frequency of locomotor oscillations. The variables βL and βR define the speeds of the left and right treadmill belts. Intermediate coefficients were used to make correspondence between α and β values and between their values and the speed of real treadmill belts in the experiments. For modeling of intact locomotion, we changed left and right α values to simulate ‘voluntary’ locomotion in the intact model via a corresponding adjustment of supraspinal drives. For modeling of locomotion in the spinal-transected model, supraspinal drives were set to 0 and we only manipulated left and right β values simulating speeds of treadmill belts.

The following values of model parameters were used for the main model (Figure 4): C=20 pF; gL=4 nS for RG half-centers and gL=1 nS for all other neurons; g¯NaP=4.4 nS; gsynE=gsynI=1 nS; ENa=50.0 mV; EEsynE=−10 mV; EEsynI=−75 mV; Vthr=−50 mV; Vmax=0 mV; V1/2,m=−40.0 mV; km=−6 mV; V1/2,h=−45.0 mV; kh=4 mV; τmax=500 ms; V1/2,τ=−45 mV; kτ=20 mV; E2o=2.5; kE1=3.12⋅10−3ms−1; kE2=0.5⋅10−3ms−1; τ1=500 ms; τ2=300 ms; thr = 0.1. All connection weights for the main model (Figure 4) are specified in Table 1.

In the models shown in Figure 3: EL=−64.0 mV; w1=w3=3.6 ; w2=3.0;w4=5.0 ; Drive-E=1.2. All other parameters are the same as in the main model, listed above.

All simulations were performed using the custom neural simulation package NSM 2.5.7. The simulation package was previously used for models of spinal circuits (McCrea and Rybak, 2008; Shevtsova et al., 2015; Zhang et al., 2022; Shevtsova et al., 2022; Danner et al., 2019; Rybak et al., 2006a; Rybak et al., 2006b; McCrea and Rybak, 2007; Zhong et al., 2012; Shevtsova, 2016). Differential equations were solved using the exponential Euler integration method with a step size of 0.1 ms. Simulation results were saved as ASCII files and represented the output functions for half-centers recorded with a precision of 0.1 ms.

The simulation results were processed using custom Matlab scripts (The Mathworks, Inc, Matlab 2023b). To assess model behavior, the activities of flexor and extensor half-centers were used to determine the onsets and offsets of flexor and extensor bursts and to calculate flexor and extensor phase durations and oscillation periods. The timing of onsets and offsets of flexor and extensor bursts was determined at a threshold level of 0.05. The oscillation period was defined as the duration between two consecutive burst onsets in the left extensor half-center. The flexor and extensor phase durations and oscillation periods were averaged over the duration of the simulation for each value of the parameter α. Duration of individual simulations depended on the value of parameter α to robustly estimate average values of burst durations and oscillation periods. For each α value, we omitted the first two transitional cycles to allow stabilization of model variables. Flexor and extensor phase durations and oscillation periods were plotted against the parameters α and β.

The simulation package NSM 2.5.7, the model configuration file necessary to create and run simulations, and the custom Matlab scripts are available at https://github.com/RybakLab/nsm (copy archived at RybakLab, 2024).