This is an interesting contribution. The strengths are in the hypothetical models 2 and 3, that propose neuromechanical mechanisms for stability in a quadruped. The main weaknesses are in the presentation of the mechanical model, some assumptions that are made within it, and a lack of context within the broader mathematical literature of legged locomotion, the comparative biomechanics, and robotics literatures.

If the mechanical model can be presented more clearly and the context expanded a bit it will be a strong contribution.

Detailed comments:
P6, Figure 2. Mice really aren't very rigid. Although in general I think this approach of attempting to add a simple mechanical model is great, some comment should probably be made to the fact that unlike a horse, mice have very bendy spines.

S2.3 onward. So is the model "2.5D"? Although I appreciate the detail in the exposition of the model, I think it could use more high level mechanical description in the beginning. How many degrees of freedom is it? Is the body frame constrained in Z? I think it is, but not sure this is clearly explained. What determines the angle of the ground reaction forces? I believe the ground reaction forces are constrained to be in the XZ plane. Is that correct? Basically, it would be good to go over the presentation of the model in a manner that summarizes the main assumptions early on, so that a mechanical modeler can understand the system quickly. I'm thinking of presnetations like those in Justin Seipel's work on SLIP models.

Eqns 1 and 2. Mice are pretty squat. How far off is the small angle approximation here? I mean, are these like 45 degree angles here?

P7 L54. Can you assume that the COM is always inside the supporting triangle? If so why is this obvious? Is taht because in the models later on legs are lifted whenever it crosses out?

Eqn 4. In line with clarify the model, a free body diagram of the center of mass would help clarify the situation. Perhaps after 3?

Eqn 8. This lambda is different than the viscous drag force lambda, right? Needs subscript, or use different greek letter?

Eqn 9. As above - what differmines the magnitue and direction of GRFs? I know "thrust" is varied as a parameter in teh model, and I believe the vertical is set equal to body wieght - if so this should be stated when describing the reaction forces to help understanding.

Eqn 11. Ok so to clarify there is a constant upward force equal to body weight at all times? For low speed locomotion in mice, do force plate recordings from the literature support this? I think it might be reasonable, but would be good to cite the literature here.

S3 Results. This seems to start early - aren't there still details of methods to be presented? Eg in 3.1 i'm just deducing that the GRFs have no lateral component and that the body frame is constrained in Z.

S3.2.1 This is where a clear exposition of assumptions made for each model would help. EG, F0 is a parameter we vary, Fweight is set equal to BW.

Fig 4. The caption for this figure has a bunch of methods detail that should probably be in the main body.

S3.2.2. here or in the introduction i think some broader robotics literature should be mentioned - espcially Owaki and Ishiguro 2017 - their robot could transition through the gaits without neuronal coupling between RGs, simply through the physics of the body influencing oscillators with force feedback onto phase.

P16: Why does the leg liftoff as soon as the COM is outside of the support triangle? This is before the balance signal with threshold 0 right?

Fig. 6. The synchronization by body rotation is great and interesting. I do wonder if squishy mice actually have this, but it's worth testing. But doesn't the rotation also accelerate teh transitions of teh other pair of limbs?

Also, i think this stabilization could be better explained with a plot of body heading veruss time. You could see the asymptotic recovery, right? As with the Full and Koditschek papers where they show the lateral leg spring model recovering from perturbation over time. Aha - I see some of these are mentioned in the discussion - great - Full and Kubow, etc. Great.

S3.2.4 at the end. Here again i think Seipel's work in the plane may be relevant, along with other Full lab members who looked at cockroach stability int eh plane.

For teh simulations, why is swing duration swept for the analysis? Shouldn't indiviudual values be set and the simulation run to determine stabiltiy, and then plotted?

P23 L34. Could you measure whether this body rotation mechanism actually happens?

Eqn 15. This is neat, a novel thing to test. I especially like that this integrated signal could be estimated from multiple sources eg vestibular and cuteanous.

Which reminds me of another work that should be cited - Art Kuos' recent paper looking at CPGs as state estimators (2022?). That should be cited in the introduction areas discussin the CPG as a state machine. The state estimator role combines the state machine and phase for transitions discussion taht's in the paper.

P28 S3.4 Owaki relevant here again.

Fig 10. Are there no mouse phase data as low speeds to overlay on Fig 10 C2?

Fig 11B. Where is the dashed black line for stride duration (period)? Was that fit so matched perfectly?

P33 L50-52. This discussion of state machine vs oscillator is confusing. Not sure what the point is. I mean, you can get state from phase, right? Cut it. Here too i thin the Ryu and Kuo 2021 paper is relevant.

S4.2 Earlier the mechnical model is presented as analytically tractible/useful. But I don't see any analysis of stability, return maps, or anything like that. I think it's very useful, but the claim of analytical help is so far not really justified. Numerically it's doing fine.

S4.4 The work of Vahedipour et al, 2018 J Biomech should be mentioned here. Although at low trotting speeds, they perturbed the balance of mice with an earthquake perturbation that changed the loads on limbs and tipped the animal, and observed that the gait moved towards bound. Does this model also have that effect? Would be interesting to note.

S4.6. One of the benefits of this model is that you now have some mechanics, yet this is not really discussed. For example, how do models 2 and 3 compare in terms of aiding with mechanical stability? The model would allow for quantifying stability margin and then comparing the different neural strategies for how much they improve stability. The approaches of Wilshin et al., 2017 Longitudinal Quasi-static stability predicts changes in dog gait on rough terrain... JEB (at walking speeds) or even earlier McGhee and Frank's quadruped walk stability for robotics are highly relevant and would strengthen the argument for this model.

Heat maps with number of steps until failure instead of duty factor would be very interesting. Though at the length of the present paper would probably be for future work.

The authors have done excellent and extensive work revising this manuscript. It is a strong contribution.