# The role of phase shifts of sensory inputs in walking revealed by means of phase reduction

- 108 Downloads

## Abstract

Detailed neural network models of animal locomotion are important means to understand the underlying mechanisms that control the coordinated movement of individual limbs. Daun-Gruhn and Tóth, Journal of Computational Neuroscience *31*(2), 43–60 (2011) constructed an inter-segmental network model of stick insect locomotion consisting of three interconnected central pattern generators (CPGs) that are associated with the protraction-retraction movements of the front, middle and hind leg. This model could reproduce the basic locomotion coordination patterns, such as tri- and tetrapod, and the transitions between them. However, the analysis of such a system is a formidable task because of its large number of variables and parameters. In this study, we employed phase reduction and averaging theory to this large network model in order to reduce it to a system of coupled phase oscillators. This enabled us to analyze the complex behavior of the system in a reduced parameter space. In this paper, we show that the reduced model reproduces the results of the original model. By analyzing the interaction of just two coupled phase oscillators, we found that the neighboring CPGs could operate within distinct regimes, depending on the phase shift between the sensory inputs from the extremities and the phases of the individual CPGs. We demonstrate that this dependence is essential to produce different coordination patterns and the transition between them. Additionally, applying averaging theory to the system of all three phase oscillators, we calculate the stable fixed points - they correspond to stable tripod or tetrapod coordination patterns and identify two ways of transition between them.

## Keywords

Central pattern generators Inter-segmental coordination Phase oscillator model Stepping patterns Transition Speed control 6-legged locomotion## Notes

### Acknowledgements

We would like to thank Philip Holmes for useful discussions in the course of the work. This research was supported by Deutsche Forschungsgemeinschaft (DFG) grants GR3690/2-1, GR3690/4-1, DA1953/5-2, and by BMBF-NSF grant 01GQ1412.

### Compliance with Ethical Standards

### Conflict of interests

The authors declare that they have no conflict of interest

## References

- Aminzare, Z., Srivastava, V., Holmes P. (2018). Gait Transitions in a Phase Oscillator Model of an Insect Central Pattern Generator.
*SIAM Journal on Applied Dynamical Systems*,*17*(1), 626–671. https://doi.org/10.1137/17M1125571.CrossRefGoogle Scholar - Ayali, A., Couzin-Fuchs, E., David, I., Gal, O., Holmes, P., Knebel, D. (2015). Sensory feedback in cockroach locomotion: current knowledge and open questions.
*Journal of Comparative Physiology A: Neuroethology, Sensory, Neural, and Behavioral Physiology*,*201*(9), 841–850. https://doi.org/10.1007/s00359-014-0968-1.CrossRefPubMedGoogle Scholar - Borgmann, A., Scharstein, H., Büschges, A. (2007). Intersegmental coordination: influence of a single walking leg on the neighboring segments in the stick insect walking system.
*Journal of Neurophysiology*,*98*(3), 1685–1696. https://doi.org/10.1152/jn.00291.2007.CrossRefPubMedGoogle Scholar - Borgmann, A., Hooper, S.L., Büschges, A. (2009). Sensory feedback induced by front-leg stepping entrains the activity of central pattern generators in caudal segments of the stick insect walking system.
*The Journal of Neuroscience*,*29*(9), 2972–2983. https://doi.org/10.1523/JNEUROSCI.3155-08.2009.CrossRefPubMedGoogle Scholar - Borgmann, A., Tóth, T.I., Gruhn, M., Daun-Gruhn, S., Büschges, A. (2011). Dominance of local sensory signals over inter-segmental effects in a motor system: experiments.
*Biological Cybernetics*,*105*(5-6), 399–411. https://doi.org/10.1007/s00422-012-0473-y.CrossRefPubMedGoogle Scholar - Brown, T.G. (1914). On the nature of the fundamental activity of the nervous centres; together with an analysis of the conditioning of rhythmic activity in progression, and a theory of the evolution of function in the nervous system.
*The Journal of Physiology*,*48*(1), 18–46. https://doi.org/10.1113/jphysiol.1914.sp001646.CrossRefPubMedPubMedCentralGoogle Scholar - Büschges, A. (2005). Sensory control and organization of neural networks mediating coordination of multisegmental organs for locomotion.
*J Neurophysiol*,*93*(3), 1127–1135. https://doi.org/93/3/1127[pii]10.1152/jn.00615.2004.CrossRefPubMedGoogle Scholar - Büschges, A., & Gruhn, M. (2007). Mechanosensory Feedback in Walking: From Joint Control to Locomotor Patterns, vol 34. https://doi.org/10.1016/S0065-2806(07)34004-6.
- Calabrese, R.L., Hill, A.A.V., Van Hooser, S.D. (2003) In Arbib, M.A. (Ed.),
*Half-center oscillators underlying rhythmic movements*, 2nd edn., (pp. 507–510). Cambridge: A Bradford Book.Google Scholar - Clewley, R. (2011). Inferring and quantifying the role of an intrinsic current in a mechanism for a half-center bursting oscillation: A dominant scale and hybrid dynamical systems analysis.
*Journal of Biological Physics*,*37*(3), 285–306. https://doi.org/10.1007/s10867-011-9220-1.CrossRefPubMedPubMedCentralGoogle Scholar - Cohen, A.H., Holmes, P.J., Rand, R.H. (1982). The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model.
*Journal of Mathematical Biology*,*13*(3), 345–369. https://doi.org/10.1007/BF00276069.CrossRefPubMedGoogle Scholar - Collins, J.J., & Stewart, I. (1993). Hexapodal gaits and coupled nonlinear oscillator models.
*Biological Cybernetics*,*68*(4), 287–298. https://doi.org/10.1007/BF00201854.CrossRefGoogle Scholar - Couzin-Fuchs, E., Kiemel, T., Gal, O., Ayali, A., Holmes, P. (2015). Intersegmental coupling and recovery from perturbations in freely running cockroaches.
*Journal of Experimental Biology*,*218*(2), 285–297. https://doi.org/10.1242/jeb.112805.CrossRefPubMedPubMedCentralGoogle Scholar - Cruse, H. (1990). What mechanisms coordinate leg movement in walking arthropods
*Trends in Neurosciences*,*13*(1), 15–21. https://doi.org/10.1016/0166-2236(90)90057-H.CrossRefPubMedGoogle Scholar - Cruse, H. (2002). The functional sense of central oscillations in walking.
*Biological Cybernetics*,*86*(4), 271–280. https://doi.org/10.1007/s00422-001-0301-2.CrossRefPubMedGoogle Scholar - Daun, S., Rubin, J.E., Rybak, I.A. (2009). Control of oscillation periods and phase durations in half-center central pattern generators: A comparative mechanistic analysis.
*Journal of Computational Neuroscience*,*27*(1), 3–36. https://doi.org/10.1007/s10827-008-0124-4.CrossRefPubMedPubMedCentralGoogle Scholar - Daun-Gruhn, S. (2011). A mathematical modeling study of inter-segmental coordination during stick insect walking.
*Journal of Computational Neuroscience*,*30*(2), 255–278. https://doi.org/10.1007/s10827-010-0254-3.CrossRefPubMedGoogle Scholar - Daun-Gruhn, S., & Tóth, T.I. (2011). An inter-segmental network model and its use in elucidating gait-switches in the stick insect.
*Journal of Computational Neuroscience*,*31*(1), 43–60. https://doi.org/10.1007/s10827-010-0300-1.CrossRefPubMedGoogle Scholar - Daun-Gruhn, S., Tóth, T.I., Borgmann, A. (2011). Dominance of local sensory signals over inter-segmental effects in a motor system: Modeling studies.
*Biological Cybernetics*,*105*(5-6), 413–426. https://doi.org/10.1007/s00422-012-0474-x.CrossRefPubMedGoogle Scholar - Delcomyn, F. (1980). Neural basis of rhythmic behavior in animals.
*Science*,*210*(4469), 492–498. https://doi.org/10.1126/science.7423199.CrossRefPubMedGoogle Scholar - Doedel, E.J., Fairgrieve, T.F., Sandstede, B., Champneys, A.R., Kuznetsov, Y.A., Wang, X. (2007). Auto-07p: Continuation and bifurcation software for ordinary differential equations. Tech. rep., California Institute of Technology, Pasadena CA 91125. http://www.macs.hw.ac.uk/gabriel/auto07/auto.html.
- Dürr, V., Schmitz, J., Cruse, H. (2004). Behaviour-based modelling of hexapod locomotion: Linking biology and technical application.
*Arthropod Structure and Development*,*33*(3), 237–250. https://doi.org/10.1016/j.asd.2004.05.004.CrossRefPubMedGoogle Scholar - Ekeberg, Ö., Blu̇mel, M., Büschges, A. (2004). Dynamic simulation of insect walking.
*Arthropod Structure and Development*,*33*(3), 287–300. https://doi.org/10.1016/j.asd.2004.05.002.CrossRefPubMedGoogle Scholar - Ermentrout, B. (1996). Type I membrances, phase resetting curves, and synchrony.
*Neural Computation*,*8*(5), 979–1001. https://doi.org/10.1162/neco.1996.8.5.979.CrossRefPubMedGoogle Scholar - Grabowska, M., Godlewska, E., Schmidt, J., Daun-Gruhn, S. (2012). Quadrupedal gaits in hexapod animals - inter-leg coordination in free-walking adult stick insects.
*Journal of Experimental Biology*,*215*(24), 4255–4266. https://doi.org/10.1242/jeb.073643.CrossRefPubMedGoogle Scholar - Grabowska, M., Tóth, T.I., Smarandache-Wellmann, C., Daun-Gruhn, S. (2015). A network model comprising 4 segmental, interconnected ganglia, and its application to simulate multi-legged locomotion in crustaceans.
*Journal of Computational Neuroscience*,*38*(3), 601–616. https://doi.org/10.1007/s10827-015-0559-3.CrossRefPubMedGoogle Scholar - Graham, D. (1972). A behavioural analysis of the temporal organisation of walking movements in the 1st instar and adult stick insect (Carausius morosus).
*Journal of Comparative Physiology*,*81*(1), 23–52. https://doi.org/10.1007/BF00693548.CrossRefGoogle Scholar - Graham, D. (1985). Pattern and Control of Walking in Insects.
*Advances in Insect Physiology*,*18*, 31–140. https://doi.org/10.1016/S0065-2806(08)60039-9.CrossRefGoogle Scholar - Guckenheimer, J., & Holmes, P. (1983).
*Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Applied Mathematical Sciences*. Berlin: Springer.CrossRefGoogle Scholar - Holmes, P., Full, R.J., Koditschek, D., Guckenheimer, J. (2006). The Dynamics of Legged Locomotion: Models, Analyses, and Challenges.
*SIAM Review*,*48*(2), 207–304. https://doi.org/10.1137/S0036144504445133.CrossRefGoogle Scholar - Hoppensteadt, F.C., & Izhikevich, E.M. (1997).
*Weakly connected neural networks, Applied Mathematical Sciences, vol 126*. Berlin: Springer. https://doi.org/10.1007/978-1-4612-1828-9.CrossRefGoogle Scholar - Izhikevich, E.M. (2000). Phase Equations for Relaxation Oscillators.
*SIAM Journal on Applied Mathematics*,*60*(5), 1789–1804. https://doi.org/10.1137/S0036139999351001.CrossRefGoogle Scholar - Izhikevich, E.M. (2007).
*Dynamical systems in neuroscience: The geometry of excitability and bursting. 1*. Cambridge: The MIT Press.Google Scholar - Jones, S.R., Mulloney, B., Kaper, T.J., Kopell, N. (2003). Coordination of cellular pattern-generating circuits that control limb movements: the sources of stable differences in intersegmental phases.
*The Journal of neuroscience*,*23*(8), 3457–3468.CrossRefPubMedGoogle Scholar - Katz, P.S. (2016). Evolution of central pattern generators and rhythmic behaviours.
*Philosophical transactions of the Royal Society of London Series B, Biological Sciences*,*371*(1685), 20150,057. https://doi.org/10.1098/rstb.2015.0057.CrossRefGoogle Scholar - Kukillaya, R.P., & Holmes, P. (2009). A model for insect locomotion in the horizontal plane: Feedforward activation of fast muscles, stability, and robustness.
*Journal of Theoretical Biology*,*261*(2), 210–226. https://doi.org/10.1016/j.jtbi.2009.07.036.CrossRefPubMedGoogle Scholar - Kukillaya, R.P., & Holmes, P.J. (2007). A hexapedal jointed-leg model for insect locomotion in the horizontal plane.
*Biological Cybernetics*,*97*(5-6), 379–395. https://doi.org/10.1007/s00422-007-0180-2.CrossRefPubMedGoogle Scholar - Kuramoto, Y. (1984).
*Chemical Oscillations, Waves, and Turbulence, vol 19*. Berlin: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-69689-3.CrossRefGoogle Scholar - Malkin, I.G. (1949). Methods of Poincar’e and Lyapunov in the theory of non-linear oscillations Moscow:. Moscow:Gostexizdat., [in Russian: Metodi Puankare i Liapunova v teorii nelineinix kolebanii].Google Scholar
- Malkin, I.G. (1959). Some problems in nonlinear oscillation theory. Moscow: Gostexizdat., [in Russian: Nekotorye zadachi teorii nelineinix kolebanii].Google Scholar
- Massarelli, N., Clapp, G., Hoffman, K., Kiemel, T. (2016). Entrainment Ranges for Chains of Forced Neural and Phase Oscillators.
*The Journal of Mathematical Neuroscience*,*6*(1), 6. https://doi.org/10.1186/s13408-016-0038-9.CrossRefPubMedGoogle Scholar - Mendes, C.S., Bartos, I., Akay, T., Mȧrka, S., Mann, R.S. (2013). Quantification of gait parameters in freely walking wild type and sensory deprived Drosophila melanogaster.
*eLife*,*2013*(2), e00,231. https://doi.org/10.7554/eLife.00231.Google Scholar - Proctor, J., & Holmes, P. (2010). Reflexes and preflexes: On the role of sensory feedback on rhythmic patterns in insect locomotion.
*Biological Cybernetics*,*102*(6), 513–531. https://doi.org/10.1007/s00422-010-0383-9.CrossRefPubMedGoogle Scholar - Proctor, J., Kukillaya, R.P., Holmes, P. (2010). A phase-reduced neuro-mechanical model for insect locomotion: feed-forward stability and proprioceptive feedback.
*Philosophical transactions Series A, Mathematical, Physical, and Engineering Sciences*,*368*(1930), 5087–5104. https://doi.org/10.1098/rsta.2010.0134.CrossRefPubMedGoogle Scholar - Roberts, A., & Roberts, L.B. (1983).
*Neural origins of rhythmic movements*. Cambridge: Cambridge University Press.Google Scholar - Schilling, M., Hoinville, T., Schmitz, J., Cruse, H. (2013). Walknet, a bio-inspired controller for hexapod walking.
*Biological Cybernetics*,*107*(4), 397–419. https://doi.org/10.1007/s00422-013-0563-5.CrossRefPubMedPubMedCentralGoogle Scholar - Skinner, F.K., Kopell, N., Mulloney, B. (1997). How does the crayfish swimmeret system work? Insights from nearest-neighbor coupled oscillator models.
*Journal of Computational Neuroscience*,*4*(2), 151–160. https://doi.org/10.1023/A:1008891328882.CrossRefPubMedGoogle Scholar - Somers, D., & Kopell, N. (1993). Rapid synchronization through fast threshold modulation.
*Biological Cybernetics*,*68*(5), 393–407. https://doi.org/10.1007/BF00198772.CrossRefPubMedGoogle Scholar - Somers, D., & Kopell, N. (1995). Waves and synchrony in networks of oscillators of relaxation and non-relaxation type.
*Physica D: Nonlinear Phenomena*,*89*(1-2), 169–183. https://doi.org/10.1016/0167-2789(95)00198-0.CrossRefGoogle Scholar - Tóth, T.I., & Daun-Gruhn, S. (2016). A three-leg model producing tetrapod and tripod coordination patterns of ipsilateral legs in the stick insect.
*Journal of Neurophysiology*,*115*(2), 887–906. https://doi.org/10.1152/jn.00693.2015.CrossRefPubMedGoogle Scholar - Tóth, T.I., Grabowska, M., Schmidt, J., Büschges, A., Daun-Gruhn, S. (2013a). A neuro-mechanical model explaining the physiological role of fast and slow muscle fibres at stop and start of stepping of an insect leg.
*PLOS ONE*,*8*(11), e78,246. https://doi.org/10.1371/journal.pone.0078246. - Tóth, T.I., Schmidt, J., Büschges, A., Daun-Gruhn, S. (2013b). A neuro-mechanical model of a single leg joint highlighting the basic physiological role of fast and slow muscle fibres of an insect muscle system.
*PLOS ONE*,*8*(11), e78,247. https://doi.org/10.1371/journal.pone.0078247. - Tóth, T.I., Grabowska, M., Rosjat, N., Hellekes, K., Borgmann, A., Daun-Gruhn, S. (2015). Investigating inter-segmental connections between thoracic ganglia in the stick insect by means of experimental and simulated phase response curves.
*Biological Cybernetics*,*109*, 349–362. https://doi.org/10.1007/s00422-015-0647-5.CrossRefPubMedGoogle Scholar - Von Twickel, A., Hild, M., Siedel, T., Patel, V., Pasemann F. (2012). Neural control of a modular multi-legged walking machine: Simulation and hardware.
*Robotics and Autonomous Systems*,*60*(2), 227–241. https://doi.org/10.1016/j.robot.2011.10.006.CrossRefGoogle Scholar - Wang, X.J., & Rinzel, J. (1992). Alternating and synchronous rhythms in reciprocally inhibitory model neurons.
*Neural Computation*,*4*(1), 84–97. https://doi.org/10.1162/neco.1992.4.1.84.CrossRefGoogle Scholar - Wendler, G. (1965). The co-ordination of walking movements in arthropods.
*Symposium of the Society for Experimental Biology*,*20*, 229–249.Google Scholar - Wosnitza, A., Bockemuhl, T., Dubbert, M., Scholz, H., Büschges, A. (2013). Inter-leg coordination in the control of walking speed in Drosophila.
*Journal of Experimental Biology*,*216*(3), 480–491. https://doi.org/10.1242/jeb.078139.CrossRefPubMedGoogle Scholar - Zhang, C., & Lewis, T.J. (2013). Phase response properties of half-center oscillators.
*Journal of Computational Neuroscience*,*35*(1), 55–74. https://doi.org/10.1007/s10827-013-0440-1.CrossRefPubMedGoogle Scholar - Zhang, C., & Lewis, T.J. (2016). Robust phase-waves in chains of half-center oscillators. Journal of Mathematical Biology. https://doi.org/10.1007/s00285-016-1066-5.