The effects of feedback on stability and maneuverability of a phase-reduced model for cockroach locomotion
Abstract
In previous work, we built a neuromechanical model for insect locomotion in the horizontal plane, containing a central pattern generator, motoneurons, muscles actuating jointed legs, and rudimentary proprioceptive feedback. This was subsequently simplified to a set of 24 phase oscillators describing motoneuronal activation of agonist–antagonist muscle pairs, which facilitates analyses and enables simulations over multi-dimensional parameter spaces. Here we use the phase-reduced model to study dynamics and stability over the typical speed range of the cockroach Blaberus discoidalis, the effects of feedback on response to perturbations, strategies for turning, and a trade-off between stability and maneuverability. We also compare model behavior with experiments on lateral perturbations, changes in body mass and moment of inertia, and climbing dynamics, and we present a simple control strategy for steering using exteroceptive feedback.
Keywords
Exteroception Feedback control Hybrid systems Neuromechanics Proprioception Stability–maneuverability trade-offNotes
Acknowledgements
This work was partially supported by NSF EF-0425878 (Frontiers in Biological Research), NSF DMS-1430077 (CRCNS U.S.-German Collaboration) and Princeton’s J. Insley Blair Pyne Fund. We thank the anonymous reviewers for their useful suggestions and for helping us to correct several errors.
References
- Ahn A, Full R (2002) A motor and a brake: two leg extensor muscles acting at the same joint manage energy differently in a running insect. J Exp Biol 205:379–389PubMedGoogle Scholar
- Ahn A, Meijer K, Full R (2006) In situ muscle power differs without varying in vitro mechanical properties in two insect leg muscles innervated by the same motor neuron. J Exp Biol 209:3370–3382CrossRefPubMedGoogle Scholar
- Altendorfer R, Moore N, Komsuoglu H, Buehler M, Brown HB Jr, McMordie D, Saranli U, Full R, Koditschek D (2001) RHex: a biologically inspired hexapod runner. Auton Robots 11:207–213CrossRefGoogle Scholar
- Brown I, Scott S, Loeb G (1995) Preflexes—-programmable high-gain zero-delay intrinsic responses of perturbed musculoskeletal systems. Soc Neurosci Abstr 21(562):9Google Scholar
- Couzin-Fuchs E, Kiemel T, Gal O, Holmes P, Ayali A (2015) Intersegmental coupling and recovery from perturbations in freely-running cockroaches. J Exp Biol 218:285–297CrossRefPubMedPubMedCentralGoogle Scholar
- Cowan N, Lee J, Full R (2006) Task-level control of rapid wall following in the american cockroach. J Exp Biol 209:1617–1629CrossRefPubMedGoogle Scholar
- David I, Holmes P, Ayali A (2016) Endogenous rhythm and pattern generating circuit interactions in cockroach motor centers. Biol Open 5:1229–1240CrossRefPubMedPubMedCentralGoogle Scholar
- Delcomyn F (1980) Neural basis of rhythmic behaviors in animals. Science 210:492–498CrossRefPubMedGoogle Scholar
- Delcomyn F (2004) Insect walking and robotics. Annu Rev Entomol 149:51–70CrossRefGoogle Scholar
- Electronic Physics Auxiliary Publication Service E (2009) See document no. e-chaoeh-19-005992 for parameter values and code documentation. http://ftp.aip.org/epaps/chaos/E-CHAOEH-19-005992/. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html
- Fuchs E, Holmes P, Kiemel T, Ayali A (2011) Intersegmental coordination of cockroach locomotion: adaptive control of centrally coupled pattern generator circuits. Front Neural Circuits 4:125PubMedPubMedCentralGoogle Scholar
- Fuchs E, Holmes P, David I, Ayali A (2012) Proprioceptive feedback reinforces centrally-generated stepping patterns in the cockroach. J Exp Biol 215:1884–1891CrossRefPubMedGoogle Scholar
- Full R, Koditschek D (1999) Templates and anchors: neuromechanical hypothesis of legged locomotion on land. J Exp Biol 202:3325–3332PubMedGoogle Scholar
- Full R, Tu M (1991) Mechanics of a rapid running insect: two-, four- and six-legged locomotion. J Exp Biol 156:215–231PubMedGoogle Scholar
- Full R, Kubow T, Schmitt J, Holmes P, Koditschek D (2002) Quantifying dynamic stability and maneuverability in legged locomotion. Integr Comp Biol 42:149–157CrossRefPubMedGoogle Scholar
- Ghigliazza R, Holmes P (2004a) A minimal model of a central pattern generator and motoneurons for insect locomotion. SIAM J Appl Dyn Syst 3(4):671–700CrossRefGoogle Scholar
- Ghigliazza R, Holmes P (2004b) Minimal models of bursting neurons: how multiple currents, conductances and timescales affect bifurcation diagrams. SIAM J Appl Dyn Syst 3(4):636–670CrossRefGoogle Scholar
- Goldman D, Chen T, Dudek D, Full R (2006) Dynamics of rapid vertical climbing in a cockroach reveals a template. J Exp Biol 209:2990–3000CrossRefPubMedGoogle Scholar
- Guckenheimer J, Holmes P (2002) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, 6th edn. Springer, BerlinGoogle Scholar
- Guckenheimer J, Johnson S (1995) Planar hybrid systems. In: Lecture notes in computer science No. 999, Springer, Berlin, pp 202–225Google Scholar
- Hill A (1938) The heat of shortening and the dynamic constants of muscle. Proc R Soc Lond B 126:136–195CrossRefGoogle Scholar
- Holmes P, Full R, Koditschek D, Guckenheimer J (2006) The dynamics of legged locomotion: models, analyses and challenges. SIAM Rev 48(2):207–304CrossRefGoogle Scholar
- Hoover A, Burden S, Fu X, Sastry S, Fearing R (2010) Bio-inspired design and dynamic maneuverabiliity of a actuated six-legged robot. In: Proceedings of IEEE international conference on biomedical robotics and biomechatronics (BIOROB), pp 869–876Google Scholar
- Jayaram K, Mongeau JM, McRae B, Full R (2010) High-speed horizontal to vertical transitions in running cockroaches reveals a principle of robustness. In: Society for Integrative and Comparative Biology. http://www.sicb.org/meetings/2010/schedule/ abstractdetails.php3?id=1109
- Jindrich D, Full R (1999) Many-legged maneuverability: dynamics of turning in hexapods. J Exp Biol 202:1603–1623PubMedGoogle Scholar
- Jindrich D, Full R (2002) Dynamic stabilization of rapid hexapedal locomotion. J Exp Biol 205:2803–2823PubMedGoogle Scholar
- Kram R, Wong B, Full R (1997) Three-dimensional kinematics and limb kinetic energy of running cockroaches. J Exp Biol 200:1919–1929PubMedGoogle Scholar
- Kubow T, Full R (1999) The role of the mechanical system in control: a hypothesis of self stabilization in hexapedal runners. Philos Trans R Soc Lond B 354:849–861CrossRefGoogle Scholar
- Kukillaya R, Holmes P (2007) A hexapedal jointed-leg model for insect locomotion in the horizontal plane. Biol Cybern 97:379–395CrossRefPubMedGoogle Scholar
- Kukillaya R, Holmes P (2009) A model for insect locomotion in the horizontal plane: feedforward activation of fast muscles, stability, and robustness. J Theor Biol 261(2):210–226CrossRefPubMedGoogle Scholar
- Kukillaya R, Proctor J, Holmes P (2009) Neuro-mechanical models for insect locomotion: stability, maneuverability, and proprioceptive feedback. CHAOS Interdiscip J Nonlinear Sci 19(2):026107CrossRefGoogle Scholar
- Lee J, Sponberg S, Loh O, Lamperski A, Full R, Cowan N (2008) Templates and anchors for antenna-based wall following in cockroaches. IEEE Trans Robot 24(1):130–143CrossRefGoogle Scholar
- Mongeau JM, Alexander T, Full R (2012) Neuromechanical feedback during dynamic recovery after a lateral perturbation in rapid running cockroaches. In: Society for Integrative and Comparative Biology. http://www.sicb.org/meetings/2012/schedule/abstractdetails.php ?id=555
- Moore T, Revzen S, Burden S, Full R (2010) Adding inertia and mass to test stability predictions in rapid running insects. In: Society for Integrative and Comparative Biology. http://www.sicb.org/meetings/2010/schedule/abstractdetails.php 3?id=1290
- Pearson K (1972) Central programming and reflex control of walking in the cockroach. J Exp Biol 56:173–193Google Scholar
- Pearson K, Iles J (1970) Discharge patterns of coxal levator and depressor motoneurones in the cockroach Periplaneta americana. J Exp Biol 52:139–165PubMedGoogle Scholar
- Pearson K, Iles J (1971) Innervation of the coxal depressor muscles in the cockroach Periplaneta americana. J Exp Biol 54:215–232PubMedGoogle Scholar
- Pearson K, Iles J (1973) Nervous mechanisms underlying intersegmental co-ordination of leg movements during walking in the cockroach. J Exp Biol 58:725–744Google Scholar
- Proctor J, Holmes P (2008) Steering by transient destabilization in piecewise-holonomic models of legged locomotion. Regul Chaotic Dyn 13(4):267–282CrossRefGoogle Scholar
- Proctor J, Holmes P (2010) Reflexes and preflexes: on the role of sensory feedback on rhythmic patterns in legged locomotion. Biol Cybern 2:513–531CrossRefGoogle Scholar
- Proctor J, Kukillaya R, Holmes P (2010) A phase-reduced neuro-mechanical model for insect locomotion: feed-forward stability and proprioceptive feedback. Philos Trans R Soc A 368:5087–5104CrossRefGoogle Scholar
- Revzen S, Burden S, Moore T, Mongeau JM, Full R (2013) Instantaneous kinematic phase reflects neuromechanical response to lateral perturbations of running cockroaches. Biol Cybern 107:179–200CrossRefPubMedGoogle Scholar
- Schmitt J, Bonnono S (2009) Dynamics and stability of lateral plane locomotion on inclines. J Theor Biol 261:598–609CrossRefPubMedGoogle Scholar
- Schmitt J, Holmes P (2000) Mechanical models for insect locomotion: dynamics and stability in the horizontal plane—I. Theory Biol Cybern 83(6):501–515CrossRefPubMedGoogle Scholar
- Schmitt J, Holmes P (2003) Mechanical models for insect locomotion: active muscles and energy losses. Biol Cybern 89(1):43–55PubMedGoogle Scholar
- Schmitt J, Garcia M, Razo RC, Holmes P, Full RJ (2002) Dynamics and stability of legged locomotion in the horizontal plane: a test case using insects. Biol Cybern 86(5):343–353CrossRefPubMedGoogle Scholar
- Sefati S, Neveln I, Roth E, Mitchell T, Snyder J, MacIver M, Fortune E, Cowan N (2013) Mutually opposing forces during locomotion can eliminate the tradeoff between maneuverability and stability. Proc Natl Acad Sci 110(47):18798–18803CrossRefPubMedGoogle Scholar
- Seipel J, Holmes P, Full R (2004) Dynamics and stability of insect locomotion: a hexapedal model for horizontal plane motion. Biol Cybern 91(2):76–90CrossRefPubMedGoogle Scholar
- Sponberg S, Full R (2008) Neuromechanical response of musculo-skeletal structures in cockroaches during rapid running on rough terrain. J Exp Biol 211:433–446CrossRefPubMedGoogle Scholar
- Sponberg S, Spence A, Mullens C, Full R (2011) A single muscle’s multifunctional control potential of body dynamics for postural control and running. Philos Trans Roy Soc B 366:1592–1605CrossRefGoogle Scholar
- Ting L, Blickhan R, Full R (1994) Dynamic and static stability in hexapedal runners. J Exp Biol 197:251–269PubMedGoogle Scholar
- Zill S, Moran D (1981a) The exoskeleton and insect proprioception I. Responses of tibial campaniform sensilla to external and muscle-generated force in the American cockroach Periplaneta americana. J Exp Biol 91:1–24Google Scholar
- Zill S, Moran D (1981b) The exoskeleton and insect proprioception III. Activity of tibial campaniform sensilla during walking in the American cockroach Periplaneta americana. J Exp Biol 94:57–75Google Scholar
- Zill S, Moran D, Varela F (1981) The exoskeleton and insect proprioception II. Reflex effects of tibial campaniform sensilla in the American cockroach Periplaneta americana. J Exp Biol 94:43–55Google Scholar