*Result*: Disorder and Homeostasis in ANIBOT A Biologically-Inspired Animal Robot.
Title:
Disorder and Homeostasis in ANIBOT A Biologically-Inspired Animal Robot.
Authors:
Castillo K; Department of Mathematics, San Diego State University, 5500 Campanile Drive, San Diego, 92182, CA, USA., Parker M; Department of Mathematics, San Diego State University, 5500 Campanile Drive, San Diego, 92182, CA, USA., Reyes N; Department of Mathematics, San Diego State University, 5500 Campanile Drive, San Diego, 92182, CA, USA., Palacios A; Department of Mathematics, San Diego State University, 5500 Campanile Drive, San Diego, 92182, CA, USA. apalacios@sdsu.edu., McGee EC; Department of Nuclear Engineering, University of California at Berkeley, 110 Sproul Hall #5800, Berkeley, 94720, CA, USA., Longhini P; Nonlinear Dynamics and Materials Research, Naval Information Warfare Center Pacific, Code 71780, 53560 Hull Street, San Diego, 92152, VA, USA., Lopez H; Nonlinear Dynamics and Materials Research, Naval Information Warfare Center Pacific, Code 71780, 53560 Hull Street, San Diego, 92152, VA, USA., Vo K; Nonlinear Dynamics and Materials Research, Naval Information Warfare Center Pacific, Code 71780, 53560 Hull Street, San Diego, 92152, VA, USA.
Source:
Bulletin of mathematical biology [Bull Math Biol] 2026 Feb 21; Vol. 88 (3). Date of Electronic Publication: 2026 Feb 21.
Publication Type:
Journal Article
Language:
English
Journal Info:
Publisher: Springer Country of Publication: United States NLM ID: 0401404 Publication Model: Electronic Cited Medium: Internet ISSN: 1522-9602 (Electronic) Linking ISSN: 00928240 NLM ISO Abbreviation: Bull Math Biol Subsets: MEDLINE
Imprint Name(s):
Publication: New York, NY : Springer
Original Publication: New York, Pergamon Press.
Original Publication: New York, Pergamon Press.
MeSH Terms:
References:
Ang J, McMillen DR (2013) Physical constraints on biological integral control design for homeostasis and sensory adaptation. Biophys J 104(2):505–515. https://doi.org/10.1016/j.bpj.2012.12.015. (PMID: 10.1016/j.bpj.2012.12.015)
Antoneli F, Golubitsky M, Jin J, Stewart I (2025) Homeostasis in gene regulatory networks. Int. J, Biomath Accepted. (PMID: 10.1142/S1793524525500706)
Aoki SK, Lillacci G, Gupta A, Baumschlager A, Schweingruber D, Khammash M (2019) A universal biomolecular integral feedback controller for robust perfect adaptation. Nature 570(7762):533–537. https://doi.org/10.1038/s41586-019-1321-1. (PMID: 10.1038/s41586-019-1321-1)
Araujo RP, Liotta LA (2018) The topological requirements for robust perfect adaptation in networks of any size. Nat Commun 9(1):1757. https://doi.org/10.1038/s41467-018-04151-6. (PMID: 10.1038/s41467-018-04151-6)
Bartussek R, Hanggi P, Jung P (1994) Stochastic resonance in optical bistable systems. Phys Rev E 49:3930–3939. (PMID: 10.1103/PhysRevE.49.3930)
Betts JG, Young KA, Wise JA, Johnson E, Poe B, Kruse DH, Korol O, Johnson JE, Womble M, DeSaix P (2017) Anatomy and Physiology. OpenStax College, Houston, TX.
Buceta J, Ibanes MI, Sancho J, Lindenberg K (2003) Noise-driven mechanism for pattern formation. Phys Rev E 67:021113. (PMID: 10.1103/PhysRevE.67.021113)
Buono PL (1998) A model of central pattern generators for quadruped locomotion. PhD thesis, University of Houston.
Buono PL, Golubitsky M (2001) Models of central pattern generators for quadruped locomotion. i. primary gaits. J Math Biol 42:291–326. (PMID: 10.1007/s002850000058)
Buono PL, Golubitsky M (2001) Models of central pattern generators for quadruped locomotion. ii. secondary gaits. J Math Biol 42:327–346. (PMID: 10.1007/s002850000073)
Doedel E, Wang X (1994)Auto94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. Applied Mathematics Report, California Institute of Technology.
Duncan W, Antoneli F, Best J, Golubitsky M, Jin J, Nijhout F, Reed M, Stewart I (2024) Homeostasis patterns. SIAM J Appl Dynamical Systems 23(3):2262–2292. (PMID: 10.1137/23M158807X)
Ferrell JEJ (2016) Perfect and near-perfect adaptation in cell signaling. Cell Syst 2(2):62–67. https://doi.org/10.1016/j.cels.2016.02.006. (PMID: 10.1016/j.cels.2016.02.006)
Gallian JA (2016) Contemporary Abstract Algebra, 9th edn. Cengage Learning, Boston, MA.
Gammaitoni L, Hänggi P, Jung P, Marchesoni F (1998) Stochastic resonance. Rev Mod Phys 70:223–287. https://doi.org/10.1103/RevModPhys.70.223. (PMID: 10.1103/RevModPhys.70.223)
Gardiner C (1985) Handbook of Stochastic Methods. Springer, Berlin.
Golubitsky M, Stewart I (2017) Homeostasis, singularities, and networks. J Math Biol 74(1–2):387–407. (PMID: 10.1007/s00285-016-1024-2)
Golubitsky M, Stewart I (2018) Homeostasis with multiple inputs. SIAM J Appl Dynamical Systems 17(2):1816–1832. (PMID: 10.1137/17M115147X)
Golubitsky M, Wang Y (2020) Infinitesimal homeostasis in three-node input-output networks. J Math Biol 80:1163–1185. (PMID: 10.1007/s00285-019-01457-x)
Golubitsky M, Stewart IN, Buono P-L, Collins JJ (1998) A modular network for legged locomotion. Physica D 115:56–72. (PMID: 10.1016/S0167-2789(97)00222-4)
Golubitsky M, Stewart IN, Buono P-L, Collins JJ (1999) Symmetry in locomotor central pattern generators and animal gaits. Nature 401(6754):693–695. (PMID: 10.1038/44416)
Golubitsky M, Stewart I (2023) Dynamics and Bifurcations in Networks. SIAM,.
Golubitsky M, Stewart IN (2002) Patterns of oscillations in coupled cell systems. In: Holmes, P., Weinstein, A. (eds.) Geometry, Mechanics, and Dynamics, p. 243. Springer, New York.
Golubitsky M, Stewart IN, Schaeffer DG (1988) Singularities and Groups in Bifurcation Theory Vol. II vol. 69. Springer, New York.
Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544. (PMID: 10.1113/jphysiol.1952.sp004764)
Ilan Y (2025) The relationship between biological noise and its application: Understanding system failures and suggesting a method to enhance functionality based on the constrained disorder principle. Biology (Basel) 14(4) https://doi.org/10.3390/biology14040349.
In V, Kho A, Longhini P, Neff JD, Palacios A, Buono PL (2022) Meet anibot: The first biologically-inspired animal robot. Int J Bif Chaos 32(1):2230001. (PMID: 10.1142/S0218127422300014)
Karin O, Swisa A, Glaser B, Dor Y, Alon U (2016) Dynamical compensation in physiological circuits. Mol Syst Biol 12(11):886. https://doi.org/10.15252/msb.20167216. (PMID: 10.15252/msb.20167216)
Khammash MH (2021) Perfect adaptation in biology Cell Syst 12(6):509–521. https://doi.org/10.1016/j.cels.2021.05.020. (PMID: 10.1016/j.cels.2021.05.020)
Kuramoto Y (2012) Chemical Oscillations, Waves, and Turbulence. Springer, New York.
Kuramoto Y, Battogtokh D (2002) Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenomena in Complex Systems 5:380–385.
Meshkat N, Sullivant S (2014) Identifiable reparamterizations of linear compartment models. J Symb Comput 63:46–67. (PMID: 10.1016/j.jsc.2013.11.002)
Meshkat N, Sullivant S, Eisenberg M (2015) Identifiability results for several classes of linear compartment models. Bull Math Biol 77(8):1620–1651. https://doi.org/10.1007/s11538-015-0098-0. (PMID: 10.1007/s11538-015-0098-0)
Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35:193–213. (PMID: 10.1016/S0006-3495(81)84782-0)
Nijhout HF, Reed MC (2014) Homeostasis and dynamic stability of the phenotype link robustness and plasticity. Integr Comp Biol 54:264–275. (PMID: 10.1093/icb/icu010)
Nijhout HF, Reed MC, Budu P, Ulrich CM (2004) A mathematical model of the folate cycle: New insights into folate homeostasis. J Biol Chem 279:55008–55016. (PMID: 10.1074/jbc.M410818200)
Omholt SW, Kefang X, Andersen O, Plathe E (1998) Description and analysis of switchlike regulatory networks exemplified by a model of cellular iron homeostasis. J Theor Biol 195:339–350. (PMID: 10.1006/jtbi.1998.0800)
Palacios A, Aven J, In V, Longhini P, Kho A, Neff J, Bulsara A (2007) Coupled-core fluxgate magnetometer: Novel configuration scheme and the effects of a noise-contaminated external signal. Phys Lett A 367:25–34. (PMID: 10.1016/j.physleta.2007.01.089)
Pilpel Y (2011) Noise in biological systems: pros, cons, and mechanisms of control. Methods Mol Biol 759:407–425. (PMID: 10.1007/978-1-61779-173-4_23)
Reed M, Best J, Golubitsky M, Stewart I, Nijhout HF (2017) Analysis of homeostatic mechanisms in biochemical networks. Bull Math Biol 79:2534–2557. (PMID: 10.1007/s11538-017-0340-z)
Rinzel J, Ermentrout GB (1989) Analysis of neural excitability and oscillations. In: Koch C, Segev I (eds) Methods in Neural Modeling: From Synapses to Networks. MIT Press, Cambridge, MA, pp 251–291.
Ryzowicz CJ, Bertram R, Karamched BR (2025) Dynamic homeostasis in relaxation and bursting oscillations. In review.
Shima SI, Kuramoto Y (2004) Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators. Phys Rev E 69:036213. (PMID: 10.1103/PhysRevE.69.036213)
Singhania R, Tyson JJ (2023) Evolutionary stability of small molecular regulatory networks that exhibit near-perfect adaptation. Biology (Basel) 12(6) https://doi.org/10.3390/biology12060841.
Sontag ED (2017) Dynamic compensation, parameter identifiability, and equivariances. PLoS Comput Biol 13(4):1005447. https://doi.org/10.1371/journal.pcbi.1005447. (PMID: 10.1371/journal.pcbi.1005447)
Strogatz S (2000) From kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143:1–20. (PMID: 10.1016/S0167-2789(00)00094-4)
Tang ZF, McMillen DR (2016) Design principles for the analysis and construction of robustly homeostatic biological networks. J Theor Biol 408:274–289. (PMID: 10.1016/j.jtbi.2016.06.036)
Tsimring LS (2014) Noise in biology Rep Prog Phys 77(2):026601. https://doi.org/10.1088/0034-4885/77/2/026601. (PMID: 10.1088/0034-4885/77/2/026601)
Vaz JR, Cortes N, Gomes JS, Jordão S, Stergiou N (2024) Stride-to-stride fluctuations and temporal patterns of muscle activity exhibit similar responses during walking to variable visual cues. J Biomech 164:111972. https://doi.org/10.1016/j.jbiomech.2024.111972. (PMID: 10.1016/j.jbiomech.2024.111972)
Wang Y, Huang Z, Antoneli F, Golubitsky M (2021) The structure of infinitesimal homeostasis in input-output networks. J Math Biol 82:62. (PMID: 10.1007/s00285-021-01614-1)
Yu Z, Thomas PJ (2022) A homeostasis criterion for limit cycle systems based on infinitesimal shape response curves. J Math Biol 84:1–23. (PMID: 10.1007/s00285-022-01724-4)
Antoneli F, Golubitsky M, Jin J, Stewart I (2025) Homeostasis in gene regulatory networks. Int. J, Biomath Accepted. (PMID: 10.1142/S1793524525500706)
Aoki SK, Lillacci G, Gupta A, Baumschlager A, Schweingruber D, Khammash M (2019) A universal biomolecular integral feedback controller for robust perfect adaptation. Nature 570(7762):533–537. https://doi.org/10.1038/s41586-019-1321-1. (PMID: 10.1038/s41586-019-1321-1)
Araujo RP, Liotta LA (2018) The topological requirements for robust perfect adaptation in networks of any size. Nat Commun 9(1):1757. https://doi.org/10.1038/s41467-018-04151-6. (PMID: 10.1038/s41467-018-04151-6)
Bartussek R, Hanggi P, Jung P (1994) Stochastic resonance in optical bistable systems. Phys Rev E 49:3930–3939. (PMID: 10.1103/PhysRevE.49.3930)
Betts JG, Young KA, Wise JA, Johnson E, Poe B, Kruse DH, Korol O, Johnson JE, Womble M, DeSaix P (2017) Anatomy and Physiology. OpenStax College, Houston, TX.
Buceta J, Ibanes MI, Sancho J, Lindenberg K (2003) Noise-driven mechanism for pattern formation. Phys Rev E 67:021113. (PMID: 10.1103/PhysRevE.67.021113)
Buono PL (1998) A model of central pattern generators for quadruped locomotion. PhD thesis, University of Houston.
Buono PL, Golubitsky M (2001) Models of central pattern generators for quadruped locomotion. i. primary gaits. J Math Biol 42:291–326. (PMID: 10.1007/s002850000058)
Buono PL, Golubitsky M (2001) Models of central pattern generators for quadruped locomotion. ii. secondary gaits. J Math Biol 42:327–346. (PMID: 10.1007/s002850000073)
Doedel E, Wang X (1994)Auto94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. Applied Mathematics Report, California Institute of Technology.
Duncan W, Antoneli F, Best J, Golubitsky M, Jin J, Nijhout F, Reed M, Stewart I (2024) Homeostasis patterns. SIAM J Appl Dynamical Systems 23(3):2262–2292. (PMID: 10.1137/23M158807X)
Ferrell JEJ (2016) Perfect and near-perfect adaptation in cell signaling. Cell Syst 2(2):62–67. https://doi.org/10.1016/j.cels.2016.02.006. (PMID: 10.1016/j.cels.2016.02.006)
Gallian JA (2016) Contemporary Abstract Algebra, 9th edn. Cengage Learning, Boston, MA.
Gammaitoni L, Hänggi P, Jung P, Marchesoni F (1998) Stochastic resonance. Rev Mod Phys 70:223–287. https://doi.org/10.1103/RevModPhys.70.223. (PMID: 10.1103/RevModPhys.70.223)
Gardiner C (1985) Handbook of Stochastic Methods. Springer, Berlin.
Golubitsky M, Stewart I (2017) Homeostasis, singularities, and networks. J Math Biol 74(1–2):387–407. (PMID: 10.1007/s00285-016-1024-2)
Golubitsky M, Stewart I (2018) Homeostasis with multiple inputs. SIAM J Appl Dynamical Systems 17(2):1816–1832. (PMID: 10.1137/17M115147X)
Golubitsky M, Wang Y (2020) Infinitesimal homeostasis in three-node input-output networks. J Math Biol 80:1163–1185. (PMID: 10.1007/s00285-019-01457-x)
Golubitsky M, Stewart IN, Buono P-L, Collins JJ (1998) A modular network for legged locomotion. Physica D 115:56–72. (PMID: 10.1016/S0167-2789(97)00222-4)
Golubitsky M, Stewart IN, Buono P-L, Collins JJ (1999) Symmetry in locomotor central pattern generators and animal gaits. Nature 401(6754):693–695. (PMID: 10.1038/44416)
Golubitsky M, Stewart I (2023) Dynamics and Bifurcations in Networks. SIAM,.
Golubitsky M, Stewart IN (2002) Patterns of oscillations in coupled cell systems. In: Holmes, P., Weinstein, A. (eds.) Geometry, Mechanics, and Dynamics, p. 243. Springer, New York.
Golubitsky M, Stewart IN, Schaeffer DG (1988) Singularities and Groups in Bifurcation Theory Vol. II vol. 69. Springer, New York.
Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544. (PMID: 10.1113/jphysiol.1952.sp004764)
Ilan Y (2025) The relationship between biological noise and its application: Understanding system failures and suggesting a method to enhance functionality based on the constrained disorder principle. Biology (Basel) 14(4) https://doi.org/10.3390/biology14040349.
In V, Kho A, Longhini P, Neff JD, Palacios A, Buono PL (2022) Meet anibot: The first biologically-inspired animal robot. Int J Bif Chaos 32(1):2230001. (PMID: 10.1142/S0218127422300014)
Karin O, Swisa A, Glaser B, Dor Y, Alon U (2016) Dynamical compensation in physiological circuits. Mol Syst Biol 12(11):886. https://doi.org/10.15252/msb.20167216. (PMID: 10.15252/msb.20167216)
Khammash MH (2021) Perfect adaptation in biology Cell Syst 12(6):509–521. https://doi.org/10.1016/j.cels.2021.05.020. (PMID: 10.1016/j.cels.2021.05.020)
Kuramoto Y (2012) Chemical Oscillations, Waves, and Turbulence. Springer, New York.
Kuramoto Y, Battogtokh D (2002) Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenomena in Complex Systems 5:380–385.
Meshkat N, Sullivant S (2014) Identifiable reparamterizations of linear compartment models. J Symb Comput 63:46–67. (PMID: 10.1016/j.jsc.2013.11.002)
Meshkat N, Sullivant S, Eisenberg M (2015) Identifiability results for several classes of linear compartment models. Bull Math Biol 77(8):1620–1651. https://doi.org/10.1007/s11538-015-0098-0. (PMID: 10.1007/s11538-015-0098-0)
Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys J 35:193–213. (PMID: 10.1016/S0006-3495(81)84782-0)
Nijhout HF, Reed MC (2014) Homeostasis and dynamic stability of the phenotype link robustness and plasticity. Integr Comp Biol 54:264–275. (PMID: 10.1093/icb/icu010)
Nijhout HF, Reed MC, Budu P, Ulrich CM (2004) A mathematical model of the folate cycle: New insights into folate homeostasis. J Biol Chem 279:55008–55016. (PMID: 10.1074/jbc.M410818200)
Omholt SW, Kefang X, Andersen O, Plathe E (1998) Description and analysis of switchlike regulatory networks exemplified by a model of cellular iron homeostasis. J Theor Biol 195:339–350. (PMID: 10.1006/jtbi.1998.0800)
Palacios A, Aven J, In V, Longhini P, Kho A, Neff J, Bulsara A (2007) Coupled-core fluxgate magnetometer: Novel configuration scheme and the effects of a noise-contaminated external signal. Phys Lett A 367:25–34. (PMID: 10.1016/j.physleta.2007.01.089)
Pilpel Y (2011) Noise in biological systems: pros, cons, and mechanisms of control. Methods Mol Biol 759:407–425. (PMID: 10.1007/978-1-61779-173-4_23)
Reed M, Best J, Golubitsky M, Stewart I, Nijhout HF (2017) Analysis of homeostatic mechanisms in biochemical networks. Bull Math Biol 79:2534–2557. (PMID: 10.1007/s11538-017-0340-z)
Rinzel J, Ermentrout GB (1989) Analysis of neural excitability and oscillations. In: Koch C, Segev I (eds) Methods in Neural Modeling: From Synapses to Networks. MIT Press, Cambridge, MA, pp 251–291.
Ryzowicz CJ, Bertram R, Karamched BR (2025) Dynamic homeostasis in relaxation and bursting oscillations. In review.
Shima SI, Kuramoto Y (2004) Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators. Phys Rev E 69:036213. (PMID: 10.1103/PhysRevE.69.036213)
Singhania R, Tyson JJ (2023) Evolutionary stability of small molecular regulatory networks that exhibit near-perfect adaptation. Biology (Basel) 12(6) https://doi.org/10.3390/biology12060841.
Sontag ED (2017) Dynamic compensation, parameter identifiability, and equivariances. PLoS Comput Biol 13(4):1005447. https://doi.org/10.1371/journal.pcbi.1005447. (PMID: 10.1371/journal.pcbi.1005447)
Strogatz S (2000) From kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143:1–20. (PMID: 10.1016/S0167-2789(00)00094-4)
Tang ZF, McMillen DR (2016) Design principles for the analysis and construction of robustly homeostatic biological networks. J Theor Biol 408:274–289. (PMID: 10.1016/j.jtbi.2016.06.036)
Tsimring LS (2014) Noise in biology Rep Prog Phys 77(2):026601. https://doi.org/10.1088/0034-4885/77/2/026601. (PMID: 10.1088/0034-4885/77/2/026601)
Vaz JR, Cortes N, Gomes JS, Jordão S, Stergiou N (2024) Stride-to-stride fluctuations and temporal patterns of muscle activity exhibit similar responses during walking to variable visual cues. J Biomech 164:111972. https://doi.org/10.1016/j.jbiomech.2024.111972. (PMID: 10.1016/j.jbiomech.2024.111972)
Wang Y, Huang Z, Antoneli F, Golubitsky M (2021) The structure of infinitesimal homeostasis in input-output networks. J Math Biol 82:62. (PMID: 10.1007/s00285-021-01614-1)
Yu Z, Thomas PJ (2022) A homeostasis criterion for limit cycle systems based on infinitesimal shape response curves. J Math Biol 84:1–23. (PMID: 10.1007/s00285-022-01724-4)
Grant Information:
N000142412547 Office of Naval Research; N66001-21-D-0041 Naval Information Warfare Center Pacific
Contributed Indexing:
Keywords: Bifurcation; Central Pattern Generator; Coupled nonlinear oscillator; Homeostasis; Network; Symmetry
Entry Date(s):
Date Created: 20260221 Date Completed: 20260221 Latest Revision: 20260309
Update Code:
20260309
DOI:
10.1007/s11538-026-01608-4
PMID:
41721944
Database:
MEDLINE
*Further Information*
*The effects of external perturbations (or disorder) in the Hopf bifurcations of a central pattern generator (CPG) network of neurons that serves as a model for the circuit realization of ANIBOT-a biologically-inspired animal robot with four legs-are studied, analytically and computationally, from the standpoint of homeostasis. In particular, we employ recent developments in the mathematical description of homeostasis, e.g., input-output functions, to explore the CPG response to perturbations of the network connectivity, the internal dynamics of the neurons, and electronic noise as it arises in the circuit realization of ANIBOT. The patterns of locomotion (Walk, Jump, Trot, Bound, Pace, and Pronk) are controlled, mainly, by the phase dynamics of a CPG network. The results show that with the exception of the Walk and Jump gaits, the phase dynamics of all other gaits exhibit perfect homeostatic responses. In addition, a distinctive feature of the network dynamics is that, under certain conditions, the external perturbations can lead to the appearance of certain patterns, which are absent in the unperturbed system, i.e., disorder-induced pattern formation.
(© 2026. The Author(s), under exclusive licence to the Society for Mathematical Biology.)*
*Declarations. Conflicts of Interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.*