Axiomathes

, Volume 16, Issue 1–2, pp 65–122 | Cite as

Complex Non-linear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks

  • I. C. Baianu
  • R. Brown
  • G. Georgescu
  • J. F. Glazebrook
Article

Abstract

A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a Łukasiewicz–Moisil Topos with an n-valued Łukasiewicz–Moisil Algebraic Logic subobject classifier description that represents non-random and non-linear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen’s (M,R)-systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occurring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, non-commutative quantum logics are considered in the context of an emerging Quantum Relational Biology.

Keywords

adjoint functors and dynamically analogous systems biogroupoids and organismal development biological principles nuclear equivalence and cell differentiation categories n-valued logics and higher dimensional algebra in neuroscience and genetics cognitive and anticipatory processes learning and quantum wave-pattern recognition colimits limits and adjointness relations in biology generalized (M,R)-systems neuro-categories and consciousness quantum automata and relational biology quantum bionetworks and their underlying quantum logics quantum computers 

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References

  1. Abbott, E. A. 1884Flatland: A Romance of Many DimensionsDoverLondonGoogle Scholar
  2. Al-Agl, , Brown, F. R., Steiner, R. 2002Multiple Categories: The Equivalence Between a Globular and Cubical ApproachAdvances in Mathematics17071118Google Scholar
  3. Anandan, J. 1992The Geometric PhaseNature360307313CrossRefGoogle Scholar
  4. Aof, M.E.-S.A., Brown, R. 1992The Holonomy Groupoid of a Locally Topological GroupoidTopology and its Applications4797113CrossRefGoogle Scholar
  5. Arbib, M. A., Manes, E. G. 1986Algebraic Approaches to Program SemanticsSpringer-VerlagBerlinGoogle Scholar
  6. Baader, F., Nipkow, T. 1998Term Rewriting and All ThatCambridge University PressCambridge UKGoogle Scholar
  7. Baars, B. J. 1988A Cognitive Theory of ConsciousnessCambridge University PressCambridge UKGoogle Scholar
  8. Baez, J.: 1995, ‘This Week’s Find in Mathematical Physics’, week 3, http://obswww.unige.ch/lbartho/TWF/week53.html
  9. Baianu, I. C., J. F. Glazebrook, and G. Georgescu: 2004, ‘Categories of Quantum Automata and N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R)-Systems and Their Higher Dimensional Algebra’, Abstract and Preprint of Report: http://www.ag.uiuc.edu/fs401/QAuto.pdf; and http://www.medical–papers.com/quantum+automata+math+categories+baianu/
  10. Baianu, I. C., R. Brown and J. F. Glazebrook: 2005, ‘Quantum Algebraic Topology and Field Theories’, (in preparation): http://www.ag.uiuc.edu/fs401/QAT.pdf
  11. Baianu, I. C., Marinescu, M. 1968Organismic Supercategories: Towards a Unified Theory of SystemsBulletin of Mathematical Biophysics30148165Google Scholar
  12. Baianu, I. C.: 1969, Theoretical and Experimental Models of Carcinogenesis, MSc. Thesis, Medical Biophysics Department & School of Physics, University of BucharestGoogle Scholar
  13. Baianu, I. C. 1970Organismic Supercategories: II. On Multistable SystemsBulletin of Mathematical Biophysics32539561Google Scholar
  14. Baianu, I. C. 1971aOrganismic Supercategories and Qualitative Dynamics of SystemsIbid.33339353Google Scholar
  15. Baianu, I. C.: 1971b, ‘Categories, Functors, Quantum Automata and Quantum Computation’, in P. Suppes (ed.), Proceedings of the 4th Intl. Congress LMPS, BucharestGoogle Scholar
  16. Baianu, I. C. 1971cResonant Transfer of Energy in OncogenesisAn. Univ. Bucharest – Physics205658Google Scholar
  17. Baianu, I. C. 1972Energetic and Categorical Considerations in EEG Interpretation’An. Univ. Bucharest – Physics216067Google Scholar
  18. Baianu, I. C. 1973Some Algebraic Properties of (M,R)-SystemsBulletin of Mathematical Biology35213217Google Scholar
  19. Baianu, I. C., Scripcariu, D. 1974On Adjoint Dynamical SystemsBulletin of Mathematical Biology36356364Google Scholar
  20. Baianu, I. C., Marinescu, M. 1974A Functorial Construction of (M,R)-Systems’Revue Roumaine de Mathematiques Pures Et Appliquees19388391Google Scholar
  21. Baianu, I. C. 1977A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear TheoryBulletin of Mathematical Biology39249258CrossRefGoogle Scholar
  22. Baianu, I. C. 1980Natural Transformations of Organismic StructuresBulletin of Mathematical Biology42431446Google Scholar
  23. Baianu, I. C.: 1983, ‘Natural Transformation Models in Molecular Biology’, Proceedings of the SIAM Natl. Meet., Denver, CO.; Eprint: http://cogprints.org/3675/; http://cogprints.org/3675/01/Naturaltransfmolbionu6.pdf.
  24. Baianu, I. C. 1984A Molecular-Set-Variable Model of Structural and Regulatory Activities in Metabolic and Genetic NetworksFASEB Proceedings43917Google Scholar
  25. Baianu, I. C.: 1987a, ‘Computer Models and Automata Theory in Biology and Medicine’ in M. Witten (ed.), Mathematical Models in Medicine, vol. 7., Pergamon Press, New York, 1513-1577; CERN Preprint No. EXT–2004–072: http://doc.cern.ch//archive/electronic/other/ext/ext-2004-072.pdf.
  26. Baianu, I. C.: 1987b, ‘Molecular Models of Genetic and Organismic Structures’, in Proceed. Relational Biology Symp., Argentina; CERN Preprint No. EXT–2004–067, http: //doc.cern.ch/archive/electronic/other/ext/ext-2004-067/ MolecularModelsICB3.doc.Google Scholar
  27. Baianu, I. C.: 2004a, ‘Quantum Interactomics and Cancer Mechanisms’, Preprint No. 00001978– http: bioline.utsc.utoronto.ca/archive/00001978/01/QuantumInteractomicsInCancer–Sept13k4E–cuteprt.pdf; http://bioline.utsc.utoronto.ca/archive/00001978/Google Scholar
  28. Baianu, I. C.: 2004b, ‘Quantum Nano–Automata (QNA): Towards Microphysical Measurements’, CERN Preprint No. EXT–2004–125, http://documents.cern.ch/cgi-bin/setlink?base=preprint& categ=ext & id=ext-2004-125Google Scholar
  29. Baianu, I. C. and V. Prisecaru: 2004, ‘Complex Systems Biology Modeling of Cancer Cell Cycling’, Preprint: arXiv q-bio/0406045.Google Scholar
  30. Baianu, I. C.: 2006, ‘Robert Rosen’s Work and Complex Systems Biology’, Axiomathes, (in this volume)Google Scholar
  31. Bak, A., R. Brown, G. Minian and T. Porter: 2004,‘ Global Actions, Groupoid Atlases and Related Topics’, http://citeseer.ist.psu.edu/bak00global.html
  32. Bell, A., Holcombe, M. 1996

    Computational Models of Cellular Processing

    Cuthbertson, R.Holcombe, M.Paton, R. eds. Computation in Cellular and Molecular Biological SystemsWorld ScientificSingapore
    Google Scholar
  33. Birkhoff, G. 1948Lattice TheoryAmerican Mathematical SocietyNew YorkGoogle Scholar
  34. Bourbaki, N. 1964Eléments de Mathématique, Livre II, AlgébreHermann, EditorParisGoogle Scholar
  35. Brown, R. 1987From Groups to Groupoids a Brief SurveyBulletin of the London Mathematical Society19113134Google Scholar
  36. Brown, R., P. J. Higgins, and R. Sivera: 2005, Non-Abelian Algebraic Topology, (new book in preparation)Google Scholar
  37. Brown, R. 2004Crossed Complexes and Homotopy Groupoids as Non-commutative tools for Higher Dimensional Local-to-Global ProblemsProceedings of the Fields Institute Workshop on Categorical Structures for Descent, Galois Theory, Hopf Algebras and Semiabelian Categories, Fields Institute Communications43101130Google Scholar
  38. Brown, R., Heyworth, A. 2000Using Rewriting Systems to Compute left Kan Extensions and Induced Actions of CategoriesJournal of Symbolic Computation29531CrossRefGoogle Scholar
  39. Brown, R., Porter, T. 2003a‘The Intuitions of Higher Dimensional Algebra for the Study of Structured Space’Revue de Synthèse124174203Seminar at the series of G. Longo Géometrie et Cognition, École Normale Supérieure, May 2001Google Scholar
  40. Brown, R. and T. Porter: 2003b, ‘Category Theory and Higher Dimensional Algebra: Potential Descriptive Tools in Neuroscience’, in N. Singh (ed.) Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, National Brain Research Centre, Conference Proceedings 1, 80–92Google Scholar
  41. Brown R., R. Paton and T. Porter: 2004, ‘Categorical Language and Hierarchical Models for Cell Systems’, in R. Paton, H. Bolouri, M. Holcombe, J. H. Parish and R. Tateson (eds.), Computation in Cells and Tissues – Perspectives and Tools of Thought, Natural Computing Series, Springer-Verlag, pp. 289–303Google Scholar
  42. Buneci, M.: 2003, Groupoid Representations, UTJ, BucharestGoogle Scholar
  43. Carnap, R. 1938The Logical Syntax of LanguageHarcourt, Brace and CoNew YorkGoogle Scholar
  44. Cordier, J.-M., Porter, T. 1989Shape Theory – Categorical Methods of ApproximationEllis HorwoodChichester, UKGoogle Scholar
  45. Dampney, C. N. G., Johnson, M. 1994

    On the Value of Commutative Diagrams in Information Modelling

    Nivat,  eds. Springer Workshops in ComputingSpringerLondon4760
    Google Scholar
  46. Jong, H., Page, M. 2000

    Qualitative Simulation of Large and Complex Genetic Regulatory Systems

    Horn, W. eds. Proceedings of the Fourteenth European Conference on Artifical Intelligence, ECAI 2000IOS PressAmsterdam141145
    Google Scholar
  47. Jong, H., Geiselmann, J., Thieffry, D. 2003

    Qualitative Modeling and Simulation of Developmental Regulatory Systems

    Kumar, S.Bentley, P. J. eds. On Growth, Form, and ComputersAcademic PressLondon109143
    Google Scholar
  48. Jong, H. 2005Qualitative Dynamics of Complex Genetic Regulatory SystemsBulletin Mathematical Biology66235257Google Scholar
  49. Dioguardi, N. 1995Fegato a Piu DimentioniEtas Libri, RCS MedecinaMilanGoogle Scholar
  50. Edelman, G. 1992Brilliant Air, Brilliant Fire – On the Matter of the MindBasic BooksNew YorkGoogle Scholar
  51. Edelman, G. 1989The Remembered PresentBasic BooksNew YorkGoogle Scholar
  52. Edelman, G., Tononi, G. 2000A Universe of ConsciousnessBasic BooksNew YorkGoogle Scholar
  53. Ehresmann, A. C., Vanbremeersch, J.-P. 1987Hierarchical Evolutive Systems: A Mathematical Model for Complex SystemsBulletin of Mathematical Biology491350(For a full list of their work see: http://perso.wanadoo.fr/vbm-ehr/Ang/Publi2T.htm)CrossRefGoogle Scholar
  54. Ehresmann, A. C. and J.-P. Vanbremeersch: 2003, A Categorical Model for Cognitive Systems up to Consciousness, Ibid. (Brown and Porter, 2003b)Google Scholar
  55. Ehresmann, A. C., Vanbremeersch, J.-P. 2002Emergence Processes up to Consciousness Using the Multiplicity Principle and Quantum PhysicsAIP Conference Proceedings627221233Google Scholar
  56. Ehresmann, C. 1965Catégories et StructuresDunodParisGoogle Scholar
  57. Ehresmann, C. 1966Trends Toward Unity in MathematicsCahiers de Topologie et Géometrie Differentielle817Google Scholar
  58. Ehresmann, C. 1967Sur les Structures AlgebriquesC.R.A.S. Paris264840843Google Scholar
  59. Eilenberg, S., MacLane, S. 1945The General Theory of Natural EquivalencesTransactions of the American Mathematcal Society58231294Google Scholar
  60. Fisher, M. J., Malcolm, G., Paton, R. C. 2000Spatio-Logical processes in Intracellular SignalingBiosystems558392CrossRefGoogle Scholar
  61. Gadducci, F. and U. Montanari: 1995, ‘Enriched Categories as Models of Computation’, in A. De Santis (ed.), Proceed. Fifth Italian Conference on Theoretical Computer Science, World Scientific, 20–42Google Scholar
  62. Georgescu, G., Popescu, D. 1968On Algebraic CategoriesRevue Roumaine De Mathematiques Pures Et Appliquees13337342Google Scholar
  63. Georgescu, G., Vraciu, C. 1970On the Characterization of Łukasiewicz–Moisil AlgebrasJournal of Algebra16486495CrossRefGoogle Scholar
  64. Georgescu, G.: 2006, ‘N–valued Logics and Łukasiewicz–Moisil Algebras’, Axiomathes (in this volume)Google Scholar
  65. Girault, F.: 1997, ‘Formalisation en Logique Lineáire du Fonctionnement des Réseaux de Petri’, Thèse, LAAS, Université Paul Sabatier ToulouseGoogle Scholar
  66. Goguen, J.: 1999, ‘An Introduction to Algebraic Semiotics with Application to User Interface Design’, in Computation for Metaphor, Analogy and Agents, Lecture Notes in AI, No. 1562, SpringerGoogle Scholar
  67. Goguen, J., Malcolm, G. 2000A Hidden Agendatheoretical Computer Science24555101CrossRefGoogle Scholar
  68. Gudder, S. 2004

    Noncommutative Probability and Applications

    Rao, M. M. eds. Real and Stochastic Analysis – New PerspectivesBirkhäuserBoston, Basel, Berlin199238
    Google Scholar
  69. Hilbert, D., Ackerman, W. 1927Grunzüge der theoretischen LogikSpringerBerlinGoogle Scholar
  70. Hughes, K. H., Macdonald, J. N. 2002Boltzmann Wavepacket Dynamics on Periodic Molecular Potential FunctionsPhysical Chemistry Chemical Physics242674273Google Scholar
  71. Jacob, F. and J. Monod: 1961a, ‘On the Regulation of Gene Activity’, in Frisch (ed.), Cold Spring Harbor Symposium Quantit. Biology 26, New York, 193–211Google Scholar
  72. Jacob, F., Monod, J. 1961bRegulation of DNA Reduplication in BacteriaIbid.28323376Google Scholar
  73. Kan, D. M. 1958Adjoint FunctorsTransactions of the American Mathematical Society87294329Google Scholar
  74. Krips, H.: 1999, ‘Measurement in Quantum Theory’, in E. N. Zalta (ed.) The Stanford Encyclopedia of Philosophy (Winter 1999 Edition)Google Scholar
  75. Krsti, S., J. Launchbury and D. Pavlovic: 2001, ‘Categories of Processes Enriched in Final Coalgebras’, SLNCS 2030, Springer, 303Google Scholar
  76. Lambek, J., Scott, P. J. 1986Introduction to Higher Order Categorical LogicCambridge University PressCambridge, UKGoogle Scholar
  77. Lawvere, F. W. 1963Functorial Semantics of Algebraic TheoriesProceedings of the National Academy of Sciences USA50869872Google Scholar
  78. Lawvere, F. W.,  et al. 1966

    ‘The Category of Categories as a Foundation for Mathematics’

    Eilenberg, S. eds. Proc. Conf. Categorical Algebra–La JollaSpringer-VerlagUSA,Berlin, Heidelberg and New York120(1965)
    Google Scholar
  79. Löfgren, L. 1968An Axiomatic Explanation of Complete Self-ReproductionBulletin of Mathematical Biophysics30317348Google Scholar
  80. MacLane, S., Moerdijk, I. 1992Sheaves in Geometry and Logic – A First Introduction to Topos TheorySpringer-VerlagNew YorkGoogle Scholar
  81. McCulloch, W., Pitts, W. 1943A Logical Calculus of Ideas Immanent in Nervous ActivityBulletin of Mathematical Biophysics5115133Google Scholar
  82. Meseguer, J. 1993

    ‘A Logical Theory of Concurrent Objects and Its Relation to the MAUDE Language’

    Agha, G.Wegner, P.Yonezawa, A. eds. Research Directions in object oriented based concurrencyMIT PressCambridge MA314390(see also http://maude.csl.sri.com/papers/).
    Google Scholar
  83. Meseguer, J. and U. Montanari: 1998, ‘Mapping Tile Logic into Rewriting Logic’, in F. Parisi-Presicce (ed.), Recent Trends in Algebraic Development Techniques, Springer LNCS 1376, pp. 62–91. (Available from http://www.di.unipi.it/ugo/wadt.ps)
  84. Mitchell, B. 1965Theory of CategoriesAcademic PressLondonGoogle Scholar
  85. Paton, R. C. 1997Glue, Verb and Text Metaphors in BiologyActa Biotheoretica45115CrossRefGoogle Scholar
  86. Paton, R. C. 2002Process, Structure and Context in Relation to Integrative BiologyBiosystems646372CrossRefGoogle Scholar
  87. Peruš, M, Bischof, H. 2003

    ‘Quantum Wave Pattern Recognition’

    Chen, K. eds. Proceedings of the 7-th Joint Conf. Information Sciencespubl. by JCIS/Assoc. for Intelligent Machinery, DurhamCary, NC USA3136(e–publ. : quant-ph/0303092)
    Google Scholar
  88. Peruš, M., Bischof, H., Kiong, L. C. 2004‘Quantum–Implemented Selective Reconstruction of High–Resolution Images’Appl. Opt.4361346138Los Alamos preprint archive, http:arxiv.org/abs/quant-ph/0401016Google Scholar
  89. Pitts, W. 1943The Linear Theory of Neuron NetworksBulletin of Mathematical Biophysics52331Google Scholar
  90. Popescu, N. 1975Abelian Categories with Applications to Rings and Modules, 2nd edn (English translation by I. C. Baianu)Academic PressNew York and LondonGoogle Scholar
  91. Porter, T. 1994aCategorical Shape Theory as a Formal Language for Pattern RecognitionAnnals of Mathematics and Artificial Intelligence102554CrossRefGoogle Scholar
  92. Porter, T.: 1994b, ‘Can Categorical Shape Theory handle Grey Level Images?’ in Shape in Picture NATO ASI Series F, Vol. 126, SpringerGoogle Scholar
  93. Rashevsky, N. 1954Topology and Life: In Search of General Mathematical Principles in Biology and SociologyBulletin of Mathematical Biophysics14317348Google Scholar
  94. Rashevsky, N. 1961Biological Epimorphism, Adequate Design and the Problem of RegenerationBulletin of Mathematical Biophysics23109113Google Scholar
  95. Rashevsky, N. 1965The Representation of Organisms in Terms of PredicatesBulletin of Mathematical Biophysics27477491Google Scholar
  96. Rashevsky, N. 1967Organismic Sets and Biological EpimorphismBulletin of Mathematical Biophysics29389393Google Scholar
  97. Rashevsky, N. 1968aNeurocybernetics as a Particular Case of General Regulatory Mechanisms in Biological and Social OrganismsConcepts de l’Age de la Science3243258Google Scholar
  98. Rashevsky, N. 1968bOrganismic Sets in Biology and SociologyBulletin of Mathematical Biophysics30246259Google Scholar
  99. Rashevsky, N. 1969Outline of a Unified Approach to Physics, Biology and SociologyBulletin of Mathematical Biophysics31159198Google Scholar
  100. Rashevsky, N. 1972Organismic SetsWilliam Clowes & SonsLondonGoogle Scholar
  101. Rayadu, P. V.: 2003. ‘Towards an Algebra of Neural Processing of Contextual Information’, Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, ibid (Brown, Porter and Paton, 2003a), pp. 110–112Google Scholar
  102. Rosen, R. 1958aA Relational Theory of Biological SystemsBulletin of Mathematical Biophysics20245260Google Scholar
  103. Rosen, R. 1958bThe Representation of Biological Systems from the Standpoint of the Theory of CategoriesBulletin of Mathematical Biophysics20317341Google Scholar
  104. Rosen, R. 1960A Quantum-Thoretic Approach to Genetic ProblemsBulletin of Mathematical Biophysics22227255Google Scholar
  105. Rosen, R. 1968aOn Analogous SystemsBulletin of Mathematical Biophysics30481492Google Scholar
  106. Rosen, R. 1968bRecent Developments in the Theory of Control and Regulation of Cellular ProcessesInternational Review of Cytology232588Google Scholar
  107. Rosen, R. 1971Some Realizations of (M,R)-Systems and Their InterpretationBulletin of Mathematical Biophysics33303319Google Scholar
  108. Rosen, R. 1973On the Dynamical Realization of (M,R)-SystemsBulletin of Mathematical Biology3519CrossRefGoogle Scholar
  109. Rosen, R. 1991Life ItselfColumbia University PressNew YorkGoogle Scholar
  110. Rosen, R. 2000Essays on Life ItselfColumbia University PressNew York(also 2nd Edn., 2004)Google Scholar
  111. Russell, B., Whitehead, A. N. 1925Principia MathematicaCambridge Univ. PressCambridge UKGoogle Scholar
  112. Russell, B. 1937Principles of Mathematics2George Allen & Unwin Ltd.LondonGoogle Scholar
  113. Schrödinger, E. 1945What is Life?Cambridge University PressCambridgeGoogle Scholar
  114. Stapp, H. P. 1999Attention, Intention and Will in Quantum PhysicsJournal of Consciousness Studies6143164Google Scholar
  115. Wallace, R. 2005Consciousness: A Mathematical Treatment of the Global Neuronal WorkspaceSpringer-VerlagBerlinGoogle Scholar
  116. Zadeh, A. L. 1965Fuzzy setsInformation Control8338353CrossRefGoogle Scholar

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Authors and Affiliations

  • I. C. Baianu
  • R. Brown
  • G. Georgescu
  • J. F. Glazebrook

There are no affiliations available

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