Abstract
Jerne's idiotypic network was previously modelled using simple proliferation dynamics and a homogeneous tree as a connection structure. The present paper studies analytically and numerically the genericity of the previous results when the network connection structure is randomized, e.g. with loops and varying connection intensities. The main feature of the dynamics is the existence of different localized attractors that can be interpreted in terms of vaccination and tolerance. This feature is preserved when loops are added to the network, with a few exceptions concerning some regular lattices. Localized attractors might be destroyed by the introduction of a continuous distribution of connection intensities. We conclude by discussing possible modifications of the elementary model that preserve localization of the attractors and functionality of the network.
Similar content being viewed by others
Literature
Atlan, H. and I. R. Cohen (Eds). 1989.Theories of Immune Networks. Berlin: Springer.
Conway, J. 1982. Game of life. InWinning Ways, E. Berkelamp, J. Conway and R. Guy (Eds), Ch. 25. New York: Academic Press.
De Boer, R. J. 1983.GRIND: great integrator differential equations. Bioinformatics Group, University of Utrecht, The Netherlands.
De Boer, R. J. 1989. Extensive percolation in reasonable idiotypie networks. InTheories of Immune Networks, H. Atlan and I. R. Cohen (Eds), pp. 26–37. Berlin: Springer.
De Boer, R. J. and P. Hogeweg. 1989a. Memony but no suppression in low-dimensional symmetric idiotypic networks.Bull. math. Biol. 51, 223–246.
De Boer, R. J. and P. Hogeweg. 1989b. Unreasonable implications of reasonable idiotypic network assumptions.Bull. math. Biol. 51, 381–408.
De Boer, R. J. and A. S. Perelson. 1991. Size and connectivity as emergent properties of a developing immune network.J. theor. Biol. 149, 381–424.
De Boer, R. J., I. G. Kevrekidis and A. S. Perelson. 1991. A simple idiotypic network model with complex dynamics.Chem. Engng Sci. 45, 2375–2382.
Farmer, J., N. H. Packard and A. S. Perelson. 1986. The immune system, adaptation and machine learning.Physica D 22, 187–204.
Holmberg, D., S. Forsgen, F. Ivars and A. Coutinho. 1984. Reactions among IgM antibodies derived from normal neonatal mice.Eur. J. Immunol. 14, 435–441.
Holmberg, D., G. Wennerstrom, L. Andrade and A. Coutinho. 1986. The high idiotypic connectivity of “natural” newborn antibodies is not found in the adult mitogen-reactive B cell repertoires.Eur. J. Immunol. 16, 82–87.
Jerne, N. 1974. Towards a network theory of the immune system.Annls Immunol. (Inst. Pasteur) 125C, 373–389.
Kearney, J., M. Vakil and N. Nicholson. 1987 Non-random VH gene expression and idiotype anti-idiotype expression in early B-cells. InEvolution and Vertebrate Immunity: The Antigen Receptor and MHC Gene Families, G. Kelsoe and D. Schulze (Eds), pp. 175–190. Austin: Texas University Press.
Kocks, C. and K. Rajewsky. 1989. Stable expression and somatic hypermutation of antibody V regions in B cells developmental pathways.Ann. Rev. Immunol. 7, 537–559.
Neumann, A. U. 1991. Dynamical phase transitions in spin models and network models of the immune response, Ph.D. thesis, Bar-Ilan University, Ramat-Gan, Israel.
Neumann, A. U. and G. Weisbuch. 1992. Window automata analysis of population dynamics in the immune system.Bull. math. Biol. 54, 21–44.
Perelson, A. S. (Ed.). 1988.Theoretical Immunology. Santa-Fe Institute Studies in the Sciences of Complexity. Redwood City, CA: Addison-Wesley.
Perelson, A. S. and G. Weisbuch. 1992. Modeling immune reactivity in secondary lymphoid organs.Bull. math. Biol. 54, 649–672.
Segel, L. A. and A. S. Perelson. 1988. Computation in shape space: a new approach to immune network theory. InTheoretical Immunology, A. S. Perelson (Ed.) Santa-Fe Institute Studies in the Sciences of Complexity, pp. 321–343. Redwood City, CA: Addison-Wesley.
Segel, L. A. and A. S. Perelson. 1991. Exploiting the diversity of time scales in the immune system: A B cell antibody model.J. statist. Phys. 63, 1113–1131.
Stewart, J. and F. Varela. 1989. Exploring the meaning of connectivity in the immune network.Immunol. Rev. 110, 37–61.
Varela, F., A. Coutinho, B. Dupire and N. Vaz. 1988. Cognitive networks: immune, neural, and otherwise. InTheoretical Immunology, A. Perelson (Ed.), Santa-Fe Institute Studies in the Sciences of Complexity pp. 359–375. Redwood City, CA: Addison-Wesley.
Weisbuch, G., R. J. De Boer and A. S. Perelson., 1990. Localized memories in idiotypic networks.J. theor. Biol. 146, 483–499.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Neumann, A.U., Weisbuch, G. Dynamics and topology of idiotypic networks. Bltn Mathcal Biology 54, 699–726 (1992). https://doi.org/10.1007/BF02459926
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02459926