Topology driven modeling the IS metaphor

Authors: Emanuela Merelli Marco Pettini Mario Rasetti

Publish Date: 2014/06/24

Volume: 14, Issue: 3, Pages: 421-430

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Abstract

In order to define a new method for analyzing the immune system within the realm of Big Data we bear on the metaphor provided by an extension of Parisi’s model based on a mean field approach The novelty is the multilinearity of the couplings in the configurational variables This peculiarity allows us to compare the partition function Z with a particular functor of topological field theory—the generating function of the Betti numbers of the state manifold of the system—which contains the same global information of the system configurations and of the data set representing them The comparison between the Betti numbers of the model and the real Betti numbers obtained from the topological analysis of phenomenological data is expected to discover hidden nary relations among idiotypes and antiidiotypes The data topological analysis will select global features reducible neither to a mere subgraph nor to a metric or vector space How the immune system reacts how it evolves how it responds to stimuli is the result of an interaction that took place among many entities constrained in specific configurations which are relational Within this metaphor the proposed method turns out to be a global topological application of the SB paradigm for modeling complex systemsThe objective pursued in this note is to frame the research on the immune system as part of data science Such research is naturally complex and articulated and our contribution intends to be here along the lines of seeing it as a viable candidate for topological data analytics and an example of the SB paradigm for modeling complex systems We recall that data science is the practice to deriving valuable insights from data by challenging all the issues related to the processing of very large data sets while Big Data is jargon to indicate such a large collection of data for example exabytes characterized by highdimensionality redundancy and noise The analysis of Big Data requires handling highdimensional vectors capable of weaning out the unimportant redundant coordinates The notion of data space its geometry and topology are the most natural tools to handle the unprecedentedly large highdimensional complex sets of data Carlsson 2009 Edelsbrunner and Harer 2010 basic ingredient of the new datadriven complexity science TOPDRIM 2012 Merelli and Rasetti 2013Topology the branch of mathematics dealing with qualitative geometric information such as connectivity classification of loops and higher dimensional manifolds studies properties of geometric objects shapes in a way which is less sensitive to metrics than geometric methods it ignores the value of distance function and replaces it with the notion of connective nearness proximity All these features make topology ideal for analysing the space of dataStarting from the notion of a mean field proposed by Parisi in his simple model for idiotypic network Parisi 1990 we propose a more sophisticated version that is multilinear in the configurational variables the antibody concentrations instead of being constant or at most linear Multilinearity allows us to recognize in the partition function Z of the model that embodies all the statistical properties of the system at equilibrium features similar to those of a particular functor of a topological field theory The latter contains indeed the same global information about the topological properties specifically its global invariants of the system configuration space and can be identified with the generating function of Betti numbers namely the Poincaré polynomial of data space Atiyah and Bott 1983 Once the homology of the space of data has been constructed and its generating cycles have been defined the related two sets of Betti numbers can be compared In this way selfconsistent information is obtained regarding 2hbox ary 3hbox ary dots nhbox ary relations among antibodies Comparison between the Betti numbers of the model and the real Betti numbers obtained by constructing the topology of phenomenological immune system space of data will unveil the hidden relations between idiotypes and antiidiotypes in particular those relations where components interact indistinctly and therefore can not be reduced to a mere subgraph but rather they bear on a new concept of interaction scalefree and metricfree The analysis of Betti numbers on phenomenological data can be dealt with techniques based on persistent homology Carlsson 2009 Petri et al 2013The challenge we are facing is to unveil whether in natural multilevel complex systems nbody interactions can drive the emergence of novel qualia in these systems In physics the interactions between material objects in real space are binary This means that mutual forces and motions are produced by twobody interactions the building blocks of any manyparticle system Thus at the atomic or molecular level description of matter living or not the total force acting on any given particle is the result of the composition of binary interactions However how can we discover if nbody interactions do exist What we are proposing here is to use the IS metaphor ie a complex system whose adaptivity is driven by data as a global topological application of the SB paradigm SB allows us to entangle in a unique model the computational component with the coordination In particular B accounts for the computation while S describes the global computation context Merelli et al 2013 The adaptation phase occurs when a machine can no longer compute in a given state of the system thus the system changes state ie the global context of computation In the IS metaphor the computation context can be identified by the global invariants while the computation with the model of interactions a sort of interactive machine Each time we discover new global invariants a new context of computation arises and with it a new IS model must be generated we call this step the adaptation phaseIn the following after giving a brief description of the antigenfree immune system and recalling Parisi’s mean field model we formally define the new topological field model and finally discuss the SB paradigm An appendix is provided with a general introduction to the fundamental tool of persistent homology and Betti numbersThe mechanism whereby the production of a given antibody elicits or suppresses the production of other antibodies that in turn elicit or suppress the production of other antibodies like a concatenation of events hints to a strict analogy of the immune system function with memory in the brain It recalls the way in which a firing neuron may induce or inhibit the firing of other neurons and so forth On the assumption that a functional network of antibodies is possible several models have been constructed among which Parisi’s model The latter studies the persistence of immune memory in the absence of any driving effect of external antigens and it offers a robust though simple theoretical framework without providing detailed description of the system Parisi 1990

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