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Topology driven modeling: the IS metaphor.

Merelli E, Pettini M, Rasetti M - Nat Comput (2015)

Bottom Line: 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 S[B] paradigm for modeling complex systems.

View Article: PubMed Central - PubMed

Affiliation: School of Science and Technology, University of Camerino, Camerino, Italy.

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 [Formula: see text] 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 n-ary relations among idiotypes and anti-idiotypes. 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 S[B] paradigm for modeling complex systems.

No MeSH data available.


Simplicial complex  from the graph
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Fig5: Simplicial complex from the graph

Mentions: A weakness of this representation is that the possible equilibrium configurations of the network are fixed, whereas we want the network to be capable of learning which antibodies should be produced without assuming that only a fraction of all antibodies have physiological relevance. Therefore, whilst we maintain the global cost function3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \displaystyle { E= \sum _{i=1}^N h_i c_i }, \, \,\, \; c_i= \Theta (h_i )\in [0,1] \; , \end{aligned}$$\end{document}E=∑i=1Nhici,ci=Θ(hi)∈[0,1],we consider in the space of antibodies , the points of which are labelled by , the graph generated by the (for simplicity we assume here that when ). We next extend to the simplicial complex , obtained from by completion, constructing the simplicial complex which has as 1-skeleton (scaffold), see Fig. 5. Each -cycle in cannot be seen as composition of two-body interactions, but represents a true -body interaction; in other words, any relationship expressed in the cycle is unique in its configuration. We denote by ) the cycles of , and by the presence or the absence of in the cycle ( if , if ) and we generalize then the standard linear form for the mean field to the form:4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} h_i=S+ \sum _{k = 1}^N \, {\mathop {\mathop {\sum }\limits _{C^{(n)} ([ \ell _1, \dots \ell _{(n+1)} ])}}\limits _{1 \le n \le N - 1}} J_{\ell _1 \dots \ell _k \dots \ell _{n+1}} \prod _{j=1}^n c_{\ell _j} \, \delta _{k, i} \; \end{aligned}$$\end{document}hi=S+∑k=1N∑C(n)([ℓ1,⋯ℓ(n+1)])1≤n≤N-1Jℓ1⋯ℓk⋯ℓn+1∏j=1ncℓjδk,iIn the partition function5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \displaystyle {Z(x) \doteq \sum _{\left\{ c_\ell \right\} } e^{-x E\left( \left\{ c_\ell \right\} \right) }} \; \; x \in \mathbb {R}, \end{aligned}$$\end{document}Z(x)≐∑cℓe-xEcℓx∈R,the sum runs over the set of all possible valuations , subdivides the set of states in classes of equivalence, giving different statistical weights—depending on a parameter —to those states which are invariant with respect to a given set of transformations. A phase transition, if any, would allow us to pass from one class of equivalence to the other when the state symmetry is (partially or fully) broken. This turns the model into a theoretical framework where, given a parameter—for example the average specific antibody concentration—we can predict when and if a configuration may break into another, giving rise to a different immunity type, i.e. change the adaptive immunity. In terms of formal language theory, going from one configuration to another belonging to a different class of equivalence has the following meaning: if we associate to the space of data a group of possible transformations preserving its topology (e.g., its mapping class group), and the related regular language, the general semantics thus naturally generated describes the set of all transformations and hence of all ‘phases’ in the form of relations.


Topology driven modeling: the IS metaphor.

Merelli E, Pettini M, Rasetti M - Nat Comput (2015)

Simplicial complex  from the graph
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4541713&req=5

Fig5: Simplicial complex from the graph
Mentions: A weakness of this representation is that the possible equilibrium configurations of the network are fixed, whereas we want the network to be capable of learning which antibodies should be produced without assuming that only a fraction of all antibodies have physiological relevance. Therefore, whilst we maintain the global cost function3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \displaystyle { E= \sum _{i=1}^N h_i c_i }, \, \,\, \; c_i= \Theta (h_i )\in [0,1] \; , \end{aligned}$$\end{document}E=∑i=1Nhici,ci=Θ(hi)∈[0,1],we consider in the space of antibodies , the points of which are labelled by , the graph generated by the (for simplicity we assume here that when ). We next extend to the simplicial complex , obtained from by completion, constructing the simplicial complex which has as 1-skeleton (scaffold), see Fig. 5. Each -cycle in cannot be seen as composition of two-body interactions, but represents a true -body interaction; in other words, any relationship expressed in the cycle is unique in its configuration. We denote by ) the cycles of , and by the presence or the absence of in the cycle ( if , if ) and we generalize then the standard linear form for the mean field to the form:4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} h_i=S+ \sum _{k = 1}^N \, {\mathop {\mathop {\sum }\limits _{C^{(n)} ([ \ell _1, \dots \ell _{(n+1)} ])}}\limits _{1 \le n \le N - 1}} J_{\ell _1 \dots \ell _k \dots \ell _{n+1}} \prod _{j=1}^n c_{\ell _j} \, \delta _{k, i} \; \end{aligned}$$\end{document}hi=S+∑k=1N∑C(n)([ℓ1,⋯ℓ(n+1)])1≤n≤N-1Jℓ1⋯ℓk⋯ℓn+1∏j=1ncℓjδk,iIn the partition function5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \displaystyle {Z(x) \doteq \sum _{\left\{ c_\ell \right\} } e^{-x E\left( \left\{ c_\ell \right\} \right) }} \; \; x \in \mathbb {R}, \end{aligned}$$\end{document}Z(x)≐∑cℓe-xEcℓx∈R,the sum runs over the set of all possible valuations , subdivides the set of states in classes of equivalence, giving different statistical weights—depending on a parameter —to those states which are invariant with respect to a given set of transformations. A phase transition, if any, would allow us to pass from one class of equivalence to the other when the state symmetry is (partially or fully) broken. This turns the model into a theoretical framework where, given a parameter—for example the average specific antibody concentration—we can predict when and if a configuration may break into another, giving rise to a different immunity type, i.e. change the adaptive immunity. In terms of formal language theory, going from one configuration to another belonging to a different class of equivalence has the following meaning: if we associate to the space of data a group of possible transformations preserving its topology (e.g., its mapping class group), and the related regular language, the general semantics thus naturally generated describes the set of all transformations and hence of all ‘phases’ in the form of relations.

Bottom Line: 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 S[B] paradigm for modeling complex systems.

View Article: PubMed Central - PubMed

Affiliation: School of Science and Technology, University of Camerino, Camerino, Italy.

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 [Formula: see text] 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 n-ary relations among idiotypes and anti-idiotypes. 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 S[B] paradigm for modeling complex systems.

No MeSH data available.