In this section, we generalize the way how Parisi’s linear model represents immunological memory by a linear mean field. The antibodies of the idiotypic cascade are denoted by \(Ab_i\); during the production of \(Ab_1\), ignited directly by the antigen, the environment of lymphocytes is modified by \(Ab_2\): the life-span of the \(Ab_1\)-producing cells and the population of helper cells specific for \(Ab_1\) increase. The symmetry of the couplings \((J_{ik}=J_{ki})\) implies that \(Ab_3\) should be rather similar to \(Ab_1\), the internal image of \(Ab_2\) should persist after it disappeared, its presence induces the survival of memory cells directed against the antigen. The process continues by iteration. In the extended model, we assume the production of \(Ab_i\) is conditioned to different extents and also by the simultaneous presence of a subset of \(2, 3, ... , N\), antibodies.
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 function
$$\begin{aligned} \displaystyle { E= \sum _{i=1}^N h_i c_i }, \, \,\, \; c_i= \Theta (h_i )\in [0,1] \; , \end{aligned}$$
(3)
we consider in the space of antibodies \(\mathcal {A}\), the points of which are labelled by \(i=1 \dots ,N \), the graph \(\mathcal {G}\) generated by the \(J_{ik} \ne 0\) (for simplicity we assume here that \(J_{ik}\in [ -1, +1 ]\) when \(J_{ik} \ne 0\)). We next extend \(\mathcal {G}\) to the simplicial complex \(\mathcal {C}\), obtained from \(\mathcal {G}\) by completion, constructing the simplicial complex \(\mathcal {C}\) which has \(\mathcal {G}\) as 1-skeleton (scaffold), see Fig. 5. Each \(n\)-cycle in \(\mathcal {C}\) cannot be seen as composition of two-body interactions, but represents a true \(n\)-body interaction; in other words, any relationship expressed in the cycle is unique in its configuration. We denote by \(C^{(n)} ([l_1, \dots l_{(n+1)} ] \)) the cycles of \(\mathcal {C}\), and by \(\delta _{k,i}\) the presence or the absence of \(i\) in the cycle (\(\delta _{k,i}=1\) if \(k=i\), \(\delta _{k,i}=0\) if \(k\ne i\)) and we generalize then the standard linear form for the mean field \(h_i\) to the form:
$$\begin{aligned} h_i=S+ \sum _{k = 1}^N \, {\mathop {\mathop {\sum }\limits _{C^{(n)} ([ \ell _1, \dots \ell _{(n+1)} ])}}\limits _{1 \le n \le N - 1}} J_{\ell _1 \dots \ell _k \dots \ell _{n+1}} \prod _{j=1}^n c_{\ell _j} \, \delta _{k, i} \; \end{aligned}$$
(4)
In the partition function
$$\begin{aligned} \displaystyle {Z(x) \doteq \sum _{\left\{ c_\ell \right\} } e^{-x E\left( \left\{ c_\ell \right\} \right) }} \; \; x \in \mathbb {R}, \end{aligned}$$
(5)
the sum runs over the set of all possible valuations \(c_\ell = 0 , 1 \; , \; \forall \ell \), subdivides the set of states in classes of equivalence, giving different statistical weights—depending on a parameter \(x \in {\mathbb {R}}\; , \; x >0\)—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.
We consider then the functor partition function, \(Z(x)\). We might of course access more information (patterns) by considering higher (\(k\)-th) order correlation functions,
$$\begin{aligned} \Gamma _k (x) \doteq \frac{1}{Z(x)} \sum _{\left\{ c_\ell \right\} } c_{\ell _1} \dots c_{\ell _k} e^{-x E\left( \left\{ c_\ell \right\} \right) } \; , \end{aligned}$$
(6)
for any given set of points \({\ell _1 \dots \ell _k } \in \mathcal {A}\). We can represent with strings of \(N\) dichotomic variables the set of \(\{c_\ell \}\), \(2^{N-1}\) possible configurations.
A crucial assumption we add to the model is that the coupling constants \(J_{\ell _1 \dots \ell _k \dots \ell _{n+1}}\) are taken to be proportional to a linear combination (with negative coefficients) of the simplex \(n\)-volume \(V^{(n)}\), the simplex corresponding to the cell defined by the set \(\bigl \{ \ell _1 , \dots , \ell _n \bigr \}\) in the cells of cycle \(C^{(n)} ([\ell _1, \dots , \ell _{(n+1)} ])\), with the volume of the cell boundary of dimension \(n-2\), weighted by the curvature at that boundary. The latter measures the ease with which the \(n\)-body interaction is favored by the manifold bending. The ensuing action is expected to measure reasonably well the probability that the \(n\)-body process described by that coupling takes place.
When the model with such interaction form is dealt with as a statistical field theory it turns out to be fully isomorphic with a Euclidean topological field theory describing a totally different physical system: gravity coupled with matter in a simplicial complex setting, consistent with general relativity. We think back to the standard example of the Ising model, which also has variables in \({\mathbb {Z}}_2\) (Parisi 1998) and recall that a statistical field theory is any model in statistical mechanics where the degrees of freedom comprise a field; i.e. the microstates of the system are expressed through field configurations. The features of the ensuing theory are quite general and far reaching. The topology of the associated moduli space depends only on the manifold genus \(g\), on the dimension \(n\) of the (vector) bundle over it used to define the field, and on the dimension \(\delta ({\mathrm{mod}} \, n)\) of the associated determinant bundle. Such space is a projective variety, smooth only if \((\delta ,n) = 1\). The recursive determination of the Betti numbers in this case is given by the Harder and Narasimhan and Atiyah and Bott recursions (Harder and Narasimhan 1975; Atiyah and Bott 1983). The former explicitly counts points of the moduli space, the latter resorts to an infinite-dimensional Morse theory with the field action functional as Morse function. These recursions lead to a closed formula for the Poincaré polynomial, i.e. for the Betti numbers of the moduli space. These implicit methods were successively made explicit (Desale and Ramanan 1975).
What is intriguing is that our field theory turns out to be isomorphic to \({\mathbb {Z}}_2\) (quantum) gravity, dealt with in nonperturbative fashion by standard Regge calculus (Regge 1961).
Let us recall here that the construction of a consistent theory of quantum gravity in the continuum is a problem in theoretical physics that has so far defied all attempts of a rigorous formulation and resolution. The only effective approach to try and obtain a non-trivial quantum theory proceeded via discretization of space-time and of the Einstein action, i.e., by replacing the space-time continuum by a combinatorial simplicial complex and deriving the action from simple physical principles.
Quantum Regge calculus, based on the well-explored classical discretization of the Einstein action due to Regge, and the essentially equivalent method of dynamical triangulations are the tools that proved most successful. Regge’s method consists in approximating Einstein’s continuum theory by a simplicial discretization of the space-time (in gravity a four-dimensional Lorentz manifold) resorting to local building blocks (simplices) and then constructing the gravitational action as the sum of a term depending on the (hyper)volumes of the different simplicial complexes and another reflecting the space-time curvature. The metric tensor associated with each simplex is expressed as a function of the squared edge lengths, which are the dynamical variables of this model. Summing over all interpolating geometries (state sum) generated by the simplicial complex construction in the embedding higher-dimensional ones (filtration), allows us to derive both the Einstein action and the equilibrium configurations simply by means of counting procedure (entropy estimate).
The \({\mathbb {Z}}_2\) version of the model is one in which representations of \(SU(2)\) labeling the edges in quantum Regge calculus are reduced to \({\mathbb {Z}}_2\). The power of the method resides in the property that the infinite degrees of freedom of Riemannian manifolds are reduced by discretization; and the theory can deal with PL spaces, described by a finite number of parameters. Moreover, for the manifolds approximated by a simplicial complex (or by dynamically triangulated random surfaces), the local coordination numbers are automatically included among the dynamical variables, leaving the quadratic link lengths \(q_\ell \), globally constrained by triangle inequalities, as true degrees of freedom.
More precisely, the model adopted here for the immune system is isomorphic to the \({\mathbb {Z}}_2\) Regge model, where the quadratic link lengths \(q_\ell \) of the simplicial complexes are restricted to take on only two values: \(q_\ell = 1 + {\mathfrak {l}} \sigma _\ell \), where \(\sigma _\ell = \pm 1 = 2 c_{\ell } - 1\). Such model has been exactly solved (in the case of quantum gravity) via the matrix model approach (Ambjørn et al. 1985) and with the help of conformal field theory (Knizhnik et al. 1988). A crucial ingredient is the choice of functional integration measure, whose behavior, with respect to diffeomorphisms, is fundamental. The very definition of diffeomorphism is a heavy constraint in constructing the PL space exactly invariant under the action of the full diffeomorphism group (Menotti 1998), and only the recent construction of a simplicial version of the mapping class group made it viable (Merelli and Rasetti 2013).
As Regge regularization leads to the usual Liouville field theory in the continuum limit based on a description of PL manifolds with deficit angles, not edge lengths, we may assume that also in our case the correct measure has to be nonlocal. Starting point for the \({\mathbb {Z}}_2\) Regge model is a discrete description of general relativity in which space-time is represented by a piecewise flat, simplicial manifold (Regge skeleton). The procedure works for any space-time dimension \(d\), metrics of arbitrary signature, and action
$$\begin{aligned} A ( \mathbf{{q}} ) = x \left( \sum _{s^d} V^{(d)} \left( s^d \right) - \zeta \sum _{s^{d-2}} {\mathfrak {d}} ( s^{d-2} ) \, V^{(d-2)} \left( s^{d-2} \right) \right) \; \end{aligned}$$
(7)
with the quadratic edge lengths \(\left\{ q_\ell \right\} \) (more precisely, the \(\sigma _{\ell }\)’s) describing the dynamics of the complex. \(x\) and \(\zeta \) denote free constants (in the discrete time picture, with uniform time step \(\tau \), energy functional and action are merely proportional). The first sum runs over all \(d\)-simplices \(s^d\) of the simplicial complex, while \(V ( s^d )\) is the \(d\)-volume of \(s^d\). The second term represents the curvature of the simplicial complex, concentrated along the \((d - 2)\)-simplices, leading to deficit angles \({\mathfrak {d}} ( s^{d-2} )\). The physical meaning of the terms entering action \(A\) is what makes it acceptable for a consistent description of the immune system with higher order (‘many body’) interactions: the lower the volumes and the higher the curvature, the lower is the action \((x, \zeta > 0)\).
At equilibrium, i.e. in the absence of an explicit time-dependence of the expectation values of the variables, the partition function for our antigen-free IS model is nothing but the field propagator of the theory, expressed via path integral
$$\begin{aligned} \displaystyle {Z = \int \mathcal{{D}} \, [\mathbf{{q}}] \, \mathrm{{e}}^{- A ( \mathbf{{q}} )}} \end{aligned}$$
(8)
Functional integration should extend over all metrics on all possible topologies, hence the path-integral approach, typically suffers from a nonuniqueness of the integration measure and a need for a nonlocal measure is advocated. The standard ‘simplicial’ measure
$$\begin{aligned} \displaystyle { \int \mathcal{{D}} \, [\mathbf{{q}}] = \prod _\ell \, \int \frac{\mathrm{{d}} q_\ell }{q_\ell ^\alpha } \, \mathcal{{F}} ( \mathbf{{q}} )}, \,\,\, \mathrm{{where}} \,\,\, \alpha \in {\mathbb {R}} \end{aligned}$$
(9)
allows exploring a family of measures, as \(\mathcal{{F}} ( \mathbf{{q}} )\) can be designed to constrain integration to those configurations which do not violate triangular inequalities, and moreover can be chosen so as to remove non realistic simplices. The characteristic partition function of the model becomes then
$$\begin{aligned} Z = \left[ \prod _\ell ^\mathcal{{N}} \int\limits _0^\infty {\mathrm{d}} q_\ell \, q_\ell ^{- \alpha } \right] \, \mathcal{{F}} ( \mathbf{{q}} ) \, \mathrm{{e}}^{- \sum _s A_s ( \mathbf{{q}} )} \; , \end{aligned}$$
where \(\mathcal{{N}}\) is the number of links and \(A_s\) is the contribution to the action of simplex \(s\).
It is worth recalling that in (Desale and Ramanan 1975) arithmetic techniques and the Weil conjecture were used, and a crucial ingredient was the property that the volume of a particular locally symmetric space attached to \(SL_n\) with respect to the canonical measure—an invariant known as the Tamagawa number of \(SL_n\)—equals 1. The simplicial volume is a homotopy invariant of oriented, closed, connected manifolds defined in terms of the singular chain complex with real coefficients. Such invariant measures the efficiency of representing the fundamental space class using singular simplices. Since the fundamental class is nothing but a generalized triangulation of the manifold, the simplicial volume can be interpreted as well both as a measure for the complexity of the manifold and as a homotopy invariant approximation of the Riemannian volume. \(Z(x)\) provides then the generating function (Poincaré polynomial) of the Betti numbers of \(\mathcal {A}\).
The final step is to compare the Betti numbers obtained empirically from the data against such generating function, thus determining [simply through the solution of a system of (non-linear) algebraic equations] the set of non-zero \(J_{\ell _1 \dots \ell _k \dots \ell _{n+1}}\). This fully determines which antibody influences which, including ‘many-body’ influences, i.e. when and if it may happen that a given set of (two or more than two) antibodies play a role only when simultaneously active.
A short discussion of Regge calculus, meant to introduce in simple way, accessible also to readers not familiar with the notion of geometry over discrete spaces (simplicial complexes), and some of the notions actually used in the derivation can be found in Battaglia and Rasetti (2003), where some of the preliminary ideas of the scheme are described, successively developed in extended way for present and other applications. As for the work in \(\mathbb {Z}_2\) quantum gravity which our generalized model of immune system is isomorphic to, a more articulated and complete set of references is available in Giulini (2007) and Bittner et al. (1999).