Figure - available from: Complexity

This content is subject to copyright. Terms and conditions apply.

# of analysis on cooperativity. (a) Klotz plot showing the saturation function Y versus the substrate concentration S (notice the logarithmic scale on the x-axis) for the relativistic (dashed line) and the classical (solid line) models. Data for Y are collected by solving numerically the self-consistent equations ((39) and (22), resp.) for J=0.9 and different values of S. Both models exhibit the sigmoidal shape typical of cooperative systems; however, the former displays a slower saturation. Analogous results are obtained for different values of J>0. (b) A Scatchard plot is built with the same collection of data by showing the ratio Y/S versus Y. Both models exhibit the concave-down shape typical of cooperative systems. However, for relatively small values of J the plot for the relativistic model is monotonically decreasing (see also Figure 3). (c) A Hill plot is built with the same collection of data by showing θ=Y/(1-Y) versus S; both observables are taken under the logarithm. When S is close to one (here S0=1) the relativistic and the classical model give overlapped curves, while when S is either very large or very small the two curves are shifted. (d) By further analyzing the plots in the previous panels we can derive estimates for the extent of cooperativity characterizing the systems. As explained in the main text, starting from data in (a) we measured the Kloshand quantifier κ=S0.9/S0.10 (⋄), by extrapolating the maximum value for data in (b) we get σ (∇), and by fitting the data in (c) at the half-saturation point we get nH (□). These estimates are obtained for both the relativistic (white symbol) and the classic (black symbols) models.

Source publication

Complex biochemical pathways or regulatory enzyme kinetics can be reduced to chains of elementary reactions, which can be described in terms of chemical kinetics. This discipline provides a set of tools for quantifying and understanding the dialogue between reactants, whose framing into a solid and consistent mathematical description is of pivotal...

## Citations

... The problem of constructing suitable partial differential equations for state functions of thermodynamic systems and the study of critical properties in terms of critical asymptotics of the solutions to these equations is an active field of research which brought further insights on a variety of classical systems, see e.g. [34][35][36][37][38][39], and appears to be promising for the study of complex systems [40,41]. Studies exploiting the Lie symmetry analysis can be therefore carried out for other systems of physical interest. ...

We consider a family of thermodynamic models such that the energy density can be expressed as an asymptotic expansion in the scale formal parameter and whose terms are suitable functions of the volume density. We examine the possibility to construct solutions for the Maxwell thermodynamic relations relying on their symmetry properties and deduce the critical properties implied in terms of the dynamics of coexistence curves in the space of thermodynamic variables.

... The problem of constructing suitable partial differential equations for state functions of thermodynamic systems and the study of critical properties in terms of critical asympotics of the solutions to these equations is an active field of research which brought further insights on a variety of classical systems, see e.g. [33][34][35][36][37][38], and appears to be promising for the study of complex systems [39,40]. Studies similar to the present work can be put forward therefore for other systems of physical interest. ...

We consider a family of thermodynamic models such that the energy density can be expressed as an asymptotic expansion in the scale formal parameter and whose terms are suitable functions of the volume density. We examine the possibility to construct solutions for the Maxwell thermodynamic relations relying on their symmetry properties and deduce the critical properties implied in terms of the the dynamics of coexistence curves in the space of thermodynamic variables.

... Note that this cost function (4) can be expanded in an alternate-sign series as thus, focusing on the attractive contributions (beyond the classical pairwise model P = 2), it is enriched by P -spin terms (with P = 6, 10, ...) that yield to further synaptic couplings where information can be stored (as recently suggested by Hopfield himself [26]), while, focusing on the repulsive contributions, it also displays P -spin terms (with P = 4, 8, ...) that favour network's pruning (as suggested, in the past, by Hopfield himself and several other authors [18,24,28,29,35] to erase spurious states). The analysis of the information processing skills of this network has been accomplished elsewhere [6,10]: we summarize it by Fig. 1, referring to the original papers for further algorithmic details, while hereafter we deepen the mathematical aspects of its statistical mechanical foundation. ...

The relativistic Hopfield model constitutes a generalization of the standard Hopfield model that is derived by the formal analogy between the statistical-mechanic framework embedding neural networks and the Lagrangian mechanics describing a fictitious single-particle motion in the space of the tuneable parameters of the network itself. In this analogy, the cost-function of the Hopfield model plays as the standard kinetic-energy term and its related Mattis overlap (naturally bounded by one) plays as the velocity. The Hamiltonian of the relativisitc model, once Taylor-expanded, results in a p-spin series with alternate signs: the attractive contributions enhance the information-storage capabilities of the network, while the repulsive contributions allow for an easier unlearning of spurious states, conferring overall more robustness to the system as a whole. Here, we do not deepen the information processing skills of this generalized Hopfield network, rather we focus on its statistical mechanical foundation. In particular, relying on Guerra's interpolation techniques, we prove the existence of the infinite-volume limit for the model free-energy and we give its explicit expression in terms of the Mattis overlaps. By extremizing the free energy over the latter, we get the generalized self-consistent equations for these overlaps as well as a picture of criticality that is further corroborated by a fluctuation analysis. These findings are in full agreement with the available previous results.

... Note that this cost function (4) can be expanded in an alternate-sign series as thus, focusing on the attractive contributions (beyond the classical pairwise model P = 2), it is enriched by P -spin terms (with P = 6, 10, ...) that yield to further synaptic couplings where information can be stored (as recently suggested by Hopfield himself [26]), while, focusing on the repulsive contributions, it also displays P -spin terms (with P = 4, 8, ...) that favour network's pruning (as suggested, in the past, by Hopfield himself and several other authors [18,24,28,29,35] to erase spurious states). The analysis of the information processing skills of this network has been accomplished elsewhere [6,10]: we summarize it by Fig. 1, referring to the original papers for further algorithmic details, while hereafter we deepen the mathematical aspects of its statistical mechanical foundation. ...

The relativistic Hopfield model constitutes a generalization of the standard Hopfield model that is derived by the formal analogy between the statistical-mechanic framework embedding neural networks and the Lagrangian mechanics describing a fictitious single-particle motion in the space of the tuneable parameters of the network itself. In this analogy the cost-function of the Hopfield model plays as the standard kinetic-energy term and its related Mattis overlap (naturally bounded by one) plays as the velocity. The Hamiltonian of the relativisitc model, once Taylor-expanded, results in a P-spin series with alternate signs: the attractive contributions enhance the information-storage capabilities of the network, while the repulsive contributions allow for an easier unlearning of spurious states, conferring overall more robustness to the system as a whole. Here we do not deepen the information processing skills of this generalized Hopfield network, rather we focus on its statistical mechanical foundation. In particular, relying on Guerra's interpolation techniques, we prove the existence of the infinite volume limit for the model free-energy and we give its explicit expression in terms of the Mattis overlaps. By extremizing the free energy over the latter we get the generalized self-consistent equations for these overlaps, as well as a picture of criticality that is further corroborated by a fluctuation analysis. These findings are in full agreement with the available previous results.