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We construct a model of cell reprogramming (the conversion of fully differentiated cells to a state of pluripotency, known as induced pluripotent stem cells, or iPSCs) which builds on key elements of cell biology viz. cell cycles and cell lineages. Although reprogramming has been demonstrated experimentally, much of the underlying processes governing cell fate decisions remain unknown. This work aims to bridge this gap by modelling cell types as a set of hierarchically related dynamical attractors representing cell cycles. Stages of the cell cycle are characterised by the configuration of gene expression levels, and reprogramming corresponds to triggering transitions between such configurations. Two mechanisms were found for reprogramming in a two level hierarchy: cycle specific perturbations and a noise-induced switching. The former corresponds to a \emph{directed} perturbation that induces a transition into a cycle state of a different cell type in the potency hierarchy (mainly a stem cell) whilst the latter is a priori undirected and could be induced, e.g., by a (stochastic) change in the cellular environment. These reprogramming protocols were found to be effective in large regimes of the parameter space and make specific predictions concerning reprogramming dynamics which are broadly in line with experimental findings.

We compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter- and intra-module degree distributions, subject only to the constraint of finite mean connectivities. We also evaluate spectra of a certain class of small-world matrices generated from random graphs by introducing short-cuts via additional random connectivity components. Both adjacency matrices and the associated graph Laplacians are investigated. For the Laplacians, we find Lifshitz type singular behaviour of the spectral density in a localised region of small $|\lambda|$ values. In the case of modular networks, we can identify contributions local densities of state from individual modules. For small-world networks, we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states, exhibiting only localised states in the band-gaps. Results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances, and with direct diagonalisation results., Comment: 18 pages, 5 figures

We present a quantum statistical analysis of a microscopic mean-field model of structural glasses at low temperatures. The model can be thought of as arising from a random Born von Karman expansion of the full interaction potential. The problem is reduced to a single-site theory formulated in terms of an imaginary-time path integral using replicas to deal with the disorder. We study the physical properties of the system in thermodynamic equilibrium and develop both perturbative and non-perturbative methods to solve the model. The perturbation theory is formulated as a loop expansion in terms of two-particle irreducible diagrams, and is carried to three-loop order in the effective action. The non-perturbative description is investigated in two ways, (i) using a static approximation, and (ii) via Quantum Monte Carlo simulations. Results for the Matsubara correlations at two-loop order perturbation theory are in good agreement with those of the Quantum Monte Carlo simulations. Characteristic low-temperature anomalies of the specific heat are reproduced, both in the non-perturbative static approximation, and from a three-loop perturbative evaluation of the free energy. In the latter case the result so far relies on using Matsubara correlations at two-loop order in the three-loop expressions for the free energy, as self-consistent Matsubara correlations at three-loop order are still unavailable. We propose to justify this by the good agreement of two-loop Matsubara correlations with those obtained non-perturbatively via Quantum Monte Carlo simulations., Comment: 13 pages, 6 figures

We compute spectra of symmetric random matrices defined on graphs exhibiting a modular structure. Modules are initially introduced as fully connected sub-units of a graph. By contrast, inter-module connectivity is taken to be incomplete. Two different types of inter-module connectivity are considered, one where the number of intermodule connections per-node diverges, and one where this number remains finite in the infinite module-size limit. In the first case, results can be understood as a perturbation of a superposition of semicircular spectral densities one would obtain for uncoupled modules. In the second case, matters can be more involved, and depend in detail on inter-module connectivities. For suitable parameters we even find near-triangular shaped spectral densities, similar to those observed in certain scale-free networks, in a system of consisting of just two coupled modules. Analytic results are presented for the infinite module-size limit; they are well corroborated by numerical simulations., Comment: 16 pages, 4 figures. to appear in J. Phys. A

We study a simple, solvable model that allows us to investigate effects of credit contagion on the default probability of individual firms, in both portfolios of firms and on an economy wide scale. While the effect of interactions may be small in typical (most probable) scenarios they are magnified, due to feedback, by situations of economic stress, which in turn leads to fatter tails in loss distributions of large loan portfolios., Comment: 11 pages, 3 figures

We study a credit risk model which captures effects of economic interactions on a firm's default probability. Economic interactions are represented as a functionally defined graph, and the existence of both cooperative, and competitive, business relations is taken into account. We provide an analytic solution of the model in a limit where the number of business relations of each company is large, but the overall fraction of the economy with which a given company interacts may be small. While the effects of economic interactions are relatively weak in typical (most probable) scenarios, they are pronounced in situations of economic stress, and thus lead to a substantial fattening of the tails of loss distributions in large loan portfolios. This manifests itself in a pronounced enhancement of the Value at Risk computed for interacting economies in comparison with their non-interacting counterparts., Comment: 24 pages, 6 figures

Alkali halide crystals doped with certain impurity ions show a low temperature behaviour, which differs significantly from that of pure crystals. The origin of these characteristic differences are tunneling centers formed by atomic or molecular impurity ions. We have investigated the dielectric susceptibility of hydroxyl ions in NaCl crystals at very low concentrations (below 30 ppm), where interactions are believed to be negligible. We find that the temperature dependence of the susceptibility is noticeably different from what one would expect for isolated defects in a symmetric environment. We propose that the origin of these deviations are random internal strains arising from imperfections of the host crystal. We will present the experimental data and a theoretical model which allows a quantitative understanding on a microscopic basis., Comment: 3 pages 3 figures, REVTeX, submitted to the proceedings of the PHONONS 2001 conference

We show that a straightforward extension of a simple learning model based on the Hebb rule, the previously introduced Association-Reinforcement-Hebb-Rule, can cope with "delayed", unspecific reinforcement also in the case of structured data and lead to perfect generalization., Comment: 13 pages, 3 figures

We investigate the non-equilibrium dynamics of spherical spin models with two-spin interactions. For the exactly solvable models of the d-dimensional spherical ferromagnet and the spherical Sherrington-Kirkpatrick model the asymptotic dynamics has for large times and for large waiting times the same formal structure. In the limit of large waiting times we find in both models an intermediate time scale, scaling as a power of the waiting time with an exponent smaller than one, and thus separating the time-translation invariant short-time dynamics from the aging regime. It is this time scale on which the fluctuation-dissipation regime is violated. Aging in these models is similar to that observed in spin glasses at the level of correlation functions, but different at the level of response functions, and thus different at the level of experimentally accessible quantities like the thermoremanent magnetization., Comment: 8 pages, 1 eps figure

The critical behaviour of the randomly spin-diluted Ising model in two space dimensions is investigated by a new method which combines a grand ensemble approach to disordered systems proposed by Morita with the phenomenological renormalization group scheme of Nightingale. Accurate approximations for the phase diagram and for the connectivity length exponent of the percolation transition are obtained. Our results suggest that the thermal phase transition of the disordered system might be different from that of the pure system: we observe a continuous variation of critical exponents with the density $\rho$ of magnetic impurities, respecting, however, weak universality in the sense that $\eta$ and $\gamma/\nu$ do {\it not\/} depend on $\rho$ while $\gamma$ and $\nu$ separately do. Our results are in qualitative and quantitative agreement with a recent Monte--Carlo study., Comment: 10 pages, LaTeX, 2 Figures available by Fax upon request from the author, TVP-94

The phenomenon of replica symmetry breaking is investigated for the retrieval phases of Hopfield-type network models. The basic calculation is done for the generalized version of the standard model introduced by Horner [1] and by Perez-Vicente and Amit [2] which can exhibit low mean levels of neural activity. For a mean activity $\bar a =1/2$ the Hopfield model is recovered. In this case, surprisingly enough, we cannot confirm the well known one step replica symmetry breaking (1RSB) result for the storage capacity which was presented by Crisanti, Amit and Gutfreund [3] ($\alpha_c^{\hbox{\mf 1RSB}}\simeq 0.144$). Rather, we find that 1RSB- and 2RSB-Ans\"atze yield only slightly increased capacities as compared to the replica symmetric value ($\alpha_c^{\hbox{\mf 1RSB}}\simeq 0.138\,186$ and $\alpha_c^{\hbox{\mf 2RSB}}\simeq 0.138\,187 $ compared to $\alpha_c^{\hbox{\mf RS}}\simeq 0.137\,905$), significantly smaller also than the value $\alpha_c^{\hbox{\mf sim}} = 0.145\pm 0.009$ reported from simulation studies. These values still lie within the recently discovered reentrant phase [4]. We conjecture that in the infinite Parisi-scheme the reentrant behaviour disappears as is the case in the SK-spin-glass model (Parisi--Toulouse-hypothesis). The same qualitative results are obtained in the low activity range., Comment: Latex file, 20 pages, 8 Figures available from the authors upon request, HD-TVP-94-4