Self-tolerance and autoimmunity in a minimal model of the idiotypic network.

We consider self-tolerance and its failure-autoimmunity-in a minimal mathematical model of the idiotypic network. A node in the network represents a clone of B-lymphocytes and its antibodies of the same idiotype which is encoded by a bitstring. The links between nodes represent possible interactions between clones of almost complementary idiotype. A clone survives only if the number of populated neighbored nodes is neither too small nor too large. The dynamics is driven by the influx of lymphocytes with randomly generated idiotype from the bone marrow. Previous work has revealed that the network evolves toward a highly organized modular architecture, characterized by groups of nodes which share statistical properties. The structural properties of the architecture can be described analytically, the statistical properties determined from simulations are confirmed by a modular mean-field theory. To model the presence of self we permanently occupy one or several nodes. These nodes influence their linked neighbors, the autoreactive clones, but are themselves not affected by idiotypic interactions. The architecture is very similar to the case without self, but organized such that the neighbors of self are only weakly occupied, thus providing self-tolerance. This supports the perspective that self-reactive clones, which regularly occur in healthy organisms, are controlled by anti-idiotypic clones. We discuss how perturbations, like an infection with foreign antigen, a change in the influx of new idiotypes, or the random removal of idiotypes, may lead to autoreactivity and devise protocols which cause a reconstitution of the self-tolerant state. The results could be helpful to understand network and probabilistic aspects of autoimmune disorders.

A simple regulatory architecture allows learning the statistical structure of a changing environment.

Bacteria live in environments that are continuously fluctuating and changing. Exploiting any predictability of such fluctuations can lead to an increased fitness. On longer timescales, bacteria can 'learn' the structure of these fluctuations through evolution. However, on shorter timescales, inferring the statistics of the environment and acting upon this information would need to be accomplished by physiological mechanisms. Here, we use a model of metabolism to show that a simple generalization of a common regulatory motif (end-product inhibition) is sufficient both for learning continuous-valued features of the statistical structure of the environment and for translating this information into predictive behavior; moreover, it accomplishes these tasks near-optimally. We discuss plausible genetic circuits that could instantiate the mechanism we describe, including one similar to the architecture of two-component signaling, and argue that the key ingredients required for such predictive behavior are readily accessible to bacteria.

Associations inferred from previous experience can help an organism predict what might happen the next time it faces a similar situation. For example, it could anticipate the presence of certain resources based on a correlated environmental cue. The complex neural circuitry of the brain allows such associations to be learned and unlearned quickly, certainly within the lifetime of an animal. In contrast, the sub-cellular regulatory circuits of bacteria are only capable of very simple information processing. Thus, in bacteria, the 'learning' of environmental patterns is believed to mostly occur by evolutionary mechanisms, over many generations. Landmann et al. used computer simulations and a simple theoretical model to show that bacteria need not be limited by the slow speed of evolutionary trial and error. A basic regulatory circuit could, theoretically, allow a bacterium to learn subtle relationships between environmental factors within its lifetime. The essential components for this simulation can all be found in bacteria ­ including a large number of 'regulators', the molecules that control the rate of biochemical processes. And indeed, some organisms often have more of these biological actors than appears to be necessary. The results of Landmann et al. provide new hypothesis for how such seemingly 'superfluous' elements might actually be useful. Knowing that a learning process is theoretically possible, experimental biologists could now test if it appears in nature. Placing bacteria in more realistic, fluctuating conditions instead of a typical stable laboratory environment could demonstrate the role of the extra regulators in helping the microorganisms to adapt by 'learning'.

Self-organized criticality in neural networks from activity-based rewiring.

Neural systems process information in a dynamical regime between silence and chaotic dynamics. This has lead to the criticality hypothesis, which suggests that neural systems reach such a state by self-organizing toward the critical point of a dynamical phase transition. Here, we study a minimal neural network model that exhibits self-organized criticality in the presence of stochastic noise using a rewiring rule which only utilizes local information. For network evolution, incoming links are added to a node or deleted, depending on the node's average activity. Based on this rewiring-rule only, the network evolves toward a critical state, showing typical power-law-distributed avalanche statistics. The observed exponents are in accord with criticality as predicted by dynamical scaling theory, as well as with the observed exponents of neural avalanches. The critical state of the model is reached autonomously without the need for parameter tuning, is independent of initial conditions, is robust under stochastic noise, and independent of details of the implementation as different variants of the model indicate. We argue that this supports the hypothesis that real neural systems may utilize such a mechanism to self-organize toward criticality, especially during early developmental stages.

Large systems of random linear equations with nonnegative solutions: Characterizing the solvable and the unsolvable phase.

Large systems of linear equations are ubiquitous in science. Quite often, e.g., when considering population dynamics or chemical networks, the solutions must be nonnegative. Recently, it has been shown that large systems of random linear equations exhibit a sharp transition from a phase, where a nonnegative solution exists with probability one, to one where typically no such solution may be found. The critical line separating the two phases was determined by combining Farkas' lemma with the replica method. Here we show that the same methods remain viable to characterize the two phases away from criticality. To this end we analytically determine the residual norm of the system in the unsolvable phase and a suitable measure of robustness of solutions in the solvable one. Our results are in very good agreement with numerical simulations.