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Register a new userRandomly Evolving Idiotypic Networks: Modular Mean Field Theory
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We develop a modular mean field theory for a minimalistic model of the idiotypic network. The model comprises the random influx of new idiotypes and a deterministic selection. It describes the evolution of the idiotypic network towards complex modular architectures, the building principles of which are known. The nodes of the network can be classified into groups of nodes, the modules, which share statistical properties. Each node experiences only the mean influence of the groups to which it is linked. Given the size of the groups and linking between them the statistical properties such as mean occupation, mean life time, and mean number of occupied neighbors are calculated for a variety of patterns and compared with simulations. For a pattern which consists of pairs of occupied nodes correlations are taken into account.
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An essential requirement for the representation of functional patterns in complex neural networks, such as the mammalian cerebral cortex, is the existence of stable regimes of network activation, typically arising from a limited parameter range. In this range of limited sustained activity (LSA), the activity of neural populations in the network persists between the extremes of either quickly dying out or activating the whole network. Hierarchical modular networks were previously found to show a wider parameter range for LSA than random or small-world networks not possessing hierarchical organization or multiple modules. Here we explored how variation in the number of hierarchical levels and modules per level influenced network dynamics and occurrence of LSA. We tested hierarchical configurations of different network sizes, approximating the large-scale networks linking cortical columns in one hemisphere of the rat, cat, or macaque monkey brain. Scaling of the network size affected the number of hierarchical levels and modules in the optimal networks, also depending on whether global edge density or the numbers of connections per node were kept constant. For constant edge density, only few network configurations, possessing an intermediate number of levels and a large number of modules, led to a large range of LSA independent of brain size. For a constant number of node connections, there was a trend for optimal configurations in larger-size networks to possess a larger number of hierarchical levels or more modules. These results may help to explain the trend to greater network complexity apparent in larger brains and may indicate that this complexity is required for maintaining stable levels of neural activation.
We consider the problem of self tolerance in the frame of a minimalistic model of the idiotypic network. A node of this network represents a population of B lymphocytes of the same idiotype which is encoded by a bit string. The links of the network connect nodes with (nearly) complementary strings. The population of a node survives if the number of occupied neighbours is not too small and not too large. There is an influx of lymphocytes with random idiotype from the bone marrow. Previous investigations have shown that this system evolves toward highly organized architectures, where the nodes can be classified into groups according to their statistical properties. The building principles of these architectures can be analytically described and the statistical results of simulations agree very well with results of a modular mean field theory. In this paper we present simulation results for the case that one or several nodes, playing the role of self, are permanently occupied. We observe that the group structure of the architecture is very similar to the case without self antigen, but organized such that the neighbours of the self are only weakly occupied, thus providing self tolerance. We also treat this situation in mean field theory which give results in good agreement with data from simulation.
We study the dynamics of a thick polar epithelium subjected to the action of both an electric and a flow field in a planar geometry. We develop a generalized continuum hydrodynamic description and describe the tissue as a two component fluid system. The cells and the interstitial fluid are the two components and we keep all terms allowed by symmetry. In particular we keep track of the cell pumping activity for both solvent flow and electric current and discuss the corresponding orders of magnitude. We study the growth dynamics of tissue slabs, their steady states and obtain the dependence of the cell velocity, net cell division rate, and cell stress on the flow strength and the applied electric field. We find that finite thickness tissue slabs exist only in a restricted region of phase space and that relatively modest electric fields or imposed external flows can induce either proliferation or death.
We study a spatial network model with exponentially distributed link-lengths on an underlying grid of points, undergoing a structural crossover from a random, ErdH{o}s--Renyi graph to a $2D$ lattice at the characteristic interaction range $zeta$. We find that, whilst far from the percolation threshold the random part of the incipient cluster scales linearly with $zeta$, close to criticality it extends in space until the universal length scale $zeta^{3/2}$ before crossing over to the spatial one. We demonstrate this {em critical stretching} phenomenon in percolation and in dynamical processes, and we discuss its implications to real-world phenomena, such as neural activation, traffic flows or epidemic spreading.
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 towards 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.
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