Advances in Complex Systems

We introduce a class of dynamical systems called Higher Order Cellular Automata (HOCA). These are based on ordinary CA, but have a hierarchical, or multi-level, structure and/or dynamics. We present a detailed formalism for HOCA and illustrate the concepts through four examples. Throughout the article we emphasize the principles and ideas behind the construction of HOCA, such that these easily can be applied to other types of dynamical systems. The article also presents new concepts and ideas for describing and studying hierarchial dynamics in general.

The design of the state observer for a kind of uncertain symmetric circulant composite systems based on regional pole assignment is studied. The "decentralized observer gain matrices" have been obtained by taking use of the special structure of the system. The central observer gain matrix for such a system has the same block symmetric circulant structure as the original system and can be constructed by the "decentralized observer gain matrices." Thus, the problem of designing state observer for such an uncertain system with n·N dimensions based on regional pole assignment can be transformed into the subproblems for N/2+1 or (N + 1)/2 uncertain systems with n dimensions. The central D-stabilizable state observer is also a symmetric circulant composite system. And the design of the observer makes best use of the advantage of interconnections of the system.

We develop a simple Keynesian type business cycle model in which heterogeneous agents are either optimistic or pessimistic. If the majority of the agents are optimistic, then consumption expenditures are high and the economy booms, otherwise consumption expenditures are low and the economy is in a recession. Within our model, the sentiment of the agents is affected by their social interactions. For instance, people regularly meet each other and thus their mood may change. Overall, our model suggests that swings in consumer confidence may generate irregular fluctuations in economic activity.

We review recent findings that, using fractal analysis, have demonstrated systematic regional and species differences in the branching complexity of neocortical pyramidal neurons. In particular, attention is focused on how fractal analysis is being applied to the study of specialization in pyramidal cell structure during the evolution of the primate cerebral cortex. These studies reveal variation in pyramidal cell phenotype that cannot be attributed solely to increasing brain volume. Moreover, the results of these studies suggest that the primate cerebral cortex is composed of neurons of different structural complexity. There is growing evidence to suggest that regional and species differences in neuronal structure influence function at both the cellular and circuit levels. These data challenge the prevailing dogma for cortical uniformity.

The formal theory of the development of early perception and motor control presented here deals with cognitive development as a mapping from a finite set of given experiences to a set of perceptual and motor-control functions. The theory involves seven constraints that uniquely define the mapping. The compatibility with observational phenomena and sufficiency of these constraints shows the validity of the theory. The principle underlying these constraints is a coding by the most efficient representation of information. The efficiency of representation is evaluated by the coding redundancy of given experiences defined as the number of real numbers that characterize experiences plus the size of the minimum continuous decoding function. The coding redundancy of experiences by the most efficient representation corresponds to the Kolmogorov complexity of the experiences. The mapping accounts for the dependence on neonatal experience of the development of perceptual and motor-control functions. This theory of development can also be seen as a metatheory of cognition that presents us a unified view of the diversity of perceptual and motor-control modules.

We investigate the behavior of a threshold model for the spreading of fads and similar phenomena in society. The model gives the fad dynamics and is intended to be confined to an underlying network structure. We investigate the whole parameter space of the fad dynamics on three types of network models. The dynamics we discover is rich and highly dependent on the underlying network structure. For some range of the parameter space, for all types of substrate networks, there are a great variety of sizes and life-lengths of the fads — what one sees in real-world social and economical systems.

RNA molecules interact by forming inter-molecular base pairs that compete with the intra-molecular base pairs of their secondary structures. We investigate the patterns of neutral mutations in RNAs whose function is the interaction with other RNAs, i.e. the co-folding with one or more other RNA molecules. We find that (i) the degree of neutrality is much smaller in interacting RNAs compared to RNAs that just have to coform to a single externally prescribed target structure, and (ii) strengthening this contraint to the conservation of the co-folded structure with two or more partners essentially eliminates neutrality. It follows that RNAs whose function depends on the formation of a specific interaction complex with a target RNA molecule will evolve much more slowly than RNAs with a function depending only on their own structure.

Despite spectacular progress in biophysics, molecular biology and biochemistry our ability to predict the dynamic behavior of multicellular systems under different conditions is very limited. An important reason for this is that still not enough is known about how cells change their physical and biological properties by genetic or metabolic regulation, and which of these changes affect the cell behavior. For this reason, it is difficult to predict the system behavior of multicellular systems in case the cell behavior changes, for example, as a consequence of regulation or differentiation. The rules that underlie the regulation processes have been determined on the time scale of evolution, by selection on the phenotypic level of cells or cell populations. We illustrate by detailed computer simulations in a multi-scale approach how cell behavior controlled by regulatory networks may emerge as a consequence of an evolutionary process, if either the cells, or populations of cells are subject to selection on particular features. We consider two examples, migration strategies of single cells searching a signal source, or aggregation of two or more cells within minimal multiscale models of biological evolution. Both can be found for example in the life cycle of the slime mold Dictyostelium discoideum. However, phenotypic changes that can lead to completely different modes of migration have also been observed in cells of multi-cellular organisms, for example, as a consequence of a specialization in stem cells or the de-differentiation in tumor cells. The regulatory networks are represented by Boolean networks and encoded by binary strings. The latter may be considered as encoding the genetic information (the genotype) and are subject to mutations and crossovers. The cell behavior reflects the phenotype. We find that cells adopt naturally observed migration strategies, controlled by networks that show robustness and redundancy. The model simplicity allow us to unambiguously analyze the regulatory networks and the resulting phenotypes by different measures and by knockouts of regulatory elements. We illustrate that in order to maintain a cells' phenotype in case of a knockout, the cell may have to be able to deal with contradictory information. In summary, both the cell phenotype as well as the emerged regulatory network behave as their biological counterparts observed in nature.

The main mechanisms that control the organization of multicellular tissues are still largely open. A commonly used tool to study basic control mechanisms are in vitro experiments in which the growth conditions can be widely varied. However, even in vitro experiments are not free from unknown or uncontrolled influences. One reason why mathematical models become more and more a popular complementary tool to experiments is that they permit the study of hypotheses free from unknown or uncontrolled influences that occur in experiments. Many model types have been considered so far to model multicellular organization ranging from detailed individual-cell based models with explicit representations of the cell shape to cellular automata models with no representation of cell shape, and continuum models, which consider a local density averaged over many individual cells. However, how the different model description may be linked, and, how a description on a coarser level may be constructed based on the knowledge of the finer, microscopic level, is still largely unknown. Here, we consider the example of monolayer growth in vitro to illustrate how, in a multi-step process starting from a single-cell based off-lattice-model that subsumes the information on the sub-cellular scale by characteristic cell-biophysical and cell-kinetic properties, a cellular automaton may be constructed whose rules have been chosen based on the findings in the off-lattice model. Finally, we use the cellular automaton model as a starting point to construct a multivariate master equation from a compartment approach from which a continuum model can be derived by a systematic coarse-graining procedure. We find that the resulting continuum equation largely captures the growth behavior of the CA model. The development of our models is guided by experimental observations on growing monolayers.