Symbolic dynamics

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In mathematics, symbolic dynamics is the practice of modelling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator.

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[edit] History

The idea goes back to Jacques Hadamard's 1898 paper on the geodesics on surfaces of negative curvature. It was applied by Marston Morse in 1921 to the construction of a nonperiodic recurrent geodesic. Related work was done by Emil Artin in 1924 (for the system now called Artin billiard), P. J. Myrberg, Paul Koebe, Jakob Nielsen, G. A. Hedlund.

The first formal treatment was developed by Morse and Hedlund in their 1938 paper. George Birkhoff, Norman Levinson and M. L. Cartwright–J. E. Littlewood have applied similar methods to qualitative analysis of nonautonomous second order differential equations.

Claude Shannon used symbolic sequences and shifts of finite type in his 1948 paper A mathematical theory of communication that gave birth to information theory.

The theory was further advanced in the 1960s and 1970s, notably, in the works of Steve Smale and his school, and of Yakov Sinai and the Soviet school of ergodic theory. A spectacular application of the methods of symbolic dynamics is Sharkovskii's theorem about periodic orbits of a continuous map of an interval into itself (1964).

[edit] Applications

Symbolic dynamics originated as a method to study general dynamical systems; now its techniques and ideas have found significant applications in data storage and transmission, linear algebra, the motions of the planets and many other areas. The distinct feature in symbolic dynamics is that time is measured in discrete intervals. So at each time interval the system is in a particular state. Each state is associated with a symbol and the evolution of the system is described by an infinite sequence of symbols - represented effectively as strings. If the system states are not inherently discrete, then the state vector must be discretized, so as to get a coarse-grained description of the system.

[edit] Hematopoietic Stem Cell Kinetics

Stem cells are generally defined as those cells in any living system that have two properties: A. Pluri-potency, i.e. the ability to differentiate into a tissue of mature uni-potent cells; B. Self-renewal capacity, i.e. the ability to maintain their own population at an approximately constant level.The most basic stem cell in higher animals is the embryonic stem cell which gives rise to other tissue stem cells such as neuronal, mesenchymal, or hematopoietic stem cells. The latter are the stem cells which generate and maintain the blood system.

Hematopoietic stem cells (HSC), like other stem cells, can not be observed directly or be isolated due to an uncertainty relationship based on the incomensurability of properties A and B above. Rather, HSC behaviors need to be inferred through indirect strategies based on the logic "If a tissue transplant yields a full hematopoietic system in an ablated host, then there was at least one HSC in the donor transplant". Sophisticated experimental methods exist to ascertain that, with a high probability, a very small number of HSC (1-2) is contained in a transplant. Therefore, it is possible to measure the number of donor-derived cells in the host over time as properties A and B are iterated in the process referred to as reconstitution.

These reconstitution kinetics are very heterogeneous. Surprisingly, using symbolic dynamics, one can show that they fall into a limited number of classes [1]. It is assumed that the hematopoietic dynamical system has three states ("+" for increased repopulation activity, "-" for decreased repopulation activity, and "~" for experimentally undetectable repopulation activity between successive measurements of the number of donor-derived cells). As the state changes are followed over at least 7 and up to 48 months, a specific sequence of symbols can be associated with each reconstitution event. Using the Hamming distance on these sequences shows then that they fall into a limited number of kinetic clusters according to a truncated power-law distribution of the form F(r) = γ r-(1/d) e-(1/λ)r between the frequency and the rank of a symbolic kinetic. This allows several conclusions. The HSC compartment is heterogeneous - there is no mother-of-all hematopoietic stem cell. Epigenetic imprinting may play an essential role in the generation of diversity of the blood system. Finally, the probability of occurrence of new patterns is very small and, thus, the above classification describes the full repertoire of HSC. Therefore, analyzing the hematopoietic system as a discrete dynamical system has played an essential role in understanding its origins which, in turn, provides valuable clues for discovering new approaches for therapeutic intervention.

[edit] See also

[edit] References

  1. ^ Sieburg, HB, Muller-Sieburg, CE. Classification of short kinetics by shape. In Silico Biol. 2004;4(2):209-17

[edit] Further reading

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