A probabilistic foundation for dynamical systems: phenomenological reasoning and principal characteristics of probabilistic evolution |
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Authors: | Emre Demiralp Metin Demiralp Luis Hernandez-Garcia |
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Institution: | 1.Department of Psychology,University of Michigan,Ann Arbor,USA;2.??stanbul Teknik üniversitesi Bili?im Enstitüsü, Maslak,Istanbul,Turkey;3.University of Michigan,Ann Arbor,USA |
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Abstract: | This paper is the second in a series of two. The first paper has been devoted to the detailed explanation of the mathematical
formulation of the underlying theoretical framework. Specifically, the first paper shows that it is possible to construct
an infinite linear ODE set, which describes a probabilistic evolution. The evolution is probabilistic because the unknowns
are expectations, with appropriate initial conditions. These equations, which we name, Probabilistic Evolution Equations (PEE)
are linear at the level of ODEs and initial conditions. In this paper, we first focus on the phenomenological reasoning that
lead us to the derivation of PEE. Second, the aspects of the PEE construction is revisited with a focus on the spectral nature
of the probabilistic evolution. Finally, we postulate fruitful avenues of research in the fields of dynamical causal modeling
in human neuroimaging and effective connectivity analysis. We believe that this final section is a prime example of how the
rigorous methods developed in the context of mathematical chemistry can be influential in other fields and disciplines. |
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Keywords: | |
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