Natural, everyday sensorimotor behaviors, such as rising from sitting, typically have an intrinsic organization of several levels of analysis. Taking this intrinsic organization as key to understanding neural dynamics is neither a top-down nor a bottom-up approach, but rather a meshing of multiple centers and levels of analysis. Motor control requires body dynamics that are consistent with physical dynamics, besides the more microscopic levels of neural dynamics. The dynamics of separate movements have been investigated as if the ends can be capped off, separated from the rest of the individual's life. Is this dynamically correct? Even chaotic behavior is deterministic. However, the mathematics of nonlinear oscillations is not all of dynamics. This paper relates Bloch's dynamical theorem to the modular, conditional approach to sensorimotor and other neural functioning.
Bloch's dynamical theorem lays a foundation for the piecewise study of structurally accurate dynamics in theoretical neurobiology. Piecewise studies can be used as a modeling option complementary to the methods of nonlinear oscillator dynamics. By applying Bloch's theorem, dynamics of movements analyzed piecewise can be extended into a smooth flow on any manifold, either as a whole or conditionally. Conditional dynamics makes dynamical modeling options explicit, often depending on what variables the organism can control, and allows one to take different modeling options at different junctures in analyzing the same phenomenon. For example, this approach allows the study of complex motor control problems to be reduced to modular constructions using singularities and flow lines. Dynamical contingencies are expressed using the mathematics of ordered structures. This paper presents Bloch's dynamical theorem and its relevance to model construction in theoretical neurobiology. Specific examples, integrated into physiological and behavioral context, are cited from the literature.
Available at: http://works.bepress.com/gin_mccollum/13/