Abstract
Dynamical systems can be coupled to other dynamical systems to cover organismic phenomenon on many scales, as they provides a general frame applicable to an ample range of phenomena. Describing a system as coupled systems highlights interfaces. Interfaces reflect the locus of organized transitions, or level crossings, places where constancy and variability merge. At interfaces there is convergence and/or divergence, so the vicissitudes of variation coalesce in meaningful averages; whereas conversely, constancy may drift and spread into variation. These transformations depend crucially on a number of processes, which are regimented by particular types of physical interactions. Variability may coalesce in averages, whereas constancy may drift. Mutatis mutandis, the same operating principles – large numbers and physical laws – means smear again in variation. In a functioning system, these two stances of constancy and variation complement and define each other.
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- 1.
Causes in a model are abstracted from the phenomenon in a particular level of analysis, and may not reflect the exact interactions of matter, but interesting slices of the phenomenon.
- 2.
This is not inherently the case, for instance, in information theory approaches, nor is it for symbolic computation (good old-fashioned artificial intelligence) approaches.
- 3.
One word about the role of parameters in a dynamical systems model. An often-voiced criticism has been that “given enough parameters, a model dances and sings (does anything).” Or “give me enough variables and I will create a green elephant”; the humorous comment is only appropriate on occasion, most often when directed towards models fitting curves. But in the case of dynamical systems, it is misapplied, a gross canard. For parameters in dynamical systems are very informative indeed. For one thing, they have the power of talking back to the phenomenon, contingent on the level of detail of the model. When a set of parameters that causes a certain model to behave as expected is found, it also says more about the phenomenon itself. Furthermore, qualitative analysis of dynamics based on parameter sets, as we will see in Sect. ??, informs about a wide breadth of dynamical behavior.
- 4.
Abduction is the process of selection of sorts (state variables, parameters, rules) of a phenomenon to construct a model.
- 5.
We know of other contributions to the muscle forces, such as bodily factors (e.g., dopamine controls muscle tonicity), whose influence may be modeled as modulatory. These factors can be modeled as parameters which may interface levels. For example, dopamine modulation of the force on a muscle may be modeled as a scaling factor of muscle-cell contraction.
- 6.
This level is not satisfactory for longer intervals, for the forces on the muscles are changing nonlinearly with a number of extra factors, such as neurotransmitters, and energy sources to the contracting cells.
- 7.
The experiments to come will draw heavily on the idea of coupled dynamics, and therefore I went to lengths to devise how to cover a natural system with coupled dynamical systems. The description was purely conceptual, but it is essentially the same kind of abstraction we employ when building models of behavior. Conclusions about the role and the forms of exchanges at interfaces is where much headway can be made in understanding the interplay between constancy and variability.
- 8.
Detail about the modeled state variables and rules can be added, of course, ad libitum. But the amount of detail introduced should be governed by knowledge gain. If our level of abstraction is that of the overall contraction force in a small time window, other variables can be excluded, such as the production and release of neurotransmitters, or neuromodulators such as noradrenaline (epinephrine). We abstract away neuromodulators, synaptic alterations (facilitation, depression), and production of neurotransmitters to better comprehend how motor commands are conveyed. The explanations of motor behavior, even after this blunt abstraction, are not impaired but strengthened, as they introduce further constraints for equivalence classes of neural behavior.
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Negrello, M. (2011). Dynamical Systems and Convergence. In: Invariants of Behavior. Springer Series in Cognitive and Neural Systems, vol 1. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8804-1_5
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DOI: https://doi.org/10.1007/978-1-4419-8804-1_5
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