Morphisms for Lumping Finite-Size Linear System Realizations of Componentized Neural Networks
Résumé
We present here a system morphism methodology to give insight into the lumping process of networks of linear systems. Mean field assumptions are summarized as an example of two particular/significant assumptions for lumping: Homogeneity and fixed-point preservation in the lumped network model. Behavioral homogeneity (all linear systems having the same state transition and parameters) and structural homogeneity (permutation-based connectivity in/between networks with a finite number of any linear state-based systems) are proved to be sufficient conditions for lumping networks using exact morphisms. Also we show how errors are handled by approximate morphisms and by lumped model behavior. The trajectory of models is decomposed into possibly dis/continuous or discrete segments of varying lengths making possible discrete event simulations and hybrid models (piecewise linear systems with discontinuities). At state level, homomorphisms between networks of detailed interacting systems and corresponding lumped network can be satisfied by finite averaging. Lumping networks allows reducing the number of components and states to obtain simulatable models. Such lumped networks can be connected together through their input/output interfaces, using an engineering approach componentizing and lumping the network. This opens interesting perspectives for analyzing real networks at computational level (computing units of computers, simulations running on these computing units, models of neural networks based on a finite number of recording electrodes, etc.). In particular, the transposition of our results to brain modeling and simulation is discussed.
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