A variational approach of some regulation problems in biology

Physiologic or morphogenetic systems rarely optimize natural explicit cost functions, i.e. based on thermodynamic or energetic criteria built from their underlying physico-chemical framework. Despite this absence, we can model these systems with simple dynamical equations (dealing with polynomial ODEs of Lienard type or rational fraction ODEs of Hill or MWC type). These models well explain the asymptotic behavior experimentally observed and offer more, i.e. an explanation of attractors in terms of singular trajectories of gradient, or hamiltonian or of mixed gradient-hamiltonian systems. This last description gives an a posteriori interpretation of the dynamics using energy cost functions based on observable variables. Examples will be studied concerning the cardio-respiratory system, the hippocampic system and the plant morphogenesis.