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Dynamic and generalized Wentzell node conditions for network equations
Author(s) -
Mugnolo Delio,
Romanelli Silvia
Publication year - 2007
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.805
Subject(s) - mathematics , semigroup , monotonic function , node (physics) , class (philosophy) , extension (predicate logic) , tree (set theory) , mathematical analysis , computer science , structural engineering , artificial intelligence , engineering , programming language
Abstract Motivated by a neurobiological problem, we discuss a class of diffusion problems on a network. The celebrated Rall lumped soma model for the spread of electrical potential in a dendritical tree prescribes that the common cable equation must be coupled with particular dynamic conditions in some nodes (the cell bodies, or somata ). We discuss the extension of this model to the case of a whole network of neurons, where the ramification nodes can be either active (with excitatory time‐dependent boundary conditions) or passive (where no dynamics take place, i.e. only Kirchhoff laws are imposed). While well‐posedness of the system has already been obtained in previous works, using abstract tools based on variational methods and semigroup theory we are able to prove several qualitative properties, including asymptotic behaviour, regularity of solutions, and monotonicity of the semigroups in dependence on the physical coefficients. Copyright © 2006 John Wiley & Sons, Ltd.

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