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Dynamic and generalized Wentzell node conditions for network equations

✍ Scribed by Delio Mugnolo; Silvia Romanelli

Publisher
John Wiley and Sons
Year
2007
Tongue
English
Weight
240 KB
Volume
30
Category
Article
ISSN
0170-4214

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✦ Synopsis

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 Β© 2006 John Wiley & Sons, Ltd.

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