Dynamic and generalized Wentzell node conditions for network equations. (English) Zbl 1163.34039
The authors use semigroup and variational methods to study diffusion problems on networks with possible dynamical boundary conditions in the nodes. The study is motivated by neuronal dendritic network. They extend the celebrated Rall lumped soma model (which uses cable equation to describe the internal diffusion of electrical potential of a neuronal dendritical tree) to a network-shaped structure of neurons. The ramification nodes in the neuronal network can be either active (with time-dependent boundary conditions) or passive (imposing only Kirchhoff laws). The setting of the problem and general method follow the article by M. K. Fijavž, D. Mugnolo, and E. Sikolya [Appl. Math. Optimization 55, No. 2, 219–240 (2007; Zbl 1121.34035)]. Besides well-posedness the authors also consider several qualitative properties of the solutions and try to connect them to some known physical phenomena that occur in neuronal networks. Besides neurobiology, presented problem has applications in quantum mechanics or engineering. A vast list of additional references is included.
Reviewer: Marjeta Kramar Fijavž (Ljubljana)
MSC:
| 34G10 | Linear differential equations in abstract spaces |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
| 47N70 | Applications of operator theory in systems, signals, circuits, and control theory |
| 34B45 | Boundary value problems on graphs and networks for ordinary differential equations |
Keywords:
networks; cable equation on networks; Wentzell and dynamic boundary conditions; operator semigroupsCitations:
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