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Some results on spectral theory over networks, applied to nerve impulse transmission. (English) Zbl 0586.35071

PolynĂ´mes orthogonaux et applications, Proc. Laguerre Symp., Bar-le- Duc/France 1984, Lect. Notes Math. 1171, 532-541 (1985).
[For the entire collection see Zbl 0572.00007.]
The author shows how to derive explicit solutions to the so-called ”linear dendritic network model”: \[ (1)\quad V^ i_ t=V^ i_{z_ iz_ i}-V^ i;\quad g_ i(V^ i_{z_ i})(1)- (V^{i+1}_{z+1})(0)=0,\quad g_ i>0;\quad V_ i(1)=V_{i+1}(0),\quad i=1,2,...;\quad V_ 1(0)=0. \] A generalization of (1) is formulated (”multilinear model”) and a characterization of the spectrum of the Laplacian of the latter is given.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35C10 Series solutions to PDEs
35G10 Initial value problems for linear higher-order PDEs

Citations:

Zbl 0572.00007