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Shape minimization of dendritic attenuation. (English) Zbl 1133.92003

Summary: What is the optimal shape of a dendrite? Of course, optimality refers to some particular criterion. We look at the case of a dendrite sealed at one end and connected at the other end to a soma. The electrical potential in the fiber follows the classical cable equations as established by W. Rall [Theory of physiological properties of dendrites. Ann. NY Acad. Sci. 96, 1071 ff (1962)]. We are interested in the shape of the dendrite which minimizes either the attenuation in time of the potential or the attenuation in space. In both cases, we prove that the cylindrical shape is optimal.

MSC:

92C20 Neural biology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
65K10 Numerical optimization and variational techniques
49N90 Applications of optimal control and differential games
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[1] Bandle, C., Von Below, J., Reichel, W.: Parabolic problems with dynamical boundary conditions: eigenvalue expansions and blow-up. Rend. Lincei Mat. Appl. 17, 35–67 (2006) · Zbl 1223.35187
[2] Bejan, A.: Shape and Structure, from Engineering to Nature. Cambridge University Press, Cambridge (2000) · Zbl 0983.74002
[3] Cox, S.J., Lipton, R.: Extremal eigenvalue problems for two-phase conductors. Arch. Ration. Mech. Anal. 136, 101–117 (1996) · Zbl 0914.49011
[4] Cox, S.J., Raol, J.H.: Recovering the passive properties of tapered dendrites from single and dual potential recordings. Math. Biosci. 190(1), 9–37 (2004) · Zbl 1049.92007
[5] Ercolano, J., Schechter, M.: Spectral theory for operators generated by elliptic boundary problems with eigenvalue parameter in boundary conditions. Commun. Pure Appl. Math. 18, 83–105 (1965) · Zbl 0138.36201
[6] Escher, J.: Quasilinear parabolic systems with dynamical boundary conditions. Commun. Partial Differ. Equ. 18, 1309–1364 (1993) · Zbl 0816.35059
[7] Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics. Birkhäuser, Basel (2006) · Zbl 1109.35081
[8] Henrot, A., Maillot, H.: Optimization of the shape, and the location of the actuators in an internal control problem. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 84(3), 737–757 (2001) · Zbl 1177.49061
[9] Hildebrandt, S., Tromba, A.: The Parsimonious Universe, Shape and Form in the Natural World. Springer, New York (1996) · Zbl 0860.01002
[10] Krein, M.G.: On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability. Am. Math. Soc. Transl. (2) 1, 163–187 (1955) · Zbl 0066.33404
[11] Privat, Y.: PhD thesis of the University of Nancy
[12] Privat, Y.: The optimal shape of a dendrite sealed at both ends, to appear · Zbl 1188.49045
[13] Rall, W.: Theory of physiological properties of dendrites. Ann. NY Acad. Sci. 96, 1071 (1962)
[14] Rall, W., Agmon-Snir, H.: Cable theory for dendritic neurons. In: Koch, C., Segev, I. (eds.) Methods in Neuronal Modeling, 2nd edn. MIT, Cambridge (1998)
[15] Scott, A.: Neuroscience: A Mathematical Primer. Springer, New York (2002) · Zbl 1018.92003
[16] Stuart, G., Spruston, N.: Determinants of voltage attenuation in neocortical pyramidal neuron dendrites. J. Neurosci. 18(10), 3501–3510 (1998)
[17] Walter, J.: Regular eigenvalue problem with eigenvalue parameter in the boundary condition. Math. Z. 133, 301–312 (1973) · Zbl 0259.47046
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