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Table 1 The six types of dendritic tapers that permit analytical solutions to a quasi-active cable equation. κ, L are positive constants and \(n\in\mathbb {Z}\). Modified from [27]. Note that outside of the taper range, \(F(X)\) increases

From: Exact solutions to cable equations in branching neurons with tapering dendrites

TypeF(X)β(X)Taper range
Exponentialexp(−2κX)\(1+\kappa^{2}\)\(\mathbb {R}\)
Hyperbolic sine\(\frac{\sinh^{2} \kappa(X-L)}{\sinh^{2} \kappa L}\)\(1+\kappa^{2}\)(−∞,L]
Hyperbolic cosine\(\frac{\cosh^{2} \kappa(X-L)}{\cosh^{2} \kappa L}\)\(1+\kappa^{2}\)(−∞,L]
Sinusoidal\(\frac{\cos^{2} \kappa(X-L)}{\cos^{2} \kappa L}\)\(1-\kappa ^{2}\)\([L+\frac{\pi}{\kappa}n,L+\frac{\pi}{\kappa}(n+\frac{1}{2})]\)
Trigonometriccos2κX\(1-\kappa^{2}\)\([\frac{\pi}{\kappa }n,\frac{\pi}{\kappa}(n+\frac{1}{2})]\)
Quadratic\((1-X/L)^{2}\)1(−∞,0]