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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

Type

F(X)

β(X)

Taper range

Exponential

exp(−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

\(\frac{\cos^{2} \kappa(X-L)}{\cos^{2} \kappa L}\)

\(1-\kappa ^{2}\)

\([L+\frac{\pi}{\kappa}n,L+\frac{\pi}{\kappa}(n+\frac{1}{2})]\)

Trigonometric

cos2κX

\(1-\kappa^{2}\)

\([\frac{\pi}{\kappa }n,\frac{\pi}{\kappa}(n+\frac{1}{2})]\)

Quadratic

\((1-X/L)^{2}\)

1

(−∞,0]