Use of Eigenslope to Estimate Fourier Coefficients for Passive Cable Models of the Neuron

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Boundary conditions for the cable equation - such as voltage-clamped or sealed cable ends, branchpoints, somatic shunts, and current clamps - result in multi-exponential series representations of the voltage or current. Each term in the series expansion is characterized by a decay rate (eigenvalue) and an initial amplitude (Fourier coefficient). The eigenvalues are determined numerically and the Fourier coefficients are subsequently given by the residues at the eigenvalues of the Laplace transform of the solution. In this paper, we introduce an alternative method for estimating the Fourier coefficients which works for all types of boundary conditions and is practical even when analytic expressions for the Fourier coefficients become intractable. It is shown that terms in the analytic expressions for the Fourier coefficients result from derivatives of the equation for the eigenvalues, and that simple numerical estimates for the amplitude coefficients are easily derived by replacing analytical derivatives by numerical eigenslope. The physical quantity represented by the slope is identified as effective neuron capacitance.