An Analytic Solution of the Cable Equation Predicts Frequency Preference of a Passive Shunt-End Cylindrical Cable in Response to Extracellular Oscillating Electric Fields

Under physiological and artificial conditions, the dendrites of neurons can be exposed to electric fields. Recent experimental studies suggested that the membrane resistivity of the distal apical dendrites of cortical and hippocampal pyramidal neurons may be significantly lower than that of the proximal dendrites and the soma. To understand the behavior of dendrites in time-varying extracellular electric fields, we analytically solved cable equations for finite cylindrical cables with and without a leak conductance attached to one end by employing the Green's function method. The solution for a cable with a leak at one end for direct-current step electric fields shows a reversal in polarization at the leaky end, as has been previously shown by employing the separation of variables method and Fourier series expansion. The solution for a cable with a leak at one end for alternating-current electric fields reveals that the leaky end shows frequency preference in the response amplitude. Our results predict that a passive dendrite with low resistivity at the distal end would show frequency preference in response to sinusoidal extracellular local field potentials. The Green's function obtained in our study can be used to calculate response for any extracellular electric field.