Theoretical analysis of the amplification of synaptic potentials by small clusters of persistent sodium channels in dendrites

We extend on the work developed by R.R. Poznanski and J. Bell from a linearized somatic persistent sodium current source to a non-linear representation of the dendritic Na+P current source associated with a small number of persistent sodium channels. The main objective is to investigate the modulation in the amplification of excitatory postsynaptic potentials (EPSPs) in dendrites studded with persistent sodium channels. The relation between membrane potential (V) and persistent sodium current density (INaP) is approximated heuristically with a sigmoidal function and the resultant cable equation is solved analytically using a regular perturbation expansion and Green’s function techniques. The transient simulated (non-evoked) response is found as a result of current injection in the form of synaptically induced voltage change located at a distance from the recording site in a cable with a uniform distribution of ion channel densities per unit length of cable (the so-called `hot-spotsʹ) and with the conductance of each hot-spot (i.e., number of channels per hot-spot) assumed to be a constant. The results show an amplification in the observed EPSPs to be compatible with the experimentally derived estimates, and in addition a saturation in the amplification is observed indicating an optimum number of ionic channels.