  The steady-state Poisson-Nernst-Planck (ssPNP) equations are an effective
model for the description of ionic transport in ion channels. It is observed
that an ion channel exhibits voltage-dependent switching between open and
closed states. Different conductance states of a channel imply that the ssPNP
equations probably have multiple solutions with different level of currents. We
propose numerical approaches to study multiple solutions to the ssPNP equations
with multiple ionic species. To find complete current-voltage (I-V ) and
current-concentration (I-C) curves, we reformulate the ssPNP equations into
four different boundary value problems (BVPs). Numerical continuation
approaches are developed to provide good initial guesses for iteratively
solving algebraic equations resulting from discretization. Numerical
continuations on V , I, and boundary concentrations result in S-shaped and
double S-shaped (I-V and I-C) curves for the ssPNP equations with multiple
species of ions. There are five solutions to the ssPNP equations with five
ionic species, when an applied voltage is given in certain intervals.
Remarkably, the current through ion channels responds hysteretically to varying
applied voltages and boundary concentrations, showing a memory effect. In
addition, we propose a useful computational approach to locate turning points
of an I-V curve. With obtained locations, we are able to determine critical
threshold values for hysteresis to occur and the interval for V in which the
ssPNP equations have multiple solutions. Our numerical results indicate that
the developed numerical approaches have a promising potential in studying
hysteretic conductance states of ion channels.
