Spine-Density Dependence of the Qualitative Behavior of a Model of a Nerve Fiber with Excitable Spines

The qualitative behaviour of solutions to a nerve fiber model with spines, which are small protrusions on the membrane, is analyzed with particular interest in conditions imposed on the spine-density function which gives nondecay of the response to a point current stimulus. The comparison principle for the initial-boundary value problem is proved. This principle implies that if a point current stimulus lifts the dendritic membrane potential then the dendritic and spine head potentials over the whole fiber will be increased. It is also shown that if the point current stimulus rests after any finite period of time then the potentials over the whole fiber will be pulled back to the rest state asymptotically.