  We deal with the periodic boundary value problem for a second-order nonlinear
ODE which includes the case of the Nagumo type equation $v_{xx} - g v + n(x)
F(v) = 0,$ previously considered by Grindrod and Sleeman and by Chen and Bell
in the study of the model of a nerve fiber with excitable spines. In a recent
work we proved a result of nonexistence of nontrivial solutions as well as a
result of existence of two positive solutions, the different situations
depending by a threshold parameter related to the integral of the weight
function $n(x).$ Here we show that the number of positive periodic solutions
may be very large for some special choices of a (large) weight $n.$ We also
obtain the existence of subharmonic solutions of any order. The proofs are
based on the Poincar\'{e} - Bikhoff fixed point theorem.
