  The connections between the $E(5)-$models (the original E(5) using an
infinite square well, $E(5)-\beta^4$, $E(5)-\beta^6$ and $E(5)-\beta^8$), based
on particular solutions of the geometrical Bohr Hamiltonian with
$\gamma$-unstable potentials, and the interacting boson model (IBM) are
explored. For that purpose, the general IBM Hamiltonian for the $U(5)-O(6)$
transition line is used and a numerical fit to the different $E(5)-$models
energies is performed, later on the obtained wavefunctions are used to
calculate B(E2) transition rates. It is shown that within the IBM one can
reproduce very well all these $E(5)-$models. The agreement is the best for
$E(5)-\beta^4$ and reduces when passing through $E(5)-\beta^6$, $E(5)-\beta^8$
and E(5), where the worst agreement is obtained (although still very good for a
restricted set of lowest lying states). The fitted IBM Hamiltonians correspond
to energy surfaces close to those expected for the critical point. A phenomenon
similar to the quasidynamical symmetry is observed.
