  In this paper we investigate certain systems of propositional intuitionistic
modal logic defined semantically in terms of neighborhood structures. We
discuss various restrictions imposed on those frames but our constant approach
is to discard superset axiom. Such assumption allows us to think about specific
modalities $\Delta, \nabla$ and new functor $\rightsquigarrow$ depending on the
notion of maximal neighborhood. We show how it is possible to treat our models
as bi-relational ones. We prove soundness and completeness of proposed
axiomatization by means of canonical model. Moreover, we show that finite model
property holds. Then we describe properties of bounded morphism, behavioral
equivalence, bisimulation and n-bisimulation. Finally, we discuss further
researches (like some interesting classical cases and particular topological
issues).
