In the  paper  [*Tame flows*][1] I have investigated a  special class of  gradient like flows.  

The  Conley index theory of such flows is particularly  easy to describe   and  leads  to a   result  that generalizes  the well known  theorem in Morse theory stating that "*crossing a critical point  amounts to attaching a cell of appropriate dimension*''; Sec. 9 and 10 of the above paper.

In Sec 11  I show that an injective function $f$ from the faces of a simplicial complex to the reals naturally  defines one such flow. The function $f$ extends naturally to a Lyapunov function of the flow.  The faces of  the complex are invariant subsets of the flow and stationary (or critical) points  of the flow are the barycenters of the faces. In particular such a function leads to  a homotopical reconstruction of the space  different from the one given by  the  simplicial decompositions.    The  attaching spaces  when crossing a critical point can be identified naturally with  the unstable variety of  that point  which is a  conical subcomplex that has an explicit combinatorial description in terms the function $f$.

The Morse-Foreman functions have one appealing property.    The Conley index of the barycenter of a Foreman non-critical face is  homotopically trivial, while the  Conley index of the barycenter of a Foreman  critical face  is of sphere  of dimension equal to the dimension $k$ of the critical  face. Crossing such a critical point  corresponds to attaching  a disk of dimension that be canonically identified with that face.

The precise details are in Sec 9-11 of the above paper.  I want to mention one other thing. In Sec 11 I tried with modest success to address one limitation of Morse-Foreman theory, namely the scarcity of Morse-Forman  functions.    Usual Morse functions on smooth manifolds are  "a dime a dozen" in the sense that generic    smooth functions  are Morse or better, yet   a smooth function is almost surely  Morse.   In the discrete  case, the probability  that a random assignments of numbers to faces yields a  discrete Morse function is   very small, in fact exponentially small in the number of faces. 

In Sec. 11 I  describe a  larger class of functions  on the set of faces of a simplicial complex that contains the discrete Morse-Foreman functions and homotopically behave like Morse-Forman functions. Unfortunately, even this larger class is rather exponentially thin.




  [1]: http://www3.nd.edu/~lnicolae/tameflow.pdf