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 Mores 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.   Its stationary (or critical) points 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  subcomplex  that can be described  explicitly in terms of 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