  Developmental dynamics of multicellular organism is a process that takes
place in a multi-stable system in which each attractor state represents a cell
type and attractor transitions correspond to cell differentiation paths. This
new understanding has revived the idea of a quasi-potential landscape, first
proposed by Waddington as a metaphor. To describe development one is interested
in the "relative stabilities" of N attractors (N>2). Existing theories of state
transition between local minima on some potential landscape deal with the exit
in the transition between a pair attractor but do not offer the notion of a
global potential function that relate more than two attractors to each other.
Several ad hoc methods have been used in systems biology to compute a landscape
in non-gradient systems, such as gene regulatory networks. Here we present an
overview of the currently available methods, discuss their limitations and
propose a new decomposition of vector fields that permit the computation of a
quasi-potential function that is equivalent to the Freidlin-Wentzell potential
but is not limited to two attractors. Several examples of decomposition are
given and the significance of such a quasi-potential function is discussed.
