Continuous analogs to discrete dynamical systems with application to modeling biological response

A general condition is formulated which establishes conditions for deriving a system of first-order differential equations which qualitatively match the behavior of a class D of discrete dynamical systems defined on a cross product {0, 1, 2, . . ., m_i -1}ⁿ, i=1, . . ., n, where {m_i} is a set of integers. The qualitative matching is defined as a homology condition between two dynamical systems. Five definitions and four theorems establish the basis for a procedure for the construction of continuous analogs which are defined by first-order differential equations, thus embedding the discrete system into a continuous one. The proposed approach is demonstrated by applying a model of generalized immune response to cognitive habit release mechanisms involved in phobia