  Adaptation to environmental change is a common property of biological
systems. Cells initially respond to external changes in the environment, but
after some time, they regain their original state. By considering an element
consisting of two variables that show such adaptation dynamics, we studied a
coupled dynamical system containing such elements to examine the diverse
dynamics in the system and classified the behaviors on the basis of the network
structure that determined the interaction among elements. For a system with two
elements, two types of behaviors, perfect adaptation and simple oscillation,
were observed. For a system with three elements, in addition to these two
types, novel types of dynamics, namely, rapid burst-type oscillation and a slow
cycle, were discovered; depending on the initial conditions, these novel types
of dynamics coexisted. These behaviors are a result of the characteristic
dynamics of each element, i.e., fast response and slow adaptation processes.
The behaviors depend on the network structure (in specific, a combination of
positive or negative feedback among elements). Cooperativity among elements due
to a positive feedback loop leads to simple oscillation, whereas frustration
involving alternating positive and negative interactions among elements leads
to the coexistence of rapid bursting oscillation and a slow cycle. These
behaviors are classified on the basis of the frustration indices defined by the
network structure. The period of the slow cycle is much longer than the
original adaptation time scale, while the burst-type oscillation is a continued
response that does not involve any adaptation. We briefly discuss the universal
applicability of our results to a network of a larger number of elements and
their possible relevance to biological systems.
