  Fluctuations in biochemical networks, e.g., in a living cell, have a complex
origin that precludes a description of such systems in terms of bipartite or
multipartite processes, as is usually done in the framework of stochastic
and/or information thermodynamics. This means that fluctuations in each
subsystem are not independent: subsystems jump simultaneously if the dynamics
is modeled as a Markov jump process, or noises are correlated for diffusion
processes. In this paper, we consider information and thermodynamic exchanges
between a pair of coupled systems that do not satisfy the bipartite property.
The generalization of information-theoretic measures, such as learning rates
and transfer entropy rates, to this situation is non-trivial and also involves
introducing several additional rates. We describe how this can be achieved in
the framework of general continuous-time Markov processes, without restricting
the study to the steady-state regime. We illustrate our general formalism on
the case of diffusion processes and derive an extension of the second law of
information thermodynamics in which the difference of transfer entropy rates in
the forward and backward time directions replaces the learning rate. As a side
result, we also generalize an important relation linking information theory and
estimation theory. To further obtain analytical expressions we treat in detail
the case of Ornstein-Uhlenbeck processes, and discuss the ability of the
various information measures to detect a directional coupling in the presence
of correlated noises. Finally, we apply our formalism to the analysis of the
directional influence between cellular processes in a concrete example, which
also requires considering the case of a non-bipartite and non-Markovian
process.
