How can elementary automata be connected to build a single-output automaton?

In automata theory, complex systems can be constructed by connecting elementary automata, which correspond to fundamental logical operations such as conjunction, disjunction, and negation.1

These elementary units, often modeled as neurons with binary states (excited or quiescent), can be linked to form larger networks. In these configurations, the output of one automaton may serve as the input for another, or multiple automata may receive inputs in parallel to produce a combined output.123

More formally, finite transducers can be decomposed into systems of series-parallel-connected automata. In a serial connection, the output of one transducer ($T_A$) acts as the input for the next ($T_B$). In a parallel connection, the input is fed simultaneously to multiple machines, and their outputs are combined. It has been demonstrated that any finite transducer can be represented as a system of such series-parallel-connected automata, where each component is either a simple two-state automaton or one whose structure is defined by a simple group of finite order.2