In this answer, I'm going to use "self" to describe the person who is empathizing and "other" as the person who is the target of that empathy (the person who's trying to be related to).
The most advance model of empathy (in terms of affective empathy and emotional contagion) that I know of is featured in "A Recipe for Empathy" by Lim et al. which is based on the idea that mirror neurons provide the body of the "other", the insula which associates the action to the emotional state and the somato-sensory cortex which represents that emotional state in the robot's own state.
She demonstrates the effectiveness of this model by first having a robot learn associations between it's own state and emotions (via motherese-like training), and then seeing if when the robot was exposed to other data, if it had the appropriate response, in terms of classification and internal-mapping. Results showed an adult-like empathy level.
In the paper, Angelica notes that one of the limitations of her model is that it does not address cognitive empathy as Hideki Kozima's model does. I will update this answer with how Hideki does this once I finish reading his paper.
Although both of these models use purely statistical models (Angelica uses Gaussian Mixture Models and Hideki uses) with only vague references to the functionality of various neural areas, they are somewhat biologically plausible, since the computations they are doing are not unfeasible for something like the Neural Engineering Fraemwork to perform.
For further information, I would recommend reading the Handbook of Research on Synthesizing Human Emotion in Intelligent Systems and the Journal on Synthetic Emotion, both which seems to contain further resources for this path of investigation.