  An appropriate level of arousal induces positive emotions, and a high arousal
potential may provoke negative emotions. To explain the effect of arousal on
emotional valence, we propose a novel mathematical framework of arousal
potential variations in the dual process of human cognition: automatic and
controlled. A suitable mathematical formulation to explain the emotions in the
dual process is still absent. Our model associates free energy with arousal
potential and its variations to explain emotional valence. Decreasing and
increasing free energy consequently induce positive and negative emotions,
respectively. We formalize a transition from the automatic to the controlled
process in the dual process as a change of Bayesian prior. Further, we model
emotional valence using free energy increase (FI) when one tries changing one's
Bayesian prior and its reduction (FR) when one succeeds in recognizing the same
stimuli with a changed prior and define three emotions: "interest,"
"confusion," and "boredom" using the variations. The results of our
mathematical analysis comparing various Gaussian model parameters reveals the
following: 1) prediction error (PR) increases FR (representing "interest") when
the first prior variance is greater than the second prior variance, 2) PR
decreases FR when the first prior variance is less than the second prior
variance, and 3) the distance between priors' means always increases FR. We
also discuss the association of the outcomes with emotions in the controlled
process. The proposed mathematical model provides a general framework for
predicting and controlling emotional valence in the dual process that varies
with viewpoint and stimuli, as well as for understanding the contradictions in
the effects of arousal on the valence.
