Free Energy in a Circumplex Model of Emotion

TL;DR

The paper develops a principled mapping of emotion into a Circumplex space by deriving valence and arousal from active-inference free-energy terms, linking $V$ and $A$ to $U$, $EU$, and posterior entropy. Valence $V$ is defined as $V = U - EU$ with $U = \log P(o_t|C)$ and $EU = \mathbb{E}_{Q(o_t|s_{t-1},\pi)}[\log P(o_t|C)]$, while arousal $A$ is $A = H[Q(s|o)]$, and both are transformed to polar coordinates $r = \sqrt{V^2 + A^2}$ and $\theta = \tan^{-1}(A/V)$. A simulated search agent with a two-factor generative model demonstrates how priors and object presence shape emotional trajectories, showing alert during search and happiness upon finding, with end states typically characterized by low arousal and neutral valence. The results illustrate that mispecified priors can induce high arousal or negative valence and underscore the role of allostasis and model fit in emotional regulation, providing a mechanistic account of emotion dynamics in simple tasks with potential applications to AI and cognitive science.

Abstract

Previous active inference accounts of emotion translate fluctuations in free energy to a sense of emotion, mainly focusing on valence. However, in affective science, emotions are often represented as multi-dimensional. In this paper, we propose to adopt a Circumplex Model of emotion by mapping emotions into a two-dimensional spectrum of valence and arousal. We show how one can derive a valence and arousal signal from an agent's expected free energy, relating arousal to the entropy of posterior beliefs and valence to utility less expected utility. Under this formulation, we simulate artificial agents engaged in a search task. We show that the manipulation of priors and object presence results in commonsense variability in emotional states.

Figures (5)

• Figure 1: Free Energy Transformed Circumplex Model. Distance to origin is emotional "intensity" and degree on the circle maps to different emotional states. In this case, we simulated an agent that stayed in a "calm" state throughout the entire trajectory.
• Figure 2: Illustration of the graph environment and the agent's factor graph. (a) Agents are located on a connected graph of locations and need to find an object that might be present at one of the locations. (b) A factor graph represents the agent's generative model. Two latent state factors that model the agent's location and the object's location, respectively, give rise to two sensory modalities through a likelihood factor: the agent's location ($A_1$) and whether the object is visible ($A_2$). The agent's location can change conditioned on move actions ($B_1$), whereas the object is kept static in our experiments ($B_2 = I$).
• Figure 3: Simulation of Scenario 3. Grayscale shows the agent's belief about the object's location, whereas red x's plot the ground truth object location. The agent's own location is marked with a blue dot. In this case, the agent first has incorrect precise prior beliefs on the object location, then they do not see the object there, and they start searching other locations until it is found.
• Figure 4: Impact of Priors on Emotional State. In Scenario 1 (left), the agent begins alert, with uniform priors. In Scenario 3 (right), the agent begins somewhere between a calm and neutral state, but they immediately become angry upon not finding the object at the location given by their prior.
• Figure 5: Impact of object presence on Emotional State. In Scenario 4 (left), the agent begins in the alert state, with "maybe here" priors. In Scenario 5 (right), the agent also begins alert, with "definitely here" priors. In Scenario 4 the agent ultimately accepts that there is no object to find and becomes calm, whereas in Scenario 5 the agent cycles between anger and depression when searching all locations and still not finding the object.