Mood is modeled as a ball of unit mass rolling on a distress-potential landscape V(x), a classic bistable ("double-well") dynamical-systems model used in computational-psychiatry work on attractor states of mood:
V(x) = a·x⁴ − b·x² + c·x
F(x) = −dV/dx = −(4a·x³ − 2b·x + c)
m·x'' = F(x) − γ·x' + η(t)
- a — catastrophizing rigidity: steepness of the wells. Higher a makes extreme thoughts more "sticky" and self-reinforcing once entered.
- b — negative belief strength: how deep and separated the two wells are. At b≈0 there is one shallow neutral basin; as b grows a second, deep negative-mood basin opens up next to the positive one.
- c — negativity bias (skew): tilts the whole landscape, making the negative well lower/easier to fall into than the positive one — the classic CBT idea of biased automatic interpretation of events.
- Reappraisal: an active correction force −k·r·x (r = reappraisal fraction) pulling the state toward neutral, plus extra damping γ — modeling the effortful, evidence-weighing work of cognitive restructuring that counteracts the bias in real time rather than removing it instantly.
- η(t): random noise representing everyday stressors; the buttons apply a one-off impulse standing in for a specific triggering event.
Watch how, at high b and c with reappraisal at 0%, the ball gets trapped in the deep negative well and small noise can't escape it (a rumination spiral). Raising reappraisal flattens the effective landscape and adds damping, so the same noise settles the ball near neutral instead — the same qualitative shift CBT aims for: not eliminating stressors, but changing how the belief system responds to them.