Competing Interests: The authors declare no conflict of interest.

Bipolar disorder is a common psychiatric dysfunction characterized by recurring episodes of mania and depression. Despite its prevalence, the causes and mechanisms of bipolar disorder remain largely unknown. Recently, theories focusing on the interaction between emotion and behavior, including those based on dysregulation of the so-called behavioral approach system (BAS), have gained popularity. Mathematical models built on this principle predict bistability in mood and do not invoke intrinsic biological rhythms that may arise from interactions between mood and expectation. Here we develop and analyze a model with clinically meaningful and modifiable parameters that incorporates the interaction between mood and expectation. Our nonlinear model exhibits a transition to limit cycle behavior when a mood-sensitivity parameter exceeds a threshold value, signaling a transition to a bipolar state. The model also predicts that asymmetry in response to positive and negative events can induce unipolar depression/mania, consistent with clinical observations. We analyze the model with asymmetric mood sensitivities and show that large unidirectional mood sensitivity can lead to bipolar disorder. Finally, we show how observed effects of lithium- and antidepressant-induced mania can be explained within the framework of our proposed model.

Bipolar disorder is characterized by cycling between manic and depressive episodes (Geller & Luby,

To understand the mechanism of bipolar disorder and accelerate the development of treatment (Geddes & Miklowitz,

Can a model exhibiting periodic mood oscillations and other observed features be derived from self-contained models that incorporate expectation and behavior? Recent psychological experiments have shown that emotion is affected by the mismatch between expectation and reality instead of the reward value (Rutledge, Skandali, Dayan, & Dolan,

In this work, we develop and analyze a variant of the models proposed by Eldar & Niv (

We propose a continuous-time model based on interactions between the dynamical variables of mood _{
m
} and _{
v
} are learning rates for mood and expectation, respectively; _{3} are linear and cubic recovery rates for mood, respectively. The perceived reality ^{−1} a mood relaxation time scale. We will see that this linear recovery term is essential for explaining the cyclothymic transition from normal to bipolar models. Finally, if mood is viewed as a physiological quantity, its magnitude should be bounded. To prevent the mood from growing indefinitely, we include a higher order nonlinear cubic term (corresponding to a quartic “potential”) in the mood equation. Thus both linear and cubic recovery terms play key roles in explaining how the bipolar disorder occurs in our model.

The reality _{
m
}(d_{
v
}
_{3}
^{3}, and a possibly nonconstant parameter _{
m
}, as we will explore later in this section. The main mechanism behind our model is that positive and negative surprises, that is, the difference between perceived reality and expectation, drive mood in corresponding directions, which in turn adjusts the perceived reality and speeds up the adaptation of expectation. In this sense, the rate of change of mood is analogous to the momentum of a damped harmonic oscillator (Eldar et al.,

Our model is actually a variant of the one proposed in Eldar et al. (_{
m
} and allows for more mathematical generality, since psychologically, the mood recovery rate may be able to vary independently from the mood learning rate. Finally, as noted earlier, we have added a cubic mood recovery term −_{3}
^{3}. This cubic suppression term and the linear decay term are essential for the system to admit limit cycle behavior that captures bipolar disorder.

The model exploits a similar central mechanism as that proposed in Eldar and Niv (_{3}
^{3}, distinguishing it from both Eldar and Niv (

Throughout this article, we will explore the effects of two forms of the reality function _{0} and a random _{
r
}. The time intervals between jumps in _{
r
} and standard deviation of the log time 1/_{
r
}. The parameters _{3}, _{
v
} are treated as positive constants throughout the article. It has also been shown that learning rates _{
m
} can be different for positive and negative events (Pulcu & Browning, _{
m
}
^{+}, _{
m
}
^{−} are positive constants. We will show in the _{
m
} (the case _{
m
}
^{+} ≠ _{
m
}
^{−}) can influence the onset of disorders. The parameters are tuned such that the time scale of mood variation matches the experimental data in Bonsall et al. (

To better connect our results with clinical observations, we calculate (QIDS-SR16) Quick Inventory of Depressive Symptomatology scores (Rush et al.,

For normal subjects, we expect that if the reality _{0} is constant, the expectation should approach _{0} and the mood will relax to zero as there are no additional stimuli; this justifies shifting _{0} → 0 without loss of generality and linearizing _{
m
}, that is, _{
m
}
^{+} = _{
m
}
^{−}, to gain insight into the basic model.

Upon linearizing _{
m
} − _{
v
} < 0 and Re(_{±}) < 0 and unstable when _{
m
} − _{
v
} > 0, corresponding to at least one eigenvalue containing a positive real part. This analysis agrees with that of Eldar et al. (_{
m
} ≫ _{
v
}. Here we focus on mood and base our study on the quantity _{
m
}, which we call the mood sensitivity parameter. In the linearly stable case, the system can support transiently oscillating behavior in mood and expectation, similar to that of a damped harmonic oscillator (Marion,

Oscillation frequencies are characterized by the imaginary part of the eigenvalues, determined by the sign of Δ. When Δ is positive, there will be no oscillation in the solutions, while negative Δ corresponds to oscillatory solutions, with oscillation frequency determined by _{
m
}, we see that Δ is a parabola with minimum at _{
m
} = _{
v
} + _{
v
}
_{
m
} increases toward the critical value _{
v
} + _{
m
} exceeds the critical value, a Hopf bifurcation occurs, the linearized dynamics become unstable, and linear analysis can no longer predict system behavior. This argument suggests that mood fluctuations even in normal (subthreshold) systems increase as the mood sensitivity increases. We verify these arguments by numerically solving _{
m
} becomes larger, as predicted by our linear analysis (

_{
m
} increases toward the critical value _{
v
} + _{
m
} = 0.3(_{
v
} + _{
m
} = 0.6(_{
v
} + _{
m
} = 0.9(_{
v
} + _{
v
} = 0.37, _{3} = 2.8 × 10^{−3} are used in all subfigures. B) The mood shows similar oscillatory behavior that becomes less damped with increasing mood sensitivity. C) When subjected to random reality events, models with large mood sensitivities exhibit larger responses in expectation. D) Similarly, the fluctuation in mood is greater in systems with larger mood sensitivity under random reality conditions. Realizations of the random reality function are generated as described in the _{
r
} = 2, _{
r
} = 1. In C) and D), mood and expectation are initialized at (

Linear stability analysis does not fully apply when the reality _{
m
}, the expectation

Once the mood sensitivity _{
m
} exceeds the threshold _{
v
} + _{
m
}
^{+} = _{
m
}
^{−} and a constant _{
m
}
^{+} = _{
m
}
^{−}, both nullclines are rotationally symmetric, allowing us to find the distance to the right edge of the boundary by setting −_{
m
} surpasses _{
v
} + _{
m
} = _{
v
} + _{
m
} still positively correlates with the mood amplitude. This prediction is verified by numerical calculations using large _{
m
} (_{
m
} ≳ _{
v
} +

_{
m
} crosses the threshold value _{
v
} + _{
m
} = 1.5(_{
v
} + _{
m
} = 0.3(_{
v
} + _{
v
} = 0.37, _{3} = 2.8 × 10^{−3} are used in all subfigures. B) Mood of bipolar subjects also persistently oscillates. C) The magnitude of mood oscillations increases as the mood sensitivity _{
m
} increases. The amplitude of oscillations obtained from numerical simulations (green stars) compares well to amplitude estimates using _{
m
} ≫ _{
v
} + _{
r
} = 2, _{
r
} = 1. For D), E), and F), the initial condition is (

While the current analysis applies only in the case of constant reality, the qualitative feature of persistent oscillations does not change even if the reality _{
m
} crosses a critical value, leading to persistent oscillations in mood and expectation qualitatively similar to those observed in mood profiles of bipolar patients.

Asymmetric response to positive and negative events and its effects on human learning have been widely reported and inferred from psychological experiments (Leppänen, _{
m
}. When the learning rate for mood ηm is asymmetric (as in _{
m
}
^{±} apply in each of the half-planes, leading to a continuous but nondifferentiable vector field. This feature complicates the linear stability analysis (see the next sub section), but it is clear that if both _{
m
}
^{+} < _{
v
} + _{
m
}
^{−} < _{
v
} + _{
m
}
^{±} such that (_{
m
}
^{+}/(_{
v
} + _{
m
}
^{−}/(_{
v
} +

_{
m
}
^{+}/(_{
v
} + _{
m
}
^{−}/(_{
v
} + _{
v
} = 1.85, _{3} = 0.014. Initial conditions are set to (

Mathematically, bipolar disorder reveals itself in the form of a limit cycle as the origin (_{
m
}
^{+} and _{
m
}
^{−}, and its delineation is more involved. Nonetheless, it is easy to show that for _{
m
}
^{+}, _{
m
}
^{−} < _{
v
} +

Consider the linearization of _{3}
^{3} term, and assume that one half-plane is stable and the other is not; for example, _{
m
}
^{+} > _{
v
} + _{
m
}
^{−} < _{
v
} + _{
v
} + _{
m
}
^{+} < _{
v
} + _{
m
}
^{+} > _{
v
} + _{
v
} + _{
m
}
^{−} > _{
v
} + _{
m
}
^{−} < _{
v
} +

_{0}, _{0}) increases or decreases in magnitude as it completes a cycle.

When spiral node behavior arises in both half-planes, the two spiral dynamics alternate along the trajectory and compete in strength. As shown in _{0}, _{0}). After a time of _{1}, _{1}). The distance to origin will change by a multiplicative factor of _{
m
}
^{+} + _{
m
}
^{−}) < 2(_{
v
} + _{
m
}
^{+} − _{
v
} − ^{2} is symmetric for _{
m
} around _{
v
} + _{
m
}
^{+} + _{
m
}
^{−}) = 2(_{
v
} +

_{
m
}
^{+} = _{
m
}
^{−} = _{
v
}. Other parameter values used in the simulations are _{
v
} = 1.48, _{3} = 2.8 × 10^{−3}. The curve _{
m
}
^{+} + _{
m
}
^{−}) = 2(_{
v
}) (red-dashed line) solves _{
m
}
^{−} = 2(_{
v
} + _{
m
}
^{+} = 0.5(_{
v
} + _{
m
}
^{−} = 0.5(_{
v
} + _{
m
}
^{+} = 2(_{
v
}+ _{
v
} = 0.37, _{3} = 2.8 × 10^{−3}. C) Under constant reality, bipolar disorder induced by asymmetry in mood sensitivities in different directions biases the mood _{
r
} = 2, _{
r
} = 1. Initial conditions: (

Bipolar disorders triggered by asymmetric mood sensitivities show oscillation in mood and expectation that are similar to those predicted in the symmetric case, but they contain systematic biases (

In this section, we explore the effects of common medications used to treat bipolar disorder. First, we want to see if our model can explain the antidepressant-induced mania seen in bipolar patients. Antidepressants are a category of medicine for treating depression disorder, and their effects on patients with depression are significant (Morris & Beck,

_{
m
} = 1.5(_{
v
} + _{
v
} = 0.37, _{3} = 2.8 × 10^{−3}; the initial conditions are (_{
m
}
^{+} = 2.25(_{
v
} + _{
v
} + _{
v
} + _{
m
}
^{+} = 1.5(_{
v
} + _{
m
}
^{−} = (_{
v
} + _{
m
}
^{+} = (_{
v
} + _{
m
}
^{−} = 1.5(_{
v
} +

The sedative effects of lithium were first discovered in 1949, but its molecular mechanisms of action have not yet been fully elucidated (Corbella & Vieta,

Existing models for bipolar disorder are based on one of two basic mechanisms: bistability and biological rhythm. Models invoking bistability assume that there are multiple stable states representing different phenotypes of depression and mania. Here variations in mood are triggered by random external perturbations arising from life events (Cochran et al., _{
m
}
^{±}, that may control a whole spectrum of states, from normal to cyclothymic personality to Type I and Type II bipolar disorders. Measuring mood sensitivity may result in a more refined method to diagnose, classify, and describe such disorders.

The perturbations from life events in biological rhythm models are usually treated as a noise term in oscillator models. We have modeled life events explicitly by a known time-dependent reality function

We also explored in detail the effects of asymmetric mood sensitivity on unipolar depression/mania and bipolar disorder. Humans are known to react differently toward positive and negative events (Pulcu & Browning,

Our work focused on the effect of mood sensitivities on unipolar depression/mania and bipolar disorder. Similar analyses can be carried out with an emphasis on, for example, the expectation learning rate _{
v
} or linear decay rate of mood

Owing to a lack of understanding of the underlying physiological mechanisms of bipolar disorder, the parameters in models, including ours, for bipolar disorder are often phenomenological and treated as fitting parameters to the experimental data. However, we have identified parameters that can be expressed in psychological terms, such as learning rate for expectation or recovery rate for mood, that can be measured by psychological experiments instead of fitting to data. For example, reaction toward events can be measured by fMRI or pupilometry (Fu et al.,

Finally, our model parameters have been assumed to be constant in time. In reality, higher order nonlinearities may arise if these physiological parameters themselves depend on mood and expectation. At the cellular level, neural synapses can be modified by the synaptic current (Fain,

Shyr-Shea Chang: Conceptualization: Equal; Formal analysis: Lead; Investigation: Lead; Visualization: Lead; Writing – original draft: Lead; Writing – review & editing: Equal. Tom Chou: Conceptualization: Equal; Funding acquisition: Lead; Supervision: Lead; Writing – review & editing: Equal.

The authors are grateful for support from the Army Research Office (W911NF-14-1-0472 and W911NF-18-1-0345) and the National Science Foundation (DMS-1516675 and DMS-1814364). SSC was supported in part by the Systems and Integrative Biology predoctoral training grant (T32GM008185).