Figure 1.
The mood and expectation of normal subjects become more oscillatory as the mood sensitivity fη m increases toward the critical value η v + k from below. A) Oscillations in expectation are highly damped for normal subjects (blue solid, fη m = 0.3(η v + k)) but become less damped when the mood sensitivity increases (green dotted, fη m = 0.6(η v + k), and red dash-dotted, fη m = 0.9(η v + k)). Since we start the solutions at (m, v) = (0, −1), the constant reality r(t > 0) = 0 represents a permanent increase in reality from r(t < 0) = −1. The numerical values η v = 0.37, f = 0.3, k = 0.37, and k 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 Mathematical Model section, with σ r = 2, k r = 1. In C) and D), mood and expectation are initialized at (m, v) = (0, 0).
Figure 2.
Our theory predicts that the onset of bipolar disorder occurs through a supercritical Hopf bifurcation as the mood sensitivity fη m crosses the threshold value η v + k and a limit cycle in mood m(t) is established. A) In a bipolar state, the expectation v(t) (dotted green) persistently oscillates, in contrast to the normal case (solid blue). We set the reality r(t > 0) = 0 and use (m, v) = (0, −1) as the initial condition. The bipolar state is modeled using fη m = 1.5(η v + k), whereas the normal state is computed using fη m = 0.3(η v + k). The numerical values η v = 0.37, f = 0.3, k = 0.37, and k 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 fη m increases. The amplitude of oscillations obtained from numerical simulations (green stars) compares well to amplitude estimates using Equation 8 (black dots) when fη m ≫ η v + k. The estimates from Equation 8 are normalized by the mean of calculated amplitudes to emphasize the match in the power law. D) Expectation v(t) in the bipolar state responds to changes in reality but remains oscillatory (green dashed). This behavior is distinct from the expectation of normal subjects (solid blue curve) that more closely follows the reality function. E) Under the same reality function as in D), the mood is much more oscillatory in the bipolar state (green dashed curve) than in the normal state (solid blue curve). F) The model predicts intermittent spikes in the QIDS score. Realizations of the reality function are generated as described in the Mathematical Model section, with σ r = 2, k r = 1. For D), E), and F), the initial condition is (m, v) = (0, 0).
Figure 3.
Response to jumps in reality with r(t) = −4 for t ∈ [0, 1) and r(t) = 4 for t ∈ [1, 2]. Here normal, manic, and depressive subjects are defined by asymmetric learning rates such that (fη m +/(η v + k), fη m −/(η v + k)) = (0.4, 0.4), (0.8, 0.1) and (0.1, 0.8), respectively. Numerical values for other parameters, common to all subjects, are η v = 1.85, f = 0.3, k = 1.85, and k 3 = 0.014. Initial conditions are set to (m, v) = (0, 0). A) The predicted expectations v of a normal subject (solid blue), a manic subject (red dash-dotted), and a depressive subject (green dotted) all attempt to follow reality (black dotted). In the depressive state, v(t) overshoots decreases in r(t), whereas expectations in the manic state overshoot rises in r(t). B) Mood levels m(t) exhibit significant systematic differences in the normal, manic, and depressive cases, showing how asymmetric mood sensitivity can lead to unipolar depression/mania when reality r(t) is changing. C) Prolonged periods of negative mood are reflected by longer periods of large QIDS scores in depressed subjects.
Figure 4.
Phase plane diagrams depicting possible scenarios of linear stability and instability. A) Linearized dynamics in the fm < v half-plane show stable node behavior, whereas fm > v half-plane supports spiral dynamics. The overall stability is determined by the stability property of the nodal half-plane, whether or not the trajectory crosses into an unstable spiral half-plane. In the illustrated example, the green rays show the stable eigendirections. B) Both half-planes support spiral dynamics: one stable, one unstable. The overall stability is determined by whether the trajectory starting at (m 0, v 0) increases or decreases in magnitude as it completes a cycle.
Figure 5.
Bipolar disorder can be triggered by large unidirectional changes in mood sensitivity, even when one of the mood sensitivities does not cross the stability threshold. A) Numerical computations were performed within the period t ∈ [0, 162.5] using r(t > 0) = 0. The stability is characterized by the standard deviation of mood when t ∈ [81.25, 162.5], and the stability boundary (white solid curve) is determined by the contour of mood variability of a model with critical mood sensitivities, that is, fη m + = fη m − = k + η v . Other parameter values used in the simulations are η v = 1.48, f = 0.3, k = 0.37, and k 3 = 2.8 × 10−3. The curve f(η m + + η m −) = 2(k + η v ) (red-dashed line) solves Equation 11 and matches well with the numerically computed stability boundary (white solid curve) when both half-planes support spirals (inside the green-dotted box). When both half-planes are stable (inside the gray-dot-dashed box), the solutions are stable, as expected, since eigenvalues in both half-planes have negative real parts. When one half-plane is an unstable spiral and the other is a stable node (upper left and lower right rectangles with one gray-dot-dashed and two green-dotted sides), the solutions are stable according to our analysis in Figure 4, consistent with the numerical results. Finally, when an unstable node is present (upper and right to green dotted lines), the system is unstable. We show that the coexistence of stable spiral and unstable node half-planes leads to instability. Stability of the case in which both stable and unstable node half-planes arise depends on initial conditions. B) Under constant reality, bipolar disorder triggered by mood sensitivity asymmetry in different directions induces different behavior in expectation v(t). Compared to the normal state (solid blue), higher negative mood sensitivity [depressive bipolar state, fη m − = 2(η v + k) and fη m + = 0.5(η v + k)] lowers expectations (green-dotted lines), while higher positive mood sensitivity [manic bipolar state, fη m − = 0.5(η v + k) and fη m + = 2(η v + k)] leads to higher expectations (red dash-dotted). Initial conditions are (m, v) = (0, −1). Parameter values used in this and in the following subfigures are η v = 0.37, f = 0.3, k = 0.37, and k 3 = 2.8 × 10−3. C) Under constant reality, bipolar disorder induced by asymmetry in mood sensitivities in different directions biases the mood m(t) in different directions. D) The biases in the asymmetry-induced oscillations in the expectation persist under random reality conditions, with depressive/manic bipolar states leading to statistically lower/higher expectations. The realization of reality is drawn as described with σ r = 2, k r = 1. Initial conditions: (m, v) = (0, 0). E) The mood trajectories m(t) show qualitatively similar biases as in B). F) Predicted QIDS scores of depressive and manic bipolar individuals are often higher than those in normal individuals.
Figure 6.
Possible effects of antidepressants and lithium on subjects with bipolar disorder, includ ing the mania-inducing effect of antidepressants and the sedative effects of lithium, are assessed in our model. A) Numerical calculation of the mood of a bipolar subject (solid blue curve) using fη m = 1.5(η v + k). At t = 9.2 weeks, within a depressive episode, the patient is treated with antidepressants, modeled by an elevation in mood (Goldbeter, 2011). Trajectories corresponding to dosages that instantaneously decrease the depression to 70% of its lowest value (green dotted), 30% of its lowest value (red dash-dotted), and 10% of its lowest value (black dotted) are shown. Note that higher doses lead to an earlier onset of mania. This antidepressant-induced mania is observed clinically (Altshuler et al., 1995; Goldberg & Truman, 2003). The numerical values for the simulations are η v = 0.37, f = 0.3, k = 0.37, and k 3 = 2.8 × 10−3; the initial conditions are (m, v) = (0, −1). B) The quick transition to a manic phase results in a depressive episode that occurs sooner than in untreated subjects, as indicated by an earlier peak in QIDS score for subjects treated with a high antidepressant dose. C) When the effect of antidepressants is modeled by an increased positive mood sensitivity, an earlier manic episode is observed with larger amplitude. The frequency of mood oscillation also increases as dosage increases. The positive mood sensitivities used in the simulations for low to high dosage are fη m + = 2.25(η v + k), 3(η v + k), 3.75(η v + k), respectively, while the negative mood sensitivities are the same as those used in A). D) The quick transition to mania also induces an earlier depressive episode, with larger QIDS score as the dosage increases. E) Simulated mood dynamics for mania-biased mood sensitivity asymmetry [red dotted, fη m + = 1.5(η v + k), fη m − = (η v + k)] and depression-biased mood sensitivity asymmetry [blue solid, fη m + = (η v + k), fη m − = 1.5(η v + k)]. The sedative effects of lithium are modeled via a symmetric 20% reduction in mood sensitivity and are implemented in our numerics at t = 27.1 weeks (black arrow). This treatment decreases oscillation amplitudes consistent with clinical observations (Phiel & Klein, 2001). F) The reduction in mood oscillation amplitudes yields smaller predicted QIDS scores.