Multiple-timescale dynamics, mixed mode oscillations and mixed affective states in a model of bipolar disorder

Abstract

Mixed affective states in bipolar disorder (BD) is a common psychiatric condition that occurs when symptoms of the two opposite poles coexist during an episode of mania or depression. A four-dimensional model by Goldbeter (Progr Biophys Mol Biol 105:119-127, 2011; Pharmacopsychiatry 46:S44-S52, 2013) rests upon the notion that manic and depressive symptoms are produced by two competing and auto-inhibited neural networks. Some of the rich dynamics that this model can produce, include complex rhythms formed by both small-amplitude (subthreshold) and large-amplitude (suprathreshold) oscillations and could correspond to mixed bipolar states. These rhythms are commonly referred to as mixed mode oscillations (MMOs) and they have already been studied in many different contexts by Bertram (Mathematical analysis of complex cellular activity, Springer, Cham, 2015), (Petrov et al. in J Chem Phys 97:6191-6198, 1992). In order to accurately explain these dynamics one has to apply a mathematical apparatus that makes full use of the timescale separation between variables. Here we apply the framework of multiple-timescale dynamics to the model of BD in order to understand the mathematical mechanisms underpinning the observed dynamics of changing mood. We show that the observed complex oscillations can be understood as MMOs due to a so-called folded-node singularity. Moreover, we explore the bifurcation structure of the system and we provide possible biological interpretations of our findings. Finally, we show the robustness of the MMOs regime to stochastic noise and we propose a minimal three-dimensional model which, with the addition of noise, exhibits similar yet purely noise-driven dynamics. The broader significance of this work is to introduce mathematical tools that could be used to analyse and potentially control future, more biologically grounded models of BD.

Keywords: Bifurcations; Bipolar disorder; Dynamical systems; Mixed affective states; Mixed mode oscillations; Multiple-timescale.

Fig. 1

A schematic representation of the model for Bipolar Disorder that uses mutual inhibition with auto-inhibition. Adapted from (Goldbeter 2011)

Fig. 2

a Time-series of M and D displaying a MMO; b Time series of $F_M$ and $F_D$ displaying a MMO. Parameter values are as in (Goldbeter , Fig.5)

Fig. 3

Mixed Mode Oscillations in the Goldbeter model, as described by the system of equation in (4) (see Appendix 1.1), with a mixed state close to the depressive state; parameter values correspond to Figure 5 in (Goldbeter 2013). Also shown are the critical manifold $S^0$, its fold curves $F^{\pm }$ and the folded-node singularity $\mathbf{fn}$. Along the MMO trajectory, slow segments are highlighted in red with single arrows, and fast segments in blue with double arrows

Fig. 4

Mixed Mode Oscillations in the Goldbeter model (4) with a mixed state close to the manic state; parameter values as in Fig. 3 except for $K_{f1}=1.293$. Also shown are the critical manifold $S^0$, its fold curves $F^{\pm }$ and the folded-node singularity $\mathbf{fn}$. Along the MMO trajectory, slow segments are highlighted by single arrows, and fast segments with double arrows

Fig. 5

a Bifurcation diagram with respect to the parameter $K_{f1}$; b Zoomed view of panel (a) (colored rectangle) highlighting 6 isolas MMOs corresponding to solutions with profile $1^1-1^6$, respectively. The label $1^s$ refers to MMO with 1 large-amplitude and s small-amplitude oscillations per period. By extension, we call $1^0$ standard BD oscillations, with only 1 frequency so no small-amplitude oscillations. c Period along all computed branches of periodic solutions. d The isola of $1^3$ MMOs shown alone so as to highlight its geometry and the fact that it is closed in parameter space. In all panels the solid lines correspond to stable equilibria, whereas the dashed lines correspond to unstable equilibria. SN: saddle-node bifurcation, H: Hopf bifurcation, Ho: homoclinic bifurcation, PD: period-doubling bifurcation

Fig. 6

Time series for variable M of stable MMO solutions for various values of $K_{f1}$, illustrating the fact that the effect of varying this parameter is to create more and more small-amplitude oscillations in between the large-amplitude oscillation displayed per period. Values of $K_{f1}$ and MMO profiles are: a $K_{f1}=0.8$ with a $1^1$ MMO; b $K_{f1}=0.79$ with a $1^2$ MMO; c $K_{f1}=0.785$ with a $1^3$ MMO; d $K_{f1}=0.782$ with a $1^4$ MMO; e $K_{f1}=0.78$ with a $1^5$ MMO; f $K_{f1}=0.779$ with a $1^6$ MMO; g $K_{f1}=0.778$ with a $1^8$ MMO; h $K_{f1}=0.777$ with a $1^{12}$ MMO

Fig. 7

Bifurcation diagram of the Desingularized Reduced System (DRS) with respect to parameter $K_{f1}$. The structure of this diagram reveals the presence of two types of equilibria of the DRS, namely folded singularities (which are not equilibria of the true slow subsystem) and true singularities (which are also equilibria of the slow subsystem). Both types of singularities meet at transcritical bifurcation points $T_i$. In the present context, such bifurcations correspond to the event where a folded node loses stability to become a folded saddle and a (true) saddle becomes a (true) node. Hence, this event marks the boundaries of the MMO regimes

Fig. 8

Chaotic attractor in system (4) obtained by direct simulation for $K_{f1}=0.78065832$

Fig. 9

a Two-parameter bifurcation diagram with respect to $V_M$ and $K_{f1}$. The black line corresponds to a family of Hopf Bifurcations (HB). The purple segment of the black line corresponds to the “isolas” that were mentioned earlier. The red line corresponds to a family of saddle-node bifurcations (SN). b The same diagram plotted in the 3D space $(K_{f1},V_M,M)$ space and superimposed on the equilibrium manifold of the system, that is, the zero set of the right-hand side of system (1)

Fig. 10

Simulation of system (4) with added Gaussian noise in the fast equations, that is, Eq. (13). Panels a1–b1 show the time series of variables D and M, respectively. Panels a2–b2 display the phase-plane projections onto the $(F_D,D)$ and onto the $(F_M,M)$ planes, respectively

Fig. 11

Noise-induced MMOs in a 3D reduced version of the Goldbeter model with added Gaussian noise in the slow variable; see Eq. (14). Panels a1–b1 show the time series of variables D and M, respectively. Panels a2–b2 display the phase-plane projections onto the $(F_D,D)$ and onto the $(F_M,M)$ planes, respectively