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.

Appendix 1: Multiple-timescale analysis of the model

1.1 Fast dynamics and fast subsystem

The full system of ODEs, written explicitly as a slow-fast system in fast-time formulation (timescale ratio parameter $\varepsilon$ multiplying the right hand side of the slow variables):

$$\begin{aligned} \begin{aligned} M'&=V_{M}\left( \frac{K_{i 1}^{2}}{K_{i 1}^{2}+D^{2}}\right) \left( \frac{K_{i 3}^{n}}{K_{i 3}^{n}+F_{M}^{n}}\right) -k_{M}\left( \frac{M}{K_{2}+M}\right) \\ D'&=V_{D}\left( \frac{K_{i 2}^{2}}{K_{i 2}^{2}+M^{2}}\right) \left( \frac{K_{i 4}^{n}}{K_{i 4}^{n}+F_{D}^{n}}\right) -k_{D}\left( \frac{D}{K_{4}+D}\right) \\ F_M'&=4\varepsilon \left( \frac{M}{K_{f 1}+M}-F_{M}\right) \\ F_D'&=\varepsilon \left( \frac{D}{K_{f 2}+D}-F_{D}\right) \end{aligned} \end{aligned}$$

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Taking the limit $\varepsilon = 0$ of the fast-time system yields the so-called fast subsystem, in which $F_{M}$ and $F_{D}$ have their dynamics frozen and are considered as parameters. The fast subsystem, that provides an approximation of the fast dynamics of the original system is:

$$\begin{aligned} \begin{aligned} M'&=V_{M}\left( \frac{K_{i 1}^{2}}{K_{i 1}^{2}+D^{2}}\right) \left( \frac{K_{i 3}^{n}}{K_{i 3}^{n}+F_{M}^{n}}\right) -k_{M}\left( \frac{M}{K_{2}+M}\right) \\ D'&=V_{D}\left( \frac{K_{i 2}^{2}}{K_{i 2}^{2}+M^{2}}\right) \left( \frac{K_{i 4}^{n}}{K_{i 4}^{n}+F_{D}^{n}}\right) -k_{D}\left( \frac{D}{K_{4}+D}\right) . \end{aligned} \end{aligned}$$

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In order to find all the equilibria of the fast subsystem, we solve the equations D’=0, M’=0. The corresponding object is called the critical manifold of the system. It is a 2D surface (equilibria of fast subsystem and phase space of the slow subsystem) that we aim to plot in the 3D phase space. By performing algebraic manipulations to the previous equations, if we take $n=1$ as in the papers by Goldbeter, we can solve for $F_{M}$:

$$\begin{aligned} F_{M}=\frac{\left( K_{2}+M\right) K_{i 1}^{2} K_{i 3} V_{M}}{M K_{M}\left( K_{i 1}^{2}+\left( \frac{K_{4}\left( \frac{K_{i 2}^{2}}{K_{i 2}^{2}+M^{2}}\right) \left( \frac{K_{i 4}}{K_{i 4}+F_{D}}\right) }{\frac{K_{D}}{V_{D}}-\left( \frac{K_{i 2}^{2}}{K_{i 2}^{2}+M^{2}}\right) \left( \frac{K_{i 4}}{K_{i 4}+F_{D}}\right) }\right) ^{2}\right) }-K_{i 3} \end{aligned}$$

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Plotting this formula for $F_{M}$ in 3D using MATLAB yields the surface in Figs. 3 and 4

1.2 Slow dynamics, slow sub-system and desingularised reduced system (DRS)

We are interested in finding the limit of the slow segments of the MMO solution, or in other words what are the dynamics on the critical manifold, as the timescale ratio parameter $\varepsilon$ tends to 0. The introduction of the so-called slow time $\tau =t\varepsilon$ allows to put the system in its slow-time parametrization.

$$\begin{aligned} \begin{aligned} \varepsilon \dot{M}&=V_{M}\left( \frac{K_{i1}^{2}}{K_{i 1}^{2}+D^{2}}\right) \left( \frac{K_{i3}^{n}}{K_{i 3}^{n}+F_{M}^{n}}\right) -k_{M}\left( \frac{M}{K_{2}+M}\right) \\ \varepsilon \dot{D}&=V_{D}\left( \frac{K_{i2}^{2}}{K_{i 2}^{2}+M^{2}}\right) \left( \frac{K_{i4}^{n}}{K_{i 4}^{n}+F_{D}^{n}}\right) -k_{D}\left( \frac{D}{K_{4}+D}\right) \\ \dot{F}_M&=4\left( \frac{M}{K_{f1}+M}-F_{M}\right) \\ \dot{F}_D&=\frac{D}{K_{f2}+D}-F_{D} \end{aligned} \end{aligned}$$

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By taking the limit $\varepsilon =0$ in the slow-time parametrization of the system, we obtain the slow subsystem that provides an approximation of the slow dynamics of the original system. The slow subsystem or reduced system (RS) is:

$$\begin{aligned} \begin{aligned} 0&=f\left( M, F_{D}, F_{M}\right) \\ \dot{F}_M&=h(M,F_M)\\ \dot{F}_D&=g(D,F_D) \end{aligned} \end{aligned}$$

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Note that the slow subsystem is a differential-algebraic system, that is, a system of two differential equations (for the original slow variables $F_{M}$ and $F_{D}$) constrained by an algebraic equation, which effectively corresponds to the critical manifold. Hence the critical manifold plays a key role in both subsystems: it is the set of equilibria of the fast subsystem, and it is the phase space of the slow subsystem. Since the algebraic constraint must be true for all time t, we can differentiate it with respect to t and we obtain:

$$\begin{aligned}&0=\frac{\partial f}{\partial M} \frac{d M}{d t}+\frac{\partial f}{\partial F_{D}} \frac{d F_{D}}{d t}+\frac{\partial f}{\partial F_{M}} \frac{d F_{m}}{d t}\;\nonumber \\&\quad \longrightarrow \quad \frac{d M}{d t}=\frac{\frac{\partial f}{\partial F_{D}} g+\frac{\partial f}{\partial F_{M}} h}{-\frac{\partial f}{\partial M}} \end{aligned}$$

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Therefore when $\frac{\partial f}{\partial M}=0$ the RS is not defined. It is worth mentioning that the condition$\frac{\partial f}{\partial M}=0$ corresponds to the fold set of the critical manifold (S-shaped surface), locally formed by two curves (only one curve with two branches that meet at a cusp point) and it can be computed by continuing the fold bifurcation points of the fast subsystem in both $F_{M}$ and $F_{D}$. This is the reason to introduce an auxiliary system called the Desingularized Reduced System (DRS) obtained by rescaling time in the RS by a factor “$-df/dM$”. If we apply the convention that the flow of the RS and of the DRS should have the same direction on the attracting sheet of the critical manifold, which corresponds to the submanifold $\{df/dM<0\}$ and opposite direction on the repelling sheet of S, which corresponds to the submanifold $\{df/dM>0\}$, then the only possible way to obtain the DRS from the RS is to introduce an auxiliary time $s=-t df/dM$ and we obtain the following Desingularized Reduced System (DRS):

$$\begin{aligned} \begin{aligned} \frac{d M}{d s}&=\frac{d f}{d F_{D}} g+\frac{d f}{d F_{M}} h=k_{1}\left( M, D, F_{D}, F_{M}\right) \\ \frac{d F_{D}}{d s}&=-\frac{d f}{d M} g=k_{2}\left( M, D, F_{D}\right) \\ \frac{d F_{M}}{d s}&=-\frac{d f}{d M} h=k_{3}\left( M, F_{D}, F_{M}\right) \end{aligned} \end{aligned}$$

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With the process of desingularization essentially, we project the points of the critical manifold (a surface) onto a plane, which removes the singularity along the fold set, and we also change the orientation of the flow along the repelling sheet. Note that the RS is a dynamical system constrained to evolve on a surface (the critical manifold), therefore it can be described with only two equations. Since we have an explicit formula for $F_{M}$ that depends only on M and $F_{D}$, we can eliminate the equation for dFm/ds. The DRS becomes:

$$\begin{aligned} \begin{aligned} \frac{d M}{d s}&=\frac{d f}{d F_{D}} g+\frac{d f}{d F_{M}} h=k_{1}\left( M, D, F_{D}, F_{M}\right) \\ \frac{d F_{D}}{d s}&=-\frac{d f}{d M} g=k_{2}\left( M, D, F_{D}\right) \end{aligned} \end{aligned}$$

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For the DRS to have an equilibrium two conditions must be satisfied: $\frac{df}{dF_D}g+\frac{df}{dF_M}h=0$ and $\frac{df}{dM}=0$ or $g=0$. The equilibria of the DRS satisfying $\frac{df}{dM}=0$ are precisely the ones that have been created by the process of desingularization allowing to pass from the RS to the DRS. Therefore, they are of interest to us since we want to understand the dynamics along the fold set of the critical manifold, where the RS is not defined. Such equilibria of the DRS are called folded equilibria for the RS. The DRS, in other words, offers a way to understand the slow flow up to the fold curve via the existence of folded equilibria, which are true equilibria of the DRS located on the fold curve. The fact that the right-hand side of the M-equation of the RS has a denominator going to 0 on the fold curve of the critical manifold makes the RS a priori undefined along this curve. However, when the term “$\frac{df}{dF_D}g+\frac{df}{dF_M}h$” also has a zero of the same order as $\frac{df}{dM}$, then $\frac{dM}{dt}$ is well defined and so is the RS. Therefore, even though the RS is undefined at most points on the fold curve, there are specific points where it is defined, and these are folded singularities. The conditions for a folded singularity $\mathbf {p}$ are the following:

• 1. $F\left( \mathbf {p}\right) =0$ (critical manifold)

• 2. $\frac{df}{dM}\left( \mathbf {p}\right) =0$ (fold curve)

• 3. $\frac{df}{dF_D}\left( \mathbf {p}\right) g\left( \mathbf {p}\right) \ +\frac{df}{dF_M}h\left( \mathbf {p}\right) =0$

It is worth noting that folded singularities are not equilibria of the RS but important points that allow a dynamical passage from the attracting onto the repelling sheet of the critical manifold. This corresponds to a canard type dynamic in the singular limit $\varepsilon =0$, and therefore the corresponding solutions of the RS are called singular canards (Desroches et al. 2012). To find the type of the folded singularity of the DRS we can compute the eigenvalues of its Jacobian matrix $\mathbf {A}$:

$$\begin{aligned} \mathbf {A}=\begin{pmatrix} \frac{d k_{1}}{d \mathrm {M}} &{} \frac{d k_{1}}{d \mathrm {F}_{D}} \\ \frac{d k_{2}}{d \mathrm {M}} &{} \frac{d k_{2}}{d \mathrm {F}_{D}} \end{pmatrix} \end{aligned}$$

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In order to find the eigenvalues for the fold point $\mathbf {p}$, we must solve $\det (\mathbf {A}-\lambda \mathbf {I})=\ 0$, with $\mathbf {A}$ evaluated at the point $\mathbf {p}$. If $s_1,\ s_2$ are the eigenvalues of the Jacobian Matrix A evaluated at the folded singularity $\mathbf {p}$, as an equilibrium of the DRS, then $\mathbf {p}$ is a folded node if $s_1<0,\ s_2<0$ and $s_1,\ s_2\ \in \mathbb {R}$.

1.3 Robustness of MMOs to noise

We add low-amplitude noise terms, of the order of $\varepsilon$, to the two fast variables:

$$\begin{aligned} \begin{aligned} dM(t)&=\textstyle \!\bigg [V_{M}\!\left( \frac{K_{i 1}^{2}}{K_{i 1}^{2}+D(t)^{2}}\right) \!\left( \frac{K_{i 3}^{n}}{K_{i 3}^{n}+F_{M}(t)^{n}}\right) -k_{M}\left( \frac{M(t)}{K_{2}+M(t)}\right) \!\bigg ]dt\\&+ {{\sigma }_{1}M(t)dW_{1}(t)} \\ dD(t)&=\textstyle \!\bigg [V_{D}\!\left( \frac{K_{i 2}^{2}}{K_{i 2}^{2}+M(t)^{2}}\right) \!\left( \frac{K_{i 4}^{n}}{K_{i 4}^{n}+F_{D}(t)^{n}}\right) -\;k_{D}\left( \frac{D(t)}{K_{4}+D(t)}\right) \!\bigg ]dt\\&+ {{\sigma }_{2}D(t)dW_{2}(t)} \\ \dot{F}_{M}(t)&= 4 \textstyle \varepsilon \left( \frac{M(t)}{K_{f 1}+M(t)}-F_{M}(t)\right) \\ \dot{F}_{D}(t)&= \textstyle \varepsilon \left( \frac{D(t)}{K_{f 2}+D(t)}-{F_{D}(t)}\right) , \end{aligned} \end{aligned}$$

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where $W_{i}(t)$ are independent standard Wiener processes (also independent of the initial conditions when the latter are stochastic), and where the structure of the drift terms (i.e. the terms in “dt”) and of the diffusion terms (i.e. the term in “$dW_{i}(t)$”) ensures that M(t) and D(t) remain positive almost surely for all $t\ge 0$: indeed, for in $M(t)=0$ (resp. $D(t)=0$) the noise term is null and the drift term (i.e. the term in “dt”) is positive in the corresponding equation.

1.4 Noise induced MMOs

Removing the variable $F_D$ and adding a noise term in the other slow variable $F_M$:

$$\begin{aligned} \begin{aligned} \dot{M}(t)&=\textstyle V_{M}\!\left( \frac{K_{i 1}^{2}}{K_{i 1}^{2}+D(t)^{2}}\right) \!\left( \frac{K_{i 3}^{n}}{K_{i 3}^{n}+F_{M}(t)^{n}}\right) -k_{M}\!\left( \frac{M}{K_{2}+M}\right) \\ dD(t)&=\textstyle \bigg [V_{D}\!\left( \frac{K_{i 2}^{2}}{K_{i 2}^{2}+M(t)^{2}}\right) \!\left( \frac{K_{i 4}^{n}}{K_{i 4}^{n}+F_{D}(t)^{n}}\right) -\;k_{D}\!\left( \frac{D}{K_{4}+D}\right) \!\bigg ]dt\\&+ {{\sigma }_{2}D(t)dW_{2}(t)} \\ \dot{F}_{M}(t)&=\textstyle \!\varepsilon \left( \frac{AM}{K_{f 1}+M(t)}-F_{M}\right) , \end{aligned} \end{aligned}$$

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where a standard Wiener process with a diffusion term $\sigma _2\,D(t)$ is added to the dynamics of D, as before, when $D(t)=0$, the diffusion term is null and the drift term is nonnegative so that D(t) stays nonnegative for all $t\ge 0$, almost surely. The thought process behind the construction of the 3D stochastic model is the following: the 2D system with only of M and D variables, with $F_M$ and $F_D$ being parameters, as described by A. Goldbeter (Goldbeter 2011, 2013) exhibits bistability between equilibria (has an S-shaped curve of equilibria) which will allow for relaxation oscillations when slow variable(s) are added (e.g. $F_M$). When slow dynamics is added for $F_M$, while $F_D$ is still a parameter, the resulting 3D system displays bistability between large-amplitude relaxation cycles and equilibria, when $F_D$ is statically varied. The system has 2 Hopf bifurcations which are both subcritical and the initially unstable branches of limit cycles become stable through saddle-node bifurcations of cycles. This system has two zones of bistability (in between the Hopf points and the saddle-node of cycles point) between equilibria and limit cycles. Then, as the noise term $\sigma _2D(t)dW_2(t)$ is introduced to the system, its effect is essentially to make the system “jitter” in the zone of bistability. Because the branch of cycles without noise grows sharply (canards), this is the reason why the noisy system exhibits noise-induced MMOs; see e.g. Simpson and Kuske (2011) for an example of analysis of such noisy complex oscillations.