Analysis of chaotic oscillations induced in two coupled Wilson–Cowan models

Abstract

Although it is known that two coupled Wilson–Cowan models with reciprocal connections induce aperiodic oscillations, little attention has been paid to the dynamical mechanism for such oscillations so far. In this study, we aim to elucidate the fundamental mechanism to induce the aperiodic oscillations in the coupled model. First, aperiodic oscillations observed are investigated for the case when the connections are unidirectional and when the input signal is a periodic oscillation. By the phase portrait analysis, we determine that the aperiodic oscillations are caused by periodically forced state transitions between a stable equilibrium and a stable limit cycle attractors around the saddle-node and saddle separatrix loop bifurcation points. It is revealed that the dynamical mechanism where the state crosses over the saddle-node and saddle separatrix loop bifurcations significantly contributes to the occurrence of chaotic oscillations forced by a periodic input. In addition, this mechanism can also give rise to chaotic oscillations in reciprocally connected Wilson–Cowan models. These results suggest that the dynamic attractor transition underlies chaotic behaviors in two coupled Wilson–Cowan oscillators.

Appendix: Summary of bifurcation analysis

The definitions and the methods of numerical analysis of each bifurcation are summarized. All the bifurcation sets shown in Fig. 2 are plotted by numerical calculation implemented by C program following the predictor–corrector method (Kuznetsov 1998). Fixed parameter values are the same as described in Sect. 2.3.

1.1 Andronov–Hopf bifurcation

Let us consider the Wilson–Cowan oscillator represented in the following form:

$$\begin{aligned} \frac{\hbox {d}E}{\hbox {d}t}&= F(E, I, P),\\ \frac{\hbox {d}I}{\hbox {d}t}&= G(E, I, Q). \end{aligned}$$

The Jacobian matrix of this system is defined as follows:

$$\begin{aligned} J(E, I, P, Q) = \left( \begin{array}{cc} \frac{\partial F(E, I, P)}{\partial E} &{} \frac{\partial F(E, I, P)}{\partial I} \\ \frac{\partial G(E, I, Q)}{\partial E} &{} \frac{\partial G(E, I, Q)}{\partial I} \end{array} \right) . \end{aligned}$$

In the two-dimensional system, the AH bifurcation occurs when two eigenvalues of the Jacobian matrix at an equilibrium change into pure imaginary numbers. This is satisfied when $\mathrm{Tr} \, J = 0$ and $\mathrm{Det} \, J > 0$, where $\mathrm{Tr}$ is the sum of diagonal elements of the matrix and $\mathrm{Det}$ is the determinant of the matrix. Therefore, the AH bifurcation set (the red solid line in Fig. 2) is a set of points which satisfy the conditions below:

$$\begin{aligned}&F(E, I, P) = 0,\\&G(E, I, Q) = 0,\\&\mathrm{Tr} \, J(E, I, P, Q) = 0,\\&\mathrm{Det} \, J(E, I, P, Q) > 0. \end{aligned}$$

If $\mathrm{Det}\,J(E, I, P, Q) < 0$, and the other conditions are satisfied, the AH bifurcation does not occur because the two eigenvalues of $J$ are not pure imaginary numbers but real numbers. Such points are called neutral saddles, at which the sum of two real eigenvalues (saddle quantity) is equal to $0$.

1.2 Saddle-node bifurcation

In the two-dimensional system, the SN bifurcation occurs when the determinant of $J$ at an equilibrium is equal to $0$. Therefore, the SN bifurcation set (the green line in Fig. 2) is a set of points which satisfy the conditions below:

$$\begin{aligned}&F(E, I, P) = 0,\\&G(E, I, Q) = 0,\\&\mathrm{Det}\,J(E, I, P, Q) = 0. \end{aligned}$$

1.3 Saddle separatrix loop bifurcation

There is a case that the stable manifold $W^s$ and unstable manifold $W^u$ conjoin in the phase space at a certain parameter value. If these manifolds both originate from the same saddle equilibrium, such a conjoint orbit is the saddle separatrix loop. The parameter values at which the saddle separatrix loop occurs are SSL bifurcation points, where a slight change in the parameter produces a stable limit cycle if the saddle quantity is less than $0$.

Let us consider a surface ${\varSigma }$ which crosses over $W^s$ originating from the saddle equilibrium. In terms of a coordinate $\xi$ on ${\varSigma }$, the origin $\xi = 0$ is assumed to correspond to the intersection of ${\varSigma }$ and $W^s$ and $\xi ^u$ is assumed to correspond to the intersection of ${\varSigma }$ and $W^u$ originating from the same saddle equilibrium. $\xi ^u$ is a function of $E$, $I$, $P$, and $Q$:

$$\begin{aligned} {\varXi } (E, I, P, Q) \equiv \xi ^u \end{aligned}$$

is called a split function. Obviously, ${\varXi } = 0$ when the saddle separatrix loop exists. Therefore, the SSL bifurcation set (the purple line in Fig. 2) is a set of points which satisfy the conditions below:

$$\begin{aligned}&F(E, I, P) = 0,\\&G(E, I, Q) = 0,\\&{\varXi } (E, I, P, Q) = 0. \end{aligned}$$

In the numerical calculation, $W^s$ and $W^u$ are replaced by the eigenvectors of $J(E, I, P, Q)$ at the saddle equilibrium point which correspond to the negative and the positive eigenvalues, respectively. Orthogonalization of the orbit linearized near the equilibrium, and each eigenvector is the condition of the existence of a homoclinic orbit.