Synchronization in repulsively coupled oscillators

We start by considering two identical HR-neurons coupled with each other through a bidirectional cross-coupling configuration $1\rightarrow 3$, i.e., the coupling is on the $1$st variable and added to the dynamics of the $3$rd variable (The coupling configuration is demonstrated schematically in the left panel of Fig. 1). Then the equation of motion governing the dynamics of the coupled system is given by,

where the subscripts $(1,2)$ indicate the oscillators, and $I=3.2$, $r=0.006$, $s=4$ are system parameters that yield chaotic behavior of uncoupled HR-neurons. $\epsilon$ $(>,\;\mbox{or}<0)$ is the real-valued constant that represents the strength of attraction or repulsion between the coupled neurons. Since our main objective is to find the region of synchronization when the neurons are repulsively coupled, we vary the coupling strength $\epsilon$ from negative to positive regime and investigate the locally stable synchronization state based on MSF approach. Figure 1 (right panel) shows the MSF of coupled HR-neuron system in Eq. (4) versus the normalized coupling parameter $K(=2\epsilon)$. One can notice that the synchronization is achievable for $K<-1.8$, but no synchrony emerges in the positive regime of the coupling strength. Therefore, with the considered cross-coupling configuration $1\rightarrow 3$, two HR neurons can achieve a stable synchronized state when they repel each other with sufficient strength.

Further, we carried out the basin stability measure [40, 41], which quantifies the volume of the basin leading to the synchronized state, to see if the attained synchronous solution in the negative coupling regime is only stable for modest perturbation from the synchronization manifold. So, we choose $\mathcal{Q}_{0}=10^{4}$ different initial states distributed randomly over the phase space volume of uncoupled HR neuron, to quantify the fraction of states leading to the synchronized state. If $\mathcal{Q}$ number of states finally arrive at the synchronization state, then the basin stability of the synchronous state is measured as $BS=\frac{\mathcal{Q}}{\mathcal{Q}_{0}}$. The range of BS is $[0,1]$, where $BS=0$ denotes that the synchronous state is unstable under any initial states, and $BS=1$ denotes that it is globally stable under any nonlocal perturbation. If $0<BS<1$, the value of BS is the likelihood of restoring the synchronized state from perturbation for a typical initial state. Figure 2 depicts the BS for the complete synchronous state, under the variation of normalized coupling strength $K(=2\epsilon)$. As observed, for $K<-1.91$, the value of BS is unity, which reflects that the complete synchronization solution is globally stable. Therefore, with the prescribed cross-coupling configuration, two HR neurons can achieve a globally stable synchronous state in the negative coupling regime.

Now, this specific choice of coupling profile for the HR-neuronal system can be made in a structured manner to achieve stable synchronization under repulsive coupling from the constant matrix that represents the linear part of the system, and consequently, we can construct generic coupling conditions for the appropriate choice of coupling profiles to accomplish complete synchrony in many pure repulsively coupled dynamical systems. In this context, the evolution of any dynamical system can be represented as,

where ${\mathbf{x}}\in\mathbb{R}^{d}$, ($d=3$ for our considered systems) is the state vector, $L$ is the $d\times d$ constant matrix attributing to the linear part of the system, $g:\mathbb{R}^{d}\to\mathbb{R}^{d}$ accounts for the nonlinear part of the system, and $P$ is a $d\times 1$ constant matrix. For the HR-neuronal system,

From the linear matrix $L$ of the HR-neuron model, we can observe that two nonzero elements- one positive and negative element $(L_{12}\;\mbox{and}\;L_{13})$ exist in the upper triangle of the linear matrix $L$ which are connected to the dynamics of 1st variable $\dot{x}_{1,2}$, and one positive element $(L_{31})$ exists in the lower triangle connected to the dynamics of 3rd variable $\dot{z}_{1,2}$. Moreover, the off-diagonal conjugate elements $L_{13}$ and $L_{31}$ are of opposite sign. This suggests the inclusion of bidirectional cross-coupling between two neurons defined by a coupling function involving $x_{1,2}$ variables to the dynamics of $z_{1,2}$ as described in the Eq. (4) since the positive element is connected to $x_{1,2}$ variable in $\dot{z}_{1,2}$ and the negative element is connected to $z_{1,2}$ variable in $\dot{x}_{1,2}$.

The above observation with the HR-neuronal model allows us to introduce a general condition for the selection of an appropriate coupling profile between two repulsively coupled dynamical systems from their corresponding linear matrix $L$. For a three-dimensional system $(d=3)$, there are nine linear coupling configurations which can be defined as $1\to 1$, $1\to 2$, $1\to 3$, $2\to 1$, $2\to 2$, $2\to 3$, $3\to 1$, $3\to 2$ and $3\to 3$, where the notation $i\to j$ describes that the coupling is on the $i$th state variables and added to the $j$th state variables. Now the question is which coupling configuration is appropriate to achieve a complete synchronization state when the systems are repulsively coupled? To answer this question, we introduce a generic procedure that the nonzero off-diagonal conjugate elements determine the choice of a suitable coupling profile. Our observation shows that mostly the bidirectional cross-coupling between two negatively coupled identical dynamical systems is responsible for the emergence of complete synchrony. Therefore, we suggest the following coupling criteria:

• •

  Suppose a nonzero element appears in the upper triangle of the linear matrix $L$, i.e., $L_{ij}\neq 0$ $(i<j)$, where $i,j=1,2,3$ and its corresponding conjugate element $L_{ji}$ in the lower triangle is also non-zero and of opposite sign. Then depending on the sign of $L_{ij}$ and $L_{ji}$, a bidirectional cross-coupling is introduced between the systems. If the element in the upper triangle is negative and its conjugate element in the lower triangle is positive (i.e., $L_{ij}<0$ $(i<j)$ and $L_{ji}>0$), then a bidirectional cross-coupling link $i\rightarrow j$ is essential. Opposite coupling configuration ($j\rightarrow i)$ is made when $L_{ij}>0$ $(i<j)$ and $L_{ji}<0$.

The statement above adequately validates our choice of bidirectional cross-coupling between two-coupled HR neurons, as shown in the previous section. We provide four more examples in support of our statement with paradigmatic Chen system [42], Sprott-I system [43, *sprottI], Rössler oscillators [45, *rossler] and NE3 systems [47, *ne3]. Further examples are demonstrated in Appendix A. The parameters for all these dynamical systems are taken in such a manner that yields chaotic behavior for the uncoupled systems. The linear matrices for the aforementioned four systems are now arranged below, from left to right, respectively,

For the Chen system, the element $L_{12}$ in the upper triangle of the linear matrix $L$ is positive in sign, and its conjugate element $L_{21}$ in the lower triangle is negative for the choice of system parameter. Hence a bidirectional cross-coupling link involving $y_{1,2}$ is to be added to the dynamics of first variables $x_{1,2}$. From the linear matrix $L$ of the Sprott-I system, we can observe that the element $L_{12}$ is negative in sign and $L_{21}$ is of positive sign. Therefore, according to the proposition, a bidirectional cross-coupling involving $x_{1,2}$ variables is to be added to the dynamics of $y_{1,2}$ for the achievement of synchrony. For the Rössler oscillator, the $L_{12}$ element of the linear matrix $L$ is $-1$ and the corresponding cross diagonal element $L_{21}$ is $1$, which are of opposite sign. So, a bidirectional cross-coupling link containing $x_{1,2}$ is to be appended to the evolution of the variables $y_{1,2}$. In a similar manner, from the linear matrix $L$ of the NE3 system, one can find that $L_{23}=1$ and $L_{32}=-1$, which are clearly of opposite signs. Therefore, a bidirectional link including $z_{1,2}$, added to the dynamics of 2nd variables $y_{1,2}$, yields a complete synchrony state in two repulsively coupled NE3 systems.

As a result, we can obtain coupling matrix $H$ to construct repulsively coupled Chen system, Sprott-I system, Rössler oscillator, and NE3 system that can achieve complete synchronization. The coupling matrices for all these four systems are represented below, from left to right respectively,

The elements of these coupling matrices in Eq. (8) are selected correspondingly:

1. 1.

   Chen system: The coupling matrix contains only one non-zero element, $H_{12}=1$. This corresponds to the bidirectional cross-coupling configuration $2\rightarrow 1$.

2. 2.

   Sprott-I system: The coupling matrix contains only one non-zero element, $H_{21}=1$, which accounts for the bidirectional cross-coupling configuration $1\rightarrow 2$.

3. 3.

   Rössler oscillator: The coupling matrix contains only one non-zero element, $H_{21}=1$. This corresponds to the bidirectional cross-coupling configuration $1\rightarrow 2$.

4. 4.

   NE3 system: The coupling matrix contains only one non-zero element, $H_{23}=1$, attributing to the bidirectional cross-coupling configuration $3\rightarrow 2$.

Now, in order to validate these coupling profiles for the emergence of complete synchrony, we perform the linear stability of the synchronization state based on the MSF scheme. Figure 3 depicts the MSF of two-coupled Chen system, Sprott-I system, Rössler oscillator and NE3 system respectively, by varying the normalized coupling strength $K(=2\epsilon)$ from negative to positive regime. In Fig. 3(a), for Chen system, the synchronization is achievable for $K<-19.25$ with $2\rightarrow 1$ coupling configuration, which clearly validates our prediction based on the linear matrices about the emergence of synchronization state in negative coupling regime. Similarly, for all the other three systems, synchrony is achievable in the negative coupling region for our predicted coupling profiles based on the elements of linear matrices $L$. This is certainly validated by their respective MSF plots represented in Fig. 3(b)-3(d). Furthermore, Fig. 3 suggests that the MSFs for these systems with considered coupling configurations exhibit different types of functional behavior. For instance, the MSF $\Psi(K)$ for Chen system and NE3 system is a monotone decreasing function that intercepts the abscissa once at some critical coupling $K_{c}<0$, resulting in an unbounded region of synchronization. Additionally, for the Sprott-I system and Rössler oscillator $\Psi(K)$ is not monotonic but intercepts the abscissa for two or more instances that yield bounded regions of synchronization. In case of the Sprott-I system, the MSF admits negative value in some range $(K_{2},K_{1})$, where $K_{2}<K_{1}<0$ and for Rössler oscillator $\Psi(K)$ possesses more than two finite crossing points across the abscissa. Therefore, analogous to the previous studies with purely attracting coupling scheme [27], for an appropriate choice of repulsive coupling configurations, two-coupled dynamical systems exhibit an unbounded and bounded region of stable synchronization state.

Now, we concentrate on network motifs, the fundamental units of many real-world networks, to see if the proposed condition for the choice of coupling configurations is effective in promoting complete synchrony in the negative coupling regime. In a network of $N$ oscillators, the dynamics of a node $i$ under bidirectional coupling configuration can be given as

Here, $F(\mathbf{x}_{i})$ represents the isolated node dynamics with $\mathbf{x}_{i}$ being the $d=3$-dimensional state vector, $\epsilon$ is the strength of bidirectional coupling between any two nodes. The $N\times N$ symmetric matrix $\mathscr{C}$ denotes the connection topology of the network; $\mathscr{C}_{ij}=1$ if any two nodes $i$ and $j$ are connected through a bidirectional link and zero otherwise. As usual, $H_{(d\times d)}$ corresponds to the coupling matrix that defines through which variables a pair of nodes are connected with each other. Complete synchronization in network (9) occurs when each node advances with the rest nodes in unison. The corresponding synchronization manifold is defined by $\mathcal{S}=\{\mathbf{x}_{1}(t)=\mathbf{x}_{2}(t)=\cdots=\mathbf{x}_{N}(t)=s(t)\}$ and the dynamics of the synchronization solution $(x(t),y(t),z(t))=s(t)$ is determined by the associated uncoupled oscillator.

By using the Linear matrix $L$ of each node, we appropriately choose the coupling matrix $H$ between any two nodes in the network. We now demonstrate the broad applicability of our proposed coupling condition through a series of examples on few network motifs that achieve complete synchrony in negative coupling regime.