Control of coherence resonance by self-induced stochastic resonance in a multiplex neural network

We consider a two-layer multiplex network of diffusively coupled FitzHugh-Nagumo (FHN) neurons in the excitable regime. We show that the phenomenon of coherence resonance (CR) in one layer can not only be controlled by the network topology, the intra- and interlayer time-delayed couplings, but also by another phenomenon, namely, self-induced stochastic resonance (SISR) in the other layer. Numerical computations show that when the layers are isolated, each of these noise-induced phenomena is weakened (strengthened) by a sparser (denser) ring network topology, stronger (weaker) intralayer coupling forces, and longer (shorter) intralayer time delays. However, CR shows a much higher sensitivity than SISR to changes in these control parameters. It is also shown, in contrast to SISR in a single isolated FHN neuron, that the maximum noise amplitude at which SISR occurs in the network of coupled FHN neurons is controllable, especially in the regime of strong coupling forces and long time delays. In order to use SISR in the first layer of the multiplex network to control CR in the second layer, we first choose the control parameters of the second layer in isolation such that in one case CR is poor and in another case, nonexistent. It is then shown that a pronounced SISR can not only significantly improve a poor CR, but can also induce a pronounced CR, which was nonexistent in the isolated second layer. In contrast to strong intralayer coupling forces, strong interlayer coupling forces are found to enhance CR, while long interlayer time delays, just as long intralayer time delays, deteriorate CR. Most importantly, we find that in a strong interlayer coupling regime, SISR in the first layer performs better than CR in enhancing CR in the second layer. But in a weak interlayer coupling regime, CR in the first layer performs better than SISR in enhancing CR in the second layer. Our results could find novel applications in noisy neural network dynamics and engineering.

Figure 1

Schematic diagram showing a multiplex network of neurons consisting of two layers. Each node in a layer is connected only to the replica node in the other layer. The lower layer (layer 2) in red represents a coupled ring network topology, where CR occurs. The upper layer (layer 1) in blue represents a coupled ring network (with possibly a different ring topology), where either SISR or CR can occur and be used together with the intra- and interlayer time-delayed coupling parameters $\{{\tau }_{{}_1},{\kappa }_{{}_1},{\tau }_{{}_2},{\kappa }_{{}_2},{\tau }_{{}_{12}},{\kappa }_{{}_{12}}\}$, to control CR in layer 2.

Figure 2

The ISI is color coded and corresponds to $\mathrm{ISI}+\xi$ with $0<\xi \ll 1$. (a) The gray region in the $({\kappa }_{{}_2},{\tau }_{{}_2})$ plane represents the excitable regime (consisting of a stable homogeneous fixed point) of the isolated layer 2 with the Hopf bifurcation parameter ${\beta }_{{}_2}=0.7500>{\beta }_{{}_h}\left({\varepsilon }_{{}_2}\right)=0.7446,n_{{}_2}=1$. (b) The excitable regime $({\kappa }_{{}_2},{\tau }_{{}_2})=(0.5,5.0)$ in (a) is independent of $N$ and $n_{{}_2}$. (c) The red region represents the oscillatory regime ${\beta }_{{}_2}=0.7400<{\beta }_{{}_h}\left({\varepsilon }_{{}_2}\right)$ and the gray region an excitable regime induced by amplitude death, $n_{{}_2}=1$. (d) The excitable regime $({\kappa }_{{}_2},{\tau }_{{}_2})=(0.75,7.0)$ in (c) is independent of $N$ and $n_{{}_2}$. (e) The oscillatory regime $({\kappa }_{{}_2},{\tau }_{{}_2})=(0.25,2.0)$ in (c) is independent of $N$ and $n_{{}_2}$. Initial conditions correspond to a spike for all neurons. Other parameters of the isolated layer 2: $N=25,{\varepsilon }_{{}_2}=0.01,\alpha =0.5,{\sigma }_{{}_2}=0.0$.

Figure 3

Coefficient of variation CV against noise amplitude ${\sigma }_{{}_2}$ of layer 2 in isolation. (a) ${\kappa }_2=0.1$; (b) ${\kappa }_2=1.0$; (c) ${\tau }_2=0.1$; (d) ${\tau }_2=7.0$. Increasing the strength of coupling force or the length of time delay shifts the CV curves up and to the right, thus weakening CR. In particular, in (b), for ${\kappa }_{{}_2}=1.0$ and ${\tau }_{{}_2}=7.0$, the red CV curve has a minimum of ${\mathrm{CV}}_{\min }=0.56$ at ${\sigma }_{{}_2}=4.6\times {10}^{-4}$ indicating a poor CR. In (d), for ${\kappa }_{{}_2}=1.5$ and ${\tau }_{{}_2}=7.0$, the pink CV curve lies entirely above the line $\mathrm{CV}=1.0$, indicating the absence of CR. Parameters of layer 2: $N=25,n_{{}_2}=1,{\beta }_{{}_2}=0.75,{\varepsilon }_{{}_2}=0.01,\alpha =0.5,{\kappa }_{{}_{12}}=0.0$.

Figure 4

Coefficient of variation CV against noise amplitude ${\sigma }_{{}_2}$ for different (local, nonlocal, global) ring network topologies in layer 2. (a) ${\kappa }_2=0.1,{\tau }_2=1.0$; in this weak coupling and short time delay regime, changing the topology ($n_{{}_2}$) has no effect on the high coherence of oscillations. (b) ${\kappa }_{{}_2}=1.0,{\tau }_2=7.0$; in this strong coupling and long time delay regime, changing the topology can significantly affect the coherence of the oscillations. In this regime, the sparser the network is, the less coherent the oscillations are. Parameters of layer 2: $N=25,{\beta }_{{}_2}=0.75,{\varepsilon }_{{}_2}=0.01,\alpha =0.5,{\kappa }_{{}_{12}}=0.0$.

Figure 5

Landscapes of the interaction potential $U_i(v_{{}_{1i}},w_{{}_{1i}})$ in Eq. (5) for a locally ($n_{{}_1}=1$) coupled ring network topology with the energy barriers indicated in the asymmetric cases (a) ($w_{{}_{1i}}=-0.25<0$) and (b) ($w_{{}_{1i}}=0.25>0$) and symmetric case (c) ($w_{{}_{1i}}=0$). The stronger the intralayer coupling force ${\kappa }_{{}_1}$ is, the deeper the energy barrier functions $\mathrm{\Delta }{U}_i^l\left(w_{{}_{1i}}\right)$ and $\mathrm{\Delta }{U}_i^r\left(w_{{}_{1i}}\right)$ are. In particular, when $w_{{}_{1i}}=0,\mathrm{\Delta }{U}_i^l\left(w_{{}_{1i}}\right)=\mathrm{\Delta }{U}_i^r\left(w_{{}_{1i}}\right)$, and both energy barriers achieve the deepest height at the strongest coupling force ${\kappa }_{{}_1}$. The saddle point and the left and right minima of the interaction potential are located at $v_{{}_{1i}}=v_m^{*}\left(w_{{}_{1i}}\right),v_{{}_{1i}}=v_l^{*}\left(w_{{}_{1i}}\right)$, and $v_{{}_{1i}}=v_r^{*}\left(w_{{}_{1i}}\right)$, respectively, i.e., $v_l^{*}\left(w_{{}_{1i}}\right)<v_m^{*}\left(w_{{}_{1i}}\right)<v_r^{*}\left(w_{{}_{1i}}\right)$.

Figure 6

Coefficient of variation CV against noise amplitude ${\sigma }_{{}_1}$ of layer 1 in isolation. (a) ${\kappa }_{{}_1}=0.1$, (b) ${\kappa }_{{}_1}=1.0$, (c) ${\tau }_{{}_1}=1.0$, (d) ${\tau }_1=7.0$. Like CR in Fig. 3, SISR is weakened by stronger coupling ${\kappa }_{{}_1}$ and longer delays ${\tau }_{{}_1}$. SISR is, however, much less sensitive to these parameters than CR. The flat bottom of the CV curves indicates that a wider range of noise amplitude can induce coherent oscillations during SISR, unlike CR in Fig. 3 with round-bottom CV curves. For the same time-delayed couplings, oscillations due to SISR are more coherent (lower CV curves) than those due to CR. In (b) and (d) with respectively a strong coupling force and long time delay, the maximum noise amplitude below which SISR occurs is controllable, but fixed in (a) and (c) with respectively a weak coupling force and short time delay. Parameters of layer 1: $N=25,n_{{}_1}=1,{\beta }_{{}_1}=0.75,{\varepsilon }_{{}_1}=0.0005,\alpha =0.5,{\kappa }_{{}_{12}}=0.0.$

Figure 7

Coefficient of variation CV against noise amplitude ${\sigma }_{{}_1}$ for different (local, nonlocal, global) ring network topologies. (a) ${\kappa }_{{}_1}=0.1,{\tau }_{{}_1}=1.0$; changing the form of the ring topology ($n_{{}_1}$) has no effect on the high coherence of oscillations in a weak coupling and short time delay regime. (b) ${\kappa }_{{}_1}=1.0,{\tau }_{{}_1}=7.0$; changing the form of the ring topology can significantly affect the coherence of the oscillations in a strong coupling and long time delay regime. In this regime, the sparser the network is, the less coherent the oscillations due to SISR are. Parameters of layer 1: $N=25,{\beta }_{{}_1}=0.75,{\varepsilon }_{{}_1}=0.0005,\alpha =0.5,{\kappa }_{{}_{12}}=0.0$.

Figure 8

Coefficient of variation CV against noise amplitude ${\sigma }_{{}_2}$ of layer 2 coupled to layer 1 in a multiplexed network. In (a) and (b) (black curves), when there is a weak interlayer coupling (${\kappa }_{{}_{12}}=0.10$) and a short interlayer time delay (${\tau }_{{}_{12}}=0.5$), a pronounced SISR in layer 1 cannot enhance a poor (or nonexistent) CR in layer 2. But with a strong interlayer coupling (${\kappa }_{{}_{12}}=0.75$) [green curves in (a) and (b)] and a short interlayer time delay (${\tau }_{{}_{12}}=0.5$), a pronounced SISR in layer 1 can enhance a poor or nonexistent CR in layer 1. Parameters of layer 1: ${\varepsilon }_{{}_1}=0.0005,{\kappa }_{{}_1}=0.1,{\tau }_{{}_1}=0.5,n_{{}_1}=12,{\sigma }_{{}_1}={\sigma }_{{}_2},{\beta }_{{}_1}=0.75,\alpha =0.5,N=25$. Parameters of layer 2: ${\varepsilon }_{{}_2}=0.01,{\tau }_{{}_2}=7.0,n_{{}_2}=1,{\beta }_{{}_2}=0.75,\alpha =0.5,N=25,{\kappa }_{{}_2}=1.0$ in (a), ${\kappa }_{{}_2}=1.5$ in (b).

Figure 9

Coefficient of variation CV against noise amplitude ${\sigma }_{{}_2}$ of layer 2 coupled to layer 1 in a multiplexed network. In (a) (${\kappa }_{{}_{12}}=0.2,{\tau }_{{}_{12}}=1.0$) and (b) (${\kappa }_{{}_{12}}=0.4,{\tau }_{{}_{12}}=1.0$), we have a weak (${\kappa }_{{}_{12}}<0.5$) interlayer coupling regime, where the CR-CR control scheme performs better than the SISR-CR control scheme in enhancing CR in layer 2 of the multiplex. In (c) (${\kappa }_{{}_{12}}=0.5,{\tau }_{{}_{12}}=1.0$) and (d) (${\kappa }_{{}_{12}}=1.0,{\tau }_{{}_{12}}=1.0$), we have a strong (${\kappa }_{{}_{12}}\ge 0.5$) interlayer coupling regime, where the SISR-CR control scheme takes over and performs better than the CR-CR control scheme. Parameters of layer 1 with CR: ${\varepsilon }_{{}_1}=0.01,{\kappa }_{{}_1}=0.1,{\tau }_{{}_1}=0.1,n_{{}_1}=12,{\sigma }_{{}_3}={\sigma }_{{}_2},{\beta }_{{}_1}=0.75,\alpha =0.5,N=25$. Parameters of layer 1 with SISR: ${\varepsilon }_{{}_1}=0.0005,{\kappa }_{{}_1}=0.1,{\tau }_{{}_1}=0.1,n_{{}_1}=12,{\sigma }_{{}_1}={\sigma }_{{}_2},{\beta }_{{}_1}=0.75,\alpha =0.5,N=25$. Parameters of layer 2: ${\varepsilon }_{{}_2}=0.01,{\kappa }_{{}_2}=1.5,{\tau }_{{}_2}=7.0,n_{{}_2}=1,{\beta }_{{}_2}=0.75,\alpha =0.5,N=25$.

Figure 10

Minimum coefficient of variation (${\mathrm{CV}}_{\min }$) against the interlayer coupling force ${\kappa }_{{}_{12}}$ and time delay ${\tau }_{{}_{12}}$ for layer 2 when coupled to layer 1 in a multiplexed network. In (a) and (b), we have a 3D plot and its 2D projection onto the parameter plane (${\kappa }_{{}_{12}},{\tau }_{{}_{12}}$) showing the enhancement performance of the CR-CR control scheme, which is good even at weak ($0.1<{\kappa }_{{}_{12}}<0.5$) interlayer coupling forces. In (c) and (d), we have a 3D plot and its 2D projection onto the parameter space (${\kappa }_{{}_{12}},{\tau }_{{}_{12}}$) showing the enhancement performance of the SISR-CR control scheme, which is poor at weak ($0.0\le {\kappa }_{{}_{12}}<0.5$) interlayer coupling forces. But at strong ($0.5\le {\kappa }_{{}_{12}}\le 1.0$) interlayer coupling forces, the SISR-CR control scheme has a good performance which is even better than that of the CR-CR control scheme, especially at longer interlayer time delays and weaker noise amplitudes. Parameters of layer 1 with CR: ${\varepsilon }_{{}_1}=0.01,{\kappa }_{{}_1}=0.1,{\tau }_{{}_1}=0.1,n_{{}_1}=12,{\sigma }_{{}_3}={\sigma }_{{}_2},{\beta }_{{}_1}=0.75,\alpha =0.5,N=25$. Parameters of layer 1 with SISR: ${\varepsilon }_{{}_1}=0.0005,{\kappa }_{{}_1}=0.1,{\tau }_{{}_1}=0.1,n_{{}_1}=12,{\sigma }_{{}_1}={\sigma }_{{}_2},{\beta }_{{}_1}=0.75,\alpha =0.5,N=25$. Parameters of layer 2: ${\varepsilon }_{{}_2}=0.01,{\kappa }_{{}_2}=1.5,{\tau }_{{}_2}=7.0,n_{{}_2}=1,{\beta }_{{}_2}=0.75,\alpha =0.5,N=25$.