Adaptive transitions in FitzHugh-Nagumo networks with Hebb-Oja coupling rules

TL;DR

Adaptive coupling is realized via Hebbian learning adjusted by the Oja rule to prevent the network link weights from growing without bounds and abrupt transitions in the asymptotic coupling strengths when the parameter related to adaptive coupling crosses from fast to slow time scales are reported.

Abstract

Adaptive coupling in networks of interacting neurons has gained recent attention due to the many applications both in biological and in artificial neural networks, where adaptive coupling or synaptic plasticity is considered as a key factor in learning processes. In the present study, we apply adaptive connectivity rules in networks of interacting FitzHugh-Nagumo oscillators. Adaptive coupling, here, is realized via Hebbian learning adjusted by the Oja rule to prevent the network link weights from growing without bounds. Numerical investigations demonstrate that during the adaptation process the FitzHugh-Nagumo network undergoes adaptive transitions realizing traveling waves, synchronized states and chimera states transiting through various multiplicities. These transitions become more evident when the time scales governing the coupling dynamics are much slower than the ones governing the nodal dynamics (nodal potentials). Namely, when the coupling time scales are slow, the network has the time to realize and demonstrate different synchronization regimes before reaching the final steady state. The transitions can be observed not only in the spacetime plots but also in the abrupt changes of the average coupling weights as the network evolves in time. Regarding the asymptotic coupling distributions, we show that the limiting average coupling strength follows an inverse power law with respect to the Oja parameter (also called "forgetting" parameter) which balances the learning growth. We also report abrupt transitions in the asymptotic coupling strengths when the parameter related to adaptive coupling crosses from fast to slow time scales. These findings are in line with previous studies on spiking neural networks.

Figures (6)

• Figure 1: FHN network with adaptive Hebb-Oja coupling: (a) Spacetime plot; the time evolution is on the x-axis, the node index, $j=1 \cdots N$, is on the y-axis, and the color represents the state of the membrane potential at time $t$, $u_j(t)$. The different synchronization regimes are clearly visible in the space time plot. (b) Evolution of the effective average coupling, $\sigma^{\rm eff}(t)$. (c) Evolution of the Kuramoto order parameter, $z(t)$. (d) Evolution of the coupling fluctuations, $D_{\sigma}(t)$. Parameter values are: $\epsilon=0.01$, $\gamma=0.5$, $R=260$, $\sigma_c=+0.2$, $\phi=\pi/2-0.1$ and the integration time step is $dt=0.001$. The parameters related to adaptivity are: $\alpha =1$ and $\tau_{\sigma}=1000$. The initial conditions for the membrane potential were randomly and homogeneously distributed in the range $-2.0 \le u_j(t=0) \le 2$.
• Figure 2: FHN network with adaptive Hebb-Oja coupling and different initial conditions with respect to Fig. \ref{['fig01']}: (a) Spacetime plot; the time evolution is on the x-axis, the node index, $j=1 \cdots N$, is on the y-axis, and the color represents the state of the membrane potential at time $t$, $u_j(t)$. The different synchronization regimes are clearly visible in the space time plot. (b) Evolution of the effective average coupling, $\sigma^{\rm eff}(t)$. (c) Evolution of the Kuramoto order parameter, $z(t)$. (d) Evolution of the coupling fluctuations, $D_{\sigma}(t)$. All parameter values are the same as in Fig. \ref{['fig01']} but the initial coupling weights $\sigma_{jk}(t=0)$ are different in the two figures.
• Figure 3: FHN network with adaptive Hebb-Oja coupling and different initial conditions with respect to Fig. \ref{['fig01']}: (a) Spacetime plot; the time evolution is on the x-axis, the node index, $j=1 \cdots N$, is on the y-axis, and the color represents the state of the membrane potential at time $t$, $u_j(t)$. The different synchronization regimes are clearly visible in the space time plot. (b) Evolution of the effective average coupling, $\sigma^{\rm eff}(t)$. (c) Evolution of the Kuramoto order parameter, $z(t)$. (d) Evolution of the coupling fluctuations, $D_{\sigma}(t)$. The control parameter is $\sigma_c=+0.1$, the Oja constant is $\alpha=0.7$ and the average initial coupling starts from $\sigma_{ij}^{\rm eff} (t = 0) = 0.3$. All other parameter values and conditions are the same as in Fig. \ref{['fig01']}.
• Figure 4: FHN network with adaptive Hebb-Oja coupling and different Oja parameters. (a) Typical temporal evolution of the coupling for different values of $\alpha$. In the case $\alpha=2.0$ (black curve) the effective coupling goes to the asymptotic state via abrupt transitions, while in case $\alpha=0.4$ (red curve) the network reaches rapidly a fluctuating asymptotic state. (b) The effective asymptotic coupling as a function of the Oja coupling $\alpha$; the black diamonds are simulation results and the red line is a fitted curve. All other parameter values are the same as in Fig. \ref{['fig01']}.
• Figure 5: FHN network with adaptive Hebb-Oja coupling and different time scales. (a) The time evolution of the average coupling strength for $\tau_{\sigma}=1$ (blue line), 10 (green line), 100 (red line) and 1000 (black line). (b) The asymptotic coupling $\sigma^{\rm eff}_{\rm asymptotic}$ (black line) and the spatial deviation $D_{\sigma}$ (red line) of the coupling around the mean as a function of the coupling time scale $\tau_{\sigma}$. The Oja parameter is set to 1 and all other parameter values are as in Fig. \ref{['fig01']}.