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

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.

• The paper demonstrates that slow Hebb-Oja adaptation drives a 1,024-node FitzHugh-Nagumo ring through abrupt transitions among incoherence, multi-headed chimeras, synchrony, and traveling waves.
• The paper finds that asymptotic effective coupling follows an inverse power law, 0.18α⁻¹·⁰⁷³, while a sharp crossover near τσ≈15 separates mixed-sign heterogeneous weights from nearly homogeneous positive coupling.
• The paper shows that coupling statistics detect adaptive transitions more clearly than the Kuramoto order parameter, and that adaptation can either organize or destabilize synchrony depending on whether coupling strengthens or decays.

This paper by Provata, Boulougouris, and Hizanidis investigates how adaptive synaptic coupling, implemented through a Hebbian learning rule regularized by Oja's forgetting term, reshapes the synchronization landscape of FitzHugh-Nagumo (FHN) oscillator networks. The central finding is that when the coupling dynamics evolve on much slower timescales than the nodal potentials, the network does not relax smoothly to its asymptotic state; instead it transits abruptly through a sequence of synchronization regimes — traveling waves, synchronized states, and chimera states of varying multiplicity — before settling. The work extends an earlier analogous study on Leaky Integrate-and-Fire networks (2602.18198) and connects adaptive-network phenomenology with the established literature on static-coupling chimeras in FHN rings.

Model and methods

The system is a ring of $N=1024$ FHN oscillators with periodic boundary conditions, each node $j$ described by a membrane potential $u_j$ and recovery variable $v_j$. Each node couples to $R$ neighbors on either side ($R=260$ or 350) via time-dependent weights $\sigma_{jk}(t)$, with a rotational cross-coupling matrix $B$ parameterized by $\phi = \pi/2 - 0.1$, a value known from prior studies to favor chimera formation. The nodal parameters $\epsilon = 0.01$ and $j$0 place each oscillator in the oscillatory regime with strongly separated membrane/recovery timescales.

Adaptivity follows

$j$1

where the first term is Hebbian learning ("neurons that fire together wire together") and the second is Oja's forgetting term, which prevents unbounded weight growth. Two control parameters govern the adaptation: the Oja parameter $j$2 ($j$3) sets the asymptotic fixed point, while $j$4 selects the coupling timescale relative to the nodal dynamics. The effective coupling is $j$5, with $j$6 fixing excitatory or inhibitory sign.

Because all measures are intrinsically transient under adaptation, the authors track the Kuramoto order parameter $j$7, the average effective coupling $j$8, and its spatial deviation $j$9, computed over the $u_j$0 active links only. Mean phase velocities are deliberately avoided since continuous weight evolution makes long-time averages unreliable. Setting the right-hand side of the adaptation equation to zero yields the mean-field prediction $u_j$1 for $u_j$2, i.e., $u_j$3, valid when the correlation condition $u_j$4 holds.

Transitions under increasing coupling

With initial weights far below their asymptotic value ($u_j$5, so $u_j$6) and slow adaptation ($u_j$7, $u_j$8), the network exhibits a clear sequence of regimes: an initial disorganized state for $u_j$9 TU; a 6-headed chimera state between roughly $v_j$0 and $v_j$1 TU; a transition at $v_j$2 TU in which incoherent domain interiors cohere while borders remain incoherent; near-full coherence around $v_j$3 TU; and finally, beyond $v_j$4 TU, an asymptotic state of two traveling incoherent domains, with the average coupling settled at $v_j$5 as predicted.

Each regime change is registered by abrupt jumps in the average effective coupling and amplified bursts in the coupling fluctuations $v_j$6; the Kuramoto order parameter rises progressively toward unity without ever cleanly separating hybrid from coherent states. A notable observation is that transitions are legible primarily in the coupling statistics rather than in $v_j$7 alone, suggesting that adaptive-weight observables are more sensitive transition detectors than conventional synchrony measures.

Repeating the protocol from a different initial condition ($v_j$8) produces a qualitatively different route — early incoherence, then a single large-coherent/small-incoherent configuration persisting to $v_j$9 TU — yet the same structural asymptotic state. The trajectory to the fixed point is thus history-dependent, while the fixed point itself is not: the asymptotic structure is dictated by the final coupling strength alone.

Destabilization under decreasing coupling

The reverse protocol starts from high weights ($R$0, $R$1, $R$2). Full synchrony is reached almost immediately ($R$3 by $R$4 TU) and persists for about 1500 TU as the couplings decay continuously. Around $R$5 TU, coupling fluctuations produced by the adaptation rule destroy full synchrony: three short-lived incoherent regions appear and are quickly replaced by a stable 2-chimera of traveling incoherent domains that persists thereafter (with a direction reversal of travel observed near $R$6 TU without other structural change).

The implication is significant: even a fully synchronized network can be destabilized into a chimera state purely by the intrinsic fluctuations of Hebb-Oja adaptation as parameters drift. This contrasts with the increasing-coupling cases, where adaptation organizes the network progressively rather than destabilizing it.

Dependence on the adaptivity parameters

Scanning $R$7 confirms the inverse power law predicted by mean-field analysis: the fitted asymptotic coupling obeys $R$8, in close agreement with the theoretical exponent $R$9 and prefactor $R=260$0. This is one of the paper's strongest quantitative results, validating the Oja fixed-point picture even along routes traversing multiple dynamical regimes.

The timescale parameter $R=260$1 produces a sharper effect. For $R=260$2, the network reaches its limiting state within 100–200 TU, with low average coupling but large temporal and spatial fluctuations; asymptotic coupling matrices retain both positive and negative link values, yielding broad, bimodal weight distributions. An abrupt transition occurs at $R=260$3: the asymptotic coupling shoots upward, temporal variations collapse, and for $R=260$4 a plateau forms with spatially homogeneous, nearly $R=260$5-like weight distributions concentrated at positive values approaching $R=260$6. Slow coupling evolution therefore allows all links to grow coherently toward the Oja fixed point, whereas fast evolution traps part of the weight distribution in the negative (inhibitory) domain.

Interpretation in terms of neural plasticity

The authors argue that large $R=260$7 is the biologically relevant regime: synaptic plasticity operates over timescales far exceeding millisecond-to-second potential dynamics, consistent with long-term potentiation and long-term depression. Under this reading, the abrupt stepwise transitions observed here suggest that learning in finite networks proceeds discontinuously rather than smoothly — an analogy the authors state explicitly but which rests on the correspondence between model timescale separation and biological plasticity, not on direct physiological evidence. They also map the increasing-coupling scenario onto synapse reinforcement during development and the decreasing-coupling scenario onto pruning or degradation associated with aging and neurodegenerative disorders; these mappings are interpretive configurations of the same equations rather than new simulations.

Limitations and open questions

The study is entirely numerical, restricted to a single ring geometry with $R=260$8, and relies on forward Euler integration (spot-checked with fourth-order Runge-Kutta); no finite-size scaling or bifurcation analysis accompanies the reported abrupt transitions. The mean-field result $R=260$9 presupposes a correlation condition on the steady-state potential products whose satisfaction the authors say can be checked numerically, but the degree to which it holds along the transient chimera regimes is not quantified. The claim that adaptive transitions may be generic receives support only from the earlier Leaky Integrate-and-Fire study, and preliminary results cited for the classical Kuramoto model point the opposite way: Hebb-Oja adaptation there does not induce multiplicity transitions because the coupling weights only weakly control chimera multiplicity, which is governed mainly by the coupling range $\sigma_{jk}(t)$0 and the phase-lag parameter untouched by the learning rule. Whether the transitions sharpen or vanish in the infinite-size limit remains explicitly open, as does the behavior in Hindmarsh-Rose, van der Pol, and Stuart-Landau networks.

Conclusion

The paper establishes that Hebb-Oja adaptivity drives FHN ring networks through abrupt, successive coherence–incoherence transitions — including chimeras of changing multiplicity — provided coupling evolves roughly two orders of magnitude slower than the fast nodal variable. Asymptotic couplings follow the predicted $\sigma_{jk}(t)$1 law, and a sharp crossover near $\sigma_{jk}(t)$2 separates heterogeneous mixed-sign weight distributions from homogeneous positive ones. The main contribution is the demonstration that timescale separation between plasticity and activity, rather than adaptivity per se, controls whether intermediate synchronization regimes are expressed at all.