Anticipatory Synchronisation in Coupled Systems

• Anticipated synchronisation is a regime where a driven system replicates the future state of a driver (y(t) ≈ x(t+τ)) using delayed negative feedback.
• Studies show it arises in diverse systems including excitable neuronal circuits, chaotic oscillators, spatially extended media, and even human EEG recordings.
• Practical applications span predictive control, extreme-event suppression, and reconciling phase lags with causal dynamics in complex systems.

Anticipated synchronisation is a counter-intuitive regime of coupled dynamics in which a driven, response, or “slave” system reproduces the future state of a driver, sender, or “master” system. In its canonical form, the anticipatory manifold is $y(t)\approx x(t+\tau)$ with $\tau>0$, so the receiver leads while remaining causally driven. This differs from complete synchronisation, $y(t)=x(t)$, and lag synchronisation, $y(t)=x(t-\Delta)$. In the standard Voss construction, anticipation is realised by unidirectional coupling together with negative delayed self-feedback in the slave, but later work established closely related anticipatory regimes in excitable systems, neuronal microcircuits, chaotic electronic circuits, spatially extended media, adaptive excitable models, and systems with distributed delay (Mayol et al., 2012, Matias et al., 2011, Ciszak et al., 2014).

1. Canonical formulation and defining criteria

The classical master–slave scheme is

$$\dot{\mathbf{x}}(t)=\mathbf{F}(\mathbf{x}(t)),\qquad \dot{\mathbf{y}}(t)=\mathbf{F}(\mathbf{y}(t))+\mathcal{K}\,[\mathbf{x}(t)-\mathbf{y}(t-\tau)],$$

with unidirectional coupling from $\mathbf{x}$ to $\mathbf{y}$ and delayed negative self-feedback in the slave. For appropriate $\tau$ and $\mathcal{K}$, the anticipated manifold $\mathbf{y}(t)=\mathbf{x}(t+\tau)$ is an exact solution; the substantive question is its stability (Ciszak et al., 2014).

In excitable FitzHugh–Nagumo master–slave systems with constant forcing, the same structure yields the exact anticipated solution

$\tau>0$0

where the slave compares the master’s present state $\tau>0$1 with its own delayed state $\tau>0$2. Under non-constant forcing this manifold is no longer exact, but approximate anticipated synchronisation can still be defined at the level of pulses and spike timing (Mayol et al., 2012).

The FitzHugh–Nagumo study quantified anticipation through three pulse-based observables: a pulse-prediction accuracy $\tau>0$3, a spurious-pulse measure $\tau>0$4, and an anticipation-time error $\tau>0$5. It defined an anticipated synchronisation region in $\tau>0$6-space by

$\tau>0$7

Within that region, slave pulses reliably precede master pulses by approximately $\tau>0$8. Outside it, AS is lost when $\tau>0$9 is too small, too large, or when $y(t)=x(t)$0 is too large (Mayol et al., 2012).

A later generalisation replaced the single discrete delay by a distributed-delay kernel,

$y(t)=x(t)$1

For generic $y(t)=x(t)$2, the simple exact manifold $y(t)=x(t)$3 is not generally available, so the relevant concept becomes approximate anticipated synchronisation, typically identified through the maximum of the cross-correlation at a positive lag $y(t)=x(t)$4 (Campo et al., 3 Mar 2026).

2. Realisations in excitable, chaotic, and spatially extended systems

In the FitzHugh–Nagumo master–slave model, the master obeys

$y(t)=x(t)$5

while the slave receives both the master drive and delayed self-feedback,

$y(t)=x(t)$6

With $y(t)=x(t)$7, $y(t)=x(t)$8, $y(t)=x(t)$9, and $y(t)=x(t-\Delta)$0, the system is excitable. Anticipated synchronisation was shown to be robust under four forcing schemes: identical white noise in master and slave, noise in master only, independent noises, and impulsive forcing in the master with constant forcing in the slave. The main result was a systematic numerical mapping of AS regions under these non-identical and stochastic forcings (Mayol et al., 2012).

In spatially extended chaos, anticipated synchronisation was demonstrated for one- and two-dimensional complex Ginzburg–Landau systems,

$y(t)=x(t-\Delta)$1

with $y(t)=x(t-\Delta)$2. The anticipated manifold $y(t)=x(t-\Delta)$3 is exact, and its stability depends strongly on the dynamical regime. In defect turbulence, bichaos, and phase turbulence, the measured linear autocorrelation times were approximately $y(t)=x(t-\Delta)$4, $y(t)=x(t-\Delta)$5, and $y(t)=x(t-\Delta)$6, while the corresponding maximal anticipation times were about $y(t)=x(t-\Delta)$7, $y(t)=x(t-\Delta)$8, and $y(t)=x(t-\Delta)$9. The largest anticipation times were obtained for complex-valued coupling constants, and the maximal anticipation time scaled with the linear autocorrelation time of the system (Ciszak et al., 2014).

Anticipation has also been realised in coupled RC phase-shift Chua circuits without explicit delay or parameter mismatch. In that construction, complete, lag, and approximate anticipating synchronisation are selected by coupling the response subsystem to different nodes of the drive circuit’s RC phase-shift network. In the anticipating configuration, the response is driven by the $$\dot{\mathbf{x}}(t)=\mathbf{F}(\mathbf{x}(t)),\qquad \dot{\mathbf{y}}(t)=\mathbf{F}(\mathbf{y}(t))+\mathcal{K}\,[\mathbf{x}(t)-\mathbf{y}(t-\tau)],$$0-variable, and the similarity function attains a minimum $$\dot{\mathbf{x}}(t)=\mathbf{F}(\mathbf{x}(t)),\qquad \dot{\mathbf{y}}(t)=\mathbf{F}(\mathbf{y}(t))+\mathcal{K}\,[\mathbf{x}(t)-\mathbf{y}(t-\tau)],$$1 at $$\dot{\mathbf{x}}(t)=\mathbf{F}(\mathbf{x}(t)),\qquad \dot{\mathbf{y}}(t)=\mathbf{F}(\mathbf{y}(t))+\mathcal{K}\,[\mathbf{x}(t)-\mathbf{y}(t-\tau)],$$2, corresponding experimentally to an anticipation time of about $$\dot{\mathbf{x}}(t)=\mathbf{F}(\mathbf{x}(t)),\qquad \dot{\mathbf{y}}(t)=\mathbf{F}(\mathbf{y}(t))+\mathcal{K}\,[\mathbf{x}(t)-\mathbf{y}(t-\tau)],$$3. This result is notable because it realises approximate AS without a variable delay line (Srinivasan et al., 2012).

3. Neuronal microcircuits and phase-resetting mechanisms

A central development in neuroscience was the replacement of explicit delayed self-feedback by a biologically plausible inhibitory loop. In the Master–Slave–Interneuron motif, the master excites the slave through an AMPA synapse, the slave excites an interneuron through AMPA, and the interneuron inhibits the slave through a GABA$$\dot{\mathbf{x}}(t)=\mathbf{F}(\mathbf{x}(t)),\qquad \dot{\mathbf{y}}(t)=\mathbf{F}(\mathbf{y}(t))+\mathcal{K}\,[\mathbf{x}(t)-\mathbf{y}(t-\tau)],$$4 synapse. Each neuron is a conductance-based Hodgkin–Huxley oscillator, and the delayed negative feedback required for AS is supplied by the interneuron rather than by an artificial delayed membrane-potential term. In this setting, a smooth transition from delayed synchronisation to anticipated synchronisation typically occurs when the inhibitory synaptic conductance $$\dot{\mathbf{x}}(t)=\mathbf{F}(\mathbf{x}(t)),\qquad \dot{\mathbf{y}}(t)=\mathbf{F}(\mathbf{y}(t))+\mathcal{K}\,[\mathbf{x}(t)-\mathbf{y}(t-\tau)],$$5 is increased, and the phenomenon remains robust when synaptic conductances, decay rates, and external currents are varied within physiological range (Matias et al., 2011).

The same general mechanism was analysed more formally in the sender–receiver–interneuron motif by means of phase-resetting curves. In that work, all neurons were Hodgkin–Huxley oscillators driven by constant current $$\dot{\mathbf{x}}(t)=\mathbf{F}(\mathbf{x}(t)),\qquad \dot{\mathbf{y}}(t)=\mathbf{F}(\mathbf{y}(t))+\mathcal{K}\,[\mathbf{x}(t)-\mathbf{y}(t-\tau)],$$6 pA, with isolated period $$\dot{\mathbf{x}}(t)=\mathbf{F}(\mathbf{x}(t)),\qquad \dot{\mathbf{y}}(t)=\mathbf{F}(\mathbf{y}(t))+\mathcal{K}\,[\mathbf{x}(t)-\mathbf{y}(t-\tau)],$$7 ms. The timing variable $$\dot{\mathbf{x}}(t)=\mathbf{F}(\mathbf{x}(t)),\qquad \dot{\mathbf{y}}(t)=\mathbf{F}(\mathbf{y}(t))+\mathcal{K}\,[\mathbf{x}(t)-\mathbf{y}(t-\tau)],$$8 distinguishes delayed synchronisation from anticipated synchronisation: $$\dot{\mathbf{x}}(t)=\mathbf{F}(\mathbf{x}(t)),\qquad \dot{\mathbf{y}}(t)=\mathbf{F}(\mathbf{y}(t))+\mathcal{K}\,[\mathbf{x}(t)-\mathbf{y}(t-\tau)],$$9 implies delayed synchronisation, while $\mathbf{x}$0 implies anticipated synchronisation. The crucial result was that a two-variable receiver PRC, $\mathbf{x}$1, accounting simultaneously for excitation from the sender and inhibition from the interneuron, is essential. The common approximation

$\mathbf{x}$2

fails for intermediate to high inhibitory conductance and misses the DS$\mathbf{x}$3AS transition. In the full Hodgkin–Huxley system, increasing $\mathbf{x}$4 drives a transition from DS for $\mathbf{x}$5 nS to AS for $\mathbf{x}$6 nS, followed by phase drift for $\mathbf{x}$7 nS (Matias et al., 2017).

These neuronal studies established two points. First, anticipation in neural systems need not rely on an explicit delay line; inhibitory microcircuits can implement the effective delayed negative feedback. Second, the sign of the phase lag is controlled by synaptic kinetics and inhibitory strength rather than by causal direction alone. The 2011 microcircuit paper further noted that the DS–AS transition inverts pre- and post-synaptic spike ordering and therefore could have consequences for spike-timing-dependent plasticity (Matias et al., 2011).

4. Anticipated synchronisation in human EEG and the phase–causality dissociation

Human EEG provides direct empirical evidence that anticipated synchronisation is not restricted to models. In alpha-band activity during a GO/NO-GO task, pairs of electrodes were analysed using coherence, phase spectra, and frequency-domain Granger causality. Anticipated synchronisation was operationally defined as a unidirectional influence from electrode $\mathbf{x}$8 to electrode $\mathbf{x}$9 together with a negative phase difference at the alpha peak, so that $\mathbf{y}$0 leads $\mathbf{y}$1 in time despite being causally influenced by $\mathbf{y}$2 (Carlos et al., 2020).

The study used 19-channel EEG, 11 participants, and the 300 ms pre-stimulus interval across 400 trials, yielding 30,000 data points per electrode per subject. Out of $\mathbf{y}$3 possible pairs per subject, there were $\mathbf{y}$4 total pairs, $\mathbf{y}$5 with an alpha-band coherence peak, $\mathbf{y}$6 with unidirectional influence, and $\mathbf{y}$7 with bidirectional influence. Among the $\mathbf{y}$8 unidirectional alpha-coherent pairs, the phase-relation counts were: zero-lag $\mathbf{y}$9, delayed synchronisation $\tau$0, anticipated synchronisation $\tau$1, and anti-phase $\tau$2. The pairs therefore exhibited in-phase, anti-phase, and out-of-phase locking, with a similar distribution of positive and negative phase differences (Carlos et al., 2020).

A concrete AS example was the pair Fz $\tau$3 Fp1, where the Granger-causal influence was from Fz to Fp1 but the phase difference at $\tau$4 Hz was $\tau$5 rad, corresponding to an effective delay of about $\tau$6 ms. The receiver electrode thus led the sender in time. The same study reported delayed synchronisation examples with positive phase lag and concluded that the human brain can display both DS and AS in the alpha band during task preparation (Carlos et al., 2020).

The primary conceptual consequence is that phase lag does not in general identify causal direction. This resolves the apparent discrepancy, already noted in monkey cortical data and theoretical neuronal motifs, between information flow and time lag: a receiver can lead in phase while remaining causally driven. A common misconception—that a negative phase lag necessarily implies reversed causality—is therefore incorrect in AS regimes (Matias et al., 2017, Carlos et al., 2020).

5. Predictive control, extreme-event suppression, and adaptive anticipatory dynamics

In excitable FitzHugh–Nagumo systems, anticipated synchronisation was converted into a control strategy. The slave’s anticipatory spikes were used as early warnings to suppress unwanted pulses in the master. Direct control applies a corrective kick when the master variable crosses a threshold $\tau$7; predict–prevent control instead applies the kick to the master when the slave crosses the threshold. For all four forcing schemes studied, the predict–prevent method produced a significantly smaller fraction of remaining pulses for a given control amplitude $\tau$8, and for a given suppression target it required a smaller $\tau$9 than direct control. The method remained superior with finite reaction time, and perfect parameter matching between master and slave was not necessary; some mismatches even improved control (Mayol et al., 2012).

A closely related predictive-control architecture was applied to rare extreme desynchronisation events in coupled electronic oscillators. There, the main system consisted of two unidirectionally coupled electronic oscillators, and an auxiliary copy was stabilised in an anticipated-synchronisation regime by delayed self-feedback. Desynchronisation amplitudes were measured by

$\mathcal{K}$0

with an analogous $\mathcal{K}$1 for the auxiliary. Large “Dragon King” events in the main system were predicted by threshold crossings in $\mathcal{K}$2, and a brief corrective reset was then applied to the main system. For deterministic mismatch-induced and stochastic noise-induced extreme events alike, the anticipatory scheme required a control amplitude $\mathcal{K}$3–$\mathcal{K}$4 smaller than direct control based on $\mathcal{K}$5 to achieve comparable suppression (Zamora-Munt et al., 2013).

A different but closely related line of work realised anticipatory dynamics in a single adaptive excitable system. In an adaptive FitzHugh–Nagumo oscillator, the excitability parameter $\mathcal{K}$6 becomes a slow variable,

$\mathcal{K}$7

and under periodic stimulation the system stores timing information about the stimulus period. After the stimulus is turned off, it produces well-timed omitted-stimulus responses whose first peak latency $\mathcal{K}$8 depends linearly on the stimulus period over an intermediate range. A related phenomenon appears in a working-memory model when the adaptive mechanism is identified with synaptic facilitation $\mathcal{K}$9. This is not the canonical master–slave AS construction, but the paper explicitly presents it as closely related anticipatory dynamics implemented through slow adaptation rather than delayed self-feedback between two coupled systems (Yang et al., 2015).

Taken together, these results suggest that anticipated synchronisation is not merely a phase relation; it can serve as a predictive layer for suppressing excitable spikes, preventing large desynchronisation bursts, or generating omitted-stimulus responses after periodic entrainment (Mayol et al., 2012, Zamora-Munt et al., 2013, Yang et al., 2015).

6. Generalisations, stability theory, and terminological boundaries

One extension replaces fixed delay by variable delay with reset. In that framework, the drive–response system is coupled through sampled delayed states updated every reset interval $\mathbf{y}(t)=\mathbf{x}(t+\tau)$0, so the response receives intermittent information about the driver. The synchronisation manifold is

$\mathbf{y}(t)=\mathbf{x}(t+\tau)$1

or equivalently $\mathbf{y}(t)=\mathbf{x}(t+\tau)$2. Anticipation occurs when $\mathbf{y}(t)=\mathbf{x}(t+\tau)$3. The stability problem reduces to a discrete error dynamics, leading to explicit coupling thresholds and a finite maximum anticipation time $\mathbf{y}(t)=\mathbf{x}(t+\tau)$4 that depends on the effective Lyapunov exponent and the reset time (Ambika et al., 2008).

A more recent extension considers distributed delay. For linear damped oscillators, the driven system can exhibit exact anticipated synchronisation up to a constant amplitude factor,

$\mathbf{y}(t)=\mathbf{x}(t+\tau)$5

where $\mathbf{y}(t)=\mathbf{x}(t+\tau)$6 and $\mathbf{y}(t)=\mathbf{x}(t+\tau)$7 are determined by the delay distribution $\mathbf{y}(t)=\mathbf{x}(t+\tau)$8. For systems with two discrete delays or a uniform delay distribution, $\mathbf{y}(t)=\mathbf{x}(t+\tau)$9 is close to the mean delay, and suitably chosen distributed delays can enlarge the stability region compared with a single-delay system having the same effective anticipation time. In multi-mode linear oscillators and in nonlinear chaotic systems such as Rössler and Lorenz, exact anticipation is replaced by robust approximate AS, again with effective anticipation times close to the mean delay (Campo et al., 3 Mar 2026).

The stability theory across the literature is uneven. The complex Ginzburg–Landau work derived approximate analytical stability boundaries for the anticipated manifold by linearising around plane waves and neglecting oscillatory terms, obtaining qualitative agreement with numerical stability maps (Ciszak et al., 2014). By contrast, the FitzHugh–Nagumo control study explicitly did not derive analytical AS boundaries and instead relied on systematic numerical characterisation in $\tau>0$00-space (Mayol et al., 2012). The EEG study also emphasised standard empirical limitations of scalp data, including volume conduction, reference choice, and the assumptions of multivariate autoregressive Granger-causality analysis (Carlos et al., 2020).

The term “anticipating synchronization” is also used in a different sense in machine-learning work on synchronization transitions. There, a parameter-aware reservoir computer is trained only on asynchronous time series and then predicts whether the system will remain asynchronous or become synchronous as a control parameter drifts. This usage concerns prediction of where synchronisation will occur in parameter space, including continuous and explosive transitions with hysteresis, and not time-leading synchronisation of one dynamical system with another (Fan et al., 2021).

The literature therefore supports three clarifications. First, anticipated synchronisation is a causal dynamical regime, not a violation of causality. Second, explicit single-delay feedback is sufficient but not necessary: RC phase-shift networks, inhibitory interneuron loops, adaptive slow variables, variable-delay reset schemes, and distributed-delay kernels can all generate anticipatory behaviour, though not always in the exact Voss sense (Srinivasan et al., 2012, Matias et al., 2011, Yang et al., 2015, Ambika et al., 2008, Campo et al., 3 Mar 2026). Third, exact anticipation is exceptional; in noisy, multimode, spatially extended, or distributed-delay systems, the empirically relevant object is often approximate anticipated synchronisation, assessed by cross-correlation, spike timing, or event prediction rather than by exact equality of trajectories.