Total magnetization as a function of time in the Ising model.

<p>Typical behavior of the total magnetization time series in a 2-dimensional Ising model. Three regimes are shown: a) <i>T</i> < <i>T</i>_<i>c</i>, b) <i>T</i> ≈ <i>T</i>_<i>c</i> and c) <i>T</i> > <i>T</i>_<i>c</i>. It is important to notice the change of scale between plots.</p

Temporal auto correlation at lag 1 as a function of temperature.

<p>Ensemble behavior of the autocorrelation function for lag <i>τ</i> = 1 as a function of temperature. Three regimes are shown, <i>T</i> < <i>T</i>_<i>c</i>, <i>T</i> ≈ <i>T</i>_<i>c</i> and <i>T</i> > <i>T</i>_<i>c</i>.</p

Enhancement of early warning properties in the Kuramoto model and in an atrial fibrillation model due to an external perturbation of the system

<div><p>When a complex dynamical system is externally disturbed, the statistical moments of signals associated to it can be affected in ways that depend on the nature and amplitude of the perturbation. In systems that exhibit phase transitions, the statistical moments can be used as Early Warnings (EW) of the transition. A natural question is thus to wonder what effect external disturbances have on the EWs of system. In this work we study the impact of external noise added to the system on the EWs, with particular focus on understanding the importance of the amplitude and complexity of the noise. We do this by analyzing the EWs of two computational models related to biology: the Kuramoto model, which is a paradigm of synchronization for biological systems, and a cellular automaton model of cardiac dynamics which has been used as a model for atrial fibrillation. For each model we first characterize the EWs. Then, we introduce external noise of varying intensity and nature to observe what effect this has on the EWs. In both cases we find that the introduction of noise amplified the EWs, with more complex noise having a greater effect. This both offers a way to improve the chance of detection of EWs in real systems and suggests that natural variability in the real world does not have a detrimental effect on EWs, but the opposite.</p></div

Temporal mean as a function of temperature.

<p>Ensemble behavior of the mean as a function of temperature. The mean corresponds to the total magnetization of the system. Three regimes are shown, <i>T</i> < <i>T</i>_<i>c</i>, <i>T</i> ≈ <i>T</i>_<i>c</i> and <i>T</i> > <i>T</i>_<i>c</i>. Note that the mean can also approach −1 at low temperatures; we only show here the positive values.</p

Spatial configurations in the Ising model.

<p>Typical spatial configurations for a 2-dimensional Ising model. Three regimes are shown: a) <i>T</i> < <i>T</i>_<i>c</i>, b) <i>T</i> ≈ <i>T</i>_<i>c</i> and c) <i>T</i> > <i>T</i>_<i>c</i>. Black squares represent spins with <i>σ</i> = +1 and white one correspond to <i>σ</i> = −1.</p

Spatial configurations in the Ising model.

<p>Typical spatial configurations for a 2-dimensional Ising model. Three regimes are shown: a) <i>T</i> < <i>T</i>_<i>c</i>, b) <i>T</i> ≈ <i>T</i>_<i>c</i> and c) <i>T</i> > <i>T</i>_<i>c</i>. Black squares represent spins with <i>σ</i> = +1 and white one correspond to <i>σ</i> = −1.</p

Lag-1 autocorrelation.

<p>At the CP the lag-1 autocorrelation approaches 1, indicating that the system has strong short-term memory. Moving away from the CP the system loses memory gradually but with different rates for each value of the perturbation amplitude <i>σ</i>, making of it a very good EW. For comparison purposes these plots were shifted so that the CP of each curve coincides, as explained before.</p

Temporal kurtosis as a function of temperature.

<p>Ensemble behavior of the kurtosis as a function of temperature. Three regimes are shown, <i>T</i> < <i>T</i>_<i>c</i>, <i>T</i> ≈ <i>T</i>_<i>c</i> and <i>T</i> > <i>T</i>_<i>c</i>.</p

Enhancement of early warning properties in the Kuramoto model and in an atrial fibrillation model due to an external perturbation of the system - Fig 2

<p>(a) An incoming pulse traveling from left to right encounters a block that fails to excite (shown in red). (b) The pulse continues normally in the upper row, but is blocked in the lower one. (c) The pulse reaches the vertical connection, which allows the pulse to travel back into the lower row (d), thus propagating backwards as shown in (e). Finally, when the backwards-traveling pulse reaches the first vertical connections, the process is repeated and this forms an elliptical pattern as shown in (f)–(h).</p

Lag-1 auto correlation for the atrial model.

<p>This parameter almost reaches unity at the critical point, albeit it remains nearly constant after the CP. For values of <i>ν</i> before the CP the lag-1 correlation is slightly smaller, but the effect may be too small to be significative.</p