El Salvador: evaluación de los daños ocasionados por el huracán Mitch, 1998: sus implicaciones para el desarrollo económico y social y el medio ambiente

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Basic properties of oscillatory phase synchronization.

<p>(A) The underlying model of this study was phase synchronization of two coupled phase-oscillators which could correspond to oscillatory signals measured from separate cortical areas. Phase-locking is the amount of consistency of instantaneous phases between the two oscillators. Phase-locking is resulting from synchronization process governed by two principal factors as described by Theory of Weakly Coupled Oscillators (TWCO): The intrinsic (natural) frequency ω and the coupling strength κ. The intrinsic frequency difference (detuning Δω) between oscillators determines the phase precession. The coupling strength κ determines the interaction strength, which is a function of the phase-relation (defined by the phase response curve, PRC). (B) The detuning Δω and the coupling strength κ defined a 2-dimensional space, in which phase-locking (gray shading) occurs within certain ranges. In a noiseless system, full synchrony (phase locking of 1) occurs in a limited area of detuning and coupling strength that appears as inverted triangle (Arnold tongue). The stronger the coupling strength, the more detuning is possible while still reaching full synchrony. Full synchrony (C) occurs if the oscillators converge on a common frequency (no phase precession). The phase-relation distribution exhibits a strong peak at the attractor phase relation. (D) Complete asynchrony is only possible when the oscillators are uncoupled. The phase-precession is smooth and the phase-difference distribution uniform. (E) The state of partial synchrony is characterized by phase-locking between 0 and 1. In most regions in the Δω vs. κ space, phase-locking might be close to 0. Yet, close to the Arnold tongue the phase-locking might still be relevant. The oscillators do not converge to a common frequency, but exhibit phase precession. The phase precession does not have a smooth trajectory, but is modulated depending on the phase-relation. This leads to non-uniform phase-difference distribution with a peak at the phase-relation in which the oscillators have the smallest frequency difference. In noisy phase-oscillatory systems, the partial synchronized regime can be the most dominant regime. (F) Because the phase-precession (= instantaneous frequency difference) is not smooth and changes as a function of phase-relation, it implies phase-relation dependent frequency modulations (PrFM). (G) We also included phase-relation dependent amplitude modulations (PrAM) due to observations in many of our neural network simulations. We assume here that PrAM, in the ranges included here, did not substantially change the phase trajectories and hence TWCO is still an adequate theoretical framework.</p

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coupled_PING_elife

Matlab simulation code of two coupled pyramidal-interneuron gamma networks (PING). It is based on the izhikevich-type neuronal model. With this code the effect of detuning (shifting preferred frequency by increasing input current) and coupling on their phase dynamics are illustrated

Analytical and numerical results of Coherence estimation of phase-locking with different levels of extrinsic (measurement) noise.

<p>In (A-B) we first show two examples. In (A) oscillator X and Y had a detuning of 3Hz and did not interact. The power spectra (middle panel) show the two power peak of the two oscillators. The coherence spectrum (right panel) was flat as expected. In (B), the oscillators did interact (κ = 1), where oscillator X influenced the phase trajectory of oscillator Y. The power spectra of oscillator Y show two extra power peaks with ± the detuning. These are the so-called modulation sidebands, well described in the cross-frequency coupling literature [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0146443#pone.0146443.ref062" target="_blank">62</a>]. Notice that one of the sidebands overlap with the power peak of oscillator X. The coherence spectrum show a strong phase-locking estimate, much higher than expected. This is because the coherence estimate reflected mostly the locking between the sideband of oscillator Y and the main power peak of oscillator X, which can be completely unrelated to the actual phase-locking of oscillators X ad Y. (C) Rendering of the Arnold tongue, shown with a 1/2 cross-section at the level of a 0.75 coupling strength, for which phase locking values are plotted as a function of positive, increasing intrinsic frequency differences between oscillators X and Y (Δω). Here, we did not add PrAM to the oscillatory signal. We compared the numerically (red dot) and analytically derived (black line) Coherence with the analytically derived true phase-locking (purple line) between two oscillators as a function of frequency detuning (Δω) and different levels of SNR. We used trial-number corrected squared coh values to minimize inflation due to a finite number of trials. In the partially synchronized states associated with different Δω values in the selected coupling condition, we observed strong deviations of Coherence from the true locking. The coh²values became more inflated with higher SNR. The numerically computed coh² matched with the analytically derived coh². (D) The impact of different levels of PrAM is shown with different level of SNR. The oscillators were uncoupled and hence asynchronous (in the condition indicated by the fat dot at the bottom of the Arnold tongue) and the true locking was therefore 0. The oscillators had a phase precession of 3Hz (chosen condition is located off the midline of the Arnold tongue). We observed strong deviations from the true locking with increasing PrAM and SNR. The numerically and analytically derived values matched.</p