Echo effect in brain networks

Echo behavior has been found in many systems and takes roles in different aspects. By checking the time series of terminating processes of dynamics events, we find that echo behavior may also exist in brain networks but its mechanism remains unclear. We here study the echo effect on real brain networks by presenting a new framework, which replaces the two perturbations in previous studies by only one perturbation and is thus convenient for theoretical analysis. We interestingly find that echo effect depends on both the phase-resetting parameter and coupling strength, and there is an optimal area of echo effect in phase diagram where the optimal phase-resetting parameter and weak coupling strength are the two necessary conditions. We further reveal that echo comes from a new mechanism of perturbation selected correlation, i.e. a reverse deduced correlation, which explains a more fundamental aspect of echo. Moreover, a theoretical analysis is provided to explain the echo effect, based on the uncoupled brain network. These findings uncover the underlying mechanism why the echoes in brain networks can only happen during the terminating processes, i.e. the condition of optimal phase-resetting or the needed perturbations is provided right before the terminating process while the condition of weak coupling is provided by the terminating process.