Spreading phenomena on networks are essential for the collective dynamics of various natural and technological systems, from information spreading in gene regulatory networks to neural circuits and from epidemics to supply networks experiencing perturbations. Still, how local disturbances spread across networks is not yet quantitatively understood. Here, we analyze generic spreading dynamics in deterministic network dynamical systems close to a given operating point. Standard dynamical systems' theory does not explicitly provide measures for arrival times and amplitudes of a transient spreading signal because it focuses on invariant sets, invariant measures, and other quantities less relevant for transient behavior. We here change the perspective and introduce formal expectation values for deterministic dynamics to work out a theory explicitly quantifying when and how strongly a perturbation initiated at one unit of a network impacts any other. The theory provides explicit timing and amplitude information as a function of the relative position of initially perturbed and responding unit as well as depending on the entire network topology.

Understanding how local perturbations induce the transient dynamics of a network of coupled units is essential to control and operate such systems. Often a perturbation initiated in one unit spreads to other units whose dynamical state they transiently alter. The maximum state changes at those units and the timings of these changes constitute key characteristics of such transient response dynamics. However, even for linear dynamical systems it is not possible to analytically determine time and amplitude of the maximal response of a unit to a perturbation. Here, we propose to extract approximate peak times and amplitudes from effective expectation values used to characterize the typical time and magnitude of the response of a unit by interpreting the system's response as a probability distribution over time. We derive analytic estimators for the peak response based on these expectation value measures in linearized systems operating close to a stable fixed point. These estimators can be expressed in terms of the inverse of the system's Jacobian. We obtain identical results with different approximations for the response dynamics, indicating that these estimators become exact in the limit of weak coupling. Furthermore, the results suggest that perturbations spread ballistically in networks with diffusive coupling.

Abstract. Electrodes for intraoperative monitoring should be reliable and should not disturb the surgeon. Since any wiring could complicate the handling of other medical instruments, new developments of autonomic, wireless electrodes are preferred. This new generation of electrodes will not only stimulate the nerve, but will monitor the action potentials as well. A failure to read nerve potentials may be indicative not just of damaged nerves, but may also result from bad electrode-nerve contact. To resolve this, we have developed electrodes which are equipped with impedance measurement features, facilitating simple connectivity checks. Due to energy constraints, the system works in the time domain using a rectangular voltage wave excitation. The voltage at the electrode is sampled and transmitted via RF link to the host computer. The frequency range covered by the excitation and sampling is between 100 Hz and 50 kHz, which is sufficient not only for detecting contact failure, but also for detecting thick layers of connective tissue between the electrode and the nerve fibre. In either instance, the surgeon can be warned of a bad electrode placement.