Properties of stationary states of delay equations with large delay and applications to laser dynamics

We consider properties of periodic solutions of the differential-delay system, which models a laser with optical feedback. In particular, we describe a set of multipliers for these solutions in the limit of large delay. As a preliminary result, we obtain conditions for stability of an equilibrium of a generic differential-delay system with fixed large delay $\tau$. We also show a connection between characteristic roots of the equilibrium and multipliers of the mapping obtained via the formal limit $\tau\to\infty$

Dynamical systems with multiple, long delayed feedbacks: Multiscale analysis and spatio-temporal equivalence

Dynamical systems with multiple, hierarchically long delayed feedback are introduced and studied. Focusing on the phenomenological model of a Stuart-Landau oscillator with two feedbacks, we show the multiscale properties of its dynamics and demonstrate them by means of a space-time representation. For sufficiently long delays, we derive a normal form describing the system close to the destabilization. The space and temporal variables, which are involved in the space-time representation, correspond to suitable timescales of the original system. The physical meaning of the results, together with the interpretation of the description at different scales, is presented and discussed. In particular, it is shown how this representation uncovers hidden multiscale patterns such as spirals or spatiotemporal chaos. The effect of the delays size and the features of the transition between small to large delays is also analyzed. Finally, we comment on the application of the method and on its extension to an arbitrary, but finite, number of delayed feedback terms

Detection and storage of multivariate temporal sequences by spiking pattern reverberators

We consider networks of spiking coincidence detectors in continuous time. A single detector is a finite state machine that emits a pulsatile signal whenever the number incoming inputs exceeds a threshold within a time window of some tolerance width. Such finite state models are well-suited for hardware implementations of neural networks, as on integrated circuits (IC) or field programmable arrays (FPGAs) but they also reflect the natural capability of many neurons to act as coincidence detectors. We pay special attention to a recurrent coupling structure, where the delays are tuned to a specific pattern. Applying this pattern as an external input leads to a self-sustained reverberation of the encoded pattern if the tuning is chosen correctly. In terms of the coupling structure, the tolerance and the refractory time of the individual coincidence detectors, we determine conditions for the uniqueness of the sustained activity, i.e., for the funcionality of the network as an unambiguous pattern detector. We also present numerical experiments, where the functionality of the proposed pattern detector is demonstrated replacing the simplistic finite state models by more realistic Hodgkin-Huxley neurons and we consider the possibility of implementing several pattern detectors using a set of shared coincidence detectors. We propose that inhibitory connections may aid to increase the precision of the pattern discrimination