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In this note we discuss the usage of the Dirac $\delta$ function in models of phase oscillators with pulsatile inputs. Many authors use a product of the delta function and the phase response curve in the right hand side of an ODE to describe the discontinuous phase dynamics in such systems. We point out that this notation has to be treated with care as it is ambiguous. We argue that the presumably most canonical interpretation does not lead to the intended behaviour in many cases., Comment: a serious flaw in the paper

Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous "jittering" regimes with non-equal interspike intervals (ISIs). Each of these regimes corresponds to a periodic solution of the system with a period roughly proportional to the delay. The number of different "jittering" solutions emerging at the bifurcation point increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how a periodic solution exhibiting several distinct ISIs can imply the existence of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback.

We study the destabilization mechanism in a unidirectional ring of identical oscillators, perturbed by the introduction of a long-range connection. It is known that for a homogeneous, unidirectional ring of identical Stuart-Landau oscillators the trivial equilibrium undergoes a sequence of Hopf bifurcations eventually leading to the coexistence of multiple stable periodic states resembling the Eckhaus scenario. We show that this destabilization scenario persists under small non-local perturbations. In this case, the Eckhaus line is modulated according to certain resonance conditions. In the case when the shortcut is strong, we show that the coexisting periodic solutions split up into two groups. The first group consists of orbits which are unstable for all parameter values, while the other one shows the classical Eckhaus behavior.

Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in the last years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. In the bifurcation point numerous regimes with non-equal interspike intervals emerge simultaneously. We show that this bifurcation is triggered by the steepness of the oscillator's phase resetting curve and that the number of the emerging, so-called "jittering" regimes grows exponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the "multi-jitter" bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit.

In this article we study networks of coupled dynamical systems with time-delayed connections. If two such networks hold different delays on the connections it is in general possible that they exhibit different dynamical behavior as well. We prove that for particular sets of delays this is not the case. To this aim we introduce a componentwise timeshift transformation (CTT) which allows to classify systems which possess equivalent dynamics, though possibly different sets of connection delays. In particular, we show for a large class of semiflows (including the case of delay differential equations) that the stability of attractors is invariant under this transformation. Moreover we show that each equivalence class which is mediated by the CTT possesses a representative system in which the number of different delays is not larger than the cycle space dimension of the underlying graph. We conclude that the 'true' dimension of the corresponding parameter space of delays is in general smaller than it appears at first glance.

Delayed interactions are a common property of coupled natural systems and therefore arise in a variety of different applications. For instance, signals in neural or laser networks propagate at finite speed giving rise to delayed connections. Such systems are often modeled by delay differential equations with discrete delays. In realistic situations, these delays are not identical on different connections. We show that by a componentwise timeshift transformation it is often possible to reduce the number of different delays and simplify the models without loss of information. We identify dynamic invariants of this transformation, determine its capabilities to reduce the number of delays and interpret these findings in terms of the topology of the underlying graph. In particular, we show that networks with identical sums of delay times along the fundamental semicycles are dynamically equivalent and we provide a normal form for these systems. We illustrate the theory using a network motif of coupled Mackey-Glass systems with 8 different time delays, which can be reduced to an equivalent motif with three delays.