Starting with a subexcitable net of FitzHugh-Nagumo elements it is shown that parameter variability (diversity) is able to induce pattern formation. These patterns are most coherent for an intermediate variability strength. This effect is similar to the well-known spatiotemporal stochastic resonance generated by additive noise in subexcitable media. Furthermore, variability is able to induce a transition to an excitable behavior of the net. This transition strongly depends on the coupling strength between the elements and it is found only for strong coupling. For weaker coupling one observes a lifetime lengthening of waves propagating through the medium, but the net stays subexcitable.

The influence of variability on the response of a net of bistable FitzHugh-Nagumo elements to a weak signal is investigated. The response of the net undergoes a resonancelike behavior due to additive variability. For an intermediate strength of additive variability the external signal is optimally enhanced in the output of the net (diversity-induced resonance). Furthermore, we show that additive noise strongly influences the diversity-induced resonance. Afterwards the interplay of additive and multiplicative variability on the response of the net is investigated. Starting with asymmetric bistable elements the enhancement of the signal is not very pronounced in the presence of additive variability. Via symmetry restoration by multiplicative variability the resonance is further enhanced. We call this phenomenon doubly diversity-induced resonance, because the interplay of both, additive and multiplicative variability, is essential to achieve the optimal enhancement of the signal. The restoration of symmetry can be explained by a systematic effect of the multiplicative variability, which changes the thresholds for the transitions between the two stable fixed points. We investigate the response to variability for globally and diffusively coupled networks and in dependency on the coupling strength.

The influence of time-delayed feedback on the dynamics of a net of oscillatory FitzHugh-Nagumo elements is investigated. We show that the global oscillation of the net can be suppressed (amplitude death) via time-delayed feedback for properly chosen delay time and feedback strength. The result of a linear stability analysis fits very well to the simulations. In the amplitude death regime, weak additive noise can induce excitation waves (noise-induced pattern formation), a fingerprint of excitable network dynamics.

The influence of time-delayed feedback on pattern formation in subexcitable media represented by a net of FitzHugh-Nagumo elements, a minimal model of neuronal dynamics, is studied. Without feedback, wave fronts die out after a short propagation length (subexcitable net dynamics). Applying time-delayed feedback with appropriate feedback parameters, pattern formation is sustained and the wave fronts may propagate through the whole net (signature of excitable behavior). The coherence of noise-induced patterns is significantly enhanced if feedback with appropriately chosen parameters is applied, and shows a resonancelike dependency on the delay time. In a next step, the transition to the excitable regime is investigated in dependence on the quota of elements, which get the feedback signal. It is sufficient to control approximately half of the elements to achieve excitable behavior. Regarding a medical application, where the external control of a neural tissue would affect not single neurons but clusters of neurons, the spatial correlation of the controlled elements is of importance. The selection of the elements, which get the feedback signal, is based on a spatially correlated random distribution. It is shown that the correlation length of this distribution affects the pattern formation.

Starting with an oscillatory net of neural elements, increasing variability induces a phase transition to excitability. This transition is explained by a systematic effect of the variability, which stabilizes the formerly unstable, spatially uniform, temporally constant solution of the net. Multiplicative noise may also influence the net in a systematic way and may thus induce a similar transition. Adding noise into the model, the interplay of noise and variability with respect to the reported transition is investigated. Finally, pattern formation in a diffusively coupled net is studied, because excitability implies the ability of pattern formation and information transmission.