Lyusternik-Schnirelman theory and eigenvalue problems for monotone potential operators

We treat the eigenvalue problem Ax = λBx, where A and B are odd potential operators, A is strictly monotone, bounded, coercive, and continuously invertible, and B is monotone and compact. A naturally defined iteration operator is employed, together with the Lyusternik-Schnirelman theory, to prove the existence of infinitely many nontrivial eigenfunctions. With the possible exception of the multiplicity assertion the results which we obtain are not new. The method which we use, however, has not been applied before to problems of this type. It exploits both the potential character and the monotonicity of the operators and makes the treatment of the infinite dimensional problem essentially as simple as that of its finite dimensional analog. This simplification results primarily from the compactness properties of the iteration operator.     

Cooperative behavior between oscillatory and excitable units: the peculiar role of positive coupling-frequency correlations

We study the collective dynamics of noise-driven excitable elements, so-called active rotators. Crucially here, the natural frequencies and the individual coupling strengths are drawn from some joint probability distribution. Combining a mean-field treatment with a Gaussian approximation allows us to find examples where the infinite-dimensional system is reduced to a few ordinary differential equations. Our focus lies in the cooperative behavior in a population consisting of two parts, where one is composed of excitable elements, while the other one contains only self-oscillatory units. Surprisingly, excitable behavior in the whole system sets in only if the excitable elements have a smaller coupling strength than the self-oscillating units. In this way positive local correlations between natural frequencies and couplings shape the global behavior of mixed populations of excitable and oscillatory elements.     

Enhancement of object detection with transcranial direct current stimulation is associated with increased attention

ABSTRACT: BACKGROUND: We previously found that transcranial Direct Current Stimulation (tDCS) improves learning and performance in a task where subjects learn to detect potential threats indicated by small target objects hidden in a complex virtual environment. In the present study, we examined the hypothesis that these effects on learning and performance are related to changes in attention. The effects of tDCS were tested for three forms of attention (alerting, orienting, and executive attention) using the Attention Network Task (ANT), which were compared with performance on the object-learning task. RESULTS: Participants received either 0.1 mA (N = 10) or 2.0 mA (N = 9) tDCS during training and were tested for performance in object-identification before training (baseline-test) and again immediately after training (immediate test). Participants next performed the Attention Networks Task (ANT), and were later tested for object-identification performance a final time (delayed test). Alerting, but not orienting or executive attention, was significantly higher for participants receiving 2.0 mA compared with 0.1 mA tDCS (p < 0.02). Furthermore, alerting scores were significantly correlated with the proportion of hits (p < 0.01) for participants receiving 2.0 mA. CONCLUSIONS: These results indicate that tDCS enhancement of performance in this task may be related in part to the enhancement of alerting attention, which may benefit the initial identification, learning and/or subsequent recognition of target objects indicating potential threats.