We obtain necessary and sufficient conditions for the existence of a center on a local center manifold for three 4-parameter families of quadratic systems on ℝ. We also give a positive answer to an open question posed in [Dias & Mello(2010)] related to similar systems.

Inferior olive (IO) neurons project to the cerebellum and contribute to motor control. They can show intriguing spatio-temporal dynamics with rhythmic and synchronized spiking. IO neurons are connected to their neighbors via gap junctions to form an electrically coupled network, and so it is considered that this coupling contributes to the characteristic dynamics of this nucleus. Here, we demonstrate that a gap junction-coupled network composed of simple conductance-based model neurons (a simplified version of a Hodgkin-Huxley type neuron) reproduce important aspects of IO activity. The simplified phenomenological model neuron facilitated the analysis of the single cell and network properties of the IO while still quantitatively reproducing the spiking patterns of complex spike activity observed by simultaneous recording in anesthetized rats. The results imply that both intrinsic bistability of each neuron and gap junction coupling among neurons play key roles in the generation of the spatio-temporal dynamics of IO neurons.

Binocular rivalry is a useful experimental paradigm to investigate aspects of neocortical dynamics related to conscious perception. Frequency-tagged EEG responses to a sine-flickered visual stimulus were contrasted between episodes of perceptual dominance, i.e., conscious perception of that stimulus and perceptual nondominance, i.e., conscious perception of a rival stimulus presented at a different frequency to the other eye. The amplitude and phase distribution of the stimulus-evoked steady-state responses depended on the stimulus modulation frequency, consistent with the presence of global resonance phenomena. At the apparent global resonance frequency, conscious perception of the stimulus modulated the steady-state response over the entire array of electrodes. These effects were significant at electrodes far from the primary visual cortex, including temporal, central, and frontal electrodes. The phase structure of the steady-state response was also investigated using coherence measures. Coherence between electrodes mostly increased during conscious perception of the stimulus. Analysis of partial coherence, removing stimulus-locked responses, indicated that synchronization of each signal to the stimulus flicker at each electrode and synchronization between signals that vary with respect to the stimulus flicker at each electrode both contribute to observed increases in coherence during conscious perception. These distinct modes of synchronization may reflect two different physiological mechanisms by which sensory signals are integrated across the cerebral cortex during conscious experience.

The spontaneous breakup of a single spiral wave of excitation into a turbulent wave pattern has been observed in both discrete element models and continuous reaction-diffusion models of spatially homogeneous 2D excitable media. These results have attracted considerable interest, since spiral breakup is thought to be an important mechanism of transition from the heart rhythm disturbance ventricular tachycardia to the fatal arrhythmia ventricular fibrillation. It is not known whether this process can occur in the absence of disease-induced spatial heterogeneity of the electrical properties of the ventricular tissue. Candidate mechanisms for spiral breakup in uniform 2D media have emerged, but the physical validity of the mechanisms and their applicability to myocardium require further scrutiny. In this letter, we examine the computer simulation results obtained in two discrete element models and show that the instability of each spiral is an artifact resulting from an unphysical dependence of wave speed on wave front curvature in the medium. We conclude that spiral breakup does not occur in these two models at the specified parameter values and that great care must be exercised in the representation of a continuous excitable medium via discrete elements.