Narrow Results

In this work, we consider a family of Lotka–Volterra maps $(x^{\prime },y^{\prime })=(x(a-x-y),bxy)$ for $a>1$ and $b>0$ which unfold a map originally proposed by Sharkosky for $a=4$ and $b=1$. Multistability is observed, and attractors may exist not only in the positive quadrant of the plane, but also in the region $y<0$. Some properties and bifurcations are described. The $x$-axis is invariant, on which the map reduces to the logistic. For any $a>1$ an interval of values for $b$ exists for which all the cycles on the $x$-axis are transversely attracting. This invariant set is the source of several kinds of bifurcations. Riddling bifurcations lead to attractors in Milnor sense, not topological but with a stable set of positive measure, which may be the unique attracting set, or coexisting with other topological attractors. The riddling and blowout bifurcations are described related to chaotic intervals on the invariant set, and these global bifurcations have different dynamic results. Chaotic intervals which are not topological attractors may have all the cycles transversely attracting and as Milnor attractors. We show that Milnor attractors may also be related to attracting cycles on the $x$-axis at the bifurcation associated with the transverse and parallel eigenvalues. We show particular examples related to topological attractors with very narrow basins of attraction, when the majority of the trajectories are divergent.

We investigate the dynamic character of a network of electrotonically coupled cells consisting of class I point neurons, in terms of a finite dimensional dynamical system. We classify a subclass of class I point neurons, called class I* point neurons. Based on this classification, we use a reduced Hindmarsh-Rose (H-R) model, which consists of two dynamical variables, to construct a network model consisting of electrotonically coupled H-R neurons. Although biologically simple, the system is sufficient to extract the essence of the complex dynamics, which the system may yield under certain physiological conditions. The network model produces a transitory behavior as well as a periodic motion and spatio-temporal chaos. The transitory dynamics that the network model exhibits is shown numerically to be chaotic itinerancy. The transitions appear between various metachronal waves and all-synchronization states. The network model shows that this transitory dynamics can be viewed as a chaotic switch between synchronized and desynchronized states. Despite the use of spatially discrete point neurons as basic elements of the network, the overall dynamics exhibits scale-free activity including various scales of spatio-temporal patterns.

The non-classical, but frequently reported behavior of GABA_A receptor-mediated excitation in mature CNS has long been regarded as a puzzle. We theorize that the function of cortical GABAergic interneurons, which might constitute a subsystem of brain's GABA interneurons, is their ability of switching from inhibitory action to excitatory action depending on the level of spatio-temporal activity in progress. From the perspective of a dynamical systems approach, such "excitatory" GABAergic responses may serve to temporarily invoke attractor-like memories when extensively activated by, for example, top-down signals as category information or attention, while an ongoing background state of GABA changes its action to inhibition, returning the dynamical nature of the memory structure back to attractor ruins.