We consider autonomous systems where two scalar differential equations are coupled with the input-output relationship of the Preisach hysteresis operator, which has an infinite-dimensional memory. A prototype system of this type is an LCR electric circuit where the inductive element has a ferromagnetic core with a hysteretic relationship between the magnetic field and the magnetization. Further examples of such systems include lumped hydrological models with two soil layers; they can also appear as a component of the recently proposed models of population dynamics. We study dynamics of such systems near an equilibrium point. In particular, we show and examine a similarity in the behaviour of trajectories between the system with the Preisach memory operator and a planar slow-fast ordinary differential equation. The nonsmooth Preisach operator introduces a singularity into the system. Furthermore, we classify the robust equilibrium points according to their stability properties. Conditions for stability, instability and partial stability are presented. A robust partially stable point simultaneously attracts many trajectories and repels many trajectories (a behaviour which is not generic for smooth ordinary differential equations). We discuss implications of such local dynamics for the excitability properties of the system.
Endothelin-1 (ET-1) is a neuroactive protein produced in most brain cell types and participates in regulation of cerebral blood flow and blood pressure. In addition to its vascular effects, ET-1 affects synaptic and nonsynaptic neuronal and glial functions. Direct application of ET-1 to the hippocampus of immature rats results in cerebral ischemia, acute seizures, and epileptogenesis. Here, we investigated whether ET-1 itself modifies the excitability of hippocampal and cortical circuitry and whether acute seizures observed in vivo are due to nonvascular actions of ET-1. We used acute hippocampal and cortical slices that were preincubated with ET-1 (20 µM) for electrophysiological recordings. None of the slices preincubated with ET-1 exhibited spontaneous epileptic activity. The slope of the stimulus intensity-evoked response (input-output) curve and shape of the evoked response did not differ between ET-1-pretreated and control groups, suggesting no changes in excitability after ET-1 treatment. The threshold for eliciting an evoked response was not significantly increased in either hippocampal or cortical regions when pretreated with ET-1. Our data suggest that acute seizures after intrahippocampal application of ET-1 in rats are likely caused by ischemia rather than by a direct action of ET-1 on brain tissue., R. Konopková ... [et al.]., and Obsahuje seznam literatury