Hybrid discrete-time neural networks

Geometric characterization of dynamical structure for neural firing activities induced by inhibitory pulse

In general, inhibitory stimuli are thought to inhibit neuronal firing, but they may actually enhance firing sometimes, such as post-inhibitory rebound spike (PIR spike) and post-inhibitory facilitation (PIF) phenomena, which play an important role in human neuronal activities. We study responses to inhibitory pulse in a classical neuron model (Quartic adaptive Integrate-and-fire model) well known to reproduce a number of biologically realistic behaviors. The three phenomena that we study are PIR, in which a neuron fires after an inhibitory pulse, and PIF, in which a subthreshold excitatory input can induce a spike if it is applied with proper timing after an inhibitory pulse, as well as period firing after inhibitory pulse. When the system features focus and saddle two equilibriums, the three phenomena will be occurred under the inhibitory pulse, while all three phenomena will not be induced when the system features node and saddle two equilibriums. Using dynamical systems theory, we explain the threshold mechanism of enhancement of neural firing response induced by inhibitory pulse and analyze the origin of these phenomena from several factors. We also describe the geometric characterization of dynamical structures of these three phenomena. This study therefore enrich the paradoxical phenomena that induced by inhibitory input and advance our understanding of its role.

Theory of hybrid dynamical systems and its applications to biological and medical systems

In this introductory article, we survey the contents of this Theme Issue. This Theme Issue deals with a fertile region of hybrid dynamical systems that are characterized by the coexistence of continuous and discrete dynamics. It is now well known that there exist many hybrid dynamical systems with discontinuities such as impact, switching, friction and sliding. The first aim of this Issue is to discuss recent developments in understanding nonlinear dynamics of hybrid dynamical systems in the two main theoretical fields of dynamical systems theory and control systems theory. A combined study of the hybrid systems dynamics in the two theoretical fields might contribute to a more comprehensive understanding of hybrid dynamical systems. In addition, mathematical modelling by hybrid dynamical systems is particularly important for understanding the nonlinear dynamics of biological and medical systems as they have many discontinuities such as threshold-triggered firing in neurons, on-off switching of gene expression by a transcription factor, division in cells and certain types of chronotherapy for prostate cancer. Hence, the second aim is to discuss recent applications of hybrid dynamical systems in biology and medicine. Thus, this Issue is not only general to serve as a survey of recent progress in hybrid systems theory but also specific to introduce interesting and stimulating applications of hybrid systems in biology and medicine. As the introduction to the topics in this Theme Issue, we provide a brief history of nonlinear dynamics and mathematical modelling, different mathematical models of hybrid dynamical systems, the relationship between dynamical systems theory and control systems theory, examples of complex behaviour in a simple neuron model and its variants, applications of hybrid dynamical systems in biology and medicine as a road map of articles in this Theme Issue and future directions of hybrid systems modelling.