Periodically kicked feedforward chains of simple excitable FitzHugh-Nagumo neurons

This article communicates results on regular depolarization cascades in periodically-kicked feedforward chains of excitable two-dimensional FitzHugh-Nagumo systems driven by sufficiently strong excitatory forcing at the front node. The study documents a parameter exploration by way of changes to the forcing period, upon which the dynamics undergoes a transition from simple depolarization to more complex behavior, including the emergence of mixed-mode oscillations. Both rigorous studies and careful numerical observations are presented. In particular, we provide rigorous proofs for existence and stability of periodic traveling waves of depolarization, as well as the existence and propagation of a simple mixed-mode oscillation that features depolarization and refraction in alternating fashion. Detailed numerical investigation reveals a mechanism for the emergence of complex mixed-mode oscillations featuring a potentially high number of large amplitude voltage spikes interspersed by an occasional small amplitude reset that fails to cross threshold. Further careful numerical investigation provides insights into the propagation of this complex phenomenology in the downstream, where we see an effective filtration property of the network; the latter amounts to a successive reduction in the complexity of mixed-mode oscillations down the chain.

Bifurcation Structures of Nested Mixed-Mode Oscillations

In recently published work [Inaba & Kousaka, 2020a; Inaba & Tsubone, 2020b], we discovered significant mixed-mode oscillation (MMO) bifurcation structures in which MMOs are nested. Simple mixed-mode oscillation-incrementing bifurcations (MMOIBs) are known to generate [Formula: see text] oscillations for successive [Formula: see text] between regions of [Formula: see text]- and [Formula: see text]-oscillations, where [Formula: see text] and [Formula: see text] are adjacent simple MMOs, e.g. [Formula: see text] and [Formula: see text], where [Formula: see text] is an integer. MMOIBs are universal phenomena of evidently strong order and have been studied extensively in chemistry, physics, and engineering. Nested MMOIBs are phenomena that are more complex, but have an even stronger order, generating chaotic MMO windows that include sequences [Formula: see text] for successive [Formula: see text], where [Formula: see text] and [Formula: see text] are adjacent MMOIB-generated MMOs, i.e. [Formula: see text] and [Formula: see text] for integer [Formula: see text]. Herein, we investigate the bifurcation structures of nested MMOIB-generated MMOs exhibited by a classical forced Bonhoeffer–van der Pol oscillator. We use numerical methods to prepare two- and one-parameter bifurcation diagrams of the system with [Formula: see text], and 3 for successive [Formula: see text] for the case [Formula: see text]. Our analysis suggests that nested MMOs could be widely observed and are clearly ordered phenomena. We then define the first return maps for nested MMOs, which elucidate the appearance of successively nested MMOIBs.

Mixed-Mode Oscillation in a Class of Delayed Feedback System and Multistability Dynamic Response

In this paper, a class of two-parameter mixed-mode oscillation with time delay under the action of amplitude modulation is studied. The investigation is from four aspects. Firstly, a parametric equation is considered as a slow variable. By the time-history diagram and phase diagram, we can find that the system generates a cluster discovery image. Secondly, the Euler method is used to discrete the system and obtain the discrete equation. Thirdly, the dynamic characteristics of the system at different time scales are discussed when the ratio of the natural frequency and the excitation frequency of the system is integer and noninteger. Fourthly, we discuss the influence of time delay on the discovery of clusters of this kind of system. The research shows that the time lag does not interfere with the influence of the cluster image, but the dynamics of the upper and lower parts of the oscillation in each period will be delayed. So, we can improve peak performance by adjusting the time lag and obtain the desired peak. Finally, we explore the multistate dynamic response of a two-dimensional nonautonomous Duffing system with higher order. According to bifurcation diagram and time-history curve, bistable state will appear in the system within the critical range. With the gradual increase of parameters, the chaotic attractor will suddenly disappear which will lead to the destruction of the bistable state.

Experimental and Numerical Observation of Successive Mixed-Mode Oscillation-Incrementing Bifurcations in an Extended Bonhoeffer–van der Pol Oscillator

Mixed-mode oscillation-incrementing bifurcations (MMOIBs) are a universal phenomenon appearing in a large class of mixed-mode oscillation (MMO)-generating dynamics that appear to be governed by asymmetric Farey arithmetic. This report presents experimental and numerical observations of the simplest MMOIBs generated by an extended Bonhoeffer–van der Pol circuit. The results indicate that maintaining a constant temperature during the circuit experiments is required to yield stable MMOs and MMOIBs. We also clarify how the firing number variations related to MMOIB phenomena follow asymmetric Farey arithmetic.