Memristor Synapse-Based Morris–Lecar Model: Bifurcation Analyses and FPGA-Based Validations for Periodic and Chaotic Bursting/Spiking Firings

Electromagnetic induction current sensed by the membrane potential in biological neurons can be characterized with a memristor synapse, which can be employed to demonstrate the real oscillating voltage patterns of Barnacle muscle fibers. This paper presents a 3D autonomous memristor synapse-based Morris–Lecar (abbreviated as m-ML) model, which is implemented through introducing a memristor synapse-based induction current to substitute the externally applied current in an existing 2D nonautonomous Morris–Lecar model. Making use of one- and two-parameter bifurcation plots and time-domain representations, diverse period-adding bifurcations as well as abundant periodic and chaotic burst firings are demonstrated. Through constructing the fold and Hopf bifurcation sets of fast spiking subsystem, bifurcation analyses of these chaotic and periodic burst firings are carried out. Moreover, the periodic and chaotic spiking firings and coexisting firing behaviors are illustrated by using one- and two-parameter bifurcation plots and local attraction basins. Finally, based on a field programmable gate array (FPGA) board, a compact digital electronic neuron is fabricated for the 3D m-ML model, from which periodic and chaotic bursting/spiking firings are experimentally measured to verify the results of the numerical simulations.

Measuring effects of different noises in a model using ISI-distance methods

This paper examines the effects of current and conductance noises in a minimal Hodgkin–Huxley type model of a cold receptor neuron. Current noise enters the membrane equation directly while conductance noise is propagated through the activation variables. Compared with common used interspike interval method, ISI-distance is a simple complementary approach to measure the different effects of current and conductance noises. ISI-distance extracts information from the interspike intervals by evaluating the ratio of instantaneous firing rates, which is parameter-free, time scale-independent and easy to visualize. Simulation results show that the most significant differences between different noise implementations in a pacemaker-like tonic firing regime at the transition to chaotic burst discharges.

Chaotic burst synchronization in a two-small-world-layer neuronal network

Chaotic burst synchronization in a two-small-world-layer neuronal network is studied in this paper. For a neuronal network coupled by two single-small-world-layer networks with link probability differences between layers, the two-layer network can achieve synchrony as the interlayer coupling strength increases. When chaotic layer network is coupled with chaotic-burst-synchronization layer network, the latter is dominant at small interlayer coupling strength, so it can make the layer with the irregular pattern show some regular and also exhibit the same pattern with the other layer. However, when chaotic layer is coupled with firing synchronization layer, the ordered layer is dominated by a disordered one with the interlayer coupling strength increasing. When the interlayer coupling strength is large enough, both networks are chaotic burst synchronization. Therefore, the synchronous states strongly depend on the interlayer coupling strength and the link probability. Moreover, the spatiotemporal pattern synchronization between the networks is robust to small noise.

CHAOTIC BURST IN THE DYNAMICS OF A CLASS OF NONCRITICALLY FINITE ENTIRE FUNCTIONS

Devaney [1991] and Devaney and Durkin [1991] exhibited the chaotic burst in the dynamics of certain critically finite entire transcendental functions such as λ exp z and iλ cos z. The Julia set of critically finite entire function Eλ(z)=λ ez for 0<λ<(1/e) is a nowhere dense subset entirely contained in the right half-plane, while it explodes and equals to the extended complex plane for λ>(1/e), a phenomena referred to as chaotic burst in the dynamics of functions in one parameter family ℰ≡{Eλ: Eλ(z)=λ exp z, λ>0}. In the present work, a class [Formula: see text] of noncritically finite entire functions is introduced and the dynamics of functions in one parameter family [Formula: see text] generated from each function g(z) in the class [Formula: see text] is investigated. It is proved that there exists a real number [Formula: see text] such that bifurcation in the dynamics of functions [Formula: see text] occurs at [Formula: see text]. Further, it is established that chaotic burst occurs in the dynamics of noncritically finite functions in one parameter family [Formula: see text] at the parameter value [Formula: see text]. Finally, certain interesting examples of the family [Formula: see text], viz. [Formula: see text], where [Formula: see text] is the well-known modified Bessel function of zero order and [Formula: see text] , where [Formula: see text] with fixed k=1, 2,… and [Formula: see text] are provided.