Third-Order Memristive Morris–Lecar Model of Barnacle Muscle Fiber

Abstract

This paper presents a detailed analysis of various oscillatory behaviors observed in relation to the calcium and potassium ions in the third-order Morris–Lecar model of giant barnacle muscle fiber. Since, both the calcium and potassium ions exhibit all of the characteristics of memristor fingerprints, we claim that the time-varying calcium and potassium ions in the third-order Morris–Lecar model are actually time-invariant calcium and potassium memristors in the third-order memristive Morris–Lecar model. We confirmed the existence of a small unstable limit cycle oscillation in both the second-order and the third-order Morris–Lecar model by numerically calculating the basin of attraction of the asymptotically stable equilibrium point associated with two subcritical Hopf bifurcation points. We also describe a comprehensive analysis of the generation of oscillations in third-order memristive Morris–Lecar model via small-signal circuit analysis and a subcritical Hopf bifurcation phenomenon.

Wave propagation in a network of extended Morris–Lecar neurons with electromagnetic induction and its local kinetics

“…The number of spikes in a bursting periodic orbit have been revealed in [12]. In [13] a 3D M-L model is formulated with memristors and basin of attraction of the asymptotically stable equilibrium point associated with two subcritical Hopf bifurcation points were separated and showed clearly the presence of an unstable limit cycle. In [14], Liu et al, investigated the M-L model developed by Prescott et al [15] with different parameter values and came up with a bistability property.…”

Modulation of contractions in the small intestine indicate desynchronization via supercritical Andronov–Hopf bifurcation

“…Another useful perspective is provided by Chua's theory of memristors, which correspond to ion channels in the biological context. Memristors can be characterised in terms of their admittance response to a stimulus current, an effect predicted from the analysis of their equilibrium as near the edge-of-chaos [45][46][47][48] . As for the biological significance of these results, if that is taken to mean function, then there is none.…”

Unstable Limit Cycles and Singular Attractors in a Two-Dimensional Memristor-Based Dynamic System

“…In order to find unstable limit cycles of a dynamic system, some feasible schemes were suggested (e.g., [ 21 ]). However, the available algorithms are either difficult to program or time-consuming to operate when applied to system (1).…”

“…In virtue of the ultra-low energy consumption and fast switching speed, they also gain popularity in many new computer architectures [ 17 ]. Some generic memristors in such as Hodgkin–Huxley Axon memristor model [ 18 , 19 ], and Morris–Lecar memristor model [ 20 , 21 ] have been identified through detailed and systematic qualitative analyses, which contribute to the penetrate complex dynamics of biological membranes. The Chua shoelace memristor [ 2 ], named as the Chua corsage memristor in [ 22 , 23 ], is also a generic memristor and possesses only one local activity domain, which presages the existence of complex dynamics in associated memristor systems [ 24 ].…”

“…The twin subcritical Hopf bifurcation domains of this system having unstable limit cycles are shown in Table 2 . Technically, there are some difficulties in finding unstable limit cycles by using the available schemes (e.g., [ 21 ]). Thus, how to find available unstable limit cycles quickly is an important and interesting issue for further exploration.…”