Collective dynamics of two coupled Hopfield inertial neurons with different activation functions: theoretical study and microcontroller implementation

Abstract

This work deals with the regular and chaotic dynamics of a system made up of two Hopfield-type neurons with two different activation functions: the hyperbolic tangent function and the Crespi function. The mathematical model is in the form of an autonomous differential system of order four with odd symmetry. The analysis highlights nine equilibrium points and four of these points experience a Hopf bifurcation at the same critical value of a control parameter which can be either the diss1ipation parameter or one of the coupling coefficients. This makes plausible the presence of four parallel bifurcation branches as well as the coexistence of multiple attractors in the behavior of the system. One of the highlights revealed in this work is the coexistence of three double-scroll type attractors of particular topology as well as the presence of a four-spiral attractor. Furthermore, the coexistence of both self-excited and hidden dynamics is also reported. All this plethora of dynamics is elucidated by making use of the usual tools for analyzing nonlinear systems such as bifurcation diagrams, the maximum of Lyapunov exponent, basins of attractions as well as phase portraits. A physical implementation of the microcontroller-based system is envisaged in order to confirm the plethora of behaviors observed theoretically.

Applying exponential unit for breaking symmetry of memristive maps

“…Researchers in this field are now exploring innovative techniques to further improve the scalability and reliability of memristor-based systems, aiming to integrate them into practical computing systems. Additionally, there is ongoing research into utilizing memristors for unconventional applications, such as cognitive computing and brain-inspired architectures, which hold great promise for the future [10][11][12][13]. As memristor technology continues to advance, it is poised to redefine the landscape of memory and computing, offering remarkable advantages that could revolutionize various industries and lead to the development of more efficient and intelligent electronic systems [14][15][16][17].…”