Relating Lurie's problem, Hopfield's network and Alzheimer's disease

Abstract

Alzheimer's disease is a degenerative brain disorder that affects millions of people around the world and still without cure. A very common application of Hopfield neural networks is to simulate a human memory as well as to evaluate problems of degeneration and memory loss. On the other hand, from the control area, one has Lurie's problem, which emerged in the 1940s and which still does not have a general solution. However many works and results came in an attempt to solve it. In this paper, the Hopfield's network is shown as a particular case of Lurie's problem, then one of the consequences of Alzheimer's disease, memory failure, is modeled using Hopfield's networks and nally a recent result of Lurie's problem is applied to the computationally modeled disease to correct the problem of memory loss. The correction is made using a controller via DK-iteration. Simulations are performed to validate the computational model of the disease and to demonstrate the effectiveness of the application of the recent Lurie's problem theorem. Therefore, in addition to the results presented, this work aims at encouraging the researches in the area, so that in the future, better diagnostic and treatment conditions will be achieved.

Analysis and synthesis of single-input-single-output Lurie type systems via H∞ mixed-sensitivity

“…Many researchers have worked on this problem since the years 1950s and 1960s, among whom can be cited as Aizerman (1947), with the well known Aizerman conjecture, and Krasovskii (1953), Popov (1961), and Kalman (1963). In a way, research on the Lurie’s problem took a bigger leap in the 1980s, when works began to appear that linked the problem to other areas and approaches such as neural networks (Liao and Yu, 2008; Pinheiro and Colón, 2019); complex network (Li et al, 2012); chaos and chaos synchronization (Kazemy and Farrokhi, 2017); convex approach to the Lurie problem (Gapski and Geromel, 1994); linear parameter varying (LPV) system (Yu and Liao, 2019); uncertain systems (Tan and Atherton, 2003); Integral Quadratic Constraints (IQC) and Zames-Falb multipliers (Carrasco et al, 2016); μ analysis (Abtahi and Yazdi, 2019; Lee and Juang, 2005); and more recently the application of Lurie’s problem in modern control systems such as Hopfield neural network controls (Pinheiro and Colón, 2021), modeling Alzheimer’s disease (Pinheiro and Colón, 2020), tracking differential extended state observer (Wang et al, 2020) and in control rates of the extended state observer (ESO) for speed control system for the pitching axis of a remote sensing camera (Liu, 2020). In addition to all these new lines of application and study of the Luries problem, its study remains current in the aeronautical field, as can be seen in Imani and Montazeri-Gh (2019).…”

“…Furthermore, Doyle (1982) also developed advanced robust control techniques using parametric uncertainty and structured singular value (SSV), which can be used for analysis and controllers design in Lurie type systems. Recent works have arisen using Doyle’s techniques (Gao et al, 2018; Manukid, 2019; Pinheiro and Colón, 2020).…”

“…In a future paper, the extension to MIMO case will be shown. Furthermore, it will be presented an application in neural networks, similarly as it was done in Pinheiro and Colón (2020) for the SSV.…”