Lag synchronization in an unidirectional ring of memristive neurons

Eur. Phys. J. Spec. Top. https://doi.org/10.1140/epjs/s11734-025-01533-y THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Regular Article Lag synchronization in an unidirectional ring of memristive neurons Vijeesh Vijayan1,2,a , Hayder Natiq3,4,b , Shaher Momani5,6,c , Viet-Thanh Pham7,d , and Matjaž Perc8,9,10,11,e 1 2 3 4 5 6 7 8 9 10 11 Center for Research, SRM Easwari Engineering College, Chennai, India Center for Cognitive Science, Trichy SRM Medical College Hospital and Research Center, Trichy, India Ministry of Higher Education and Scientific Research, Baghdad 10024, Iraq Department of Computer Technology Engineering, College of Information Technology, Imam Ja’afar Al-Sadiq University, Baghdad, Iraq Nonlinear Dynamics Research Center (NDRC), Ajman University, 20550 Ajman, United Arab Emirates Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam Faculty of Natural Sciences and Mathematics, University of Maribor, 2000 Maribor, Slovenia Community Healthcare Center Dr. Adolf Drolc Maribor, 2000 Maribor, Slovenia University College, Korea University, 02841 Seoul, Republic of Korea Department of Physics, Kyung Hee University, Seoul 02447, Republic of Korea Received 22 January 2025 / Accepted 16 February 2025 © The Author(s) 2025 Abstract In recent years, the study of neuronal models has provided significant insights into brain dynamics and neurological disorders. Map-based neuronal models, such as the Rulkov map, have gained considerable popularity due to their computational efficiency and ability to replicate complex neuronal dynamics. We thus here study the collective dynamics of an unidirectional ring network composed of three memristive Rulkov maps, with particular emphasis on synchronization patterns and their dependence on coupling types. By employing electrical and memristive/field couplings, we analyze the emergence of complete synchronization, lag synchronization, and phase synchronization under varying coupling strengths. Our findings highlight how diffusive-based synaptic pathways modulate synchronization and collective behavior in the network. The presented results also offer new perspectives on the role of coupling functions in shaping neuronal synchronization, and they reveal their deeper implications for understanding pathological brain states and for designing neuromorphic systems. 1 Introduction In recent years, mathematical neuronal models have become a powerful tool for analyzing brain function and understanding various neurological disorders. Since neurons are the fundamental components of the brain, several map-based [1–3] and flow-based [4–7] neuronal models have been developed to simulate their dynamics. Map-based models, in particular, have gained significant attention due to their computational efficiency and simplicity [8, 9]. Among these, the Rulkov map [10] stands out as one of the most widely used map-based neuronal models and has been applied extensively in the literature [11–13]. Coupled neuronal models are known to exhibit fascinating collective dynamics, one of which is synchronization [14, 15]. Notably, pathological neuronal synchronization has been implicated in the progression of several neurological disorders, including Alzheimer’s disease [16] and Parkinson’s disease [17]. In networks of oscillators, various forms of synchronization have been observed, such as complete synchronization [18, 19], cluster synchronization [20, 21], lag synchronization [9, 22], phase synchronization [23, 24], relay synchronization [25], and chimera [26, a e-mail: e-mail: c e-mail: d e-mail: e e-mail: (corresponding author) b 0123456789().: V,-vol 123 Eur. Phys. J. Spec. Top. 27]. The type of synchronization that emerges is influenced by numerous factors, including the coupling function, coupling strength, network topology, and the intrinsic dynamics of individual neurons [28, 29]. The choice of coupling type plays a crucial role in determining the collective dynamics of coupled neuronal networks. Different types of couplings, such as electrical [30], chemical [11], and field couplings [31], have been explored extensively to investigate their impact on synchronization and other collective behaviors. Each type of coupling introduces distinct interactions between neurons, leading to diverse dynamical patterns in the network. In 1971, the memristor, short for memory resistor, was proposed by Leon Chua as the fourth fundamental circuit element, alongside the resistor, capacitor, and inductor. The memristor defines the relationship between electrical charge and magnetic flux [32]. This unique component has been employed to model the effects of electromagnetic induction on neurons and is also used as a synaptic element in neuronal networks [33–35]. Various types of memristors have been introduced, characterized by their memductance and memristance properties [36]. Several researchers have explored the dynamics of memristive systems by combining memristors with established mathematical models. For instance, Peng et al. developed a memristive Hénon map and analyzed its dynamics using bifurcation diagrams, Lyapunov exponent spectra, and entropy complexity algorithms. Compared to the original Hénon map, the memristive version exhibited higher complexity and larger chaotic regions [37]. Similarly, Li et al. incorporated cosine memristance into the linear, Tent, Sine, and logistic maps to design their memristive counterparts. Their analysis revealed that all four memristive maps could generate hyperchaotic behavior, and a hardware platform was constructed to implement these models [38]. In another study, Bao et al. proposed a memristive logistic map and examined its dynamics using bifurcation and Lyapunov exponent analysis. Their findings showed that the inclusion of a memristor enhanced the chaotic complexity of the logistic map, resulting in hyperchaotic behavior [39]. Furthermore, Li et al. studied the effects of magnetic induction by introducing a hyperbolic tangent memductance to construct a memristive Rulkov (m-Rulkov) model. Complex dynamics such as hyperchaos and transient chaotic bursting activity were observed, and a hardware implementation of the proposed model was developed [35]. Liu et al. designed another memristive Rulkov model and used entropy complexity algorithms and bifurcation diagrams to explore its dynamics. The memristive model demonstrated a broader range of complexity compared to the original Rulkov map [12]. Mehrabbeik et al. analyzed a ring network of m-Rulkov maps coupled with electrical, chemical, and electrochemical synapses. They observed various collective dynamics, including synchronization, chimera states, amplitude death, and lagged-phase synchronization, by varying the coupling strength [8]. Sriram et al. constructed a regular network of memristive Chialvo maps and explored its dynamics under di (...truncated)