Applied Mathematics and Computation 350 (2019) 217–223 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Synchronizability of two neurons with switching in the coupling Fatemeh Parastesh a , Hamed Azarnoush a , Sajad Jafari a , Boshra Hatef b , Matjaž Perc c,d,* , Robert Repnik c a Department of Biomedical Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran b Neuroscience Research Center, Baqiyatallah University of Medical Sciences, Tehran, Iran c Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000, Maribor, Slovenia d Complexity Science Hub Vienna, Josefstädterstraße 39, A-1080, Vienna, Austria a r t i c l e i n f o Keywords: Synchronization Neuronal network Switching coupling Ephaptic coupling Master stability function a b s t r a c t Time-varying networks are prominently present in the nervous system, where the connec- tions between neurons vary according to the activity of pre and post synapses. Using this as motivation, we here consider a system of two ephaptically coupled neurons with a pe- riodically time-varying link between the two membrane potentials. We derive the master stability function in dependence on the membrane potential coupling strength and in de- pendence on the ephaptic coupling strength. We first determine coupling ranges at which the two neurons are synchronizable if the link between them is static. In doing so, we find that the ephaptic coupling has a negligible impact on synchronization, while the cou- pling through the membrane potential has a strong effect. We then consider switching in the coupling, determining the average synchronization error for a fixed ephaptic coupling strength and different membrane potential coupling strengths, and for various switching frequencies. We find the values of the switching coefficient that can synchronize the two neurons for each of the considered switching frequencies, and finally, we also use the basin stability method to determine the global stability of synchronization. © 2019 Elsevier Inc. All rights reserved. 1. Introduction Complex networks are helpful tools for clarifying different characteristics of the complex systems [1–4]. Synchronization is a universal phenomenon in complex networks which has been investigated in different physical, chemical and biological networks [5–9]. In the past years, synchronous activity has been discussed extensively in neuronal networks [10–13]. It has been revealed that synchronization of bursting neurons is associated with various cognitive functions and pathological brain states. As examples we can mention Parkinson’s disease, essential tremor and epilepsies. Also synchronization of bursting neurons has an important role in processing information of healthy neural tissues [14,15]. The connections between network components are of great significance [16]. Majority of the previous research has stud- ied the synchronization phenomenon in the networks with static coupling [17–19]. But in reality the networks, e.g., bio- logical, epidemiological and social networks, have time varying coupling schemes [20]. In recent years, a few studies have considered the network topologies which vary by time [20–24]. In 2004, Belykh et al. [25] proposed blinking network that * Corresponding author at: Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia. E-mail addresses: matjaz.perc@um.si (M. Perc), robert.repnik@um.si (R. Repnik). https://doi.org/10.1016/j.amc.2019.01.011 0096-3003/© 2019 Elsevier Inc. All rights reserved.