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Abstract

Anomalous phase synchronization describes a synchronization phenomenon occurring even for the weakly coupled network and characterized by a non-monotonous dependence of the synchronization strength on the coupling strength. Its existence may support a theoretical framework to some neurological diseases, such as Parkinson’s and some episodes of seizure behavior generated by epilepsy. Despite the success of controlling or suppressing the anomalous phase synchronization in neural networks applying external perturbations or inducing ambient changes, the origin of the anomalous phase synchronization as well as the mechanisms behind the suppression is not completely known. Here, we consider networks composed of N=2000 coupled neurons in a small-world topology for two well known neuron models, namely, the Hodgkin-Huxley-like and the Hindmarsh-Rose models, both displaying the anomalous phase synchronization regime. We show that the anomalous phase synchronization may be related to the individual behavior of the coupled neurons; particularly, we identify a strong correlation between the behavior of the inter-bursting-intervals of the neurons, what we call neuron variability, to the ability of the network to depict anomalous phase synchronization. We corroborate the ideas showing that external perturbations or ambient parameter changes that eliminate anomalous phase synchronization and at the same time promote small changes in the individual dynamics of the neurons, such that an increasing individual variability of neurons implies a decrease of anomalous phase synchronization. Finally, we demonstrate that this effect can be quantified using a well known recurrence quantifier, the “determinism.” Moreover, the results obtained by the determinism are based on only the mean field potential of the network, turning these measures more suitable to be used in experimental situations. Neural phase synchronization (PS) is one of the most important problems in neural systems since it is related to how information is processed in the brain. A large number of works focus on how the connectivity structure (which, in general, is neither random nor regular) of a network is related to PS. It is known that some neural (sub)networks in the brain have small-world properties, allowing a short mean path length and at the same time a large clustering coefficient. PS occurring even for weak coupling and under small-world regimes may be related to a disorder of the nervous system, like Parkinson’s disease. In this paper, we aim at studying how anomalous PS phenomenon of small-world networks composed of bursting HodgkinHuxley-type and HindmarshRose neuron models are related to the individual behavior of the neurons. To perform the analysis, we use the Kuramoto order parameter and the recurrence quantifier, the so called “determinism.” The determinism is computed using just the mean field potential of the network and leads to similar results to those obtained using the order parameter that must be computed based on the signal of each neuron on the networks.