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The population model of Wilson-Cowan is perhaps the most popular in the history of computational neuroscience. It embraces the nonlinear mean field dynamics of excitatory and inhibitory neuronal populations provided via a temporal coarse-graining technique. The traditional Wilson-Cowan equations exhibit either steady-state regimes or else limit cycle competitions for an appropriate range of parameters. As these equations lower the resolution of the neural system and obscure vital information, we assess the validity of mass-type model approximations for complex neural behaviors. Using a large-scale network of Hodgkin-Huxley style neurons, we derive implicit average population dynamics based on mean field assumptions. Our comparison of the microscopic neural activity with the macroscopic temporal profiles reveals dependency on the binary state of interacting subpopulations and the random property of the structural network at the Hopf bifurcation points when different synaptic weights are considered. For substantial configurations of stimulus intensity, our model provides further estimates of the neural population's dynamics official, ranging from simple periodic to quasi-periodic and aperiodic patterns, as well as phase transition regimes. While this shows its great potential for studying the collective behavior of individual neurons particularly concentrating on the occurrence of bifurcation phenomena, we must accept a quite limited accuracy of the Wilson-Cowan approximations-at least in some parameter regimes. Additionally, we report that the complexity and temporal diversity of neural dynamics, especially in terms of limit cycle trajectory, and synchronization can be induced by either small heterogeneity in the degree of various types of local excitatory connectivity or considerable diversity in the external drive to the excitatory pool., Comment: 25 pages, 10 figures

We investigate the dynamical properties of a network of coupled neural oscillators with identical Jansen-Rit masses. The connections between nodes follow the regular Watts-Strogatz topology. Each node receives deterministic external input plus internal input based on the output signal from the neighbors. This paper aims to change these two inputs and analyze the generated results. First, we attempt to analyze the model using the mean-field approximation, i.e., identical inputs for all nodes. Then, this assumption is relaxed. A more detailed analysis of these two states is discussed. We use the Pearson correlation coefficient to measure the amount of synchronization. As a result of the mean-field approach, we find that despite observed changes in behavior, there is no phase transition. Surprisingly, both the first (discontinuous) and second (continuous) phase transition occurs by relaxing the mean-field assumption. We also demonstrate how changes in input can result in pathological oscillations similar to those observed in epilepsy. Results show that coupled Jansen-Rit masses emerge a variety of behaviors influenced by various external and internal inputs. Moreover, our findings indicate that delta waves can be generated by altering these inputs in a network of Jansen-Rit neural mass models, which has not been previously observed in a single Jansen-Rit neural mass model analysis. \\ Overall, a wide range of behavioral patterns, including epilepsy, healthy behavior, and the transition between synchrony, and asynchrony are examined comprehensively in this paper. Moreover, this work highlights the putative contribution of external and internal inputs in studying the phase transition and synchronization of neural mass models.