Role of coupling delay in oscillatory activity in autonomous networks of excitable neurons with dissipation (Q6549986)

In the present work consider the FitzHugh-Nagumo oscillator, which represents one of the simplest neuron models and is widely used in numerical simulations. This two-variable system is a paradigmatic model for neural excitability and is described as follows: \begin{align*} \N\varepsilon \frac{dx}{dt}= x-\frac{x^3}{3}-y\\\N\frac{dy}{dt}= \gamma x-y+\beta\N\end{align*} \Nwhere \( x \) is the fast variable (activator) and represents the voltage across the cell membrane and \( y \) is the slow recovery variable (inhibitor), which corresponds to the recovery state of the resting membrane of a neuron. All the control parameters are dimension- less. The small parameter \( \varepsilon > 0 \) is the ratio of the activator to inhibitor time scales. The parameter \( \beta \) determines the asymmetry, and the parameter \( \gamma \) is responsible for dissipation in the neuron. Then extending the model and our numerical simulation to a ring network with local and nonlocal coupling topology between the neurons. Study the interplay between dissipation in the single unit, the delay time, the coupling parameters, and initial conditions in forming the spatiotemporal dynamics of the networks. The numerical simulations performed and the results obtained can be straightforwardly generalized to the dynamics of delay-coupled simplified FitzHugh-Nagumo models. Providing recent references and test equation for numerical justification.