Influence of extrinsic inputs and synaptic gains on dynamics of Wendling's neural mass model: A bifurcation analysis

We analyze the neurodynamics attributed by a model proposed by Wendling and co-workers (2002) [Wendling, F., Bartolomei, F., Bellanger, J.J. & Chauvel, P. (2002) Epileptic fast activity can be explained by a model of impaired GABAergic dendritic inhibition. Eur. J. Neurosci., 15, 1499.] to explain several different types of electroencephalographic activities. We could find three principal types of steady states when the system parameters change slowly: (i) the model produce a constant output when it is under a state of stable equilibrium point with a constant input. If a small perturbation is introduced (e.g., noisy input), the output changes into noise without oscillatory components, which is related to the normal background activity or low-voltage rapid activity, (ii) Hopf bifurcations lead to stable limit cycles, which we call Hopf cycles. The model generates a rhythmic oscillating output when it is under a state of Hopf cycles, which is related to slow rhythmic activity or slow quasi-sinusoidal activity, (iii) global bifurcations lead to homoclinic limit cycles that appear suddenly at high amplitude, which we call spike cycles. In general, the spike cycles are not harmonic but they have a spike-like appearance (anharmonic oscillation). The model produces a spike-like output when it is under a state of spike cycles, which is related to the sustained discharge of spikes. Finally, the bifurcation analysis demonstrates the influence of the interaction between the excitatory and inhibitory synaptic gains on the dynamics.