Dynamical principles underlying song degradation in birdsong neural circuit

Our previous work has shown that AFP could adjust synaptic conductance of RA network with its firing pattern changed from bursting to tonic firing, eventually leading to song degradation. However, it is difficult to explore the dynamic of RA network due to its high dimensionality and complexity. To address this issue, we use the mean field theory proposed by Wilten Nicola to reduce RA network to a system of switching ordinary differential equations. Numerical simulations prove that the mean field model could qualitatively and quantitatively describe the behavior of nucleus RA. As a result, we perform the non-smooth bifurcation analysis over the mean field model to understand the bifurcations that RA network displays. Then we determine that there is a subcritical-Andronov-Hopf bifurcation at a particular applied current, which causes the emergence of the bistability with current increased. The bistability consists of a stable node and a stable limit cycle, corresponding to tonic firing and bursting of the full network, respectively. With these properties, we can explain the dynamical principles of AFP's adjustment on RA during song degradation. Furthermore, our results would effectively promote the combination of dynamical analysis and neural network modeling based on the mean field theory.