Analysis to dynamics of complex electrical activities in a wilson model of brain neocortical neuron using fast-slow variable dissection with two slow variables

The neocortex of the brain plays the most important role in achieving functions of the brain via the electrical activities of neurons. Understanding the transition regularity of firing patterns and underlying dynamics of firing patterns of neurons can help to identify the brain functions and to treat some brain diseases. Different neocortical neurons exhibit regular spiking (RS), fast spiking (FS), intrinsic bursting (IB), and continuous bursting (CB), which play vital roles and wide range of functions. Fast-slow variable dissection method combined with bifurcation analysis has been an effective method to identify underlying dynamical mechanism of spiking and bursting modulated by a single slow variable. The spiking is related to the stable limit cycle of the fast subsystem, and the bursting is associated with the transitions or bifurcations between the stable limit cycle and resting state of the fast subsystem. Such underlying dynamics of bursting have been widely used to distinguish different bursting patterns and identify complex dynamics of bursting modulated by various different factors such as synaptic current, autaptic current, and stimulations applied at a suitable phase related to the bifurcations, which play important roles in the real nervous system to regulate neural firing behaviors. Unfortunately, the bursting of neocortical neuronal model (Wilson model) is modulated by two slow variables, the gating variable of calcium-activated potassium channel H and the gating variable of T-type calcium channel T. H is slower than T. Then, the underlying dynamical mechanism of the IB and CB of the neocortical neurons cannot be acquired with fast-slow variable dissection method when H is taken as the sole slow variable, due to that the fast-subsystem contains the slow variable T. In the present paper, we use the fast-slow variable dissection method with two slow variables (H and T) to analyze the bursting patterns. The bifurcations of the fast subsystem, and the intersections between the bifurcation curves and the phase trajectory of bursting in the parameter plane (H, T) are acquired. Due to that neither of the two slow variables of the bursting is sufficiently slow, the bifurcations of only some intersections are related to the bursting behaviors whereas others not. Then, the position relationship between the bifurcation curves and bursting trajectory in the three-dimensional space (H, T, V) (V is membrane potential of bursting) is further acquired, from which the bifurcations related to bursting behaviors are acquired and bifurcations not associated with the bursting behaviors are excluded. The start and termination phases of the burst of the IB are related to the saddle-node on invariant circle (SNIC) bifurcation, and not to the saddle-node (SN) bifurcation. The start and termination phases of the burst of the CB are related to the SNIC bifurcation and the supercritical Andronov-Hopf (SupHopf) bifurcation, respectively, and not to the SN bifurcation. The results present a comprehensive and deep understanding of the underlying dynamics of bursting patterns in the neocortical neurons, which lay the foundation for regulating the firing patterns of the neocortical neurons.