Recent advances in the experimental and theoretical study of dynamics of neuronal electrical firing activities are reviewed. Firstly, some experimental phenomena of neuronal irregular firing patterns, especially chaotic and stochastic firing patterns, are presented, and practical nonlinear time analysis methods are introduced to distinguish deterministic and stochastic mechanism in time series. Secondly, the dynamics of electrical firing activities in a single neuron is concerned, namely, fast-slow dynamics analysis for classification and mechanism of various bursting patterns, one-or two-parameter bifurcation analysis for transitions of firing patterns, and stochastic dynamics of firing activities (stochastic and coherence resonances, integer multiple and other firing patterns induced by noise, etc.).types of synchronization of coupled neurons with electrical and chemical synapses are discussed. As noise and time delay are inevitable in nervous systems, it is found that noise and time delay may induce or enhance synchronization and change firing patterns of coupled neurons. Noise-induced resonance and spatiotemporal patterns in coupled neuronal networks are also demonstrated. Finally, some prospects are presented for future research. In consequence, the idea and methods of nonlinear dynamics are of great significance in exploration of dynamic processes and physiological functions of nervous systems.

a) Input Video (b) Dynamic View Synthesis (c) Motion Editing Deformation Mesh 3D Gaussians Deformation Fields Figure 1: We propose the Mesh-adsorbed Gaussian Splatting (MaGS), a unified framework for reconstructing and simulating a dynamic 3D object from a monocular video.

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Gaussian white noise (GWN), as an intrinsic noise source, can give rise to various firing activities at the rest state before a supercritical or subcritical Hopf bifurcation (supH or subH) in the Chay system without or with external current input, when V K , V C , λ n and I are considered as changeable control parameters. These firing activities are closely related to the global bifurcation mechanism of the whole system and the fast/slow dynamical subsystems, and can be tackled by means of bifurcation analysis.GWN can induce some typical bursting phenomena in the stochastic Chay system. Firstly, integer multiple "fold/homoclinic" or "circle/homoclinic" bursting due to GWN, with only one spike per burst, can arise from rest states before both subH and supH (with respect to the parameter V K ), and their respective trajectories have the same shape and property. However, less spikes appear and their peaks are lower before supH, comparing with those before subH. Secondly, a "fold/fold" point-point hysteresis loop bursting due to GWN is generated before supH (with respect to the parameter V C ) on the upper branch of a "Z"-shaped bifurcation curve between two fold bifurcations of the fast system. Thirdly, at a rest state before subH (with respect to the additional current I) situated on the lower branch of a "S"-shaped bifurcation curve between two fold bifurcations of the fast system, a GWN-induced firing pattern appears and is classified as "Hopf/homoclinic" bursting via "fold/homoclinic" point-point hysteresis loop.GWN-induced firing activities other than bursting can also be observed in the stochastic Chay system. For example, sometimes GWN-induced continuous spiking without any particular shape may arise at a rest state before supH (with respect to the parameter V K ) for certain values of parameters. Moreover, under the situation that a stable node and a stable focus coexist before subH (with respect to the parameter I) and the attractive region of the stable node is larger than that of the stable focus, GWN only provoke random subthreshold impulsing oscillation near the stable node.