The nervous system is composed of a large number of neurons, and the electrical activities of neurons can present multiple modes during the signal transmission between neurons by changing intrinsic bifurcation parameters or under appropriate external forcing. In this review, the dynamics for neuron, neuronal network is introduced, for example, the mode transition in electrical activity, functional role of autapse connection, bifurcation verification in biological experiments, interaction between neuron and astrocyte, noise effect, coherence resonance, pattern formation and selection in network of neurons. Finally, some open problems in this field such as electromagnetic radiation on electrical activities of neuron, energy consumption in neurons are presented.
Autapse is a type of synapse that connects axon and dendrites of the same neuron, and the effect is often detected by close-loop feedback in axonal action potentials to the owned dendritic tree. An artificial autapse was introduced into the Hindmarsh-Rose neuron model, and a regular network was designed to detect the regular pattern formation induced by autapse. It was found that target wave emerged in the network even when only a single autapse was considered. By increasing the(autapse density) number of neurons with autapse, for example, a regular area(2×2, 3×3, 4×4, 5×5 neurons) under autapse induced target wave by selecting the feedback gain and time-delay in autapse. Spiral waves were also observed under optimized feedback gain and time delay in autapses because of coherence-like resonance in the network induced by some electric autapses connected to some neurons. This confirmed that the electric autapse has a critical role in exciting and regulating the collective behaviors of neurons by generating stable regular waves(target waves, spiral waves) in the network. The wave length of the induced travelling wave(target wave, spiral wave), because of local effect of autapse, was also calculated to understand the waveprofile in the network of neurons.
A nonlinear circuit can be designed by using inductor, resistor, capacitor and other electric devices, and the electromagnetic field energy can be released from the circuit in the oscillating state. The generation of spikes or bursting states in neurons could be energetically a costly process. Based on the Helmholtz＇s theorem, a Hamilton energy function is defined to detect the energy shift induced by transition of electric modes in a Hindmarsh–Rose neuron. It is found that the energy storage is dependent on the external forcing, and energy release is associated with the electric mode. As a result, the bursting state and chaotic state could be helpful to release the energy in the neuron quickly.
Autapse connected to the neuron can change the electric activity of neuron. The effect of autapse on neuronal activity is often described by adding an additive forcing current along a close loop, which is described by a time-delayed feedback on the membrane potential. Neuron often responds to electric autapse forcing sensitively and quickly, while the chemical autapse changes the electric activity of neuron slowly. By applying external forcing, a shift transition of electric activity can be more easily induced by the electric autapse than the chemical autapse. Our results confirm that chemical autapse can enhance and/or suppress the transition of electric activity with excitable or inhibitory type driven by electric autapse, vice versa. It indicates that an appropriate switch-off-on for autapse can make the neuron give different types of response to external forcing. Particularly, cooperation and competition between chemical and electric autapse help neuron response to external forcing in the most reliable way.
