Physical Foundations of Neurophysiology: Non-linear Models of Action Potential Propagation and Synaptic Transmission

The problem of synaptic transmission (generally saying, the problem of intercellular interaction) in neurophysiology, for which there are electrical and chemical aspects, is as important for living nature as the problem of intermolecular interaction for inanimate nature. The electrical aspect of intercellular interaction is associated with the processes of generation and propagation of the action potentialAction potential (spike) along the axon. The chemical aspect is due to the chemical reactions of the interaction of mediators and receptors in the synaptic clefts (synapses) that separate the cell body, axons and dendrites. Here, in Sects. 4.2 and 4.3, the general provisions and sequential equations of the Hodgkin-Huxley modelHodgkin-Huxley model (1963 Nobel Prize in Physiology or Medicine) will be considered. In addition, this chapter will propose new nonlinear kinetic modelsNonlinear kinetic models of synaptic transmissionSynaptic transmission of a nerve impulse, as well as obtain analytical solutions to the corresponding systems of differential equations and analyze the consequences that follow from the analysis of singular points in accordance with the classifications of Poincaré and Lyapunov. As an application of our theoretical approach to the problem of synaptic transmission in cholinergic synapsesCholinergic synapse, the systems of two (or three) nonlinear differential equations will be proposed to find the change of order parameters (ruling modes) such as concentrations of acetylcholine (ACh), acetylcholine-receptor (AChR) complexes, and acetylcholinesterase (AChE). The main findings of our studies are as follows: the linear size of the activation zone is evaluated as the correlation length for a AChR binary mixture (Sects. 4.2.2 and 4.3.2); the stationary states and types of singular points are studied for all models of synaptic transmission (Sects. 4.2.4 and 4.3.3); the nonlinear kinetic model of synaptic transmission with three concentrations of ACh, AChR, and AChE demonstrates a strange-attractorStrange attractor behavior (Sects. 4.2.4 and 4.3.3).