Difficulties in Mathematical Modelling of Control Processes in One-type Neuron Populations. Andrey Pokrovsky, Oleg Sotnikov

Abstract. Geometry of a neuron is similar to a tree with branches of different diameters. There is a thin isolating cellular membrane instead of a bark of the tree. The intracellular plasma and extracellular liquid have different electric potentials. There are several types of ionic channels put in cellular membrane: channels controlled by electric field and chemically controlled channels (controlled by mediators). A rise and propagation of neuronal spikes are defined by electrically controlled channels. Chemically controlled channels are means of interactions between neurons. Equations of Hodgkin-Huxley type on geometrical graph-"tree" are used as a mathematical model of electrical state of a neuron. Such (or simplified) models are used in modelling of neural networks up to-date. However, morphology bibliography and experimental data of professor Sotnikov show that some processes of neighbouring neurons have connections like pores or electrical junctions (gap- and tight-junctions) in some structures of nerve system. It means that information is operated on some random neuronal clusters but not on a single neurons. This case of mathematical modelling gives rise to several new problems: 1) modelling of distribution of a number of neurons in a cluster, a model of "average" cluster; 2) generation of blocks of pulses in clusters; 3) modelling of retrograde spreading of pulses by dendrites; 4) functions of off-synaptic receptors and secondary messengers in controlling of dendrites electrical state. There is a fundamental problem to discuss: why pores between dendrites (and, therefore clusters) in some structures are numerous but in neighbouring structures of the same brain they are seldom or never exist (Example: fascia dentata and CA3 field in hippocampus)? What does it mean in the concept of informational mechanisms?

Keywords . Mathematical modelling, control signals, pores, membrane perforation, clusters of neurons, neuron syncytium.

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