Difficulties in Mathematical Modelling of Control Processes in One-type Neuron Populations Pokrovsky A.N., Sotnikov O.S. Проблемы математического моделирования процессов управления популяцией однотипных нейронов А.Н.Покровский, О.С.Сотников Санкт-Петербургский гос. университет, Институт физиологии им. И.П. Павлова РАН

I. Neurons There are roughly neurons in a human brain.

Схематическое изображение нейрона V Intracellular potential φ Extracellular potential

Notations: V - Intracellular potential, φ - Extracellular potential Geometrical model of a neuron: geometry graph (tree) Г 0 Branches : lines (of Г 0 ). Nodes: points (nodes of Г 0 ). Electrical model of a neuron : Currents along branches i(x,t) ; Currents across branches through surface I(x,t) Diffusion model: concentrations p(x,t).

Equations on the branches (of graph Г 0 ): Conditions in points of branching : 1) continuity by х of V(x,t), p(x,t); 2) The sum of currents i(x,t) and flours p(x,t) into the node is equal zero.

II. Sincitial connections of neurons. Fig. 1 [1]. Pores between two axons and between three dendrites. Arrows – the pores; С – soma of the neuron. El. microscope. Ув [1]. O. S. Sotnikov. Statics and structural kinetic of living asynaptic dendrites. St.-Petersburg, «NAUKA», с.

Fig. 2. Pores (arrows) near axon-dendrit synapses. а,б – variants of structures. El. microscope. Ув

Fig. 3. Forms of inter-neurons connections. а – chemical synapse; б-в – electrical contacts; г – cito-plasmic sincitium. Arrows – perforations. Down – geometrical model for electrical (б, в, г) and chemical (г) signals.

Fig. 4 Different inter-neuronal connections: а – between processes of neurons; б – between soma of neurons; в – between axon and dendrite in the synapse. б б а а с в в Doun: geometry models

Fig. 5 [1]. One neuron. Faze contrast, об. 20, ок. 10.

Fig. 6 [1]. Contacts of neurons. Faze contrast, об. 20, ок. 10.

III. Equatios for clusters of neurons Several neurons with connections by pores are named cluster; denote as Г р. Geometry model – geometrical graph. Several neurons with connections by electrical contacts and by pores are named electrical cluster; denote as Г Е. Geometry model – geometrical graph.

Equations for Г р (diffusion) Equations for Г E (electrical cluster) Conditions in nodes: 1) continuous by х V(x,t), p(x,t); 2) Sum of currents i(x,t) and flours p(x,t), into the node is equal zero.

Graphs Г р and Г E differ ! граф Г р к виду Г E и только после этого интегрировать уравнения.

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