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Проблемы математического моделирования процессов управления популяцией однотипных нейронов

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

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Проблемы математического моделирования процессов управления популяцией однотипных нейронов

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  1. Difficulties in Mathematical Modelling of Control Processes in One-type Neuron PopulationsPokrovsky 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

  2. I. Neurons There are roughlyneurons in a human brain.

  3. Схематическое изображение нейрона Intracellular potential V φ Extracellularpotential

  4. Notations: V - Intracellular potential, φ - Extracellularpotential • Geometrical model of a neuron:geometry graph (tree) Г0 • Branches :lines (ofГ0) . • Nodes: points (nodesofГ0 ). • Electricalmodel of a neuron : • Currents along branchesi(x,t) ; • Currents across branchesthrough surface I(x,t) • Diffusion model: concentrations p(x,t) .

  5. Equationson thebranches (ofgraphГ0): • Conditions in points of branching : 1)continuity byхofV(x,t), p(x,t); • 2)The sum of currentsi(x,t) andfloursp(x,t) into the node is equal zero.

  6. II. Sincitialconnections of neurons. • Fig. 1 [1]. Pores between two axonsandbetween three dendrites. • Arrows – the pores; С – somaof the neuron.El.microscope. Ув. 30000. • [1]. O.S.Sotnikov. Staticsandstructural kineticof living asynaptic dendrites.St.-Petersburg, «NAUKA», 2008. - 397 с.

  7. Fig. 2. Pores (arrows) nearaxon-dendrit synapses. а,б – variantsof structures.El.microscope.Ув. 40000.

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

  9. Fig. 4 • Different inter-neuronal connections: • а – between processes of neurons; • б – betweensoma of neurons; • в – between axon and dendrite in the synapse. Doun:geometrymodels а б в с

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

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

  12. III. Equatios for clustersof neurons • Several neurons with connections by pores are named cluster;denote as Гр . • Geometrymodel – geometricalgraph. • Several neurons with connections by electrical contacts and by poresare named electrical cluster;denote as ГЕ . • Geometrymodel – geometricalgraph.

  13. Equations for Гр (diffusion) Equations forГE (electrical cluster) • Conditions in nodes: 1)continuous by хV(x,t), p(x,t); • 2)Sumof currents i(x,t) and floursp(x,t) , into the node is equal zero.

  14. GraphsГр andГEdiffer!граф Гр к виду ГEи только после этого интегрировать уравнения.

  15. END

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