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

I. Neurons

There are roughly neurons in a human brain.

1010

Схематическое изображение нейрона

1110V

Intracellular potential

φ

Extracellular potential

Notations: V - Intracellular potential, φ - Extracellular potential

• Geometrical model of a neuron: geometry graph (tree) Г0

• Branches : lines (of Г0) .

• Nodes: points (nodes of Г0 ).

• 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 Г0):

• 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.

)(,K,...,k,x);t,x(V)x(s)t,x(i kx 11

)(.IIIII))t,x(V)x(s)(x(l)t,x(i)x(l)t,x(I sLKNaCxxx 211

)();VV(gI);VV(qpgI

);VV(qqgI));t,x()t,x(V(CI

LLLKKK

NaNaNattC

3433

231

)(.)V)tt,x()tt,x(V)(tt,x(g)xx(I sn,n,n,t

ssn,

4

)(.)t,x(q))]t,x()t,x(V())t,x()t,x(V([))t,x()t,x(V())t,x(q( iiiiti 5

)6(.),()()))((()(11111

txcxxppxsDpxxt

)7(,),((;)))((()( 23122222 txpfppppxsDp xxt

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. 30000.

• [1]. O.S. Sotnikov. Statics and structural kinetic of living asynaptic dendrites. St.-Petersburg, «NAUKA», 2008. - 397 с.

Fig. 2. Pores (arrows) near axon-dendrit synapses. а,б – variants of structures. El. microscope. Ув. 40000.

• 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.

ббаа с вв

Doun: geometry models

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 Гр .

• Geometry model – geometrical graph.

• Several neurons with connections by electrical contacts and by pores are named electrical cluster; denote as ГЕ .

• Geometry model – geometrical graph.

Equations for Гр (diffusion)

Equations for ГE (electrical cluster) )(,K,...,k,x);t,x(V)x(s)t,x(i kx 11

)(.IIIII))t,x(V)x(s)(x(l)t,x(i)x(l)t,x(I sLKNaCxxx 211

)(.)V)tt,x()tt,x(V)(tt,x(g)xx(I sn,n,n,t

ssn,

4

)(.)t,x(q))]t,x()t,x(V())t,x()t,x(V([))t,x()t,x(V())t,x(q( iiiiti 5

)7(,),((;)))((()( 23122222 txpfppppxsDp xxt

)6(.),()()))((()(11111

txcxxppxsDpxxt

)();VV(gI);VV(qpgI

);VV(qqgI));t,x()t,x(V(CI

LLLKKK

NaNaNattC

3433

231

• 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 Гр and ГE differ !

граф Гр к виду ГE и только после этого интегрировать уравнения.

END