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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
10
There are roughly 10
in a human brain.
neurons
Схематическое изображение нейрона
Intracellular potential
1011
V
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):
i( x ,t )   s( x )Vx ( x ,t ); x   k , k  1,..., K ,
(1)
I ( x ,t )  l 1( x )ix ( x ,t )   l 1( x )( s( x )Vx ( x ,t ))x  I C  I Na  I K  I L  I s .
I C  C( Vt ( x ,t )   t ( x ,t ));
I K  g K p3 q34 ( V    VK );
I Na  g Na q13 q2 ( V    VNa );
I L  g L ( V    VL );
( 3)
I s    ( x  x ) gs ( x ,t  t ,n )( V ( x ,t  t ,n )   ( x ,t  t ,n )  Vs ) .

(2)
(4)
t ,n
( qi ( x ,t ))t  i ( V ( x ,t )   ( x ,t ))  [ i ( V ( x ,t )   ( x ,t ))  i ( V ( x ,t )   ( x ,t ))] qi ( x ,t ) .
( p1 ) t  D1 ( s( x)( p1 ) x ) x   1 p1     ( x  x )c( x , t ).
( p2 )t  D2 (s( x)( p2 ) x ) x   2 p2  p1;
•
•
p3  f ( p2 ( x, t ) ,
( 5)
(6)
(7)
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.
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;
Doun: geometry
models
• в – between axon and dendrite in the synapse.
а
б
с
в
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)
( p1 ) t  D1 ( s( x)( p1 ) x ) x   1 p1     ( x  x )c( x , t ).
( p2 )t  D2 (s( x)( p2 ) x ) x   2 p2  p1;
(6)
p3  f ( p2 ( x, t ) ,
(7)
Equations for ГE (electrical cluster)
i( x ,t )   s( x )Vx ( x ,t ); x   k , k  1,..., K ,
I ( x ,t )  l 1( x )ix ( x ,t )   l 1( x )( s( x )Vx ( x ,t ))x  I C  I Na  I K  I L  I s .
I C  C( Vt ( x ,t )   t ( x ,t ));
I Na  g Na q13 q2 ( V    VNa );
I K  g K p3 q34 ( V    VK );
I L  g L ( V    VL );
I s    ( x  x ) gs ( x ,t  t ,n )( V ( x ,t  t ,n )   ( x ,t  t ,n )  Vs ) .

(2)
( 3)
(4)
t ,n
( qi ( x ,t ))t  i ( V ( x ,t )   ( x ,t ))  [ i ( V ( x ,t )   ( x ,t ))  i ( V ( x ,t )   ( x ,t ))] qi ( x ,t ) .
•
•
(1)
( 5)
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
граф Гр к виду
ГE
and
ГE
differ !
и только после этого интегрировать уравнения.
END