'''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.  

'''References.''' 

 
1.  Sotnikov O.S.: Structural dynamix of living asynaptic dendrites. Ed. "Nauka", St-Petersburg, 2008. 397 p. (in 
Rush).  

2.  Radchenko A.N.: INFORMATION MECHANISMS OF THE BRAIN: Associative   memory, Quasiholographic 
properties, EEG-activity, Sleep.   St.-Petersburg, Helicon Plus, 2007. 240 p. (in Rush). 

3.  Pokrovsky A.N.: Two ways of chemical control of  a neuron. In: International Conference DIFFERENTIAL 
EQUATIONS and   TOPOLOGY, ABSTRACTS: 173-174. Moscow, June 17-22, 2008. (in Rush). 

4.  Pokornyi Yu.V., Penkin O.M., Pryadiev V.L., Borovskikh A.V., Lazarev K.P., Shabrov S.A. : Differential 
equations on  geometrical graphs.  FIZMATLIT, Moscow, 2004. 272 p. (in Rush).  

5.  Penkin O.M. : About a geometrical  approach to multistructures and some qualitative prorerties of solutions. { 
Partial   Differential Equations on Multistructures, ed. by F. Ali Mehmeti, J. von Belov and S. Nicaise / Lect. 
Notes Pure Appl. Math.} V. 219.:183-192, 2001. 

6.  Hodgkin A,. Huxley A. A quantitativ description of membrane currents and its application to conduction and 
exitation in  nerve. {J. Physiol.},  117(2): 500, 1952.