Modelling Biological Neurons in Terms of Electrical Circuits - Advanced Models of Neural Networks - page 16

Fig. 1.10 ( a ) Real structure
of a dendrite, ( b )
representation of a dendrite
through its segmentation in
smaller parts, each one
characterized by uniform
potential

a

b

1.7 Modelling Dendrites in Terms of Electrical Circuits

1.7.1 Dendrites

The greater part of the cumulative surface of the membrane in neural cells is
occupied by dendrites. Dendrites permit the creation of connections between
different neurons. Moreover, dendrites direct signals of a specific voltage amplitude
from the neighboring neurons to the soma of a particular neuron. The dendrites, as
parts of the membrane, have ionic channels which open or close according to the
change of the membrane's potential [ 16 , 65 ]. As a final result, dendrites define the
response of the neurons to synaptic inputs (voltages V.x;t/;Fig. 1.10 ).
For the study of the voltage dynamics in dendrites, an established approach is to
partition their tree-like structure in smaller parts which are characterized by spatially
uniform voltage potential, i.e. @V.x;t/

@x D 0. It is considered that there exist differences
in the value and distribution of the voltage V.x;t/, in different parts of the dendrites
(Fig. 1.11 ).
For each smaller part in the dendrite that is characterized by spatially uniform
potential, the major parameters are: the cylinder's radius ˛ i , the length L i ,the
membrane's potential V i , the capacitance (normalized per unit of surface) c i , and
the resistance of the membrane (normalized per unit of surface) r L i .Itisassumed
that in each compartment there is an electrode to which external current I electrode is
applied.
From the analysis of the circuit of the elementary part of the dendrite one has

I cap C I ion D I long C I electrode (1.59)

where I cap and I ion are the capacitance and ionic currents per unit area of membrane
for segment i, and additionally

I cap D C i dV i

dt ;I ion D

V i
r i M

(1.60)