Information Technology Reference
In-Depth Information
Chapter 1
Modelling Biological Neurons in Terms
of Electrical Circuits
Abstract
The functioning of the cells membrane can be represented as an electric
circuit. To this end: (1) the ion channels are represented as resistors, (2) the gradients
of the ions concentration are represented as voltage sources, (3) the capability of
the membrane for charge storage is represented as a capacitor. It is considered
that the neurons have the shape of a long cylinder, or of a cable with specific
radius. The Hodgkin-Huxley model is obtained from a modification of the cable's
PDE, which describes the change of the voltage along dendrites axis. Cable's
equation comprises as inputs the currents which are developed in the ions channels.
Other models of reduced dimensionality that describe voltage variations along the
neuron's membrane are the FitzHugh-Nagumo model and the Morris-Lecar model.
Cable's equation is also shown to be suitable for describing voltage variations along
dendrites. Finally, the various types of ionic channels across the neurons' membrane
are analyzed.
1.1
The Hodgkin-Huxley Equations
In neural cells there is a voltage difference appearing between the inner and the
outer side of the membrane
V
M
D V
in
V
out
(1.1)
where V
in
is the voltage at the inner part of the cell, V
out
is the voltage at the outer
part of the cell, and V
M
is the membrane potential. The term “resting potential”
describes the membrane's voltage when the cell is in rest and has an approximate
value of 70 mV.
In the neural cell two types of current are developed: The incoming current is
associated with the inflow of Na
C
ions and has as a result to raise the cell's potential
(approaching to 0 mV). The outgoing current is associated with the outflow of K
C
ions, or with the inflow of Cl
ions and has as a result to reduce significantly the
cell's potential.
