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Neuron
A neuron is a nerve cell which receives information from other neurons
via its dendrites. These electrical signals are added together in
the cell nucleus, and then relayed to other neurons via the axon.
The single idealized nerve circuit looks like a low pass RC-circuit
filter.
The variables used to described the simple abstract nerve cell are
current I, voltage V, and resistence R found in Ohm's law, I = V/R,
and also capacitance C found in the RC-circuit equation written as:
C(t) dV(t)/dt - I(t) = 0
The equation above explicitly describes the variable C, V, and I
as functions of time, t.
The biological neuron contains properties of an electrical capacitor
and battery. The capacitor allows alternating current to be filtered, and
the battery enables the storage of electricity. The capacitance, C, of a
nerve circuit is written as:
C = Q/V
where Q is the electrical charge, V is the voltage.
This equation could also be written as:
Q = C V.
You can intuitively feel that the capacitance should go up as the
charge increases. The current in a simple nerve circuit can be written
as the rate of flow of electrical charge:
I(t) = dQ(t) / dt.
Substituting for Q:
I(t) = C(t) dV(t)/dt.
This is the RC-circuit equation. It describes the simple neuron in
general terms except at the synapse where the function C(t) becomes
complex. The description at this complex discontinuity or synapse can
be constructed using the delta function.
Although the neural circuit functions as a low pass filter, it is
still a discontinuous circuit at the synaptic junctions where the circuit
becomes unstable. When a synapse occurs a cascaded electro-chemical
discharge or current flows across the junction. This chaotic
synaptic discharge is controlled by counter forces in the circuit called
the circuit impedance. For simple low frequency electronic circuits
the intrinsic reaction against this discharge are the other capacitors
in the circuit. In a neural network, the reactance again the cascading
synapses are the neurons in the circuit themselves. This is called
neural inhibition [1] in the cell assembly model described by Donald Hebb.
As a note, there's another quantity called inductance which becomes
a factor in high-frequency circuits. In biological networks, the
relative energy output per nerve cell is small. That is, neural
electrical circuits oscillate at a low frequency under 250 hertz [2].
Recent studies of the ultra high gamma waves, probably the fastest
oscillations in the brain with the exception of the ear, have shown
that these relatively high frequency waves are used in neural
synchronization. The potential induction of electrical fields
effecting neighborhood cells in neural circuits in synchronizing
synapses is theoretically possible because circuit frequencies over
60 hertz are high enough for the induction effect (the telephone
tap) to be measurable.
The neural voltages depends upon the synaptic conductances or
capacitance at the dendrites. The firing threshold depends
upon the membrane conductance in the nerve cell nucleus.
There is a more detailed theoretical model of the neuron as a circuit
called the integrate-and-fire [3] model. This model will become more
important as experimentalists are able to measure the pulse trains of
neurons in various context, and build a database of pulse train
sequences emitted by neurons. Since neurons act together to perform
a task, this model would require a coupling of integrate-and-fire equations
to describe multiple neurons acting together.
References
1.
Moshe Abeles, Corticonics; Neural Circuits of the Cerebral Cortex,
Cambridge University Press, 1991.
2.
Magnetic Resonance Imaging of Oscillating Electrical Currents,
Nicholas W. Halpern-Manners, Vikram S. Bajaj, Thomas Z. Teisseyre,
and Alexander Pines,
PNAS, May 11, 2010, vol. 107, no.19
MRI of Oscillating Electrical Currents:
3.
Models of Individual Neurons, Christop Koch, Sep 1997
www.klab.caltech.edu/~koch/biophysics-book/neuron_models.ps
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