Gobierno de
la ciudad de Buenos Aires
Hospital Neuropsiquiátrico
"Dr. José Tiburcio Borda"
Laboratorio
de Investigaciones Electroneurobiológicas
y
Revista
Electroneurobiología
ISSN: 0328-0446
The Nervous Principle:
Active versus
Passive Electric Processes In Neurons
by
Danko Dimchev Georgiev, M.D.
Division
of Electron Microscopy, Medical University of Varna, Varna 9000, BULGARIA
Correspondencia / Contact: dankomed@SoftHome.net
Electroneurobiología 2004; 12 (2), pp. 169-230; URL: http://electroneubio.secyt.gov.ar/index2.htm
Copyright
© 2004 del autor / by the author. Esta es una investigación
de acceso público; su copia exacta y redistribución por cualquier medio están
permitidas bajo la condición de conservar esta noticia y la referencia completa
a su publicación incluyendo la URL original (ver arriba). / This is an Open
Access article: verbatim copying and redistribution of this article are
permitted in all media for any purpose, provided this notice is preserved along
with the article's full citation and original URL (above).
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Abstract
This essay presents in
the first section a comprehensive introduction to classical electrodynamics. The
reader is acquainted with some basic concepts like right-handed coordinate
system, vector calculus, particle and field fluxes, and learns how to calculate
electric and magnetic field strengths in different neuronal compartments.
Then the exposition comes to explain the basic difference
between a passive and an active neural electric process; a brief historical
perspective on the nervous principle is also provided. A thorough description is
supplied of the nonlinear mechanism generating action potentials in different
compartments, with focus on dendritic electroneurobiology. Concurrently, the
electric field intensity and magnetic flux density are estimated for each
neuronal compartment.
Observations are then discussed, succinctly as the
calculated results and experimental data square. Local neuronal magnetic flux
density is less than 1/300 of the Earth’s magnetic
field, explaining why any neuronal magnetic signal would be suffocated by the
surrounding noise. In contrast the electric field carries biologically important
information and thus, as it is well known, acts upon voltage-gated transmembrane
ion channels that generate neuronal action potentials. Though the transmembrane
difference in electric field intensity climbs to ten million volts per meter,
the intensity of the electric field is estimated to be only ten volts per meter
inside the neuronal cytoplasm.
Principio del funcionamiento nervioso: oposición de procesos eléctricos activos y pasivos en las neuronas
Sumario
Este trabajo presenta en su primera
sección una introducción general a la electrodinámica clásica. Cubre los
temas electroneurobiológicos introductorios de la mayoría de los cursos de
neurociencias, asegurando ante todo la familiaridad del estudiante con los
sistemas de coordenadas a mano derecha, así como con el cálculo de vectores y
con los flujos de partículas y de campo. En esta sección el lector aprende a
calcular las intensidades de campo eléctricas y magnéticas dentro de los
diferentes compartimientos neuronales.
Luego la exposición se aboca a
explicar la esencial diferencia entre procesos neuroeléctricos pasivos y
activos; se provee también una breve perspectiva histórica sobre el principio
o fundamento de la función neural. Proporciónase una descripción detallada de
los mecanismos no lineares que generan potenciales de acción en los diferentes
compartimientos, con énfasis en la electroneurobiología dendrítica.
Concurrentemente se estiman la intensidad de campo eléctrico y la densidad de
flujo magnético para cada compartimiento neuronal.
Las observaciones son entonces analizadas, sucintamente por cuanto los resultados calculados cuadran bien con los datos experimentales. La densidad local del flujo magnético es menos que 1/300 de la del campo magnético terrestre, lo que explica por qué cualquier señal magnética útil es sofocada por el ruido ambiental. En contraste, el campo eléctrico porta información biológicamente relevante y, como es muy bien sabido, actúa sobre canales iónicos transmembranales abiertos y cerrados por voltaje, que controlan el potencial de acción de la célula. Aunque la diferencia en la intensidad del campo eléctrico a través de la membrana asciende a diez millones de voltios por metro y aun más, la intensidad del campo eléctrico se estima en sólo diez voltios por metro dentro del citoplasma neuronal.
Нейронный принцип: сравнительное описание активных и пассивных электрическиx процесcов
Резюме
Первая
часть
представляет
собой
краткое
введение в
класическую
электродинамику.
Здесь
приведены
общие
положения,
излагаемые
во многих
учебниках
физики:
правоориентированная
координатная
система,
векторноe
исчисление,
поток из
частиц и
полевый
поток. В этой
части
читатель
знакомится
со способами
вычисления
интенсивности
электрического
и магнитного
поля в
различных
структурах
нервной
клетки.
Во
второй части
объясняется
различие
между
активными и
пассивными
электрическими
процессами в
нейронах. Эта
проблема
рассматривается
также в
историческом
аспекте.
Представлено
подробное
описание
нелинейных
механизмов
генерации
действующих
потенциалов
в отдельных
структура
нервной
клетки, и
особенно
электронейробиологии
дендритов.
Для каждого
органа
клетки (дендриты,
сома, аксон)
вычислены
интенсивности
электрического
и магнитного
поля.
Полученные результаты соответствуют экспериментальным данным. Плотность локального магнитого потока нейронов составляет менее 1/300 плотности магнитного потока Земли. Поэтому шум среды подавляет магнитный сигнал нейрона. Напротив, электрическое поле несет биологически значимую информацию и оказывает влияние на зависящие от разности потенциалов ионные каналы которые генерируют действующие потенциалы нейронов. Несмотря на то, что трансмембранное электрическое поле достигает 10 миллионов В/м, в нейронной протоплазме интенсивность электрического поля составляет лишь 10 В/м.
Table of Contents
1.1 Right-handed coordinate systems
2 Electric and magnetic fields in neurons
2.1 Passive electric properties – cable equation
2.1.1 Spread of voltage in space and time
2.1.2 Assessment of the electric field intensity
2.1.3 Propagation of local electric currents
2.2 Active electric properties – the action potential
2.2.1 Nernst equation and diffusion potentials
2.2.2 Resting membrane potential
2.2.3 Generation of the action potential
2.3.1 Electric intensity in dendritic cytoplasm
2.3.2 Electric currents in dendrites
2.3.3 Magnetic flux density in dendritic cytoplasm
2.3.4 Active dendritic properties
2.5.1 The Hodgkin-Huxley model of axonal firing
2.5.2 Passive axonal properties
2.5.3 Electric intensity in the axonal cytoplasm
2.5.4 Magnetic flux density in axonal cytoplasm
2.6 Electric fields in membranes
In order to investigate the electromagnetic field structure
in neurons it behooves to be acquainted with the basic mathematical definitions
and physical postulates in classical electrodynamics. Before anything else, it
is worth pointing out that a quantity is either a vector or a scalar. Scalars
are quantities fully described by a magnitude alone. Vectors are quantities
fully described by both a magnitude and a direction. Because we will work mostly
with vectors we have to define what is positive normal to a given surface s,
what is the positive direction of a given contour Γ and what is a
right-handed coordinate system.
Right-handed coordinate system Oxyz is such a system in
which if the z-axis points toward your face the counterclockwise rotation of the
Ox axis to the Oy axis has the shortest possible path. The positive normal +n of
given surface s closed by contour Γ is collinear with the Oz axis of
right-handed coordinate system Oxyz whose x- and y-axis lie in the plane of the
surface. The positive direction of the contour Γ is the direction in which
the rotation of x-axis to the y-axis has the shortest possible path (Zlatev,
1972).
FIG 1 Left: Direction of the positive normal +n and the positive
direction of the contour Γ. Right: Right handed coordinate system Oxyz.
After that, for working with vectors it should be noted
that there are two types of multiplication of vectors - the dot product and the
cross product. Geometrically, the dot product of two vectors is the magnitude of
one times the projection of the other along the first. The symbol used to
represent this operation is a small dot at middle height (·), which is where
the name dot product comes from. Since this product has magnitude only, it is
also known as the scalar product:
where b
is the angle between the two vectors.
Geometrically, the cross product of two vectors is the area
of the parallelogram between them. The symbol used to represent this operation
is a large diagonal cross (×), which is where the name cross product comes
from. Since this product has magnitude and direction, it is also known as the
vector product:
where the vector 
is a unit vector perpendicular to
the plane formed by the two vectors. The direction of
is determined by the right hand
rule.
The right hand rule says that if you hold your right hand
out flat with your fingers pointing in the direction of the first vector and
orient your palm so that you can fold your fingers in the direction of the
second vector, then your thumb will point in the direction of the cross product.
The gradient 
is a vector operator called Del or
Nabla (Morse & Feshbach, 1953; Arfken,
1985; Kaplan, 1991; Schey,
1997). It is denoted as:
f = grad(f)
The
gradient vector is pointing toward the higher values of f, with magnitude equal
to the rate of change of values. The direction of f is the orientation in which the directional derivative has the largest value
and |f| is the value of that directional derivative. The directional derivative
uf(x0,y0,z0) is the rate at which
the function f(x,y,z) changes at a point (x0,y0,z0)
in the direction u.
and 
is the unit vector (Weisstein,
2003).
The particle flux is a scalar physical quantity
defined by the expression:
,
where
dV denotes a volume segment with
length dl that is filled with fluid that for
time
dt passes with velocity v trough any
cross section s of
dV; β is the is the angle between the vectors 
and 
. It is worth to remind that
has the direction of the positive
normal +n, and its magnitude is proportional to the surface area s. Simple
substitution of the expression for
dV
into the expression for 
gives us
Thus
we have obtained that the particle flux is a scalar product of two vectors - the
particle velocity vector
and the surface vector
. In electrodynamics, ion currents in electrolytes and the currents composed of
electrons are the particle fluxes of top neurobiological interest, but
quasiparticles such as solitons and phonons are also modeled.
We
could define an analogous scalar quantity when we investigate physical fields,
e.g. the field of electromagnetic force or electromagnetic field. There we can
define the field flux as a scalar product of the field intensity 
through surface .
The
electromagnetic force field is composed from the forces of electric and magnetic
fields, whose different causal actions can be nonetheless described as mediated
by a single sort of microphysical change-causing energy packets, called photons.
Taking the photons’ action collectively – like as floods can be described by
neglecting the swervings of individual water molecules – the electric field
could be described via the vector field of electric intensity
. Electric intensity is defined as the ratio of the electric force 
acting upon a charged body and the
charge q of the body:
It should be noted that the electric field is a potential
field – that is, the work WΓ along closed contour Γ with
any length l is zero:
Every point in the electric field has an electric potential
V defined with the specific (for unit charge) work needed to carry a charge from
this point to infinity. The electric potential of point c of a given electric
field has potential V defined by:
where V∞= K = 0. The electric potential
difference between two points 1 and 2 defines voltage V, whose synonyms are
electric potential, electromotive force, potential, potential difference, and
potential drop:
The link between the electric intensity
and the gradient of the voltage V is:
Another vector, not directly measurable, that describes the
electric field is the vector of electric induction
. For isotropic dielectrics electric induction is defined as:
where ε is the electric permittivity of the
dielectric. The electric permittivity of the vacuum is denoted as ε0 =
8.84×10-12 F/m.
The Maxwell’s law for the electric flux ФD
of the vector of electric induction says that ФD
through any closed surface s is equal to the located in the space region s
charge q that excites the electric field. This could be expressed
mathematically:
If the normal +n of the surface s and the vector
form angle b,
then the flux ФD could be defined as:
FIG 2 The flux ФD of the vector of the
electric induction
through surface s.
From the Maxwell’s law we could easily derive the
Gauss’ theorem:
where ФE is the flux of the vector of the
electric intensity
through the closed surface s.
It is important to note that the full electric flux ФD
could be concentrated only in a small region Δs of the closed surface s, so
in such cases we could approximate:
In other cases, when the electric field is not concentrated
in such a small region but we are interested in knowing the partial electric
flux ΔФD through partial surface Δs for which is
responsible electric charge Δq, it is appropriate to use the formula:
The electric current
i
, that is the flux of physical charges, could be defined by using both scalar
and vector quantities (Zlatev,
1972):
where
is the density of the electric
current. As a scalar quantity the current density J is defined by the following
formula:
where sn is the cross section of the current
flux ФJ. It is useful to note that usually by means of i it is
denoted the flow of positive charges. In the description, the flow of negative
charges could be easily replaced by a positive current with equal magnitude but
opposite direction. Sometimes, however, we would like to underline the nature of
the charges in the current. To this purpose we will use vectors with indices,
e.g.
or
, where the direction of the vectors coincides with the direction of motion of
the negative or positive charges.
If we have a cable and a current flowing through it,
according to Ohm's law the current i is proportional to the voltage V and
conductance G and inversely proportional to the resistance R:
where ρ is the specific resistance for
the media, γ is the specific conductance, l is the length of the
cable and s is its cross section.
The
magnetic field is the second component of the electromagnetic field and is
described by the vector of magnetic induction
(also known as: magnetic field
strength or magnetic flux density) that is perpendicular to the vector of the
electric intensity
. The magnetic field does only act on moving charges. It manifests itself via
the magnetic force
acting upon flowing currents inside
the region where the magnetic field is distributed. From Laplace’s law it is
known that the magnetic force
, which acts upon an electric current-conveying cable immersed in a magnetic
field with magnetic induction
, is equal to the vector product:
If we have a magnetic dipole, the direction of the vector
of magnetic induction is from the south pole (S) to the north pole (N) inside
the dipole – and from N to S outside it.
The magnetic field could be excited either via changes in
an existing electric field
or by a flowing electric current
i.
In the first case the magnetic induction is defined by the Ampere’s law (in
case J = 0):
In the second case, if we have a cable with current i, it
will generate a magnetic field with magnetic induction
whose lines of force have the
direction of rotation of a right-handed screw piercing in the direction of the
current i.
FIG 3 Direction of the lines of magnetic induction around the
path axis with current (a) and along the axis of contour with current (b). The
current i by convention denotes the flux of positive charges.
The
total electromagnetic field manifests itself with a resultant electromagnetic
force
defined by the Coulomb-Lorentz
formula:
where
is the velocity of the charge q.
If we have magnetically isotropic media, then we could
define another vector describing the magnetic field. It is called magnetic
intensity
, tantamount to
where μ is the magnetic permeability of the media. The
magnetic permeability of the vacuum is denoted with μ0 = 4π×10-7
H/m.
The circulation of the vector of magnetic intensity along
the closed contour Γ1 with length l, which interweaves in its
core the contour Γ2 with current i flowing through Γ2,
is defined by the formula:
It can be seen that the magnetic field is a non-potential
field, since the lines of field intensity
are closed and do always interweave
the contour with the excitatory current i. The circulation of the vector
will be zero only along the closed
contours which do not interweave in their cores any current i (Zlatev,
1972).
FIG 4 The circulation of the vector of magnetic intensity
along the closed contour Γ1
equals the current i flowing through the interweaved contour Γ2.
Analogously to defining the flux ФD of the
vector of the electric induction
we can define the flux ФB
of the vector of the magnetic induction
:
It is useful to know that the change in magnetic flux
generates induced voltage V according to the Lenz’s law:
Thus the Lenz’s law shows that there will be induced
voltage (and therefore electric current) if there is a static cable inside
changing magnetic field:
or if the cable is moving inside a static magnetic field:
The full magnetic flux ФL of the magnetic
field self-induced by a contour with current i is called self-induced flux. The
self-induced flux is a linear function of the current:
where L is scalar known as self-inductance and depends only
on the magnetic permittivity μ of the media and the geometric parameters
ΠL that determine the size and the shape of the contour:
L
= f (m
, PL)
Self-induced voltage appears on electric wires every time
that there is a change of the current i – and this self-induced voltage
opposes to the change of the current:
We have up till now presented the basic principles of
electromagnetism. In order to summarize them it is useful to write down the
Maxwell’s equations. Although these equations have been worked out more than a
century ago they present in concise form the whole electrodynamics. We will
consider two cases: (i) in the absence of magnetic or polarizable media and (ii)
with magnetic and/or polarizable media.
In absence of magnetic or polarizable media the equations
can be written in both forms, i.e. in integral or differential form. They will
be listed on the following table.
Table 1
Maxwell’s equations in the absence of magnetic or polarizable media.
|
Laws |
Integral form |
Differential form |
|
Gauss’ law for electricity |