Download A comparision of Hodgkin-Huxley and soliton neural theories

Adv. Radio Sci., 8, 75–79, 2010
www.adv-radio-sci.net/8/75/2010/
doi:10.5194/ars-8-75-2010
© Author(s) 2010. CC Attribution 3.0 License.
Advances in
Radio Science
A comparision of Hodgkin-Huxley and soliton neural theories
R. Appali, S. Petersen, and U. van Rienen
Institute of General Electrical Engineering, Chair of Electromagnetic Field Theory, University of Rostock,
Justus-von-Liebig-Weg 2, 18059 Rostock, Germany
Abstract. Hodgkin and Huxley were the pioneers to abstract
biological neuron as an electric circuit and nerve signal as
the voltage impulse. The Hodgkin-Huxley theory (Hodgkin
and Huxley, 1952) has set the direction and defined the goals
for much of the ensuing research in biophysics. However,
in 2005, T. Heimburg and A. D. Jackson, biophysicists from
Copenhagen proposed a new neural theory called Soliton theory (Heimburg and Jackson, 2005). In this theory, the nerve
conduction is proposed as a density wave.
In this paper, Hodgkin-Huxley and Soliton theories are described and a theoretical comparison has been carried out
throughout the analysis of the theories and models.
1
Introduction
Nerves are the information transmitter lines from the brain to
organs. The nerves are described as the bundle of discrete individual cells called neurons. These individual cells transmit
the information as signals.
The main parts of a neuron are the cell body (soma), signal receiving ends (dendrites), signal transmitter (axon), and
connecting ends to other neurons or glands (synapses). The
scientific model of the neuron was first given in electrical
terms by Hodgkin and Huxley (Hodgkin and Huxley, 1952).
Hodgkin and Huxley conducted the experiments on nonmyelinated giant axon of squid to study the neuronal properties. The Hodgkin-Huxley equations (Hodgkin and Huxley,
1952) are the starting point for detailed neuronal models.
2
The Hodgkin-Huxley theory
The theory was the summation of active experimental and
theoretical association of Hodgkin and Huxley. Their five
publications in 1952 in the Journal of Physiology are quite
Correspondence to: R. Appali
([email protected])
renowned. The four key research developments which led to
the theory were: (Häusser, 2000)
– Firstly, Cole and Curtis demonstrated that the action potential is associated with a large increase in membrane
conductance (Cole and Curtis, 1939).
– Secondly, Hodgkin and Huxley made the first intracellular recording of an action potential (Hodgkin and Huxley, 1952). This demonstrated that the action potential exceeds zero mV, denying Bernstein’s hypothesis
(Bernstein, 1902) that the underlying increase in membrane permeability is non-selective.
– Hodgkin and Katz (Hodgkin and Katz, 1949) explained
the overshooting action potential as the result of an increase in sodium permeability approving the work of
Overton (Overton, 1902).
– Finally, Hodgkin, Huxley and Katz developed the
voltage-clamp circuit to facilitate quantitative measurement of ionic currents from squid axon.
The step depolarization of squid axon triggering an inward
current followed by an outward current was then proved by
Hodgkin and Huxley. With the aid of ionic substitution, they
demonstrated that this net current could be separated into
two distinct components, a fast inward current carried by
Na+ ions, and a more slowly activating outward current carried by K+ ions. Using ingenious voltage-clamp protocols,
they concluded that these two currents result from independent permeability mechanisms for Na+ and K+ with conductance changing as a function of time and membrane potential.
This striking conceptual breakthrough was then termed as the
“ionic hypothesis” (Häusser, 2000).
The three different types of ion current, viz., sodium,
potassium, and a leak current that consists mainly of Cl−
ions contribute to the voltage signal in the neuron. The flow
of these ions through the cell membrane is controlled by their
respective voltage-dependent ion channels. The leak current
takes care of other channel types which are not described in
particular. The most remarkable achievement of this theory
Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.
the generality
of thehowever,
approach.
Thus,
ncentration
in extracellular
The Nernst
Huxley,
1952), liquid.
the very
first complete indicating
respectively.
Normally,
some
of the
the
H-H
model
links
the
microscopic
level
of
ion
tential generated
by
the
difference
in
ion
description of the excitability of single neuron.
channels are blocked. The probability that a
channelschannel
to the macroscopic
of currents
and
ncentrations is represented by a battery. The
is open is level
described
by additional
76
R.
Appali
et
al.:
A
comparision
of
Hodgkin-Huxley
and
soliton
neural
theories
action potentials
(see
variables m,
n, fig.
and 4).
h. The combined action of m
uivalent circuit
is shown
2.1 The
Model:in fig. 2.
and h controls the Na+ channels. The K+ gates
are controlled by n. Hodgkin and Huxley
formulated the three ionic components as
Outside
Na+
du
C = gNam3h( u ENa ) + gK n4 ( u EK )
+
C
+
+
+
+
+
dt
g
g
Na
k
gL
+ gL ( u EL ) + I( t ).
V
(1)
Inside
K+
The parameters ENa, EK, and EL are the reversal
potentials. Reversal potentials and conductances
are empirical parameters. So, it is demonstrated
EL
ENa
Ek
Fig. 1. Schematic model of Hodgkin-Huxley theory (modified from
that the neurons transmit information by firing
Figure 1: Schematic model of Hodgkin-Huxley
Gerstner et al., 2002).
and propagating electrical action potentials along
theory (modified from (Gerstner et al., 2002))
their axons. Usually, motor neurons possess a
Intracellular
sheathpotential
around their
axonsfrom
actingthe
as an
The Hodgkin-Huxley model (H-H model) can be Figure myelin
3: Action
adapted
Fig. 3. Action
potential
adapted
from
the
original
paper
of
Hodgkin
insulator and enhancing the speed of signal
the help
of fig.model
1. The semi- original
gure 2: explained
Equivalentwith
circuit
of H-H
paper of Hodgkin and Huxley
and Huxley.
transmission.
permeable
cell
membrane
separates
the
interior
odgkin and Huxley, 1952)
The model can reproduce and explain a wide
of the cell from the extracellular liquid and acts
acterized
by their
resistance
equivalently,
by their
con- and
range
of data
fromor,
squid
axon like
the shape
as a capacitor. If an input current I(t) is injected
ductance.
The
leakage
channel
is
described
by
a
voltagepropagation of the action potential (see fig. 3), its
into the cell, it may add further charge on the
independent
the conductance
of the
other
2
sharpconductance
threshold,andrefractory
period,
anode-break
capacitor, or leak through the channels in the cell ion channels
is voltage and time dependent.
excitation, accommodation and sub threshold
membrane. Because of active ion transport
If all channels are open, they transmit currents with a maxoscillations.
can also
describe
through the cell membrane, the ion concentration imum conductance
gNaThe
or gKmodel
, respectively.
Normally,
how- many
channel
types with
smallTheparameter
changes
of the channels
are blocked.
probability that
inside the cell is different from that of the ion ever, some
a
channel
is
open
is
described
by
additional
variables
m,
n,Thus,
indicating
the
generality
of
the
approach.
concentration in extracellular liquid. The Nernst
+
and
h.
The
combined
action
of
m
and
h
controls
the
Na
the H-H +model links the microscopic level of ion
potential generated by the difference in ion
channels. The K togates
controlled bylevel
n. Hodgkin
and and
the are
macroscopic
of currents
concentrations is represented by a battery. The Huxleychannels
formulated the three ionic components as
action potentials (see fig. 4).
equivalent circuit is shown in fig. 2.
I
Extracellular
Extracellular
I of H-H model (Hodgkin and Huxley,
Fig. 2. Equivalent circuit
1952).
C
was the empirical representation of the experimental data in
gNa
gk 1952), the very
a quantitativegLmodel (Hodgkin
and Huxley,
first complete description of the excitability of single neuron.
V
2.1
The model
EL
ENa (H-H model) Ecan
The Hodgkin-Huxley
model
k be explained
with the help of Fig. 1. The semi-permeable cell membrane
separates the interior of the cell from the extracellular liquid
and acts as a capacitor. If an input current I (t) is injected into
Intracellular
the cell, it may add further charge on the capacitor C, or leak
through the channels in the cell membrane. Because of active
Figure
Equivalent
circuit of
H-H
model
ion
transport2:through
the cell membrane,
the ion
concentra(Hodgkin
1952)
tion
inside theand
cellHuxley,
is different
from that of the ion concentration in extracellular liquid. The Nernst potential generated
by the difference in ion concentrations is represented by a
battery. The equivalent circuit is shown in Fig. 2.
As mentioned above, the Hodgkin-Huxley model describes three types of channels. All channels may be charAdv. Radio Sci., 8, 75–79, 2010
C
du
= gNa m3 h(u − ENa ) + gK n4 (u − EK )
dt
+gL (u − EL ) + I (t).
(1)
The parameters ENa , EK , and EL are the reversal potentials.
Reversal potentials and conductances are empirical parameters. So, it is demonstrated that the neurons transmit information by firing and propagating electrical action potentials
along their axons. Usually, motor neurons possess a myelin
sheath around their axons acting as an insulator and enhancing the speed of signal transmission.
The model can reproduce and explain a wide range of data
from squid axon like the shape and propagation of the action potential (see Fig. 3), its sharp threshold, refractory period, anode-break excitation, accommodation and sub threshold oscillations. The model can also describe many channel
Figure 3: Action potential adapted from
types with small parameter changes indicating the generality
original Thus,
paperthe
ofH-H
Hodgkin
and the
Huxley
of the approach.
model links
microscopic
level of ion channels to the macroscopic level of currents and
action potentials (see Fig. 4).
2
www.adv-radio-sci.net/8/75/2010/
the
for the generation and propagation of solitons.
Pure lipid membranes exhibit “non-linearity” (:
T. Heimburg and A. D. Jackson proposed an
maximally compressible close to the transition
alternative theory in 2005. The theory is a
and decreased compressibility with increasing
theoryof Hodgkin-Huxley
of nerve and
pulse
R. thermodynamic
Appali et al.: A comparision
soliton neural theories
77
density
or
pressure)
and
“dispersion”
(:
lipid
propagation. This theory is based on the lipid
compressibility
is
frequency
dependent
both
for
characteristic of transition from a fluid to a gel
area and volume compressibility) (Heimburg and
state over a range of several degrees, slightly
Jackson, 2005; Heimburg and Jackson, 2007a).
below the body temperature (Heimburg and
Both the effects of non-linearity and dispersion
Jackson, 2005 and references there in). That is,
produce a self-sustaining and localized density
when a biological membrane is compressed, the
pulse with a moving segment of the nerve
fluid membrane gradually reaches to a denser gel
membrane in the gel state (see fig. 6). This pulse
state. Now, a sufficient pressure/ compression
is called a soliton (Heimburg and Jackson,
either in membrane area or volume will bring a
2007a). The solitary wave maintains its shape
transition to the gel state (Heimburg, 2007a).
while it travels at a constant speed less than the
This transition is associated with the release of
sound velocity in the lipid membrane. Solitons
heat (Heimburg, 2007a; Heimburg, 2007b). The
Figure
4:
Separation
of
ionic
conductances
underlying
the
action
in H-H
model.
can potential
propagate
over
longModified
distances without loss of
relation between
heat underlying
capacity,the action
density,
Fig. 4. Separation
of ionic
conductances
potential in H-H model. Modified from Hodgkin and Huxley (1952).
from
(Hodgkin
and Huxley,
1952)
energy (Heimburg and Jackson, 2005). The pulse
compression and temperature are shown in fig. 5.
is also called
adiabatic
pulse, because no
Lipid membranes
have the as
properties
required
3. Soliton Theory
for the energy
generation is
and propagation
lost to of
thesolitons.
environment during
Pure lipid membranes exhibit “non-linearity” (:
T. Heimburg and A. D. Jackson proposed an
propagation (Heimburg and Jackson, 2007b).
maximally compressible close to the transition
alternative theory in 2005. The theory is a
The soliton
modewithofincreasing
pulse propagation is
and decreased
compressibility
thermodynamic theory of nerve pulse
density
or
pressure)
and
“dispersion”
(:
propagation. This theory is based on the lipid
basically different from the lipid
AP given by the H-H
compressibility is frequency dependent both for
characteristic of transition from a fluid to a gel
model
(Anderson
et
al.,
2009).
area and volume compressibility) (Heimburg and
state over a range of several degrees, slightly
is associated
with a transient
Jackson,The
2005;solitary
Heimburgwave
and Jackson,
2007a).
below the body temperature (Heimburg and
Both
the
effects
of
non-linearity
and
dispersion
Jackson, 2005 and references there in). That is,
voltage change due to the transient change in
produce a self-sustaining and localized density
when a biological membrane is compressed, the
statea of
membrane.
This
the membrane
pulse with
moving
segment of
the affects
nerve
fluid membrane gradually reaches to a denser gel
potential
difference.
Especially,
it
changes both
membrane in the gel state (see fig. 6). This pulse
state. Now, a sufficient pressure/ compression
is
called
a
soliton
(Heimburg
and
Jackson,
either in membrane area or volume will bring a
the area and thickness of membrane and thereby
2007a). The solitary wave maintains its shape
transition to the gel state (Heimburg, 2007a).
(Heimburg
2007a).
while itcapacitance
travels at a constant
speed lessand
thanJackson,
the
This transition is associated with the release of
These inchanges
to be
as a voltage pulse and
sound velocity
the lipid seem
membrane.
Solitons
heat (Heimburg, 2007a; Heimburg, 2007b). The
can propagate
without loss
of
relation between heat capacity, density,
leads overtolonga distances
capacitive
current
due to the
energy (Heimburg and Jackson, 2005). The pulse
compression and temperature are shown in fig. 5.
unbalanced
charge
of
membrane.
The
reversible
is also model
called ofasmembrane
adiabatic lipid
pulse,
because no theory (adapted
Figure 6: Schematic
thermodynamic
from (Anderso
observed
during
action
potential
is proposed
energy
is lost
tomodel
the of
environment
during
Fig. 6.heat
Schematic
membrane
lipid
thermodynamic
theory
propagation
This theoryenthalpy
also explainsi.e.
the effect of
Thus the
heat flow
is correlated
with theand
changes
(adapted
Anderson
et
2009).
as from
a(Heimburg
result
ofal.,Jackson,
lipid 2007b).
melting
Figure
Heatlateral
capacity,
lateral
and
Fig.
5. Heat 5:
capacity,
density and
lateral density
compressibility
The soliton
mode of pulse propagationon is
the nerve pulse propagation. Acco
in membrane density
and potential.
production
of
latent
heat
during
lipid
transition
as Heimburg
a function
of basically different from the AP given by the theory,
as lateral
a function ofcompressibility
temperature (adapted from
and Jackson,
H-H
the anesthetics in the memb
from
fluid
to
gel
state
and
the fluid-gel
reabsorption
of temp
2005).
model
(Anderson
et
al.,
2009).
temperature (adapted from (Heimburg
and
the
transition
3.1 Soliton equation
TheLipid
solitary
wave
isthe
associated
with
a transient
membranes
have
the
properties
required
for
the
genphenomenon
known
as
freezing
poin
heat
as
system
returns
to
the
fluid
state
Jackson, 2005)
change
due
to the transient
in lipid membranes
(Heimburg
and Jackson, 2007c). So i
eration(Anderson
and
propagation
of2009).
Pure
Hydrodynamic voltage
equation
for the
propagation
ofsolitons.
a change
et
al.,
state of membrane. This affects the membrane
to force the
membrane
density pulse in exhibit
the presence of dispersion
for a
(maximally
compressible
close
to the through th
potential “non-linearity”
difference. Especially,
it changes
both
transition,
thereby
action potentia
cylindrical membrane
along
x-axis
is
(Heimburg
transition
decreased
compressibility
with increasing the
denthe
area andand
thickness
of membrane
and thereby
is suppressed. It is noted that fre
T. Heimburg and A. D. Jackson proposed an alternative
the- 2005)
and Jackson,
given
by
sity or pressure)
and
(lipid compressibility is
capacitance
(Heimburg
and“dispersion”
Jackson, 2007a).
depression is a general ther
ory in 2005. The theory is a thermodynamic theory of nerve
These
changes
seem to beboth
as a for
voltage
and
frequency
dependent
areapulse
and phenomenon
volume
compressibil3
and not chemical
2
4
pulse propagation. This theory is based on the lipid
leads
to
a
capacitive
current
due
to
the
∂ charac∂
∂
ity)
and
Jackson,
and
Jackson,
2 ∂ (Heimburg
A
A
A (2) 2005; Heimburg
(Anderson
et al.,
2009).
[
]
Δ
ρ
=
c
Δ
ρ
h
Δ
ρ
∂x
unbalanced
charge
of
membrane.
The
reversible
teristic of transition from a fluid to a gel state over
∂ t 2 a range ∂ x 2007a).
∂x 4
heat observed during action potential is proposed
of several degrees, slightly below the body temperature (He4.i.e.
Comparison
as Both
a result
lipidofmelting
enthalpy
the of
effects
non-linearity
and
dispersion produce
Figure
5:
Heat
capacity,
lateral
density
and
imburg and Jackson, 2005, and references there After
in). That
transformation
x-v.t),lipid
production
of latent and
heat(z=
during
transition
lateral compressibility as a function ofthe co-ordinate
a
self-sustaining
localized
density
pulse
with a moving
is, when a biological membrane is compressed,bythe
fluid
the propagation
thethe
time
The
differences between both
from
fluid of
to the
gelvelocity,
state and
reabsorption
of major
temperature (adapted from (Heimburg introducing
and
segment
nerve
membrane
in
the
gel
state
(see Fig. 6).
membrane gradually
reaches to a denser gel state.
Now, equation
independent
thereturns
propagation
are: (Hodgkin and Hux
heat asdescribing
the system
to the fluid theories
state
Jackson, 2005)
This
called
a solitonand
(HeimburgHeimburg
and Jackson,
densityarea
excitation
is pulse
given
as
(Heimburg
and 2007a).
Jackson, 2005; He
(Anderson
et is
al.,
2009).
a sufficient pressure/ compression either in membrane
The
solitary
wave
maintains
its
shape
while
it
travels
at a et al., 2009
Jackson,
2007a)
Jackson,
2007a;
Anderson
or volume will bring a transition to the gel state (Heim-
3
Soliton theory
constant speed less than the sound velocity in the lipid memburg, 2007a). This transition is associated with the ∂release
2
∂
2
A
2
A
H-H distances
The actionwithout
potential is due to th
over long
]
Δρ =3 [(c 0brane.
v relation
+ pΔρ A +Solitons
q( Δρ A ) 2 +can
...) ∂ ∂propagate
z Δρ
of heat (Heimburg, 2007a; Heimburg, 2007b). The
∂z
∂z 2
current
conducted
loss
of
energy
(Heimburg
and
Jackson,
2005).
The
pulse
is across the m
between heat capacity, density, compression and temperature
∂4
A
(3) because no energy
ion channels
that
act as resisto
-as
h adiabatic
Δ
ρ
also
called
pulse,
is
lost
to
are shown in Fig. 5.
∂z 4
Where
the environment during propagation (Heimburg
Jackson,
S The and
nerve
pulse is a sol
ν is propagation velocity
A
2007b).
The
soliton
mode
of
pulse
propagation
is
basically
generated by the lipid transi
ρ is lateral membrane density
www.adv-radio-sci.net/8/75/2010/
membrane.
c is sound velocity in fluid phase of
membrane
Adv. Radio
Sci.,Nerve
8, 75–79,
2010
H-H
pulse propagation
is then
p and q are the parameters determined
as
a
moving
self
regenerating
p
from
sound
velocity
and
density
capacitive discharges through th
dependence
resistors followed by capacitor
h is the parameter to set the linear scale
78
R. Appali et al.: A comparision of Hodgkin-Huxley and soliton neural theories
different from the AP given by the H-H model (Anderson et
al., 2009).
The solitary wave is associated with a transient voltage
change due to the transient change in state of membrane.
This affects the membrane potential difference. Especially,
it changes both the area and thickness of membrane and
thereby capacitance (Heimburg and Jackson, 2007a).
These changes seem to be as a voltage pulse and leads to
a capacitive current due to the unbalanced charge of membrane. The reversible heat observed during action potential
is proposed as a result of lipid melting enthalpy i.e. production of latent heat during lipid transition from fluid to gel
state and the reabsorption of heat as the system returns to the
fluid state (Anderson et al., 2009).
Thus the heat flow is correlated with the changes in membrane density and potential.
3.1
4
Comparison
The major differences between both the neural theories are
(Hodgkin and Huxley, 1952; Heimburg and Jackson, 2005;
Heimburg and Jackson, 2007a; Anderson et al., 2009):
H-H The action potential is due to the electric current conducted across the membrane by ion channels that act as
resistors.
S The nerve pulse is a solitary wave generated by the lipid
transition in the membrane.
H-H Nerve pulse propagation is then explained as a moving self regenerating pulse of local capacitive discharges
through the resistors followed by capacitor recharging.
S The propagation is due to the combined effect of nonlinearity and dispersion.
Soliton equation
H-H The theory is pure electrical with ionic hypothesis.
Hydrodynamic equation for the propagation of a density
pulse in the presence of dispersion for a cylindrical membrane along x-axis is (Heimburg and Jackson, 2005) given
by
i
∂2
∂ h 2∂
∂4
A
A
1ρ
=
c
/
1ρ
−
h
1ρ A
∂x
∂x
∂t 2
∂x 4
S The theory is physical/ mechanical i.e. based on thermodynamics.
H-H The nerve signal or action potential is electrical pulse.
S The nerve signal is propagating adiabatic electromechanical pulse.
(2)
After the co-ordinate transformation (z = x − v · t), by introducing the propagation velocity, the time independent equation describing the propagation density excitation is given as
(Heimburg and Jackson, 2007a)
H-H The pulse propagation dissipates energy in the form of
heat.
S The propagating pulse does not dissipate heat energy.
H-H The nerve’s physical changes are not explained.
i
∂2
∂ h 2
v 2 2 1ρ A =
c0 + p1ρ A + q(1ρ A )2 + ... ∂ ∂z1ρ A
∂z
∂z
∂4
(3)
−h 4 1ρ A
∂z
Where
ν is propagation velocity
ρ A is lateral membrane density
c is sound velocity in fluid phase of membrane
p and q are the parameters determined from sound velocity and density dependence
h is the parameter to set the linear scale of propagating
pulse.
This theory also explains the effect of anesthetics on the
nerve pulse propagation. According to this theory, the anesthetics in the membrane lower the fluid-gel transition temperature, a phenomenon known as freezing point depression
(Heimburg and Jackson, 2007c). So it is difficult to force the
membrane through the fluid-gel transition, thereby the action
potential generation is suppressed. It is noted that freezing
point depression is a general thermodynamic phenomenon
and not chemical in nature (Anderson et al., 2009).
Adv. Radio Sci., 8, 75–79, 2010
S The nerve changes dimensions and gets thicker during
nerve pulse. The voltage pulse accompanies the propagating signal, as a result of these transient geometric
changes of the membrane.
Both the theories explain the existence of voltage pulse in
nerve pulse propagation. However, as an information transmitting pulse in H-H model and as a by-product signal in
soliton theory.
5
Discussion
Hodgkin-Huxley model of nerve pulse propagation is undoubtedly a remarkable achievement in the field of physiology. However, the theory could not explain the physical
phenomena such as reversible heat changes, density changes
and geometrical changes observed in the experiments (Iwasa
et al., 1980; Tasaki, 1982, 1999; Tasaki et al., 1989, Tasaki
and Byrne, 1992).
Soliton model is an alternative model, convincingly explaining some of the unexplained observations in the experiments. However, there are several other questions that this
www.adv-radio-sci.net/8/75/2010/
R. Appali et al.: A comparision of Hodgkin-Huxley and soliton neural theories
has to answer like ion flow involvement in nerve signal propagation as stated by H-H model and also the faster propagation in myelinated nerves than in unmyelinated.
6
Context
This is a theoretical comparison of both the theories. This
has been carried out to analyze and choose a detailed neuron
model which fits in for the goals of our project. We are examining the soliton theory to model and simulate the action
potential and the coupling of electrode for a detailed study of
the neuron.
Acknowledgements. We are grateful to DFG (German Science
Foundation) for funding our project in the Research Training Group
1505/1.
References
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88, 104–113, 2009.
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Cole, K. S. and Curtis, H. J.: Electric impedance of the squid giant
axon during activity, J. Gen. Physiol., 22, 649–670, 1939.
Gerstner, W. and Kistler, M. W.: Spiking neuron models. Single
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