Download Neuron, a simpler theory.

Neuron, a simpler theory.
Solving the Hodgkin-Huxley uncertainties
“Everything should be made as simple as possible, but not one bit simpler.”
“The most incomprehensible thing about the world is that it is at all comprehensible.”
Albert EINSTEIN
Bernard DELALANDE
[email protected]
Neuron physiology is actually described using the well known models created by Hodgkin-Huxley
(HH) and their derivatives which came later, the Fitzhugh-Nagumo (FN) equations. These attempts to
understand the internal functioning of the neuron are unfortunately far from the strict Nature's
observation. These equations, particularly, are unable to explain an unidirectional propagation of action
potential nor they can explain accommodation or the acceleration seen in axons. These equations are
unable to create an action potential without an external application of current.
Because neuron is a cell and cell is an entity, it is reasonable to think that it uses the same means all
along its length. This paper is about an alternative theory that explains how a neuron may function.
The exposed theory is simpler than the previous ones but explains more accurately the observed
phenomenons.
Cell considerations and derivative constraints
Neuron is the longest cell of human body. Some neurons may have more than a meter length. If we
could magnify a single C fibre of 1 µm diameter to 1 cm (x 10,000) then our little cell will be 10 km
long! The size of neuron imposes an elegant and useful maintenance of the cell and because neuron is
also a communication cell, the maintenance has to be done while the cell is communicating. These
constraints were solved by Nature. It is logical to think that communication is done superficially
because we found “active” and fast ions channels in the membrane. This implies automatically that
“feeding” is done more deeply since it is also logical to think that these two functions (which may be
done in opposite direction) need mandatory different pathways.
A/ Impossible mixing
Nobody actually denies that some “electrical” properties of neuron are results from physical movements
of ions between the extracellular and internal milieu. This simple affirmation may already discard the
electrical functioning of neuron.
The following table show some known major differences between neuron “electrical” properties and
electricity.
Neuron
Electricity
Equality
Electrical existence
Physical presence of ions
Electronic imbalance in circuit
NO
Speed phenomenon
< 250 ms
300,000,000 ms
NO
Local circuit theory
Electrons?
Electrons
???
Ions movements
YES
NO
NO
-1
-1
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Neuron
Electricity
Equality
Saltatory conduction
Electrons?
Electrons
???
Action potential
Ions flux
Electrons
NO
Circuit
Arbitrary
Hard-wired
NO
Tab.1 Neuron vs Electricity
It is easy to conclude that neuron is all but electrical. (1)
B/ Concentrations discordances
The previous models use heavily the gradients concentrations in their equations. That makes sense.
We have measured these concentrations many times and a ionic concentration is a ratio established
between a known number of molecules/ions and a defined volume. If we examine a free endings C
fibre tree (fig.1) , the concentrations are based on a finite and concrete number mi of ions in a
defined volume of solvent, it is possible to conclude that numbers are different from site A, B or C
because the shapes/volumes aren't the same.
Fig 1 C Fibre endings
For an example the extracellular potassium concentrations may be equal:
[ K OutA ]=[ K OutB ]=[ K OutC ]
2 a
because the following relation are true
m A m B mC
= =
VA VB VC
2 b
But inequalities stand;
m A m B mC
V A V BV C
2 c
2 d 
Since the HH model uses currents sources based upon the assumption of ideal/perfect electrical
components and ions concentrations, it may fails or engages the model in divergent behaviours in
regard of real neurons.
C/ Superficial “concentrations”
It had been observed during action potential that ions flux are made superficially. Hodgkin and Huxley
used some radioactive ions and noted that there is only few ions involved during AP and it was done
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close to the membrane. R Mackinnon [1] concluded in the same way and some theories [2] are unable
to stand without a superficial presence of ions. There is absolutely no reason to contradict these
findings. A Neuron theory must follow the facts.
If we limit ions exchanges to a superficial area of axon (fig.2) then we may change accordingly our
hypothesis about ions concentration.
Fig.2 A piece of axon with the active layer T r
Let us introduce a thickness variable T r (thickness of reaction). We may hypothesise that it is a
constant. In that way, our previous concentrations will act on reduced volumes and numbers. Is it really
a good idea which reflects the facts?
If we examine a standard graph of HH model versus experimental trace (fig.3) thus we see large
differences. The model diverges in speed and the curve is too weak.
Fig.3 HH curve versus experimental one
How is it possible to solve this riddle without changing concentrations and ions channels numbers? [3].
If ions chambers are reduced thus concentrations across the membrane will change more quickly in all
positive and negative ways. Many biologists would argue that concentrations remains stable. If it is the
case, the only conclusion possible become that concentrations are not involved in actions potential,
only few ions. Thus it is not possible to maintain equations based upon such concentrations.
If we consider that action potential is the result of a limited number of ions then it augments the
available amplitude and give some “negativeness” to K+. Reducing the available volumes is like
applying a scaling factor in time (x) and amplitude variable (y). Since the hypothesis, of a limited
thickness of reaction, fits closer to facts then it is retained in our theory.
Ions movements, in axons, take place in a superficial (and constant?) thickness Er. (3)
D/ Visible but virtual amplitudes.
If we represent a cut-away of an axon (our hypothesis) with some ions (for clarity) we are seeing:
The resting potential is assumed normally in the reduced volumes and ions, are normally allocated all
around the membrane (inside and outside) by diffusion and repulsion (Einstein-Stokes' law). A visible
conclusion is considered from the plausible drawing (fig.4);
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Fig.4 Axon cut- away (resting potential)
Each piece of membrane surface supports only a low number of ions. The resting potential is
proportional to the available ions and their concentrations and the circumference/radius of axon. Then
it is possible, with same concentrations all along of axon, with a varying shape, to observe a varying
resting potential or action potential [5] according to the radius of the site [6]. This simple relation
implies already an amplification of action potential varying with the shape of axon.
V maxC = f  RC , m Koc , m Kic , m Naoc , m Naic 
4
E/ Linear is non-linear
It is largely admitted that ions pass through ions channels (opened) in a random manner (Brownian
agitation). This fact permits us to elucidate partly a known property of action potential duration. It is
well known that for small diameter fibres, APs are longer than for large ones.
We can take an example with two fibres of different diameters but with a linear repartition of ions
channels on the membrane. These “slices” are of course largely virtual (fig.5) but confirm observations
made on axons [6].
Fig.5 Two axons with a linear ions channel repartition
There is an higher probability that channels are opened at the same time with larger axon, then the
probability that one ion goes through a channel is higher with a larger axon. Since more ions will flow
more quickly through an increased number of channels [7] thus ions flux is higher with larger radius
even if ions channels concentration is the same.
A linear change of radius may produce a non-linear variation of incoming/outgoing ions.
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F/ Neuron uses solitons
There is no doubt that action potential is a soliton. It is a self propagated wave but it is much more. A
soliton is limited and very constrained. Actually we are unable to stand a valid soliton equation which
works in all the lenght of the axon and its dendritic trees. In a soliton, the wave is quite constant and
collisions of solitons don't summarize but cross and follow their ways. Neuron uses a “super” soliton
that is able to add the encountering solitons, accelerates and even doing back propagation.
That is very far from the compartmented solutions given by HH model of FH one.
A better way is to solve the mystery using simple equations that reflect the known facts. A better way is
done by avoidance of “electrical” means.
We will use membrane, ions (as atoms), water since ions are solvated and of course the marvels of ions
channels. It is then possible to create a recursive equation which enable an elegant model that works in
all areas of a neuron.
G/ Recording problem
If we suppose the recorded electric curves as valuable we mustn't forget that a soliton is a dynamic
phenomenon. Recording a dynamic phenomenon that is travelling under the electrodes may lead to
weird interpretations and false curves.
Fig.6 Recording problem of action potential
Electrodes may record some “past” and “future” events in the time line of the wave (fig.6). It is
important to keep in mind this problem.
H/ Ions solvation
Neurons and extracellular sites are electrolytes that is water and solvated ions. Ions and water are mixed
in a process named solvation. Ions and water form bounds because ions are charged particles and they
need to find an equilibrium.
Na and K ions are very small particles and their diameter are for sodium and for potassium.
When ions are associated with water molecules, the resulting compound is of a bigger size and volume
varies accordingly.
Na compound is 13 bigger in volume.
K compound is 11 bigger in volume.
Na+water > K+water.
Roderick Mackinnon has demonstrated that ions are crossing membrane, via ion channels, in a
partially dehydrated form [1] and they regain their complete hydration in all other cases.
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Ions that cross membrane are gaining volume with water compounds.
I/ Ions viscosity
We have seen that water compounds make with ions, larger molecules (fig.7). It is known that larger
molecules have increased viscosity by the Stokes' law.
Fig.7 Na+ and K+ water compounds.
F =6 pi RnVc
5
This viscosity is directly proportional to the radius of the sphere/compound.
In biological electrolytes, ions aren't moving at the same speed.
Na ions move slower than potassium ones when they are solvated.
J/ Diameter and membrane thickness
Another non-linear relation is introduced by diameter and membrane thickness [8].
It is known and accepted that axons have different shape and size that is varying from their “beginning”
to their “endings”. It is also stated that small unmyelinated fibres have a conduction velocity which
varies approximately in relation of the square root of their radius.
Fig.8 Diameter and membrane ratio
The cited paper [8] introduces for axons, the notion of thin wallet elastic tube. It is true that thickness
of membrane is quite constant but axon diameter is variable. It is then plausible, if we consider an
hydraulic component in action potential propagation, that the varying ratio may change directly the
speed of propagation.
But this hypothesis seems not to work because a small axon may be more rapid than a larger one since
the tube is more rigid than the latter.
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Fig.9 Diameter and conduction velocity
It is true and also false at the same time (fig. 9).
• Speed is increased more quickly for small axons than large ones. The slope of the curve decreases
with diameter. It fits thus the observations.
• Conduction velocity evolutes in a linear manner for myelinated axons.
Many problems remain with this alternative model since it supposes a continuous flow of viscous water
but avoids the intakes and outtakes of ions Na+ and K+. The models bypasses also the hydration of
ions that plays a large role in our model. The model is too mechanical and doesn't bring solutions for
unidirectional propagated waves and their initiations.
K/ Model failures
The HH model was created in 1952 and was a major improvement over the previous ones. Of course,
it is a largely simplified neuron model but helped us to track the cell's activity.
Its major problem is that it is unable to carry a recursive function for propagated action potential and
does not explain the observed unidirectional propagation. It is necessary to compartment/insulate the
model to produce a bit of movement.
The second major problem with HH model is it takes the “current” sources as separated phenomenons
without any relation/links. It is a sequential model that relates observations but do not explain them
really with their intricate connections. One would argue that a connection is not really established
between K+ and Na+. Many examples show that removing one component stops the activity. [9], [10],
[11], [12].
A third problem comes with simplification of incoming and outgoing ions processes. HH model uses a
single complex relation but it had been observed than ions do not flow in the same manner and speed
through the membrane. It was possible to hypothesise that these two functions were linked but
separated.
Because the HH model is a stable and static one, it was not really intended to characterize a solitonic
solution. Time variable is used to describe the variation of a curve amplitude but has no relation with
displacement or movement of this one.
L/ Components of a solution?
Well, we have now some facts that involves new relations and reactions:
We have a volume change every time an ion crosses the membrane.
We have repulsion and diffusion.
We have a change of viscosity coming with each ion.
We have water molecules and elastic tubes and pressure.
There is also two major kind of ions which have difference in shape and speed.
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It seems important to solve the Nature's reason of different ions used in its “super” solitonic solution.
We could have a single ionic hypothesis where Na+ is the only available “engine”. If we know without
any contest that Na+ flows inside the axon thus there is few possible explanations;
•
Na+ is incoming because the external side is responsible of this motion.
•
Na+ is incoming because the internal side is responsible.
In all cases, Na+ is changing and it may be a relation with ions flux and Na+ quantity. Such relations
do not work at all because they are diverging by nature. The result of our single ionic hypothesis ends
with all ions on a side.
Nature reinitializes the “system” and makes it functioning in a well definite range. This becomes
possible at least with two ions, linked together (fig .10). Adding a third one (like Ca++) is a good way
to create pacemakers behaviours.
Fig.10 Ionic relationship
It is not fortuitous that we find a relation that already exists. The Na/K pump is constructed with the
same ions. And it is another clue favoring a strong association of these ions in the action potential
process.
There are definitive proves based on observations;
Fig.11 Ionic relationship proves
The conductances curves (fig. 11) are the most powerful example of this intricate relation between
sodium and potassium;
• Na is entering when many K+ are present inside
• K is outgoing when Na is present inside
These mandatory steps are liked in a closed loop and a solitonic wave is a based upon a recursive
formulation. A stable recursive program is a closed loop.
There is more Na channels than K ones. If we give to K+ ions the ability to open Na+ channels and
because there is much more potassium inside, thus it is possible that K+ opens the sodium channel. If
we consider there is 10/50 times more Na+ channels than for potassium then we can make a small
drawing showing this repartition (fig. 12). It is conceivable that K ions move over few Na channels
before outgoing its own?
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Fig.12 Na vs K density. (Na/K=40)
M/ A “super” soliton solution
All these elements do not really help us to construct a set of equations for a solution.
Another way to solve the riddle is done by sampling the action potential travelling wave (fig. 13).
Fig.13 Solitonic action potential
I represented a membrane with three samples of the travelling AP over it. The sample rate must be
small enough for having overlapping curves. That is achieved taking a small interval time a.
We can see curves that are sequenced in time varying the interval a. All these curves are in fact only one
separated in position, by the interval. They are snapshots of action potential.
One may say that curve point A is located on a “present” windowed curve (in black) and points B and
t 1a and
t 12 a because in a soliton
C are future locations of A at times
V t 1 =V t 1a=V t 12 a (points A, B and C are on an horizontal line, at a same amplitude
and same time location within a curve). One may say that curves are transmitted without deformation
by simple displacement on a X axis.
But if we accept the future locations of the curves and thus their existence, we must accept the
membrane has an amplitude A at location x 1 , an amplitude B' at x 2 and C' at x 3 , at time
t 1 because there is a simple causation effect relation. Thus we are knowing already the future
amplitudes of membrane and it gives us by simple deduction the “engine” of action potential.
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Here is now a possible explanation of movement (table 2):
x1
x2
x3
Amplitude
+++ ++
+
# ions
+++ ++
+
# Na+
+++ ++
+
# K+
+++ +++ +++
Volume
+++ +
+
Viscosity
+++ +
+
Repulsion
+++ +
Ct
Possible motion
Right Right Ct
Tab.2 Curves values
And show on a graph these variations for clarity (fig 14)
Fig.14 Variations at different locations
It is now visible that volume, viscosity and repulsions forces impose an unidirectional propagation of
action potential. There is no alternative issue but it doesn't mean fortunately that antidromic or backpropagation is impossible. In contrary, it is fully supported by this theory but it might occur only in
well defined circumstances.
Of course this theory discard totally the “local circuit” theory applied for action transmission potential
but this new one is only based upon facts and not on arbitrary suppositions. If the “local circuit” theory
is disabled then it becomes impossible to explain either the saltatory conduction since it is a derivation
of the primer.
But does the myelin wrappings fit better in our findings? We know that it is more ions channels at
Ranvier's node and turns of myelin are done for some good reason.
The official explanation is that it insulates the internal milieu from external one (fig.14). That's just
true but in a total different way that the one given to us. It increases the membrane rigidity allowing a
longer transmission of ions solvated wave.
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Fig.15 Myelin sheath
We have exactly a logarithmic decreasing of pressure [8] and ions concentration (fig.16). And it is
exactly what we are seeing in observations.
Fig.16 Myelin sheath
It is only necessary that the starting pressure assumed at node of Ranvier will be enough high to push
the ions in the good direction. It is well known that ions channels density is higher at these sites and
much higher than those encountered with unmyelinited fibres.
N/ Does it fit the facts?
If we summarize the properties that are given by the theory versus the HH model then we have;
New Model
HH Model
Universal
YES
NO
Continuous
YES
NO (compartmented)
Solitonic
YES
NO
Branching
YES
LOW
Trees propagated
YES
LOW
Local current
YES
NO (impossible)
Saltatory propagation
YES
NO (impossible)
Amplification
YES
NO
Acceleration
YES
NO
Neural network
YES
LOW
Fits facts
YES
LOW
Compatible with HH one
YES
NA
O/ Conclusion
This new theory based on observed phenomenons and solitonic theory provides a simpler explanation
about neuron behaviours and it enhances the previous ones solving some of uncertainties carried by
older models.
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