1. Soliton solutions in neurological systems
Roberto D. Ortiz∗
Department of Physics, Texas A&M University, College Station, TX 77845
(Dated: April 29, 2015)
Soliton studies are a growing branch of mathematics, physics and biology that focuses on modeling
systems of partial differential equations. Historically the neuron model for signal propagation is seen
as action potential voltage gates, however recent modeling of neuron-lik lipid membrane systems
using partial differential equations have resulted in soliton-like solutions. We numerically confirm
that the proposed neuron signaling solutions exhibit all the characteristics of a soliton wave form
with variable amplitudes dependent on the velocity of the propagating wave.
I. INTRODUCTION
Before classifying a solution or phenomena as a soli-
ton there are special attributes that must be met. The
name soliton represents a classification of solutions given
to partial differential equations(PDE’s) exhibiting special
wave propagation. These particular solutions are the re-
sult of the cancellation of non-linear and dispersive forces
which creates a self-reinforcing behavior while keeping
the shape and propagation velocity constant. Also dur-
ing collisions with other solitons the shape and speed of
propagation must remain the same post collision with the
exception of a phase shift.
In this paper we apply the conditions necessary to clas-
sify a phenomena as a soliton on a recent analytical so-
lution for neuron pathway signaling seen in reference[1].
Before delving into the biophysics portion of this paper
it is important to review and understand what a soliton
would look like in a sample physics system.
A common example of soliton like behavior is the prop-
agation of a light pulse through a medium. An incom-
ing light pulse is made up of many different frequencies
of light. Different frequencies of light travel at different
velocities(dispersion) however the refractive index of the
medium material will generate a non-linear Kerr effect[2].
The Kerr effect modifies the refractive index in propor-
tion to the amplitude of the light moving through. In
special cases of frequency and amplitude the effects of dis-
persion and the Kerr effect cancel out allowing for a wave
to continue propagating without a change in shape de-
spite varying frequencies. Essentially this solution to the
wave equation acts as a particle moving through space be-
cause the function maintains it’s initial properties when
moving through space.
One of the most interesting traits of a soliton is the
ability to interact with other solitons and still retain the
initial speed and shape after collisions[3]. The next sec-
tion will extensively cover the formation and differences
between kink and antikink solitons.
∗send correspondence to: Ortiz2011@tamu.edu
II. SINE-GORDON SOLITON FORMATION
The sine-Gordon PDE is a special case of the Klein-
Gordon wave equation at large amplitudes. The wave
equation takes on the following form:
∂2
f
∂t2
−
∂2
f
∂x2
+ sin(f) = 0 (1)
where the function f(x, t) is a function of position and
time. When given initial conditions and boundary con-
ditions this PDE can be solved and the solution is of the
form:
f(x, t) = 4Arctan e(γ(x−vt))
(2)
γ =
1
√
1 − v2
(3)
v is the speed of the soliton solution. Using (2) we can
find how the soliton propagates through space by taking
a partial derivative of f(x, t) with respect to x, giving
the following equation after simplifying to a hyperbolic
secant
∂f
∂x
= g(x, t) =
2
√
1 − v2
Sech
x − vt
√
1 − v2
(4)
FIG. 1: The space derivative of the sine-Gordon equation
given in 4 for the kink soliton solution.

2. 2
Plotted in Mathematica (4) gives a wave like shape
seen in Fig. 1. It is important to note that when we take
the negative root of γ in (3) the resulting ∂fa
∂x will create
an antikink of the form
∂fa
∂x
= ga(x, t) =
2
√
1 − v2
sech
vt − x
√
1 − v2
(5)
which is also plotted in Mathematica as Fig. 2.
FIG. 2: The space derivative of the sine-Gordon equation
given in (5) for the antikink soliton solution.
Both the kink and antikink solutions have a distinct
and similar ∂f
∂x shape at all points in time satisfying the
first characteristic of a soliton which is to retain a con-
stant shape and velocity. Secondly the waves are local-
ized to a region meaning they decay to zero quickly as
x goes away from the center of the shape. The last at-
tribute can be seen in Fig. 3 as both the kink and an-
tikink solitons collide but remain unchanged after colli-
sion.
FIG. 3: The collision between the kink and antikink solitons
show that after the collision the solitons retain their initial
shape and speed. The ability to collide and remain unaffected
is a clearly elastic property that soliton researchers consider
very particle-like.
III. SOLITON SOLUTION IN NEURON SIGNAL
PROPAGATION
Neurological signaling has been characterized for the
last 60 years according to the Hodgkin-Huxley model of
voltage gates and action potentials(akin to an electrical
signal) [4]. The acceptance of this model was univer-
sal and resulted in a nobel price in physiology to both
Hodgkin and Huxley ten years after it’s emergence. To
uproot such an established model the soliton wave prop-
agation hypothesis led principally by Thomas Heimburg
in reference[1] must be rigorously tested and evaluated.
Heimburg proposes that neuron signaling is due to soli-
tary sound propagation with a nonlinear term due to
compressibility factors of the neuron membranes and a
dispersion term from the compressibilities dependence on
frequency and pulse velocity.
To evaluate this new hypothesis we will numerically
integrate the proposed soliton like solution found in the
equation from [1] labeled here as (6). After the inte-
gration we can examine and propose what behavior is
soliton-like or otherwise.
h
∂∆ρA
∂z
2
= (c2
0 − v2
)(∆ρA
)2
+
p
3
(∆ρA
)3
+
q
6
(∆ρA
)4
(6)
where h=2 is a scaling factor. ∆ρA
is the sound prop-
agation function that has soliton-like behavior and is de-
pendent on z where z = x−v∗t. The factors c0, p, and q
are all factors of the propagation medium, we have chosen
the data for the lung surfactant given in the Heimburg
reference [1]
c0 = 171.4 (7)
p = −6.86
c2
0
ρA
0
(8)
q = −32.32
c2
0
(ρA
0 )2
(9)
ρA
0 = 4.107 × 10−3
(10)
ρA
0 is the equilibrium lateral density of the medium and
c0 is the equilibrium sound velocity. Solving for ∆ρA
analytically we get
z+ =
−p
q
1 +
v2 − v2
min
1 − v2
min
(11)
z− =
−p
q
1 −
v2 − v2
min
1 − v2
min
(12)
∆ρA
(z) =
2z+z−
(z+ + z−) (z+ − z−) Cosh[z
√
1 − v2]
(13)
Equations (12) & (13) are positive and negative roots of
the PDE (6), both are dependent on the velocity of the
sound wave and the equilibrium pulse velocity vmin = .87
given by the properties of the membrane[1]. Equation

3. 3
FIG. 4: Evaluation of equation (13) for various velocities.
As velocities approach 1 the max amplitude of the wave de-
creases. This relationship will be explored further in the pa-
per.
(13) can then be evaluated at different velocities v seen
in Fig. 4. For our purposes we choose to keep ∆ρ(z) a
function of x and t to observe if the shape or velocity of
the solution varies with time which leads to Fig. 5.
FIG. 5: Varying velocity plots for neuron sound propagation.
Shows time evolution of sound waves is constant and the am-
plitudes of different velocity waves increases as v decreases.
The shown values of v for Fig. 5 all keep their shape
and locality consistent through time confirming the first
and second characteristics of a soliton-like solution. The
lower velocities have larger maximum values accompa-
nied by a broader width of the soliton. The velocities
were chosen based on how close they are in relation to
vmin from the first term in (6) while also maintaining
v < 1 to ensure a real number from (3).
For final consideration of soliton-like solutions we
choose to graph and solve for the collision between 2 neu-
ron signal pulses graphed in Fig. 6. As seen in the figure
there is no large change of velocity or shape after collision
between two pulses confirming that the proposed neuron
signaling passes the last attribute for a soliton-like solu-
tion.
FIG. 6: Collision between 2 soliton-like functions of ∆ρ with
v = .88. The initial shape and speed of the solitons are unaf-
fected after collision.
IV. DISCUSSION
We will first highlight the differences between the
Hodgkin-Huxley(H-H) model and the proposed neuron
signaling model and then focus on the validity of the
soliton model from the calculations from the previous
sections.
The H-H model explains neurological signals as an elec-
trical system made up of ion channels, linear and non-
linear voltage gates, a lipid membrane capacitance, and
electrochemical gradients[5]. Essentially this creates a
system of 4 non-linear differential equations that can be
solved numerically and not analytically [4]. The H-H
model equations were chosen to fit experimental data and
does so incredibly well. However, this model does not
give a thermodynamic picture of the activity of the cell
which is an important concept in cell membrane proper-
ties.
In the H-H model as in an electrical system there will
be heat created whenever a pulse is generated according
to basic principles of electronics. This is proven exper-
imentally [6, 7], however the data also exhibits a tem-
perature cooling after the signal propagation which is
completely uncharacteristic of an electrical model which
can only generate heat [8]. The temperature differ-
ence is more than enough to influence protein and lipid
membrane structure because initial unperturbed states

4. 4
of these elements are close to the melting point phase
transition [9, 10].
A change in phase can result in large changes in
the lateral compressibility factor(κA
S ) and thus lateral
densities(ρA
) of lipid membranes can fluctuate allow-
ing for the possibility of effective ”waves” propagating
through the membranes [11]. The equation for sound
propagation through a medium depends on the compress-
ibility and can be expressed as
∂2
∆ρA
∂t2
=
∂
∂x
1
κA
S ρA
∂∆ρA
∂t2
(14)
The change in κA
S due to the temperature change asso-
ciated with a pulse will create the non-linear effects seen
in the right hand side of (6). Since the lipid membrane
contains many ionic proteins and chemicals the physical
shifting of the membrane as a whole would result in the
electrical signal seen in experimental data[12]. This idea
of varying lateral compressibility factors and densities is
what drives the neuron soliton model evaluated in this
paper.
When the neuron soliton PDE is evaluated the solu-
tion exhibits a soliton-like structure similar to a Gaus-
sian seen in Fig. 4. The solution has a Sech dependence
as in the sin-Gordon model so a soliton shape is under-
standable. As the velocity of the pulse approaches the
value of the equilibrium pulse velocity vmin, the PDE
solution increases in amplitude and broadens in width.
This shape remains consistent with the pulse propaga-
tion seen in Fig. 5 for multiple velocities. The solution
contains a Cosh behavior which will naturally show a
wave form as long as v < 1 and the PDE roots are real,
requiring v > vmin.
The collision between two solutions in Fig.6 shows that
the waves do in fact retain their shape and velocity after
collision with each other. The analytic solution to the
PDE has no factor dependent on the collision so equal
initial and final states are expected. This is a property
shared by solitons and is very similar to the sin-Gordon
2 soliton collision in Fig. 3. Recently(2014) there has
been an experiment which found that neuron pulses do
not change their initial velocity and shape after collision
demonstrating soliton behavior in lobsters [13]. More
studies need to be done experimentally and the compu-
tational evaluation done in this paper can be used to
explore further soliton-like phenomena.
V. CONCLUSION
This paper finds that the proposed neuron signaling
propagation found in Heimburg’s paper [1] does exhibit
the characteristics of a soliton-like solution akin to a sin-
Gordon soliton wave. The shape and velocity of the neu-
ron soliton is consistent through time evolution and and
for velocities close to the equilibrium pulse velocity vmin
the wave takes on a localized form around the maximum
signal amplitude. Lastly when 2 neuron solitons collide
the initial and final wave shapes and velocities are iden-
tical.
In principle, the confirmation of quasi-1D soliton like
solutions theoretically must be accompanied by rigorous
experimental data. Verifying experimentally that neuron
signal pulses show soliton properties will be needed to so-
lidify a neurological signaling model to overturn the 60
year standing Hodgkin-Huxley model. The implications
of a new theory would change protein signaling pathway
models leading to interesting topics on how various pro-
tein pathways can remain unchanged when colliding with
each other.
Acknowledgments
This work evaluated the neuron soliton signaling pro-
posed in the research paper [1] by Heimburg and Jackson.
We thank Ben Schroeder for Mathematica syntax assis-
tance, and Richard Lawrence for critical reading of the
original manuscript. Thanks to Texas A&M University
for access to online research papers used as referencs for
this work. Lastly we thank Professor Helmut Katzgraber
for introducing the paper proposal over the branch of
mathematical equations known as solitons.
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