# Soliton model in neuroscience

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The soliton model in neuroscience is a recently developed model that attempts to explain how signals are conducted within neurons. It proposes that the signals travel along the cell's membrane in the form of certain kinds of sound (or density) pulses known as solitons. As such the model presents a direct challenge to the widely accepted Hodgkin–Huxley model which proposes that signals travel as action potentials: voltage-gated ion channels in the membrane open and allow ions to rush into the cell, thereby leading to the opening of other nearby ion channels and thus propagating the signal in an essentially electrical manner.

## History

The soliton model was developed beginning in 2005 by Thomas Heimburg and Andrew D. Jackson,[1][2][3] both at the Niels Bohr Institute of the University of Copenhagen. Heimburg heads the institute's Membrane Biophysics Group and as of early 2007 all published articles on the model come from this group.

## Justification

The model starts with the observation that cell membranes always have a freezing point (the temperature below which the consistency changes from fluid to gel-like) only slightly below the organism's body temperature, and this allows for the propagation of solitons. It has been known for several decades that an action potential traveling along a neuron results in a slight increase in temperature followed by a decrease in temperature.[4] No net heat is released during the overall pulse. The decrease during the second phase of the action potential is not explained by the Hodgkin–Huxley model (electrical charges traveling through a resistor always produce heat), but traveling solitons do not lose energy in this way and the observed temperature profile is consistent with the soliton model. Further, it has been observed that a signal traveling along a neuron results in a slight local thickening of the membrane and a force acting outwards;[5] this effect is not explained by the Hodgkin–Huxley model but is clearly consistent with the Soliton model.

It is undeniable that an electrical signal can be observed when an action potential propagates along a neuron. The soliton model explains this as follows: the traveling soliton locally changes density and thickness of the membrane, and since the membrane contains many charged and polar substances, this will result in an electrical effect, akin to piezoelectricity.

## Formalism

The soliton representing the action potential of nerves is the solution of the partial differential equation

$\frac{\partial^2 \Delta \rho}{\partial t^2} = \frac{\partial}{\partial x} \left[\left(c_0^2 + p\Delta \rho + q\Delta \rho^2\right)\frac{\partial \Delta \rho}{\partial x}\right] - h\frac{\partial^4 \Delta\rho}{\partial x^4},$

where t is time and x is the position along the nerve axon. Δρ is the change in membrane density under the influence of the action potential, c0 is the sound velocity of the nerve membrane, p and q describe the nature of the phase transition and thereby the nonlinearity of the elastic constants of the nerve membrane. The parameters c0, p and q are dictated by the thermodynamic properties of the nerve membrane and cannot be adjusted freely. They have to be determined experimentally. The parameter h describes the frequency dependence of the sound velocity of the membrane (dispersion relation). The above equation does not contain any fit parameters. It is formally related to the Boussinesq approximation (water waves) for solitons in water canals. The solutions of the above equation possess a limiting maximum amplitude and a minimum propagation velocity that is similar to the pulse velocity in myelinated nerves. There also exist periodic solutions that display hyperpolarization and refractory periods.[6]

## Role of ion channels

The soliton model explains several aspects of the action potential, which are not explained by Hodgkin–Huxley. Since it is of thermodynamic nature it does not address the properties of single macromolecules like ion channel proteins on a molecular scale. It is rather assumed that their properties are implicitly contained in the macroscopic thermodynamic properties of the nerve membranes. The model is therefore neither in conflict with the action of membrane proteins nor with pharmacology. The soliton model predicts membrane current fluctuations during the action potential. These currents are of similar appearance as those reported for ion channel proteins.[7] They are thought to be caused by lipid membrane pores spontaneously generated by the thermal fluctuations, which are at the core of the soliton model.

## Application to anesthesia

The authors claim that their model explains the previously obscure mode of action of numerous anesthetics. The Meyer-Overton observation holds that the strength of a wide variety of chemically diverse anesthetics is proportional to their lipid solubility, suggesting that they do not act by binding to specific proteins such as ion channels but instead by dissolving in and changing the properties of the lipid membrane. Dissolving substances in the membrane lowers the membrane's freezing point, and the resulting larger difference between body temperature and freezing point inhibits the propagation of solitons.[8] By increasing pressure, lowering pH or lowering temperature, this difference can be restored back to normal, which should cancel the action of anesthetics: this is indeed observed. The amount of pressure needed to cancel the action of an anesthetic of a given lipid solubility can be computed from the soliton model and agrees reasonably well with experimental observations.