Peng Wang(王鵬)

School of Civil Engineering and Architecture,University of Jinan,Jinan 250022,China

Keywords: Hamilton’s principle, Noether theorem, fractional derivative, multiscale electromechanical coupling,neuron membrane

1.Introduction

The action potential(AP)has been considered to be only an electric signal, and the well known Hodgkin–Huxley (H–H) model[1]is the foundation of this viewpoint.However, a recent experiment[2]found that the AP propagation accompanies mechanical deformation like swelling or contraction.So the standpoint of AP being a mechano-electrophysiological coupling process was proposed.[3]Engelbrechtet al.[4]gave a continuum mechanics model to explain this mechanoelectrical coupling phenomenon, Chenet al.[5]gave a finite element computation model and Drapaca[6]proposed an electromechanical model using Hamilton’s principle,and gave its Euler–Lagrange equation.Later, she extended this model to a fractional derivative scenario.[7]However, due to the complexity and nonlinearity, it is hard to solve the mathematical equations of these models.Symmetry is an important tool in finding the first integral, differential equation reduction and classification,[8]so we can apply symmetry in electromechanical coupling equations to find their conserved quantities,further to reduce orders of the differential equations.However,to our knowledge, the Noether symmetry and conserved quantities of neuron dynamics have not been investigated.

Symmetry is a higher rule in physics.Since Noether revealed the relations between symmetry and conserved quantities in her eminent paper,[9]Noether theorem has had various extensions and has been applied in physics and mechanics.[10–22]Fractional calculus is increasingly important as it can more exactly describe complex phenomena in science and engineering.Fractional Noether theorem has been extended to the fractional Lagrange system, Hamilton system, generalized Hamilton system, Birkhoff system, and so on.[23–33]However,there is no research on fractional Noether symmetry in mechano-electrophysiological coupling equations of neuron dynamics.Considering the visco-elasticity of neuron membranes, we will adopt a fractional derivative of variable orders.In this paper we will generalize fractional Noether theorem to neuron dynamics.

The structure of this paper is as follows.In Section 2 we will introduce a variable order fractional mechanoelectrophysiological model of a neuron membrane with Riemann–Liouville fractional derivative, deduce its variable order fractional Lagrange equation,and show that the variable order fractional H–H equation can be deduced from the variable order fractional Lagrange equation.In Section 3 variable order fractional Noether symmetry and conserved quantities of the neuron dynamics are studied,and the criterion of variable order fractional Noether symmetry and the form of variable order fractional conserved quantities are given.In Section 4 we will specifically discuss the deduced variable fractional conserved quantities in various conditions.The final section is the conclusion.

2.The variable order fractional Lagrange equation of neuron membrane dynamics

Conduction and processing of information are the characteristics of neurons.Neurons make use of electric signals to conduct information.As we know, action potential plays a key role in producing electric signals in neurons.However,experimental observations prove that action potential production is not only because of ion transmission across neuron membranes, but also accompanied with deformation of the neuron membrane, a multiscale mechanics and electrophysiology coupling process.Recently,Drapaca[7]proposed a new fractional multiscale mechano-electrophysiological coupling model using Hamilton’s principle, however its numerical solutions are in fact uncoupled.In this paper we will revisit the model,but study the more general case using Noether symmetry analysis.

2.1.The model

The membrane of a neuron consists of a phospholipid bilayer embedded channel protein.The propagation of electric signals in the neuron system is achieved by producing action potential accompanied with an ion channel open or shut.The action potential can induce deformation of the neuron membrane,and inversely the deformation of the neuron membrane can also induce action potential.We can use the Hodgkin–Huxley model to describe the electric process and a linear visco-elastic Kelvin–Voigt model to describe the mechanical process,and Maxwell models to control the activations of the Na+and K+channels, and the inactivation of the Na+channel, respectively.The coupling process is connected through membrane capacitance and displacement (see Fig.1).As we know, the neuron can be considered as an axis-symmetric cylinder with circular cross section, so we can study half the neuron by symmetry.

Fig.1.Schematic of the mechano-electrophysiological coupling model of an axon membrane.The axon is considered as an axis-symmetric homogeneous circular cylinder with intracellular space filled with axoplasm, and the outer layer is the membrane.Using the symmetry and homogeneity of the column we only need to study half of the axon.At cellular scale we model the intracellular space as a viscoelastic material using the Kelvin model connected with axon capacitor(the dotted box ①),where(x2,x3,x3)denote the deformation motion of the cytoskeleton with different ionic channels, which correspond to the m,n,h gate. x1 is the displacement of the membrane.Mechanical motion or electrical stimuli can trigger the circuit.At subcellular scale ionic exchange obeys fractional Hodgkin–Huxley equations(the dotted box ②)and is controlled by Maxwell models.

We can express the macro-mechanical kinetic energyT=m1(0(t)x1)2, wherem1denotes the half constant mass of the neuron,Ais the cross-sectional area,andx1is the macroscopic (cell level) displacement that depends on time.In order to use the nonconservative Hamilton principle proposed in Ref.[34], we use the left and right Riemann–Liouville fractional derivative of variable orders with 0≤α(t)<1

However, definitions (1) and (2) will introduce additional memory effects for piecewise uniform accelerated motions.[35]Fortunately,when 0≤α(t)<1 the approximations

In this paper we will use a Riemann–Liouville fractional derivative of variable orders with 0≤α(t)<1,|dα/dt|?1,t ∈(0,a).

2.2.The fractional Lagrange equation of variable orders

Based on the electric charge conservation law, we have a holonomic constraint to charge:eC ?eNa?eK?el=0, so the freedom of the coupling system is 10.Introduce generalized coordinates to express universally the spatial variables and electrical variables,qs(s= 1,...,10), whereq1=x1,q2=x2,q3=x3,q4=x4,q5=eNa,q6=eK,q7=el,q8= ?x2,q9= ?x3,q10= ?x4.The Lagrangian of the neuronal mechanoelectrophysiological model is

The virtual work of nonconservative generalized forces is

The Hamilton principle of the nonconservative mechanoelectrophysiological system of the axon membrane is

By expanding the above equation, and using the communication relation0i(t)δ=δ0(t)which holds for the holonomic constrained system, and the relation of the fractional integral by parts[23–25,34]

and the end points relationsδq(0)=0,δq(a)=0,we can get the fractional multi-scale mechano-electrophysiological coupling Lagrange equation of neuronal membrane dynamics

whereψ=ψm+ψe.The coupling equation of motion describes the changes of ions between the outer layer and intracellular space and deformation of the neuron membrane.

Whenψ=Qs=0, we can get the fractional Lagrange equation of a conservative system

Putting the expression of LagrangianLinto Eq.(10), we can get the Euler–Lagrange differential equations

whereV=Ui=qC/Cis the potential of the capacitor.

Here the Euler–Lagrange equations are different from the equations in Ref.[7], because we suppose the elastic parameters of the cytoskeleton depend on the macro-deformation of the membrane.Kirchhoff’s current law demands0γ(t)(qC+qNa+qK+ql)=I,[7]whereIis a known external electric current applied on the membrane.

Putting Eqs.(17)–(19) into Kirchhoff’s current law, the fractional Hodgkin–Huxley equation of the membrane potential can be found

Because of lack of information on the mechanotransduction process of axons,the expressions ofm2,m3,m4,k2,k3,k4,η2,η3,η4,Qsare difficult to know,so in Ref.[7],the author made a simplification to neglect them by a scales comparison.However, these factors should affect the characteristics of axon action.So, in the following study we will treat the general cases by Noether symmetry analysis to study these neglected parameters and how they affect conserved quantities and the expression of these parameters, which may be useful for numerical solutions of the mechano-electrophysiological coupling equations of axons.

3.Variable order fractional Noether theorem of neuronal membrane dynamics

We introduce a one-parameter infinitesimal Lie transformation group in space(t,qs,˙qs)

whereεis an infinitesimal parameter,andξ0(t,q, ˙q),ξs(t,q, ˙q)are infinitesimal transformation generators.The infinitesimal generator vector

is the operator for the infinitesimal generator of the oneparameter Lie group of transformations(24)in space(t,q, ˙q).The first prolongation of the infinitesimal generator vector[34]is

which defines a first extended one-parameter Lie group of transformation in space (t,q, ˙q) by partial derivatives, (˙)means first derivative tot.

The Hamilton action is

Under the infinitesimal transformation, the curveγis transformed to curveγ?.The corresponding Hamilton action is transformed to

The variation ?Sof Hamilton actionSis the main linear part of the differenceS(γ?)?S(γ)to infinitesimal parameterε,and we have

where ?denotes anisochronous variation, andδdenotes isochronous variation.Expanding the above equation,we have

and using the relation

wherefis an arbitrary function oft,we can get

Put the infinitesimal transformation Eq.(24)into Eq.(30),and the following expression can be obtained:

Definition 1If the variation of the Hamilton action satisfies

the infinitesimal transformation(24)is a Noether symmetrical transformation.

Based on definition 1,we can get the Noether symmetry criterion.

Criterion 1If the infinitesimal generatorsξ0(t,q,),ξs(t,q,)satisfy

the transformation invariance is named Noether symmetry,which is also called variational symmetry.For Noether symmetry we can deduce the invariant.

Theorem 1For the Lagrange system (11), if the generatorsξ0(t,q, ˙q),ξs(t,q, ˙q)of the infinitesimal transformations have Noether symmetry(37),there exist fractional conserved quantities as

which are called variable order fractional Noether conserved quantities.We can verify this theorem by defniingIN =0,which denotes thatINis a conserved quantity.[26]

In the proving process of this theorem we have used condition(37)and variable order fractional Lagrange equation(11).In fact we can generalize the Noether symmetry to nonconservative dynamical systems.

Definition 2If the Hamilton action is generalized quasiinvariant under the infinitesimal transformation group,that is,the variation satisfies

the infinitesimal transformation (24) is a generalized quasisymmetrical transformation, whereGN(t,q,0i(t)q) is a gauge function,andδqsis the sum of the virtual work of the generalized non-conservative force.

Based on definition 2,we can get the generalized Noether symmetry criterion.

Criterion 2If there exists a gauge functionGN(t,q,0i(t)q) that makes the infinitesimal generatorsξ0(t,q,),ξs(t,q, ˙q)satisfy

the infinitesimal transformation is named variable order fractional quasi-Noether symmetry.The Noether symmetry can always lead to conserved quantities.

Theorem 2For the Lagrange equation Eq.(10) of neuronal membrane dynamics, if the infinitesimal generatorsξ0(t,q,),ξs(s,q,) satisfy equation (41) (criterion 2), the system has the following first integrals:

which are variable order fractional Noether conserved quantities.

Proof

In the proving process of this proposition we have used condition(41)and variable order fractional Lagrange equation(10).

4.Solutions of Noether symmetry generators and conserved quantities

Putting the exact form of LagrangianL(6)and dissipative functionψinto the Noether identity Eq.(41),we have

Next let us discuss the structures of Noether conserved quantities when external nonpotential forcesQs ／= 0 (s=1,2,3,4,8,9,10).We will show that the forms of generalized potential forces impact the integrability of the system.In the following we use the approximate relation (3) to simplify the calculus,and for convenience we still use the symbols0i(t),ti(t).

and we can work out the gauge function

The corresponding Noether conserved quantities are

and we can work out the gauge function

The corresponding Noether conserved quantity is trivial withIN=0.

Ifk1= const.and the generalized nonpotential forces have formsQ8=η20(t)q8,Q9=η30(t)q9,Q10=η40(t)q10,we have solutions

and we can work out the gauge function

The corresponding Noether conserved quantity is

which denotes the micro mechanical potential energy.

Ifk1=const.,C=const.,we have solutions

and we can work out gauge functions respectively

The corresponding conserved quantities are

which denote the sum of the electric energy and the micro mechanical potential energy.

In this section we have discussed the effects of parametersk(q2,q3,q4),C(q1) and non-potential forcesQson the forms of Noether conserved quantities.From the above calculations we can conclude that the Noether symmetry and Noether conserved quantities are strongly determined by nonpotential forces and the material parameters.We did not give all the conserved quantities for the neuron membrane dynamics model, because we can see the conserved quantities are strongly dependent on the external non-conservative forces and characteristic parameters of the neuron membrane.But we give general formulations(41)and(42)to calculate its conserved quantities.

5.Conclusion

Considering a neuron axon without myelin as an axissymmetrical cylinder with homogeneous deformation in a neuron membrane, we unified the deformation and electrophysiological multiscale coupling process through Hamilton’s principle in generalized coordinates with fractional derivative of variable orders.The Lagrange equation (10) of the variable order fractional multiscale mechano-electrophysiological model of the neuron membrane is given through which we can deduce the variable order fractional differential equations of motion found in Ref.[7] and the variable order fractional Hodgkin–Huxley equation.The variable order fractional Noether symmetry criterion (37) and (41) and Noether conserved quantities(38)and(42)are given under Lie group transformations.Through the variable order fractional Noether criterion we work out some solutions that correspond to Noether conserved quantities under different external stimuli.The results show that Noether conserved quantities closely depend on the external nonconservative forces and material parameters of the neuron, which need to be further verified by experiments.Other symmetrical methods,like Lie symmetry and Mei symmetry can also be applied to this system,and these will be studied in the future.

Acknowledgment

Project supported by the National Natural Science Foundation of China(Grant Nos.12272148 and 11772141).