A model for the nerve impulse propagation using two first-order differential equations

Volume 98A, number 1,2

PHYSICS LETTERS

3 October 1983

A MODEL FOR THE NERVE IMPULSE PROPAGATION USING TWO FIRST-ORDER DIFFERENTIAL EQUATIONS K. RAJAGOPAL Department ofPhysics, Bharathidasan University, Tiruchirapalli 620 023, Tamilnadu, India Received 10 January 1983

A simple model is proposed for the nerve impulse propagation using two first-order differential equations, which are found to be completely integrable. The solutions have been obtained explicitly in terms of elliptic functions to the propagating waves and pulses. This model also predicts linearity of the frequency—current relationship.

The phenomenological Hodgkin—Huxley model for the propagation of a voltage pulse along the membrane of the neuronal axons [1,2] is a satisfactory, empirically supported theory. Even though numerical computations of the partial differential equations of the model show remarkable agreement with experimental results, they are difficult to handle analytically. As these equations are found to be highly nonlinear, linearizations are not ideal approximations. FitzHugh [3] then introduced a second-order model which fairly predicts several results observed experimentally by Hodgkin and Huxley. Recently Hindmarsh and Rose [4] put forward a model which predicts a new property, a long interspike interval that was not shown clearly by the previous models, All the three models discussed above were found to be completely non-integrable and it is the aim of the present paper to suggest a model, which is cornpletely integrable, for the nerve impulse propagation using two first-order differential equations. Although this model seems to be a generalization

nomenological variables n and h respectively; and X could be identified with the Hodgkin—Huxley phenomenological variable m),then ~ = arn’~ Y —

V\

—

/

J

‘

where a is a constant. The rate of change of the intrinsic current is assumed to be given by = b Y 2 —

where b is a constant. Experimental observations of Hindmarsh and Rose [4] have suggested a cubic nonlinear form forf(X). Also, they have obtained that the time course of the applying stimulus current Z(t) depends nonlinearly on the membrane potential. In view of their observations, in the present case is taken 13 = k ~ k X3 —

~‘ ‘~

of the FitzHugh model, it can be developed from first principles. For a space-clamped axon if it is assumed that the rate of change of the membrane potential ~t’ depends lmearly on Z and Y, and depends nonlinearly on the membrane potential X itself (here Z represents any current stimulus applied through the electrode to the axon; Y represents the intrinsic current which could be identified with the Hodgkin—Huxley phe-

0.031-9163/83/0000—0000/s 03.00 © 1983 North-Holland

~‘

where

1

2

‘

and k

2 are arbitrary constants. Also the time course of the applying stimulus current is taken as Z(t)=yX—~X

(4

where y and r~are constants with the values ‘y k1 + b/a and i~= k2.

Considering eq. (1), integrating once with respect to t and putting (2), (3) and (4) therein, we have 77

Volume 98A, number 1,2

Y

=

PHYSICS LETTERS

3]= aX

ab [

—

fIX3

(5)

,

7X — r~X where a = ab’~and 13

ab~.One can obtain the solution to the cubic nonlinear differential equation (5) in terms of elliptic functions [5,6]. There are different types of solutions possible, depending upon the sign and magnitude of a and i3 in eq. (5). For example,

a>0,f3>0:

x2

1 and is neglected for the present discussion. The results obtained so far are in good agreement on actual plotting for suitable values of a and 13 with the numerical results of ref. [7] which shows the existence of periodic wave trains as well as solitary pulses. It is worth noting to add that the frequency— current relationship could also be obtained using this

(i) X = C cn(Xt + 0, k), k2 = 13C2/2(—cs + fIC2), =a+~,2 ICI~(2a/I3)1’2 (6)

model. Multiplying eq. (1) byb and(2) bya and adding we obtain bS’+a)~’=abZ. (10)

(ii) X = C dn(Xt + 0, k),

Integrating the above, we have

,

X2=j3C2/2

k2

ICl~(2a/!3)”2

=

2(13C2

—

a)/13C’2, (7)

,

a <0, I3 <0 and a =

(11)

the duration of the spike, the frequency of the osdillation is given by

—Ial, 13 = —till:

(iii)X=C sn(Xt+ 0,k), =

bEiX+aL~Y=abZt,

where and ~Y are changes values of X and Y ~X, and assuming thatthe they occurininthe time t. Ignoring

and

k2

3 October 1983

131C2/(2lal

—

113IC2), x2

f =

—

I/31C2/2

,

(a/13)112~C~(2a/I3)”2

(8)

1/i’ =abZ/(bL~X+aL\Y).

(12)

We observe thatf has a linear relation with Z which agrees exactly with Hindmarsh and Rose’s numerical predictions.

where k2 is the square of the modulus of the elliptic function. Similar forms can be given for the cases a < 0,13>0 and a > 0, i3 < 0. In eqs. (6)—(8), 0 is the mitial phase. Without loss of generality one can choose the initial phase 0 = 0. In the present discussion the solutions (6) and (7) are of much interest since solitary pulse solutions could be obtained by letting the modulus of the elliptic function k 1. While in the present case solutions (6) and (7) become

References

[6]

A.L. Hodgkin and A.F. Huxley, J. Physiol. 116 (1952) 473, 497 A.L. Hodgkin and A.F.Huxley,J. Physiol. 117 (1952) 500. R. FitzHugh, Biophys. J. 1(1961)445. J.L. Hindmarsh and R.M. Rose, Nature 296 (1982) 162. P.M. Mathews and M. Lakshmanan, Ann. Phys. (N.Y.) 79 (1973) 171. M. Lakshmanan and K. Rajagopal, Phys. Lett. 82A

X ~(2a/13)~’2sech a”2t as k 1 the solution (8) gives a tanh-function solution as k

[7]

(1981) 266. J.W. Cooley and F.A. Dodge, Biophys. J. 6 (1966) 583.

[1]

449, [2] [3]

[4] [5]

—~

78

(9) -~