Bulletin of Mathematical Biology, Vol. 43, pp. 21 32 Pergamon Press Ltd. 1981. Printed in Great Britain :© Society for Mathematical Biology

0090-8240/8 I/0101-0021 S02.00/0

NONLINEAR BIOLOGICAL OSCILLATORS : ACTION- ' ANGLE VARIABLE APPROACH WITH AN APPLICATION TO COWAN'S M O D E L OF NEUROELECTRIC ACTIVITY.

I I VIRENDRA SINGH

Theoretical Physics Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

We obtain within the action angle variable approach new expressions, involving the Dirac delta function, for time periods and time averages of dynamical variables which are useful for nonlinear biological oscillator problems. We combine these with Laplace transformation techniques for evaluating the required perturbation expansions. The radii of convergence of these series are determined through a complex variable approach. The method is powerful enough to yield explicit results for such systems as the two species Volterra model, Goodwin's model of protein synthesis etc. and as an illustration, is applied here to_ Cowan's model of neuroelectric activity. We also point out the usefulness of tl~e action integral in the case where parameters occurring in dynamics have slow time variations.

1. Introduction. A number of interesting biological problems lead to a consideration of nonlinear oscillators (Goel et al., 1971; Pavilidis, 1973; Rescigno and Richardson 1973; Waltman, 1964). Some interesting exa- mples of this are provided by the Lotka Volterra model of population dynamics (Lotka 1924; Volterra 1931), the Goodwin model for a control equation for protein synthesis (Goodwin, 1963) and Cowan's model of neuroelectric activity (Cowan, 1968, 1972). Existence of non-linearity precludes, in general, any analytical results for these problems and one has to resort to various approximations or numerical methods (Minorsky 1962, Hirsch and Smale 1974). It would be rather pleasing if useful analytical methods were available for such problems.

For the two species Volterra system, Frame (1974) succeeded in obtaining a systematic convergent power series expansion for the time period of oscillation. His work was a tour de force and appears to depend

21

22 VIRENDRA SINGH

on very specific aspects of the Volterra system. It does not appear possible to generalise it to other non-linear systems. For the Goodwin model similar results were obtained by us earlier (Singh 1977). Analysing the reasons for a successful t reatment of these two examples we have been able to devise a method which is rather powerful and leads to these earlier results in a rather effortless manner. The method is applicable to a large class of non-linear oscillators. We however require that the "Hamiltonian" formalism be applicable to our dynamical problem. The method, making an efficient use of the action integral, can be used to obtain the time period of oscillation as well as for obtaining time averages of various interesting dynamical observables. The radius of convergence of the needed power series expansions are obtained by a consideration of singularities of functions of complex variables which are defined by integrals via methods familiar in high energy physics. As a demonstrat ion of the method we apply it to Cowan's model of neuroelectric activity which has not previously been solved analytically.

We may note here that Hamiltonian non-linear oscillators provide a rather ideaiised model of biological phenomena. The more realistic mathematical models are often not of the Hamiltonian variety. In parti- cular Cowan's model of neuroelectric activity (Cowan, 1968, 1972), which is discussed in the present paper, has itself been modified to a non- conservative one (Wilson and Cowan, 1973). We believe however that as a preliminary to the study of more realistic systems, and sometimes in their own right, non-linear Hamil tonian systems still deserve some study.

2. Action-angle variable approach. 2.1. Hamiltonian equations of motion. Consider a non-linear oscillator,

with two degrees of freedom, which is described by a Hamiltonian function H(p,q) of two variables p, q and satisfies the Hamiltonian equations of motion

dp/dt = - [OH(p, q)/gq], dq/dt = - [OH(p, q)/gp].

It can be shown that H(p,q) is a constant of motion. We shall deal only with those systems for which phase space, i.e. (p,q) plane trajectory given by H ( p , q ) = E is a closed, non-self-interacting curve (Fig. 1). The constant of motion of E shall be referred to as"energy" . The motion would be then periodic with time period T(E) (see e.g. Goldstein 1959).

It is known that for such periodic systems, if we are interested only in frequency of motion or in time averages of various dynamical quantities, then in principle the best approach is via acti0n-angle variables. This

NONLINEAR BIOLOGICAL OSCILLATORS

P h

~ , q ) = E

23

Figure l. The trajectory in the phase plane, i.e. (p,q) plane, of the system for "energy" E. The action integral J(E) is the area of the region enclosed by the

trajectory H(p, q) =E.

approach was widely used in the days of old quantum theory in the early twenties and therefore some of the best descriptions of it are found in texts dealing with old quantum theory (Born 1927, Tomonaga 1962).

In practice, however, the expressions involved remain purely formal and hard to make use of. We shall first obtain an elegant expression (2.6) for time period T(E) involving the Dirac delta function. Surprisingly we have not come across this expression before in the literature and we attribute this to the circumstance that the Dirac delta function was introduced in the literature after the decline of old quantum theory. Evaluation of this expression looks even more difficult but we shall see that a judicious use of the Laplace transform allows us t~ evaluate it in most cases of interest. A similar technique works for other quantities of interest.

2.2. Period of oscillation and time averages. Consider the action in- tegral J(E) for "energy" E, given by the area enclosed by the trajectory of the system in the phase space (i.e. the (p,q) plane) and given by the expression

J (E) = ~p(J~, q) dq, (2.1)

where p(E, q) is obtained by inverting

HIp(E, q), q] =E. (2.2)

We then have an important result of classical dynamics which expresses the time period T(E) of the motion in terms of the action integral through

T (E) = [dJ (E)/dE3. (2.3)

24 VIRENDRA SINGH

The usual expression for T(E), following from (2.1) and (2.3) is (Tomonaga 1962, Appendix VIII);

T(E) = [(OH~@)2 + (OH/~q)2]~j, (2.4)

wher6 ds is a line element of the phase space trajectory of the system and integration is over one complete cycle.

It is to be noted that for non-linear systems in general the inversion required to obtain p(E,q) by solving (2.2) is in general not possible analytically, The expression (2.4), while formally correct and exact, cannot in general be used. Our first task is to recast (2.1) and (2.3) into forms which do not require inversion of (2.2) for p(E, q).

We now note that we can rewrite J(E) as a double integral, instead of the present single integral (2.1),

J ( E ) = f fdpdqO[E-H(p,q)-l, (2.5)

where O(x) is the Heaviside step function of argument x, given b y

0 ( x ) = l if x > 0

= 0 if x < 0 .

We now use a result f rom distribution theory (Fr iedman 1956);

dO(x)/dx = 6 (x),

where 6(x) is the Dirac delta distribution, to obtain

T ( E ) = f f d p d q 6 [ E - H ( p , q ) ] . (2.6)

O u r next task is to give a practical me thod which can be used to evaluate this elegant expression for T(E). We find the Laplace t ransform of T(E) to be a very useful tool for this purpose. Consider

t o o

L(x) = dE T(E)e -ex . ', (2.7) 0

Using (2.6) we obtain

N O N L I N E A R B I O L O G I C A L O S C I L L A T O R S 25

L ( x ) = dp dqe -mp'q)x (2.8) oo oo

If H (p, q) = H 1 (P) + H2 (q)then

where

L(x ) = L 1 (x)L 2 (x)

f oo

Li(x) = dye n,(y)x - - o f )

We have found that for the non-linear models of the two species Volterra system (Goodwin, 1963; Cowan, 1968, 1972 etc.) the needed Laplace transformation L(x) can be written down analytically. One then obtains the time period by taking the inverse Laplace transformation. In particular asymptotic behaviour of L(x) for x ~ oo determines the small E behaviour of T(E).

It may be that in a different non-linear problem another transform such as the Mellin transform would be more useful than the Laplace transform.

The time average ( f (p, q)) of a dynamical quantity f (p, q) over complete periods is similarly given by

( f (p ,q ) )T (E)= f fdpdqf(p, q)6[E-H(p,q)] (2.9)

and can be evaluated by techniques similar to those used for T(E). 2.3. Radius of convergence for power series in E of T(E) and other time

averages. For this purpose we use formal expressions such as (2.4) for T(E) which express it as a function of E through an integral. While these expressions are not at all suitable for evaluating the various terms of the power series in E, they are useful for extending the defininition of these functions to a complex variable z such that

5 ~ T ( z )=T(E) for real E. ~ E + i0

Now let the singularity of T(z), which is nearest to z=0 , be given by Z=Zo in the complex z-plane. The radius of convergence of T(E) as a power series in E would be given by

I l<lzol

26 VIRENDRA SINGH

For determination of z o we can use the methods developed recently in high energy physics (Eden et al., 1966). This approach to a determination of the radius of convergence dispenses with explicit determination of the coef- ficients of the power series and messy estimates on numbers such as Bernoulli numbers, used for example by Frame (1974) for the two species Volterra model.

3. Application to Cowan's Model of Neuroelectric Activity. 3.1. The equations of the model. This model is somewhat Similar to the

two species Volterra model where thalamic cells "feed" predatory cortical cells. The equations describing the model are

d N t / d t = [ a I - al2N2]N 1 (1 - N 1 )

dN2 /d t= - l [ a 2 -a21N1]N2(1 - N 2 ) , (3.1)

with a~, a2, a12 , a21 > 0 where N 1 and N 2 refer to neuroelectric activity of the relevant cells.

To obtain a Hamil tonian description of this system we define the generalised momentum p and coordinate q by

nl =a2/a21, n2 =al /a l2 ,N1 = (n 1 ealV/1 +n 1 ea'P),

g 2 = (n2eazq/1 + n2 ea2q). (3.2)

With these definitions the equations of motion (3.1) can be re-expressed as

where

an'd

dp/d t= -OH/Oq and dq/d t= +aH/Op,

H (p, q) = h(p; n l ,a 1) + h(q; nz, a2).

nah (s; n, a) = ln(1 + n e as ) - n a s + (1 - n) In (1 - n). (3.3)

The constants in h(s;n,a) minimum is at zero.

3.2. Time period T(E). Section 2.2,

have been fixed in such a way that their

We have , using the formalism developed in

L ( x ) = dEe-E~T(E)=l ( x ;n l , a l ) l ( x ;n2 , a2), 0

(3.4)

where

We therefore have,

NONLINEAR BIOLOGICAL OSCILLATORS 27

f clo

l(x;n,a)= dse -~h(~;",")

l(x;n,a)= dsexp - [ln(l +ne"~)-nas+ (1-n)ln(1-n)] . --0(3

(3.5

To evaluate to u given by

this integral we try the change of integration variable

s = in u - ( l /a) In n

to obtain

Using a result given in Magnus and Oberhettinger (1954, p. 4) we obtain

#ka( I '

where 7(z) is related to gamma function F(z) by

V(z) = z =-~ e - =,,/(27c)7(z). (3.7a)

We thus have

For large z values the asymptotic behaviour of 7(z) is given by

n=N ( ( ) " - }BZn I)nLO(1/zZN)I , (3.7C)

where Bzn are the Bernoulli numbers, given by

F2(2n)!3 ~

28 V I R E N D R A S I N G H

We thus finally have

dEe_E~T(E) = x / a i ) , _--n 1__ x/a2 x 2 , 0 1 a l

(3.8)

where

T(O) = (2xl~/[ala2 (1 - n 1 ) (1 - n 2)-1) (3 .9 )

is, as we shall see, the period for E = 0 and

fl(a, b) = [y(a)7(b )/y(a + b)].

The power series expansion of T(E) around E = 0 is easy to work out by using the asymptotic expansion of the right hand side of (3.8) which can in turn be worked out by using (3.7c) to any desired order. We have, for x ---> (]~ ~

f l dEe-~:<T(E)=(T(O)/x)[1Ar(Cllx)-i-(C2/x2)-~(C3/x3)-}- " ' ' ]

where

C,= (al/12)[( l~nl)-n, l+ (ae/12){(~)-n21.

c2=½(cl)2;

[1-139 [ 3 _ 1 _ [ . n 3 ) 3 1 } + Z ~ I ( l _ n l ) C3 =tk -i 4 l ' ' t , - . , + ala21288[(1/1 - nl) - nl] [ (1/1 - n2) - n2] C1,....

which on taking the inverse Laplace transformation leads to

(3.10)

T(E) = T ( 0 ) [ 1 + C1E + C2(E2/2!)+ C3(E3/3 !)+...]. (3.11)

The radius of convergence of this series will be calculated now. 3.3. Radius of convergence of power series in E for T(E).

phase plane trajectory with "energy" E we have Along the

H(p, q) = h(p; n l , a I ) q- h(q; n2, a2) =E.

We label points along the trajectory by a coordinate 0 defined by

N O N L I N E A R BIOLOGICAL OSCILLATORS 29

h(p; nl, a I ) = H1 (p) = E cos20

h(q ; n2, a2 )= H 2 (q) = E sin20 (3.12)

2 ~ 2 0 ~ 0 .

F rom (3.12)

[ d H ~ (p)] d p / d t = - 2 E cos 0 sin O(dO/dt).

Using the Hamil tonian equations of motion, we get

dHI (P) ] [ dH2(q)] Esin20(dO/dt). (3.13) - ~ JL ~ J=

It therefore follows that

T(E)=E fI={[dH1 d0s in20 (p)/d~p] - [d-H 2 (q)/dq] j "

To obtain the radius of convergence we extend this definition of T(E) to a function of a complex variable z th rough

T(z) = z (q)/dq] }' (3.14) [ [dH1 (P)/dp~-dH 2

where we now stipulate

H I (p) = z cos20

H 2 (q) = z sin20

and study the singularities in the complex z-plane of the function T(z). Now for a fixed 0,

[dH 1 (q)/dp] dp/dz = cos20.

Hence p as a function of complex variable z is singular at points at which

dH 1 (p)/dp = O,

30 VIRENDRA SINGH

i.e. ( 1 - n l ) e ~ l V = 1,

i.e. a l p = - l n ( 1 - n l ) + 2 m ~ i m = 0 , + _ 1 , + 2 , . . . .

For these values of p, the complex variable z takes the values

z = - 2m~i/a 1 cos20.

Similarly q as a funct ion of the complex variable z is singular for

z=-(2m'~i /a2s in20) where m' = 0, -+ l, _+ 2, . . ..

Since 0 varies f rom 0 to 2re, the singulari ty nearest to the origin is at

iz = minl-(2rc/a 1 ), (2~/a 2 )].

Hence the power series for T(E) in E given by (3.11) converges for

IEI < 2~ min [(1/a 1 ), (1/a 2 )].

3.4. Time averages. Let us denote the time average of N't" F,n,,(E). We have

F m , ( E ) T ( E ) = f f d p d q N ' l ' N " 2 6 [ E - H ( p , q ) ] .

Taking the Laplace t r ans fo rmat ion

f ~ = F,~(x ; nl, a 1 )F,(x; n2, a2 ) , dE e- EXFmn(E )Y (E, ) o

where

Fj(x;n ,a)= ds(ne°S/l +ne~S)iexp[-xh(s;n,a)].

To evaluate this integral we again make the subst i tu t ion

s = l n u - ( 1 / a ) l n n , i.e. n e ~ = u ~

to obtain

(3.15)

N~ by

(3.16)

(3.17)

N O N L I N E A R B I O L O G I C A L O S C I L L A T O R S 31

Fi(x;n,a)=n-X/ . ( l_n) ~/.(1-./.) o k( l ~u"~+~7~")J

= n-'~/" (1-n)-x/ , (1 _,/,)~F[j + (x/a)]F[ (x/a)(l-n/n)]_; F[j(x/na)] 3

[ F(x/a) J ( F ~ ) ] • (1 -n/n)]. (3.18)

The rest of the calculation proceeds in exactly the same fashion as for T(E) and hence need not be given.

4. Slow Time Variation of Parameters. It may happen that various parameters occurring in the equations of motion are not strictly constants but vary extremely slowly. This may, for example, be due to external influences which vary slowly with time. The theory of adfabatic invariants is helpful in such cases. For example, if for a simple harmonic oscillator t h e spring constant changes slowly, then one can show that the ratio of energy to its frequency remains constant.

In the problems we have been discussing here, the action integral J(E) can be shown to be an adiabatic invariant (Tomonaga, 1962). Once the problem has been solved for T(E) it is very simple to obtain J(E) via

I E

J(E) = dE'T(E'). 0

We also have the useful relation

x dE e-~XJ(E) = dEe F'XT(E). 0 0

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Caianiello, pp. 181-188. Berlin: Springer. 1972. "Stochastic Models of Neuroelectric Activity." In Towards a Theoretical

Biology, Ed. C. H. Waddington, Vol. 4, p. 169-188. Edinburgh University Press. Eden, R. J., P. V. Landshoff, D. Olive and J. C. Polkinghorne, 1966. Analytic S-matrix.

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32 VIRENDRA SINGH

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Friedman, B. 1956. Principles and Techniques of Applied Mathematics, Chapter 3, New York: John Wiley.

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RECEIVED 6-19-79

REVISED 1-7-80