Biological Cybernetics

Biol. Cybernetics27, 155--163 (1977)

9 by Springer-Verlag 1977

Limit Cycle Model for Brain Waves F. Kaiser Institute of TheoreticalPhysics,Universityof Stuttgart, Stuttgart, FRG

Abstract. The extraordinary high sensitivity of certain

biological systems to weak electromagnetic signals motivated Fr6hlich to establish his brain wave model. Although the basic assumptions of such a model are still highly speculative, it seems worth while to investigate the quantitative dynamics of it. In this model coherent excitations in the 10 I1 Hz region in combination with a collective enzyme-substrate reaction create and stabilize a limit cycle oscillation. In the present paper the existence of the limit cycle and its collapse due to external stimulation is studied. The bifurcation of this periodically oscillating resting state is analysed and the stability of additional steady states is proven. Approximations are discussed and a correspondence between the limit cycle model and nerve impulse generating models is established.

I. Introduction

During the Neuroscience Program Work Session "On the Action of Weak Electric Fields on the Brain" (MIT, November 1974) it has been proposed by H. Fr/Shlich that in the Greater Membrane a collective enzyme substrate interaction could lead to a limit cycle connected with a corresponding electric interaction. The required long range interactions would be carried by the conjectured coherent excitation in the 1011 Hz region (Fr6hlich, 1968, 1972, 1975a, b). The frequency of the limit cycle may, however, well be in the 10 Hz region and refers to an activation-deactivation cycle of the enzyme. In this way large regions of the enzymesubstrate system in this model would oscillate coherently. The electric interaction arises from the proposed high electric dipoie moment of the activated enzymes. However, it should be emphasized that the model heavily is based on general results of the physics of dielectric materials. The assumed consequences o f these

physical laws in biological matter are still highly speculative. Among these consequences is the spectral location (1011Hz) of the oscillations as well as the possibility of long range coherence and the rise of collective behaviour. The model may be viewed as a first approach to find a physical basis for the interaction of tissue with weak electromagnetic fields. The relation of the supposed collective enzymatic reaction to any reactions known from neurophysiology and biology is far from being established. Although the basis altogether is rather speculative, the model may serve as a starting point for a search which focuses on properties of the membrane surface in its longitudinal extensions rather than on the transmembrane properties. Up to know only the latter has been a center of interest. As yet there is only little experimental evidence for Fr6hlich's initial assumptions, but it seems to grow. Devyatkov (1974) reports on the influence ofmm-band electromagnetic radiation (1011 Hz region) on biological objects. Experimental support is given for Fr6hlich's conjecture of a metastable state in enzymes (Kollias and Melander, 1976). Both results have not been reproduced to the author's knowledge. However, there is hope that in the near future new experimental results become available, since several groups started investigations on the basic questions. It is the purpose of the present investigation, to evaluate quantitatively the dynamical properties of Fr6hlich's model. In this model the limit cycle (LC) interacts with an external electric field. In (Fr6hlich, 1977) it has been conjectured that if such a field acts sufficiently long, the limit cycle may collapse. This would lead to an electric signal much stronger than the externally applied field and might thus generate a nerve signal although the applied field would be too weak to do so. In this manner we have a possibility to describe the high sensitivity of biological systems to weak elec-

156

~N

~,

II. Physical Background and the Model Fr6hlich has suggested that collective enzymatic reactions take place in the "Greater Membrane" provided it is supplied with substrate molecules. Following the suggestions of Fr6hlich we consider a population of enzyme molecules of which N are in the excited polar state and Z are not excited. S is the number of substrate molecules. Both the enzymes and the substrate show long range selective interactions which tend to increase their number by influx. For this enzyme-substrate reaction Fr6hlich has given the following system of nonlinear differential equations (Fr/Shlich, 1977):

Fig. 1. Steady states and cycles for (4) end (5) (qualitative behaviour only). The arrows indicate the direction of the time evolution of the system. For physical reasons only the positive phase plane (N, S >0) is allowed

tromagnetic signals. The LC stores the energy of the incoming weak electric signal until finally the collapse occurs.

The present paper represents a first stage in which the mathematical properties of the limit cycle are investigated. It is found that in the case of a nearly time independent external field there is a wide region for the relevant parameters, for which the resting state of the membrane is an oscillating one. If the applied field exceeds a certain critical value the limit cycle oscillation breaks down. A stable non-oscillating steady state is then built up. This behaviour can be viewed as a transport phase transition. The transition occurs at the point of bifurcation from a stable oscillating situation to a stable non-oscillating one (Andronov et al., 1973). There is a great literature dealing with pulse propagations in membranes. All calculations are based on mathematical models for the physiological processes. The most famous formulation are the Hodgkin-Huxley Equations (Hodgkin and Huxley, 1952) for the description of the membrane current and the excitation in nerves and some simplifications of it (e.g. Fitz-Hugh, 1961 ; Nagumo et al., 1962; Green and Sleeman, 1974). In Fitz-Hugh's and related equations the resting state of the nerve membrane is a unique nonoscillating stable state. If an external stimulus exceeds a threshold value then an onset of solitary pulses and of travelling waves (train o fimpulses) sets in. In our model, on the contrary, the onset of propagating pulses is determined by the collapse of an oscillating resting state. After the collapse the situation changes drastically. In order to describe spatial non-homogenous solutions (i.e. propagating pulses) diffusive processes have to be added to the other processes. In a forthcoming paper we shall discuss the space dependent problem. There we also compare our equations with those of Fitz-Hugh.

dtN = aNZS - fiN

(1)

d,S = 7 S - ccNZS

(2)

d,Z = f i N - ),(Z - A) - aNZS

(3)

(Z,N are the numbers of unexcited and excited enzymes, S the number of substrate molecules.) It is assumed that the rate of increase of the activated enzymes is proportional to their own concentration N, to the concentration of unexcited enzyme molecules Z and to the number of substrate molecules S. Each transition from the nonpolar (or weakly polar) ground state of the enzyme to the highly polar excited state leads to the chemical destruction of a substrate molecule. Furtheron, there are spontaneous transitions from the excited to the ground states. yS and ;~(Z-A) result fi'om the long range interaction, where A is the equilibrium concentration of the unexcited enzyme molecules in the absence of excited enzymes and substrates (N = S = 0). To simplify the above system of nonlinear differential equations, we assume that the equilibrium of the nonexcited enzyme concentration is reached very fast. Then Z can be treated as time independent. This procedure may be viewed as an adiabatic elimination process of the fast variable. Equation (3) is then irrelevant and (1) and (2) are the well known Lotka-Volterra equations (Volterra, 1931). din = - fiN + cANS

(4)

dtS =~;S- c~ANS.

(5)

(The number o f activated enzymes would correspond to the predator concentration, the substrate molecules to the prey population.) The behaviour of these two equations is well understood. We find two steady states

(ss): N O= 0 ;

S O= 0

SS 1

(6)

N O=y/c~A ;

S O=fl/o~A,

SS 2

(7)

157

The first one is a saddle point type of solution (in principle unstable), the second singular point is a center, which is orbitally stable. We can integrate (4) and (5). The result is (8)

N r S p = ce~A(N +S).

It represents an infinite set of closed curves in phase space (N-S-space), the so called cycles enclosing the steady state 2. This is illustrated in Figure 1. The cycles represent orbitally stable solutions which are periodic in time with an approximated circular frequency coo = 1//~. By this frequency the substrate and enzyme concentrations oscillate around their static equilibrium. We now introduce a linear transformation of variables : N = ~7 + v

s= ~

(9)

+~

(lO)

by the ions of the system. However, the time dependent fraction v should not be completely screened. This then gives rise to a macroscopic oscillating polarization P. The system in turn exhibits a tendency towards a ferroelectric instability. In this way a rate of change d t P is induced in proportion to P. The process is governed by an activation energy U that is proportional to P2ax _ p 2 , where Pm~x is the polarization of the ferroelectric state. Electric resistances against the system's tendency to become ferroelectric also have to be accounted for and give a further contribution - d 2 p (i.e. a relaxation term). With these two terms and assuming the macroscopic polarization P to be proportional to the time dependent number v of excited enzyme molecules we obtain a total additional nonlinear term of the form

This "dielectric" term must be added to the "chemical" expressions in (11) so that finally we have dtv

which, instead of (4) and (5) yields the equations

(15)

d, v = ( c2 e - r2~2 - dZ)v .

= ?or + (c2e -r2v2- d2)v + o~Arrv

(16) (17)

d,v = 7 a + o:Arrv

(11)

d~a = - f l v - o~Arrv .

d t ~ = - f l v - c~Arrv .

(12)

The combined interactions considered here lead to equilibrium states which may easily be influenced by electric fields as well as by changes of the chemical situations. Since the polarization now has been incorporated, it is a matter of consistency to include also the interaction with an electric field. We do this by a further term on the right hand side of (16). Such a field indeed exists. It consists of two parts: one field which is due to thermal fluctuations and another one which is externally applied. Both fields are time dependent and both are expected to contribute to the steady state and stability conditions. The mathematical complications inherent in (16) and (17) are particularly transparent if one derives a second order differential equation for the variable v by elimination of the variable a. The result is

In the new coordinate system the steady state solutions are v o = -?/c~A ;

a o = -fl/o:A

SS 1

(13)

vo = 0 ;

~0 = 0 .

SS 2

(14)

By the linear transformation the stability of the steady states is not changed (i.e. SS 1 is a saddle point, SS 2 is a center). The reaction kinetic equations are now supplemented by additional terms, which represent the high dipole moment o f the activated enzyme. Following Fr/Shlich's ideas we may assume that large regions of the system of proteins, substrates, ions and structured water are activated by the chemical energy available from substrate-enzyme reactions. These regions are expected to oscillate coherently with a frequency of about 1011Hz (Fr6hlich, 1968). By means of the interaction chemical oscillations in the number of substrate and activated enzyme molecules with the very low frequency ] / / ~ might be established. This oscillation also represents an electric oscillation via the high dipole moment of the excited enzymes. The oscillation frequency is determined by the rate y of the long range attraction of the substrate particles (autokatalytic reaction) and by the decay rate fl of the excited enzyme molecules. This frequency may be very low, possibly in the 20 Hz region which has been found in the brain. The electric dipole moment o f the time independent fraction N o of excited enzyme molecules is likely to be screened

~A

d

2

1

dttv - 7 +~Avv ( t v ) + ~ + o~Av 9 { - c~A(Tv + c~Av z + F )

+ c2(? -- 2 7 F Z v z -- 2c~AFZv 3) 9 e - r 2 v 2 _ d27}dt v + ( c ~ A F - fiy)v + ~A(cZe-r2~2_

d 2 _ fl)v 2 + d t F .

(18)

Our present investigations will be restricted to the situation where either F is nearly independent of time or F depends on time but the equations are linearized. Both cases lead to considerable mathematical simplifications. The electric field may be decomposed in two

158 Y

III. Steady States and Periodic Solutions in the Quasistatic Case

VI1)

We have to solve (16) and (17): dry = ?o + (c2e- r2~ _ d2)v + ~Aav + F d,G= -- & - - . A G v .

%

First we look for steady states and prove their stability by Ljapunovs first method. The system (16) and (17) is readily seen to have SS 1 :

{v~= _ 3/c~A GO

y~l): C2d~+/~;

as a steady state solution (referred to as steady State 1). v o is calculated from

y~='/~-F+(d~+/b% ~A

y~3~: c2>d2+3 '

F - ~?fl +(c2e -r2v~ - d 2 _ /3)Vo= 0

Fig. 2. Graphic determination of the position and the stability of steady State 1, restricted to the case 73>c~AF

parts F(t) = F R(t ) + Fs(t )

(20)

which can be done graphically in a rather satisfying way. The zeros are given as points of intersection of the two curves (see Fig. 2)

?/3 - F +(d2 + /3)Vo

2a = ~

(21)

Yz = roe- r ~ .

(22)

(19)

where F R is the random field, F s the external systematic field. We first assume that the time average of the thermal field cancels:

(G(O),=0. Nevertheless, it has been proposed by Fr6hlich that even if ( F2(t) )t >>( F~(t) ) t

one may expect that weak external fields cause significant nerve pulses. The random field contributes to the behaviour of the system only in the absence of the external field. It then gives rise to occasional nerve impulses, a situation which is widely known from "in vivo" experiments. The restriction to an almost time independent external field practically includes the two limiting cases of very slow and very fast variing (averaged) fields. If F is slowly variable in comparison to all other processes the field might be considered constant with respect to time and enters the equations by its instantaneous value. If, on the other side, the field F changes very fast the system's variables do not remarkably change within one period and we can replace the nonconstant field by its steady state or time average value (adiabatic elimination of the extreme fast variable). A detailed discussion is only possible if one knowns the time scales for the relevant chemical and electrical processes.

We have to linearize the system around this singular point, i.e. we expand the nonlinear terms in our equations around this steady state (Vo, %) and drop all but the linear terms. This gives a sequence of linear differential equations. The eigenvalues of the determinant given by the coefficients of these linear equations are easily found. They read

21 = c Z e - r 2 ~ ( 1 - 2 F 2 v ~ ) - d e - f l = - ~ l - d 2 - / 3

(23)

22 = - ~Av 0 .

(24)

22 and 22 determine the stability of the steady state and also of the nonlinear system if one is close enough to the steady state. Several cases must be discussed : For v o > 0 and ~/- d 2 -/3 > 0 the SS 1 is a saddle point, whereas for ~/-d2-fi 0 the SS 1 is an unstable node whereas for t / - d 2 - f i < 0 the SS1 is a saddle point [details of the stability analysis and the relevant terminology may be found e.g. in (Andronov et al., 1966)3. We can reformulate the stability conditions as derivations of the two curves cited above : for vo > 0 we have a saddle point type of solutions if Y2 > Y'~ and a stable node if y~ o

T

B>o

in the entire phase space. We arrive at the result that limit cycles can exist, but their amplitudes must exceed certain critical amplitudes. This is shown in Figure 5. It should be noticed that we have here one of the desired results: the conditions for the existence of a stable limit cycle are completely independent of the external field F. Second, we show that the conditions of Hopf-bifurcation are fullfilled (Hopf, 1942; Marsden, 1976). Rewriting (25)

b

21/2= 89

s Fig. 5. The regionof a possibleexistenceof limit cycles.The limitcycle must surround the unstable focus, which lies within the hatched regionon the a-axisand must crossat least one of the two linesa and b

z

c~Af7

++_b/(cZ-d2-~AF)Z-4(fly-eAF))

(28)

the conditions that the SS2 is a stable focus are

/ < 4(fly-o~aF). At the bifurcation point the steady state gets unstable with respect to the parameter F when one increases F beyond Fcrit=Y(cZ-d2)/o:A. The eigenvalues still remain complex, but for F = Fcrit we have

\\

Re(21) = Re().z)=0 d v Re(21)[~=~c = - ~ A y - 1 + 0 , /~.. stabte /" "- lim.il

F c ---Fcrit.

cycte

(29)

Thus the conditions of the Hopf-bifurcation are fullf~led, i.e. periodic solutions do exist which bifurcate at S Fig. 6. Phase plane diagram. The trajectoriesinside the limit cycle spiral from the unstable focus(SS2)to the stable limit cycle,whereas all the trajectorieslying outside of the limit cyclespiral towards the limit cycle. SS1 is the unstable saddle point. The arrows on the trajectories define the time evolutionof the system

the whole behaviour inherent in this multiple steady state situation are a matter of laborous computation.

Let us consider now the limit cycles in more detail. First we apply the Bendixson theorem which can rule out the possibility of limit cycles. We have to study the function

B = dfl, v + d~dta.

(26)

If the function B does not change its sign in a certain phase space element O, then there is no limit cycle which lies entirely in #2. We have calculated the function r2v2(1 - - 2 F 2 v 2) - - d 2 q-

The possibility of the existence of limit cycles, the real existence of periodic solutions and the existence of an unstable focus in this region suggests the existence of at least one stable limit cycle. To evaluate it we have integrated the differential equations numerically. The phase plane diagram and especially the limit cycle itself, the steady states and the trajectories are drawn in Figure 6. In a first approximation the frequency of the limit cycle is given by 09 = Im(,~1/2)

IV. Existence of Limit Cycles

B = c2e-

F=F e

aA ( a - v)

(27)

=Im 89

2- ~;F)2-4(,y-o~AF

)

(30)

with

ca > d2 . The external field changes the position, frequency and amplitude of the limit cycle. It can be shown (by numerical integration) that the amplitude of the limit cycle does not change re-

161 O.)2

markably. However, if the bifurcation point is reached the amplitude necessarily goes down to zero (see Fig. 7b).

0~o2 _

!

I

UNIll

V. Discussion of the Results

o~AF I;Fc

V

~AF the steady State 2 is an 7 unstable focus (UF). I f F increases, bifurcation occurs at F~ = 7(c 2 - dZ)/eA and for F > F~ the steady State 2 is a stable focus"

N

I

1. For F > 0 and c 2 - d 2 >

UN

SF

UF+SLC

ISN I I I I I

I

a

UF + SLC

v

I

~SF. F = Fc

o

With the necessary restriction f17 > e A F (i.e. S = O) one finds in addition, that the steady State 1 consists only of a single saddle point. This follows from the fact that at the bifurcation point given by c 2 - d 2 = ~AF and for fi? Y > ~AF one has c 2 - d 2 - fl < 0. The above situation is supposed to be the most interesting one : The system exhibits self excited oscillations (SLC) around an unstable state. All trajectories in the positive plane of the phase space tend to this limit cycle. With increasing field F the limit cycle and the unstable focus is shifted to the left of the phase plane, the frequency of the oscillation decreases. If F approaches its critical value Fcfit, the limit cycle collapses and the only solution is a stable focus, which again is shifted to smaller values of a 0 with increasing field F (Fig. 8c). It should be noted that under certain conditions we can perhaps find a distinctly different behaviour: a stable focus surrounded by an unstable limit cycle and both surrounded by a stable limit cycle due to the scheme UF + SLC

F=Fc

~SF + ULC + SLC.

c~AF 2. For F > 0 and c 2 - d 2 < - we have no bifurcation 7

but only the transition SF

,

F = Fe

SN

for the steady State 2 (Fig. 8c). eAF 3. For F < 0 and c: - d 2 > - we have no bifurcation. 7 The unstable focus with its stable limit cycle remains unchanged with increasing field F: UF + S L C ~ UF + SLC.

The steady State 1 is again a saddle point (Fig. 8a).

~AF

b

7

Fig. 7a. Approximate frequency of the limit cycle, coo = Im 1 / - f i 7 is the frequency of the Lotka-Volterra cycles, coc is the frequency where bifurcation from U F + S L C to S F takes place, i.e. for co < coc the limit cycle collapses. (The notation is the same as in Figure 3). b Amplitude a of the limit cycle as a function of the applied field F

4. For F_Tfl (F large, positive). In this region the steady State 2 is shifted to values S O< 0. For physical reasons this steady State 2 must be excluded. Moreover this steady state is completely unstable (saddle point). Nevertheless, the steady State 1 exists, but then a o = -fl/c~A and therefore So=0. This steady state is a unique or a threefold one (see Fig. 4b). In none of the cases the necessary conditions for the existence of a limit cycle are fullfilled. 6. Further changes in the dynamic behaviour may occur if one considers the fact that the parameters c 2 and F 2 are functions of temperature T. In this case both the

162

N

iV

........ SP

r--X . . . . -D- d

I b

,

~2-~,42~--~,-~ ~,AF F _~_~_F_ c2-d2~o

c2- d2o ?'

Fig. 8a--c. Phase plane diagrams (schematic) for different situations. All parameters are fixed, F changes in the way shown in the diagram. (For explanation see text)

position of steady states and their behaviour with respect to stability are influenced by changes of T. The situation becomes rather complicated. In most of the cases the system changes with temperature in the same way as with the external field. Temperature changes may amplify or weaken the effect of the external field. Vl. Relation to Other Pulse Generating Models Since 1952 when Hodgkin and Huxley (Hodgkin and Huxley, 1952) constructed their famous model for the generation and propagation o f impulses on axons many simplifying models have been established (Fitzhugh, 1961 ; Nagumo et al., 1962; Green and Sleeman, 1974). These model systems have been able to exhibit many of the typical effects known from experimental data (Holden, 1976; Hadeler et al., 1976; Hastings, 1975). The brain wave model of Fr6hlich displays many of the solutions found for the above nerve membrane model equations. However, one should keep in mind that this model has not been created for nerve impulses but for brain wave description.

Let us briefly comment on this surprising situation. The case c~A-~0 (linearisation with respect to N, S) leads to the fact that the steady State 2 is either a stable focus or a stable node (c z - d 2 < 0) or either an unstable focus or an unstable node (c a - d 2 > 0). In the latter case the conditions of the Li6nard-Theorem (Rosen, 1970) are fullfilled, i.e. a stable limit cycle does exist. Thus, the system may be considered as a modified Van der Pol oscillator. Here, the field F only shifts the position of the steady state, but we have no bifurcation and therefore no transition from one state to an other one. Moreover, the steady State i does not exist. The result is that if there exists a stable oscillation (limit cycle) it cannot be destroyed. If one introduces an additional damping term in the second equation (i.e. dta = - f l y - 8 0 9 one gets a system of equations that displays similar solutions as the Fitz-Hugh equations. Details of this modified Bonhoeffer-Van der Pol system are given in a forthcoming paper. Restricting to cubic non-linearities in the expansion of the exponential we get equations that have exactly the same structure as Fitz-Hugh's equations. This is an interesting intercorrelation, since it is well known that these equations describe the nerve pulse propagation in a rather satisfying way. In order to look for the system's behaviour after the collapse of the limit cycle has occured we have to include the space dependence of the variables. The discussion of the resulting nonlinear parabolic differential equations is a very cumbersome task. Details and some results with respect to traveling waves will be published (Kaiser, 1977). It turns out that the situation with four steady states (see Fig. 4b) seems to be the most interesting one in this case. VH. Summary and Outlook The limit cycle model for brain waves shows a great variety of steady state solutions. Above all the required behaviour, i.e. the collapse of a limit cycle with increasing external field F, is displayed. However, one should keep in mind that the steady state behaviour has only been calculated for a time averaged field (quasistatic situation). The limit cycle condition (c 2 > d 2) exhibits the inherent balance between the system's tendency to a cooperative behaviour (i.e. ferroelectricity) and its dissipative processes (i.e. electric resistances). Without a more detailed information about the collective substrate-enzyme reaction one cannot state which of the steady state solutions is the relevant one. Time dependent external fields must be taken into account as well as the space dependence of the chemical variables. Thus, the onset of propagating pulses after the limit cycle collapse as a function of both intensity and frequency of external fields must be considered.

163

Further studies on the energy storage mechanism inherent in the limit cycle are greatly desirable as well as an analysis of correlations between brain waves and nerve action. The latter process has actually been found (Elul, 1974) on studying experimentally the relation between neuronal waves and brain waves (EEG). A possible explanation for this mechanism is given by our brain wave model. The limit cycle corresponds to a coherent oscillation of large regions in the Greater Membrane. Limit cycle interactions could lead to a collective behaviour of large regions in the brain. In addition these coupled limit cycles may regulate and controlle the onset of nerve action. Although this model is far from being proved in an experimental fashion, it seems well suited as a starting point to derive biological behaviour from a physical basis. However, the increasing number of experimental results (Fr6hlich, 1976) is in a good agreement with the basic ideas leading to Fr6hlich's model and with our very first results. Most of the work still remains to be done. Acknowledgement. The author would like to thank Prof. H. Fr6hlich for encouragement and many stimulating discussions and Prof. M. Wagner for a critical reading of the manuscript.

References Andronov, A.A., Vitt, A.A., Khaikin, S. E. : Theory of oscillators. Oxford: Pergamon Press 1970 Andronov, A. A., Leontovich, E. A., Gordon,I. I., Maier, A. G. : Theory of bifurcation of dynamic systems on a plane. New York : Wiley 1973 Deviatkov, N.D.: Influence of millimeter-band electromagnetic radiation on biological objects. Sov. Phys. USPEKHI 16, 568--579 (1974) Elul,P. : Relation of neuronal waves to EEG. Neurosci. Res. Prog. Bull. 12, 97--101 (1974) Fitzhugh, R. : Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445~466 (1961) Fr6hlich, H. : Long range coherence and energy storage in biological systems. Int. J. Quant. Chem. 2, 641--649 (1968) Fr6hlich, H. : Selective long range dispersion forces between large systems. Phys. Lett 39A, 153--154 (1972)

Note added in proof New experimental results seem to support the existence of coherent vibrations in the 1011-1012Hz region. S. J. Webb has found resonances between 10 ~1 and 1012 Hz in active bacterial cells by Raman spectroscopy (private communication, to be published), W. Grundler and F. Keilmann have found a resonant growth rate response of yeast cells when irradiated by weak microwaves in the 42 GHz region, (private communication, in print in Phys. Lett.)

FrShlich, H. : Evidence for condensation-like excitation of coherent modes in biological systems. Phys. Lett. 51A, 21--22 (1975a) FrShlich,H. : The extraordinary dielectric properties of biological materials and the action of enzymes. Proc. Nat. Acad. Sci. USA 72, 4211---4215 (1975b) Fr6hlich, H. : Long range coherence in biological systems. Lectures delivered at the symposium on interdisciplinary aspects of modern physics, Parma, Italy, may 1976 (to appear in Nuovo Cim. Suppl.) Fr/Shlich,H. : Possibilities of long and short range electric interactions o f biological systems. In : interaction of weak electric and magnetic fields with the brain. Adey, W. R., Bawin, S. M. eds. Neuroscience Res. Progr. Bull. 15, 67--72 (1977) Green, M.W., Sleeman, B. D. :On Fitzhugh's nerve axon equations. J. Math. Biol. 1, 153 163 (1974) Hadeler, K,P., an der Heiden, U., Schumacher, K. : Generation of the nervous impulse and periodic oscillations. Biol. Cybernetics 23, 211--218 (1976) Hastings, S.P.: Some mathematical problems from neurobiology. Amer. Math. Mthly. 82, 881--895 (1975) Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.) 117, 500--544 (1952) Holden, A.V.: Response of excitable membrane models to a cyclic input. Biol. Cybernetics 21, 1 7 (1976) Hopf, E.: Abzweigung einer periodischen L6sung yon einer station~iren L6sung eines Differentialsystems. Berichte der Math. Phys. Klasse der S~ichsischen Akademie der Wissenschaften, Leipzig, XCIV, 1 (1942) Kaiser, Ft. : Traveling wave solutions for brain wave models (to be published) Kollias,N., Melander, W.R.: Laser-induced stimulation of chymotrypsin activity, Phys. Lett. 57 A, 102--103 (1976) Marsden,J. E., McCracken, M. : Applied mathematical sciences Vol. 19. Berlin-Heidelberg-New York: Springer 1976 Nagumo, J., Arimoto, S., Yoshizawa, S. : An active pulse transmission line simulating nerve axons. Proc. IRE, 2061--2071 (1962) Rosen, R. : Dynamical system theory in biology. New York:Wiley I970 Volterra,V. : Lecons sur la theorie mathematique de la lutte pour la vie. Paris: Gauthier-Villars 1931 Received : May 27, 1977

Dr. F. Kaiser Inst. f. Theor. Physik der Universit~t P faffenwaldring 57 D-7000 Stuttgart 57 Federal Republic of Germany