On Zeeman's equations for the nerve impulse

Bulletin of Mathematical Biology, Vol. 43, No. 3, pp. 279-325, 1981 Printed in Great Britain

0092-8242/81/030279-47502.00/0 Pergamon Press Ltd.

�9 1981 Society for Mathematical Biology

O N Z E E M A N ' S E Q U A T I O N S F O R T H E N E R V E I M P U L S E

�9 IAN STEWART Mathematics Institute, University of Warwick, Coventry, CV4 7AL, England

and

A. E. R. WOODCOCK t Thompson Biology Laboratory, Williams College, Williamstown, MA 01267, U.S.A.

A model Ior the nerve impulse due to Zeeman (1972) and based on catastrophe theory is compared with alternative models and criticisms of Zeeman's model by Sussmann and Zahler (1977, 1978) are assessed. The criticisms of Zeeman's motivation for his model are found to carry some weight.

Sussmann and Zahler (1977, 1978) list numerous features of Zeeman's model which, they state, are not in agreement with experiment. These statements as they stand are largely erroneous, and the model still remains to be tested by a critical series of experiments.

However, a detailed analysis reveals defects in Zeeman's model, not among those claimed by Sussmann and Zahler, showing that the explicit equations of the model cannot be correct. The possibility of a modified approach along similar lines and its ultimate adoption remains open.

1. Introduction. Elec t rophys io logica l investigations, beginning with those of Berns te in (1902) and leading to the celebrated w o rk of H o d g k i n and H u x l e y (1952a, b , c , d ) on the giant axon of the squid Loligo (reviewed for example in Cole, 1968; Hodgk in , 1964, 1976) have p rov ided s t rong suppor t for " ionic" or " m e m b r a n e " models of the nerve impulse.

The nerve impulse is a free runn ing e lec t rochemical event t r iggered by an a de qua t e st imulus in an exci table m e m b r a n e system. The act ivi ty is of the "a l l -or -no th ing" type: be low a th resho ld level of s t imulus intensity, no th ing appears to h a p p e n - - t h e m e m b r a n e re turns to rest. Howeve r , at th resho ld levels and above the m e m b r a n e exhibits a d rama t i c change in its

t A d d r e s s for correspondence.

279

280 I. STEWART AND A. E. R. W O O D C O C K

electrical properties and then returns to equilibrium. These properties led Crick (in Zeeman, 1972) to suggest that a test of the suitability of applied catastrophe theory might focus on its use as a model of the nerve impulse. Its success would be measured by comparison with the classic model of the nerve impulse developed by Hodgkin and Huxley (1952a, b, c, d; Hodgkin, 1964). In response to this challenge, Zeeman (1972) proposed a model for the nerve impulse motivated by the cusp catastrophe and based on a system of differential equations which differed in several respects from the Hodgkin and Huxley model.

Zeeman's (1972) proposal was extensively criticized by Sussmann and Zahler (1977, 1978), Zahler and Sussmann (1977), and Zahler (1978). These proposals have also been critically discussed by Paterson (1977) and by Deakin (personal communication). In this paper we shall examine both Zeeman's model and these criticisms. We find that while Sussmann and Zahler's analysis of the motivating arguments behind the model raises some substantial problems, their discussion of the model and its implications is largely incorrect. However, Zeeman's model (1972), as it stands, has a number of faults not observed by Sussmann and Zahler: we shall discuss these, and briefly consider whether they might be avoided by a modified model.

2. Physical and Chemical Bases of the Nerve Impulse. Hodgkin and Huxley (1952a, b, c, d) observed that an electrical potential difference exists across the unstimulated nerve membrane, the inside being electronegative with respect to the outside. This difference, the membrane resting potential, seems to be caused both by metabolically maintained concentration gradients of sodium and potassium ions and by a differential membrane conductance to these ions. Stimulation of the nerve above a threshold level produces a nerve impulse or action potential characterized by a transient reversal of the membrane potential (Figure 1). Using a technique called voltage clamping, Hodgkin and Huxley determined that sodium ions flowed into the nerve during the early part of the impulse and potassium ions flowed out during the later part (Figure 2). The nerve membrane has a greater conductance for sodium ions at the start of the impulse and a greater conductance for potassium ions during the remainder of the impulse.

3. The Hodgkin and Huxley Model. Hodgkin and Huxley (1952a, b,c,d) used data obtained from many physiological experiments to construct a mathematical model of the nerve impulse.

This model involves a system of ordinary and partial differential and other equations reflecting assumed properties of the nerve membrane, and

ON ZEEMAN'S EQUATIONS FOR THE NERVE IMPULSE 281

A > E ~>

100

5 0

- H:H

0 I

2msec 1

(a)

1 oo I Expt .

0 2 rnsee I

(b)

101

v

- E C Z

- - 0

0-1

T arrival of propagation wave

2 m s e c

I

Figure 1. The action potential: a comparison of an experimentally measured propagated action potential ("Expt') with the theoretical predictions of Hodgkin and Huxley ("H:H") and Zeeman ("ECZ"). The Hodgkin and Huxley curve is drawn after Hodgkin and Huxley (1952d). The Zeeman curve is determined before the trigger point, J, by a propagation wave and afterwards by the local nerve impulse equations [equations (11)-(13)]. It is computed in the assumption (which we later show to be invalid) that the "singular" approximation to his equation (11) is applicable. (After Zeeman, 1972.)

(Hodgkin and Huxley, 1952d; Zeeman, 1972. Reproduced with permission.) B M B - - B


Sodium conductance Potassium conductance

mmh~

o _ ~ r

o.-

o

rnmho/cm 2

I

26 1 26 o-- '

I [ 1 1 I I 1 I 0 2 4 0 2 4 6 8 msec msec

,O

Figure 2. Voltage clamp data: changes of sodium and potassium conductance with different membrane depolarizations. Open circles are experimental estimates; the smooth curves are theoretical predictions. The numbers associated with each curve show the membrane clamp depolarization in mV.

(After Hodgkin and Huxley, 1952d. Reproduced with permission.)

was constructed ad hoc to fit experimental data. In the Hodgkin and Huxley model the nerve impulse is assumed to propagate along the axon according to a partial differential cable equation. Hodgkin and Huxley (1952d) claim a favorable comparison between the measured conduction velocity of the impulse of 21.2 m/sec for a particular nerve and the value of 18.8 m/sec that they predict from their analysis.

Innumerable modifications have been made to the Hodgkin and Huxley equations either to incorporate new experimental results or to aid in their analysis. Here the fundamental mathematical interest is to derive from the equations the existence and form of a travelling wave solution, where the wave maintains the same temporal profile as it transverses the axon, and to


investigate its stability. This type of problem has been the subject of extensive analysis, and some recent results are given in Baumann and Bond (1978); Carpenter (1976, 1977); Conley and Smoller (1976); Evans and Feroe (1977); Feroe (1978); Hassard (1978); Hassard and Wan (1978); and Hunter et al. (1975). A novel alternative to the Hodgkin-Huxley equations, which fail to predict correctly the results of prepulsing experiments, occurs in Bell and Cook (1979).

4. A Geometrical Interpretation of the Hodgkin and Huxley Equations. FitzHugh (1960, 1961) has used a "geometrical" approach to produce a model of the nerve impulse which he called the Bonhoeffer-Van der Pol model. He considered the use of this approach was not an attempt to produce an "accurate quantitative model" (of the nerve impulse), but rather an exhibition of the "basic dynamic interrelationships" responsible for the properties of the impulse (see Figure 3, for example). Indeed, the precise algebraic form of FitzHugh's (1960, 1961) equations, which he chose as the simplest possible, did not seem to be important since the equations "could be changed without altering the general properties" of the impulse behavior (FitzHugh, 1961, p. 464).

Y

O,

-I -2

ABSOLUTELY / . ~ ' ~ REFRACTORy'

SELF-EXCITATORY \

-'l 6 x i

Figure 3. The Bonhoeffer-Van der Pol model of the action potential: phase plane and physiological state diagram. The broken lines are the 2 = 0 and 3;'=0 nullclines, the dotted lines the locus of initial conditions following instantaneous z shocks at rest. (a=0.7, b=0.8, c=3.0, z=0.0.) For further

details see the text. (After FitzHugh, 1961. Reproduced with permission.)

5. The Zeeman Model. Zeeman (1972) was interested, as FitzHugh (1961) had been, in "modelling the dynamics [of the nerve impulse] (which is relatively simple) rather than the biochemistry (which is relatively complicated)" (Zeeman, 1972, p. 8). Indeed, he thought that the use of such a dynamical approach, if borne out experimentally, might aid in


understanding the underlying biochemistry and electrophysiology of the nerve impulse. Zeeman (1972) sought, as Hodgkin and Huxley (1952d), and FitzHugh (1960, 1961) had before, the simplest set of equations that would serve as an adequate model of the nerve impulse. The major difference between Zeeman's approach and the earlier models is his attempt to derive, by a combination of mathematical analysis and heuristically motivated mathematical hypotheses, a description of the simplest plausible local behavior of the nerve impulse. His chosen motivation leads to a description in terms of trajectories on one of the elementary catastrophe manifolds--here smooth surfaces with only fold and cusp singularities of projection (Thom, 1969, 1972, 1975; Poston and Stewart, 1978). The cusp catastrophe surface represents "the most complicated" thing that can happen locally (Zeeman, 1972, p. 29). More accurately, it is the most complicated thing that need happen locally: anything more complicated would require special justification (Thom, 1977; Poston, 1978; Arnol'd, 1972). Thus Zeeman rules out other types of surfaces (a "looped hosepipe" for example) because these are "topologically . . . and biologically more complicated" than necessary (Zeeman, 1972, p. 31). Zeeman concludes that the nerve impulse can be described qualitatively by trajectories on the cusp catastrophe manifold (Figure 4).

Zeeman does not give full details of the mathematical justification for using the cusp catastrophe; but he refers to Thorn's theorem classifying potential functions as the main step in this justification. The argument for this step presumably is the following: to obtain the equation of the slow manifold from a fast equation;

2 =f(x , a, b); (1)

we set 2 = 0, so that:

0 =f(x , a, b). (2)

Thom's theorem applies directly to equations of the form (2), as well as to cases where the right~hand side is the gradient of a potential (Poston and Stewart, 1978). ]-In fact with a single state variable (x) a potential can always be constructed by integration, but this is not necessary for this approach to apply.] For a more extensive mathematical discussion, see Takens (1976). For an explanation of Zeeman's justification of a single cusp, see Poston (1978), Zeeman (in press). As the discussion in Zeeman (1972) clearly indicates, he is seeking the simplest plausible model, not proving it is the only one possible: this is somewhat obscured in the critical writings of Sussmann and Zahler (1977, 1978).


>re, Vr z,~z )

RELATIVE REFRACTORY

ACTION POTENTIAL

ABSOLUTE REFRACTORY

X : FN(SODIUM CONDUCTANCE)

Figure 4. The nerve impulse: a qualitative model of the nerve impulse drawn on the cusp catastrophe manifold. Functions of the potassium conductivity and membrane potential nerve as control parameter axes. An increase in membrane potential can lead to an action potential [path: (a, b,c)] followed by a return to equilibrium [path: (c,d,e,a)]. Position "d" represents an absolute refractory state [path: (d,f) produces no qualitative behavior change], and position "e" a relative refractory state [path: (e, g, h) represents a discontinuity in behavior]. The path: (a,e,d,c) could represent the phenomenon of "accommodation"

(Hodgkin, 1964). (For further details, see the text.)

An im por t an t ques t ion addressed nei ther by Z e e m a n n o r by his critics was po in ted out to us by T im Pos ton . T o wha t extent does the b i furca t ion t he o r y of the s teady-s ta te equa t i on (2) reflect the behav io r of the original equa t ion (1)? Zeeman ' s discussion concen t ra tes solely on the equa t ion of the slow manifo ld [here equa t ion (2)]. O n e i m p o r t a n t p h e n o m e n o n that this a p p r o a c h fails to detect is the occur rence of H o p f b i furca t ion (Marsden and McCracken , 1976; Hopf , 1942) in which a limit cycle

286 I. STEWART AND A. E. R. WO O D C O C K

appears surrounding the "steady-state" solution. This will not happen if x is only one-dimensional; but in Zeeman's methodology the reduction to one dimension of state is accomplished after the deduction of a cusp, by way of the Splitting Lemma (Poston and Stewart, 1978), so the possibility of high-dimensional x should not be ruled out in advance. A case in which Hopf bifurcation does occur in conjunction with cusp geometry for steady- state solutions of equations of practical significance is studied in Uppal, Ray, and Poore (1976). One of the defects of some early work in catastrophe theory is a failure to adequately discuss the relation between the quasistatics of the catastrophe and the dynamics of the explicit equations. This is, however, a sin of omission, since information on the steady-state solutions is often needed as a prelude to finding the full dynamic behavior of a system.

In Zeeman's (1972) model the propagated impulse is "factorized" (Zeeman does not use this word, but it expresses the idea) into two essentially independent parts. The first is a propagation wave along the nerve axon, travelling at uniform velocity according to a simple partial differential equation. The second is the local action at a given point on the axon. During propagation the propagation wave dominates until the membrane potential reaches a certain threshold value, the trigger voltage; this "switches on" the local activity at each point in turn along the axon. Once this local action has been triggered its effect dominates that of the propagation wave. The relation between the spreading wave of local activity and the propagation wave is similar to that between "primary" and "secondary" waves in Zeeman (1974), although the mechanism for the wave itself is not. The propagation velocity of an impulse in Zeeman's (1972) model, therefore, depends essentially on the rate at which the invading current from an adjacent, active, segment of the nerve causes an initially inactive segment to reach threshold. An obvious omission from Zeeman's (1972) model is a mechanism through which the local membrane events override and, therefore, turn off those of the propagation wave.

The existence of a travelling wave is an automatic consequence of Zeeman's (1972) model. This is true independently of the equations used to describe either the local action or the propagation wave: it requires only the uniform motion of a propagated voltage rising to the trigger level. Thus, Zeeman builds into his model conditions which directly imply a travelling wave, whereas in the Hodgkin and Huxley model this is deduced (with some difficulty) from the equations.

There are some physiological reasons for trying to "factorize" the activity of the axon in this way. Zeeman (1972) discusses some of them on pp. 47-48: the main one is the observation, due to Hodgkin (1964), that "nerve conduction resembles the burning of a fuse of gunpowder rather


than the propagation of an electric signal along a cable", with most of the energy being released locally rather than travelling with the impulse. Further Zeeman remarks that the analogy is misleading in one respect: the timing. Propagation along the axon is much swifter than the local release of energy (unlike burning gunpowder in a fuse). Zeeman argues that this adds further justification to "factorizing" the action into a travelling threshold and a local action, because the threshold has passed a long way down the axon very rapidly after triggering the local action, hence need not interfere with it.

This argument assumes some mechanism to override the propagation wave, since it is the whole wave, and not just its travelling threshold, that can interfere. Without such a mechanism it is less reasonable to think of the propagation wave as a pulse, since equation (14), the solution to Zeeman's partial differential equation for the propagation wave, shows exponential growth if not overridden. Thus the argument is available only if the "factorization" is assumed to have taken place, and amounts more to a verification that it is reasonably self-consistent. Such partly heuristic separations and reductions are standard scientific practice (compare such respectable devices as treating one of a pair of colliding atomic nuclei as lacking internal structure). They are often, as here, part of the process of constructing a model to obtain equations, rather than an approximation in solving them. Zeeman's exploitation of the fast/slow dichotomy in seeking the solutions of his explicit equations (Section 5) is the latter type of approximation, and certainly lacks rigour. But to the construction of a model, rigour is not one of the tests that can easily be applied, since it is relevant only to purely deductive arguments.

Sussmann and Zahler (1978, p. 181) contest Zeeman's argument saying "Zeeman argues that the propagating wave traverses a 1 mm length of axon 100 times faster than the duration of the complete local action potential, and this justifies the separation of local and global phenomena. Although '10 times faster' would be a fairer estimate (he has simultaneously exaggerated the speed of propagation in the squid giant axon and underestimated the duration of the action potential), the basic fact is that there is no rigorous way to justify the proposed method of solution [suggesting the existence of a travelling wave-]." (Our italics.)

It is instructive to compare the above with the relevant passage from Zeeman (1972, pp. 48-49), which reads:

"Suppose we have a segment of axon 1000 times as long as it is wide, for example, 1 # in diameter and 1 mm in length. Suppose that the velocity of propagation is 100m/sec. Then the propagation wave traverses this segment in 10-2msec. Meanwhile the complete action potential takes about 1 msec, which is 100 times as long .... Of course in different organs

288 I. STEWART AND A. E. R. WOODCOCK

and different species these times vary-- for example . . . [in] a giant squid axon . . . the velocity is only about 20 re~see." (Our italics.)

6. Local Dynamics of the Zeeman Model. Most of Zeeman's analysis concerns the local dynamic. He discusses the possible types of differential equation that may give rise to:

( a ) a threshold, (b) an action, (c) a smooth return to equilibrium.

This discussion first gives heuristic motivation for, and subsequently assumes, that the equations involve a slow manifold and a fast foliation, almost everywhere transverse to the slow manifold (Zeeman, 1972). The system is assumed to be displaced from equilibrium; it then homes in rapidly to the slow manifold along the fast foliation, and then follows a trajectory governed by the slow equations. The full system of equations is "stiff" in the sense of numerical analysis. This property, which c'an make numerical treatment awkward, is useful in a topological approach by way of the ',singular" approximation in which the ratio of "fast" to "slow" tends to infinity.

Using these ideas Zeeman obtains what he states to be the "simplest" model of this type, namely a spiral flow on a cusp catastrophe (Figure 5). He cites one explicit set of equations, (3), (4), and (5), for such a flow:

52 = - (x 3 + ax + b)

c~ =- - 2a - 2x

(3)

(4)

(5)

Here x is the fast variable, ~ is small, and the fast foliation consists of lines parallel to the x-axis (Zeeman, 1972). Zeeman points out that the introduction of equations (4) and (5) "changes the nature of the variables, a and b, [normally regarded as parameters or controls] by giving them a dynamic role . . . . " Furthermore, the appearance of the term x in these equations "can be regarded as a form of feedback on the cusp catastrophe" (Zeeman, 1972, p. 32). The slow manifold is obtained by making 2--0, that is,

x3 +ax+b=O, (6)

which is the equation for the canonical cusp catastrophe manifold (Thorn, 1972; Poston and Stewart, 1978).


- ' - . ~ - " . - " V on .' ~1 ! ~, t ' , / I A a p ; j l i I', ', 7 /

/ ) - " ! I f l i y/s~ow

s j

sodium conductance

g,.

plane of

clamp

Vc.-• b, V 1" potential

clamp potential

Figure 5. Spiral flows, voltage clamp data, and the cusp catastrophe: functions of the potassium conductance and membrane potential parametrize the control space of the cusp; a function of the sodium conductance is the behavior variable. Switching on the clamp moves the nerve membrane from its equilibrium point E to the clamp plane. The point representing the state of the nerve follows paths FG (rapid increase in sodium conductance), and GH (slow decrease in sodium conductance to equilibrium). Position H is the clamp equilibrium. Release of the clamp will permit the state point to follow the spiral flow pattern to equilibrium E. (After Zeeman, 1972. Reproduced with

permission.)

Sussmann and Zahler (1977, 1978) refer to these explicit equat ions as " M ode l I" a l though it is unclear tha t Zeeman is proposing them as other than an example of a general type of model . In Zeeman ' s paper they are "Example 8". The only serious model in the paper is what Sussmann and Zahler (1977, 1978) call " M o d e l II". Zeeman (1972, p. 32) warns the reader that " there is noth ing unique about the slow equat ions", [of Example 8]; and also that : "Example 8 is too e lementary to predict the correct quant i ta t ive features [of the nerve impulse]" . (p. 34.)


The main difficulty with Zeeman's argument is that it is true only of equations of fast/slow type, and then only if the fast flow is markedly faster than the slow flow over the whole range of variables considered and the fast foliation is essentially parallel. Zeeman argues for the latter in terms of energetics of the biochemistry, but it is unclear how much force the argument has. Sussmann and Zahler (1977, 1978) correctly point out that Zeeman's Lemma 3 relies on such a hypothesis. This lemma states that in two dimensions we cannot have qualities (a), (b), and (c) all at once: a smooth return is impossible. If we assume that Zeeman's discussion is a tacit formulation of (a) and (b) in mathematical terms, then with the hypotheses stated above the lemma is rigorous enough; but these hypotheses are not stated explicitly by Zeeman.

If the speed of the "fast" flow is sufficiently variable it is possible to obtain a smooth return in two dimensions, as shown by Sussmann and Zahler's Figure 20a. In a very similar way FitzHugh (1961, Figure 1; here reproduced as Figure 3) obtains a fast action and smooth return in two dimensions by incorporating a short segment of slower flow with the "action".

Sussmann and Zahler's Figure 20b (1977, 1978) is not as relevant: in rigorous terms even a very small jump is still a jump; whereas on a scale that renders so small a jump of no practical significance a model based on their Figure 20b would be unstable relative to perturbations of comparable magnitude, hence effectively structurally unstable. That is, it is technically structurally stable, but its "domain of stability" is extremely small. It is therefore unlikely to be observed in practice.

Thus, while Zeeman's motivation may be held as making the model plausible, it does not amount to rigorous proof that it is necessary [for, even if we allow the hypotheses above, there is no reason to suppose that these hypotheses are necessarily the correct mathematical formulation of properties (a), (b) and (c)]. There are further problems with the motivation: for example the existence of an action is held to imply the local occurrence of a large and a small eigenvalue (hence the whole fast/slow phenomenonology); but Zeeman (1972) suggests that this local phenomenon is not in fact sufficient to justify the term "action". But in a non-local context it ma) not be necessary either, because the distinction between fast and slow ma~ not be sufficiently great over the entire region in which the action takes place: eigenvalues at a particular point may lose their significance elsewhere. Indeed, the "large" eigenvalue is necessarily zero at a fold point, and the model requires the existence of folds.

7. Fast~Slow Flows. It is harder to concur with Sussmann and Zahler's opinion that the fast/slow distinction is incompatible with topological

ON ZEEMAN'S E Q U A T I O N S FOR THE NERVE I M P U L S E 291

methods, except "for the limiting singular theory obtained by letting e--*0" (1978, p. 170). Certain caution is indicated in modelling a fast variation by a discontinuous jump, and without careful analysis this is at best a heuristic step; but it is one commonly made. It is of course usually disguised as the assumption that the case of small e is approximately that of the singular case. This assumption is frequently made in electronic engineering, for example: the archetypal instance is the Van der Pol oscillator (Van der Pol, 1926; Andronov et al., 1966).

Zeeman's motivation, however, is entirely phrased in terms of the singular case. That is, he assumes that his equations, for small e, behave approximately like the singular case, and deduces his cusp manifold from that hypothesis. This makes most of the discussion of Section 9 of Sussmann and Zahler a little beside the point: what they describe as "confusion" between fast flow and jumps is the usual modelling hypothesis of much of applied mathematics, that a rapid variation may be modelled by a discontinuity. A shockwave, for example, has a fine structure across the shock front with a rapid, but continuous, variation; by Sussman and Zahler's standards the common model of a step function amounts to "confusion" and should not be used. Similarly, their objection (1978, p. 168) that catastrophe theorists "prove theorems about the technical 'jump', and claim that these say something about jumps in our everyday sense" is an objection to the standard practice of interpreting the real-world phenomena corresponding to a choice of model. As Sussmann and Zahler take pains to point out elsewhere, the model is not the same as the reality. But to obtain results it is necessary to pass from one to the other!

In using this modelling hypothesis it is necessary to check that the final equations arrived at have small enough e for the assumption to be valid. It is extremely hard to deal with this problem in general in a rigorous way, as noted by Takens (1976), but it can in principle be handled numerically in any particular instance. For some interesting rigorous results on more conventional nerve impulse equations, making heavy use of the fast/slow distinction and topological arguments, see Carpenter (1976). The extent to . which the fast/slow distinction as a modelling hypothesis is commonly used in this area may be judged by Carpenter's remark that "two slow (or fast) variables may be slow or fast relative to one another".

In fact a more pertinent criticism of Zeeman's final equations, "Model II", was made (verbally) by Pierre de la Harpe in 1974. In these equations e is 0.8. This is somewhat large, and while one cannot make definite assertions without numerical checks, it is unclear that Zeeman's explicit equations actually belong to the class of equations that he motivates. Inasmuch as heuristic motivating arguments are common in experimental science, subject only to the test of observation, it becomes much more

292 I. STEWART AND A. E, R. W O O D C O C K

crucial to look at the final model than at the motivation. For this reason we shall refrain from further discussion of the motivation, even though there remains room for disagreement with several of Sussmann and Zahler's contentions (see Paterson, 1977 for some pertinent observations on the effect of scale changes on the fast/slow distinction, in rebuttal of some of their statements). Their general point, that Zeeman's argument does not amount to rigorous proof of the necessity for a cusp catastrophe model, remains valid; though we doubt that Zeeman intended to make such a strong claim.

The model nevertheless remains a reasonable one to try: it would be hard to find any scientific model that can rigorously be proved the only one possible. It becomes unreasonable only when serious discrepancies with experiment are found. We show in section 11 that the supposed discrepancies with experiment claimed by Sussmann and Zahler (1977, 1978) have no basis in fact. However, in their criticism they do not make explicit reference to the large size of ~ in Zeeman's model [al though their passing remark on p. 169 "often it (e) is not even small" may refer to this. If it does, they explore it no further.] As we shall see, it signals a serious difficulty: Zeeman's equations do not, in fact, appear to belong to the class that he motivates, nor do they behave in the way that this motivation assumes.

8. Zeeman's Quantitative Equations. Zeeman refines his qualitative model to give a system of explicit equations, which Sussmann and Zahler call "Model II". It is the only specific model advocated in the paper, and then only with diffidence: Zeeman discusses the difficulties of this procedure and the unlikelihood of obtaining good results at the first attempt, and remarks "we do not hold a very strong brief for our explicit equations" ip. 601. In fact, his main concern is with method. At any rate, only this model may be legitimately compared with quantitative experimental data.

Zeeman specifies that the fast (x) and slow (a,h) variables are coordinates of the canonical cusp catastrophe manifold. The dependence of the membrane potential on the b variable is given by equation ~7t:

V= (20b+ 16)mY, 17)

and the potassium, sodium, and chloride ionic conduclanccs by tile following relationships:

gK = 2,38 [a + 0.5] + mmho/cm 2 18)

gNa = (4[x + 0.5] _ )2 mmho/cm 2 (9)

gcl = 0.15 mmho/cm 2 . (10)

Here


[Y]+=I/2(y+IY [) bq_ =1/2 (y-lyl).

Zeeman's explicit equations are as follows:

2 = - 1.25 (x 3 +ax + b) (11)

d=[x+O.O6(a+O.5)](x-l.5a-l.67)[O.O54(b-0.8)2+0.75] (12)

/) = - gK(b + 1.4)-- gNa(b -- 4.95)-- gcl(b - 0.15). (13)

The fast equation (11) for 2 is the canonical cusp catastrophe, and in the nota t ion used above we have e = 1/1.25=0.8. The equat ion (12) for ci is an ad hoc equation with its terms chosen to fit the voltage c lamp data. The slow equat ion (30) for b is a modification of an electrical equat ion used by Hodgkin and Huxley (1952d).

Sussmann and Zahler (1978, p. 176) raise an interesting question regarding equations (8) and (9) which " . . . display gN, and gK as non- smooth functions of x and a. This is inconsistent with the use of Thorn's t h e o r e m . . . " .

Strictly speaking, this is untrue. Thorn's theorem deals with smooth transformations of variables; here a,b,x. It does not require that every physical variable in the problem be smoothly related to the canonical variables. There are a number of examples in the physical literature that illustrate this. For example, in Gilmore 's application of catast rophe theory to lasers (Poston and Stewart, 1978, Chapter 15) two physical variables are considered: the ampli tude and intensity. These are related by intensity

(amplitude) 2, and this t ransformat ion is not a diffeomorphism at zero (since its inverse, the square root, is not differentiable there). It is impor tant not to confuse t ransformations allowed within the theory and transformations encountered when it is applied. Hence, as a technical objection to the mathemat ics as applied, Sussmann and Zahler 's remark is incorrect.

Having said that, it is at least an aesthetic defect of Zeeman's model that non-smooth functions' are used here; and permitt ing them earlier might alter the motivat ing arguments further. It is possible to find smooth functions arbitrarily close to those chosen by Zeeman (1972), using modifications by a "bump function" (Br6cker and Lander, 1975) to smooth them near the origin. While this avoids the letter of the criticism we see little advantage in carrying it out for what is a debatable problem in a minor modelling step. Further , the non-smooth nature of Zeeman's


functions arises from a "switching" effect at a threshold: it is probably better to make these thresholds explicit.

9. Zeeman' s Model: The Action Potential. In Zeeman's model the propagated action potent ial-- that is, the graph of depolarization against time at a fixed point on the axon, as in Figure 1--is produced in two stages.

(a) The propagation wave along the axoplasm grows exponentially until it reaches a, threshold value V, it at time Tcrit (our notation).

(b) The llocal equations (11)-(13) then override the propagation wave and subsequent changes in V, a, b, and x are derived from their solutions.

In keeping with his motivational arguments, Zeeman assumes the system closely approximates the "singular" case with e=0. Hence by t=T~rit the fast equation for x takes the system to the trigger point

J(a, b,x)= ( -0 .45 , 1.2, - 1.2)

having started at the equilibrium point

E(a ,b ,x )=( -0 .4 , -0.8,1.07).

There is a slow adrift in a, the b-equation is overriden, and the x-value jumps instantly to the lower sheet of the surface.

Assuming this to be the case, we have computed the shape of the action potential taking T~rit =0.5 msec. The result shown in Figure 6 is in excellent

V (my)

100

t (msec) ! 2

Figure 6. The propagated action potential: computer calculations of the action potential in the singular approximation appear to support Zeeman's (1972) hand calculations (but see Figure 11). The state point begins at equilibrium coordinates (x=l .07 , a = - 0 . 4 , b = - 0 . 8 ) , and is displaced to the trigger point J by the propagation wave. In this calculation it is assumed, following Zeeman (1972), that the fast equation acts to keep the state point on thc slow manifold. Therefore, the coordinates of J (x = - 1.2, a = -0 .45, b = 1.2) ~t which the local equations (11), (12) and (13) dominate are solutions of: x 3

+ a x + b = 0, the cusp catastrophe manifold equation.


agreement with Zeeman's Figure 26 and the data of Figure 1 above. Likewise, we have computed trajectories associated with the cusp model for both the clamped (Figure 7) and unclamped (Figures 8 and 9) nerve membrane. The trajectories displayed in Figure 7 were obtained by solution of equations (11), (12), and (13) with /~=0 and with a series of

B = FN(MEMBRANE POTENTIAL)

g |

14G

X ~ FN($OD|UM CONDUCTANCE)

Figure 7. The voltage clamp: computer solutions of Zeeman's (1972) local membrane equations (11), (12) and (13) with b set equal to zero. Paths represent trajectories on the voltage clamp plane for a series of different clamp potentials. They, in general, qualitatively resemble the path: FGH (Figure 5). The simple cusp catastrophe manifold and clamp trajectories are plotted in orthographic projection. Trajectories were computed using modified Runge-

Kutta-Gill numerical integration.


imposed clamp potentials. Most of these trajectories qualitatively resemble the single voltage clamp trajectory described by Zeeman (1972, Figure 23). Our computations of the effect of a series of different initial displacements of the unclamped membrane potential (shown here as Figure 8 and in

B " FN(MEHBRANE POTENTIAL)

n <

EOUILIBRIUM J [ J I [ J I ] [ I /J ] I X ] \

X " FN($ODIUM CONDUCTANCE)

Figure 8. Flows on the slow manifold: state point trajectories derived as solutions of the local nerve impulse equations (11), (12) and (13) with different starting values. These trajectories reveal equilibrium- and saddle-points and a threshold for self-sustaining action. The simple cusp catastrophe manifold and state point trajectories are plotted in orthographic projection. Trajectories were computed using modified Rung~Kut ta -Gi l l numerical integration. For further

details, see the text.


greater detail in Figure 9) agree qualitatively with those computed by Zeeman (1972, Figure 25, reproduced here as Figure 10).

However, Zeeman's value e = 0.8 prompts the question whether it is valid to assume the singular solution. Is the fast equation fast enough to move the system near to J in the time available? This is a more specific point than

t h e general dissatisfaction with fast/slow approximations expressed by Sussmann and Zahler (1977, 1978), and they make little mention of it; also they do not observe the following phenomena.

B = FN(MEMBRANE POTENTIAL)

I L

"z

o

/i

X = FN(SODIUM CONDUCTANCE,)

Figure 9. Flows on the slow manifold: a portion of Figure 8 drawn to a larger scale to present the equilibrium- and saddle-points and the threshold in

more detail. For further details, see Figure 8 and the text.

298 l. STEWART AND A. E. R. WOODCOCK

8 ! I

-2

J~

equilibrium threshold trigger

saddle

- 2 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 | I I I I t I I

! I I

V, Vc, V~, V(mV)

Figure 10. Flows on the slow manifold: hand-computed trajectories qualitatively resembling those obtained by numerical integration using a modified Runge-Kutta-Gil l program (see Figures 8 and 9). For further details

see the text. (From Zeeman, 1972. Reproduced with permission,)

We have plotted the action potential V on the following assumptions, appropriate to the explicit equations (11), (12), and (13) rather than the singular case.

(a) Prior to time T~,t =0.5 msec, the potential V is given by Zeeman's equation

where the constants are as listed in Zeeman (1972) pp. 56-57, and A is chosen so that V~rit =40 mV, the value of the trigger voltage (Zeeman, 1972) at time T~rit. (Zeeman does not appear to state explicitly the value of T~rit,


although his Figure 26 indicates that it must be very close to 0.5msec: very small changes in this value are unimportant.)

(b) During this period the propagation wave overrides the b-equation, but leaves the ~- and d-equations operative. [Recall that V=(20b+16) , and the stipulation that both the propagation wave and the b-equation cannot act at the same time.]

(c) After time T~rit the local b-equation overrides the propagation wave. (d) The system then follows the local equations and returns to

equilibrium. This seems to be the only reasonable interpretation of Zeeman's

intentions: we must allow x and a to follow the local equation, since he does; and we must switch V-equations at time To,it.

That this series of events is visibly not a good approximation to the observed behavior of the nerve impulse is clear from the shape of the action potential (Figure 11) computed on this basis. There is a period of about 1.5msec, during which V s!owly decreases, starting at time T, it; eventually the impulse is triggered, but the rise is smaller in amplitude than it should be.

- V ( m y )

lO0

j - 2

Figure. 11. The propagated action potential: computer calculations using the Euler Method, which suffices here and is simple to use, of the action potential profile predicted by Zeeman's (1972) local nerve impulse equations (11), (12) and (13). These equations become operative after the propagation wave has moved the state of the membrane to the trigger point J. However, unlike the computations in Figure 6 no extra assumption is made that the state point moves very rapidly between layers of the slow manifold. Under these circumstances operation of the "fast" equation (with e=0.8) alone requires a period of about 1.5 msec to produce a smaller than expected action potential.

For further details, see the text.


This occurs precisely because the 2-equation is not fast enough. Figure 12 shows a projection in the (b,x)-plane, and Figure 13 the (a,x) plane, of the relevant trajectory, with each dot representing a time interval of 0.02 msec. The solid curve is the section a = - 0 . 4 of the cusp manifold.

During the first 0.5msec, when b is increasing exponentially, the A- equation has insufficient time to operate. In fact the exponential growth confines the point to a region very near E until the last 0.1 msec of this initial triggering period. When the A-equation switches on, the system does indeed home vertically towards J (which is actually a little surprising) but very slowly, so that the impulse is not triggered at time T , it, but at about t =2.0msec. The "extra" time is required so that the "fast" equation can drift the system slowly down; this takes some 1.5 msec. which is actually longer than the duration of the nerve impulse.

Note that this does not provide evidence against the use of jump approximations altogether, as may be seen by permitting ~ to become

i ". t

ltion wove ~- �9 .

if !g g

I [ bl 5

i

Figure 12. The propagated action potential: trajectory of the state point on the (x,b) plane of the cusp catastrophe (at a = - 0 . 4 ) based on Figure 11. Action of the propagation and local equations (11), (12) and (13) does not move the state point to the trigger point J on the slow manifold at a = - 0 . 4 . Each dot represents a time interval of 0.02 msec. For further details, see the text.


2-

%

"X

" , . . . . o . .

' * . , . ~ 1 7 6

I !

.... " - . . . . �9

" ' . o . . . . . . . , �9 �9 �9

Figure 13. The propagated action potential: trajectory of the state point from Figure l l on the (x,a) plane of the cusp catastrophe. For further details, see

Figures 11 and 12 and the text.

much smaller. We replace Zeeman's x-equation by"

~:2 = - (x 3 + ax + b) (15)

and leave the rest unchanged. With ~:=0.02 the corresponding trajectory is as shown in Figures 14 and 15. The point does indeed move very rapidly to J, as suggested by the ' j ump" model, and thereafter stays close to the slow manifold. For an even smaller ~: the sudden downward motion is even more rapid, to an extent that makes computat ion considerably harder. This is because the equations are "stiff': we see that for numerical purposes the singular approximation can be better behaved than the exact equations with very small ~.

Even making ~: small does not regain Zeeman's Figure 26 for the action potential, however. With c=0.40 (Figure 16) the "horizontal" portion of the computed action potential is reduced. However, even with e =0.05 (Figure 17) a noticeable "kink" still exists, although this profile is a


2Q

I [ " I { l l I I I I bl ............it... 5

Figure 14. The propagated action potential: trajectory of the state point on the (x,b) plane of the cusp catastrophe. Speeding up the fast equation by reducing e from 0.8 to 0.02 has the effect of moving the state point to the trigger point J on the slow manifold. At J, the local equations (11), (12) and (13) dominate and return the state point to equilibrium E. For further details,

see Figures 11, 12 and 13 and the text.

much better approximation to Figure 6 than is Figure 11. For an acceptable wave profile we probably need e=0.01 or less; though some improvement is possible if we "smooth" the transition from the propagation wave to the local equation--a procedure that also makes good physiological sense. For example we can smoothly reduce the contribution of the propagation wave, while increasing that of the local equations, over a short time interval.

Without this type of modification, however, we have seen that Zeeman's equations (11), (12), and (13) fail to correctly predict the profile of the propagated action potential: instead they generate an action potential curve with a long latency (1.5 msec), a smooth rise to a peak, and a somewhat linear recovery to rest (Figure 11). This shape, however, is qualitatively similar to a membrane (or non-propagated) action potential generated by a stimulus just above threshold (see, for example Hodgkin and Huxley,


v(m,,)

100

2'

I

j -

".%,

"X

I

t . . - /

.~176176176176176176176176 ~'~176176176176176176 ...... ~176 ~

Figure 15. The propagated action potential: trajectory of the state point on the (x,a) plane of the cusp catastrophe. For further details, see Figure 14 and

the text.

3 4 ! !

t (mser !

Figure 16. The propagated action potential: use of the local nerve impulse equations with e=0.4 (which improves the fit to the voltage clamp data) does not produce an action potential resembling that obtained experimentally. For

further details, see Figure 1 and the text.

304 1. STEWART AND A. E. R. WO O D C O C K

V (my)

100

t (msec)

Figure 17. The propagated action potential: even with e=0.05 the computed action potential profile is not acceptable when compared with experiment. For

further details, see Figures 1 and 16, and the text.

1952d, p. 525). In this case the initial stimulation is followed by a period of several milliseconds before the action potential appears.

"Membrane" action potentials are those in which "the membrane potential is uniform at each instant, over the whole length of the fibre . . . there is no current along the [axon]" (Hodgkin and Huxley, 1952d, p. 519). It is under these conditions that Zeeman's local equations (11)-(13) are operative. The effect of the relatively large value of e (~=0.8) in Zeeman's equations is to reduce the rate at which the membrane potential will change. In the singular case this will occur instantaneously. Here effect of the invasion of the propagated part of the electrical activity [equation (14)] resembles that of the imposed stimulus required to generate a membrane action potential.

This similarity in wave form is seen even more clearly in comparisons of Figure 11 with membrane action potentials measured and computed by Adelman and FitzHugh (1975, p. 1324). Adelman and FitzHugh produced important modifications of Hodgkin and Huxley's (1952d) equations which make provision for the following observations: the presence of an external myelin sheath prevents the rapid diffusion of extracellular accumulations of potassium ions from the squid nerve (Adelman et al., 1973). This reduces the effective concentration difference of potassium ions across the nerve membrane as the action potential proceeds to completion. It also affects the process of sodium ion inactivation responsible for stopping the inward flow of sodium ions (Adelman and Palti, 1969). Under these circumstances Adelman and FitzHugh (1975) record action potentials which are very similar in shape to those predicted by their modification of Hodgkin and


Huxley's equ~ttions. They are also qualitatively similar to Figures 11 and 16 of this paper.

An understanding of the nature of e in Zeeman's equation (11) (e=0.8) and its possible relationship to modelling some properties of the nerve axon may possibly come from studying Ram6n et al. (1975). In a series of experiments Ram6n and his colleagues cooled a segment of a squid giant axon and then observed the behavior of a propagated action potential as it moved from a warm segment (25~ into this cool (3~ region. On either side of the cool region the propagated action potential closely resembled that described by Hodgkin and Huxley (reproduced here as part of Figure 1). However, propagated action potentials recorded from the cool region showed a gradual rise to a plateau depolarization of some 25mV above the resting potential for about 0.5 msec. This was followed by a smooth rise to a peak and an approximately linear recovery to rest (Ram6n et al., 1975, Figure 8). Under these conditions the cool temperature appears to slow down the normal process of action potential propagation and produce a response that partially resembles a membrane action potential produced by a stimulus just above threshold. Therefore, it is tempting to suggest that the value of e may reflect the rate of some temperature- dependent membrane process. When slowed by a decrease in temperature this proces w produces action potential curves resembling those obtained from equations (11)-(13) with e equal to 0.8. Decreasing the value of ~, or speeding up the equations, produces curves (Figures 16 and 17) which are closer to those of the normal propagated action potential (Figure 1).

We are not asserting that Zeeman's equations in their original form necessarily model any of these phenomena. But the qualitative similarity suggests that suitable modifications of those equations may be worth pursuing in appropriate circumstances.

10. Zeeman's Model: Voltage Clamp Experiments. In the voltage clamp experiment (Hodgkin and Huxley, 1952d; Hodgkin, 1964; Cole, 1968) a segment of axon is isolated and the potential difference across the membrane is clamped at various fixed voltages. The flow of sodium and potassium ions across the membrane is then measured. Figure 2 shows experimental data (circles) and the values predicted by the Hodgk in - Huxley theory, for the potassium and sodium conductance of the nerve membrane (after Hodgkin, 1964).

Zeeman discusses the qualitative fit of his general type of model to such data, and uses voltage-clamp data to construct his "Model II", but he does not explicitly exhibit the predicted curves for potassium and sodium conductances (gK and gN,)- We have computed these; for the depolarizations given in Figure 2 we obtain Figure 18. The fit is

306 I. STEWART AND A. E. R. WO O D C O C K

qualitatively good, but not entirely satisfactory. By making e=0.4 we obtain a better fit, as in Figure 19. The sodium conductance probably could be made to fit better still by scaling its functional relationship to x [equation (9)].

We saw above that the error in Zeeman's computat ion of the action potential may be corrected by making e very small. To fit the voltage clamp data also requires a smaller value of e. However, for e =0.4 the predicted action potential profile (Figure 16) is still unsatisfactory. If we decrease ~ further, there is no apparent problem with potassium conductance; but the sodium conductance rises with something close to a jump discontinuity, and thereafter decreases monotonically. For example, with e=0.05, the voltage-clamp results are as shown in Figure 20, for a depolarization Vr This does not compare well either with the experimental data or the theoretical curves of Hodgkin and Huxley (Figure 2).

This sudden jump in gNa is especially obvious if we consider the "singular" jump approximation: the system instantaneously jumps from the upper surface (zero sodium conductance) to the lower (non-zero), and then rises slowly along the slow manifold, decreasing the conductance. This shows that it will not be easy to correct this problem by changing the functional relation between x and gNa, since a similar discontinuity will arise.

Thus Zeeman's equations (with minor modifications) give reasonably good agreement with the voltage clamp data but not the action potential; modifying them by making e small gives a good agreement with the action potential but not the voltage clamp data for sodium conductance.

11. Other Criticisms. Sussmann and Zahler (1978, p. 182) have heavily criticized Zeeman's model, stating that "the catastrophe theory models are capable of explaining neither the voltage-clamp data nor the propagated action potential". We assume they mean "describing" rather than "explaining" since the nature of a model is to provide an analogy rather than an "explanation". According to our analysis, above, the true situation is that it can describe either one successfully, but not (at least without serious modification) both. The question, whether further modification is worth attempting, is partly contingent on how accurate the remainder of Sussmann and Zahler's criticism is, since if correct their work is a severe indictment of the whole approach. For this reason we must examine their criticisms in detail.

l l(a). Discontinuous response to continuous change of clamp voltage. In Section 11(1) Sussmann and Zahler (1978, p. 176) compare the "recorded


response-Current curves for different [-voltage] clamp values". They claim, citing experimental evidence (Cole, 1968; Hodgkin, Huxley, and Katz, 1952, pp. 431-434), that "the curves change continuously". In Zeeman's models they state that: "when the clamp is turned on the state point jumps either to the upper or the lower attractor surface of the smooth manifold, depending on whether V~ [-the clamp voltage] is smaller or larger than a certain critical value V ~ [-a term introduced by Sussmann and Zahler, not by Zeeman]. Thus the behavior changes abruptly as V~ passes V ~ unlike the experimental result" (pp. 176-177).

Sussmann and Zahler do not state explicitly the value of V ~ but in their subsequent discussion it seems to refer to Zeeman's threshold value. (It definitely cannot refer to any other value since the response varies continuously at all other values.) Thus for "Model II" V ~ is close to 18 mV: the point at which the fold line is crossed when a stays at its equilibrium value of - 0 . 4 and b increases.

We are mystified by these comments. Some of the experimental results quoted (Hodgkin, Huxley, and Katz, 1952, pp. 433 434) refer to the "time course of [-the] membrane potential following a short shock", not to the behavior of a clamped axon. Furthermore, these results imply the existence of a critical depolarization for the production of the action potential. Indeed, to quote Hodgkin et al. (1952), "If the depolarization was more than 15 mV . . . the response became regenerative and produced an action potential of about 100mV. If it was less it was followed by a subthreshold response . . . If the potential was displaced to the threshold level, it might remain in a state of unstable equilibrium for considerable periods of time." (p. 434.) This behavior is qualitatively similar to that shown in Figure 11 of this paper, which is, of course, computed from Zeeman's equations (11)-(13). A possible explanation for this confusion is that Sussmann and Zahler quoted the wrong page numbers in their discussion.

Sussmann and Zahler's (1978) remarks on the effect of voltage clamping on potassium and sodium conductivities further promote this feeling of mystification: on p. 173 they correctly quote Zeeman's remark about this effect (a fast increase and then a slow decrease in sodium conductance and a delayed, but slowly increasing, potassium conductance) based on Zeeman's Figure 23. However, by p. 177 they claim that "the model [-referring to their Figure 18] predicts that the steady state value of gK will be small if V~ < V ~ and much larger if V~ > V~ it also forecasts that g~: will increase almost immediately (no 'delay') when the clamp is turned on, if V~ < V ~ Both of these predictions are qualitatively very different from experimental results." Zeeman (1972) makes no such claim, and numerical solutions of Zeeman's quantitative equations (11), (12), and (13), called


"Model II" by Sussmann andZahle r (see Figures 18 and 21) do not appear to exhibit this (supposed) behavior.

On "Model II", which is the only one appropriate for comparison with quantitative data (as is necessary since the value of V~ affects the relevance of the data), the only possible candidate for V ~ is the threshold value of 18inV. This is outside the range of the experimental voltage clamp data cited (both by Zeeman and by Sussmann and Zahler). These data are therefore irrelevant to Sussmann and Zahler's contention that the predicted discontinuity is "unlike the experimental result" (1978, p. 177): over this range of values, Zeeman's model predicts c o n t i n u o u s variation. Their argument is comparable to an attempt to prove that water does not freeze based on the continuity of its properties between 10~ and 90~

Zeeman's (1972) Figure 24, shown here as Figure 2, refers only to V~ >=26mV, and the other references cited (Hodgkin and Huxley, 1952d, for example) contain such a small amount of data for V~< 18 mV that it would be extremely difficult to decide the question of continuity or discontinuity with any confidence. An analysis of the data presented by Hodgkin and Huxley (1952d, Figure 3) seems to suggest a continuous increase in the maximum value of potassium conductance obtained during voltage clamps in the range 6-38 mV. Further, the vertical scale on the graphs involved is such that the discontinuity , if it occurred, could possibly be so small as to be difficult to detect.

Numerical plots of solutions to Zeeman's equations, using a greatly magnified scale, are shown in Figure 21 for a variety of values of V~ on either side of threshold. The sequence of graphs obtained appears to show a c o n t i n u o u s variation within the first 6 msec.

However, a rigorous analysis (which we omit for reasons of space) shows that, for example, the limiting value of gK as t approaches infinity is a discontinuous function of V~. However the size of the discontinuity is less than 0.60mmho/cm 2 and this is probably too small to be detected in practice" it is certainly too small to detect on the scale of the data cited on the numerical plots shown. In these circumstances the word "abruptly" is ill chosen.

Zeeman's "threshold" value of 18 mV is of course very close to the value 15mV cited by Hodgkin, Huxley, and Katz (1952); and Zeeman (1972) explains on pp. 48 and 56 of his paper the involvement of the saddle point, as well as the fold, in modelling threshold effects.

As Figures 18 and 21 show, the claim that "gK will increase almost immediately . . . when the clamp is turned on" is wrong. The variable a that defines gK is a slow variable in the model, and behaves as this designation indicates. Possibly Sussmann and Zahler mean gNa rather than gK, which does increase abruptly (see Section 10) when the clamp is turned

109

ON ZEEMAN'S EQUATIONS FOR THE NERVE lMPULSE

f 30 mmho/cm2

gK

5 msec II I i I I I i I

309

8 8

6 3

3 8 - ~ �9 �9

" u J J d I I I I I

Figure 18. The voltage clamp: changes in sodium and potassium conductance with different membrane depolarizations predicted by solution of the local nerve impulse equations (11), (12) and (13) with equation (13), (b) set equal to zero. Here, 5=0.8. Sodium and potassium conductances were calculated by substitution of solutions for x, a and b into equations (8) and (9) respectively. Numbers associated with each curve show the assumed membrane clamp depolarization. Conductance changes appear to be delayed when compared

with the data in Figure 2. For further details, see the text.

on in "Model I", provided the clamp exceeds threshold, though not in "Model II" when e = 0 . 8 (Figures 18 and 20).

However, these computations of potassium conductance as a function of membrane depolarization deduced from Zeeman's equations do reveal a rather different and possibly serious problem for his model. They predict (Figure 21) that depolarization from rest to between 10 and 18mV, for example, reduces the potassium conductance to zero for many msecs. For larger depolarizations the period of zero potassium conductance becomes


109

2 �9 30 mmho/r

gNa

5msec

gK

88

63 S i I I I i i

2 6

Figure 19. The voltage clamp: sodium and potassium conductances computed as in Figure 18, but with e set equal to 0.4 appears to produce a closer agreement with the experimental data in Figure 2. However, the curves computed from solution of equations (11), (12) and (13) exhibit a significant delay in the rise of the sodium conductance. Numbers associated with each curve show the assumed membrane clamp depolarization. For further details,

see Figures 2, 18 and the text.

shorter and eventually, above 30 mV, disappears as the response becomes dominated by a secondary increase in conductance (Figure 21). This predicted behavior marks a clear distinction between the Zeeman (1972) and Hodgkin and Huxley (1952d) models. It is also at variance with what might naively be expected from a consideration of the equilibrium conditions of the potassium ions associated with the nerve membrane. At the resting potential the concentration gradient of potassium ions across the nerve membrane is within 10 to 15mV of the ionic equilibrium potential computed from electrochemical considerations. The membrane exhibits a small but measurable potassium conductance under these


20 -

mmho/cm 2

gK 10

�9 ' I

f t (msec)

Figure 20. The voltage clamp: sodium and potassium conductance computed as in Figure 18 but with e set equal to 0.05. Assumed membrane depolarization 52mV. While the change in potassium conductance appears to be qualitatively acceptable when compared with experiment (Figure 2, for example), the almost discontinuous change in sodium conductance appears markedly different from

the experimental data. For further details, see Figures 2, 18, 19 and the text.

circumstances. Clamping the nerve membrane at a potential even further from the potassium ion equilibrium potential should, following this argument, cause an even greater electrical imbalance for these ions.

This should cause an increase in potassium conductance, provided that the flow of ions is not prevented by some other process. The marked difference in the changes in potassium conductance predicted by Hodgkin and Huxley (1952d), and by Zeeman (1972) at small depolarizations, might therefore serve as a real basis for comparing these models with experimental data. While some small depolarization-Voltage clamp data does exist (for example Hodgkin and Huxley, 1952a; Chandler and Meves, 1965) it is not very extensive or conclusive. Resolution of this question, therefore, will have to await the provision of further data. In the event that this prediction of the Zeeman (1972) model is not borne out by suitable experimental observations, modification of its equations might rectify the problem and serve as a basis for the model's refinement. Such modification appears feasible, in contrast to the problems raised in Section 9.

However, because of the accumulation of extracellular potassium ions (see Adelman and FitzHugh, 1975, for example) these subthreshold clamp experiments might produce results which might be difficult to interpret (Dodge, personal communication). A more realistic approach might follow the procedures of Mauro et al. (1970). They studied the effect of small transient depolarizing clamps on the membrane potential in both experimental and theoretical studies of the squid axon. In particular,


1 m m h o / c m 2 /

26~'~ ,

22 ~ '~ ,

18 ~ ' ~ ,

5 111 s e c

/ /

14

-----w---, I0

L 6

2

Figure 21. The voltage clamp: changes in potassium conductance for relatively small membrane depolarizations computed as in Figure 18, with e set equal to 0.8. Vertical Scale: 1 unit equals lmmho/cm 1. (Note change of this scale compared with Figures 18, 19 and 20.) Numbers associated with each curve show the assumed membrane clamp depolarization. The curves predict that membrane clamps of between 10 to 18mV reduce the potassium conductance to zero for at least 5msec; a clamp at 30mV shuts off the conductance for about 1 msec. Above 30mV the two portions of the curve fuse as the delayed, rapidly rising, phase of the conductance change dominates. For

further details, see Figures 18, 19, 20 and the text.


further investigation of Zeeman's model could study its predictions of nerve membrane behavior in response to transient depolarizing clamps.

11(b). Independence of potassium and sodium channels. Section 10(2) of Sussmann and Zahler (1977, 1978) is concerned with the question of the independence of the membrane conductance pathways for potassium and sodium. They contrast Zeeman's (1972) proposal that the "potassium conductance cannot begin to rise until the sodium conductance has reached its peak", during the propagation of the nerve impulse, with his observation that "Hodgkin and Huxley assume that the two permeabilities are indepenaent of one another". Now Hodgkin and Huxley (1952d, p. 530) say the "rapid rise [in membrane potential] is due almost entirely to sodium conductance, but after the peak the potassium conductance takes a progressively larger share until, by the beginning of the positive phase, the sodium conductance has become negligible". However, from Hodgkin and Huxley's (1952d) data sodium and potassium conductances both clearly depend on the existing value of the membrane potential. Therefore, since a change in one of the two ionic conductances leading to a change in membrane potential will influence the conductance of the other ion, the conductances are, in this sense, interdependent.

Sussmann and Zahler (1978, p. 178) cite a review by Ehrenstein and Lecar (1972) of data obtained "both by use of radioactive potassium and the selective neurotoxins TTX and TEA" as a support for the claim that the two permeabilities are independent. However, since these toxins may interfere with many membrane transport processes, even perhaps differentially with the same process, their use cannot provide unequivocal evidence about the nature of ionic movement across the nerve membrane. Indeed, Chandler and Meves (1965) indicate that one of these toxins (TTX) can influence both sodium inflow and early, transient, movements of potassium ions in the clamped squid nerve membrane. Sussmann and Zahler's unqualified claim that Zeeman's prediction (of sodium and potassium interdependence) "contradicts experiment" (1978, p. 178) is not true. The nature of ionic flows associated with the nerve membrane has been the subject of extensive research since the work of Hodgkin and Huxley (see Meves, 1978; Hille, 1976; Noble 1966; and Roy, 1975 for reviews). Recent investigations have reported an interaction between sodium and potassium flows in the nerve membrane (French and Wells, 1977); a change in sodium ionic membrane selectivity caused by changes in internal potassium concentration (Cahalan and Begenisich, 1976); an effect of external potassium ion accumulations on sodium inactivation (Adelman and Palti, 1969); and even the possibility of multiple sodium channels (Chandler and Meves, 1965; Bezanilla and Armstrong, 1977; and Armstrong and Bezanilla, 1977). The situation is obviously very BMB--C


complicated and it seems to be too early to make any conclusive statements about the interdependence of sodium and potassium movement. It is certainly very premature to make the statement--as Sussmann and Zahler (1978, p. 178) do - - tha t a claim based on sodium and potassium interdependence "weakens the theoretical underpinnings of the catastrophe theory models."

11(c). Shut-off of the voltage clamp (Model I). In Section 10(3.) Sussmann and Zahler discuss "Model I" in relation to a shut-off of the voltage clamp. As noted earlier, the force of this criticism depends on whether Zeeman's "Example 8" is intended as a serious model, or merely as a representative of a class of possible models. If the latter, then Sussmann and Zahler's argument is inappropriate, and indeed affords the equations of Zeeman's "Example 8" a prominence that their context suggests was not intended. Although we think the latter more likely, let us assume the former.

Zeeman describes what would happen, on this model, if the voltage clamp were shut off and the voltage allowed to vary at will. Sussmann and Zahler contrast this with an experimental procedure in which the voltage is reclamped to zero. Zeeman (1972, p. 50) never asserts that his description relates to any experimental procedure, but only suggests that "if the clamp is released then the state returns to equilibrium E along the dotted flow lines."

According to Sussmann and Zahler (1978), reclamping to zero leads us to consider the phase plane b=0 , in which the phase portrait is Figure 22(a). They observe that as well as the equilibrium point E there is another possible equilibrium at the origin 0, which is "stable on one side and unstable on the other (p. 179). Hence the system returns not to E as Zeeman asserts [-referring not to the voltage reclamped to zero but to a freely varying voltage], but to the origin." While this is not the situation that Zeeman discusses it is a valid point to raise, since the model is required to agree with any experimental procedure, including one not envisaged by Zeeman (1972).

It is, however, only true of models of type I for which the equilibrium point is at b=0 , with arbitrary choice of a. Negative b [-as Zeeman (1972) chooses for Model II and his Figure 23] leads to a different phase portrait with only E as equilibrium. Figure 22(b) shows the relevant portrait for "Model II" with b = - 0 . 8 , and this is typical. Having noted this we may ask whether Zeeman's location of the equilibrium point in his Figure 23 is indeed an error, as Sussmann and Zahler assert, or a step towards Model II by refining Type 1 models to avoid the problem noted by them. This is unclear; but what is clear is that Model II, the only serious quantitative model, avoids this criticism entirely.

O N Z E E M A N ' S E Q U A T I O N S F O R T H E N E R V E I M P U L S E 315

I I 't

,,

4" /

J / t

I

1

I I

I I I

/ /

I I t I

I

~, ,~ ,,.,,,, ~ ~ _

/ /

)~ / j _ _ j \

\ " I I

~ t t L I

initial posit ion in this region

Figure 22. The voltage clamp: (a) The (x,a) phase plane of a fast cusp gradient dynamic plus show manifold flow at (b=0) (Redrawn from Sussmann and Zahler, 1978). Sussmann and Zahler (1978) claim that the flow lines in this figure support their, notion that restoration of a voltage clamp to zero would cause the state point to move to the origin (0) rather than the equilibrium point E. (b) The corresponding (x,a) phase plane for b<0. This is the (x,a) plane passing through the equilibrium point in Zeeman's (1972) voltage clamp model, "Model II", though not for his "Example 8", "Model I". The initial voltage clamp (path FGH, Figure 5) has moved the state point to some positive value of "a" (a function of the potassium conductance). In cases (a) with noise, and (b) with or without noise, simple removal of the clamp (Zeeman, 1972) without "returning the clamp voltage to zero" (Sussmann and Zahler, 1978), will generally return the state point to the equilibrium, E, not to the origin 0. Reclamping to the resting potential will also force the state point to E. For

further details, see Figures 5, 7 and the text.

A realistic analysis, even for M o d e l I, mus t take accoun t of possible noise in the system. Sussmann and Zah le r address this poin t as follows: "subject to the r a n d o m f luc tuat ions to be expected in a physical system, the s ta te poin t will r e tu rn ei ther to E, or to the origin, a t r a n d o m " (p. 179). W e c a nno t fol low this: the origin, as they have noted, is uns table and hence c a nno t occur as a rest pos i t ion given noise. Deak in has in formed us tha t he and Pa te r son reached the same conc lus ion in 1977. The on ly stable


equilibrium is E, hence in a noisy system the only rest position is E. This agrees with Zeeman's assertion (even though that was made for a different circumstance).

Note that for b---0 the return to E will often involve a period of time spent near 0, where the flow is very slow (by virtue of the singularity). Also for b > 0 [rather than b < 0 as in Figure 22(b)-] there is an additional equilibrium other than E or 0, which is technically stable; however, it has a very small basin of attraction and hence will not be observed in practice if b is small.

Sussmann and Zahler (1978) claim the same incorrect result for Ibl small; and even in their terms this is wrong for b<0, as Figure 22(b) shows.

11(d). The applied current (Model 11). I n Section 10(4) Sussmann and Zahler (1978) discuss the applied current in Zeeman's Model II. They imply that the model based "on the assumption I = 0 , where 1 is the total outward flow of current across the membrane . . . cannot describe the vol tage clamp experiment". Here they fail to understand that the assumption I = 0 refers only to the derivation of the local equations where non-zero 1 would imply that charge accrues locally at all points. Furthermore Hodgkin and Huxley (1952d, p. 519) use this same criterion ( I =0 ) to describe the "membrane" action potential in which "the membrane potential is uniform . . . over the whole length of the fibre". These aspects of the Zeeman and Hodgkin and Huxley models have similar bases. In the case of the voltage clamp experiment in Zeeman's model the applied external voltage overrides the /;-equation (13), which is the one that assumes 1=0, and clamps b to a fixed value; only x and a vary. Therefore, as Paterson (1977)notes, in Zeeman's model of the voltage clamp experiment the assumption that I = 0 is not required.

11(e). Velocity of the impulse. Hodgkin and Huxley (1952d, p. 523) determine the velocity of the propagated impulse by a process of elimination: the "correct value brings V (determined by solution of the appropriate equation) back to zero (the resting condition) when the action potential is over".

Note that this procedure shows that a travelling wave is possible rather than necessary, but predicts its velocity if it occurs. Note also the sensitivity of the equation to this velocity: anything other than the unique "correct" value leads to unbounded solutions for the membrane potential V.

Zeeman computes the velocity of the impulse by a different method. It is an automatic feature of the "factorization" into a propagatio n wave and a local action, assumed in this model, that there is a travelling wave whose velocity is that of the propagation wave. Hence the velocity may be


computed by using the functional form for the propagation wave, given by Zeeman's Theorem 3. By using experimental values of the voltage and its rate of change at a known time, Zeeman computes the velocity to be 0 = 21.9 m/sec.

Sussmann and Zahler (1978, p. 180) describe this computat ion as "heuristic", which it is not: it is a rigorous (and easy) deduction from Zeeman's equation [reproduced here as equation (14) of the propagation wave as fitted to the experimental data]. Sussmann and Zahler (1977, 1978) seem to have confused Zeeman's rigorous analysis within a heuristically motivated model with a heuristic deduction within a different model. They remark that the existence of a travelling wave is not guaranteed, in general. Now for the customary models this is t r ue - - and a major mathematical effort in these models is indeed in making this deduction. But for Zeeman's model the travelling wave is a trivial consequence of the "factorization". As the threshold traverses the membrane at each point of the axon it generates a series of (local) action potentials "staggered" in time according to when the threshold arrives. Because this local response is the same at all points, it has the same appearance as a travelling wave. To our mind, this is not an especially attractive feature of the model: it makes travelling waves "too easy" to produce, and begs a number of important questions. However, one can hardly deny that Zeeman's model involves a travelling wave, just because it seems to obtain one too easily.

Sussmann and Zahler (1978) note early on in Section 10 that, using Zeeman's procedure, "one . . . calculates 0 [the propagation velocity] =21.9 m/sec" and that Zeeman claims this "agrees well with the observed velocity of 21.2m/sec" (p. 175). They then claim that this computat ion "has nothing to do with Zeeman's cusp models" (p. 180) [which is fair enough in the sense that only the equation for /~, which is Zeeman's modification of one of Hodgkin and Huxley's (1952d) equations, is required for this purpose]. However, by Section 10(6) they have recast this observation as "we have seen that the value 0=21 .gm/sec cannot be deduced from Zeeman's Model II" (p. 181). What we have seen is that Zeeman's deduction of this value, while following from his equations, does not make essential use of the catastrophe-theoretic features of his model. To infer that this value is not a consequence of the model is thus incorrect.

They then embark on a computation, to be described below, whose conclusion is that no travelling wave may be obtained from Zeeman's model. Now we have already seen that a travelling wave is an automatic consequence of Zeeman's model. It follows that a computat ional method which concludes that one does not exist is either wrong, or applied to the wrong model.


In fact, Sussmann and Zahler (1978) merely combine Zeeman's local equations with the cable equation. They then follow Hodgkin and Huxley's methods of obtaining a solution of this equation, failing to realize that the local equation is inoperative for the first 0.5msec, and subsequently overrides the propagation wave. This procedure they describe as "the correct way to obtain the propagated action potential" (p. 181). In effect, Sussmann and Zahler analyze what would happen in Zeeman's equations if we were to ignore the switching, keeping the local equations running the whole time and replacing Zeeman's propagation wave by Hodgkin and Huxley's cable equation. That effects inconsistent with experiment may be derived by combining only a part of Zeeman's model with part of Hodgkin and Huxley's in this way can hardly be considered a valid criticism of Zeeman's model.

In a summary of their critique Zahler and Sussmann (1977) state (under the heading: "careless discussion of evidence") that: "the catastrophe theory nerve-impulse models . . . lead to the wrong propagation speed for the action potential". This is untrue, and does not even represent correctly the conclusions of their own analysis of the above "hybrid" of Zeeman's model and Hodgkin and Huxley's (which finds not an incorrect velocity, but the non-existence of any travelling wave). This same summary repeats the allegations of disagreement with voltage-clamp data [see Sections l l(a) and ll(c)] and incorrectly refers to the independence of sodium and potassium channels as "universally accepted" ]-see Section l l(b)].

There is, however, a criticism that c a n be made along these lines. It concerns the role played by the cable equation in the Hodgkin and Huxley model (1952d) to describe the interaction between nearby segments of axon. Zeeman tells us how the interaction occurs in two circumstances: the voltage clamp experiment (no interaction) and the propagated action potential (interaction simulated by the propagation wave). His model does not specify how neighboring segments interact under other circumstances. By contrast, Hodgkin and Huxley's use of the cable question is applicable to any initial conditions along the axon; their theory thus gives predictions under a wider range of conditions.

Zeeman's (1972) model would therefore be greatly improved if it were supplemented by a prescription for modelling this interaction under arbitrary initial conditions. The obvious way to do this is to mimic Hodgkin and Huxley and add in the cable equation: what Sussmann and Zahler's (1977, 1978) computations appear to show is that this procedure does not work. Sussmann and Zahler have not "tested Model II" (p. 181) as they state: they have. tested their own attempt to modify it (though this might not be apparent to the casual reader). That this attempt fails is no fault of Zeeman's model.


The existence of bounded solutions of the Hodgkin and Huxley (1952d) equations depend very critically on values of the velocity term (0). In some cases (FitzHugh and Antosiewitz, 1959) it was necessary to interpolate between values of 0 that differed only by one ten-millionth of one per cent to obtain bounded solutions. To support their rejection of Zeeman's (1972) model for the propagated action potential is it necessary to prove that no

value of 0 (in a reasonable range including Zeeman's value 21.9m/sec) leads to bounded solutions. Interpolation from Sussmann and Zahler's Table 1 suggests this to be the case--but it is desirable to confirm the validity of such interpolation, perhaps by using very small increments of 0.

Lieberstein (1967a, 1967b), and Paterson (1977) have discussed the reason for this extreme sensitivity of the Hodgkin and Huxley (1952d) equations to 0. Lieberstein (1967b) notes that local solutions of these equations are the sum of two exponentials and attributes this sensitivity to truncation errors introduced by Hodgkin and Huxley (1952d). He cautions against the "casual dropping (of) terms from differential equations based on the assessment that shows them to be dominated numerically by other terms in the differential equations". Under similar circumstances, Poston and Stewart (1978) recommend that "when truncation introduces degeneracies it is only prudent to try putting the higher terms back".

The stability problems of the original Hodgkin and Huxley equations may be overcome by equations (Lieberstein, 1967a, 1967b) which incorporates a line inductance term, "even if this term is very small". These equations are of a hyperbolic rather than the parabolic type used by Hodgkin and Huxley (1952d) (Lieberstein, 1967b). It would be desirable to analyze a combination of local equations of Zeeman's (1972) type with this modified cable equation, before concluding that Zeeman's proposed approach cannot be made to work.

12. Conclusions. (1) Sussmann and Zahler's (1977, 1978) criticisms of the derivation of Zeeman's model carry some weight, but are not of themselves conclusive evidence against it since it remains a reasonable possibility (subject to experimental testing) once the necessary hypotheses are spelled out.

(2) Their criticisms of the predictions of the model, in detail, are almost entirely incorrect, and appear to be based on a fundamental misunderstanding of the mechanisms of the model. Claimed discrepancies between the predictions of Zeeman's model and the results of experiment are largely based on incorrect statements of its predictions, inappropriate experiments, or irrelevent ranges of data; .and no mention is made of conflicting evidence which might support Zeeman's model in cases where the existing results are inconclusive.


(3) However, there is a fault not noticed by Sussmann and Zahler (1977, 1978) (although their observations concerning motivation might suggest investigating it), namely that e =0.8 is too large. We have solved Zeeman's equations [here (11)-(14)] for several non-singular cases, that is, with e r for the propagated action potential. These solutions produce action potential curves which more closely resemble a membrane action potential generated by a just supra-threshold stimulation than the curve of a propagated action potential. We note that ~ may be interpreted to reflect some temperature-dependent membrane process. We have described the effect of changing the values of e on the shape of the action potential

i

curves. (4) We note that Zeeman's model predicts that the membrane potassium

conductance decreases ~o zero for low clamp voltages, a behavior which is also not described by Sussmann and Zahler (1977, 1978). While other values of ~ (e = 0.02) produce a more suitable propagated action potential curve, a value of e=0.05 produces what appears to be an abnormal curve for the membrane sodium conductance.

(5) Certain features of Zeeman's model,i such as the saddlepoint, have independent interest; and it remains possible that some model based on a travelling threshold may be constructed that will avoid the above problems. Whether such a model will have any connection with catastrophe theory is unclear. The specific choice of fast and slow variables made in Zeeman's w o r k is, as Sussmann and Zahler suggest, debatable; this may indicate that quite drastic changes are required for any hope of success.

(6) In passing we note the status of this model in regard to "catastrophe theory". According to Sussmann and Zahler it is a model considered extremely important by "catastrophists", and one of the best exhibits in favor of the applicability of that subject. As evidence they quote a statement made by Abraham in 1972 that "this is one of the most impressive applications of catastrophe theory". This is asserted to be a "good example of how truth is transformed into untruth by repetition". Poston (1978) notes that no repetitions of this statement are cited by Sussmann and Zahler; that he pointed out possible defects in Zeeman's model in 1975 (the question of e=0.8 being raised by de la Harpe in 1974); that Stewart (1975) states that there are difficulties with this model. The nerve impulse model is not currently considered to be an important application of catastrophe theory, although features of its methodology have led to interesting work [-see (8) below]. Better evidence of its applicability as such (though, of course, not to biology) is the large number of highly successful applications that now exist in the physical sciences (Poston and Stewart, 1978; Zeeman et al., 1977).

ON ZEEMAN'S E Q U A T I O N S F O R THE NERVE I M P U L S E 321

(7) A major problem faced by Zeeman's model is its failure to seriously challenge the successes of the Hodgkin and Huxley and other models. This coupled with the fact that no experimental observation as yet has shown Zeeman's model to be better than other models is a serious obstacle to its adoption, even ignoring the problems raised in (3) above.

(8) Despite its faults, Zeeman's (1972) paper is not without value. To our knowledge no criticism has been made of the theory of the heartbeat, an integral part of the paper. The methodology suggested by Zeeman has been used by Merrill (1977) to study the humoral immune response, and by Varela et al. (1977) to describe the control of ciliary movements-- though the latter is more an application of the Van der Pol equation than of catastrophe theory. Zeeman's use of fast/slow flows on catastrophe manifolds stimulated the work of Takens (1976) on constrained differential equations, and Taken's results have in turn been applied by Plant (1977) to nerve impulse equations similar to those introduced by FitzHugh (1960, 1961).

(9) In the heat of debate it is easy to lose sight of original objectives. While popularizations of "catastrophe theory" have tended to concentrate on the elementary catastrophes, the programme envisaged by Thom (1975) included the whole field of bifurcation in continuous systems, there called "generalized" or non-elementary catastrophes. Their importance has repeatedly been emphasized in the literature (for example, see Thom, 1977). Probably the simplest of these is the Hopf bifurcation, well known to applied mathematicians, and to which both Thorn (1975) and Zeeman (1965) refer.

Thom's contention that methods from topological dynamics, applied to bifurcations, should prove relevant to biology, is borne out by much recent work. Whether such methods are "catastrophe theory" is a matter of definition" in the current controversy it is of course in the interests of opponents to restrict the definition to as narrow a field as possible, and of proponents to broaden it--tendencies which are evident in the literature of the debate. Whatever the outcome of this argument over terminology, one fact emerges on which both factions seem to agree: biology (and science in general) requires more than simply the elementary catastrophes for its models [Although the elementary catastrophes both remain important as the best developed and most accessible examples of this kind of thinking, and have numerous applications in their own right (Poston and Stewart, 1978)]. Topological dynamics, bifurcation theory, and singularity theory (elementary catastrophe theory) are beginning to appear as different, but often overlapping, aspects of the same broad picture (see Golubitsky and Schaeffer, 1979a, b for example, or the review by Marsden, 1978). It is this broad picture that Thorn (1976, 1977) means by "catastrophe theory". But


it is the deve lopmen t of this picture, and no t the ch am p io n in g of any par t icu la r piece of technique, tha t is impor tan t .

Michae l Deak in (Monash )~ -wi th w h o m we discussed a pre l iminary draf t of this p a p e r - - b r o u g h t to ou r a t t en t ion the u n d e r g r a d u a t e essay by Douglas P a t e r s o n (1977) on Zeeman ' s mode l and its cri t icism by Sussmann and Zahler . T im P o s t o n (Battelle, Geneva) , and Chr i s tophe r Z e e m a n (Warwick) have also rendered us mos t valuable critical assistance. Freder ick D o d g e (IBM, Y o r k t o w n Heights) p rov ided us with i m p o r t a n t in fo rmat ion ab o u t sub th resho ld vol tage c lamping exper iments . We th an k Richard F i t z H u g h (Nat iona l Inst i tutes of Heal th , Bethesda, MD.) and Raphae l Zahler (Yale) for their helpful review of this paper . This paper was wri t ten while I.N.S. was a visitor at the Univers i ty of Connec t icu t , Storrs , Connect icut , U.S.A. We are grateful to the Univers i ty of Connec t i cu t for the facilities provided. We are also grateful for the helpful secretarial assistance of Judi th C o u n t e r (Williams).

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Documents

On Zeeman's equations for the nerve impulse