Krnjevic & Miledi, 1959; Naess & Storm-Mathisen, 1955) it is unlikely that such. Sydney, Australia 2036

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1 J. Phyriol. (1985), 367, pp With 5 text-figures Printed in Great Britain ACTIVATION OF THE HUMAN DIAPHRAGM DURING MAXIMAL STATIC EFFORTS BY S. C. GANDEVIA AND D. K. McKENZIE From the Unit of Clinical Neurophysiology, The Prince Henry Hospital and Department of Respiratory Medicine, Prince of Wales Hospital, School of Medicine, University of New South Wales, Sydney, Australia 2036 (Received 2 October 1984) SUMMARY 1. Voluntary activation of the human diaphragm was assessed in four subjects by interpolation of supramaximal stimuli (one to four shocks, interstimulus interval 20 ms) to one phrenic nerve during graded static inspiratory and expulsive efforts at functional residual capacity. 2. There was an inverse relationship between the size of the voluntarily generated pressure and the size of the electrically evoked change in pressure. Each subject activated the diaphragm fully during brief (2-5 s) maximal inspiratory and expulsive efforts, as judged by the failure of supramaximal stimuli to augment the voluntarily maintained pressure. 3. During prolonged inspiratory efforts or following fatigue of the diaphragm produced by a sustained contraction each subject was able to activate the diaphragm maximally but did not do so on all occasions. INTRODUCTION During voluntary contractions, most co-operative subjects can activate maximally distal muscles of the limbs (Merton, 1954; Belanger & McComas, 1981 ; Merton, Hill & Morton, 1981; Grimby, Hannerz & Hedman, 1981; Bigland-Ritchie, Kukulka, Lippold & Woods, 1982; for reviews see Bigland-Ritchie, 1981; Marsden, Meadows & Merton, 1983) despite occasional reports to the contrary (Ikai, Yabe & Ishii, 1967; Bigland-Ritchie, Jones, Hosking & Edwards, 1978). Moreover, the decline in force during sustained maximal contractions of limb muscles (i.e. fatigue) can be unaltered by supramaximal stimulation of the motor nerve or the muscle itself (Merton, 1954; Bigland-Ritchie, Jones & Woods, 1979; Merton et al. 1981; Bigland-Ritchie, 1981; Bigland-Ritchie et al. 1982; but cf. Grimby et al. 1981). Although the neuromuscular junction may fail to transmit action potentials during repetitive stimulation (e.g. Krnjevic & Miledi, 1959; Naess & Storm-Mathisen, 1955) it is unlikely that such failure is significant during voluntary contractions of limb muscles (Bigland-Ritchie, 1981; Bigland-Ritchie et al. 1982; Bellemare, Woods, Johansson & Bigland-Ritchie, 1983). Stimulation of the human motor cortex fails to restore force during fatiguing

2 46 S. C. GANDEVIA AND D. K. McKENZIE voluntary contraction (Merton et al. 1981). Thus it is probable that force is limited by the contractile apparatus during sustained maximal voluntary contractions of distal muscles. The extent of voluntary activation of the respiratory muscles during maximal contractions is unclear. It has long been held that maximal static respiratory efforts are limited by reflex inhibition (Craig, 1960; Agostoni & Rahn, 1960; Grassino, Goldman, Mead & Sears, 1978; Gibson, Clark & Pride, 1981) or by general discomfort and haemodynamic changes (Milic-Emili, Orzalesi, Cook & Turner, 1964). Differences in the electromyographic (e.m.g.) activity of the diaphragm (obtained with oesophageal electrodes) during maximal static inspiratory and expulsive efforts led to the suggestion that the diaphragm is submaximally activated during inspiratory efforts (Agostoni & Mead, 1964). Submaximal voluntary activation ofthe diaphragm has also been postulated to account for the broad plateau of the 'length-tension' curve of the human diaphragm measured during maximal static contractions (Braun, Arora & Rochester, 1982). During repeated maximal static contractions (each of 5 or 30 s duration), the inspiratory muscles of healthy subjects are relatively resistant to the development of fatigue compared with the expiratory muscles or the muscles acting at the elbow joint (Gandevia, McKenzie & Neering, 1983). Fatigue here is considered to be a failure to produce force. This finding cannot be explained by a preponderance of fatigueresistant fibre types (I and IIA) in the inspiratory muscles (McKenzie, Gandevia & Shorey, 1983). However, the resistance to fatigue would not be surprising if the inspiratory muscles could not be fully activated during maximal voluntary contractions, as if some capacity to generate force were 'held in reserve' and only used when some muscle fibres failed to produce force. Indeed, the apparent resistance to fatigue during maximal bilateral contractions of quadriceps femoris is due to submaximal activation of the relevant motoneurones (Vandervoort, Sale & Moroz, 1984). Inability to activate both halves of the diaphragm fully at the onset of a maximal effort could produce an apparent resistance to fatigue of the inspiratory muscles (Gandevia et al. 1983). This study reports the use of twitch interpolation during graded voluntary contractions to investigate the extent of central activation of the diaphragm. If any motor units are not fully activated during maximal voluntary efforts, then supramaximal stimulation of the appropriate motor nerve should recruit them and produce a detectable twitch (Merton, 1954; see also Belanger & McComas, 1981; Bigland-Ritchie, 1981). A brief account of some of this work has appeared (McKenzie & Gandevia, 1984). METHODS Subjects Experiments were performed on four healthy male subjects (the authors and two volunteers who were not told of the hypothesis under investigation). Each subject was studied on three to five occasions. Subjects gave informed consent and the project was approved by the appropriate ethics committees. Experimental protocol Experiments were performed with the subject seated in a warm room breathing through a mouth-piece connected to a shutter and a pneumotachograph. Oesophageal, gastric, and mouth

3 ACTIVATION OF HUMAN DIAPHRAGM pressures were recorded. Transdiaphragmatic pressure (the difference between gastric and oesophageal pressure) was also monitored. Tidal breathing (integrated flow) was monitored on an oscilloscope to enable the shutter to be closed at the subject's end-expiratory level (functional residual capacity, f.r.c.). In some experiments the antero-posterior diameters of the chest and abdomen were monitored using pairs of linearized magnetometers at the height of the nipples and just above the umbilicus. The subject viewed the diameters as an X- Y plot so that contractions could be commenced at the same thoraco-abdominal configuration (Konno & Mead, 1967). In other studies subjects wore a moulded abdominal cast to reduce shortening of the diaphragm. 47 E.m.g. Pga Poes -mt: 1 ~~~~~~Stimulator probe Neck brace - Moving stage Oesophageal balloon -l -- Electrodes Stabilizing balloon _ Gastric balloon - Fig. 1. To stimulate the phrenic nerve an adjustable probe electrode (mounted on a moving stage) is attached to a rigid neck brace. A multilumen catheter (with 'stabilizing' balloon at the gastro-oesophageal junction) records gastric and esophageal pressures (Pga and P1es' via balloons as indicated) and the electrical activity of the diaphragm (e.m.g., via electrodes as indicated). Mouth pressure (Pm) was measured as indicated. In some experiments a moulded abdominal cast was used to limit displacement of the diaphragm and in other experiments pairs of magnetometers (not shown) were used to monitor antero-posterior diameters of the chest and abdomen. The phrenic nerve was stimulated at the level of the cricoid cartilage (Sarnoff, Sarnoff & Whittenberger, 1951) with an adjustable probe electrode (tip diameter 2'5 mm). The electrode was mounted on a moveable stage attached to a firm neck brace (Fig. 1). The anode was fixed to the manubrium and a large electrode on the anterior chest wall grounded the subject. Rectangular pulses ( ,ss duration) were delivered from an isolated constant voltage (or current) source while the electrode position was adjusted carefully so that supramaximal shocks (as judged by the amplitude of the diaphragmatic compound muscle action potential at f.r.c.) could be delivered to

4 48 S. C. GANDEVIA AND D. K. McKENZIE the phrenic nerve with minimal spread to the brachial plexus. The electrode was then tightened in the assembly to prevent movement during maximal voluntary inspiratory contractions when the sternomastoid and the scalene muscles are active. As judged by the stimulus intensities required for threshold activation and for a maximal action potential the electrode rarely required further adjustment for periods of up to 2 h. Shocks delivered during strong contractions were times the stimulus intensity required to produce a muscle action potential of maximal amplitude at rest at f.r.c. During repetitions of maximal contractions with interpolated shocks we always checked that the potentials could not be increased by further increases in stimulus intensity. The subject performed a series of graded inspiratory efforts against a closed airway with supramaximal stimuli (one to four shocks, interstimulus interval 20 ins) delivered to one phrenic nerve after 1-5 s. The airway (or oesophageal) pressure at the time of stimulation ('initial voluntary pressure') and the change in pressure evoked by stimulation ('evoked pressure') were measured. Stimuli were delivered early in these contractions because one of the protocols which was used to measure the endurance of the inspiratory muscles consisted of repeated static efforts of 5 s duration (Gandevia et al. 1983). Evoked pressure changes (including transdiaphragmatic pressure) were also measured from oscilloscopic records at high gain. In some experiments data were recorded on tape for subsequent analysis. On a separate day subjects performed series of graded expulsive efforts with interpolation of supramaximal shocks to the phrenic nerve during the first 5 s. During an expulsive effort simultaneous contraction of the diaphragm and abdominal muscles elevates gastric pressure which was used as an index of the contractile force of the diaphragm. Oesophageal pressure was often displayed to the subject to enable him to keep his glottis open, thus preventing recruitment of expiratory intercostal muscles. Despite this feed-back subjects were often unable to prevent glottic closure, especially during strong expulsive efforts. No attempt was made to maintain an identical chest-wall configuration between or during inspiratory and expulsive efforts but all contractions were commenced at the relaxation point near functional residual capacity. To determine whether there was any decline in central activation of the diaphragm during sustained maximal efforts, subjects also performed several prolonged maximal inspiratory contractions (15-30 s). Supramaximal stimuli were delivered to the phrenic nerve after s when the inspiratory pressure had declined. In three subjects activation was studied before and after diaphragmatic fatigue produced by a prolonged ( s) nearly maximal expulsive effort performed with the glottis open and the subject breathing. This protocol did not require breath-holding and it avoided unnecessary movements of the neck brace and stimulating electrode which might occur with repeated inspiratory manoeuvres. Recording of diaphragmatic contraction and e.m.g. The strength of diaphragmatic contraction during static inspiratory and expulsive efforts was inferred from changes in airway, oesophageal and gastric pressures. Gastro-oesophageal catheters (Fig. 1) were constructed from multilumen Swan Ganz thermodilution catheters (2-3 mm o.d.) to enable simultaneous recording of diaphragmatic e.m.g. and (via respiratory balloons secured over the catheter) gastric and oesophageal pressures. The catheters were similar to those described previously (McKenzie & Gandevia, 1985) but, in addition, were stabilized at the gastro-oesophageal junction by a third balloon (4 cm circumference) inflated with ml of air. A weight was attached to the external portion of the catheter, 10 cm from the nares, so that the catheter was free to move with the diaphragm (confirmed in one subject at fluoroscopy). The respiratory balloons were inflated with air (oesophageal 0-6 ml; gastric 1-5 ml) and connected to separate differential pressure-transducers (Statham PM 131 TO). The 95 % response time to a square-wave pressure change for the balloon-catheter-transducer systems was about 30 ins. Electromyographic activity of the diaphragm (dominated by the crural portion) was monitored with Ag-AgCl ring electrodes which encircled the catheter 1 and 6 cm proximal to the stabilizing balloon at the gastrooesophageal junction. The distal electrode was positioned close to the level of the motor point of the diaphragmatic crus (McKenzie & Gandevia, 1985). All e.m.g. signals were amplified times and filtered (bandwidth 3-2 Hz-3 2 khz). Measurements of the compound action potentials included amplitude (from peak to peak) and area. When single shocks were delivered both the area under the initial negative phase and that under the biphasic response were measured but when trains of stimuli were delivered only the area of the negative phase could be measured (see Fig. 2). Areas were measured with a digital planimeter.

5 ACTIVATION OF HUMAN DIAPHRAGM 49 RESULTS Inspiratory efforts Maximal inspiratory pressures attained at f.r.c. ranged from -117 to cmh2o (mean cmh2o) while the maximal change in pressure evoked by a train of four supramaximal shocks (inter-stimulus interval 20 ms) to one phrenic nerve ranged from cmh20 (mean 29 cmh20). Thus, for inspiratory pressure measured at the mouth, the largest evoked response was an average of 23 % of the maximal voluntary pressure. Mouth pressure. 30% 50% 60% 80% 100% E.m.g. 100 cm H20 2s 0 5 mv 10 ms Fig. 2. Above: mouth pressure (negative) during graded inspiratory efforts (30-100% maximal voluntary effort) at functional residual capacity (f.r.c.) terminated by supramaximal stimulation of one phrenic nerve (four shocks, interstimulus interval 20 ms, stimulus intensity times the level required to produce a maximal response). Results from one experiment in one subject. During maximal voluntary effort electrical stimulation (at arrow) produced no change in mouth pressure. Below: for four of the trains of shocks delivered above, the first two compound action potentials are superimposed. The action potential at 50 % of maximal effort was contaminated by the electrocardiogram and is not shown. The action potentials remained maximal. The maximal inspiratory responses evoked by unilateral stimuli occurred at voluntary inspiratory pressures ranging from 10-50% of maximum. This wide range presumably reflects variability between subjects in the effective compliance of the respiratory system. For trains of supramaximal stimuli delivered during brief contractions of 2-5 s duration there was an inverse relationship between the voluntarily maintained inspiratory pressure and the pressure change evoked by the trains of stimuli (Figs. 2 and 3). There was no consistent mechanical response to trains of four shocks during maximal voluntary contractions (even when the records were examined at high gain). Although the absolute change in pressure produced by single

6 50 S. C. GANDEVIA AND D. K. McKENZIE supramaximal shocks was smaller than that evoked by trains of stimuli, a similar inverse relationship between the initial voluntary pressure and the evoked pressure was found (Fig. 3). Supramaximal stimuli (one-four shocks) were also delivered to the phrenic nerve during prolonged maximal inspiratory efforts of s duration when the voluntary pressure had declined to a mean of 81 % of maximum (range %). All subjects were able to maintain maximal voluntary activation of the diaphragm (as judged by the absence of a mechanical response) but did not always do so: for each subject, in up to half these prolonged contractions, the stimuli produced a mechanical response indicating submaximal voluntary activation. By contrast, in brief maximal con %\KXx x x " E 50C~~~ 0 r CU~~~~~~~~~~~% BE co Q._ By 100- *10 OA s% 0 :%%0 *v. *"-"";" '4 0 * %' r I I 00 *0 0 i C~~~~~~~~ 50.- h ok '. 10; r -0-97,o oo Initial voluntary Pm (% max.) Fig. 3. Results from four subjects illustrate the inverse relationship between the voluntarily maintained inspiratory pressure (at the mouth: Pm) and the change in mouth pressure evoked by maximal stimulation of the phrenic nerve. Circles represent results obtained when trains of four shocks were used (interstimulus interval 20 ms, stimulus intensity 15-3 times the level required to produce a maximal response). Similar relationships were obtained when single supramaximal shocks were used (illustrated by the crosses for one subject in the top left panel). tractions submaximal activation occurred infrequently (in less than one in five contractions in any subject). Fig. 5A shows an attempted maximal inspiratory effort which was clearly submaximal: the pressure increased voluntarily after extra encouragement and it increased in response to supramaximal electrical stimulation.

7 A B Poes lga \~~~~~~~~~ ACTIVATION OF HUMAN DIAPHRAGM Gastric pressure 0 5 s I Ex. (glottis open) Ilnsp. J L J L J t K 15% 30% 60% 80% -100% 1 s Ex. (glottis closed) C.m.a.p. C.m.a.p. i ]---- >VX 10 Ms ] Control 15% 100% 51 Fig. 4. A, a series of graded expulsive efforts ( % maximal voluntary contraction), performed at f.r.c., with single supramaximal shocks delivered to the phrenic nerve is shown on the left (calibration: 100 cmh20). The compound muscle action potentials (c.m.a.p.s) on the right were recorded during control conditions (upper trace) and during the five graded contractions shown on the left (maximal effort, lower trace). Note that there is an artifactual change in shape of the action potential between the control and that recorded at 30 % maximum: the action potential recorded during maximal effort did not increase when the stimulus intensity was increased twofold. Calibrations: 1 mv, 10 ms. B, maximal activation of the diaphragm during three different voluntary manoeuvres: expulsive effort (ex.) with glottis closed (left panels), expulsive effort with glottis open (middle panels) and inspiratory effort (insp., right panels). Calibration for oesophageal pressure (Poesy upper records) and gastric pressure (Pga middle records): 100 cmh20- Single supramaximal stimuli (at arrows) produced a transient decrease in the voluntarily maintained pressure. C.m.a.p.s (lower records) were of maximal amplitude (calibration: 1 mv). Expulsive efforts Interpolated shocks were also delivered to the phrenic nerve during series ofgraded expulsive efforts (Fig. 4A). An inverse relationship, similar to that demonstrated for inspiratory efforts, was found between the initial voluntary gastric pressure and the pressure change evoked by supramaximal stimulation of one phrenic nerve. The maximal gastric pressures achieved during these expulsive manoeuvres averaged 210 cmh2o (range: cmh2o) while the maximal change in gastric pressure evoked by a single supramaximal shock was a mean of 51 cmh2o. All subjects were able to activate the diaphragm maximally during expulsive efforts whether the glottis was closed or open although the maximal gastric pressure was smaller with the glottis open (Fig. 4B).

8 52 S. C. GANDEVIA AND D. K. McKENZIE Activation of the fatigued diaphragm In addition to assessment of voluntary activation during prolonged inspiratory contractions (see above) activation of the diaphragm was studied before and immediately after diaphragmatic fatigue was induced by a strong sustained expulsive manoeuvre ( s duration) during which the subjects continued to breathe. The subsequent maximal inspiratory effort was on average 73 % of the unfatigued maximum (mean often tests in three subjects). All subjects were capable of activating fully the fatigued diaphragm so that there was no increase in pressure during the interpolated shocks (Fig. 5). In two subjects activation was complete in two of three attempted contractions and in the third it was complete in two of four attempts. A B C j j J X jcmh20 1 s Aig \ JlmVot1mv 10 ms Fig. 5. A, an example of submaximal activation of the diaphragm during an attempted maximal inspiratory effort when the inspiratory pressure was increased voluntarily after extra encouragement (small arrow) and in response to a supramaximal shock to one phrenic nerve (large arrow). B, maximal activation of the diaphragm before diaphragmatic fatigue. C, maximal activation of the diaphragm during a brief maximal inspiratory effort performed immediately (within 2 s) after fatigue was induced (see Methods). Muscle action potentials (below) recorded during each of these manoeuvres confirmed that the stimuli remained maximal and provided no evidence for failure at the neuromuscular junction during fat igue. These three contractions were recorded from one subject in one experimental session. Diaphragmatic e.m.g. A critical factor in these studies was the assumption that the stimuli were supramaximal for all phrenic motor axons. An adjustable neck brace on which the stimulating probe was mounted ensured that the stimulus intensity required to produce a maximal compound action potential remained constant for prolonged periods (Fig. 1, see Methods). However, the recording electrodes may move in relation to the active fibres during the manoeuvres. Artifactual changes in the maximal compound muscle action potential occur during changes in lung volume and changes in rib-cage and abdominal shape (Gandevia & McKenzie, 1985).

9 ACTIVATION OF HUMAN DIAPHRAGM In preliminary experiments, when the catheter was taped to the nose and thoracoabdominal configuration was not monitored, the evoked muscle action potential often decreased by up to 50 % in amplitude during inspiratory efforts begun at f.r.c. compared with controls (at f.r.c.). However, the potentials were clearly maximal because no enlargement occurred with further increases in stimulus intensity. In these studies the maximal muscle action potential did not change in size as the initial voluntary pressures increased from about 50 % of maximum (when a large change in pressure was evoked) to a maximal contraction (Fig. 2). In subsequent studies a third 'stabilizing' balloon on the gastro-oesophageal catheter and control of initial thoraco-abdominal configuration reduced, but did not eliminate, these recording difficulties. When maximal inspiratory efforts were performed with the rib cage and abdomen maintained in a configuration similar to that during relaxation at f.r.c. action potentials were often % larger in amplitude than controls. This increase in size of action potentials has been noted in studies of limb muscles (Bigland-Ritchie et al. 1982). With graded expulsive efforts there was a change in shape of action potentials between those recorded during weak contractions and controls (Fig. 4A). However, minimal change (in amplitude or area) occurred between contractions at 30-50% of maximum and during maximal efforts (Fig. 4A). There was no evidence that failure of transmission at the neuromuscular junction could explain the degree of diaphragmatic fatigue produced by prolonged effort (Fig. 5). In all experiments maximal efforts were always repeated with a two-four times increase in stimulus intensity to check that the stimulus level remained supramaximal. 53 DISCUSSION This study indicates that co-operative subjects can activate the diaphragm maximally during voluntary inspiratory efforts and expulsive manoeuvres performed with or without closure of the glottis. When the diaphragm was fatigued maximal activation did not always occur. Twitch interpolation can be used to study voluntary activation of muscles such as the diaphragm but there are potential difficulties. The active tension generated by the diaphragm cannot be measured directly. Transdiaphragmatic pressure is a relative measure of the total tension (active and passive) in the diaphragm but, during maximal inspiratory efforts, the passive tension produced indirectly by contraction of the abdominal muscles is extremely variable (de Troyer & Estenne, 1981; Gibson et al. 1981). Similarly, during expulsive efforts, glottic closure and recruitment of expiratory intercostal muscles may alter the transdiaphragmatic pressure without influencing the active state of the diaphragm (for example, see Fig. 4B). The use of mouth (or oesophageal) pressure as an index of inspiratory force, and gastric pressure as an index of expulsive force resulted in inverse relationships between the initial voluntary pressure and the pressure evoked by supramaximal nerve stimulation which have a similar shape to those reported for limb muscles (Merton, 1954; Belanger & McComas, 1981). Although the compliance of the respiratory system results in relatively small mechanical responses to single unilateral stimuli applied to the phrenic nerve, trains of four shocks ensured that the maximal evoked response was

10 54 54 S. C. GANDEVIA AND D. K. McKENZIE more than 20 % of the maximal voluntary pressure. This value is greater than that obtained with twitch interpolation in other muscles (Merton, 1954; Belanger & McComas, 1981; Bellemare et al. 1983). In some subjects the diaphragm may be maximally activated at submaximal inspiratory pressures possibly because of late recruitment of accessory inspiratory muscles at high pressures. A second technical problem is that static contractions of the diaphragm are not truly isometric and it is impossible to ensure that each contraction is performed with the diaphragm and other muscles of the rib cage and abdomen at identical muscle lengths. When the lung volume and the thoraco-abdominal configuration at which the contractions began were controlled some variation in the size of the mechanical response still occurred. However, this scatter of data points was comparable to that found for the muscles which act at the ankle joint (Belanger & McComas, 1981). In the present study displacement of the stimulating electrode was minimized for inspiratory efforts by using a probe stimulator fixed to a rigid neck brace (see Methods), and it did not occur during expulsive efforts because the muscles of the neck remained relaxed. That the muscle action potentials did not increase when the stimulus intensity was substantially increased during maximal inspiratory or expulsive efforts was the criterion used to check that the stimuli remained maximal for all the motor axons in the phrenic nerve. However, the diaphragm is relatively inaccessible for recording e.m.g. and oesophageal electrodes move in relation to the crural (and other) fibres of the diaphragm. Artifactual changes in amplitude of the action potential occur with changes in lung volume and thoraco-abdominal configuration in man (Gandevia & McKenzie, 1985). In the present study contractions were performed at f.r.c. with some at a constant thoraco-abdominal configuration. The failure of electrical stimulation to produce a mechanical response during maximal contractions was not due to failure of excitation of the muscle fibres. Bellemare & Bigland-Ritchie (1984b) have reported that subjects were able to activate the diaphragm fully during attempts to produce maximal transdiaphragmatic pressure. However, while breathing through inspiratory resistances the diaphragm was not maximally activated indicating 'central fatigue' (Bellemare & Bigland-Ritchie, 1984a). By contrast, in the present study subjects were often able to activate the diaphragm fully in brief inspiratory contractions when the diaphragm was fatigued. Similarly, during prolonged static contractions maximal activation was possible but did not always occur. The reasons for this variation in the ability to activate the diaphragm are unclear but may reflect differences in motivation and respiratory sensations during the different manoeuvres. The demonstration that the diaphragm can be activated maximally during static efforts supports the previous suggestion (Gandevia et al. 1983) that the relative resistance to fatigue of the inspiratory muscles may be due to local factors (such as blood flow and intrinsic muscle properties), not to failure of neural drive. If maximal activation of the diaphragm is secured in brief but not all prolonged contractions then a fatigue test using brief contractions has theoretical advantages. This work was supported by the National Health and Medical Research Council of Australia and The Asthma Foundation of New South Wales. The authors are indebted to Drs D. I. McCloskey and D. Burke for valuable advice and criticism.

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