Upsala J Med Sci 85: 265-282, 1980 The Teorell Membrane Oscillator--a Complete Nerve Model Ulrich F. Franck Technical University, Aachen, Germany ABSTRACT The membrane oscillator discovered by T. Teorell in1954 (10) is one of the most remarkable kinetic systems exhibiting spontaneous periodic behaviour under constant environmental conditions. It can be shown and demonstrated experimentally that the intrin- sic reason for the occurrence of oscillations in the Teorell oscillator, like in other physicochemical and biological oscil- latory systems, is an appropriate antagonistic actiGn of a la- bilizing positive and a recovering stabilizing negative feedback. Each kind of feedback causes a group of characteristic temporal phenomena, which are observed in all oscillatory systems inclu- ding the living excitable nerve such as: instability, bistability, excitability, propagation of excitation and recovery, refractori- ness, abolition, accommodation etc. . Introduction: In Gottingen in 7951 Torsten Teorell presented at the 50th Hauptversammlung of the Deutsche Bunsen-Gesellschaft a most memo- rable lecture: "On the quantitative treatment of membrane per- meability". This lecture and its publication(9) i n i t i a t e d a s t r o n g new development in membrane science, and it was quite a convin- cing confirmation of the rightness and fruitfulness of these new conceptions when in 1954 his discovery of spontaneous membrane flux oscillations in a ion-exchange membrane system became known. Though 2 6 years have passed since that event, the Teorell membrane oscillator has thoroughly retained its importance and actuality not only as an exceptional ion-exchange membrane system but also as a general kinetic model for all physicochemical and biological oscillatory systems due to its remarkable transparent kinetics. 265 1 ) N o n l i n e a r P r o p e r t i e s of Ion-exchange Membranes A s i s well-known, i n t h e T e o r e l l membrane o s c i l l a t o r t h r e e f o r c e s a r e e f f e c t i v e s i m u l t a n e o u s l y ( d i f f e r i n g p o t e n t i a l , p r e s s u r e and' s a l t c o n c e n t r a t i o n on b o t h s i d e s o f t h e membrane) c a u s i n g t h r e e k i n d s of f l u x e s a c r o s s t h e membrane ( e l e c t r i c c u r r e n t , volume f l u x o f f l u i d and s a l t f l u x ) , which a r e m u t u a l l y c o u p l e d t o a h i g h d e g r e e . The e s s e n t i a l k i n e t i c f e a t u r e of t h e T e o r e l l o s c i l l a t o r i s i t s p r e s s u r e d e p e n d e n t non - monotonic c u r r e n t - v o l t a g e c h a r a c t e r i s t i c (12) ( F i g . I ) , g i v i n g r i s e t o i t s i n s t a b i l i t y p r o p e r t y , which i s one of t h e n e c e s s a r y p r e r e q u i s i t e s f o r t h e Occurrence Of o s c i l l a t i o n s . With r e s p e c t t o t h e d i r e c t i o n s of t h e d r i v i n g f o r c e s and t h e s i g n of t h e membrane c h a r g e f o r m a l l y e i g h t cases of c o m b i n a t i o n are p o s s i b l e . Because, however, n o n l i n e a r c u r r e n t - v o l t a g e c h a r a c t e r i s - t i c s o n l y o c c u r i n t h e T e o r e l l system under c o n d i t i o n s of opposing e f f e c t s of v o l t a g e and h y d r o s t a t i c p r e s s u r e , o n l y t h e f o u r cases shown i n F i g . 2 remain t o be c o n s i d e r e d h e r e . The pronounced v o l t a g e dependence of t h e membrane r e s i s t a n c e i s c a u s e d by t h e e l e c t r o o s m o t i c volume f l u x , which c a r r i e s e i t h e r c o n d u c t i n g s a l t i n t o o r o u t of t h e p o r o u s ion-exchange membrane. I n t h e f o u r c a s e s i n q u e s t i o n t y p i c a l n o n l i n e a r c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s a r i s e a s a consequence of r e s i s t a n c e t r a n s i t i o n s o c c u r r i n g i n t h e v o l t a g e r a n g e where a r e v e r s a l o f t h e volume f l u x t a k e s p l a c e e x h i b i t i n g e i t h e r non-monotonic N-shaped o r mo- n o t o n i c f -shaped c u r v e s . F i g . 3 g i v e s t y p i c a l e x p e r i m e n t a l m e a - s u r e m e n t s of s u c h n o n l i n e a r c h a r a c t e r i s t i c s c o n c e r n i n g c o a r s e - g r a i n e d c a t i o n - and anion-exchange membranes whose h i g h p o r o s i t y a l l o w s s t r o n g c o n v e c t i o n a l f l u x e s . I t may be mentioned h e r e , t h a t a l s o i n n o n - c o n v e c t i o n a l membranes n o n l i n e a r c h a r a c t e r i s t i c s of N- and / - t y p e a r e t o be e x p e c t e d namely under c o n d i t i o n s where v o l t a g e d e t e r m i n i n g p r o p e r t i e s of t h e membrane depend upon t h e v o l t a g e i t s e l f . T h i s i s t h e case f o r i n s t a n c e when t h e membrane c h a r g e ( ? ) o r i t s s i g n ( 0 ) o r t h e se- l e c t i v i t y change s t r o n g l y w i t h i n a s u f f i c i e n t l y narrow r a n g e of t h e membrane p o t e n t i a l ( F i g . 4 ) . 266 a b C F i g . 1 The T e o r e l l membrane o s c i l l a t o r a ) s e t - u p ; b) c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s ( p a r a m e t e r : P); c ) o s c i l l a t i o n s of membrane p o t e n t i a l and h y d r o s t a t i c p r e s s u r e d i f f e r e n c e P ( 1 2 , 1, 5 ) . Electroosmotic S y m PO~OW Cat$--eschange Membrane , PDTDUS Anmn-erchange Membrane F i g . 2 a ) s e t - u p ; b) membrane r e s i s t a n c e R a s a f u n c t i o n of t h e membrane N o n l i n e a r e l e c t r o o s m o t i c ' membrane s y s t e m s p o t e n t i a i E ; c ) t t c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s ; Glass-sinter 0 : -1 Anion-exchanger grains w: +I F i g . 3 Examples of n o n l i n e a r c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s of p o r o u s ion-exchange membranes ( p a r a m e t e r : P h y d r o s t a t i c p r e s s u r e - mm column of w a t e r ) ( l ) . 267 -qf -qJ- I IE - I I ' I I F i g . 4 Non-convectional membrane s y s t e m s (X: membrane c h a r g e ; w : s i g n of membrane c h a r g e ; g : conductivity) !raw -5a7 0 ,500 +?Om F i g . 5 Membrane c o n d u c t a n c e a s a f u n c t i o n of membrane c h a r g e ( T . T e o r e l l 1951 ) (9). I n 1951 T . T e o r e l l has shown t h a t t h e c o n d u c t i v i t y o f a n i o n - exchange membrane depends on t h e c o n c e n t r a t i o n of i t s f i x e d i o n c h a r g e ? ( F i g . 5 ) . Hence t h e c o n d u c t i v i t y of a membrane e x h i b i - t i n g v o l t a g e dependence of ? i s a f u n c t i o n of t h e membrane v o l - t a g e t o o . The t h i r d c a s e of F i g . 4 ( v o l t a g e d e p e n d i n g s e l e c t i v i t y ) c o r r e s - ponds t o t h e well-known c o n c e p t of t h e Hodgkin-Huxley t h e o r y of e x c i t a b l e b i o l o g i c a l t i s s u e s . 2) The Feedback Concept of N o n l i n e a r P h y s i c o c h e m i c a l Systems The main c o n c e r n of t h i s t e x t i s t o show t h a t a l l k i n e t i c a l pheno- mena r e s u l t i n g from n o n l i n e a r f l u x - f o r c e c h a r a c t e r i s t i c s a r e b r o u g h t f o r t h i n a c t u a l f a c t by f e e d b a c k mechanisms which a l l p h y s i c o c h e m i c a l o s c i l l a t o r y s y s t e m s i n c l u d i n g t h e T e o r e l l o s - c i l l a t o r and t h e e x c i t a b l e n e r v e have i n common ( 3 , 6 ) . 268 B e f o r e d o i n g s o , w e have t o l o o k a l i t t l e c l o s e r i n t o f e e d b a c k p r o c e s s e s o c c u r r i n g i n p h y s i c o c h e m i c a l s y s t e m s . A s w e l l known f e e d b a c k a r i s e s when a p r o c e s s a c t s k i n e t i c a l l y upon i t s e l f . I t c o n s i s t s t h e r e f o r e b a s i c a l l y i n a c l o s e d c h a i n of a c t i o n c a u s i n g t h e known e f f e c t s of " s e l f - e n h a n c e m e n t " i n case of p o s i t i v e f e e d b a c k o r " s e l f - i n h i b i t i o n " i n c a s e of n e g a t i v e f e e d b a c k . Such p r o c e s s e s of s e l f - i n f l u e n c e may a r i s e i n two ways ( F i g . 6 ) : n 1 +Tk+ I d + X L P Systemic leedbach F i g . 6 "Non-systemic" and " s y s t e m i c " f e e d b a c k i n physicochemi- c a l s y s t e m s . I f t h e out.put of a t r a n s m i s s i o n system ( t h a t may be an e l e c t r o n i c a m p l i f i e r , a c h e m i c a l r e a c t i o n , a n e l e c t r o d e o r a membrane s y s t e m e t c . ) a c t s k i n e t i c a l l y upon i t s cwn i n p u t , t h e n a f e e d b a c k s i t u a - t i o n a r i s e s i n which t h e o u t p u t - " e f f e c t " i n f l u e n c e s i t s own " c a u s e " . T h i s mode of f e e d b a c k h a s no e f f e c t upon t h e p r o p e r t i e s of t h e t r a n s m i s s i o n system. F o r t h i s r e a s o n it s h a l l b e d e s i g n a t e d h e r e a s "non-systemic " f e e d b a c k ( 4 ) . I n p h y s i c o c h e m i c a l and b i o l o g i c a l s y s t e m s , however, most f e e d b a c k mechanisms a c t n o t upon t h e i n p u t b u t i n s t e a d upon t h e p r o p e r t i e s of t h e t r a n s m i s s i o n s y s t e m . T h i s k i n d of f e e d b a c k c o r r e s p o n d i n g l y s h a l l be d e s i g n a t e d a s " s y s t e m i c " f eedback . Feedback l o o p s i n open s y s t e m s may c o n c e r n t h e f o r m a t i o n o r t h e consumption of t h e k i n e t i c s p e c i e s X and may be - a s a l r e a d y mentioned - of p o s i t i v e o r n e g a t i v e n a t u r e . I n t h i s way f o u r d i f - f e r e n t f e e d b a c k s i t u a t i o n s are p o s s i b l e b e i n g c a l l e d : backward a c t i v a t i o n , forward i n h i b i t i o n , backward i n h i b i t i o n and f o r w a r d a c t i v a t i o n ( F i g . 7 ) ( 7 ) . 269 backward activation forward inhibition backward lnhibltlon forward activation Positive feedback Negative feedback F i g . 7 P o s i t i v e and n e g a t i v e f e e d b a c k i n open s y s t e m s . O s c i l l a t o r y s y s t e m s c o n t a i n a t l e a s t two s i m u l t a n e o u s p r o c e s s e s . By mutual ( c r o s s - ) c o u p l i n g between t h e s e p r o c e s s e s 1 6 d i f f e r e n t p o s s i b i l i t i e s have t o be t a k e n i n t o c o n s i d e r a t i o n ( F i g . 8 ) . I n t h e i r o v e r - a l l e f f e c t s t h e y l e a d t o f o u r c a s e s e a c h f o r backward a c t i v a t i o n , f o r w a r d i n h i b i t i o n , backward i n h i b i t i o n and f o r w a r d a c t i v a t i o n ( 6 ) - F i g . 8 Feedback by c o u p l i n g of two s i m u l t a n e o u s p r o c e s s e s ( 6 ) . I n m o s t r e a l i s t i c f e e d b a c k s y s t e m s t h e t y p e s of e f f e c t i v e f e e d - back p r o c e s s e s c a n be found o u t d i r e c t l y by e x p e r i m e n t a l methods. Because a l l c o u p l i n g o r f e e d b a c k mechanisms a r e r e p r e s e n t e d by r e a l r e a c t i o n s o r t r a n s p o r t a t i o n p r o c e s s e s of d e f i n e d r a t e s , a l l f e e d b a c k p r o c e s s e s n e c e s s a r i l y need t i m e f o r t h e i r p r o c e e d i n g . The t i m e d e l a y i n p h y s i c o c h e m i c a l f e e d b a c k l o o p s v a r i e s i n a l a r g e scale of magnitude. I n t h e case of t h e T e o r e l l s y s t e m i t r a n g e s i n m i n u t e s , i n t h e case of t h e n e r v e membrane i n m i l l i s e c o n d e s . 3 ) The Temporal B e h a v i o u r o f Feedback s y s t e m s The t e m p o r a l b e h a v i o u r of p o s i t i v e and n e g a t i v e f e e d b a c k s y s t e m s i s of e s s e n t i a l i m p o r t a n c e f o r t h e o c c u r r e n c e and t h e t e m p o r a l p a t t e r n of p h y s i c o c h e m i c a l o s c i l l a t i o n s . 270 .- 0 63 0 0 F i g . 9 N o n l i n e a r f l u x - f o r c e c h a r a c t e r i s t i c s a s a r e s u l t of f e e d b a c k a c t i o n ( 4 ) . F i g . 9 i l l u s t r a t e s t h e g e n e r a l t e m p o r a l b e h a v i o u r of t h e f o u r modes of c o u p l i n g l e a d i n g t o p o s i t i v e o r n e g a t i v e f e e d b a c k . Assuming f o r t h e s a k e of s i m p l i c i t y t h a t t h e f o r m a t i o n and con- sumption o f t h e k i n e t i c a l s p e c i e s X are p r o c e s s e s of f i r s t o r d e r , a p l o t o f v e r s u s X g i v e s t w o s t r a i g h t l i n e s which i n t e r s e c t a t t h e s t e a d y s t a t e v a l u e X s t . T h e i r s l o p e s c o r r e s p o n d d i r e c t l y t o t h e r e l e v a n t r a t e c o n s t a n t k o r t h e c o n d u c t a n c e g . S e l f - c o u p l i n g a l t e r s t h e s l o p e of t h e k i n e t i c c h a r a c t e r i s t i c be- l o n g i n g t o t h e f e e d b a c k p r o c e s s i n q u e s t i o n . I n case of f o r w a r d p o s i t i v e f e e d b a c k it i s g e t t i n g f l a t t e r w i t h i n c r e a s i n g X and it i s g e t t i n g s t e e p e r i n c a s e of n e g a t i v e f e e d b a c k . F i g . 9 shows i n p a r t i c u l a r how non-monotonic N-shaped c u r v e s re- s u l t from p o s i t i v e f e e d b a c k and monotonic I - s h a p e d c u r v e s from n e g a t i v e f e e d b a c k . P r o v i d e d t h s t t h e p o s i t i v e f e e d b a c k i s s u f f i c i e n t l y s t r o n g t h r e e i n t e r s e c t i o n - r e p r e s e n t i n g s t e a d y s t a t e s - may o c c u r between t h e c u r v e s of consumption of X e x h i b i t i n g b i s t a b i l i t y and i n s t a b i l i t y r e s p . . P o s i t i v e f e e d b a c k l a b i l i z e s t h e system, and it i s t h e i n t r i n s i c r e a s o n f o r a l l i n s t a b i l i t y phenomena of a u t o c a t a l y t i c and o t h e r s e l f - e n h a n c i n g s y s t e m s . 27 1 I n c o n t r a s t n e g a t i v e f e e d b a c k h a s a s t a b i l i z i n g e f f e c t . I t g e n e r a - t e s a l w a y s one s t a b l e s t a t e o n l y . The bottom row o f g r a p h s shows t h e "dynamic d i a g r a m s " which d e s - c r i b e t h e o v e r - a l l t e m p o r a l v a r i a t i o n of X a s a f u n c t i o n of X it- ... s e l f . The i n s e r t e d a r r o w s i n d i c a t e t h e d i r e c t i o n s o f t h e r e s u l t i n g t e m p o r a l v a r i a t i o n s , and which i l l u s t r a t e how s t a b i l i t y and i n - s t a b i l i t y a r i s e from t h e v a r i a t i o n s i n t h e immediate n e i g h b o u r - hood of a p a r t i c u l a r s t e a d y s t a t e . F o r p o s i t i v e a s w e l l as f o r n e g a t i v e f e e d b a c k c h a r a c t e r i s t i c msdes of t e m p o r a l b e h a v i o u r r e s u l t a s a d i r e c t consequence of t h e f e e d b a c k d e l a y . Concerning i o n i c membranes, a l r e a d y i n 1951 T . T e o r e l l p o i n t e d o u t i n t h e p a p e r c a p a c i t a n c e " a l r e a d y mentioned t h a t i n s u c h membranes "pseudo- and " p s e u d o - i n d u c t a n c e ' ' b e h a v i o u r are o b s e r v e d (Fig.10). Kupuztfw ,, Ku,ztbw -indukhu " Sponnung (zstmp _ _ _ _ . indukfiv " Zeitlicher Spannungsverlauf verschiedener S ysteme Elektroly- ten-Membran (nach Einschalten eines konstanten Stromes). Das System zeigt scheinbnre Kapazitat. Induktivitat bzw. Kapazitjt + lnduktivitat F i g . 1 0 P s e u d o - c a p a c i t i v e and p s e u d o - i n d u c t a n c e b e h a v i o u r o f ion-exchange membranes ( T . T e o r e l l 1 9 5 1 ) ( 9 ) . F i g . 1 1 Temporal b e h a v i o u r o f f e e d b a c k s y s t e m s compared w i t h e l e c t r i c c a p a c i t a n c e and i n d u c t a n c e c i r c u i t s ( 6 ) . 272 Fig. 11 i l l u s t r a t e s pattern r e s e m b l i n g t h a t of a c a p a c i t y . Systems c o n t a i n i n g n e g a t i v e f e e d b a c k behave c o r r e s p o n d i n g l y l i k e i n d u c t a n c e s e x h i b i t i n g o v e r - s h o o t phenomena which i n f a c t are r e c o v e r y phenomena as a c o n s e - quence o f t h e s t a b i l i z i n g a c t i v i t y of t h e n e g a t i v e f e e d b a c k . Now w e a r e i n a p o s i t i o n t o i d e n t i f y t h e n a t u r e of f e e d b a c k l o o p s e x i s t i n g i n a p a r t i c u l a r s y s t e m by s t u d y i n g i t s t e m p o r a l b e h a v i o u r . t h a t p o s i t i v e feedback a c t u a l l y l e a d s t o t e m p o r a l I n t h i s c o n t e x t it might b e s t a t e d t h a t from t h e f e e d b a c k p o i n t of view it i s e v i d e n t t h a t t h e t e m p o r a l b e h a v i o u r of p h y s i c o c h e m i c a l s y s t e m s i s q u i t e t h e same a s it i s well-known from f e e d b a c k i n e l e c t r i c c i r c u i t s o r t e c h n i c a l c o n t r o l s y s t e m s , and i n a c t u a l f a c t t h e o c c u r r e n c e of o s c i l l a t i o n s i n p h y s i c o c h e m i c a l and b i o l o g i c a l s y s t e m s o b v i o u s l y i s b a s e d on t h e same dynamical p r i n c i p l e s a s it i s t h e case i n e l e c t r i c a l s y s t e m s whose o s c i l l a t o r y b e h a v i o u r a s a r e s u l t of f e e d b a c k a c t i o n i s q u i t e s e l f - e v i d e n t f o r u s . 4 ) The P r i n c i p l e of A n t a g o n i s t i c Feedback. I t i s a h i g h l y i n t e r e s t i n g f a c t t h a t a l l p h y s i c o c h e m i c a l o s c i l l a - t o r y s y s t e m s found so f a r c l e a r l y e x h i b i t p o s i t i v e and n e g a t i v e f e e d b a c k s i m u l t a n e o u s l y . I t i s most l i k e l y t h a t o s c i l l a t i o n s i n such s y s t e m s a r e t h e r e s u l t of a g e n e r a l k i n e t i c p r i n c i p l e which i s r e p r e s e n t e d i n F i g . 1 2 i n form of a g e n e r a l f e e d b a c k p a t t e r n . I t may b e d e s i g n a t e d h e r e a s t h e " P r i n c i p l e o f a n t a g o n i s t i c f e e d b a c k of p h y s i c o c h e m i c a l o s c i l - 1 a t o r s l 1 ( 3 ) . F i g . 1 2 A n t a g o n i s t i c f e e d b a c k i n o s c i l l a t o r y open s y s t e m s c o n t a i n i n g two s i m u l t a n e o u s p r o c e s s e s (3). According t o t h i s p r i n c i p l e t h e o s c i l l a t i o n s a r e u n d e r s t o o d a s a consequence o f an a n t a g o n i s t i c i n t e r a c t i o n o f a r e l a t i v e l y f a s t a c t i n g p o s i t i v e f e e d b a c k o f l a b i l i z i n g t e n d e n c y and a s l o w e r ac- t i n g n e g a t i v e f e e d b a c k of s t a b i l i z i n g r e c o v e r i n g t e n d e n c y . T h i s c o n c e p t of o s c i l l a t o r y s y s t e m s c o n s i s t i n g of two d i s t i n c t l o o p s of p o s i t i v e and n e g a t i v e f e e d b a c k r e q u i r e s a t l e a s t two k i n e t i c a l v a r i a b l e s , one f o r e a c h f e e d b a c k mechanism. Each l o o p , however, may c o n t a i n s e v e r a l v a r i a b l e s i n series. A l l t h e s e va- r i a b l e s p a r t i c i p a t e i n t h e o v e r - a l l o s c i l l a t o r y p r o c e s s . I n most r e a l cases of o s c i l l a t i o n s it i s p o s s i b l e t o f i n d o u t from t h e r e c o r d i n g s t o which c l a s s of f e e d b a c k l o o p a p a r t i c u l a r o s c i l l a - t i n g v a r i a b l e b e l o n g s ( F i g . 1 3 ) . A l s o a g e n e r a l c l a s s i f i c a t i o n of t h e t e m p o r a l phenomena of f e e d b a c k s y s t e m s can b e g i v e n now w i t h r e s p e c t t o t h e n a t u r e of t h e f e e d b a c k b e i n g r e s p o n s i b l e f o r a p a r - t i c u l a r phenomenon ( F i g . 1 4 ) . Teorell Membrane B 2 Reaction Bray Reaction I r o n In HNO, Variable 01 Br- ' ~ l - ~ l l ) ) J $ ! ~ - - poslttve - - - - - feedback - t- t - t - t - negative Variableof cl?+w 'w,,,!ozw iw leedback t - t - 1- t - F i g . 1 3 S i m u l t a n e o u s o s c i l l o g r a m s of v a r i a b l e s b e l o n g i n g t o d i f f e r e n t f e e d b a c k l o o p s . V a r i a b l e s i n v o l v e d i n p o s i t i v e f e e d - back l o o p s e x h i b i t f l i p - f l o p - t y p e o s c i l l o g r a m s showing i n s t a - b i l i t y f e a t u r e s ( - ) . O s c i l l o g r a m s o f v a r i a b l e s b e l o n g i n g t o n e g a t i v e f e e d b a c k l o o p s u s u a l l y have s i m p l e saw-tooth s h a p e w i t h o u t marks of i n s t a b i l i t y (B.Z.: Belousov - Z h a b o t i n s k y r e a c t i o n ) ( 6 ) . - MI.- F i g . 1 4 C l a s s i f i c a t i o n of t e m p o r a l phenomena of f e e d b a c k s y s t e m s (4). 274 Positive Feedback -X- rn -E- f3 lo l m ( 9 m ) forward inhibition t- , I f 9, I\- -.+ L a b i l i z a t i o n - : -:- : , E * t- dLb $-BJ Pseudo Capactlance - dt c' ._ - - - + V < Increasing voltage inhlbits the membraneconductance by concentratmn profile deformation - Negative Feedback j : i t - t- forward activation -E- E ---?- - S t a b i l i z a t i o n t- Pseudo-Inductance E - lncrea~ing voltage actovates the membrane conductance by increaslng hydrostatic pressure F i g . 1 5 P o s i t i v e and n e g a t i v e feedback a r i s i n g i n t h e Teorel.1 system. 275 5 ) The Feedback S i t u a t i o n i n t h e T e o r e l l Membrane O s c i l l a t o r L e t u s now l o o k a t t h e T e o r e l l membrane system from t h e f e e d b a c k p o i n t of view. T h e r e i s c l e a r l y a p o s i t i v e f e e d b a c k l o o p p r o v a b l e w i t h r e s p e c t t o t h e transmembrane p o t e n t i a l , and which m a n i f e s t s i t s e l f i n t h e v o l t a g e dependence of t h e membrane r e s i s t a n c e ( F i g . 1 5 ) . The " f o r m a t i o n " and "consumption" of t h e k i n e t i c a l s p e c i e s X c o n c e r n s p r i m a r i l y t h e s a l t c o n t e n t i n s i d e t h e membrane a s a r e s u l t of t h e volume f l u x d r i v e n by t h e e l e c t r o o s m o t i c e f f e c t of t h e membrane p o t e n t i a l i n s i d e t h e ion-exchange membrane. Because t h e s a l t c o n t e n t of t h e membrane d e t e r m i n s t h e membrane r e s i s t a n c e , t h e r e e x i s t under c u r r e n t clamp c o n d i t i o n s a d i r e c t r e l a t i o n s h i p between t h e s a l t c o n t e n t and t h e membrane p o t e n t i a l . T h e r e f o r e w e may r e p l a c e X d i r e c t l y by t h e v o l t a g e E a s t h e cha- r a c t e r i s t i c k i n e t i c a l v a r i a b l e o f t h e p o s i t i v e f e e d b a c k l o o p . I t i s e a s y t o show t h a t i n t h e T e o r e l l membrane s y s t e m t h i s l o o p i s r e a l i z e d by a f o r w a r d i n h i b i t i o n c o n s t e l l a t i o n . Here, a n i n c r e a - s i n g v o l t a g e i n h i b i t s t h e membrane c o n d u c t a n c e by a c o r r e s p o n d i n g d e f o r m a t i o n of t h e c o n c e n t r a t i o n p r o f i l e . The p o s i t i v e f e e d b a c k l e a d s t o p s e u d o - c a p a c i t i v e b e h a v i o u r , t o a non-monotonic c u r r e n t - v o l t a g e c h a r a c t e r i s t i c and t o a c o r r e s p o n d i n g dynamic r e l a t i o n s h i p between - and E . dE d t The n e g a t i v e f e e d b a c k of t h e T e o r e l l o s c i l l a t o r ( F i g . 1 5 ) comes a b o u t as a r e s u l t o f t h e volume f l u x dependence o f t h e h y d r o s t a - t i c p r e s s u r e , which i s - b e s i d e s t h e membrane p o t e n t i a l - t h e o t h e r d r i v i n g f o r c e o f t h e volume f l u x . The n e g a t i v e f e e d b a c k l o o p r e p r e s e n t s h e r e a t y p i c a l example of mutual c o u p l i n g o f two s i m u l t a n e o u s p r o c e s s e s whose o v e r - a l l e f f e c t c o n c e r n s a f o r w a r d a c t i v a t i o n . I n c r e a s i n g v o l t a g e a c t i v a t e s h e r e t h e membrane c o n d u c t a n c e by i n c r e a s i n g h y d r o s t a t i c p r e s s u r e . 6 ) The T e o r e l l Membrane O s c i l l a t o r a s a "Complete Nerve Analogue". S u r v e y i n g t h e l i s t of f e e d b a c k phenomena of F i g . 1 4 w e a r e s t r o n g l y reminded of t h e phenomenology of t h e e x c i t a b l e n e r v e and o t h e r b i o l o g i c a l e x c i t a b l e t i s s u e s . Tab. 1 sums up t h e p r o p e r t i e s which are c o n s i d e r e d a s f u n d a m e n t a l phenomena of t h e n e r v e membrane. 276 p h y s i c o c h e m i c a l s y s t e m s e x h i b i t i n g t h e s e p r o p e r t i e s a r e d e s i g n a t e d as"comp1ete n e r v e models" o r " n e r v e a n a l o g u e s " . Table 1. F u n d a m e n t a l E l e c t r o p h y s i o l o g i c a l P r o p e r t i e s o f E x c i t a b l e Tissues I n s t a b i l i t y : I m p u l s e Response: P r o p a g a t i o n : R h y t h m i c a l A c t i v i t y : B i s t a b i l i t y E x c i t a b i l i t y ( T r i g g e r a b i l i t y ) T h r e s h o l d Phenomena A b o l i t i o n E x c i t a b l e A c t i o n p o t e n t i a l S p o n t a n e o u s R e c o v e r y o f E x c i t a t i o n R e f rac t o r i n e s s Accommodation O v e r s h o o t Phenomena Two-way P r o p a g a t i o n of E x c i t a t i o n P r o p a g a t i o n o f A c t i o n p o t e n t i a l T r i g g e r a b i l i t y o f P r o p a g a t i o n Phenomena S a l t a t o r i c P r o p a g a t i o n O s c i l l a t o r y Phenomena Obviously t h e n e r v e r e p r e s e n t s k i n e t i c a l l y a s y s t e m of a n t a g o n i s - t i c f e e d b a c k t o o . Hence w e may d e f i n e a c o m p l e t e n e r v e a n a l o g u e b r i e f l y as a system e x h i b i t i n g a n t a g o n i s t i c f e e d b a c k a s o u t l i n e d h e r e . There a r e s e v e r a l a r t i c l e s p u b l i s h e d by T . T e o r e l l t w e n t y y e a r s ago which c l e a r l y show t h a t a l r e a d y i n t h e f i f t i e s he w a s q u i t e aware and c o n v i n c e d t h a t h i s membrane o s c i l l a t o r i n a c t u a l f a c t r e p r e s e n t s a c o m p l e t e n e r v e a n a l o g u e (11 - I S ) I n p a r t i c u l a r by a i d of a n a n a l o g u e c o m p u t a t i o n program d e r i v e d from t h e s e t - u p and t h e d a t a of t h e membrane o s c i l l a t o r T e o r e l l d e m o n s t r a t e d c o n v i n c i n g l y a g r e a t d e a l o f t h e s e e x c i t a b i l i t y phe- nomena. But a l s o under s u i t a b l e c o n d i t i o n s t h e s e phenomena c a n b e d e m o n s t r a t e d i n t h e r e a l T e o r e l l membrane s y s t e m . Because t h e a u t h o r d o e s n o t know e x a c t l y which o f t h e s e e x p e r i - ments have a l r e a d y been c a r r i e d o u t by T . T e o r e l l h i m s e l f , some of t h e a u t h o r ' s r e s u l t s may be g i v e n h e r e i n o r d e r t o show t h e e f f i c i e n c y and t h e c o m p l e t e n e s s of t h e n e r v e a n a l o g u e p r o p e r t i e s of T e o r e l l ' s membrane system. Tab. 2 g i v e s a l i s t of m a t e r i a l s s u i t a b l e f o r membranes i n t h e T e o r e l l o s c i l l a t o r . Most of t h e e x p e r i m e n t s shown i n t h e f o l l o - wing p i c t u r e s have been c a r r i e d o u t by means of D u r a n - s i n t e r g l a s s 277 2 - membranes of a t h i c k n e s s of . 7 mm and a n a r e a of .3 c m ( X : m o l / l , w i d t h of p o r e s : ~9 p). l o - * Table 2 . Membrane M a t e r i a l s S u i t a b l e f o r T e o r e l l - O s c i l l a t i o n s P o r c e l a i n ( T e o r e l l ) - 7 r P o r o u s G l a s s - s i n t e r ( T e o r e l l ) 11.I P o r o u s D u r a n g l a s s - s i n t e r 9 r P o r o u s P o l y v i n y l c h l o r i d e ( ' P o r v i c ' ) 4 p 2 . 5 p N u c l e p o r F i l t e r (Meares & P a g e ) I P Q u a r t z Powder 5 v P o r o u s P o l y s t y r o l ( F l e x o l i t h ' ) A120 3-Po wde r 21.1 C a t i o n - e x c h a n g e r G r a i n s ( T e o r e l l ) Anion-exchanger G r a i n s E- cation-exchange anion-xchsnge membrane membrane F i g . 1 6 T h r e s h o l d b e h a v i o u r of F i g . 1 7 A c t i o n p o t e n t i a l re- t h e TNA ( T e o r e l l Nerve Analogue). s p o n s e of t h e TPJA. F i g . 1 6 shows t h e well-known t e m p o r a l p a t t e r n of t h e t h r e s h o l d b e h a v i o u r of s u c h a n e x c i t a b l e D u r a n - g l a s s membrane. I n t h i s case t h e compartments are v i r t u a l l y u n l i m i t e d so t h a t no o s c i l l a - t i o n s can o c c u r . 278 E x c i t a b i l i t y i n s u c h a b i s t a b l e s y s t e m i s , a s well-known, a two- s i d e d phenomenon w i t h r e s p e c t t o b o t h s t a b l e s t a t e s , which are c o m p l e t e l y e q u i v a l e n t i n t h e k i n e t i c a l s e n s e . S u p e r - t h r e s h o l d s t i m u l i s h i f t t h e membrane r e s i s t a n c e beyond t h e v a l u e o f t h e un- s t a b l e c o n c e n t r a t i o n p r o f i l e , s u b t h r e s h o l d s t i m u l i don't reach it. Simply by l i m i t a t i o n of one compartment f i t t e d w i t h a n o v e r f l o w t h e T e o r e l l system e x h i b i t s t r i g g e r a b l e a c t i o n p o t e n t i a l s ( F i g . 1 7 ) . I t depends upon on which s i d e of t h e s e t - u p t h i s compartment l i m i - t a t i o n i s made whether a n upward o r a downward a c t i o n p o t e n t i a l o c c u r s . The r e c o v e r y i s t h e r e s u l t of t h e v a r i a t i o n of t h e hydro- s t a t i c p r e s s u r e c a u s e d by t h e volume f l u x d u r i n g t h e p e r i o d of e x c i t a t i o n . The o v e r f l o w p r e v e n t s r e p e t i t i v e e x c i t a t i o n . F i g . 18 R e f r a c t o r i n e s s of :TI.-- + t h e TNA. F i g . 1 9 A b o l i t i o n o f ac- t i o n p o t e n t i a l of t h e TNA. F i g . 18 d e m o n s t r a t e s a t t h e same membrane s y s t e m t h e phenomenon of r e f r a c t o r i n e s s by a p p l y i n g o f two s t i m u l i of t h e same s t r e n g t h w i t h i n c r e a s i n g t e m p o r a l d i s t a n c e s . With d e c a y i n g r e f r a c t o r i n e s s - r e a l i z e d by d e c a y i n g h y d r o s t a t i c p r e s s u r e , t h e s y s t e m r e g a i n s i t s e x c i t a b i l i t y . F i g . 1 9 d e m o n s t r a t e s t h e phenomenon of a b o l i t i o n of t h e a c t i o n p o t e n t i a l by c o u n t e r a c t i n g s t i m u l i . I n c a s e of t h e e x c i t a b l e n e r v e I. T a s a k i h a s d e m o n s t r a t e d a b o l i t i o n i n 1 9 5 6 (8). A l s o t h e phenomenon of e x c i t a t i o n p r o p a g a t i o n c a n be d e m o n s t r a t e d by means of T e o r e l l ' s membrane system ( F i g . 2 0 ) . P r o p a g a t i o n i s a 279 d i r e c t consequence o f t h e b i s t a b i l i t y of t h e system c o n t a i n i n g s t r o n g p o s i t i v e f e e d b a c k . The p r o p a g a t i o n may c o n c e r n t h e c o n v e r - s i o n of t h e u p p e r s t a t e i n t o t h e lower s t a t e o r t h e r e v e r s e d p r o - cess. A l s o s a l t a t o n i c p r o p a g a t i o n c a n b e r e a l i z e d i n T e o r e l l ' s n e r v e model ( 2 ) . @ membrane F i g . 2 0 P r o p a g a t i o n of F i g . 2 1 A p r o j e c t of a T N K s t a t e t r a n s i t i o n s ( 2 ) e x h i b i t i n g a c t i o n p o t e n t i a l a ) from lower t o u p p e r s t a t e ; p r o p a g a t i o n . b ) from upper t o l o w e r s t a t e ; p r o p a g a t i o n of a c t i o n p o t e n t i a l s s h o u l d be d e m o n s t r a t a b l e t o o by means of t h e T e o r e l l system. F i g . 2 1 shows a p r o j e c t of a s e t - u p which s h o u l d e x h i b i t a c t i o n p o t e n t i a l p r o p a g a t i o n . It c o n s i s t s of a n a r r a y of compartments bounded by i o n i c c o n d u c t i n g w a l l s on b o t h s i d e s of t h e membrane a l l o w i n g t h a t l o c a l i z e d h y d r o s t a t i c p r e s s u r e c a n be m a i n t a i n e d . T h e r e i s no d o u b t t h a t t h i s model w i l l work and it c a n b e p r e d i c t e d on t h e b a s i s of t h e e x p e r i e n c e s w i t h t h e T e o r e l l s y s t e m t h a t i n t h i s n e r v e a n a l o g u e a wave of a c t i o n p o t e n t i a l accompanied by a wave of h y d r o s t a t i c p r e s s u r e and a d o u b l e wave o f eddy c u r r e n t s would t r a v e l a l o n g t h e membrane a f t e r a l o c a l s t i m u l a t e d e x c i t a t i o n . T h i s p r o j e c t , whose r e a l i z a t i o n would b e v e r y l a b o r i o u s , i s men- t i o n e d h e r e i n o r d e r t o show t h a t t h e m i s s i n g of a c t i o n p o t e n t i a l 280 propagation experiments is no disproof of the completeness of the nerve analogue property of Teorell's membrane system. Fig. 22 Synopsis of the general phenomenology 0 s systems con- taining antagonistic feedback. Concluding a synopsis of physicochemical oscillations are shown in Fig. 22. These systems which are regarded as nerve models are of extremely different substantial nature, but they have obvious- ly one property in common: the antagonistic feedback. From this point of view it is no more surprising that they exhibit neverthe- less an identic phenomenology. Among these systems Teorell's mem- brane oscillator certainly is the system being best understood now due its simplicity and transparent kinetics. REFERENCES 1. Franck, U.F.: uber das elektrochemische Verhalten von porosen Ionenaustauschermembranen. Ber. Bunsenges. Phys.Chem.67: 657-671, 1963 nen. Ber. Bunsenges. Phys.Chem. 71: 789-799, 1967 cillatory systems Faraday Symp. Chem.Soc. 9: 137-149, 1974 Angew.Chem. Int.Ed.Eng1. 17: 1-15, 1978 in coarse-grained ion-exchanger membranes. Electrochimica Acta 23: 1081-1091, 1978 2. Franck, U.F.: Phanomene an biologischen und kunstlichen Membra- 3. Franck, U.F.: Kinetic feedback processes in physicochemical os- 4. Franck, U.F.: Chemical Oscillations 5. Franck, U.F.: A quantitative treatment of oscillatory phenomena 28 1 6. Franck, U.F.: Feedback kinetics in physicochemical oscillators. 7. Higgins, J.: Oscillating reactions 8. Tasaki I.: J.G.Physio1. 39: 377, 1956 9. Teorell, T.: Zur quantitativen Behandlung der Membranpermeabi- 10.Teore11, T: Expt.Cel1 Res., 1955, suppl. 3, 339 11.Teorel1, T: Biophysical aspects on mechanical stimulation of 12.Teorel1, T: Oscillatory electrophoresis in ion exchange mem- 13.Teorel1, T: Excitability phenomena in artificial membranes. Bio- Ber.Bunsenges.Phys.Chem.84: 334-341, 1980 1nd.Eng.Chem. 59: 19-62, 1967 litat. 2.f.Elektrochem. 55: 460-469, 1951 excitable tissues, Acta Soc.Med.Upsa1iensi.s 64: 341-352, 1959 branes. Arkiv for Kemi 18(22): 401-408, 1961 physical J. 2: 27-52, 1962 Received 80 10 17 Address for reprint requests: Ullrich Franck Institut fur Physikalische Chemie, Tempelgraben 59, Aac hen , Germany 282