Bulletin of Mathematical Biology VoL 53, No. 3, pp. 327 343, 1991. printed in Great Britain.
0092-8240/91 $3.00 + 0.00 Pergamon Press plc © 1991 Society for Mathematical Biology
AUTOCATALYTIC MEMBRANE CONDUCTANCE MEMORY
AND
STEVEN JAFFE
Department of Mathematics, University of Southern. California, Los Angeles, CA 90089-1113, U.S.A. (E.mail : sjaff [email protected] ) A basic characteristic of biological memory is that it has a graded duration, which, even for socalled short-term memory, can vary from minutes to days (i.e. over about three orders of magnitude), depending on the training protocol, which one can think of as determining the "strength" of the memory. Furthermore, the molecular analysis of simple learning in invertebrates has revealed many examples where "learning" is produced by a decrease in an appropriate membrane conductance. This paper provides a quantitative analysis of a simple kinetic scheme whereby a conductance decrease can be produced by repetitive nerve impulses, with a duration that varies with stimulus frequency. The simplest model considered is based on the actual kinetics of the naturally-occurring ionophore Monazomycin. This model yields durations ranging only ove r a factor of about 10, for reasonable parameter values. However, a simple modification of the model yields memory durations ranging over three or more orders of magnitude. We also show that Monazomycin-like kinetics can appear as the result of a combination of simple uni- and bi-molecular reactions, thus making more plausible the possibility that the effects described here may operate in actual biological systems.
1. Introduction. A basic characteristic of biological memory is that it has a graded duration, which, even for so-called short-term memory, can vary from minutes to days (i.e. over about three orders of magnitude), depending on the training protocol, which one can think of as determining the "strength" of the memory. Furthermore, the molecular analysis of simple learning in invertebrates has revealed many examples where "learning" is produced by a decrease in an appropriate membrane conductance (Hawkins, 1983). This paper provides a quantitative analysis of a simple kinetic scheme whereby a conductance decrease can be produced by repetitive nerve impulses, with a duration that varies with stimulus frequency. The simplest model considered is based on the actual kinetics of the naturally-occurring ionophore Monazomycin. This model yields durations ranging only over a factor of about 10, for reasonable parameter values. However, a simple modification of the model yields memory durations ranging over three or more orders of magnitude. We also show that Monazomycin-like kinetics can appear as the result of a combination of simple uni- and bi-molecular reactions, thus making more 327
328
S. J A F F E
plausible the possibility that the effects described here may operate in actual biological systems. Monazomycin (MA) is an antibiotic that forms ion channels in thin lipid membranes. Muller et al. (1981) have shown that the induced conductance: (a) is strongly voltage-dependent; (b) has unusual, autocatalytic, kinetics. Muller and Peskin (1981) note that these properties produce memory-like effects and suggest ways that such molecules, incorporated into nerve membranes, could serve as a molecular basis for some forms of biological memory. The basic observation is that, because of (a), a nerve impulse will produce a transient conductance change, and because of (b) the time-constant for recovery to equilibrium will depend on the size and direction of the conductance change. Specifically, we show that a conductance decrease is produced rapidly and recovers slowly, while a conductance increase is slow to produce and recovers rapidly. We therefore suggest that a nerve-impulseinduced decrease in a MA-like conductance has the right qualitative properties to serve as a substrate for biological memory. It is interesting to note that those biological examples where memory processes have been analysed at the molecular level--such as habituation and sensitization in Aplysia, and operant conditioning in locusts (Woollacott and Hoyle, 1977; Hoyle, 1986)-invariably involve a membrane conductance decrease (Hawkins, 1983). For example, in Aplysia, habituation is produced by a presynaptic decrease in C a 2 + conductance (Klein et al., 1980). Sensitization, although physiologically opposite in effect, corresponds not to a Ca 2 + conductance increase, but to a decrease in K + conductance (Kandel and Schwartz, 1982). In this note, we give a quantitative description of a MA-like conductance subjected to a periodic train of nerve impulses. We derive the "dose-response" function, relating stimulus frequency to steady-state conductance change; the "strength-duration" function, relating the magnitude of the conductance to the time-scale of the subsequent recovery to equilibrium; and the function relating memory strength to the time-scale for approach to the steady-state. We present a modification of the MA-like kinetics that prolongs the memory. We also note that a MA-like conductance can appear as a consequence of a combination of simple uni- and bi-molecular kinetics. 2. The Kinetics. is:
Let 9 be the MA-like conductance. The basic kinetic equation g
B
(1) where B is constant and A and 9~ are strongly voltage-dependent: A = ~ e ~v
9~=~
e uv.
(2)
AUTOCATALYTIC M E M B R A N E C O N D U C T A N C E A N D M E M O R Y
329
For our purposes the exact form of (2) is not important; we need only to suppose that a nerve impulse, which increases V, will decrease go; thus we assume in (2) that # < 0 . * It is almost immediate from (1) that a conductance decrease will last longer than an increase. The quantitative analysis is based on the observation, due to Leibniz, that the change of variable:
y=g -~,
(3)
turns (1) into the linear differential equation:
.9= --BA(y-yo~),
where yo~:=g~ B.
(4)
If A, y~ are constant, the solution is:
y(t)=ay(O)+b,
where a=e -BA', b=(1-a)yo~.
(5)
Even ifA and Yoo are not constant, the solution has the form of (5), but with a and b now functionals of A(-0' y~(-r), r e [ 0 , t]. Therefore a nerve impulse, viewed as a fixed waveform V(t) of duration T, can be replaced by a voltage step, with suitably averaged values of A and goo, in so far as its effect on g(t), t/> Tis concerned. The assumption that the shape of a nerve impulse is not affected by the changing conductance g is, of course, a simplification that cannot hold exactly, but we adopt it as a reasonable first approximation. We therefore treat a nerve impulse as a square pulse of duration Te from the "resting potential" Vr to a level Vo> ~ . Let the corresponding values of A and g® be denoted by Ar, A e #sod gr, g~; recall that we assume g~ < gr. With this simplification, we obtain tlie following results.
2.1. Recovery half-time. Define tmem=tmcm(go)to be the time taken to recover half-way from an initial conductance go to the resting conductance gr, with V = V~. T h e n g(tmem)=l(go+gr) and we find from (5) that:
1 (
tmem----~-~rlOg(l+o-)+~rlOg 2B
(l+a-1)nJ ,
(6)
where a: =gr/go; we call a the strength of the "memory" goNote that the second term in (6) vanishes for B = 1. Even for B # 1 it is bounded and thus not important for the qualitative effects which we will discuss. Therefore from now on we will simplify the discussion by taking B = 1, so (6) becomes: * For MA itself, # > 0 ; however # would be < 0 for a molecule with the opposite charge (see Muller and Peskin, 1981).
330
S. JAFFE
1 log(1 + a), tmem -- Ar
a = gr, go
(6')
corresponding to the kinetic equation:
A
(1')
g
Thus for a conductance decrease (o- > 1) tme m witl be large, approaching oo logarithmically with a, while a conductance decrease (o- < 1) is short-lived, tmom approaching 0 as a ~ 0 . This is the basic "memory" effect. For .comparison, note that for simple first-order kinetics, 0 = Ar(g~ --g), the recovery half-time would be log 2/A r, independent of go. The logarithmic dependence of tmom on a is rather weak. For M A itself, the ratio gr/9~ associated with a change in V the size of a nerve impulse (about 100 mV) is typically about 108, which still gives a value of tmom only about 27 times the value (log 2)/A~ associated with an infinitesimal displacement from gr" This is a rather small "dynamic range" compared to the behavior of biological models, such as habituation in Aplysia, where the duration of the m e m o r y varies from minutes to days. We suggest in Section 4 below a modification of (1) that leads to a m u c h stronger dependence of tmom on a and hence an extended dynamic range.
2.2. Response to a pulse train. Consider a train of impulses of duration Te at V = Ve separated by intervals of length Tr at V-- Vr (Fig. 1). The conductance g will approach an oscillatory steady-state between levels gmln and gm,x. We will use 9rain (or rather amem, defined as gffgmin) to measure the strength of the steady-state memory.
°! gmaR groin
vLv t
N
pl
>. :uo!l~lou ~u!aoHoj oql osn o ~
',o2q/v
'~_(~Z + ~ ) - - f £ouonboJj sninm!ls jo smao~ u! s~inso.I oq~ ssoadxo OSlV oAk •jios].t u.to£moz~uoIAI £q pogs!l~s o ~ ~ q l s.tolom~J~d oq] uo suo.r~!puoo aopun p.qgA S.~ lOS pUOOOS o q z "o~ugI o!tueu£p o:~ae[ ~ oJnsuo leql s~olotue~ed oql uo suo!l!puoo ~opun p.~[vh st los ls.KJ OtLL "oz ~ £ ) ° L ' ~ j j o uo.tl:)unj g sg "°~1 pup wo,,~ aoj oglntuaoJ o:~gtu!xo.~ddg jo s~os OA~:~OA~ •OU OAk (r) (t)X pup ' mo~ (zfl (~)1 g I110 R I pup ~°"1 jo sonigA gu!puodsoaaoo oq~ olouop [IVa oA~,pup (z)~1! ilgo 'tuntu!xgtu s]~. o~ ,mo~(~).o~,! [[go ' t u n t m .m u ~ sl~ .tuoJj SOUgA . m°~o . '°,L O1 ~ tuo~j SOLt~A ~,Z sV ~E
X~IOIAI~IA~ (INV ~tDNVZDfl(INOD ~tNVSflIAI~IIAIDIZXHVIVDOZffIV
334
S. J A F F E
(22)
tm~, __ t,,e,,(1) ~ A f 1 log(heTe) - 1.
R e m a r k s . (A1) says that the effect of a single pulse is small. On the other hand, (A2) says that recovery from a pulse is slow, so that the cumulative effect of a train of pulses can be large. Under (A3), which is satisfied by MA, note that the dependence of ame., on T~, as given by (20) and (21), becomes independent of'0"m, x . P r o o f Define W (1), W (2) as the values of W for T r = O O and Tr=To, respectively. Define: o~:= A~Te
fl:=Ar/A ~
x:=AeTr,
o~1 and W (2) ~ 1. Now: e-" W = (1-e-PX) 1 - e -~'
e-" 1 - e -~'
W (1) - - -
W (2) = (1 --e -~p)
e-" - . l-e-"
-
W(1)-, oo means a-,O, and now W(2)--,0 means fl--,0; these translate into (A1) and (A2). Given a, fl-,0 we have: W(1)= ~- a(1 + O(~)),
(23)
W (2) = fl(1 + O(~)),
(24)
which gives by (15): (1)
O'mo m
_
-- (1 + aO'm,0 (1 + O(~)), which gives (18),
(2) __ O'me m - - O ' m a x ( 1 -]- 0 ( ~ ) ) ,
which gives (19),
(25) (26)
and: O". . . .
(1)
O'me m
(1 - e -p~) (1 + 0(~)) + 5~O'max (l_e-pX)(l+O(a))+c~ (l+~am.~)(l+O(~))"
(27)
If we now assume (A3) then a a m . ~ ~ and (27) immediately gives (20). We get (21) from (25) and (26). Noting that amer,(1)-, ~ by (25), formula (11) now gives (22). • In order to compare the assumptions made above to the behavior of MA itself, we briefly summarize the data in Muller and Peskin (1981 ). First, # and 2 in (2) have the same sign, so the assumption that ge < gf also implies A~ < Ar,
AUTOCATALYTIC MEMBRANE CONDUCTANCE AND MEMORY
335
contrary to (A2). If we take Vo- V~= 100 mV then the measured values of/~ and 2 imply that am, x = g~/9~ ~ l0 s and A e / A r ,~ 10- 4. (However, it should be noted that this represents a considerable extrapolation from the data, since measurements of goo and A were m a d e over only about two orders of magnitude.) The values of A, which depend on the concentration of MA in solution as well as on V, were measured between 10- 3 and 10 - 5 ms - 1; thus for Te -- 1 ms, the width of a typical nerve impulse, approximations (A1) and (A3) are reasonable. (A2) is not satisfied; instead we have the opposite condition: (A2')
A e ~ A r.
Although (A2) does not hold, we still have the weaker condition that: (A4)
A r r e ~0~, 1 - e that:
AND MEMORY
337
~ > 1 - e - P L so that when fl--, oc with a f l ~ 0 , (38) implies
(~) m O'me
l --e -#x
uniformly for x >~a,
which gives (32). Since Vme m ' r ~"~ ~'mem"r(1)>~ 1 by (28), (11) gives tmem ~ At-- 1 log O'mem, from which (33) follows. Equation (34) is obtained from (3.3) by using 1 - - e - A ~ T ~ . - ~ A r T r for ArT~~ 1, and A~T~,..A~f -~ for T~>>To, and translating the conditions on T~ into conditions o n f . • Example. Take A~=0.1 s - I , A e = 1 0 -4 s -1, O-max= 10 8, T = 1 0 - 3 s, which are realistic values for MA. Then from (35): (1)m ~,.~ l0 log 10 = 23 S, tree
~Z)m ~ 10 log 105 = 115 S, tree
and (34) becomes: tmem~23 + 10 1og(10f) (S), Figure 2 compares the exact value for tree m (curve A) and the approximation (34) (curve B). Curve C shows tmem when Ao is changed from 10- 4 to 10- 3 s - 1; the effect is very nearly just a vertical translation, as predicted by approximation (33).
140 120 100 E
8o 60 C m
40 A 20
10.3
1; "2
1£"1
1
1; 1
1£ 2
? 103 (2T.)-1
stimulus frequency (see "1)
Figure 2. Comparison of exact (6') and approximate (34) formulas for memory duratj,on treem as a function of stimulus frequency. At=0.1 s -t, A~=10 -4 s -t, am,X= 10s, To= 10-3 s. Curve A: exact. Curve B: approximate. Curve C: exact with A~=10 3s-t
338
S. J A F F E
Thus the approximation (34) is seen to be quite good over the full range of frequencies A r ~
0092-8240/91 $3.00 + 0.00 Pergamon Press plc © 1991 Society for Mathematical Biology
AUTOCATALYTIC MEMBRANE CONDUCTANCE MEMORY
AND
STEVEN JAFFE
Department of Mathematics, University of Southern. California, Los Angeles, CA 90089-1113, U.S.A. (E.mail : sjaff [email protected] ) A basic characteristic of biological memory is that it has a graded duration, which, even for socalled short-term memory, can vary from minutes to days (i.e. over about three orders of magnitude), depending on the training protocol, which one can think of as determining the "strength" of the memory. Furthermore, the molecular analysis of simple learning in invertebrates has revealed many examples where "learning" is produced by a decrease in an appropriate membrane conductance. This paper provides a quantitative analysis of a simple kinetic scheme whereby a conductance decrease can be produced by repetitive nerve impulses, with a duration that varies with stimulus frequency. The simplest model considered is based on the actual kinetics of the naturally-occurring ionophore Monazomycin. This model yields durations ranging only ove r a factor of about 10, for reasonable parameter values. However, a simple modification of the model yields memory durations ranging over three or more orders of magnitude. We also show that Monazomycin-like kinetics can appear as the result of a combination of simple uni- and bi-molecular reactions, thus making more plausible the possibility that the effects described here may operate in actual biological systems.
1. Introduction. A basic characteristic of biological memory is that it has a graded duration, which, even for so-called short-term memory, can vary from minutes to days (i.e. over about three orders of magnitude), depending on the training protocol, which one can think of as determining the "strength" of the memory. Furthermore, the molecular analysis of simple learning in invertebrates has revealed many examples where "learning" is produced by a decrease in an appropriate membrane conductance (Hawkins, 1983). This paper provides a quantitative analysis of a simple kinetic scheme whereby a conductance decrease can be produced by repetitive nerve impulses, with a duration that varies with stimulus frequency. The simplest model considered is based on the actual kinetics of the naturally-occurring ionophore Monazomycin. This model yields durations ranging only over a factor of about 10, for reasonable parameter values. However, a simple modification of the model yields memory durations ranging over three or more orders of magnitude. We also show that Monazomycin-like kinetics can appear as the result of a combination of simple uni- and bi-molecular reactions, thus making more 327
328
S. J A F F E
plausible the possibility that the effects described here may operate in actual biological systems. Monazomycin (MA) is an antibiotic that forms ion channels in thin lipid membranes. Muller et al. (1981) have shown that the induced conductance: (a) is strongly voltage-dependent; (b) has unusual, autocatalytic, kinetics. Muller and Peskin (1981) note that these properties produce memory-like effects and suggest ways that such molecules, incorporated into nerve membranes, could serve as a molecular basis for some forms of biological memory. The basic observation is that, because of (a), a nerve impulse will produce a transient conductance change, and because of (b) the time-constant for recovery to equilibrium will depend on the size and direction of the conductance change. Specifically, we show that a conductance decrease is produced rapidly and recovers slowly, while a conductance increase is slow to produce and recovers rapidly. We therefore suggest that a nerve-impulseinduced decrease in a MA-like conductance has the right qualitative properties to serve as a substrate for biological memory. It is interesting to note that those biological examples where memory processes have been analysed at the molecular level--such as habituation and sensitization in Aplysia, and operant conditioning in locusts (Woollacott and Hoyle, 1977; Hoyle, 1986)-invariably involve a membrane conductance decrease (Hawkins, 1983). For example, in Aplysia, habituation is produced by a presynaptic decrease in C a 2 + conductance (Klein et al., 1980). Sensitization, although physiologically opposite in effect, corresponds not to a Ca 2 + conductance increase, but to a decrease in K + conductance (Kandel and Schwartz, 1982). In this note, we give a quantitative description of a MA-like conductance subjected to a periodic train of nerve impulses. We derive the "dose-response" function, relating stimulus frequency to steady-state conductance change; the "strength-duration" function, relating the magnitude of the conductance to the time-scale of the subsequent recovery to equilibrium; and the function relating memory strength to the time-scale for approach to the steady-state. We present a modification of the MA-like kinetics that prolongs the memory. We also note that a MA-like conductance can appear as a consequence of a combination of simple uni- and bi-molecular kinetics. 2. The Kinetics. is:
Let 9 be the MA-like conductance. The basic kinetic equation g
B
(1) where B is constant and A and 9~ are strongly voltage-dependent: A = ~ e ~v
9~=~
e uv.
(2)
AUTOCATALYTIC M E M B R A N E C O N D U C T A N C E A N D M E M O R Y
329
For our purposes the exact form of (2) is not important; we need only to suppose that a nerve impulse, which increases V, will decrease go; thus we assume in (2) that # < 0 . * It is almost immediate from (1) that a conductance decrease will last longer than an increase. The quantitative analysis is based on the observation, due to Leibniz, that the change of variable:
y=g -~,
(3)
turns (1) into the linear differential equation:
.9= --BA(y-yo~),
where yo~:=g~ B.
(4)
If A, y~ are constant, the solution is:
y(t)=ay(O)+b,
where a=e -BA', b=(1-a)yo~.
(5)
Even ifA and Yoo are not constant, the solution has the form of (5), but with a and b now functionals of A(-0' y~(-r), r e [ 0 , t]. Therefore a nerve impulse, viewed as a fixed waveform V(t) of duration T, can be replaced by a voltage step, with suitably averaged values of A and goo, in so far as its effect on g(t), t/> Tis concerned. The assumption that the shape of a nerve impulse is not affected by the changing conductance g is, of course, a simplification that cannot hold exactly, but we adopt it as a reasonable first approximation. We therefore treat a nerve impulse as a square pulse of duration Te from the "resting potential" Vr to a level Vo> ~ . Let the corresponding values of A and g® be denoted by Ar, A e #sod gr, g~; recall that we assume g~ < gr. With this simplification, we obtain tlie following results.
2.1. Recovery half-time. Define tmem=tmcm(go)to be the time taken to recover half-way from an initial conductance go to the resting conductance gr, with V = V~. T h e n g(tmem)=l(go+gr) and we find from (5) that:
1 (
tmem----~-~rlOg(l+o-)+~rlOg 2B
(l+a-1)nJ ,
(6)
where a: =gr/go; we call a the strength of the "memory" goNote that the second term in (6) vanishes for B = 1. Even for B # 1 it is bounded and thus not important for the qualitative effects which we will discuss. Therefore from now on we will simplify the discussion by taking B = 1, so (6) becomes: * For MA itself, # > 0 ; however # would be < 0 for a molecule with the opposite charge (see Muller and Peskin, 1981).
330
S. JAFFE
1 log(1 + a), tmem -- Ar
a = gr, go
(6')
corresponding to the kinetic equation:
A
(1')
g
Thus for a conductance decrease (o- > 1) tme m witl be large, approaching oo logarithmically with a, while a conductance decrease (o- < 1) is short-lived, tmom approaching 0 as a ~ 0 . This is the basic "memory" effect. For .comparison, note that for simple first-order kinetics, 0 = Ar(g~ --g), the recovery half-time would be log 2/A r, independent of go. The logarithmic dependence of tmom on a is rather weak. For M A itself, the ratio gr/9~ associated with a change in V the size of a nerve impulse (about 100 mV) is typically about 108, which still gives a value of tmom only about 27 times the value (log 2)/A~ associated with an infinitesimal displacement from gr" This is a rather small "dynamic range" compared to the behavior of biological models, such as habituation in Aplysia, where the duration of the m e m o r y varies from minutes to days. We suggest in Section 4 below a modification of (1) that leads to a m u c h stronger dependence of tmom on a and hence an extended dynamic range.
2.2. Response to a pulse train. Consider a train of impulses of duration Te at V = Ve separated by intervals of length Tr at V-- Vr (Fig. 1). The conductance g will approach an oscillatory steady-state between levels gmln and gm,x. We will use 9rain (or rather amem, defined as gffgmin) to measure the strength of the steady-state memory.
°! gmaR groin
vLv t
N
pl
>. :uo!l~lou ~u!aoHoj oql osn o ~
',o2q/v
'~_(~Z + ~ ) - - f £ouonboJj sninm!ls jo smao~ u! s~inso.I oq~ ssoadxo OSlV oAk •jios].t u.to£moz~uoIAI £q pogs!l~s o ~ ~ q l s.tolom~J~d oq] uo suo.r~!puoo aopun p.qgA S.~ lOS pUOOOS o q z "o~ugI o!tueu£p o:~ae[ ~ oJnsuo leql s~olotue~ed oql uo suo!l!puoo ~opun p.~[vh st los ls.KJ OtLL "oz ~ £ ) ° L ' ~ j j o uo.tl:)unj g sg "°~1 pup wo,,~ aoj oglntuaoJ o:~gtu!xo.~ddg jo s~os OA~:~OA~ •OU OAk (r) (t)X pup ' mo~ (zfl (~)1 g I110 R I pup ~°"1 jo sonigA gu!puodsoaaoo oq~ olouop [IVa oA~,pup (z)~1! ilgo 'tuntu!xgtu s]~. o~ ,mo~(~).o~,! [[go ' t u n t m .m u ~ sl~ .tuoJj SOUgA . m°~o . '°,L O1 ~ tuo~j SOLt~A ~,Z sV ~E
X~IOIAI~IA~ (INV ~tDNVZDfl(INOD ~tNVSflIAI~IIAIDIZXHVIVDOZffIV
334
S. J A F F E
(22)
tm~, __ t,,e,,(1) ~ A f 1 log(heTe) - 1.
R e m a r k s . (A1) says that the effect of a single pulse is small. On the other hand, (A2) says that recovery from a pulse is slow, so that the cumulative effect of a train of pulses can be large. Under (A3), which is satisfied by MA, note that the dependence of ame., on T~, as given by (20) and (21), becomes independent of'0"m, x . P r o o f Define W (1), W (2) as the values of W for T r = O O and Tr=To, respectively. Define: o~:= A~Te
fl:=Ar/A ~
x:=AeTr,
o~1 and W (2) ~ 1. Now: e-" W = (1-e-PX) 1 - e -~'
e-" 1 - e -~'
W (1) - - -
W (2) = (1 --e -~p)
e-" - . l-e-"
-
W(1)-, oo means a-,O, and now W(2)--,0 means fl--,0; these translate into (A1) and (A2). Given a, fl-,0 we have: W(1)= ~- a(1 + O(~)),
(23)
W (2) = fl(1 + O(~)),
(24)
which gives by (15): (1)
O'mo m
_
-- (1 + aO'm,0 (1 + O(~)), which gives (18),
(2) __ O'me m - - O ' m a x ( 1 -]- 0 ( ~ ) ) ,
which gives (19),
(25) (26)
and: O". . . .
(1)
O'me m
(1 - e -p~) (1 + 0(~)) + 5~O'max (l_e-pX)(l+O(a))+c~ (l+~am.~)(l+O(~))"
(27)
If we now assume (A3) then a a m . ~ ~ and (27) immediately gives (20). We get (21) from (25) and (26). Noting that amer,(1)-, ~ by (25), formula (11) now gives (22). • In order to compare the assumptions made above to the behavior of MA itself, we briefly summarize the data in Muller and Peskin (1981 ). First, # and 2 in (2) have the same sign, so the assumption that ge < gf also implies A~ < Ar,
AUTOCATALYTIC MEMBRANE CONDUCTANCE AND MEMORY
335
contrary to (A2). If we take Vo- V~= 100 mV then the measured values of/~ and 2 imply that am, x = g~/9~ ~ l0 s and A e / A r ,~ 10- 4. (However, it should be noted that this represents a considerable extrapolation from the data, since measurements of goo and A were m a d e over only about two orders of magnitude.) The values of A, which depend on the concentration of MA in solution as well as on V, were measured between 10- 3 and 10 - 5 ms - 1; thus for Te -- 1 ms, the width of a typical nerve impulse, approximations (A1) and (A3) are reasonable. (A2) is not satisfied; instead we have the opposite condition: (A2')
A e ~ A r.
Although (A2) does not hold, we still have the weaker condition that: (A4)
A r r e ~0~, 1 - e that:
AND MEMORY
337
~ > 1 - e - P L so that when fl--, oc with a f l ~ 0 , (38) implies
(~) m O'me
l --e -#x
uniformly for x >~a,
which gives (32). Since Vme m ' r ~"~ ~'mem"r(1)>~ 1 by (28), (11) gives tmem ~ At-- 1 log O'mem, from which (33) follows. Equation (34) is obtained from (3.3) by using 1 - - e - A ~ T ~ . - ~ A r T r for ArT~~ 1, and A~T~,..A~f -~ for T~>>To, and translating the conditions on T~ into conditions o n f . • Example. Take A~=0.1 s - I , A e = 1 0 -4 s -1, O-max= 10 8, T = 1 0 - 3 s, which are realistic values for MA. Then from (35): (1)m ~,.~ l0 log 10 = 23 S, tree
~Z)m ~ 10 log 105 = 115 S, tree
and (34) becomes: tmem~23 + 10 1og(10f) (S), Figure 2 compares the exact value for tree m (curve A) and the approximation (34) (curve B). Curve C shows tmem when Ao is changed from 10- 4 to 10- 3 s - 1; the effect is very nearly just a vertical translation, as predicted by approximation (33).
140 120 100 E
8o 60 C m
40 A 20
10.3
1; "2
1£"1
1
1; 1
1£ 2
? 103 (2T.)-1
stimulus frequency (see "1)
Figure 2. Comparison of exact (6') and approximate (34) formulas for memory duratj,on treem as a function of stimulus frequency. At=0.1 s -t, A~=10 -4 s -t, am,X= 10s, To= 10-3 s. Curve A: exact. Curve B: approximate. Curve C: exact with A~=10 3s-t
338
S. J A F F E
Thus the approximation (34) is seen to be quite good over the full range of frequencies A r ~
