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The epileptic neuron: I
BULLETIN OF MATHEMATICALBIOLOGY
voL~
35, 1973
THE
EPILEPTIC
NEURON:
I
• N. gAs~.vsxY
Mathematical models of neurons are studied which exhibit spontaneous and repetitive firings in the absence of normally occurring substances. The activity of such neurons could result in the symptoms of epilepsy. The identification of such substances would be important in the t r e a t m e n t of the disease as well as in the understanding of their mode of action. The importance of the geometrical and physical parameters for the generation of spontaneous repetitive discharges is brought out.
In a previous paper (Rashevsky, 1972), it was shown that if the decay coefficients lc and m of the excitatory and inhibitory substances are considered as due to the binding of the excitatory and inhibitory substances b y substances S and Q, respectively, these being substances which are present in every normal healthy individual, then if those substances are absent in an individual, the neurons of such an individual will fire spontaneously and repetitively, thus showing the principal symptoms of epilepsy. This point of view leads to a possible approach for treatment of epilepsies. Instead of giving the patient anticonvulsant drugs or sedatives which are foreign to a normal organism, we m a y be able to effect a cure by administering to a patient the substances S and Q which are present in every healthy individual. To do so requires first of all the identification of those substances S and O. I n addition, the development of specific procedures may undoutedly be helped b y a theoretical insight into their mode of operation. In the previous paper (gashevsky, 1972), hereinafter referred to as I, a discussion is given of the kinetics of the interaction of the excitatory and inhibitory 709
710
N. RASI-IEVSKY
substances w i t h S and Q, as if the process was taking place in a homogenous system. B u t the brain is a n y t h i n g b u t homogenous. All substances are either f o r m e d inside the n e u r o n and diffuse outward, or t h e y are formed in t h e endocrine glands, t h e blood stream, or less likely, in t h e interstitial liquid a n d t h e n diffuse into t h e neuron. The m e t h o d s for obtaining those substances S and Q would be quite different in t h e t h r e e cases, of course. I f t h e y are inside the neurons one would have to t a k e recourse to brain extracts, p r o b a b l y from higher anthropoids which are occasionally subject to epilepsy. This would be done u n d e r t h e assumption t h a t S and Q are t h e same in h u m a n s and in anthropoids, just as, for example, the t h y r o i d h o r m o n e is t h e same in h u m a n s and in m a m mals. If, on t h e other hand, the p r o d u c t i o n of S and Q occurs in the blood, t h e n we m i g h t even obtain t h e m f r o m h e a l t h y non-epileptic humans. A p r e l i m i n a r y s t u d y of the effects of the size, shape and physical p a r a m e t e r s of a n e u r o n on rates of reactions of different substances which are either produced or consumed in it, has been carried o u t (I~ashevsky, 1973, hereinafter referred to as I I ) . I t was p o i n t e d out t h a t t h e discussions of s t a t i o n a r y states are all p a r t i c u l a r cases of a more general differential equation. W e r e p r o d u c e t h a t e q u a t i o n here for convenience, the notations being the same: d.~ ~ Co d-t = q A -
(1)
I n this expression, A of the soma, A s, is given b y (24) of I I , ~2
= lS-
a
+
(2)
and A~ for the axon is given b y =
+
(3)
Now let us r e t u r n to (2) and (3) of I. dxl/dt
= al -
kxl
dx2/dt
= a 2 + a21xl
-
a12x2, -
m x 2.
(¢) (5)
These equations will hold for a homogenous system. I n a neuron, all will d e p e n d on w h e t h e r the p r o d u c t i o n of x 1 and x 2 takes place inside or outside of the neuron. (For b r e v i t y we shall designate as xl and x2 t h e actual substances, although in the e q u a t i o n x 1 and x 2 are their concentrations in g c m - 3.) T h e results will also depend on the A's, each substance having a different A. Consider first t h a t the p r o d u c t i o n of x 1 and x 2 takes place inside the neuron.
THE
EPILEPTIC NEURON:
I
711
Since w e eventually must deal with oscillatoryeases, w e cannot use the expressions for the stationary states found in II, but must use (I). The rate of production ql of x I is n o w equal, as seen from (4),to:
ql
=
~1
-
kxl
-
(6)
a12x2.
Introducing this into (1), and denoting b y a bar the average concentrations, we find, after rearranging: dxl
(
dt=
Xol~
khl + 1
al -b A1 ]
A1
x 1 - al2x 2.
(7)
mA~ + 1 A2 x2"
(8)
Similarly, from (1) and (5) we obtain: dx~ ( xo2 ~ dg = _a2 + A 2 ] + a21xl
Consider also the case that the substances S and Q are formed at corresponding constant rates qs and qo. On one hand they diffuse outward, on the other hand they disappear by combining with x 1 and x2, respectively. In the previous study (Equation (1) of I), we assumed that k = l~'cs so that the rate of decay of xl is ~x 1 = ]c'xlc s,
or
]~ = /~'Cs.
(9)
mx2 = m ' x 2 % ,
or
m = m'%.
(10)
Similarly, we have
Allowing for stoichiometric factors 71 and y2, the rates of disappearance of cs and c o due to their combining with x~ and x 2, respectively, are: 71k'xlcs
and
(11)
72m'x~c o.
Hence, purely biochemically, the net rates of production of c s and co are, respectively qs - 711c'xlcs
and
qo - 7am'xa% •
Those values of "net productions" must be introduced for g into (1). thus obtain, again using bars to denote averages: d-cs
Cos _ 71~,~1-~s dt = qs + ~
(12) We
Cs As
(13)
co AQ
(14)
and d~o
cos dt = qo + Ao
7am'~2c°
712
N.: RASHEVSKY
E q u a t i o n s (7), (8), (13) and (14) form a system of coupled differential equations which govern the variation of 31, x2, cs and co" Because of the presence in (13) and (14) of the products x l c s and 52~Q, t h e system is nonlinear and, as with all nonlinear systems, it is difficult to handle. W e shall simplify the problem b y first considering a limiting case. L e t us consider the case where there are m a n y more of S t h a n of 31 and of Q t h a n of 32. I n t h a t case all molecules of ~1 m a y become bound, t h a t is, t h e y will disappear as such, y e t the consideration Us remains practically constant. The same holds in relation to 32 and Q. I n t h a t case the system (7), (8), (13) and (14) becomes linear, since now cs and ~o are constant. The first two equations of t h e system, n a m e l y (7) and (8), do not contain either cz of ~Q. The quantities in parentheses of all four equations are constant. U n d e r these assumptions, cs and cQ are unaffected b y x 1 and x 2. T h e y are p r o d u c e d in the cell and diffuse o u t w a r d according to (1). Thus t h e y will e v e n t u a l l y reach the s t a t i o n a r y state ~8 ---- COS -[- q s A s ;
cQ = Coo -]- qQAQ.
(15)
The coefficients k and m in (7) and (8) now become, because of (9) and (10) ]c = k'(Cos + qsAs);
m = m'(coQ + qQAQ).
(16)
T h e y are constants which depend on the geometric and physical characteristics of t h e cell because of A s and AQ. T h e y do n o t vanish except when cos = COQ -~ qsqQ "~ O, or if, for finite values of qs and qQ, A s = AQ = 0. As we h a v e seen in I, the vanishing of k and m are necessary and sufficient for the system (2) and (3) of I to give spontaneous u n d a m p e d oscillations, thus producing spontaneous repetitive discharges. E q u a t i o n s (7) and (8) of the present r e p o r t are different, because of the A's, from (2) and (3) of I. I f we now m a k e qs = qQ = 0, t h a t is, if the p r o d u c t i o n of S and Q stops, t h e n according to (9) and (10) k = m = 0. The same will h a p p e n if in A s and AQ we m a k e the permeabilities h s and h o zero, and also p u t Cos = c0Q = 0, for t h e n it follows f r o m (24) and (25) o f / / , A s = AQ = oo. W e may, of course also p u t D s and DQ equal to zero. This, in a semiliquid m e d i u m of the cytoplasm, is unlikely, however. On the other hand, it is well k n o w n t h a t biological m e m b r a n e s are completely impermeable to some substances. Still, even if we make ]c = m = 0, b y introducing the above assumptions, our E q u a t i o n s (7) and (8) will r e m a i n different from (15) and (16) of I. I f we again use the t r a n s f o r m a t i o n (18) of I, we find, from (4) and (5): dxt dt
xl -
A 1
a12xa;
(17)
THE EPILEPTIC
dx2 dt
NEURON:
I
713
x2 = a2ixl -
(18)
-~2"
I n t r o d u c i n g the variables Yl and Y2 which represent the deviation f r o m t h e s t a t i o n a r y state, and p u t t i n g Yl = xi - x~; Y2 = Y2 - x*, where x~ and x* are t h e s t a t i o n a r y state values, we have Yi = C1 eVt;
Y2 = C~ e v~.
(19)
where v is either of t h e two conjugate i m a g i n a r y roots of the scalar equation:
v =
2A1A2
+ 2a4[
)k~-A; ]
- 4 a12a2i + ~
.
(20)
Since A i >1 0 and A2 > 0, therefore if the expression u n d e r the radical in (20) is negative, or i.e. if
t h e n t h e s t a t i o n a r y state is approached tt~'ough d a m p e d oscillations. I f C1 and C2 are large e n o u g h so t h a t the amplitude of x i - x 2 exceeds the depolarization threshold h a of t h e membrane, we shall h a v e a short train of spikes. T h e n u m b e r of spikes in the t r a i n is the greater, t h e smaller the value of (A~ + As)/A1A 2. If, on the other hand, the i n e q u a l i t y sign in (2I) is reversed, t h e n the s t a t i o n a r y state is approached a s y m p t o t i c a l l y exponentially. Incidentally, the first case m i g h t correspond, as we shall see, to a petit mM. F o r a grand mal, however, we m u s t have:
Ai + As
=
I
A--: +
1
=
o.
(22)
Only t h e n is v a p u r e l y i m a g i n a r y number. R e q u i r e m e n t (22) can be fulfilled only if b o t h Ai and A2 = oe. This requires t h a t b o t h the permeability hi for x i and the p e r m e a b i l i t y h~ for x 2 are infinite. Thus the necessary and sufficient condition for an indefinite train of spont a n e o u s l y generated spikes is t h a t the substances x 1 and x 2 are not only p r o d u c e d inside the neuron, b u t t h a t the m e m b r a n e is completely impermeable to t h e m . Neither x i nor x 2 can leave the neuron. I n the case of a t r a i n of d a m p e d oscillations, a finite b u t smaller A i or A,, means a longer t r a i n of spikes. I n I I we h a v e seen t h a t for the axon the values of A are from t h r e e to twenty-five times smaller t h a n for the soma. Hence, longer trains of spikes are p r o b a b l y generated in the dendrites and axon, r a t h e r t h a n in the soma. This example illustrates the i m p o r t a n c e of the geometrical
714
N. RASHEVSKY
a n d physical p a r a m e t e r s for t h e g e n e r a t i o n of s p o n t a n e o u s r e p e t i t i v e discharges. This w o r k was done for a n d s u p p o r t e d b y t h e J. M. l~ichards L a b o r a t o r y , Grosse P o i n t e P a r k , Michigan.
LITERATURE l~ashevsky, N. 1972. "Suggestions for a Mathematical Model of a Pathological Neuron with Spontaneous Repetitive Discharges." Bull. Math. Biophysics, 34, 223-230. 1973. "The Diffusion and Metabolic Differences Between Soma and Axon of a Neuron." Bull. Ma$h. Biology, 35, 421-429.
voL~
35, 1973
THE
EPILEPTIC
NEURON:
I
• N. gAs~.vsxY
Mathematical models of neurons are studied which exhibit spontaneous and repetitive firings in the absence of normally occurring substances. The activity of such neurons could result in the symptoms of epilepsy. The identification of such substances would be important in the t r e a t m e n t of the disease as well as in the understanding of their mode of action. The importance of the geometrical and physical parameters for the generation of spontaneous repetitive discharges is brought out.
In a previous paper (Rashevsky, 1972), it was shown that if the decay coefficients lc and m of the excitatory and inhibitory substances are considered as due to the binding of the excitatory and inhibitory substances b y substances S and Q, respectively, these being substances which are present in every normal healthy individual, then if those substances are absent in an individual, the neurons of such an individual will fire spontaneously and repetitively, thus showing the principal symptoms of epilepsy. This point of view leads to a possible approach for treatment of epilepsies. Instead of giving the patient anticonvulsant drugs or sedatives which are foreign to a normal organism, we m a y be able to effect a cure by administering to a patient the substances S and Q which are present in every healthy individual. To do so requires first of all the identification of those substances S and O. I n addition, the development of specific procedures may undoutedly be helped b y a theoretical insight into their mode of operation. In the previous paper (gashevsky, 1972), hereinafter referred to as I, a discussion is given of the kinetics of the interaction of the excitatory and inhibitory 709
710
N. RASI-IEVSKY
substances w i t h S and Q, as if the process was taking place in a homogenous system. B u t the brain is a n y t h i n g b u t homogenous. All substances are either f o r m e d inside the n e u r o n and diffuse outward, or t h e y are formed in t h e endocrine glands, t h e blood stream, or less likely, in t h e interstitial liquid a n d t h e n diffuse into t h e neuron. The m e t h o d s for obtaining those substances S and Q would be quite different in t h e t h r e e cases, of course. I f t h e y are inside the neurons one would have to t a k e recourse to brain extracts, p r o b a b l y from higher anthropoids which are occasionally subject to epilepsy. This would be done u n d e r t h e assumption t h a t S and Q are t h e same in h u m a n s and in anthropoids, just as, for example, the t h y r o i d h o r m o n e is t h e same in h u m a n s and in m a m mals. If, on t h e other hand, the p r o d u c t i o n of S and Q occurs in the blood, t h e n we m i g h t even obtain t h e m f r o m h e a l t h y non-epileptic humans. A p r e l i m i n a r y s t u d y of the effects of the size, shape and physical p a r a m e t e r s of a n e u r o n on rates of reactions of different substances which are either produced or consumed in it, has been carried o u t (I~ashevsky, 1973, hereinafter referred to as I I ) . I t was p o i n t e d out t h a t t h e discussions of s t a t i o n a r y states are all p a r t i c u l a r cases of a more general differential equation. W e r e p r o d u c e t h a t e q u a t i o n here for convenience, the notations being the same: d.~ ~ Co d-t = q A -
(1)
I n this expression, A of the soma, A s, is given b y (24) of I I , ~2
= lS-
a
+
(2)
and A~ for the axon is given b y =
+
(3)
Now let us r e t u r n to (2) and (3) of I. dxl/dt
= al -
kxl
dx2/dt
= a 2 + a21xl
-
a12x2, -
m x 2.
(¢) (5)
These equations will hold for a homogenous system. I n a neuron, all will d e p e n d on w h e t h e r the p r o d u c t i o n of x 1 and x 2 takes place inside or outside of the neuron. (For b r e v i t y we shall designate as xl and x2 t h e actual substances, although in the e q u a t i o n x 1 and x 2 are their concentrations in g c m - 3.) T h e results will also depend on the A's, each substance having a different A. Consider first t h a t the p r o d u c t i o n of x 1 and x 2 takes place inside the neuron.
THE
EPILEPTIC NEURON:
I
711
Since w e eventually must deal with oscillatoryeases, w e cannot use the expressions for the stationary states found in II, but must use (I). The rate of production ql of x I is n o w equal, as seen from (4),to:
ql
=
~1
-
kxl
-
(6)
a12x2.
Introducing this into (1), and denoting b y a bar the average concentrations, we find, after rearranging: dxl
(
dt=
Xol~
khl + 1
al -b A1 ]
A1
x 1 - al2x 2.
(7)
mA~ + 1 A2 x2"
(8)
Similarly, from (1) and (5) we obtain: dx~ ( xo2 ~ dg = _a2 + A 2 ] + a21xl
Consider also the case that the substances S and Q are formed at corresponding constant rates qs and qo. On one hand they diffuse outward, on the other hand they disappear by combining with x 1 and x2, respectively. In the previous study (Equation (1) of I), we assumed that k = l~'cs so that the rate of decay of xl is ~x 1 = ]c'xlc s,
or
]~ = /~'Cs.
(9)
mx2 = m ' x 2 % ,
or
m = m'%.
(10)
Similarly, we have
Allowing for stoichiometric factors 71 and y2, the rates of disappearance of cs and c o due to their combining with x~ and x 2, respectively, are: 71k'xlcs
and
(11)
72m'x~c o.
Hence, purely biochemically, the net rates of production of c s and co are, respectively qs - 711c'xlcs
and
qo - 7am'xa% •
Those values of "net productions" must be introduced for g into (1). thus obtain, again using bars to denote averages: d-cs
Cos _ 71~,~1-~s dt = qs + ~
(12) We
Cs As
(13)
co AQ
(14)
and d~o
cos dt = qo + Ao
7am'~2c°
712
N.: RASHEVSKY
E q u a t i o n s (7), (8), (13) and (14) form a system of coupled differential equations which govern the variation of 31, x2, cs and co" Because of the presence in (13) and (14) of the products x l c s and 52~Q, t h e system is nonlinear and, as with all nonlinear systems, it is difficult to handle. W e shall simplify the problem b y first considering a limiting case. L e t us consider the case where there are m a n y more of S t h a n of 31 and of Q t h a n of 32. I n t h a t case all molecules of ~1 m a y become bound, t h a t is, t h e y will disappear as such, y e t the consideration Us remains practically constant. The same holds in relation to 32 and Q. I n t h a t case the system (7), (8), (13) and (14) becomes linear, since now cs and ~o are constant. The first two equations of t h e system, n a m e l y (7) and (8), do not contain either cz of ~Q. The quantities in parentheses of all four equations are constant. U n d e r these assumptions, cs and cQ are unaffected b y x 1 and x 2. T h e y are p r o d u c e d in the cell and diffuse o u t w a r d according to (1). Thus t h e y will e v e n t u a l l y reach the s t a t i o n a r y state ~8 ---- COS -[- q s A s ;
cQ = Coo -]- qQAQ.
(15)
The coefficients k and m in (7) and (8) now become, because of (9) and (10) ]c = k'(Cos + qsAs);
m = m'(coQ + qQAQ).
(16)
T h e y are constants which depend on the geometric and physical characteristics of t h e cell because of A s and AQ. T h e y do n o t vanish except when cos = COQ -~ qsqQ "~ O, or if, for finite values of qs and qQ, A s = AQ = 0. As we h a v e seen in I, the vanishing of k and m are necessary and sufficient for the system (2) and (3) of I to give spontaneous u n d a m p e d oscillations, thus producing spontaneous repetitive discharges. E q u a t i o n s (7) and (8) of the present r e p o r t are different, because of the A's, from (2) and (3) of I. I f we now m a k e qs = qQ = 0, t h a t is, if the p r o d u c t i o n of S and Q stops, t h e n according to (9) and (10) k = m = 0. The same will h a p p e n if in A s and AQ we m a k e the permeabilities h s and h o zero, and also p u t Cos = c0Q = 0, for t h e n it follows f r o m (24) and (25) o f / / , A s = AQ = oo. W e may, of course also p u t D s and DQ equal to zero. This, in a semiliquid m e d i u m of the cytoplasm, is unlikely, however. On the other hand, it is well k n o w n t h a t biological m e m b r a n e s are completely impermeable to some substances. Still, even if we make ]c = m = 0, b y introducing the above assumptions, our E q u a t i o n s (7) and (8) will r e m a i n different from (15) and (16) of I. I f we again use the t r a n s f o r m a t i o n (18) of I, we find, from (4) and (5): dxt dt
xl -
A 1
a12xa;
(17)
THE EPILEPTIC
dx2 dt
NEURON:
I
713
x2 = a2ixl -
(18)
-~2"
I n t r o d u c i n g the variables Yl and Y2 which represent the deviation f r o m t h e s t a t i o n a r y state, and p u t t i n g Yl = xi - x~; Y2 = Y2 - x*, where x~ and x* are t h e s t a t i o n a r y state values, we have Yi = C1 eVt;
Y2 = C~ e v~.
(19)
where v is either of t h e two conjugate i m a g i n a r y roots of the scalar equation:
v =
2A1A2
+ 2a4[
)k~-A; ]
- 4 a12a2i + ~
.
(20)
Since A i >1 0 and A2 > 0, therefore if the expression u n d e r the radical in (20) is negative, or i.e. if
t h e n t h e s t a t i o n a r y state is approached tt~'ough d a m p e d oscillations. I f C1 and C2 are large e n o u g h so t h a t the amplitude of x i - x 2 exceeds the depolarization threshold h a of t h e membrane, we shall h a v e a short train of spikes. T h e n u m b e r of spikes in the t r a i n is the greater, t h e smaller the value of (A~ + As)/A1A 2. If, on the other hand, the i n e q u a l i t y sign in (2I) is reversed, t h e n the s t a t i o n a r y state is approached a s y m p t o t i c a l l y exponentially. Incidentally, the first case m i g h t correspond, as we shall see, to a petit mM. F o r a grand mal, however, we m u s t have:
Ai + As
=
I
A--: +
1
=
o.
(22)
Only t h e n is v a p u r e l y i m a g i n a r y number. R e q u i r e m e n t (22) can be fulfilled only if b o t h Ai and A2 = oe. This requires t h a t b o t h the permeability hi for x i and the p e r m e a b i l i t y h~ for x 2 are infinite. Thus the necessary and sufficient condition for an indefinite train of spont a n e o u s l y generated spikes is t h a t the substances x 1 and x 2 are not only p r o d u c e d inside the neuron, b u t t h a t the m e m b r a n e is completely impermeable to t h e m . Neither x i nor x 2 can leave the neuron. I n the case of a t r a i n of d a m p e d oscillations, a finite b u t smaller A i or A,, means a longer t r a i n of spikes. I n I I we h a v e seen t h a t for the axon the values of A are from t h r e e to twenty-five times smaller t h a n for the soma. Hence, longer trains of spikes are p r o b a b l y generated in the dendrites and axon, r a t h e r t h a n in the soma. This example illustrates the i m p o r t a n c e of the geometrical
714
N. RASHEVSKY
a n d physical p a r a m e t e r s for t h e g e n e r a t i o n of s p o n t a n e o u s r e p e t i t i v e discharges. This w o r k was done for a n d s u p p o r t e d b y t h e J. M. l~ichards L a b o r a t o r y , Grosse P o i n t e P a r k , Michigan.
LITERATURE l~ashevsky, N. 1972. "Suggestions for a Mathematical Model of a Pathological Neuron with Spontaneous Repetitive Discharges." Bull. Math. Biophysics, 34, 223-230. 1973. "The Diffusion and Metabolic Differences Between Soma and Axon of a Neuron." Bull. Ma$h. Biology, 35, 421-429.