Catastrophe Theory of Dopaminergic Transmission: …behavioralhealth2000.com/wp-content/uploads/sites/3661/...The model is internally consistent with known clinical effects of drugs

J. theor. Biol. (1981) 92,373-400

Catastrophe Theory of Dopaminergic Transmission: A Revised Dopamine Hypothesis of Schizophrenia

ROY KING, JOACHIMD. RAEsEt AND JACK D. BARCHAS

Nancy Pritzker Laboratory of Behavioral Neurochemistry, Depart- ment of Psychiatry and Behavioral Sciences, Stanford University

School of Medicine Stanford, California 94305, U.S.A.

(Received 2 October 1980, and in revised form 10 April 1981)

Mathematical modeling of experimentally observed parameters of dopaminergic neuronal activity suggests the occurrence of multiple equilibrium states in neurons characterized by certain precisely defined properties of the tyrosine hydroxylating system. These equilibria may become unstable under certain conditions and transitions between multiple states are predicted. In addition, modeling of the spatial interactions of dopamine neurons within a neural net leads to domain wall soliton-like solutions of neuronal firing. In the discrete spatial case, these equations are isomorphic to those of the Ising model of phase transitions in lattice spins.

The hypothesis is proposed that the occurrence of multiple stable equilibrium states rather than excessive dopaminergic transmission forms the pathophysiological basis of the schizophrenic thought disorder.

The model is internally consistent with known clinical effects of drugs such as neuroleptics, reserpine and amphetamine. In agreement with postmortem and other studies, the theory predicts the lack of increased concentrations of dopamine or its major metabolite, homovanillic acid, in brain and cerebrospinal fluid of schizophrenics.

The mathematical model is compatible with the theory that postulates an attention deficit as an underlying mechanism of schizophrenic psychosis and allows for a possible genetic heterogeneity of the disease.

1. Introduction

The dopamine hypothesis of schizophrenia (Carlsson & Lindquist, 1963; Horn & Snyder, 1971) postulates excessive transmission through dopaminergic neurons as cause of the disease. The main support for this theory has come from two findings: (1) amphetamine, a dopamine releasing agent, can cause a paranoid psychosis in humans (Angrist & Sathananthan, 1974; Janowsky, El-Yousef & Davis, 1973) and (2) antipsychotic drugs

tAddress reprint requests to: Joachim D. Raese, M.D., Nancy Pritzker Laboratory of Behavioral Neurochemistry, Department of Psychiatry and Behavioral Sciences, Stanford University School of Medicine, Stanford, California 94305, U.S.A.

373

OOZZ-5193/81/200373+28$02.00/0 @ 1981 Academic Press Inc. (London) Ltd.

374 R. KING ET AL

bind to central dopamine receptors as antagonists with affinities correlating closely with their clinical potencies (Creese, Burt & Snyder, 1976; Snyder. Creese & Burt, 1975). Furthermore, Lu-methyl-para-tyrosine. a drug which by inhibiting tyrosine hydroxylase (EC 1.14.16.2) the initial and rate- limiting step in catecholamine biosynthesis (Nagatsu, Levitt & Udenfriend, 1964; Levitt et al., 1965) lowers brain catecholamine concentration has been shown to reduce the dose of antipsychotics required to produce clinical inprovement (Carlsson et al. 1972). There is, however, no evidence for increased dopamine turnover in the schizophrenic brain. In fact, normal levels of the dopamine (DA): metabolite homovanillic acid (HVA) have been found in postmortem brain tissue of chronic schizophrenics (Bacopoulos et al., 1977) and in cerebrospinal fluid (CSF) (Persson & Roos, 1969).

In this report we outline a restatement of the DA hypothesis in terms of a catastrophe theory (Thorn, 1975) modeling of dopaminergic neuronal function. This model generates concrete predictions about the pathophys- iology of schizophrenia which can be tested. It suggests, furthermore, that it is not an increase in DA transmission as the basis for psychosis but rather a failure of correct spatial or temporal distribution of transmission in dopaminergic neurons.

2. The Hypothesis

We suggest that schizophrenia is a state defined by the occurrence of multiple equilibrium states susceptible to “catastrophic” transitions between two stable equilibrium states of dopaminergic neuronal activity. This can be conceptualized to occur in either temporal or spatial (or both) distribution, i.e. a total population of neurons may be forced into a dysrhyth- mic, erratic firing pattern, or the activity of subpopulations of neurons in a nucleus may dissociate into widely divergent impulse rates. A net increase in total DA turnover is not expected as a result of this functional state.

(AI PHYSIOLOGY OF NIGRO-STRIATAL DOPAMINE NEIJRONS

The following considerations are based on a unique property of the physiology of dopaminergic neurons. Electrical stimulation of the firing rate of nigro-striatal neurons results in increased dopamine synthesis and turnover (Nowycky & Roth, 1978; Roth Salzman & Nowycky, 1978). This

+ Abbreviations: dopamine-DA; homovanillic acid-HVA; cerebrospinal fluid-CSF: tyrosine hydroxylase-TH; multiple equilibrium states-MES; monoamine oxidase-MAO: alpha-methylparatyrosine-a-MPT.

DOPAMINE HYPOTHESIS OF SCHIZOPHRENIA 375

can also be achieved by blocking striatal postsynaptic DA receptors with neuroleptic drugs which results in an activation mediated by a striato-nigral feedback loop (Anden, 1972; Zivkovic, Guidotti & Costa, 1974). Surpris- ingly, a decrease or shutdown of the firing rate by the administration of y-butyrolactone or transsection of the neuronal pathway also results in a dramatic increase of striatal DA synthesis rate which persists despite synaptosomal DA accumulation (Anden, Magnusson & Stock, 1973; Wal- ters, Roth & Aghajanian, 1973; Kehr et al., 1972). This surprising observa- tion has been made in a number of different laboratories not only for nigro-striatal but also for dopaminergic neurons in the medial forebrain, the nucleus accumbens and olfactory tubercles (reviewed in Costa et al., 1975). This behavior seems to be unique to dopaminergic neurons. The DA synthesis rate can be plotted as a function of the neuronal firing rate as shown in Fig. 1. The biphasic dependency of dopamine synthesis on

x x’

FIG. 1. DA-synthesis (6) in nigrostriatal neurons as a function of firing rate (x’). The functon giving A2 for x satisfies the condition 4(0)>41$(x) required for the occurrence of multiple equilibrium states (see text).

impulse flow forms the basis for the mathematical formulations below. We have proven mathematically that a broad class of functions 4(x) will yield qualitatively very similar results. The proof is added to the manuscript as an appendix. The model only requires the existence of a negative feedback loop and a biphasic relationship of firing and DA synthesis rates, which is characterized by a certain minimum difference between 4(O) and 4(f) (see below). The model presented here is based on characteristics of nigro-striatal neurons. It is also applicable, however, to mesolimbic and cortical DA projections which have similar properties. The DA synthesis

376 R. KING ET AI..

rate is controlled through regulation of tyrosine hydroxylase, the rate- limiting enzyme in the synthesis of DA (Nagatsu et al., 1964; Levitt et al., 1965). In uiuo, the enzyme operates in the presence of such severely subsaturating pteridine cofactor concentrations (Lovenberg & Bruckwick, 1975) that probably only 5% of its potential activity is expressed under control conditions. We and others have shown that direct phosphorylation of TH by cyclic AMP dependent protein kinase leads to a dramatic activation of the enzyme which is associated with a 4-5 fold increase of its affinity for the pteridine cofactor and a several fold decrease in its affinity for the competitive feedback inhibitor DA (Raese er al., 1979a; Edelman et al., 1978; Vulliet, Langan & Weiner, 1979). We have also demonstrated (Raese et al., 19796) that TH can be phosphorylated and activated by a recently discovered cyclic AMP independent protein kinase, partially purified from rat brain, which has recently been shown to be activated by micromolar concentrations of calcium in the presence of phospholipids (Takai et al., 1979). These findings suggest that a mechanism may exist for a calcium sensitive phosphorylation of TH, which could directly link depolarization induced neuronal calcium influx and TH activation. There is evidence that TH phosphorylation may occur in viva (Letendre, MacDonnell & Guroffi, 1977; Waymire et al., 1979) suggesting that TH activation by impulse flow may be mediated via calcium-or cyclic AMP dependent phosphorylation of the enzyme.

I B) MATHEMATICAL F0RMUL.A I‘ION AND RESUl.TS

There is a current resurgence of non-linear methods in biology. These techniques utilizing catastrophe theory (Thorn, 1975) have been applied to the neurophysiology of the Wake-NREM-REM cycle (McCarley & Hobson, 1975), embryological development (Thorn, 1975), and, on a rather descriptive level, the study of anorexia nervosa (Zeeman, 1977), manic- depressive illness (Johnson, 1978), and schizophrenia (Woodcock & Davis, 1978). Our analysis focuses upon the following relevant features of dopaminergic transmission.

(1) Dopaminergic neuronal activity is modulated through feedback inhibition of firing, either mediated through other neurons (long loop) or by dopamine itself via dendrites impinging on neighboring DA cell bodies (Nowycky & Roth, 1978; Anden, 1972).

(2) DA is taken up from the synaptic cleft via a presynaptic pump. (3) DA synthesis in the presynaptic terminal is increased by increased

firing of the DA neuron and, paradoxically also by reduced firing as outlined above.


From these properties, we can derive the important variables and their differential equations. Let x’ be the firing rate of a DA neuron and let y’ be the concentration of DA in the synaptic cleft. We assume a linear relationship between x’ and y’:

x1=8-fly’. (1)

Here S is a measure of firing in the absence of feedback regulation and p is a measure of feedback (both pre- and post-synaptic). We assume /3 = VI/K1 where Kr is a binding constant for pre(post)-synaptic inhibition and VI is a constant of proportionality. We make the following assumptions about the dynamics of y’:

>“= cYM’x’-py’. (2)

Here p denotes the removal of synaptic DA by the combined effects of reuptake and metabolic breakdown, (Y, the fraction of DA released per impulse from the functional synaptic stores (Costa et al., 1975), and M’ is the concentration of DA in those stores.

We assume an equilibrium relationship between M’ and N’ (N’ is the concentration of DA in the pre-synaptic cytoplasm) of the form:

M’ = KN’. (3)

Also the kinetics of N’ are assumed to be governed by:

I+’ = 4(x’, y’) - dN’ (4)

where 4 is a synthesis rate and d a metabolic rate for N’. We allow 4 to be of the form:

(5)

to correspond qualitatively to experimental results (see Fig. 1). (2 is equal to the firing rate for which DA synthesis is minimal.) The quadratic in x’ is a consequence of the property of DA neurons that for both low and high firing rates, synthesis is increased. The hyperbolic in y’ allows for saturation of the inhibition of DA synthesis is by external DA.

The time orders of equations (2) and (3) are vastly different: l/p is of the order of seconds (Iversen ef al., 1975) while l/d is of the order of 10 min (Costa et al., 1975). Therefore, we limit our time interval to minutes and set equation (2) to equilibrium. Thus (YM’x’ -py’ = 0 and from equations (1) and (3):

y’= aKN’S Pa aKBN’ + p ’ X’=paKN’+p* (6)

378 R. KING ET Al.

Now equation (4) becomes:

i+ = qs(x’(N’), Y’(N’)) -dN’ = (L(N’). (7)

By studying the equilibrium points of equation (7) and their stability, we enter the realm of catastrophe theory. For certain values of the internal parameters LY, p, 6, p, Kz, etc., equation (7) will have a single equilibrium point and equation (7) will behave as though it were linear. But for other values of the internal parameters equation (7) will give rise to multiple equilibrium states (MES). We postulate that the occurrence of these multiple equilibrium states may give rise to some of the clinical phenomena we find in schizophrenia.

If we set equation (7) = 0, then N’ = d(.\-‘, y’)/n, or substituting this into equations (2) and (3) yields:

or

thus

CXK -+!i.x’,r-p~‘=0

x’[A+B(x’-.$]Kz pd K,+y’

-,K y’ = 0,

?!+l pdK2 \’ = () -7 .ucrK ’

(9)

110)

Now let

Then equation (10) becomes:

x[A +i(x - l)?]

Y+l -p’y =o. (11)

Substituting x = s’- by into equation (11) and solving for x we find:


This is a cubic equation with either one or three real roots. If it has three roots, then it can be shown that under the differential equation (7), two will be stable and one will be unstable. If it has one root, the equation will be stable. This property of the cubic indicates that our equation belongs to the “cusp” catastrophe, the simplest catastrophe of discontinuous changed. We have solved equation (12) numerically for a wide range of values for p, 6 and g In addition, we calculated and plotted the significant observable (output) variables of the DA neuron, namely:

I

x =?- the firing rate i

y’ = K2y, the synaptic concentration of DA

Yetr = py’ = &Z, the pre(post)-synaptic inhibitory effect of DA

$J = dN, the synthesis (turnover) rate of presynaptic DA.

These results are shown in Figs 2-5.

1 2 3 4 5 6

FIG. 2. The effect of the combined effects of DA uptake and degradation (6) on the firing of DA neurons (x). 8 is the spontaneous firing rate in the absence of feedback. The following values for p were chosen: curve 1 i = 0.04; curve 2 6 = O-09; curve 3 b = 0.39. Calculation was done using equation (12) with A = 0.001, p = 10 and B = 0.10.

380 R. KING ET AL.

I I 1 I I I I 0 1 2 3 4 5 6

8

FIG. 3. DA synthesis rate (~$1 as a function of firing rate. Parameters of 6 for curves 1-3 are identical to those given in the legend to Fig. 2

6t

. II lom2-’ I I 1 I I I I

0 1 2 3 4 5 6

FIG. 4. The synaptic self-inhibitory effect of DA as a function of firing rate. Parameter of p’ for curves l-3 are identical to those of Figs 2 and 3.


0 1 2 3 4 5 6

s

FIG. 5. Effect of DA receptor blockade on the DA concentration in the synaptic cleft. Curve 1 control; curve 3’ blockade of DA receptors. Y’ denotes the DA concentration in the synaptic cleft. A value for 6 of 0.04 was chosen.

In Fig. 2 we have plotted x vs. s’ (d is the firing rate of dopaminergic neurons in absence of feedback) for three values of p’. In curve 1 of Fig. 2, we see that for s’ between 1.3 and 3.3 there are three values of x for each L?. The upper and lower limbs are stable while the middle is unstable. For curve 2 with an increased p’, the multistable interval in g, is decreased. Finally, for curve 3 with an even larger p’; there are no MES present. Figure 3 plots q5 vs. c!? We similarly find the MES in curves 1 and 2 but not in 3. Notice also _that there is, if one compares curves 1 and 3 during the MES interval of 6, no particular increase or decrease of DA turnover. This agrees with the experimental results indicating no significant differen- ces in DA metabolites between schizophrenics and controls (see Dis- cussion). Figure 4 shows the effect of these different values of p on Yen vs. c?. One sees that in addition to the MES’s in curves 1 and 2, there is a net decrease in Y,,, as one increases p’. In Fig. 5 we have plotted the effect of a dopamine receptor blocker on y’ (y’ is the DA concentration in the synaptic cleft). For curve 3’, we have increased K1 (the binding constant

382 R. KING ETAL

for the post-synaptic receptor) and kept the ratio K2/K1 constant. Notice the large increase in y’, the synaptic concentration of DA and the disappear- ance of the MES. Curve 3 of Fig. 4 shows the effect of DA blockade on Y,,. There is also a net decrease in Y,.,.

FIG. 6. Three-dimensional representation of stable and unstable states of DA neuron. From trisection of axes F = 1 O-“‘5 to10~“5;~=0,~:r=Otol~h;~=10;B=~~~10.A-0~001. For explanation of curves 1. 2, 3 see text.

Figure 6 is a computer generated three dimensional plot of solutions to equation (12). It is homologous to Thorn’s cusp catastrophe surface. Curve 1 is outside the fold; hence shows only a single equilibrium. Curve 3 lies within the fold and demonstrates hysteresis. Curve 2 denotes the boundary between the stable and unstable regimes.

One can also derive formulae giving criteria for the existence of the MES when K2 is large. For taking the limit as K2 -+ co, equation (121 becomes

(13)

The discriminant of equation (13) determines the number of real solutions to that cubic equation. If we let

(14)

and

(15)

DOPAMINE HYPOTHESIS OF SCHIZOPHRENIA

Then the discriminant of equation (13) becomes:

383

If D > 0, there will be one real root. If D < 0, there will be three real roots. A necessary condition for D < 0 is a < 0, i.e.

(16)

So A/6 - 3 must be necessarily less than zero since G/‘lr3B is always positive. This will be satisfied, since 4(x) = A + B(,V -x)*, if and only if 4(O) > 44,(F). In addition, the smaller

- F=P= pdK,

B/3 BaKf2V1

is, the greater the range of values of & which will give rise to MES. We can now list the parameters which will decrease F, and thus increase the incidence of MES.

(1) &p = decreased DA reuptake, (2) ld = decreased MAO activity, (3) JK1 = hypersensitivity of post-synaptic DA receptors, (4) TCY = increased DA release (e.g. amphetamine), (5) fK = increased concentration of DA in synaptic stores.

Likewise, increasing p and K1 (e.g. through phenothiazines) and decreasing K (e.g. through reserpine) or B (through CY -MPT) all will increase F and tend to eliminate the MES.

We extend our theory to account for the spatial properties of dopaminergic firing within the substantia nigra; first we solve the continuum case, then we pursue in more detail the discrete spatial model. Experimental evidence shows that DA neurons have local dendrito-dendritic synaptic connections and project topographically to the caudate nucleus, which in turn reprojects topographically to the substantia nigra (Dray, 1980). These interactions are thought to be inhibitory. Adding these interactions to our formalism, we see that:

c

x’(u) = S(u)- J p(u - v)y’(v) d”v R”

(17)

jt'(u) = aM'(u)x'(u)-py'(u) (18)

&f’(u) = 4(“(‘), Y’(“))-dM’(u) K (19)

384 R. KING ETAI.

where /? is a kernal describing the local inhibition by a neuron at point UFR" on a neuron at L~FR”. Furthermore, we will assume that the process is at equilibrium;

~(,~',y')=A+B(.~'~-.i:)' 1

P(LL -LJ) =pe hll, L

and that the interaction is one-dimensional, then:

I

ixi

x’(u) = S(u)- be *,,, .( Ly4(X’(L’)J

dKp x’(c) dc

-3c 120)

a non-linear integral equation. Operating on both sides with respect to the operator

we have

or substituting p = 2b/A :

I

d’S 3 d“6 +dlc’=A-[Y(s’)]+iT. 122)

Equation (22) is formally identical to the time independent Ginzburg- Landau or generalized Klein-Gordon equation (Ginzburg & Landau, 1950). Assuming S(U) = 6 = constant, it can be integrated to give

1 dx”

!i du (-) =/a(x)-WaN]

where

thus

This solution describes a series of static “kink-solitons” (Bishop, 1979) or domain walls whose behavior between the walls is near the stable solutions to the non-spatial cubic equation Y(x’) = 0 analyzed in part 1


FIG. 7. Spatial variation of firing demonstrating soliton domain wall. X” is the firing rate of high firing cells; Xd is the firing rate of low firing cells and u is the distance in arbitrary units.

(see Fig. 7). These solitons-like solutions in the continuous cases suggest pursuing the analogy with condensed matter physics (e.g. ferromagnets) still further. If u and z7 are purely discrete variables i, j then equations (l), (2) and (3) at equilibrium become:

xl =s-py; - 1 P(i-j)yj =&-py; j#i

y! =&4(x;, Y;) where

Si = S - 1 p(i -j)yj. j#i

(25)

We now assume S, xl, yI are random fluctuating variables. Figure 2 demonstrated the MES case relating a folded curve S, and x’. Notice that there are two equilibria x, and xd for S1 < Si < &, i.e. the upper and lower legs of the curve.

Statistical mechanics offers us methods for approximately solving the random equations (25) and (26). These are coupled equations with each neuron i surrounded by a random environment. If the neuron’s environment tends to be on the lower leg of the curve, then the xi’s will be small and yj’s large for j near i. This results in & being smaller and neuron i will tend to be driven into the slower firing state. Similarly, if the surrounding neurons are firing rapidly, neuron i will also tend to be drawn into the more rapid state. Thus the dopamine nucleus is locally co-operative in the multi- equilibrium states even though in the single equilibrium state, the feedback

386 R. KING E7-AL

-py is negative and there is no co-operativity. To solve more explicitly equations (25) and (261 we shall approximate the interactions by the molecular (mean) field method (Brout. 1965). This method assumes that the random fluctuating neighboring interactions can be reduced to an average neighborhood interaction and has been successfully applied to such co-operative physical systems as ferromagnets, gas-liquid condensation, or superconductivity (Brout, 19651.

Here we assume J’ = (1.:) = the average value of y: so

Si=S-C P(i-j)v=S-Iv&=6 where M= C B(i--iI/@. ,#I ,*I

We assume that S(t) is a stationary gaussian-Markov process (Ornstein- Uhlenbeck process (Bharachu-Reid, 19601 with mean (6(t)) = 6 and covari- ante fl:

(6(0)6(f))-s2 = w-‘e l’.

In order to study the dynamics of our system, we need to calculate the rates of jumps from upper to lower limbs of the curve and vice versa. One can show (Mandl, 1968) that if z(t) is an Ornstein-Uhlenbeck process, with mean 0, then the first passage time to N > 0, i.e.

Min {x(t) 2 a) tends to be exponentially distributed with rate cu(a I = l/y(a) where

for a >> u

2ir ?,2,,,2 (r CT3 z--e‘ [ -+,+. ’ .I.

r ll (1

Now, if

r,, = rate of flip from x,/ to x,, and

rLf = rate of flip from ,Y,, to x,,

then

rd (Y(S)-S,)

r, a(S2-S’)

128)

(291

(30)

where


#=(&+&)/2 and A=&-&.

Now, if pyu = S’--x, and pyd = S’ - xd,

pU = probability that the DA neuron is in the “up” state and Pd = probability that the DA neuron is in the “down” state then:

If Y=P”yu+&iyd=LyU+L

ru + rd r, + rd yd’ (31)

2y-h-yd z=

Yd-yu

then by equations (30) and (31) one can show that

z = tanh A(&6) 1 &&+AflMy A/3Mwz T+-ln-

2Cr 2 &--6 -TF+TF 1 (32)

where - Yd+Yu

W=Yd-Yu,Y= 2 .

Define

and w’ = /3w = x, -xd then equation (32) becomes

z=tanh AMw’

H(6)+-pz . 3

This equation is identical to the self-consistent field equation for a ferromagnet or for condensation of a gas (Brout, 1965). The variable z which varies between - 1 and 1 is the analogue of the magnetization, H(S) is the external magnetic field and AMw’/4cr2 is the interaction energy J divided by kT.

For AMw’/4a2> 1 and H(S) = 0, equation (33), has two roots at fzo corresponding to a nucleus with predominantly fast firing or slow firing DA neurons. For AMw’/402< 1 equation (33) has only the root z = 0, an equal distribution of fast and slow neurons. Small changes in H(6) if AMw’/402 = 1 can cause sudden changes in the global equilibrium of the system, e.g. z. + - zo or vice versa. Figure 8 illustrates a computer generated

388 R. KING ET AL

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,.... . :.::::::.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . : .t

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . : : : : : . : : : : : : : : : : : : : : :

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

: . . . . . . . . . . . . . . . . . . . . . . . . . : ;

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , , : . : : : : : : : : : : : : : : : : : : : : : : : . : : : : : . : : : : : : : : : : : : : : : :

: . . . . . . . . . . . . . . . . . . . . . . :

: . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I. ,:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

: : . . : : . . . . . . . . . . . . . . . . . .

. : ..‘. ‘..““.“‘:::::::::::::::::::’ : : : : : : : : , . : : : : . : : : : : : : : : : : : . : : : : : : : , : : : : : : : : : : : : : : : : : : : : : ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , , , . , : . . : : . : : : : : : : : : . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

,,,,_,,

, . . . . . . . . . . . . . . . . . . . ;

. . . . . . . . . . . . . . . . . . . . . . . : : . : : : : : : . : . . . . . . . . . . . . . . . . . . . ..~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ . . . . . . . . . . . . . ,

, . . . . . . . . . . . . . . . . . . : : . . : : : : : : : : : : : : : . : : : : : : : : : : :

z : . : : : : . : : : : : : : : : : : : : : .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

i . . . . . . . . : . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

: : : : : : : : : : : : : : : . . : : : . : : . . . : : : : : : . : . : : : : : : : ; ,__,,,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :

:I::’ . . . . . . . . . . . . . . . , . . . , . : . : : : : . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

: ” . . . : : : : : : : : : : : : : : : : : . . . . . . . . . . . . . . . . . . . . . . . . . . .

. : : : : : : : : : : : : : : : : : : : : : : a . . . . . . ,

: . . . . . . . . . . . . . . . . . . . . . . . .

. . . _),_,,,,_ , , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,

. . . : : . . : . . . . . : . . . . :

. . . . . . . . . . . . : .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t =95 t = 145


presentation of locally cohesive clusters for different times predicted by the Ising model for ferromagnetization.

Let us calculate the parameter AMw’/4~r* for the specific model 4(x’, y’) =A +B(x’-a>*. Then from equation (13)

( A x3-2x2+ 1+7+ pd

BZ nKfiBT* ’ >

This would give a curve like that of Fig. (2) if

where

da’ + cyK@Bf3- . (34)

and

p2:2A 2 pd pdcY 27 3%%(rK~BP2-aK~BZ3*

If we let

F= pd A aK/3Bf2

and G=- BX2

then for the two limits of the curve S1 and S2

S,=(&+G)/F+$+2F”* (: : G-$)3’2

and

&=(&+G)F+$-2F”* (;+q3’“*

Then

A=S2-6,=4Fr’* (: I G;i)“*

(35)

(36)

FIG. 8. Monte Carlo simulation of kinetic Ising model for sudden reversal of magnetic field at various times for a temperature below the critical point. This illustration is analogous to cluster formation in the substantia nigra in response to a sudden change in external stimulation, if the conditions for MES are met. See text.

390

or

R. KING ET Al..

If F, G<< 1, then

Therefore

hlkhl Mu-’ Sz7=----- 4a- 27Fc"

One sees that all the manipulations that decrease F will increase the value of S and drive the system below its “critical temperature.” In addition, the value of A4 increases, i.e. the strength of the increasing neighbors will also lower the apparent temperature.

In the language of critical phenomena, the transition from the “normal” to the “schizophrenic” state is a second order phase transition producing symmetry breaking into fast and the slow firing states. In addition there is a first order phase transition separating the fast and the slow firing modes. Since the process is identical to the kinetic Ising model (Binder, 1973) one would expect the formation of clusters of high or low firing neurons (Binder, 1976). We show in this paper that such an equilibrium configuration is possible in the one-dimensional case with its “kink” solutions. Near the transition between normalcy and multiple equilibrium states one would find an increase in long range spatio-temporal clustering and fluctuations of firing.

3. Discussion

The mathematical modeling of functional parameters of dopaminergic neurons as outlined above suggest the occurrence of several simultaneous equilibrium states with the potential for transitions between these states corresponding to the simplest of the seven elementary catastrophes (Thorn, 1975), the so-called “cusp” catastrophe. Changes in several parameters will increase the likelihood of transitions between MES. This is predicted for increasing values of & for example by stress, which is known to increase the firing rate of catecholaminergic neurons. Similarly, a decrease of the firing rate below a critical value will increase the probability of transitions between MES (see Fig. 3, for values for s’ of 1.5 and 3). The time frame of these transitions is governed by equation (4), which deals with the


synthesis and degradation of DA. Since the activation of TH by impulse flow takes place within a few minutes, the transitions between MES occur in a similar time frame following a critical perturbation. We hypothesize that the occurrence of MES may form the pathophysiological basis of some aspects of the schizophrenic thought disorder. This hypothesis generates specific predictions about the biochemical mechanism(s) involved. First, the activation response of TH must reach activity levels of at least 4 fold above control or more (see Fig. 1). Furthermore, the area of the unstable equilibrium state correlates positively with increasing degrees of TH activation. Second, abnormally low MAO activity is predicted to lead to the occurrence of MES if the TH activation response meets the critical value of 4 fold.

An intriguing consequence of the coexistence of MES is the great variety of possible spatial arrangements of neurons which are locked into divergent firing rates in accordance with their MES status. Our interpretation of the spatial properties of dopaminergic firing in terms of both static “kink solitons” (Bishop, 1979) and the kinetic Ising model of thermodynamics (Binder, 1976) predicts the formation of coherent clusters of cells locked into high or low firing states. Thus, a dopaminergic nucleus, i.e. the S. nigru may fragment into several “nuclei” in terms of the frequency domain of firing which are controlled by anatomical or functional order. Furthermore, these clusters may dissolve, reorganize, and/or move within the neural net. It is easy to see that this phenomenon results in asymmetric and erratic dopaminergic input to the terminal fields in the frontal cortex, extrapyramidal system and limbic areas. Such inhomogeneous spatial distribution can be visualized using the analogy of boiling water where we observe the coexistence of gas and liquid phases in a given volume. The multiple equilibrium state will only be entered if the dopamine synthesis rate (4) as a function of firing rate (x’) satisfies the equation 4(O) 244(f) (see Fig. 1). Experimental data obtained for rat striatum indicate that blocking of impulse flow by administration of y-butyrolactone (GBL) results in fact in a 4-5 fold increase in DA synthesis (Nowycky & Roth, 1978). Similar degrees of activation of DA synthesis by GBL have also been found for cortical DA neurons (Kehr et al., 1972). We have shown that the increase of DA synthesis as a consequence of decreased neuronal firing is not dependent on any particular function of C&(X) or reuptake function p(y) in order to allow for the mathematical treatment described in Results (see Appendix). Rather, broad classes of functions 4(x) and p(y) (or shapes of the curve in Fig. 1) are permitted as long as a critical value of 4 for low values of x is obtained. The theory is therefore critically dependent on only one observed property unique to dopaminergic neurons,

392 R. KING ET AL.

i.e. the biphasic stimulation of dopamine synthesis by low and high firing rates. The theory is internally consistent with known clinical effects of several drugs. Improvement is predicted as a results of reserpine or neuroleptic drug treatment and worsening of the clinical states by the administration of dopaminergic stimulants such as amphetamine. Further- more, the model does not predict an increase in the concentrations of DA or its metabolites HVA in brain tissue or CSF. This latter finding is consistent with previous work on postmortem brain and CSF of schizophrenics (Bacopoulos et al., 1977; Persson & Roos, 1969). In addition, tyrosine hydroxylase levels are expected to be normal in dopamine- containing brain areas, which is also in agreement with postmortem data (Barchas, Elliott & Berger, 1978). The model is consistent, however, with the prediction of an increased response of the enzyme to activating conditions, e.g. exposure to protein kinases and MgATP, which lead to a reduction of its Km for pteridine cofactor. In fact, the likelihood of transitions between equilibrium states correlates positively with increasing degrees of tyrosine hydroxylase activation. The in oivo phosphorylation of the enzyme in response to alterations in impulse flow has not yet been demonstrated. The kinetic changes, however, are identical to those pro- duced by in vitro phosphorylation of the enzyme. It would be worthwhile to re-examine human postmortem brain tissue with regard to the response of tyrosine hydroxylase to phosphorylating conditions.

Several predictions emerge from our mathematical model, some of which can be tested experimentally. If we can assume that the amphetamine psychosis reproduces at least some elements of the schizophrenic process, one can study the effect of chronic amphetamine administration of the activity of DA nuclei (i.e. S. nigra) in animals. Cluster formation can be detected by coherence pattern in simultaneously recording electrodes. This effect should be reversable by neuroleptic drugs. We have carried out experiments utilizing cats with electrodes chronically implanted in the S. nigra and amygdala and were able to observe hypersynchrony and increased coherence which could be normalized by haloperidol. This finding is consistent with our model. In addition, we have preliminary evidence that schizophrenic patients have a defect in their ability to synchronize the so-called oscillatory potential (OP’s) elicited from the retina by repeated light flashes. These OP’s are critically controlled by DA in amacrine cells. This increased variance in response to light flashes in schizophrenics may be due to a “critical slowing” in the dissolution of metastable clusters in dopaminergic amacrine cells (Raese, King & Barchas, to be submitted). We are pleased to note that others (Mandell, 1980) are also investigating statistical mechanical models of mental illness.


Dopamine has been implicated as a transmitter involved in mediating attention (Matthysse, 1978) and the theory has been proposed that some aspects of schizophrenic thought disorder may be the result of an attention deficit (Holzman, Levy & Proctor, 1978). The model proposed in this paper is not only compatible with the attention deficit theory, but also suggests a pathophysiological mechanism for it, since a fragmentation of dopaminergic transmission would prevent sustained attention.

A second psychophysiological consequence of fragmented DA transmission could be a disturbance of affect. DA has been shown to be involved in the modulation of affective states as indicated by the role of DA in the “reward systems” of rat brain (Stein, 1978). Perturbation of dopaminergic transmissions in such a system could account for the anhedonia and affective blunting of affect in schizophrenia. It is also interesting to note that some patients suffering from Parkinson’s disease, who receive long-term L- DOPA treatment, experience sudden (within minutes) transitions from hypo- or akinetic to hyperkinetic states and vice versa (Barbeau, 1974).

Finally, it should be pointed out that the hypothesis suggested in this paper is compatible with a possible biochemical heterogeneity of schizophrenia. Increased susceptibility of TH to activating mechanisms, deficient DA reuptake or metabolic breakdown by MAO, and increased postsynaptic DA receptor number and/or sensitivity could all lead to the common pathway of transitions between multiple equilibrium states of dopaminergic activity.

We thank Charles Restivo for his expert help in writing the computer programs. We also immensely appreciate Dr. Kurt Binder who supplied us with the Monte Carlo simulations to Fig. 8. This work was supported by NIMH Program-Project Grant, MH 23861, and by the Office of Naval Research.

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APPENDIX

Theorem : 1. Suppose a(x) is monotonically increasing; C”(R); and (Y(O) = 0,

a’(p)>O. 2. Suppose 4 (A, x) 2 0 is C”(R’); 4(A, x) and a&(A, ,Y) are convex in

A, for all xlf; &+(A,f)=O; &&&(A,~)>O, q%(A,,y)z4(A,i) for all A andforX;~(O,X)=~(O,k)forallx;~(A,j)=~(~)forallA.

3. Assume 31(6, p, x) E C”(R3) s.t. Vx > OVy > 0 and Vm < 036, p s.t. y = 1(S, /3, x) and m = SJ(S, /3, ,Y). Suppose furthermore that I(S, p, x) is monotonically decreasing to 0 as x + co .

Then if $(A, 8, j3, x) = a(x)4(A, x) - f(6, p, x), we can prove that:

3A1, B, 5, s.t. Vm = I’(& b, jJ), VA > A13x1, ,y2, x3 s.t..

0 <XI <x2 <X <x3 and +A(XI) = 4dx2) = 44dx3) = 0.

Proof: Step 1: (We denote 4 (x) = 4 (1, ,Y) and suppress S, p, A where obvious in $; (Y, q!~

and I) 1,4(o) = -I(O) < 0. Also

$A(x) = -1(o) +x[$‘(~A)] for some 0 < (A < x (Al)

by the mean-value theorem. By the convexity of 4 and 4’ we have

diA)<q ( ‘) + 1 -a ‘#&A)

and

396 R. KING ETAL..

But ~cJ(~A) =4(X) and bb(la) = 0 so that inequalities imply for A> 0 that

~A(~A)~(~-A)~(X)+A~(~A) (A21

4,6Ka) 2 &‘KA). (A31

Now by equation (Al) and the definition of JI and the mean-value theorem:

+AI~‘(~A)[~(~A)-~(X)I+~(~A)~‘(~A)}I (A41

using inequalitites (2) and (3) and given that LY’, (Y are > 0. Define p(x) = Q’(Xh#JW -l’(x), ~~x~=~y’~x~C~~x~-~~X~l+~~x~~‘~x~ P and n are C”(R) and p(O) = -l’(O) > 0 and ~(0) = (4(O) - 4(i))a’(O) > 0. Now choose a1 s.t.

Xf(O, &L P(O)

P(X)>--- T(O)

2 ’ 7r(X)>---

2 and S,<X.

Then since 0 C LA < x, we find that

Substituting these into inequality (5):

GA(X)?-[(0)+X

Let x = S,/2 we see

or

A>I(w61Pw4~A, dOhf1

1

4


Step 2: Choose g, p’ s.t. 1cs: I43 = 2 ~(fM(X), m = /ycq fi,f) then

J/CA, s: fi, c?) = --aW4(2)<0 Step 3: Choose x’ s.t.

this is possible since

Therefore combining steps 1, 2 and 3, we find

+bA(i) < 0, and $A(x’) > 0.

Thus

3x1, x2, x3, o<X1<~<X2<~~X3~X1S.t.

clA(xl)=~A(X2)=~ACY3)=0

A > AI, by the intermediate-value theorem. We generalize our dynamics to the following:

y = h (6, p, x), h monotonically 4 to 0, S = external stimulation and p = feedback parameter. .

it =~(Xb(Yw-P(Y) (A3

M=KN bw

I+ = @Ak>r(y) - dN (A7)

where p is monotonically fp(0) = 0, s is semi-monotonically l, s(O) = 1 and r is semi-monotonically J; r(0) = 1; (Y is monotonically T; (Y(O) = 0. At equilibrium we get

o = dy)r(y)K

d ol(xhbA(x)-$ & I .

(A9)

398 R. KING ET AL

Define

then

d,lA h’(6,8, x) . -!?-EL+e <() s’r sr’ sr 1

since s’, r’, 5 0, p’ > 0 and h’ < 0.

Now define ~(6,p,A,~)=cu(x)~~(x)-1(6,8,)(‘)

$ satisfies the conditions of the above theorem if 38, p, s.t. 1(6, /?Z, x-1 > a(j)qb(i). In such a case we have xl, x2, xs s.t. +a(xr) = $a(x~) = $A(XJ) = 0. These are the three equilibrium pts of our dynamics. Now we must consider the stability of equation (A7) at these pts. We assume equation (A5) is at equilibrium.

Now equations (A5) and (A7) can be written utilizing ~1 = h(6, 0, x) as

3 =&(,‘)N-p(y) (AlO)

ti=zo+dN. (All)

Case 1: Assume equation (All) is at equilibrium and equation (AlO) is dynamic, then

q= G(Y)Z(Y) d

-po’)=K(y)

and

K’(y)1 equilihrlum =

Case 2: Assume equation (AlO) is at equilibrium and equation (Al 1) is dynamic, then

dp(y) &r=z(y)--= cu(y)

g(N); N=P cu

and

But

~~=~=(~/~)s(y)+u(y)s’(1.)<0 and p’>O

DOPAMINE HYPOTHESIS OF SCHIZOPHRENIA

so dN/dy > 0.

399

=t &‘+z&&‘)=~K’ a{

dp = 6

since - da? I

2. equilibrium

therefore

g’W K’

equilibrium = I G W/4) eg

and

Now

sgn WL, = sgn (KL. 6412)

(Y(X)S(YK K(y)= d 4dx)r(y)-dy)

= s(y)&(y% (/YMak) - I(& P, XH

s(y)rbK K(y)= d WA, 4 P, x)1

=f(yMx)~

so

=f(YM’(x)l~‘(x).

Therefore

sg:n K’leg = -sgn $leg. (A13)

Combining equation (A12) and (A13) we have

sgn WL, = -en (+‘Lg . 3w g’l,, 5 0

sgn g’l,, 2 0 and sgn g’l,, 5 0 since 9 is increasing at ,yl, decreasing at ,y2 and increasing at x3. Therefore VE 3g*, fi = g*(N) is stable at x1, unstable at x2 and stable at x3

lk-g*ll<s llg-g*ll=suNp{l(g-g*)(N)I+I(g’-g’*)(N)l}.

400 R. KING ET AL

NOTE ADDED IN PROOF

We have recently proved (P. Hagan & R. King, unpublished) for non-symmetric input 6, that the domain wall solution illustrated in Fig. 7 is in fact a solitary wave. It moves to the right or to the left depending upon whether the high firing or low firing state is preferred, respectively.

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