Catastrophe Theory of Dopaminergic Transmission: A Revised ...

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,

Department 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

Behavioral

University

requests to: Joachim

D. Raese, M.D.,

Nancy Pritzker

Laboratory

of

Neurochemistry,

Department

of Psychiatry

and Behavioral

Sciences, Stanford

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 ratelimiting 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 pathophysiology 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 dysrhythmic, 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;

tyrosine

hydroxylase-TH;

multiple

alpha-methylparatyrosine-a-MPT.

homovanillic

equilibrium

acid-HVA;

states-MES;

cerebrospinal

fluid-CSF:

monoamine

oxidase-MAO:

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). Surprisingly, 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; Walters, Roth & Aghajanian, 1973; Kehr et al., 1972). This surprising observation 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 ratelimiting 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 associatedwith 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), manicdepressive 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.

DOPAMINE

HYPOTHESIS

OF

SCHIZOPHRENIA

377

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¡¯

where 4 is a synthesis rate and d a metabolic

be of the form:

(4)

rate for N¡¯. We allow 4 to

(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)

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