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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