World Journal of Neuroscience, 2012, 2, 183-186 WJNS
http://dx.doi.org/10.4236/wjns.2012.24028 Published Online November 2012 (http://www.SciRP.org/journal/wjns/)
A nonlinear neural population coding theory of quantum
cognition and decision making
Taiki Takahashi1, Taksu Cheon2
1Department of Behavioral Science, Center for Experimental Research in Social Sciences, Hokkaido University, Sapporo, Japan
2Laboratory of Physics, Kochi University of Technology, Kochi, Japan
Received 15 August 2012; revised 27 September 2012; accepted 12 October 2012
ABSTRACT
Mathematical frameworks of quantum theory have
recently been adopted in cognitive and behavioral
sciences, to explain the violations of normative deci-
sion theory and anomalies in cognition. However, to
date, no study has attempted to explore neural im-
plementations of such “quantum-like” information
processing in the brain. This study demonstrates that
neural population coding of information with nonlin-
ear neural response functions can account for such
“quantum” information processing in decision-mak-
ing and cognition. It is also shown that quantum deci-
sion theory is a special case of more general popula-
tion vector cording theory. Future applications of the
present theory in the rapidly evolving field of “psy-
chophysical neuroecon omics” ar e al so dis cussed .
Keywords: Quantum Probability; Psychophysics;
Neuroeconomics; Population Coding
1. INTRODUCTION
Recent years witnessed a rapid growing of the appli-
cations of mathematical frameworks of quantum physics
[1-3] and quantum information theory [4] in psychology,
cognitive science, behavioral science, and economics
[5-11]. One of the advantages of the utilization of “quan-
tum” theoretical frameworks is that quantum probability
theory can describe the violation of some classical pro-
bability laws such as the law of total probability [10].
Although we claimed the importance of the explorations
of possible cognitive mechanisms underlying such
“quantum”—like behavior in human decision and cog-
nition [7,8], no study to date addressed this issue. This
point is important, because it is not very probable that
quantum mechanical effects appear in the brain under
normal physical conditions at body temperature [12].
We now show a simple example of the violation of
classical probability laws in human decision making and
cognition (corresponding to the violation of the Sure-
thing Principle in decision theory proposed by [13]). The
law of total probability states that





  
101011
||PaPbPa bPbPa b
1
(1-1)
where


Px
i(, ;0,xabi1)

 is a probability at which
event i
x
occurs and


Px|y
ij
is a conditional probability
of eventi
x
given event
j
y(,; 0,yabj1).
Mathematical psychologists Tversky and colleagues ex-
perimentally demonstrated that humans violate the law of
total probability in their probabilistic choice (referred to
as “disjunction effect”, [14,15]) and probability judgment
(referred to as “conjunction fallacy”, [16]). The simplest
quantum formalism for modeling the violation of the law
of total probability is [9]:






 

 
101011
10 10 11
||
2||
PaPbPa bPbPa b
Pa PbPa bPa b
1
cos


 (1-2)
Where

is a “quantum phase” parameterizing the de-
gree of the violation (“quantum interference”) of the law
of total probability (see [7,8], for more complete quan-
tum formalism with “composite” system setting). It is to
be noted that Eq.1-2 is obtained through “Born’s rule” in
quantum probability theory (see [1], for a standard re-
ference of quantum theory).
2. POPULATION CODING THEORY OF
QUANTUM COGNITION AND
DECISION-MAKING
In neuroscience of sensory and motor systems, it has
been established that information of sensory input and
motor output is, in many cases, encoded at the neuronal
population level rather than individual single neuron
level [17-20]. Let us start our current investigation into
the neural foundation of quantum decision theory from
this empirical observation. Suppose that (scalar) physical
input (or output) signal (from sensory organs or to motor
systems) activates a neuronal ensemble (a population
vector) consisting of n neurons, even when the intensity
(magnitude) of input/output signal is a single real scalar

T. Takahashi, T. Cheon / World Journal of Neuroscience 2 (2012) 183-186 184
parameter. The physical input to (or output from) the
ensemble of n neurons (“population state vector”) can be
expressed in a vector form as:

12
,,,
n
pxpx pxpx (2-1)
where 1is a real number which indicates the input to
single neuron i in the neural population (ensemble)
consisting of the n neurons.
px
The important point here is that even when the signal
is a (real) scalar number (e.g. time, probability, the in-
tensity of sound and light), the corresponding input to the
neural ensemble system is in a vector form (of which
components are real numbers) in Eq.2-1, in mathema-
tical terms.
Then, the n neurons’ response vector for the neural
population is

12 1
( ),(),,() (,,)
nn
,
x
fpxfpxfpxxx (2-2)
where each componenti

i
f
px x(i is a positive in-
teger no larger than n) of the vector
x
is (real scalar)
each single neuron’s response (e.g., a change in firing
rate [Hz], the concentration of the product of induced
biochemical reactions [nmol/L], note thati
x
could be
negative) of neuron i in the neural population consisting
of n neurons. Notably, the function f is generally non-
linear (see standard neuroscience textbooks such as [21]).
We can now assume that the real scalar “intensity”
of the neuronal ensemble’s response to(a total sum of
all the n neurons’ activations) is:
Φ
px


11
nn
i
ii
pxxf px

 


i

(2-3)
This assumption is natural in that the result of the
neural population activity may be, for instance, an in-
crease in the level of a specific type of some neuro-
transmitters (e.g., glutamate, GABA, dopamine and sero-
tonin) or hormones (cortisol, testosterone, and oxytocin)
in the brain region containing the neural ensemble con-
sisting of n neurons.
Note that, as accumulating neurophysiological and psy-
chophysical evidence suggests, this intensity of neural
populational response may linearly correspond to sub-
jective (or psychological) intensityof the physical
(scalar) stimulus input [19]:

 
px kpx (2-4)
Let us then consider two distinct types of the popu-
lation state vectors (input or output vectors to the neural
ensemble, note that this “state” is a state of input/output,
not the state of neuronal responses)and and a
linear combination of the two population state vectors
(with real weighting coefficients and ):
px
c
py
1
c2
12
:pzc pxcpy (2-5)
Here we can ask, what is the physical intensity of the
neuronal ensemble’s response to the linearly-combined
input (or output) state vector? The answer is: pz

 

12
12
1
n
ii
i
pzc pxcpz
cpxcpy

 

 (2-6)
Here we should notice that


12 12
ccc pxcpypxpy 

(2-7)
The reason for this inequality is that f is generally a
non-linear function. As we will see later, the violation of
the linear additivity of neural ensemble’s response shown
by Eq.2-7 is the mathematical root of the “quantum in-
terference” in human decision-making and cognition.
For more intuitive understanding, let us assume that f
is a power function corresponding to a psychophysical
quantity (such as subjective probability and preference):


s
f
xx

. In other words, we here assume Stevens’
power law in psychophysics [22]. Then, Eq.2-3 reduces
to

12 1
,,,
s
n
n
i
px pxpxpx



i (2-8)
Then, the subjective quantity from the input (or output)
state vector is (from Eq.2-4)



12
1
(, ,,
n
s
n
i
i
pxkpx pxpx
kpx



)
(2-9)
When we consider the special case of 2s

with
setting 1k

(without losing generality, because psy-
chophysical quantity

is in an arbitrary unit), Eq.2-9
reduces to
 
2
1
n
i
i
pxpx px

 
 (2-10)
Here we adopted the standard definition of the “norm”
(length) of the vector: px in linear algebra. In this
case when s = 2, let us again consider a linear combi-
nation (“superposed”) neural populational input (or out-
put) state 12 pzcpxcpy

 (defined in Eq.2-5). The
subjective quantity (with Stevens’ exponent s = 2) induced
by this superposed state is:

 
22
12 1
22
22,
pzcpxcpycpx
cpy pxpy
 

(the last term is an inner product of and ).
px py
 

22 22
12
12
2c
cpx cpy
ccpx py os


 (2-11)
where

is an angle between vectorsand . px py
When we put


1,pzP a
 
10
,cPb
2
 
2
21
,cPb


2Ppaxb10
, and


11
2Ppayb, Eq.2-11 is the same as Eq.1-2 which

T. Takahashi, T. Cheon / World Journal of Neuroscience 2 (2012) 183-186 185
often appears in quantum decision theory [9,10]. Taken
together, it can be concluded that so-called “quantum de-
cision theory” is a special case of more general nonlinear
population coding theory of neural information (i.e.,
Eq.2-1 and 2-3) in which Stevens’ exponent is fixed at s
= 2. When s is an integer larger than 2, there appears
more interference terms. It should further be noted that
the present theory removes the necessity of quantum
physical effect (and associated complex-numbered vec-
tors in the Hilbert space) in the human brain in explain-
ing the seemingly “quantum-like” phenomena in human
cognition and decision making. Also, psychophysical
experiment demonstrated that subjective intensity of muscle
force follows Stevens’ power law with the exponent
(
1.7s
http://www.cis.rit.edu/people/faculty/montag/vandplite/
pages/chap_6 /ch6p10.ht ml) which is close to 2, support-
ing our present hypothesis on human choice behavior.
3. IMPLICATIONS OF THE PRESENT
THEORY TO NEUROECONOMICS
AND DECISION NEUROSCIENCE
Rapid advances in neuroeconomics suggest the impor-
tance of psychophysical considerations for proper the-
ories in decision neuroscience (we can call it “psycho-
physical neuroeconomics”, [23-26]). For instance, ano-
malies in human decision making (i.e., deviations from
normative decision theory or axioms in microeconomics)
such as preference reversal over time in intertemporal
choice has been explained by nonlinearity of subjective
time in terms of physical time [25-28]. Therefore, future
studies in neuroeconomics and decision neuroscience
should incorporate the nonlinearity arising from popu-
lation vector cording of decision parameters (e.g., utility
function, psychological time, subjective probability, pro-
bability weighting function), by combining neuroeco-
nomic theory and quantum theory of cognition and de-
cision.
4. ACKNOWLEDGEMENTS
The research reported in this paper was supported by a grant from the
Grant-in-Aid for Scientific Research (Innovative Areas, 23118001;
Adolescent Mind & Self-Regulation) from the Ministry of Education,
Culture, Sports, Science and Technology of Japan.
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