A nonlinear neural population coding theory of quantum cognition and decision making

doi:10.4236/wjns.2012.24028

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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/)

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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

Email: [email protected]

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)

where





i(, ;0,xabi1)



 is a probability at which

event i

occurs and





Px|y

is a conditional probability

of eventi

given event

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

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:



,,,

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.

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

( ),(),,() (,,)

fpxfpxfpxxx (2-2)

where each componenti



px x(i is a positive in-

teger no larger than n) of the vector

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

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:





pxxf px



 





(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 ):

: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:



 



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):





xx



. In other words, we here assume Stevens’

power law in psychophysics [22]. Then, Eq.2-3 reduces



12 1

,,,

px pxpxpx







i

(2-8)

Then, the subjective quantity from the input (or output)

state vector is (from Eq.2-4)







(, ,,

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



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:



 

12 1

22,

pzcpxcpycpx

cpy pxpy

 



(the last term is an inner product of and ).

px py

 



cpx cpy

ccpx py os





 (2-11)

where



is an angle between vectorsand . px py

When we put





1,pzP a



,cPb

 

,cPb





2Ppaxb10

, and





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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