# Thermodynamic properties and Hilbert space of human brain

Dongmei Shi1,2,3 Meng Li4 Martin Walter4 Hamid R. Noori1,5,6 1 Max Planck
Institute for Biological Cybernetics, Tübingen 72076 Germany,
2 Institut für Neuromodulation and Neurotechnologie, Universitätsklinikum
Tübingen, Tübingen 72076, Germany,
3 Department of Physical Science and Technology, Bohai University, Jinzhou
121000, China,
4 Klinik für Psychiatrie und Psychotherapie, Universitätsklinikum Jena, Jena
07743 Germany
5 McGovern Institute for Brain Research, Massachusetts Institute of Technology
Cambridge, MA 02139, USA,
6 Laboratory of Multiscale Dynamics of Brain Disorders, Center for Excellence
in Brain Science and Intelligence Technology Institute of Neuroscience,
Chinese Academy of Sciences, Shanghai 200031, China

###### Abstract

Any macro system consists of a lot of micro particles. According to
statistical physics the macro properties of a system are realized as the
statistical average of the corresponding micro properties. In our study, a
model based on ensemble theory from statistical physics is proposed.
Specifically, the functional connectivity (FC) patterns confirmed by LEiDA are
taken as the micro states of a system, and static functional connectivity
(SFC) is seen as the macro state. When SFC can be written as the linear
combination of these FC patterns, it is realized that these FC patterns are
valid micro states of which the statistical results of relevant behaviors can
describe the corresponding properties of SFC. In this case, the thermodynamic
functions in ensemble theory are expressed in terms of theses micro states. We
applied the model to study the biological effect of ketamine on brain, and
proved by the maximum work principles that compared to the control group, the
capability of work done of brain that has been injected ketamine has declined
significantly. Moreover, the mechanic operator of brain is further studied and
a Hilbert space which is spanned by the eigenvectors of the operator is
gained. The confirmation of mechanic operator and Hilbert space has opened big
probabilities with quantum mechanics of studying brain system, which would
unfold a new era in relevant studies of neuroscience.

##  I Introduction

Human brain is a complex system consisting of functionally and structurally
interconnected regions. It is an advanced functional system that performs
diverse cognitive functions in a constantly changing environment brain1 ;
brain2 . Even when no task is performed, the brain displays the spontaneous
waxing and waning of meaningful functional networks on a slow time scale rest1
; rest2 ; rest3 . Understanding the relation between brain architecture and
function is a central challenge in neuroscience.

In the past decades, structural and functional neuroimaging studies have
provided a lot of knowledge about primate and human brain rest2 ; neuim1 ;
neuim2 ; neuim3 ; neuim4 ; neuim5 ; neuim6 . In the meanwhile, computational
network analysis has provided insight into the organization of large-scale
cortical connectivity in several species, including rat, cat and macaque
monkey cnet1 ; cnet2 ; cnet3 ; cnet4 . In human brain, the topology of FC
patterns has been widely investigated FC1 ; FC2 ; FC3 ; FC4 , and key
attributes of these patterns have been characterized across different
conditions of rest or cognitive load. FC FCdef1 ; FCdef2  is defined as the
temporal dependence of neuronal activity patterns of anatomically sperated
brains. Many studies have shown the feasibility of examining FC between brain
regions as the level of coactivation of functional MRI time series measured
during rest FCdef3 . The resting-state activity has been proposed reflecting
the spontaneous activation and deactivation of different network
configurations supported rest4 ; rest5 ; rest6 ; rest7 , resulting in a
constant reconfiguration of FC patterns over time. However, this pattern can’t
be captured by the traditionaly static functional connectivity SFC analysis
for which BOLD signal correlation is computed over the entire recording
session asfc1 ; asfc2 . New methods have been proposed to calculate the FC
patterns namely phase coherence connectivity or multiplication of temporal
derivatives which allows for a higher temporal resolution, but is more
susceptible to high-frequency noise fluctuations newm1 ; newm2 ; newm3 ; newm4
; newm5 . To overcome this difficulty, J. Cabral $et$ $al$ cabral  focused on
the dominant FC patterns captured by the leading eigenvector of BOLD phase
coherence matrices, investigating how the switching behavior of resting-state
FC patterns related with the cognitive performance. Recently, Deco $et$ $al$
neuim6  proposed a possible definition of brain state defined as an ensemble
of ’Metastable substates’ which are actually originated from the FC patterns.

Inspired by the relation that SFC is presented by the linear combination of FC
patterns from the Cabral’s work cabral , we proposed a model in which the FC
patterns are defined as the microstates and SFC is seen as the macrostate of a
system. When the linear relation between SFC and FC patterns is satisfied, the
thermodynamic properties of SFC can be described by FC patterns by means of
ensemble theory. In Deco’s neuim6  work, a brain state is defined as an
ensemble of FC patterns (or ’Metastable substates’) from a macroscopic scale,
whereas in present study is defined as an FC pattern from the microscopic
scale. These two definitions can be connected by statistical physics which
provides disciplines to describe the macro properties of a system by analyzing
the corresponding micro properties, which is meanwhile the basic idea of our
model. We applied the model to study the ketamine effect on human brain and
compared the results with those of controlled group. Significant differences
have been found between the two groups. In addition, work done and heat
conduction happened to the brain are studied by means of the first law of
thermodynamics. We further investigated the mechanic operator of brain system
and built a Hilbert space which is spanned by the eigenvectors of the
operator. This work has provide new research perspectives of studying
neuroscience from which thermodyanmics, statistical physics and quantum
mechanics could be applied. Moreover, because our model mostly depends on the
properties of the time series, besides the brain system it can be extensively
applied to other complex systems as long as the time series meet the
conditions specified in our model.

The data adopted in our work are collected from 31 participants from the
kematine group, and 30 participants from the controlled group in which the
placebo is injected. Each participant experiences three stages: Baseline on
which the subjects have not been injected the ketamine (or placebo), Stage1 on
which the subjects have been injected the ketamine (or placebo) after one
hour, and Stage2 on which the subjects have been injected the ketamine
(placebo) after 24 hours.

##  II Experimental Data

  1. 1.

Study design and Participant

The data set was collected from a completed randomised, double-blind, placebo-
controlled study in university of Magdeburg (Magdeburg, Germany), which was
designed to investigate the biological effects of ketamine (more details in
the registered clinical trial at the European Union Drug Regulating
Authorities Clinical Trials (No. 2010-023414-31). The study design has been
reported in our previous studies data1 ; data3 . In a brief, three
measurements at baseline, 1h and 24h post-infusion were collected in each
participant. Following baseline fMRI scan, each participant were randomised by
age and sex to a 40 min continuous infusion of either (R,S)-ketamine (0.5
mg$/$kg) or 0.9$\%$ saline via an infusion pump outside the MRI scanner. The
second MR scan was started at 1h post-infusion and the resting state fMRI scan
was performed approximated around 100 min post-infusion. The third fMRI scan
were acquired 24h after their associated baseline measurements. During
ketamine administration, participants may experience transient dissociation
and hallucination, as well as increased heart beats and blood pressure. It is
well-know that the half-life time of ketamine is around 80 mins. Here, the
proposed framework was applied to investigate the dynamics of brain states and
the mechanic operator of the evolution from baseline to reconfiguration and
normalization in the end in this study.

A group of 80 healthy volunteers (33 female, mean age $\pm$ SD: 25.9 $\pm$ 5.3
years) was recruited and screened by the mini-international neuropsychiatric
interview (MINI, German Version 5.0.0), physical and ophthalmic examination,
blood laboratory tests, and electrocardiography in Magdeburg, Germany. All
participants were in good physical health and no pregnancy, left-handedness,
and any MRI-incompatible devices. Current or former history of substance use
disorder was excluded and a drug screening was performed. The study was
approved by the Institutional Review Board of the Otto-von-Guericke-University
Magdeburg, Germany. All subjects gave written informed consent and received
financial reimbursement.

  2. 2.

MRI scan and Resting-State Functional Connectivity calculation

All scans were acquired on a 7 T MR scanner (Siemens Healthcare, Erlangen,
Germany) using a 32-channel head array coil (for RS-fMRI: 280 time points, TE
$=$ 22 ms, TR $=$ 2.8 s, flip angle $=80^{\circ}$, 62 contiguous axial slices,
bandwidth = 2246 Hz/pixel, acquisition matrix = 106 $\times$ 106, voxel size
$=2\times 2\times 2mm^{3}$; T1- weighted MRI: 3 D-MPRAGE sequence, TE $=$ 2.73
ms, TR = 2300 ms, T1 = 1050 ms, flip angle = $5^{\circ}$, bandwidth = 150
Hz$/$pixel, acquisition matrix = $320\times 320$, voxel size = $0.8\times
0.8\times 0.8mm^{3}$).

In order to exclude anatomical abnormalities and abnormal frontal signal drop
out, individual EPI data were visually inspected by two independent raters and
an additional point-spread function (PSF) mapping protocol was scanned for EPI
distortion correction. Twenty-three EPI scans out of 240 total scans were
discarded and resulted in 19 subjects (7 female, age 27.1 $\pm$ 6.9 years)
with a low-quality EPI in one or more sessions.

Next, the pipeline developed in the 1000 Functional Connectomes Project (https
://www. nitrc.org/projects/fcon$\\_$1000/) with few modifications were
performed for data preprocessing. The major steps of this well-established
pipeline included brain extraction, time slicing, realignment, nuisance
regression, normalization, temporal filtering with a bandpass filter (0.01–0.1
Hz), and spatial smoothing (FWHM 6 mm). The framewise displacement was
calculated and added as additional confound along with the motion parameters
(three translations, three rotations), as well as the mean signals of white
matter and cerebrospinal fluid (CSF), using a multiple linear regression for
each subject. With the proposed image in standard MNI space, the parcellation
scheme with 200 ROIs by Schaefer and colleagues data2  was applied in this
study. In each subject, the average time course within each parcel was
extracted and a 200$\times$200 adjacent matrix was created.

##  III Methods

###  III.1 Confirming the brain state by LEiDA

The functional connectivity FC patterns are defined as the micro states of
brain, and determined by LEiDA model cabral ; neuim6 . Precisely, the dynamic
functoinal connectivity (dFC) is first calculated, which is a phase coherence
matrix and defined as

$dFC(n,p,t)=cos(\theta(n,t)-\theta(p,t)).$ (1)

$n$ (or $p$) denotes the brain area $n$ (or $p$), and $\theta(n,t)$ (or
$\theta(p,t)$) represents the phase function of brain area $n$ (or $p$) at
time point t. At each point in time, each brain area has a corresponding phase
function which is gained by Hilbert Transform newm1 ; newm2 ; newm3 ; newm4 ;
newm5 . Apparently, dFC is a dynamic function describing the functional
coherence between the brain areas at each time point.

Considering that the dFC at each time point is undirected and symmetric across
the diagonal, the dimension of dFC can be largely reduced by calculating its
leading eigenvector $v$ that captures the instantaneous dominant connectivity
pattern of dFC. Upon computing the leading eigenvector of the dFC, the
recurrent FC patterns are identified by applying the $k-$means clustering
algorithem km1 ; km2 ; km3 ; km4 . For further details regarding LEiDA, please
read cabral .

###  III.2 Equilibrium ensemble theory model

Compared to the dynamic functional connectivity dFC, the static functional
connectivity (SFC) for which the correlation between the brain areas is
computed over the entire recording session, therefore describes an average
effect of the functional coherence between the brain areas. That the FC
patterns describe the functional coherence between the brain regions during
the recording session whereas SFC describes the average effect of the
functional correlation between the brain areas, permits us to take SFC as the
statitical average of relevant performance of FC patterns.

Assumptions: In our study, it is realized that the FC patterns gained by LEiDA
are valid to present the statistical properties of SFC when the similarity
shown in equation (2) is no less than $80\%$. $w_{i}=v_{i}*v_{i}^{T}$ is the
outer product of FC pattern $v_{i}$, and $p_{i}$ is the corresponding
occurrence frequency. $S_{imi}$ denotes the similarity that is measured as the
Pearson correlation ($P_{corr}$) between SFC and the weighted sum of $w_{i}$.
Before we introduce the Ensemble theory model, some assumptions have to be
presented.

$S_{imi}=P_{corr}(SFC,\sum_{i}p_{i}w_{i})$ (2)

  1. 1.

The resting state of brain is taken as an equilibrium on which the brain does
not process any external promoted tasks. Macroscopically, the brain state is
static (e.g., sleeping), however microscopically the brain’s activities always
present.

  2. 2.

FC patterns are defined as the micro states of brain, whereas SFC is taken as
brain’s macro state.

  3. 3.

When the condition $S_{imi}\geq 0.80$ meets, $\\{v_{i},p_{i}|i=1,...,n\\}$
correspond to the valid micro states of which the statistical average of
performance can describe the properties of macrostate SFC.

  4. 4.

The leading eigenvalue of $w_{i}=v_{i}*v_{i}^{T}$ is set to represent the
system’s $E_{i}$ when the system stays in state $v_{i}$.

For the assumption of energy $E$, we have tried three different parameters.
Besides the leading eigenvalue of $v*v_{T}$, other parameters including the
average of eigenvalues of dFCs’ leading eigenvectors which belong to the same
cluster and the average of corresponding Bold signals, are also evaluated. It
has been proven that these three parameters genergally showed consistent
results.Without losing the generality, the leading eigenvalue of $v*v_{T}$ is
finally taken as the system’s energy in our model.

Equilibrium ensemble theory (EET): In statistical physics, an ensemble is an
idealization consisting of a large number of virtual copies of a real system.
Considered all the copies at once, each copy of them represents a possible
microstate that the real system might be in. A thermodynamic ensemble studies
a system when it stays in the statistical equilibrium ensem1 ; ensem2 ; ensem3
. When the system is realized to be with determined particles N, volume V and
temperature T, the Canonical ensemble theory is applicable, and the
corresponding partition function $Z$ is written as

$Z=\sum_{i=1}e^{-\beta E_{i}}.$ (3)

Where $E_{i}$ is the energy of system when the system stays in the micro state
$v_{i}$. $\beta=\frac{1}{\kappa T}$, where $\kappa$ is the Boltzmann constant
$1.38*10^{-23}J/K$ boltz_c , T is the absolute temperature with the unit K
kelwin .

Considering that $\kappa$ is very small and $E_{i}$ is also a small value
(which is represented by the leading eigenvalue and normally $<3$) , in order
to make $lnZ$ meaningful, we have modified $\kappa$ to another constant
$R=\kappa N_{a}\rho=8.31J/K$. $N_{a}$ is the Avogadro constant
($6.02*10^{23}/mol$), and $\rho=1$ mol. The constants $\kappa$ and $R$ have
the consistant unit. Accordingly, the partition function is rewritten as

$Z=\sum_{i=1}e^{-\gamma E_{i}},$ (4)

where $\gamma=\frac{1}{RT}$. Partition function $Z$ describes the statistical
properties of the system, based on which other thermodynamic functions can be
presented. Accordingly, interal energy $U$, thermodynamic entropy $S$ and
Helmholtz free energy $F$ are written as

$U=-\frac{\partial lnZ}{\partial\gamma},$ (5) $S=R(lnZ-\gamma\frac{\partial
lnZ}{\partial\gamma}),$ (6)

and

$F=-RTlnZ.$ (7)

##  IV Results and Discussion

###  IV.1 Thermodynamic properties of brain

Based on the data provided (See ’Experimental Data’), we use phase coherence
connectivity to obtain the dFC, with the size $N\times N\times T$, where
$N=200$ is the number of parcellated brain regions, and $T=276$ is the number
of recording frames. The micro states are confirmed by applying the $k-means$
clustering analysis on all the leading eigenvectors acorss time points and
subjects (i.e., $276\times 31=8556$ leading eigenvectors for ketamine group).
By doing this, on one hand, as many as significant microstates are collected,
on the other hand, the statistics is therefore more meaningful. The parameter
$k$ in $k-means$ clustering changes from 2 to 50, repeating each 30 times.
Each thermodynamic property gained below is the average across these 30
calculations.

$S_{imi}$ is firstly calculated for both groups. It is found that
$\overline{S_{imi}}\geq 0.80$ on each stage, which means that the $SFC$ can be
presented by the linear combination of the micro states. Furthermore, the
brain system on each stage can be seen with determined particals $N$, volume
$V$ and temperature $T$, accordingly the Canonical ensemble theory based on
EET model described above is applied. $T$ is further seen not to change across
the stages, and set 1$K$ without losing the generality.

The thermodynamic properties: Internal energy $E$, entropy $S$ and $F$ for
ketamine group and the controlled group are shown in table 1 and table 2,
respectively.

In table 1, one can see that compared to the baseline, the internal energy
does not change significantly, whereas the entropy has increased on stage1 and
stage2, and the free energy has decreased. According to Boltzmann formula of
entropy, the entropy can be also written as $S=kln\Omega$, where $\Omega$
corresponds to the number of microstates. Therefore, one can conclude that
more microstates in brain have emerged after the injection of ketamine. Tabel
2 shows the corresponding results for the controlled group. One can see that
the internal energy and entropy show the similar trends with those shown in
the ketamine group, i.e., compared to the baseline, the internal energy does
not show significant differences on stage1 and stage2, entropy has increased
on stage1 and stage2, and the $F$ decreased.

Aparently, though the entropy or the free energy has changes after the subject
has been injected the ketamine, we can not say that these changes are caused
by the ketamine, since similar changes also happened to the controlled group.
Therefore, the work done of brain is further investigated.

Maximum work principle: According to the Maximum work principle mwp1 , when a
system experiences an isothermal process from state 1 to state 2, the work
done of this system maximally is the decrement of free energy, i.e.,

$\nabla=F_{1}-F_{2}\geq-W.$ (8)

$-W$ is the work done of the system. For our case, during the process from
baseline to stage1 or stage2, the temperature of the brain or the environment
does not have very significant changes, therefore, the process can be seen as
an isothermal process. According to the results shown in the above tables, we
gain that, for the ketamine group, the decrement of free energy between
baseline and stage1 is $\nabla_{ketamine}^{1}=F_{base}-F_{s1}=1.3$, the
decrement of free energy between baseline and stage2 is
$\nabla_{ketamine}^{2}=F_{base}-F_{s2}=1.71$; for the controlled group, the
corresponding results are $\nabla_{placebo}^{1}=4.63$ and
$\nabla_{placebo}^{2}=3.41$. Thus it can be concluded that the maximal work
that the brain can do measured from the ketamine group is significantly less
than that of the brain measured from the controlled group. If we undersetand
the maximal work that brain can do as the work cability of brain to perform
tasks, the results tell us that the work cability of brain measured from the
ketamine group is obviously weaker than that from the controlled group.

Table 1: Thermodynamic Functions of Ketamine group Stages : Functions | Internal Energy $\overline{U}$ | Entropy $\overline{S}$ | Free energy $\overline{F}$
---|---|---|---
Baseline | 0.79 | 27.33 | -26.54
Stage 1 | 0.78 | 28.63 | -27.84
Stage 2 | 0.79 | 29.04 | -28.25
Table 2: Thermodynamic Functions of Placebo group Stages : Functions | Internal Energy $\overline{U}$ | Entropy $\overline{S}$ | Free energy $\overline{F}$
---|---|---|---
Baseline | 0.79 | 25.90 | -25.14
Stage 1 | 0.80 | 30.57 | -29.77
Stage 2 | 0.78 | 29.33 | -28.55

###  IV.2 Heat conduction of the brain

In this part, we have studied the heat conduction happened to the brain
studied in ketmaine group.

From the first law of thermodynamics,

$\triangle{U}=Q+W.$ (9)

Where $U$ is the internal energy, and $\triangle{U}$ denotes the increment of
$U$. $Q$ denotes the heat that the system has absorbed, and $W$ indicates the
work that the environment has done to the system. From the result in part A,
the internal energy does not change significantly, so we have
$\triangle{U}=0$. Accordingly,

$W=-Q.$ (10)

From the Clausius entropy expression clau1 ; clau2 ; clau3 ; clau4 , for any
reversible process from state a to state b, the increment of entropy
$\triangle{S}$ is expressed as

$\triangle{S}=S_{b}-S_{a}=\int_{a}^{b}\frac{dQ}{T}.$ (11)

Since the entropy S is a state function, the increment $\triangle{S}$ only
depends on the initial state and terminal state. Therefore, theoretically we
can always create a reversible process that has the same initial state and the
same terminal state, and then apply the formula to study $\triangle{S}$. Let
us take baseline as the initial state $a$, ketamine stage (stage 1 or stage 2)
as the terminal state $b$, and $T$ is seen invariable during the reversible
process. Then the increment $\triangle{S}$ is

$\triangle{S}=\int_{a}^{b}\frac{dQ}{T}=\frac{1}{T}\int_{a}^{b}dQ=\frac{Q}{T},$

then we have

$Q=T\triangle{S}.$

From the results of S on stage 1 or stage 2, we have $\triangle{S}>0$.
Together with equation $(10)$, we gain

$Q>0,W(=-Q)<0.$

The results indicate that during the process from baseline to ketamine stage,
brain absorbs heat from the external environment, and meanwhile does positive
work to the external environment. The environment includes any other systems
which have the interactions with the brain system.

###  IV.3 Discussions

From table 1 one can see that although some thermodynamic properties have
changed, the changes are not significant. For example, the entropy S has
increased by 1.3 on stage1, and 1.71 on stage2 compared to that on baseline,
respectively. Since SFC represents macro state, we further studied the
similarity between the SFCs from any two different stages. The similarity is
measured as the Pearson correlation. It is found that the similarity between
SFC from baseline and SFC from any other stage is $\approx 0.95$, which means
that the macro states of these three stages are all like, so that the
macroproperties from different stages don’t have distinct differences. The
reason why it shows similar macrostate on each stage, is perhaps that the
ketamine concentration in the body has largely decreased even after one hour,
so that the brain has recoved to a large extent. However, through Maximum work
principle, significant difference between the ketamine groupe and controlled
group has been presented by the decrement of $F$, though the existence of big
similarity between macrostates.

##  V Mechanic operator and Hilbert space of brain

###  V.1 Confirming the mechanic operator $\mathcal{A}$

In quantum mechanics, a mixed quantum state $\Phi$ can be described as

$\Phi\to\begin{cases}\phi_{1},&f_{1}\\\ \vdots&\vdots\\\ \phi_{i},&f_{i}\\\
\vdots&\vdots\\\ \phi_{k},&f_{k}\\\ \end{cases}$ (12)

Where $\phi_{i}$ is the ${\it i}$th pure state, and $f_{i}$ is the
corresponding probability with which the system stays in $\phi_{i}$. $f_{i}$
meets

$\sum_{i}f_{i}=1$ (13)

In our study, the FC states $\\{v_{i}|i=1,…,k\\}$ are taken as the microstates
and the corresponding probabilities $\\{p_{i}|i=1,…,k\\}$ meet the condition
shown in equation (13). Let us assume that there exists a mixed quantum state
which is described by the pure state $\\{v_{i}|i=1,…,k\\}$, and there is a
mechanic operator $\mathcal{A}$ that is correlated to $\\{v_{i}|i=1,…,k\\}$.
In order to confirm this operator, we have the following hypotheses:

  1. 1.

The pure states $\\{v_{i}|i=1,…,k\\}$ have close relations with
$\mathcal{A}$’s characteristic functions $\Psi=\\{\psi_{i}|i=1,…,k\\}$ . When
k = 200 (the number of parcellated brain regions), the characteristic
functions of A can be gained by Schmidt process performed on
$\\{v_{i}|i=1,…,k\\}$.

  2. 2.

The system’s energies $\\{E_{i}|i=1,…,k\\}$ when the system stays in
$\\{v_{i}|i=1,…,k\\}$, have close relations with the characteristic values
$\Lambda$ of $\mathcal{A}$ . When k = 200, $\\{E_{i}|i=1,…,k\\}$ are the
corresponding characteristic values
$\Lambda=\\{\lambda_{i}=E_{i}|i=1,…,200\\}$.

Based on the above analyses, we confirm the mechanic operator $\mathcal{A}$ by
the following steps.

  1. 1.

Step 1: Looking for a mixed state $\Phi$ which has 200 pure states. In the
meanwhile, the relation of $\overline{S_{imi}}$ no less than 0.8 still holds.

  2. 2.

Step 2: Verifying if these 200 states $\\{v_{i}|i=1,…,200\\}$ are linearly
uncorrelated. If they are linearly uncorrelated, we confirm the characteristic
functions $\Psi$ by performing the Schmidt Process on the states
$\\{v_{i}|i=1,…,200\\}$ to gain the charateristic functions
$\\{\psi_{i}|i=1,…,200\\}$; Otherwise, if they are linearly correlated, one
should go back to step 1 till all the 200 pure states are linearly
uncorrelated.

  3. 3.

Step 3: According to linear algebra, the mechanic operator $\mathcal{A}$ can
be determined by

$\mathcal{A}=R_{A}U_{\Lambda}R_{A}^{-1},$ (14)

where $R_{A}$ is a matrix (full rank) whose columns are the different vectors
$\\{\psi_{i}|i=1,…,200\\}$ from $\Psi$; $U_{\Lambda}$ is a diagonal matrix in
which the diagonal elements are the corresponding eigenvalues; and
$R_{A}^{-1}$ is the inverse matrix of $R_{A}$.

By applying equilibrium ensemble theory, we have

  1. 1.

On each stage, a mixed state $\Phi$ with 200 linearly uncorrelated pure states
is found, and accordingly the corresponding mechanic operator is confirmed for
each stage.

  2. 2.

Let us write the mixed states of baseline, stage 1 and stage 2 as $\Phi_{0}$,
$\Phi_{1}$ and $\Phi_{2}$, respectively, and write the corresponding mechanic
operators as $\mathcal{A}_{0}$, $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$,
respectively. It has been testified that for any ${\it i,}$ ${\it j}$, and
$i\neq j(i,j\in 0,1,2)$,

$\Phi_{i}\neq\Phi_{j},\mathcal{A}_{i}\neq\mathcal{A}_{j}.$

In short, the operators are confirmed for each stage, and any two of the
operators are different. In the following, we will discuss the relations of
the three mechanic operators.

###  V.2 The relation of the mechanic operators between different stages

####  V.2.1 Similarity matrix of the mechanic operator

Firstly, let us see the operator $\mathcal{A}_{0}$ when the brain stays on the
baseline. $\mathcal{A}_{0}$ is gained by

$\mathcal{A}_{0}=R_{0}U_{0}R_{0}^{-1},$

so from $\mathcal{A}_{0}$, we have

$U_{0}=R_{0}^{-1}\mathcal{A}_{0}R_{0}.$ (15)

According to the linear algebra, two $n\times n$ matrices M and N are called
similar if there exists an invertible $n\times n$ matrix $\Gamma$ such that

$N=\Gamma^{-1}M\Gamma.$ (16)

Therefore, based on the equation (15) of $U_{0}$, it is concluded that $U_{0}$
is the similarity matrix of $\mathcal{A}_{0}$, i.e.,

$\mathcal{A}_{0}\sim U_{0}$

In the similar way, on stage 1 and stage 2 we have

$\mathcal{A}_{1}\sim U_{1},\mathcal{A}_{2}\sim U_{2}.$

where $U_{1}$ and $U_{2}$ are the corresponding diagonal matrices of
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$.

####  V.2.2 The relation of eigenvalue spectrums of $\mathcal{A}_{0}$,
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$

The eigenvalue spectrums of $\mathcal{A}_{0}$, $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ correspond to the elements of main diagonals of $U_{0}$,
$U_{1}$ and $U_{2}$, respectively. During the process of confirming the pure
states (FC patterns), a different run may induce a different form of the
operator. Therefore, on each stage I repeat 20 times runs, and then have
gained 20 operators with corresponding 20 eigenvalue spectrums. I made a
comparison of eigenvalue spectrum between different runs, and the
corresponding results are shown in Fig. 1.

Figure 1: (Left) Eigenvalue spectrums of the 60 opertors, and (Right) the mean
of eigenvalue spectrum $\overline{E}$ by averaging these 60 eigenvalue
spectrum with the standard deviation $\sigma$.

From Fig.1, one can see that in general all the spectrums show a consistent
trend and the dispersion $\sigma$ is significantly small. Hence, it can be
concluded that the eigenvalue spectrums can be seen to keep invariable,
regardless of the run or the stage, i.e.,

$U_{0}=U_{1}=U_{2}.$ (17)

####  V.2.3 The relation of mechanic operators of $\mathcal{A}_{0}$,
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$

From part $1$, we got

$\mathcal{A}_{0}\sim U_{0},\mathcal{A}_{1}\sim U_{1},\mathcal{A}_{2}\sim
U_{2},$

and from part $2$ we got

$U_{0}=U_{1}=U_{2}$

Based on the transitivity property and symmetry property of the similarity
matrix, we have

$\mathcal{A}_{0}\sim\mathcal{A}_{1}\sim\mathcal{A}_{2}$ (18)

Therefore, one can conclude that any two of these three operators are the
similarity matrices to each other. It means that, mechanic operators
$\mathcal{A}_{0}$, $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ can be seen as
three different representations of the same linear map under three different
bases.

###  V.3 Hilbert space of human brain

The confirmation of mechanic operator allows us to build a Hilbert space that
is spanned by the operator’s eigenvectors. In Hilbert space, any brain state
can be presented as a linear combination of the base vectors, in the form of a
vector in which the elements are expansion coefficients. Since the base
vectors are actually derived from the microstates of brain, therefore can be
seen as the fundamental constituents of brain state. Then the studies related
to brain state can be refocused on the studies of fundamental states. For
example, the comparison between two brain states in Hilbert space is actually
the analysis of how the brain state is presented by the funtamental states. In
general, the confirmation of mechanic operator has opened up big probabilies
with quantum mechanics to investigate human brain. Besides brain state, state
transition, phase transition, etc., could be investigated by means of relevant
theories from quantum mechanics. Due to the space limitations, further study
on Hilbert space based on the current data is not processed. However, the
study from this perspective has been already started, as another important
research project.

##  VI Discussions and Conclusions

In the present work, we proposed a model based on statistical physics. FC
pattern is defined as the microstate of brain, and SFC is taken as the
macrostate which can be seen as the statistical average of the micro
propterties. From the Equilibrium ensemble theory, the corresponding
thermodynamic properties are presented as functions of microstates. We applied
this model to investigate the biological effect of ketamine on brain by
analyzing the thermodynamic properties. The Maximum work principle based on
the decrement of free energy has explained the distinct difference between
ketamine and controlled group: the maximal work that the brain can do
evaluated from the controlled group is remarkably bigger than that evaluated
from the ketamine group. The Work done of brain is realized (suggested)
relating to the amount of tasks the brain can perform parallelled, and how
efficient the task is performed. Hence it is concluded that the capability of
work done of brain from controlled group is remarkably bigger that that from
the ketamine group. Moreover, the first law of thermodynamics has been applied
to investigate the work done and heat conduction of brain.

Taking the microstate of brain as a mixed quantum state, the mechanic operator
is futher investigated. Based on the data from ketamine group, three different
mechanic operators for three stages have been confirmed, which are proven
actually the different representations of a same linear map under different
bases. This result could be related to a fact that the macrostates of brain on
differents stages are basically consistent. The study on the mechanic operator
brings a new research perspective for the study of neuroscience: on one hand,
the confirmation of mechanic operator enables us to build a Hilber space in
which brain states as well as other properties can be studied by applying
quantum mechanics; on the hand hand, besides the operator we have gained in
current study, there exist other operators which describe different dynamic
properties of brain system.

Our mode has provided a new insight into understanding the human brain, and
could be also extensively applied to the diagnosis of some mental diseases,
e.g., by monitoring the changes of thermodynamic funtions. In the meanwhile,
since the model is purely depending on the property of time series, besides
brain system it is still feasible for other systems when the relevant time
series meet the conditions specified in the model.

## Acknowledgment

We would like to thank Han Ping (Department of Physical Science and
Technology, Bohai University) and Joana Cabral (Department of Psychiatry,
University of Oxford) for very valuable discussions.

This work was supported by the Bundesministerium für Bildung und Forschung
(e:Med program: FKZ: 01ZX1311A fund  and FKZ: 01ZX1503), and the European
$Union^{\prime}s$ Horizon 2020 Program 668863-SyBil-AA, National Natural
Science Foundation of China (Grants No. 11305017), and Specialized Foundation
for Theoretical Physics of China (Grant No. 11247239). AL was supported by a
Sofja Kovalevskaja award from the Alexander von Humboldt Foundation and by the
Institutional Strategy of the University of Tübingen (DFG, ZUK 63).

## References

  * (1) Park, H. J., and Friston, K. (2013). Structural and functional brain networks: from connections to cognition. Science, 342(6158).
  * (2) Fornito, A., Zalesky, A., and Breakspear, M. (2015). The connectomics of brain disorders. Nature Reviews Neuroscience, 16(3), 159-172.
  * (3) Beckmann, C. F., DeLuca, M., Devlin, J. T., and Smith, S. M. (2005). Investigations into resting-state connectivity using independent component analysis. Philosophical Transactions of the Royal Society B: Biological Sciences, 360(1457), 1001-1013.
  * (4) Damoiseaux, J. S., Rombouts, S. A. R. B., Barkhof, F., Scheltens, P., Stam, C. J., Smith, S. M., and Beckmann, C. F. (2006). Consistent resting-state networks,(2). Proc. Natl. Acad. Sci, 103.
  * (5) Smith, S. M. et al. Proceedings of the National Academy of Sciences of the United States of America 109, 3131–3136, doi:10.1073/pnas.1121329109 (2012).
  * (6) Biswal, B., Zerrin Yetkin, F., Haughton, V. M., and Hyde, J. S. (1995). Functional connectivity in the motor cortex of resting human brain using echo‐planar MRI. Magnetic resonance in medicine, 34(4), 537-541.
  * (7) Greicius, M. D., Krasnow, B., Reiss, A. L., and Menon, V. (2003). Functional connectivity in the resting brain: a network analysis of the default mode hypothesis. Proceedings of the National Academy of Sciences, 100(1), 253-258.
  * (8) Salvador, R., Suckling, J., Coleman, M. R., Pickard, J. D., Menon, D., and Bullmore, E. D. (2005). Neurophysiological architecture of functional magnetic resonance images of human brain. Cerebral cortex, 15(9), 1332-1342.
  * (9) Naros, G., Lehnertz, T., Leão, M. T., Ziemann, U., and Gharabaghi, A. (2020). Brain state-dependent gain modulation of corticospinal output in the active motor system. Cerebral Cortex, 30(1), 371-381.
  * (10) Vidaurre, D., Hunt, L. T., Quinn, A. J., Hunt, B. A., Brookes, M. J., Nobre, A. C., and Woolrich, M. W. (2018). Spontaneous cortical activity transiently organises into frequency specific phase-coupling networks. Nature communications, 9(1), 1-13.
  * (11) Deco, G., Cruzat, J., Cabral, J., Tagliazucchi, E., Laufs, H., Logothetis, N. K., and Kringelbach, M. L. (2019). Awakening: Predicting external stimulation to force transitions between different brain states. Proceedings of the National Academy of Sciences, 116(36), 18088-18097.
  * (12) Sporns, O., Chialvo, D. R., Kaiser, M., and Hilgetag, C. C. (2004). Organization, development and function of complex brain networks. Trends in cognitive sciences, 8(9), 418-425.
  * (13) Burns, G. A., and Young, M. P. (2000). Analysis of the connectional organization of neural systems associated with the hippocampus in rats. Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, 355(1393), 55-70.
  * (14) Scannell, J. W., Burns, G. A. P. C., Hilgetag, C. C., O’Neil, M. A., and Young, M. P. (1999). The connectional organization of the cortico-thalamic system of the cat. Cerebral Cortex, 9(3), 277-299.
  * (15) Felleman, D. J., and Van Essen, D. C. (1991). Distributed hierarchical processing in the primate cerebral cortex. In Cereb cortex. 1: 1–47.
  * (16) Achard, S., Salvador, R., Whitcher, B., Suckling, J., and Bullmore, E. D. (2006). A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs. Journal of Neuroscience, 26(1), 63-72.
  * (17) Stam, C. J., Jones, B. F., Nolte, G., Breakspear, M., and Scheltens, P. (2007). Small-world networks and functional connectivity in Alzheimer’s disease. Cerebral cortex, 17(1), 92-99.
  * (18) Bassett, D. S., Meyer-Lindenberg, A., Achard, S., Duke, T., and Bullmore, E. (2006). Adaptive reconfiguration of fractal small-world human brain functional networks. Proceedings of the National Academy of Sciences, 103(51), 19518-19523.
  * (19) Achard, S., and Bullmore, E. (2007). Efficiency and cost of economical brain functional networks. PLoS Comput Biol, 3(2), e17.
  * (20) Aertsen, A. M., Gerstein, G. L., Habib, M. K., and Palm, G. (1989). Dynamics of neuronal firing correlation: modulation of” effective connectivity”. Journal of neurophysiology, 61(5), 900-917.
  * (21) Friston, K. J., Frith, C. D., Liddle, P. F., and Frackowiak, R. S. J. (1993). Functional connectivity: the principal-component analysis of large (PET) data sets. Journal of Cerebral Blood Flow and Metabolism, 13(1), 5-14.
  * (22) Lowe, M. J., Dzemidzic, M., Lurito, J. T., Mathews, V. P., and Phillips, M. D. (2000). Correlations in low-frequency BOLD fluctuations reflect cortico-cortical connections. Neuroimage, 12(5), 582-587.
  * (23) Hansen, E. C., Battaglia, D., Spiegler, A., Deco, G., and Jirsa, V. K. (2015). Functional connectivity dynamics: modeling the switching behavior of the resting state. Neuroimage, 105, 525-535.
  * (24) Deco, G., Jirsa, V. K., and McIntosh, A. R. (2011). Emerging concepts for the dynamical organization of resting-state activity in the brain. Nature Reviews Neuroscience, 12(1), 43-56.
  * (25) Cabral, J., Kringelbach, M. L.,and Deco, G. (2014). Exploring the network dynamics underlying brain activity during rest. Progress in neurobiology, 114, 102-131.
  * (26) Cabral, J., Kringelbach, M. L., and Deco, G. (2017). Functional connectivity dynamically evolves on multiple time-scales over a static structural connectome: Models and mechanisms. NeuroImage, 160, 84-96.
  * (27) Smith, S. M., Vidaurre, D., Beckmann, C. F., Glasser, M. F., Jenkinson, M., Miller, K. L., … and Barch, D. M. (2013). Functional connectomics from resting-state fMRI. Trends in cognitive sciences, 17(12), 666-682.
  * (28) Sporns, O., Tononi, G., and Edelman, G. M. (2000). Theoretical neuroanatomy: relating anatomical and functional connectivity in graphs and cortical connection matrices. Cerebral cortex, 10(2), 127-141.
  * (29) Deco, G., Cabral, J., Woolrich, M. W., Stevner, A. B., Van Hartevelt, T. J., and Kringelbach, M. L. (2017). Single or multiple frequency generators in on-going brain activity: A mechanistic whole-brain model of empirical MEG data. Neuroimage, 152, 538-550.
  * (30) Shine, J. M., Oluwasanmi, K., Bell, P. T., Gorgolewski, K. J., Gilat, M., and Poldrack, R. A. (2015). Estimation of dynamic functional connectivity using Multiplicative Analytical Coupling. Neuroimage, 122, 399-407.
  * (31) Glerean, E., Salmi, J., Lahnakoski, J. M., Jääskeläinen, I. P., and Sams, M. (2012). Functional magnetic resonance imaging phase synchronization as a measure of dynamic functional connectivity. Brain connectivity, 2(2), 91-101.
  * (32) Ponce-Alvarez, A., Deco, G., Hagmann, P., Romani, G. L., Mantini, D., and Corbetta, M. (2015). Resting-state temporal synchronization networks emerge from connectivity topology and heterogeneity. PLoS Comput Biol, 11(2), e1004100.
  * (33) Deco, G., and Kringelbach, M. L. (2016). Metastability and coherence: extending the communication through coherence hypothesis using a whole-brain computational perspective. Trends in neurosciences, 39(3), 125-135.
  * (34) Cabral, J., Vidaurre, D., Marques, P., Magalhães, R., Moreira, P. S., Soares, J. M., … and Kringelbach, M. L. (2017). Cognitive performance in healthy older adults relates to spontaneous switching between states of functional connectivity during rest. Scientific reports, 7(1), 1-13.
  * (35) MacQueen, J. (1967, June). Some methods for classification and analysis of multivariate observations. In Proceedings of the fifth Berkeley symposium on mathematical statistics and probability (Vol. 1, No. 14, pp. 281-297).
  * (36) Lloyd, S. P. Least square quantization in PCM. Bell Telephone Laboratories Paper. 1957. Published in journal much later: Lloyd., S. P. Least squares quantization in PCM (PDF). IEEE Transactions on Information Theory. 1982, 28 (2): 129–137.
  * (37) Forgey, E. (1965). Cluster analysis of multivariate data: Efficiency vs. interpretability of classification. Biometrics, 21(3), 768-769.
  * (38) Pelleg, D., and Moore, A. (1999, August). Accelerating exact k-means algorithms with geometric reasoning. In Proceedings of the fifth ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 277-281).
  * (39) Gibbs, J. W. (1902). Elementary principles in statistical mechanics: developed with especial reference to the rational foundations of thermodynamics. C. Scribner’s sons.
  * (40) Kittel, Charles, Herbert Kroemer (1980). Thermal Physics, Second Edition. San Francisco: W.H. Freeman and Company. pp. 31 ff. ISBN 0-7167-1088-9.
  * (41) Landau, L.D., Lifshitz, E.M. (1980). Statistical Physics. Pergamon Press. pp. 9 ff. ISBN 0-08-023038-5.
  * (42) Richard Feynman (1970). The Feynman Lectures on Physics Vol I. Addison Wesley Longman. ISBN 978-0-201-02115-8.
  * (43) Rankine, W. J. M., ”A manual of the steam engine and other prime movers”, Richard Griffin and Co., London (1859), p. 306–307.
  * (44) Van’t Hoff, J.H. (1896). Studies in Chemical Dynamics (Chemical Equilibrium, pgs. 143-54; Affinity, pgs. 229-36; Determination of the Work Done by Affinity, pgs. 237-41; Applications, 241-50; Electrical Work Performed by a Chemical Change, pgs. 251-73; Principle of maximum work, pgs. 224-28, 241-42); Revised and Enlarged by Ernst Cohen (trans. Thomas Ewan). 286-pgs. London: Williams and Norgate.
  * (45) Clausius, R. (1850). Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen. Annalen der Physik, 155(3), 368-397.
  * (46) Clausius, R. (1850). Über die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen. Annalen der Physik, 155(3), 368-397..
  * (47) Atkins, Peter; Julio De Paula (2006). Physical Chemistry, 8th ed. Oxford University Press. p. 79. ISBN 978-0-19-870072-2.
  * (48) Engel, Thomas; Philip Reid (2006). Physical Chemistry. Pearson Benjamin Cummings. p. 86. ISBN 978-0-8053-3842-3.
  * (49) Lieb, E. H., and Yngvason, J. (1999). The physics and mathematics of the second law of thermodynamics. Physics Reports, 310(1), 1-96.
  * (50) Li M, Woelfer M, Colic L, Safron A, Chang C, Heinze H-J, Speck O, Mayberg HS, Biswal BB, Salvadore G, Fejtova A, Walter M (2018) Default mode network connectivity change corresponds to ketamine’s delayed glutamatergic effects. Eur Arch Psychiatry Clin Neurosci Available at: https://doi.org/10.1007/s00406-018-0942-y [Accessed November 25, 2018].
  * (51) Schaefer, A., Kong, R., Gordon, E. M., Laumann, T. O., Zuo, X. N., Holmes, A. J., … and Yeo, B. T. (2018). Local-global parcellation of the human cerebral cortex from intrinsic functional connectivity MRI. Cerebral cortex, 28(9), 3095-3114.
  * (52) Woelfer, M., Li, M., Colic, L., Liebe, T., Di, X., Biswal, B., … and Walter, M. (2019). Ketamine-induced changes in plasma brain-derived neurotrophic factor (BDNF) levels are associated with the resting-state functional connectivity of the prefrontal cortex. The World Journal of Biological Psychiatry, 1-15.
  * (53) Bremner, J. D., Krystal, J. H., Putnam, F. W., Southwick, S. M., Marmar, C., Charney, D. S., and Mazure, C. M. (1998). Measurement of dissociative states with the clinician‐administered dissociative states scale (CADSS). Journal of Traumatic Stress: Official Publication of The International Society for Traumatic Stress Studies, 11(1), 125-136.
  * (54) Spanagel R., Durstewitz D., Hansson A., Heinz A., Kiefer F., Köhr G., Matthäus F., Nöthen M. M., Noori H. R., Obermayer K., Rietschel M., Schloss P., Scholz H., Schumann G., Smolka M., Sommer W., Vengeliene V., Walter H., Wurst W., Zimmermann U.S.; Addiction GWAS Resource Group, Stringer S., Smits Y., Derks E. M., (2013) A systems medicine research approach for studying alcohol addiction. Addiction Biology 18: 883-96.