# Brain Model Based on the Canonical Ensemble with Functional MRI: A Thermodynamic Exploration of the Neural System

Authors: Chenxi Zhou; Bin Yang; Wenliang Fan; Wei Li
Journal: Complexity
 (2021)

---

## Abstract

Objective. System modeling is an important method to study the working mechanisms of the brain. This study attempted to build a model of the brain from the perspective of thermodynamics at the system level, which brought a new perspective to brain modeling. Approach. Regarding brain regions as systems, voxels as particles, and intensity of signals as energy of particles, the thermodynamic model of the brain was built based on the canonical ensemble theory. Two pairs of activated regions and two pairs of inactivated brain regions were selected for comparison in this study, and the thermodynamic properties based on the proposed model were analyzed. In addition, the thermodynamic properties were extracted as input features for the detection of Alzheimer’s disease. Main Results. The experimental results verified the assumption that the brain follows thermodynamic laws. This demonstrated the feasibility and rationality of the proposed brain thermodynamic modeling method, indicating that thermodynamic parameters drawn from our model can be applied to describe the state of the neural system. Meanwhile, the brain thermodynamic model achieved good accuracy in the detection of Alzheimer’s disease, suggesting the potential application of thermodynamic models in auxiliary diagnosis. Significance. (1) In the previous studies, only some thermodynamic parameters in physics were analogized and applied to brain image analysis, while, in this study, a complete system model of the brain was proposed through the principles of thermodynamics. And, based on the neural system models proposed, thermodynamic parameters were obtained to describe the observation and evolution of the neural system. (2) Based on the proposed thermodynamic models, we found and confirmed that the neural system also follows the laws of thermodynamics: the activation of system always leads to increased internal energy, increased free energy, and decreased entropy as what is discovered in many other systems besides classic thermodynamic system. (3) The detection of neural disease was demonstrated to benefit from the thermodynamic model, which confirmed that the thermodynamic model proposed can indeed describe the evolution of the neural system diseases. And it further implied the immense potential of thermodynamics in auxiliary diagnosis.


---

## Body

## 1. Introduction

The human brain is the most complex organ with unlimited potential. To further understand the physiological principle and working mechanism of the brain, scientists have tried to build models of the brain from different perspectives and then observe and study system states and characteristics. Therefore, dynamics applied in modeling neural systems, called neurodynamics, is an important theory. It attempts to build neural system models based on dynamics at different levels, from microscopicion channels to macroscopic connections of the cerebral cortex. Then, system states and dynamic characteristics can be observed to explore the working mechanism of the brain. Deco and Rolls [1] studied the spiking mechanisms of synapses and neurons under biased competition using dynamic analysis. Based on neurodynamics, Quyen [2] developed, a comprehensive framework to analyze the spatiotemporal characteristics of the brain on a large scale. Heller and Casey [3] studied the temporal development of adolescents in emotion by building a corresponding model with neurodynamics and explored the relationship between temporal dynamic changes in adolescents with emotional and mood/anxiety disorders. Amari and Maginu [4] built an autocorrelation associative memory model based on statistical neurodynamics to analyze nonequilibrium dynamical behaviors in the recall process. By simplifying the KIII model, Harter and Kozma [5] built a KA model of aperiodic dynamics observed in cortical systems to understand intelligent behavior in biological agents.The brain network is also significant in modeling neural systems, which are based on function integration and differentiation mechanisms in neurophysiology. The brain network includes the structural brain network and the functional brain network. The structural brain network describes the anatomical links between units, such as the white matter fiber tracts [6]. The functional brain network focuses on describing the interactions between units, for example, the functional correlation [7] or the autocoherent oscillations of inhibitory neurons [8]. Brain network modeling has also been applied to research on pathologic mechanisms of various neural diseases. Xiang et al. [9] explored the changes in the shortest paths and clustering coefficients of functional brain networks in Alzheimer’s disease. Dubbelink et al. [10] studied the changes in brain network topology in Parkinson’s disease using magnetoencephalography. Jeong et al. [11] studied the difference in functional brain networks between the epileptic and control groups using global mutual information and global efficiency of different bands with whole-brain magnetoencephalography. The brain network allowed people to diversify out of previous research focusing on one specific independent object, with a view to understand the physiological principle and working mechanism of the neural system at different levels from the perspective of the collaborative working mechanism of the brain.As mentioned above, the existing works in brain modeling generally focused on the neurodynamics or brain network. While the brain is a multilayered and multidimensional complex system, it is risky to study the brain only from a single level or perspective. In fact, the characterization and understanding of the brain remain inadequate. We still need to try new ways to model and analyze the brain.Thermodynamics is one of the most important aspects of physics. It focuses on the laws and physical properties of thermal motion and the evolution process of macroscopic matter systems consisting of microscopic particles. Scientists have tried to extend some concepts or theories of thermodynamics into other research fields for general physical state analysis at the system level. In mechanical engineering, Zhang [12] established a theory of nonequilibrium thermodynamics for studying the ThermoPoro mechanical modeling of saturated clays. Albertin et al. [13] used computational thermodynamics to optimize the hardness and wear resistance of high-chromium cast iron. Based on statistical thermodynamics, Kamiyama et al. [14] proposed a “Hakoniwa” method to predict the properties of materials composed of different types of atoms by calculating the atomic energy. In astronomy, Setare and Sheykhi [15] studied the interaction between viscous dark energy and dark matter in the RSII brane world with thermodynamics. Whitehouse and Bate [16] extended thermodynamics to research the collapse of molecular cloud cores and proposed a three-dimensional algorithm to explore the thermodynamic properties during star formation. In biology, from a statistical thermodynamic point of view, Guo and Brooks [17] proposed a method that made it possible to calculate all thermodynamic properties of a protein model with a specific structure and the free energy surface and compaction processes of the protein model proposed were characterized. Based on the thermodynamics, Fischer et al. [18] generated a distribution of the mixed-canonical ensemble corresponding to different temperatures, and the crossing of energy barriers of RNA was observed by sampling from this distribution. These studies suggest that thermodynamic theory is fundamental and universal to some extent. In other words, its ideas and methods not only allow for research on traditional thermodynamic systems, but are also suitable for system modeling in other fields. Furthermore, thermodynamics can often bring new analytical perspectives for understanding and interpreting systems in other subject areas.Because of many meaningful results of the researches with thermodynamics applied to those areas, some scientists have tried to evaluate or describe the properties of the brain by defining or using parameters derived from thermodynamics. Zhang et al. [19] used Tsallis entropy from discrete wavelet packet transforming the analysis of brain images and identified glioma, meningioma, and other neural diseases. Wang et al. [20] explored the influence of occupational factors on brain complexity by calculating the brain entropy of seafarers and nonseafarers. Lebedev et al. [21] evaluated the impact of lysergic acid diethylamide on personality using a mixed-effects model based on changes in brain entropy and observed personality during follow-up. Coopting the concept of free energy in informatics, Friston [22] proposed the free energy principle, holding that each biological self-organizing system in equilibrium will minimize the free energy to avoid the “surprise.” Friston and Buzsaki [23] attempted to explain the optimization and control mechanisms as well as the functional differentiation of the brain with the proposed free energy principle theoretically. Freeman and Vitiello [24] defined the square of EEG amplitudes as the rate of free energy dissipation to measure the amount of work performed. These studies showed that it is feasible to observe or analyze the brain with specific properties derived from thermodynamics, such as information entropy and free energy in information theory. However, these studies are limited to borrowing concepts from thermodynamics for signal analysis.Different form the previous studies mentioned above which only defined some parameters learned from the thermodynamic theory to analyze the brain image, this article attempted to construct a thermodynamic model of the brain at the system level based on the canonical ensemble theory. Based on the proposed model, the thermodynamic parameters were obtained to describe the observation and evolution of the neural system from the perspective of thermodynamics. Through experiments, we verified the rationality and feasibility of the model proposed. The benefit of thermodynamic model towards the detection of neural disease was found, which implied the immense potential of thermodynamics in auxiliary diagnosis. This research tried to explore and characterize the working mechanism of the neural system from a new perspective in order to shed new light on the understanding of the brain.


## 2. Materials and Methods

### 2.1. Data Acquisition

The data used in this study were obtained from Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (https://adni.loni.usc.edu/). The ADNI was launched in 2003 as a public-private partnership led by principal investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography, other biological markers, and clinical and neuropsychological assessments can be combined to measure the progression of mild cognitive impairment and early Alzheimer’s disease (AD). For up-to-date information, see https://www.adni-info.org. One hundred and sixteen patients with AD (age: 74.6 ± 7.5) and 174 healthy subjects (age: 75.5 ± 6.1) were recruited from ADNI as the AD group and NC (normal control) group, respectively. Demographic information of the subjects is shown in Table 1. There was no significant difference between the two groups in terms of age or sex.Table 1
Demographic characteristics of subjects.
GroupNumberSex (male/female)Age (year)AD11655/6174.6 ± 7.5NC17497/7775.5 ± 6.1


### 2.2. Data Preprocessing

The resting state functional MRI (rs-fMRI) images used in this study were scanned using 3T Philips scanners. The specific operation parameters were as follows: slice thickness, 3.3 mm; number of slices, 48; TR/TE, 3000 ms/30 ms; flip angle, 80°; and imaging matrix, 64 × 64. Each series contained 140 volumes. Data preprocessing was performed using SPM8 and DPARSF [25]. The process was as follows: first, the first ten frames were discarded for magnetization equilibrium. Slicing timing and realigning were performed on the time series. Subjects with head rotation exceeding 2°or head translation exceeding 2 mm were excluded. The Montreal Neurological Institute standard human brain template was used to normalize all the corrected image data. Then, the images were smoothed using a 4 × 4 × 4 Gaussian kernel to decrease spatial noise. The global mean signal was removed to reduce nonneuronal signal fluctuations. The whole brain was divided into 90 regions using an automated anatomical labeling (AAL) template.


### 2.3. Denoising with Point Process Method

The blood oxygen level-dependent (BOLD) signal has been demonstrated to be a description of the hemodynamic response to neural stimulation [26]. Therefore, scientists attempted to search for extreme points of the BOLD signal corresponding to neural stimulation and applied these points to simulate a clear BOLD signal combined with a standard hemodynamic equation [27]. Point process analysis holds that there are some significant feature points in the time series of a complex event and the noise can be decreased to highlight the essential nature by extracting these feature points [28]. Therefore, this study used the point process method to remove noise in the BOLD signal for a high signal-to-noise ratio.The BOLD signal was standardized first. In the time series, the maximum and minimum points were recorded, and the subsequent extreme points were selected as a pair in chronological order. Then, an impulse sequence was obtained by calculating the amplitude increment of every point pair per unit time. Subsequently, we convolved this impulse sequence with the hemodynamic response function proposed by Cohen [29], and a clear BOLD signal was reconstructed.


### 2.4. Statistical Thermodynamic Modeling Method

The canonical ensemble is an important concept in statistical thermodynamics. In statistical thermodynamics, as the system has many different microscopic states under given macroscopic conditions, the measurement of a system is the average of multiple measurements of the system over time. To obtain the measurement at one point, we can measure many identical systems that are under the same macroscopic state at that point and replace the time-average measurement with the average of a large number of identical systems at the same point. The set of a large number of identical systems that are under the same macroscopic state is called an ensemble. Under this premise, the average measurement over time can be replaced by the ensemble average. For an isolated system in equilibrium, in which the energy, volume, and number of particles have all been given, as the probability of possible microscopic states of the system follows the microcanonical distribution, we can analyze the system using the microcanonical ensemble theory. However, it is difficult for most systems to ensure the energy necessary for the microcanonical ensemble analysis. Therefore, researchers generally study a closed system with a given temperature, volume, and number of particles, in which the probability of possible microscopic states of the system follows the canonical distribution. Then, we applied a canonical ensemble to analyze the thermodynamic properties of the system.Some scientists extended the canonical ensemble or its related theory to the research field of electrochemistry [30], black holes [31–33], biology [18, 34], and so on [35, 36], and the achievements in these fields indicated that the canonical ensemble is universal, but not limited to traditional thermodynamic systems. Thus, we attempted to introduce it to the study of neuroscience. Research has shown that the temperature of the human brain is generally maintained at 36.9 ± 0.4°C [37]. In addition, the volumes and voxels of different brain functional regions are certain for each subject in neuroimaging. If we consider the voxels as particles inside the brain region, the number of particles can be determined. Meanwhile, brain regions do not exchange voxels with the outside; in other words, there is no “matter” exchange in the brain region. Then, the brain region basically meets the requirements of a closed system. Therefore, we believe that the brain region can be regarded as a thermodynamic system whose microscopic states conform to the canonical distribution. Then, we can build a brain thermodynamic model based on the theoretical framework of the canonical ensemble.Based on the analysis above, this study proposes a brain thermodynamic model. We regarded the brain functional region as a system consisting of nearly independent particles. Each voxel in neuroimaging corresponds to one nearly independent particle. The BOLD signal intensity of fMRI reflects the metabolic intensity, and a higher BOLD signal indicates a higher metabolic level, leading to an increase in oxygen consumption and an increase in energy consumption. Therefore, the energyE of the voxel was defined as the amplitude of the corresponding reconstructed BOLD signal in this study, and the physiologically activated intensity of each voxel can be evaluated by energy E. Because the relevant computational data had been standardized, all calculated results were dimensionless.Each frame of voxel is considered to be a microscopic state of it, andZi is the partition function of the microscopic state of the voxel at the i-th frame, which can be calculated as(1)Zi=exp−βEi,where Ei is the energy of the voxel at the i-th frame and the specific expression of β is as follows:(2)β=1kT,where k is the Boltzmann constant in thermodynamics, which is equal to 1 to simplify the calculation in this study. The thermodynamic temperature T is 310 K, which is equal to 37°C, the theoretical temperature of the human brain. The whole brain can represent the environment of brain regions.If we consider each time point of the BOLD time series to one voxel as a microscopic state, the partition function of the voxel can be calculated by summing all microscopic states, and the computational formula is described as(3)Zr=∑iZi=∑iexp−βEi,where Zr is the partition function of the voxel and ∑iZi is the sum of its microscopic states at all points in the BOLD time series. Owing to the indistinguishability of identical particles and ignoring the weak interactions among particles, we regard a voxel as a subensemble of the brain region. Therefore, the partition function of the brain region is a combination of all voxels inside. The partition function of the brain region, Zbr, can be derived from Zr:(4)Zbr=1N!∏rZr,where ∏rZr is the quadrature of the partition function of all voxels in the brain region and N is the number of voxels.Based on the theoretical framework of the canonical ensemble, we constructed a brain thermodynamic model regarding brain regions as systems, voxels as particles, the intensity of reconstructed BOLD signals as the energy of particles, and the points of fMRI time series as different microscopic states. We referred to this model as the brain thermodynamic model (BrainTDM). This model attempted to describe the working mechanism of the neural system from a thermodynamic point of view.Based on the BrainTDM proposed in this paper, the internal energy, free energy, and entropy of the neural system can be defined to evaluate the thermodynamic characteristics of brain regions.The internal energy,U, of the brain region can be calculated as(5)U=∑rUr=∑r∑iEiexp−Ei/kTZr.The free energy,F, of the brain region can be calculated as(6)F=U−TS=−kTlnZbr=−kTlog10Zbrlog10e,where e is the Euler number, approximately equal to 2.71828.The entropy,S, of the brain region can be calculated as(7)S=klnZbr−kβ∂∂βlnZbr,U=−∂∂βlnZbr,F=−kTlnZbr,S=klnZbr−kβ∂∂βlnZbr=−FT+UT=U−FT.For a specific brain region, internal energy represents the statistical average energy of all microscopic states in the system, free energy represents the energy of the brain region that could be used to perform external work, and entropy represents chaos of the brain region after being affected by the external environment. Based on thermodynamics, we supposed that when the brain region was activated and consumed external energy to do work, the energy it contained and the energy to do work would both increase and its internal state could be more orderly compared with inactivated brain regions.


### 2.5. Experimental Design

In this experiment, the fMRI of the brain was processed using SPM8 and divided into 90 brain regions with the AAL template. The point process method was applied to reconstruct clear BOLD signals as inputs for the following model. In modeling, brain regions were regarded as systems, voxels were regarded as particles, and the intensity of BOLD signals was regarded as the energy of the particles. Then, a brain thermodynamic model was built based on the ensemble theory. The related thermodynamic parameters were calculated, including the partition functionZ, internal energy U, free energy F, and entropy S. More analyses were performed to explore the potential applications of this model.


#### 2.5.1. Experimental Paradigm I

In thermal physics, once the thermodynamic system consumes external energy to perform work, the entropy decreases, and the internal energy as well as the free energy increases. We assume that the brain system also follows this rule. That is, when the brain regions are activated with neurons that consume external energy to fire synchronously and sequentially, the thermodynamic parameters of brain regions will change in the same way. In experiment paradigm I, we evaluated the proposed modeling method by analyzing the thermodynamic parameter features of activated and inactivated cerebral regions. Because the default mode network (DMN) is most commonly shown to be activated when a person is not focused on the outside world and the brain is at wakeful rest [38, 39], we selected the medial prefrontal cortex (mPFC), which is a key brain region of the default network as the object of the study. Meanwhile, considering the constant sound stimulation during fMRI scanning, we also selected the Heschl gyrus (HES), which is mainly located in the primary auditory cortex, as the object of the study [40]. Furthermore, we selected the precentral gyrus (PreCG), which is also referred to as the primary motor region or primary motor cortex that belongs to the task-positive network, and the olfactory cortex (OLF), which is a key component of the limbic system, as the controls [39]. The brain regions selected for the analysis are listed in Table 2.Table 2
Selected brain regions. Activated/inactivated indicates that the brain regions were activated/inactivated during fMRI scanning.
Activated brain regionsInactivated brain regionsLeft/right medial prefrontal cortexLeft/right precentral gyrusLeft/right Heschl gyrusLeft/right olfactory cortexBy considering brain regions as systems, voxels as particles, and the intensity of reconstructed BOLD signals as the energy of particles, we constructed the brain thermodynamic models of the selected regions. The thermodynamic parameters of different brain regions were calculated based on the constructed models, including the partition function, internal energy, free energy, and entropy. By analyzing the differences between the thermodynamic parameters of the activated and inactivated brain regions, we attempted to determine whether the neural system also follows the laws of thermodynamics.


#### 2.5.2. Experimental Paradigm II

Experiment paradigm II aimed to explore the potential applications of the brain thermodynamic model proposed in computer-aided diagnosis (CAD) or other realistic scenarios. For this purpose, we tried to recognize subjects with AD, taking parameters derived from the brain thermodynamic model as input features. We performed the same classification task with parameters based on the traditional brain network model as a comparison. The specific experiments were performed as follows:(a)
Thermodynamic Parameters—Kendall (TP-Kendall): this experiment constructed the brain thermodynamic model by considering brain regions as systems and voxels as particles. Then, four thermodynamic parameters can be obtained based on this model, namely, the partition function, internal energy, free energy, and entropy. Because 90 brain regions were divided from the cerebral cortex in the AAL template, 360 thermodynamic parameters could be derived as alternative features. According to the cross-validation scheme, 72 parameters with the largest Kendall tau rank correlation coefficients of the training set were chosen and then the same set of features as the test set was used as the input features of the classifiers.(b)
Thermodynamic Parameters—Expert (TP-Expert): the modeling process of this experiment was the same as that of TP-Kendall, except that we merely chose the thermodynamic parameters from 18 brain regions highly associated with AD based on expertise as features for classification. We also obtained 72 features, as each region had four thermodynamic parameters.(c)
Brain Network—Kendall (BN-Kendall): this experiment built the functional brain network model traditionally using Pearson’s correlation coefficient to obtain link intensities between 90 brain regions, and 4005 link strengths between brain regions were calculated. Then, according to the cross-validation scheme, 72 link strengths with the largest Kendall tau rank correlation coefficient over the 4005 link strengths of the training set were chosen and the same set of features was used in the test set, which were the input features of the classifiers.All three experiments selected 72 features for AD detection with the KNN classifier.


## 2.1. Data Acquisition

The data used in this study were obtained from Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (https://adni.loni.usc.edu/). The ADNI was launched in 2003 as a public-private partnership led by principal investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography, other biological markers, and clinical and neuropsychological assessments can be combined to measure the progression of mild cognitive impairment and early Alzheimer’s disease (AD). For up-to-date information, see https://www.adni-info.org. One hundred and sixteen patients with AD (age: 74.6 ± 7.5) and 174 healthy subjects (age: 75.5 ± 6.1) were recruited from ADNI as the AD group and NC (normal control) group, respectively. Demographic information of the subjects is shown in Table 1. There was no significant difference between the two groups in terms of age or sex.Table 1
Demographic characteristics of subjects.
GroupNumberSex (male/female)Age (year)AD11655/6174.6 ± 7.5NC17497/7775.5 ± 6.1


## 2.2. Data Preprocessing

The resting state functional MRI (rs-fMRI) images used in this study were scanned using 3T Philips scanners. The specific operation parameters were as follows: slice thickness, 3.3 mm; number of slices, 48; TR/TE, 3000 ms/30 ms; flip angle, 80°; and imaging matrix, 64 × 64. Each series contained 140 volumes. Data preprocessing was performed using SPM8 and DPARSF [25]. The process was as follows: first, the first ten frames were discarded for magnetization equilibrium. Slicing timing and realigning were performed on the time series. Subjects with head rotation exceeding 2°or head translation exceeding 2 mm were excluded. The Montreal Neurological Institute standard human brain template was used to normalize all the corrected image data. Then, the images were smoothed using a 4 × 4 × 4 Gaussian kernel to decrease spatial noise. The global mean signal was removed to reduce nonneuronal signal fluctuations. The whole brain was divided into 90 regions using an automated anatomical labeling (AAL) template.


## 2.3. Denoising with Point Process Method

The blood oxygen level-dependent (BOLD) signal has been demonstrated to be a description of the hemodynamic response to neural stimulation [26]. Therefore, scientists attempted to search for extreme points of the BOLD signal corresponding to neural stimulation and applied these points to simulate a clear BOLD signal combined with a standard hemodynamic equation [27]. Point process analysis holds that there are some significant feature points in the time series of a complex event and the noise can be decreased to highlight the essential nature by extracting these feature points [28]. Therefore, this study used the point process method to remove noise in the BOLD signal for a high signal-to-noise ratio.The BOLD signal was standardized first. In the time series, the maximum and minimum points were recorded, and the subsequent extreme points were selected as a pair in chronological order. Then, an impulse sequence was obtained by calculating the amplitude increment of every point pair per unit time. Subsequently, we convolved this impulse sequence with the hemodynamic response function proposed by Cohen [29], and a clear BOLD signal was reconstructed.


## 2.4. Statistical Thermodynamic Modeling Method

The canonical ensemble is an important concept in statistical thermodynamics. In statistical thermodynamics, as the system has many different microscopic states under given macroscopic conditions, the measurement of a system is the average of multiple measurements of the system over time. To obtain the measurement at one point, we can measure many identical systems that are under the same macroscopic state at that point and replace the time-average measurement with the average of a large number of identical systems at the same point. The set of a large number of identical systems that are under the same macroscopic state is called an ensemble. Under this premise, the average measurement over time can be replaced by the ensemble average. For an isolated system in equilibrium, in which the energy, volume, and number of particles have all been given, as the probability of possible microscopic states of the system follows the microcanonical distribution, we can analyze the system using the microcanonical ensemble theory. However, it is difficult for most systems to ensure the energy necessary for the microcanonical ensemble analysis. Therefore, researchers generally study a closed system with a given temperature, volume, and number of particles, in which the probability of possible microscopic states of the system follows the canonical distribution. Then, we applied a canonical ensemble to analyze the thermodynamic properties of the system.Some scientists extended the canonical ensemble or its related theory to the research field of electrochemistry [30], black holes [31–33], biology [18, 34], and so on [35, 36], and the achievements in these fields indicated that the canonical ensemble is universal, but not limited to traditional thermodynamic systems. Thus, we attempted to introduce it to the study of neuroscience. Research has shown that the temperature of the human brain is generally maintained at 36.9 ± 0.4°C [37]. In addition, the volumes and voxels of different brain functional regions are certain for each subject in neuroimaging. If we consider the voxels as particles inside the brain region, the number of particles can be determined. Meanwhile, brain regions do not exchange voxels with the outside; in other words, there is no “matter” exchange in the brain region. Then, the brain region basically meets the requirements of a closed system. Therefore, we believe that the brain region can be regarded as a thermodynamic system whose microscopic states conform to the canonical distribution. Then, we can build a brain thermodynamic model based on the theoretical framework of the canonical ensemble.Based on the analysis above, this study proposes a brain thermodynamic model. We regarded the brain functional region as a system consisting of nearly independent particles. Each voxel in neuroimaging corresponds to one nearly independent particle. The BOLD signal intensity of fMRI reflects the metabolic intensity, and a higher BOLD signal indicates a higher metabolic level, leading to an increase in oxygen consumption and an increase in energy consumption. Therefore, the energyE of the voxel was defined as the amplitude of the corresponding reconstructed BOLD signal in this study, and the physiologically activated intensity of each voxel can be evaluated by energy E. Because the relevant computational data had been standardized, all calculated results were dimensionless.Each frame of voxel is considered to be a microscopic state of it, andZi is the partition function of the microscopic state of the voxel at the i-th frame, which can be calculated as(1)Zi=exp−βEi,where Ei is the energy of the voxel at the i-th frame and the specific expression of β is as follows:(2)β=1kT,where k is the Boltzmann constant in thermodynamics, which is equal to 1 to simplify the calculation in this study. The thermodynamic temperature T is 310 K, which is equal to 37°C, the theoretical temperature of the human brain. The whole brain can represent the environment of brain regions.If we consider each time point of the BOLD time series to one voxel as a microscopic state, the partition function of the voxel can be calculated by summing all microscopic states, and the computational formula is described as(3)Zr=∑iZi=∑iexp−βEi,where Zr is the partition function of the voxel and ∑iZi is the sum of its microscopic states at all points in the BOLD time series. Owing to the indistinguishability of identical particles and ignoring the weak interactions among particles, we regard a voxel as a subensemble of the brain region. Therefore, the partition function of the brain region is a combination of all voxels inside. The partition function of the brain region, Zbr, can be derived from Zr:(4)Zbr=1N!∏rZr,where ∏rZr is the quadrature of the partition function of all voxels in the brain region and N is the number of voxels.Based on the theoretical framework of the canonical ensemble, we constructed a brain thermodynamic model regarding brain regions as systems, voxels as particles, the intensity of reconstructed BOLD signals as the energy of particles, and the points of fMRI time series as different microscopic states. We referred to this model as the brain thermodynamic model (BrainTDM). This model attempted to describe the working mechanism of the neural system from a thermodynamic point of view.Based on the BrainTDM proposed in this paper, the internal energy, free energy, and entropy of the neural system can be defined to evaluate the thermodynamic characteristics of brain regions.The internal energy,U, of the brain region can be calculated as(5)U=∑rUr=∑r∑iEiexp−Ei/kTZr.The free energy,F, of the brain region can be calculated as(6)F=U−TS=−kTlnZbr=−kTlog10Zbrlog10e,where e is the Euler number, approximately equal to 2.71828.The entropy,S, of the brain region can be calculated as(7)S=klnZbr−kβ∂∂βlnZbr,U=−∂∂βlnZbr,F=−kTlnZbr,S=klnZbr−kβ∂∂βlnZbr=−FT+UT=U−FT.For a specific brain region, internal energy represents the statistical average energy of all microscopic states in the system, free energy represents the energy of the brain region that could be used to perform external work, and entropy represents chaos of the brain region after being affected by the external environment. Based on thermodynamics, we supposed that when the brain region was activated and consumed external energy to do work, the energy it contained and the energy to do work would both increase and its internal state could be more orderly compared with inactivated brain regions.


## 2.5. Experimental Design

In this experiment, the fMRI of the brain was processed using SPM8 and divided into 90 brain regions with the AAL template. The point process method was applied to reconstruct clear BOLD signals as inputs for the following model. In modeling, brain regions were regarded as systems, voxels were regarded as particles, and the intensity of BOLD signals was regarded as the energy of the particles. Then, a brain thermodynamic model was built based on the ensemble theory. The related thermodynamic parameters were calculated, including the partition functionZ, internal energy U, free energy F, and entropy S. More analyses were performed to explore the potential applications of this model.


### 2.5.1. Experimental Paradigm I

In thermal physics, once the thermodynamic system consumes external energy to perform work, the entropy decreases, and the internal energy as well as the free energy increases. We assume that the brain system also follows this rule. That is, when the brain regions are activated with neurons that consume external energy to fire synchronously and sequentially, the thermodynamic parameters of brain regions will change in the same way. In experiment paradigm I, we evaluated the proposed modeling method by analyzing the thermodynamic parameter features of activated and inactivated cerebral regions. Because the default mode network (DMN) is most commonly shown to be activated when a person is not focused on the outside world and the brain is at wakeful rest [38, 39], we selected the medial prefrontal cortex (mPFC), which is a key brain region of the default network as the object of the study. Meanwhile, considering the constant sound stimulation during fMRI scanning, we also selected the Heschl gyrus (HES), which is mainly located in the primary auditory cortex, as the object of the study [40]. Furthermore, we selected the precentral gyrus (PreCG), which is also referred to as the primary motor region or primary motor cortex that belongs to the task-positive network, and the olfactory cortex (OLF), which is a key component of the limbic system, as the controls [39]. The brain regions selected for the analysis are listed in Table 2.Table 2
Selected brain regions. Activated/inactivated indicates that the brain regions were activated/inactivated during fMRI scanning.
Activated brain regionsInactivated brain regionsLeft/right medial prefrontal cortexLeft/right precentral gyrusLeft/right Heschl gyrusLeft/right olfactory cortexBy considering brain regions as systems, voxels as particles, and the intensity of reconstructed BOLD signals as the energy of particles, we constructed the brain thermodynamic models of the selected regions. The thermodynamic parameters of different brain regions were calculated based on the constructed models, including the partition function, internal energy, free energy, and entropy. By analyzing the differences between the thermodynamic parameters of the activated and inactivated brain regions, we attempted to determine whether the neural system also follows the laws of thermodynamics.


### 2.5.2. Experimental Paradigm II

Experiment paradigm II aimed to explore the potential applications of the brain thermodynamic model proposed in computer-aided diagnosis (CAD) or other realistic scenarios. For this purpose, we tried to recognize subjects with AD, taking parameters derived from the brain thermodynamic model as input features. We performed the same classification task with parameters based on the traditional brain network model as a comparison. The specific experiments were performed as follows:(a)
Thermodynamic Parameters—Kendall (TP-Kendall): this experiment constructed the brain thermodynamic model by considering brain regions as systems and voxels as particles. Then, four thermodynamic parameters can be obtained based on this model, namely, the partition function, internal energy, free energy, and entropy. Because 90 brain regions were divided from the cerebral cortex in the AAL template, 360 thermodynamic parameters could be derived as alternative features. According to the cross-validation scheme, 72 parameters with the largest Kendall tau rank correlation coefficients of the training set were chosen and then the same set of features as the test set was used as the input features of the classifiers.(b)
Thermodynamic Parameters—Expert (TP-Expert): the modeling process of this experiment was the same as that of TP-Kendall, except that we merely chose the thermodynamic parameters from 18 brain regions highly associated with AD based on expertise as features for classification. We also obtained 72 features, as each region had four thermodynamic parameters.(c)
Brain Network—Kendall (BN-Kendall): this experiment built the functional brain network model traditionally using Pearson’s correlation coefficient to obtain link intensities between 90 brain regions, and 4005 link strengths between brain regions were calculated. Then, according to the cross-validation scheme, 72 link strengths with the largest Kendall tau rank correlation coefficient over the 4005 link strengths of the training set were chosen and the same set of features was used in the test set, which were the input features of the classifiers.All three experiments selected 72 features for AD detection with the KNN classifier.


## 2.5.1. Experimental Paradigm I

In thermal physics, once the thermodynamic system consumes external energy to perform work, the entropy decreases, and the internal energy as well as the free energy increases. We assume that the brain system also follows this rule. That is, when the brain regions are activated with neurons that consume external energy to fire synchronously and sequentially, the thermodynamic parameters of brain regions will change in the same way. In experiment paradigm I, we evaluated the proposed modeling method by analyzing the thermodynamic parameter features of activated and inactivated cerebral regions. Because the default mode network (DMN) is most commonly shown to be activated when a person is not focused on the outside world and the brain is at wakeful rest [38, 39], we selected the medial prefrontal cortex (mPFC), which is a key brain region of the default network as the object of the study. Meanwhile, considering the constant sound stimulation during fMRI scanning, we also selected the Heschl gyrus (HES), which is mainly located in the primary auditory cortex, as the object of the study [40]. Furthermore, we selected the precentral gyrus (PreCG), which is also referred to as the primary motor region or primary motor cortex that belongs to the task-positive network, and the olfactory cortex (OLF), which is a key component of the limbic system, as the controls [39]. The brain regions selected for the analysis are listed in Table 2.Table 2
Selected brain regions. Activated/inactivated indicates that the brain regions were activated/inactivated during fMRI scanning.
Activated brain regionsInactivated brain regionsLeft/right medial prefrontal cortexLeft/right precentral gyrusLeft/right Heschl gyrusLeft/right olfactory cortexBy considering brain regions as systems, voxels as particles, and the intensity of reconstructed BOLD signals as the energy of particles, we constructed the brain thermodynamic models of the selected regions. The thermodynamic parameters of different brain regions were calculated based on the constructed models, including the partition function, internal energy, free energy, and entropy. By analyzing the differences between the thermodynamic parameters of the activated and inactivated brain regions, we attempted to determine whether the neural system also follows the laws of thermodynamics.


## 2.5.2. Experimental Paradigm II

Experiment paradigm II aimed to explore the potential applications of the brain thermodynamic model proposed in computer-aided diagnosis (CAD) or other realistic scenarios. For this purpose, we tried to recognize subjects with AD, taking parameters derived from the brain thermodynamic model as input features. We performed the same classification task with parameters based on the traditional brain network model as a comparison. The specific experiments were performed as follows:(a)
Thermodynamic Parameters—Kendall (TP-Kendall): this experiment constructed the brain thermodynamic model by considering brain regions as systems and voxels as particles. Then, four thermodynamic parameters can be obtained based on this model, namely, the partition function, internal energy, free energy, and entropy. Because 90 brain regions were divided from the cerebral cortex in the AAL template, 360 thermodynamic parameters could be derived as alternative features. According to the cross-validation scheme, 72 parameters with the largest Kendall tau rank correlation coefficients of the training set were chosen and then the same set of features as the test set was used as the input features of the classifiers.(b)
Thermodynamic Parameters—Expert (TP-Expert): the modeling process of this experiment was the same as that of TP-Kendall, except that we merely chose the thermodynamic parameters from 18 brain regions highly associated with AD based on expertise as features for classification. We also obtained 72 features, as each region had four thermodynamic parameters.(c)
Brain Network—Kendall (BN-Kendall): this experiment built the functional brain network model traditionally using Pearson’s correlation coefficient to obtain link intensities between 90 brain regions, and 4005 link strengths between brain regions were calculated. Then, according to the cross-validation scheme, 72 link strengths with the largest Kendall tau rank correlation coefficient over the 4005 link strengths of the training set were chosen and the same set of features was used in the test set, which were the input features of the classifiers.All three experiments selected 72 features for AD detection with the KNN classifier.


## 3. Results

Based on the brain thermodynamic model proposed in this paper, we built models of neural systems in experimental paradigms I and II, and the results are shown. To avoid the influence of accidental factors, a tenfold cross-validation strategy was applied to all classification tasks.


### 3.1. Experimental Paradigm I

Based on the modeling method proposed in this paper, we built a brain thermodynamic model of selected brain regions and obtained the energy of activated and inactivated brain regions for each subject. In the NC group, we calculated the average energy of each brain region separately and then obtained the corresponding time series trajectory, as shown in Figure1. The average energy of activated brain regions, mPFC/HES, was significantly higher than that of inactivated brain regions, PreCG/OLF.Figure 1
Average energy trajectories of activated and inactivated brain regions. The red curves represent the average energyE of activated brain regions mPFC.L/mPFC.R/HES.L/HES.R, and the blue curves represent the average energy E of inactivated brain regions PreCG.L/PreCG.R/OLF.L/OLF.R.Based on the model built, the thermodynamic parameters of each brain region were calculated, including the partition function, internal energy, free energy, and entropy. As the value of the partition function was too large, we took the logarithm in the calculation of this parameter. We calculated the mean of the thermodynamic parameters of different brain regions separately over the NC group, and the results are shown in Figure2. There were no significant differences between activated brain regions, mPFC, and HES in these thermodynamic parameters. We observed the same phenomenon between inactivated brain regions, PreCG, and OLF. However, there was a significant difference between the activated and inactivated brain regions. The mean values of partition function and entropy in activated brain regions were significantly lower than those in inactivated brain regions, and the internal energy and free energy were the exact opposite.Figure 2
Thermodynamic parameters of activated and inactivated brain regions. Left and right represent left brain regions and right brain regions, respectively.At-test was applied to the thermodynamic parameters of different brain regions, and the results are shown in Tables 3–6. There were significant differences between the activated and inactivated brain regions in all four thermodynamic parameters: partition function, internal energy, free energy, and entropy. There were no significant differences almost in all four thermodynamic parameters among the four activated brain regions; the same result was observed among the four inactivated brain regions.Table 3
T test for lgZ.
Brain regionmPFC.LmPFC.RHES.LHES.RPreCG.LPreCG.ROLF.LOLF.RmPFC.L1mPFC.R0.251HES.L0.050.011HES.R0.580.960.071PreCG.L0.000.000.000.001PreCG.R0.000.000.000.000.131OLF.L0.000.000.000.000.090.541OLF.R0.000.000.000.000.960.220.131L and R represent left and right brain regions, respectively.P value < 0.05 indicates a significant difference.Table 4
T test for U.
Brain regionmPFC.LmPFC.RHES.LHES.RPreCG.LPreCG.ROLF.LOLF.RmPFC.L1mPFC.R0.671HES.L0.010.011HES.R0.260.150.231PreCG.L0.000.000.000.001PreCG.R0.000.000.000.000.151OLF.L0.000.000.000.000.000.031OLF.R0.000.000.000.000.120.700.101L and R represent left and right brain regions, respectively.P value < 0.05 indicates a significant difference.Table 5
T test for F.
Brain regionmPFC.LmPFC.RHES.LHES.RPreCG.LPreCG.ROLF.LOLF.RmPFC.L1mPFC.R0.251HES.L0.050.011HES.R50.580.960.071PreCG.L0.000.000.000.001PreCG.R0.000.000.000.000.131OLF.L0.000.000.000.000.090.541OLF.R0.000.000.000.000.960.220.131L and R represent left and right brain regions, respectively.P value < 0.05 indicates a significant difference.Table 6
T test for S.
Brain regionmPFC.LmPFC.RHES.LHES.RPreCG.LPreCG.ROLF.LOLF.RmPFC.L1mPFC.R0.111HES.L0.180.031HES.R0.190.530.051PreCG.L0.000.000.000.001PreCG.R0.000.000.000.000.151OLF.L0.000.000.000.000.820.251OLF.R0.000.000.000.000.070.010.221L and R represent left and right brain regions, respectively.P value < 0.05 indicates a significant difference.Furthermore, we attempted to classify brain regions into activated and inactivated regions using thermodynamic parameters as input features with the KNN classifier. Three types of distance measurements were taken in the KNN: correlation distance, cosine distance, and Euclidean distance. The results are presented in Table7. The classification accuracy fluctuated around 88%, reaching the highest value of 88.66% (cosine distance).Table 7
Accuracy of classification for activated/inactivated brain regions (%).
DistanceCorrelationCosineEuclideanAccuracy87.30 ± 0.6388.66 ± 0.6488.57 ± 0.70Correlation, cosine, and Euclidean represent correlation distance, cosine distance, and Euclidean distance.


### 3.2. Experiment Paradigm II

In experiment paradigm II, we tried to apply the brain thermodynamic model proposed to the CAD of AD. For comparison, we designed three experiments, namely, TP-Kendall, TP-Expert, and BN-Kendall. These experiments also used KNN as a classifier based on the correlation distance, cosine distance, and Euclidean distance. The results are presented in Table8. The accuracy of TP-Kendall ranged from 77.14% to 80.47%, with an average of 79.32% and a maximum of 80.47%; the accuracy of TP-Expert ranged from 70.16% to 72.36%, with an average of 71.02% and a maximum of 72.36%, while that of BN-Kendall ranged from 70.51% to 73.28%, with an average of 71.79% and a maximum of 73.28%. Therefore, regardless of whether the Kendall coefficient or expertise was used for feature selection, the recognition accuracies of the proposed brain thermodynamic model were not inferior to those of the brain network model. In particular, with the same feature selection method, the accuracy of AD recognition based on the thermodynamic model (TP-Kendall) was 7.53% higher than that of the brain network model (BN-Kendall). More specifically, the specificity and sensitivity of TP-Kendall were also 7.60% and 7.43% higher than those of BN-Kendall, respectively, which meant that the thermodynamic model was better for identifying both healthy subjects and patients with AD. In addition, this study focused on thermodynamic modeling, rather than the relationship between specific brain regions and AD, so the brain regions to which the selected thermodynamic parameters belonged would not be discussed.Table 8
Results of the classification for AD/NC (%).
Feature selectionDistanceCorrelationCosineEuclideanTP-KendallSpecificity84.93 ± 2.2185.18 ± 2.1879.77 ± 2.41Sensitivity73.48 ± 2.4573.41 ± 2.5773.20 ± 2.38Accuracy80.35 ± 1.7180.47 ± 1.7277.14 ± 1.66TP-ExpertSpecificity73.32 ± 1.3071.81 ± 1.3371.73 ± 1.17Sensitivity70.91 ± 1.9167.69 ± 2.0368.74 ± 1.94Accuracy72.36 ± 1.0070.16 ± 1.0270.53 ± 1.01BN-KendallSpecificity73.55 ± 2.5977.99 ± 2.5275.55 ± 2.60Sensitivity65.96 ± 3.8166.22 ± 3.1965.62 ± 3.58Accuracy70.51 ± 2.0473.28 ± 1.8471.58 ± 1.95TP-Kendall, TP-Expert, and BN-Kendall represent three different methods of feature selection designed in Section2.5.2. Correlation, cosine, and Euclidean represent correlation distance, cosine distance, and Euclidean distance.In this study, both TP-Kendall and BN-Kendall experiments used Kendall for feature selection. Based on these two experiments, we analyzed the impact of different input features on AD recognition using the KNN classifier as described above. The results are presented in Figure3. Regardless of the number of input features, the accuracy of AD recognition with thermodynamic parameters from the proposed model was always higher than that with link strengths from the brain network model. Furthermore, the accuracy of AD recognition based on the brain thermodynamic model reached 86.34% with only 360 input features in the correlation distance. However, the classification accuracy based on the brain network model peaked only at 85.82% with 4005 input features in the correlation distance.Figure 3
Impact of the feature number on accuracy of AD/NC classification.


## 3.1. Experimental Paradigm I

Based on the modeling method proposed in this paper, we built a brain thermodynamic model of selected brain regions and obtained the energy of activated and inactivated brain regions for each subject. In the NC group, we calculated the average energy of each brain region separately and then obtained the corresponding time series trajectory, as shown in Figure1. The average energy of activated brain regions, mPFC/HES, was significantly higher than that of inactivated brain regions, PreCG/OLF.Figure 1
Average energy trajectories of activated and inactivated brain regions. The red curves represent the average energyE of activated brain regions mPFC.L/mPFC.R/HES.L/HES.R, and the blue curves represent the average energy E of inactivated brain regions PreCG.L/PreCG.R/OLF.L/OLF.R.Based on the model built, the thermodynamic parameters of each brain region were calculated, including the partition function, internal energy, free energy, and entropy. As the value of the partition function was too large, we took the logarithm in the calculation of this parameter. We calculated the mean of the thermodynamic parameters of different brain regions separately over the NC group, and the results are shown in Figure2. There were no significant differences between activated brain regions, mPFC, and HES in these thermodynamic parameters. We observed the same phenomenon between inactivated brain regions, PreCG, and OLF. However, there was a significant difference between the activated and inactivated brain regions. The mean values of partition function and entropy in activated brain regions were significantly lower than those in inactivated brain regions, and the internal energy and free energy were the exact opposite.Figure 2
Thermodynamic parameters of activated and inactivated brain regions. Left and right represent left brain regions and right brain regions, respectively.At-test was applied to the thermodynamic parameters of different brain regions, and the results are shown in Tables 3–6. There were significant differences between the activated and inactivated brain regions in all four thermodynamic parameters: partition function, internal energy, free energy, and entropy. There were no significant differences almost in all four thermodynamic parameters among the four activated brain regions; the same result was observed among the four inactivated brain regions.Table 3
T test for lgZ.
Brain regionmPFC.LmPFC.RHES.LHES.RPreCG.LPreCG.ROLF.LOLF.RmPFC.L1mPFC.R0.251HES.L0.050.011HES.R0.580.960.071PreCG.L0.000.000.000.001PreCG.R0.000.000.000.000.131OLF.L0.000.000.000.000.090.541OLF.R0.000.000.000.000.960.220.131L and R represent left and right brain regions, respectively.P value < 0.05 indicates a significant difference.Table 4
T test for U.
Brain regionmPFC.LmPFC.RHES.LHES.RPreCG.LPreCG.ROLF.LOLF.RmPFC.L1mPFC.R0.671HES.L0.010.011HES.R0.260.150.231PreCG.L0.000.000.000.001PreCG.R0.000.000.000.000.151OLF.L0.000.000.000.000.000.031OLF.R0.000.000.000.000.120.700.101L and R represent left and right brain regions, respectively.P value < 0.05 indicates a significant difference.Table 5
T test for F.
Brain regionmPFC.LmPFC.RHES.LHES.RPreCG.LPreCG.ROLF.LOLF.RmPFC.L1mPFC.R0.251HES.L0.050.011HES.R50.580.960.071PreCG.L0.000.000.000.001PreCG.R0.000.000.000.000.131OLF.L0.000.000.000.000.090.541OLF.R0.000.000.000.000.960.220.131L and R represent left and right brain regions, respectively.P value < 0.05 indicates a significant difference.Table 6
T test for S.
Brain regionmPFC.LmPFC.RHES.LHES.RPreCG.LPreCG.ROLF.LOLF.RmPFC.L1mPFC.R0.111HES.L0.180.031HES.R0.190.530.051PreCG.L0.000.000.000.001PreCG.R0.000.000.000.000.151OLF.L0.000.000.000.000.820.251OLF.R0.000.000.000.000.070.010.221L and R represent left and right brain regions, respectively.P value < 0.05 indicates a significant difference.Furthermore, we attempted to classify brain regions into activated and inactivated regions using thermodynamic parameters as input features with the KNN classifier. Three types of distance measurements were taken in the KNN: correlation distance, cosine distance, and Euclidean distance. The results are presented in Table7. The classification accuracy fluctuated around 88%, reaching the highest value of 88.66% (cosine distance).Table 7
Accuracy of classification for activated/inactivated brain regions (%).
DistanceCorrelationCosineEuclideanAccuracy87.30 ± 0.6388.66 ± 0.6488.57 ± 0.70Correlation, cosine, and Euclidean represent correlation distance, cosine distance, and Euclidean distance.


## 3.2. Experiment Paradigm II

In experiment paradigm II, we tried to apply the brain thermodynamic model proposed to the CAD of AD. For comparison, we designed three experiments, namely, TP-Kendall, TP-Expert, and BN-Kendall. These experiments also used KNN as a classifier based on the correlation distance, cosine distance, and Euclidean distance. The results are presented in Table8. The accuracy of TP-Kendall ranged from 77.14% to 80.47%, with an average of 79.32% and a maximum of 80.47%; the accuracy of TP-Expert ranged from 70.16% to 72.36%, with an average of 71.02% and a maximum of 72.36%, while that of BN-Kendall ranged from 70.51% to 73.28%, with an average of 71.79% and a maximum of 73.28%. Therefore, regardless of whether the Kendall coefficient or expertise was used for feature selection, the recognition accuracies of the proposed brain thermodynamic model were not inferior to those of the brain network model. In particular, with the same feature selection method, the accuracy of AD recognition based on the thermodynamic model (TP-Kendall) was 7.53% higher than that of the brain network model (BN-Kendall). More specifically, the specificity and sensitivity of TP-Kendall were also 7.60% and 7.43% higher than those of BN-Kendall, respectively, which meant that the thermodynamic model was better for identifying both healthy subjects and patients with AD. In addition, this study focused on thermodynamic modeling, rather than the relationship between specific brain regions and AD, so the brain regions to which the selected thermodynamic parameters belonged would not be discussed.Table 8
Results of the classification for AD/NC (%).
Feature selectionDistanceCorrelationCosineEuclideanTP-KendallSpecificity84.93 ± 2.2185.18 ± 2.1879.77 ± 2.41Sensitivity73.48 ± 2.4573.41 ± 2.5773.20 ± 2.38Accuracy80.35 ± 1.7180.47 ± 1.7277.14 ± 1.66TP-ExpertSpecificity73.32 ± 1.3071.81 ± 1.3371.73 ± 1.17Sensitivity70.91 ± 1.9167.69 ± 2.0368.74 ± 1.94Accuracy72.36 ± 1.0070.16 ± 1.0270.53 ± 1.01BN-KendallSpecificity73.55 ± 2.5977.99 ± 2.5275.55 ± 2.60Sensitivity65.96 ± 3.8166.22 ± 3.1965.62 ± 3.58Accuracy70.51 ± 2.0473.28 ± 1.8471.58 ± 1.95TP-Kendall, TP-Expert, and BN-Kendall represent three different methods of feature selection designed in Section2.5.2. Correlation, cosine, and Euclidean represent correlation distance, cosine distance, and Euclidean distance.In this study, both TP-Kendall and BN-Kendall experiments used Kendall for feature selection. Based on these two experiments, we analyzed the impact of different input features on AD recognition using the KNN classifier as described above. The results are presented in Figure3. Regardless of the number of input features, the accuracy of AD recognition with thermodynamic parameters from the proposed model was always higher than that with link strengths from the brain network model. Furthermore, the accuracy of AD recognition based on the brain thermodynamic model reached 86.34% with only 360 input features in the correlation distance. However, the classification accuracy based on the brain network model peaked only at 85.82% with 4005 input features in the correlation distance.Figure 3
Impact of the feature number on accuracy of AD/NC classification.


## 4. Discussion

The brain is the most complex system in the world. Brain modeling is an effective way to explore the work mechanism of the brain and is one of the hottest research topics all the time. Some scientists have attempted to model the brain based on neurodynamic principles and methods, from the microscopic ion channel layer to the macroscopic neural network layer [41]. Most initial studies have focused on the basic working mechanism of neurons or neural connectivity [1, 2, 42]. Later, scientists began to use neurodynamic methods to explain some complicated brain functions, such as emotion, language acquisition, and language comprehension [3, 43, 44], and then developed a method to analyze the pathological mechanisms of epilepsy, AD, and other neural system diseases [45, 46]. In addition, from another perspective, in brain differentiation and integration, scientists have proposed modeling methods for brain networks. Early research has mainly focused on exploring how to construct brain networks from the neuron layer to the functional cortex layer. The structure equation, causality, correlation, and consistency were applied to the definition of the structure or functional connectivity of the brain [6, 47]. Moreover, scientists have used brain networks to explore the impact of brain lesions on coupling brain regions [48] and the pathological mechanism of neural system diseases, such as brachial plexus injury and AD [49, 50]. Some studies have attempted to extract the characteristics of the brain network for the identification, prediction, and prognosis of diseases and obtained very good results [51–53].These modeling methods for the brain have obtained many meaningful results. However, the brain is a typical multidimensional complex system, and the understanding of the brain is inadequate, especially from the point of view of the system.Thermodynamics, a major branch of physics, mainly studies the thermal properties of an object from the perspective of energy conversion at the macrolevel. Based on the observed phenomenon in the experiment, thermodynamics applies mathematical modeling methods to draw relevant conclusions by logical deduction. Thus, it belongs to phenomenological theory, indicating that the conclusions drawn from this study are highly reliable and universal. Therefore, researchers have always tried to extend and apply related mature theories and concepts of thermodynamics to system modeling and analysis in other research fields, such as mechanical engineering [12–14], astronomy [15, 16], biology [17, 18], economics [54, 55], and other nonclassical physical fields [56–60], in order to realize generalized physical state analysis of objects at the system level in different areas. In this paper, we assumed that the energy conversion of the cerebral cortex is the physical foundation of the brain to implement various complex functionalities. Thus, we assumed that the brain also follows the related macroscopic laws of energy conversion in thermodynamics. When neurons are activated, the brain also consumes the external energy input to work as thermodynamic systems, which leads to increased internal energy, increased free energy, and decreased entropy of the system. Based on the above assumption, we proposed a thermodynamic model of the brain for the first time. Using brain regions as systems, voxels as particles, and the intensity of BOLD signals as the energy of particles, this method built the BrainTDM and tried to explain the work mechanism of the brain based on the canonical ensemble theory from the perspective of energy conversion in thermodynamics.In experimental paradigm I, we selected two pairs of activated brain regions and two pairs of inactivated brain regions in the resting state as objects. Then, the BrainTDMs of the regions were built and the thermodynamic parameters were calculated. The descriptive statistics and thet test results of them were shown in Figure 2 and Tables 3–6, which demonstrated that the internal energy and free energy of activated brain regions (mPFC and HES) were all much higher than those of the inactivated regions (PreCG and OLF), while the opposite was true for the partition function and entropy. This validates the assumption that the brain obeys the laws of thermodynamics at the system level. When activated, the neurons of the specific region burn energy to generate electrical impulses for information transmission, which is doing work just as those thermodynamic systems, leading to the corresponding thermodynamic parameter change of regions: internal energy and free energy increase and entropy decrease. In other words, the brain also follows the laws of energy conversion in thermodynamic systems at the macroscopic level. This was the most important discovery of this study. The results demonstrated that the brain thermodynamic model proposed in this paper is workable and constructing a brain model based on the canonical ensemble theory of thermodynamics is feasible.We also observed that the energy of activated brain regions was significantly higher than that of inactivated brain regions as shown in Figure1. This is consistent with the common notion that the activation levels of the mPFC and HES were significantly higher than those of PreCG and OLF in the resting state. For activated regions, the enhancement of DMN activation in the resting state has been demonstrated in many studies, which is consistent with the observation of fMRI imaging in experiments. The activation of the auditory related HES was due to the continuous noise from nuclear magnetic resonance equipment in the experiment. For inactivated regions, PreCG and OLF are related to the control of movement and olfaction, respectively, and were considered to be inactivated due to the resting state. In addition, we tried to classify the brain regions into activated and inactivated regions using the KNN classifier using thermodynamic parameters as input features. The experimental results in Table 7 showed that the classification accuracy remained at approximately 88% and reached the highest 88.66% in three types of distance measurements, correlation distance, cosine distance, and Euclidean distance, which indicated that thermodynamic parameters could actually reflect the differences in the energy conversion state between activated and inactivated brain regions. In thermodynamics, the internal energy, free energy, entropy, and other thermodynamic parameters are regarded as the common state functions describing the state of the system. This study implied that thermodynamic parameters obtained from the proposed model could also be used as state functions of brain regions to characterize the activated state of brain regions.Experimental paradigm II attempted to apply the brain thermodynamic model proposed to discriminate diseases of the neural system. With the same feature selection method, the detection of AD based on thermodynamic parameters from the proposed model achieved better results than using link strengths from the traditional brain network model, which was shown in Table8. According to Figure 3, the accuracy of AD recognition based on the brain thermodynamic model reached 86.34% with only 360 input features. This demonstrated that brain thermodynamic parameters contain some pathological information of neural diseases in nature. This information must be essential and critical in describing the change of brain with diseases, as Figure 3 showed that we could obtain 68.10% accuracy of AD detection with only 18 thermodynamic parameters as input features in cosine distance. The above results indicate that the brain thermodynamic model proposed in this paper not only explains the basic working mechanism of the neural system from thermodynamics, but also has the potential to be applied to the recognition and prediction of neural system diseases. On the other hand, the experimental results also implied that the detection of neural system diseases may benefit from the laws of energy conversion in neural systems.


## 5. Conclusion

In the study of brain modeling, scientists generally focused on neurodynamics or brain network. While brain is a multilayered and multidimensional complex system, we attempted to explore the brain from a new perspective. Drawing on the application of thermodynamics in other fields, we proposed a model of the brain on the system level based on thermodynamics, instead of defining or using some parameters borrowed from thermodynamic parameters. Specifically, this study mapped the neural system to the thermodynamic system by taking brain regions as systems, voxels as particles, and the intensity of BOLD signals as the energy of particles. Based on the canonical ensemble theory, the BrainTDM was built to explore the work mechanism of the brain. The experiment results demonstrated the feasibility and rationality of modeling the neural system from the perspective of thermodynamics, and, on the other hand, they verified the hypothesis that the brain also follows the laws of thermodynamics. In addition, the study also indicated the positive effects of the laws of energy conversion on the detection of neural system diseases, implying that the potential of the model can be applied to auxiliary diagnosis. However, the fMRI only describes the metabolism activities of brain on a macrolevel (brain regions), and we hope that there will be possible validation of the proposed thermodynamic model from microlevel in the future, such as the microscopic signals (spikes of neuron) from an implantable brain-computer interface. Furthermore, some working mechanisms of the brain may be discovered by combining the thermodynamic properties of macroscopic and microscopic neural signals.



---

*Source: 9961864-2021-12-21.xml*