Long article review: entropy, free energy, symmetry and dynamics in the brain

Thesis title ：

Entropy, Free Energy, Symmetry and Dynamics in the Brain

Thesis link ：https://iopscience.iop.org/article/10.1088/2632-072X/ac4bec/meta

 

Abstract ：

Neuroscience is the home of concepts and theories rooted in various fields , Including information theory 、 Dynamical system theory and Cognitive Psychology . But not all fields can be connected coherently , Some concepts are incommensurable , And domain specific terms create obstacles to integration . For all that , Conceptual integration is still a form of understanding that provides intuition and consolidation , There is no conceptual integration , Progress will be headless . This paper focuses on the integration of deterministic and stochastic processes within the framework of information theory , Thus, the information entropy and free energy are connected with the emergence dynamics mechanism and self-organization mechanism in the brain network . We identified the neuronal population （neuronal populations） Basic properties of , This attribute causes the equivariant matrix to appear in the network , And in this network , Complex behaviors can be represented by structured flows on manifolds , So as to establish an internal model related to brain function theory . We propose a neural mechanism to produce internal models from the symmetry breaking of brain network connections . Emerging ideas illustrate how free energy is related to internal models , And how they are produced in the neural basement .

  introduction

Prediction code （Predictive coding） It is one of the most influential brain function theories in contemporary times [1-3]. The theory is based on the intuition that the brain operates as a Bayesian inference system , In this way, the internal generation model can predict the external world . These predictions are constantly compared with sensory input , Form prediction errors and update internal models （ See the picture 1）. From a theoretical point of view , It is so fascinating to express brain function in the form of predictive coding , Because it solves too many esoteric concepts in different fields . In this way, there is an opportunity to link abstract concepts into an integrated framework , These abstract concepts such as dynamics 、 Deterministic action and stochastic action 、 Emergence 、 Self organizing [4-6]、 Information 、 entropy 、 Free energy [7,8]、 Steady state and so on . This cross domain integration is consistent with an intuitive understanding of these complex abstract concepts , Although it is usually reserved for ease of handling , All levels of complexity for a given concept are not described equivalently . for example , Use simple internal models in predictive coding theory （ For example, when making decisions , Dynamics is reduced to transformation ）, But simple models are difficult to generalize to more complex behaviors . When the focus is on the reasoning part of the process , This approach is taken for granted , And we don't want to do this . In fact, we emphasize the neural basis of internal models in brain activation and the theory of complex behavior emergence . however , Considering the concept of information theory （ Especially entropy and free energy ） Importance , In predictive coding , Such an attempt requires them to integrate with the probability distribution function and related deterministic and stochastic effects existing in contemporary brain network models （ See the same view [9-11]）.

chart 1： Bayesian brain hypothesis （Bayesian brain hypothesis） An illustration of the reasoning process of . The generative model on the left side of the figure shows internal neurodynamics , Through perception and action （ On the right side of the picture ） Exchange information with the outside world to update the model .

Karl Friston It is the first time to put forward the principle of taking free energy as a brain function [12-14], This paper expounds the self adaptation in Mathematics 、 How self-organizing systems resist natural （ Thermodynamic ） Disorderly tendency . Over time , The principle of free energy has been derived from the Helmholtz machine （Helmholtz machine） Developed from the concept of free energy used in , It is used to explain the cortical response in the context of predictive coding , And gradually developed into the general principles of agents , This is also called active reasoning [15]. Bayesian inference process and maximum information criterion （maximum information principle） In fact, both of them can be restated as the problem of free energy minimization . Although these concepts about free energy are used in two related frameworks , But they are not completely equivalent . Ambiguous or because of such facts , That is, their general form is similar to the Helmholtz free energy of thermodynamics （Helmholtz free energy）, But it comes from two different reasoning routes （ See [15]）. The first is the so-called “ Free energy from constraints ”, Corresponding to the minimum free energy under the maximum information criterion , Represents the trade-off between deterministic constraints and random actions [7]. Our main consideration is this type of free energy and its constraints . The other is variational free energy （variational free energy） And it is related to Bayesian brain hypothesis . This concept of free energy stems from the restatement of Bayesian rules , That is, to find the minimum relative entropy （KL- The divergence ） The probability distribution of this optimization problem , The relative entropy represents the error deviating from the exact Bayesian posterior .

Make sure the constraint is Structured flows on manifolds （structured flows on manifolds, SFMs）[16,17] The frame is expressed in the form of dynamics , Among them, structured flow belongs to the low dimensional dynamic system generated in the network , Therefore, it is the main candidate to represent the internal model in brain theory . Through the mediation of probability distribution , Empirical functions （ Such as discharge rate 、 energy 、 Variance and so on ） The correlations on both contain free energy and SFMs The connection between the two , among , Probability distribution is formed under the interaction of certainty and randomness in the system .Ilya Prigogine When explaining the meaning of entropy , The close relationship between these functions is explained in detail . ad locum , Time （ The concept of ） Beyond the concept of repetition and degradation , Reach the concept of constructive irreversibility , As life system shows , Through the exchange of entropy with its environment, we can perpetuate ourselves . Biology is believed to need to engrave irreversible time on matter . In the context of neuroscience , This reminds us of Ingvar The assumption of , That is, the brain has the ability to simulate through its time polarized structure “ Future memory ”[20], And the brain can maintain and navigate the past distributed in different brain regions 、 Present and future experience .
Although the mathematical formula of entropy first appeared in classical thermodynamics （ Involving things like heat 、 Macroscopic quantities such as temperature and energy exchange ） In the background of , Later statistical mechanics expressed entropy as the probability logarithmic function of the system in different possible microscopic states . The function form of the latter is the same as Shannon information entropy , among , The probability in Shannon information entropy expression is the probability of different possible values of variables （ see 2.2 section ）. Such as Edwin T Jaynes stay 1957 As Nian said , At a deeper level , As a measure of uncertainty expressed in probability distribution , These two concepts of entropy are closely linked ; In two cases , It is regarded as the prediction problem of probability distribution constrained by observation , The probability distribution with maximum entropy is the only unbiased choice [7].

The above brief discussion on the meaning of entropy , stay Hermann Haken Established collaborators （synergetics） Find a theoretical framework in , The framework formally integrates away from the balance system （far-from-equilibrium systems） Medium dissipative structure （dissipative structures） Emerging mathematical forms . Nonlinearity and instability lead to emergence and complexity mechanism , Entropy and fluctuation lead to irreversibility and unpredictability . The essence of this dynamics naturally leads to the concept of probability , It makes up for our inability to accurately depict the unique trajectory of the system . Synergetics have always been the driving force to apply these principles to other fields , Especially life science and neuroscience . Clarify the duality of deterministic and stochastic effects and elaborate on how they occur in the brain , It will create a stage for our vision . therefore , The goal of this article is to unravel these complex relationships , So as to unify the seemingly unrelated brain dynamics model framework .

Considering the complementarity of these concepts , And for the convenience of readers who are interested in macro brain dynamics modeling from different backgrounds , Let's take a step back , Review some basic concepts related to probability theory and information theory .

  The information theoretic framework of the brain —— System evolution modeling

Edwin T Jaynes emphasize , The major conceptual progress of information theory lies in a clear quantity , That is, the information entropy of uncertainty quantity is expressed by probability distribution , It intuitively reflects that the wide distribution represents more uncertainty than the peak distribution , At the same time, all other conditions consistent with this intuition are met [7]. Without any information , The corresponding probability distribution has no information at all and the entropy information disappears . With some deterministic constraints , Such as the measured value of the average value of physical observation , Using the Lagrange equation of the first kind （Lagrange's theory of first kind）, We can solve the corresponding maximum information entropy under this constraint （ Equivalent to minimizing free energy [7,12]） Probability distribution of . In that sense , The consideration of entropy takes precedence over the discussion of deterministic effects , And it is concluded that it should conform to the maximum entropy distribution , The conclusion is based on the fact , That is, the maximum entropy distribution conforms to the results of all deterministic effects , But beyond that , I can't deny other influences like missing information . When entropy is the main concept , Free energy 、 Probability distribution function and SFMs The relationship between other related quantities is naturally established . These relationships express themselves in the real world through relevance , At the physical level, it is also available by measuring the function of system state variables , And in principle, it is allowed to systematically estimate all parameters of the system .

2.1. Predictive coding and its modeling framework

The content of prediction coding and its related concepts , Essentially based on three core equations ：

The first equation establishes a simplified form of Bayesian theorem , In terms of probability distribution function p To said , among p(x,y) It's a state variable x and y The joint probability of ,p(y|x) Is a given variable x State variables y Conditional probability of . In the Bayesian framework , Parameters and state variables have similar status in a certain sense , That is, they can be described by distribution and enter the probability function in the form of parameters p in . for example , The given parameters k, state x and y The joint probability of can be written as p(x,y|k), Determine a given set of parameter values k Under the condition of , Get a set of data (x,y) The possibility of , among k The prior distribution of is p(k). A priori represents our understanding of the model and initial values .

The second equation , Langevin equation （ Technically, there are more precise forms of calculus that can be used in, for example [4] Find , The assumption here is ITO calculus ） Determine the generation model , The brain activity at the neurogenic level consists of N Dimensional state vector Q=(x,y,...)∈R^N  Express ,f(Q,k) Represents deterministic impact , Expressed as state based Q And parameters k（ Or a set of parameters {k}） Of M Dimensional flow vector f.v∈R^N The fluctuation effect is determined , It is usually assumed that <v_i(t)v_j(t')>=cδ_ijδ(t-t') The Gaussian white noise of , among δ_ij yes Kronecker-delta function ,δ(t-t') yes Dirac function . Noise effects （ Including multiplicative noise or colored noise ） There may be a more general formula , Please refer to the relevant literature by yourself .

The third equation establishes the observation model , Through the forward model h(Q) And measurement noise w Put the source activity Q(t) And the sensor signal obtained by experiment Z(t) Connect . For EEG measurement ,h yes Maxwell The gain matrix determined by the equation ; For fMRI measurements ,h By neurovascular coupling and hemodynamics Ballon–Windkessel The model gives . In the current situation , The observation model is irrelevant , But it is very important in real world applications , And it is usually the main pollution factor in model inversion and parameter estimation . For completeness , We mention these engineering problems here , But for the sake of simplicity , hypothesis h It is an identity operation with zero measurement noise , therefore Z=Q.

Predictive coding involves a major research field in behavioral neuroscience [21,22], Especially dedicated to perception - Ecological psychology of action and dynamic system research （ecological psychology）[23,24]. Here ,James J Gibson Emphasize the importance of environment [25], In particular, how the environment of an organism provides it with the perception of various behaviors . This perception - The action cycle closely matches the prediction and update cycle of the internally generated model in the prediction coding . But ecological psychology emphasizes one nuance , That is, ecologically available information —— Relative to the external or internal feeling —— Cause perception - The emergence of action dynamics .Scott Kelso And colleagues have made great contributions to the formalization of this framework , And developed an experimental paradigm , The dynamic characteristics of the internal model used to test the coordination between organisms and the environment [26-28]. These paradigms are theoretically influenced by Hermann Haken Synergetics [4] Inspired by the , Synergetics has always been the basis of self-organization theory . This has led to a lot of research focusing on the transition between perception and action states , Including the modeling of Bi sensory and multi sensory motor coordination [29-32] And system experimental test [33-44]. These methods are then extended to a wider range of paradigms [45-53], It aims to extract the main characteristics of the interaction between organisms and the environment [22]. The substantial evidence produced by many works is conducive to the dynamic description of behavior , These evidences can be classified as SFMs frame , For behavior [48] And the brain [54-56] Medium function perception - Action variables .

2.2. Maximum information principle

We calculate the corresponding probability distribution p_i, So as to understand the process of determining certainty and randomness , This process determines the state variables x Discrete values of x_i. Shannon proved such a remarkable fact , That is, there is a quantity of information H(p₁,...p_n), The uncertainty expressed by these probability distributions can be uniquely measured [8]. He pointed out in his initial proof , Requirements of three basic conditions , Especially the combination law of events and probability , Naturally, the following expression is formed .
 
ad locum ,K Is a positive constant . measure H Directly corresponding to the expression of entropy in statistical mechanics , It's called information entropy [7]. The discussion of its attributes is usually carried out from the perspective of the detailed probability related to the amount of information available to the observer . Without any differential information , Laplace's principle of insufficient reason （principle of insufficient reason） Assign equal probability to two events . This subjective School of thought believes that probability is the expression of human ignorance , Express our expectations about whether the event will happen . This idea is the basis of predictive coding theory and its explanation of the cognitive process of brain production . The objective School of thought is rooted in Physics , It is believed that probability is the objective attribute of events , In principle, this attribute can always be measured by the frequency of events in random experiments . Here , By studying the deterministic and stochastic processes under predictive coding , We hope to remain neutral on this point . The reference to the role of certainty and randomness usually belongs to the language of the objective School , among , The subsequent interpretation within the framework of predictive coding is part of the Subjective School . The generation model in predictive coding represents two types of actions through Langevin equation and forms a probability function , such , Through empirical measure function <g(x)>, It provides us with a way to obtain this information . among , Angle brackets indicate the expected value .

g(x) Correlation and normalization requirements Σ_ip_i=1, Express the constraints caused by certainty and randomness , Under these constraints, information entropy has a maximum . In addition to the assignment that maximizes the information entropy , Any other assignment will introduce other deterministic deviations or arbitrary assumptions , And we didn't make these assumptions . This insight is Edwin T Jaynes The essence of maximum information criterion [7]. therefore , Adopt the method commonly used in classical mechanics , introduce Lagrange Parameters λ_i, We maximize information entropy H Then we get the following formula ：

The expression of probability distribution is generalized as event x_i Multi measure function of g_i(x_i). The entropy of this distribution S Maximize , Write it down as ：

The relationship between empirical measure and maximum information entropy is determined by partition function Z determine （ Such as status and （Zustandssumme） In the ）：

Contact the following ：

Measured by experience <g_j(x)> The obtained correlation is related to the stationary probability distribution function through these equations , And further linked to the generation model expressed by Langevin equation （ That is, deterministic influence ）.