10.1.1.87.6917physics of life from first principles.pdf

8/7/2019 10.1.1.87.6917Physics of Life from First Principles.pdf

http://slidepdf.com/reader/full/1011876917physics-of-life-from-first-principlespdf 1/86

EJTP 4, No. 16(II) (2007) 11–96 Electronic Journal of Theoretical Physics

Physics of Life from First Principles

Michail Zak∗Jet Propulsion Laboratory California Institute of Technology,

Pasadena, CA 91109 USA

Received 25 January 2007, Accepted 15 May 2007, Published 20 December 2007Abstract: The objective of this work is to extend the First Principles of Newtonian mechanics to include modeling of behavior of

Livings. One of the most fundamental problems associated with modeling life is to understand a mechanism of progressive evolution of

complexity typical for living systems. It has been recently recognized that the evolution of living systems is progressive in a sense that it is

directed to the highest levels of complexity if the complexity is measured by an irreducible number of different parts that interact in a well-

regulated fashion. Such a property is not consistent with the behavior of isolated Newtonian systems that cannot increase their complexity

without external forces. Indeed, the solutions to the models based upon dissipative Newtonian dynamics eventually approach attractors

where the evolution stops, while these attractors dwell on the subspaces of lower dimensionality, and therefore, of the lower complexity. If

thermal forces are added to mechanical ones, the Newtonian dynamics is extended to the Langevin dynamics combining both mechanics

and thermodynamics effects; it is represented by stochastic differential equations that can be utilized for more advanced models in which

randomness stands for multi-choice patterns of behavior typical for living systems. However, even those models do not capture the main

property of living systems, i.e. their ability to evolve towards increase of complexity without external forces. Indeed, the Langevin dynamics

is complemented by the corresponding diffusion equation that describes the evolution of the distribution of the probability density over the

state variables; in case of an isolated system, the entropy of the probability density cannot decrease, and that expresses the second law of

thermodynamics. From the viewpoint of complexity, this means that the state variables of the underlying system eventually start behavingin a uniform fashion with lesser distinguished features, i.e. with lower complexity. Reconciliation of evolution of life with the second law of

thermodynamics is the central problem addressed in this paper . It is solved via introduction of the First Principle for modeling behavior of living

systems. The structure of the model is quantum-inspired: it acquires the topology of the Madelung equation in which the quantum potential

is replaced with the information potential. As a result, the model captures the most fundamental property of life: the progressive evolution,

i.e. the ability to evolve from disorder to order without any external interference. The mathematical structure of the model can be obtained

from the Newtonian equations of motion (representing the motor dynamics) coupled with the corresponding Liouville equation (representing

the mental dynamics) via information forces. The unlimited capacity for increase of complexity is provided by interaction of the system with

its mental images via chains of reections: What do you think I think you think. . . ?. All these specic non-Newtonian properties equip the

model with the levels of complexity that match the complexity of life, and that makes the model applicable for description of behaviors of

ecological, social and economics systems.

c Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Physics of life, Biophysics, Complex SystemsPACS (2006): 87.10.+e, 89.75.k, 89.75.Fb, 87.16.Ac

“Life is to create order in the disordered environment against the second law of thermodynamics”. E. Schr¨odinger, 1945.

1. Introduction

It does not take much knowledge or experience to distinguish a living matter frominanimate in day-to-day situations. Paradoxically, there is no formal denition of life thatwould be free of exceptions and counter-examples. There are at least two reasons for that.

∗ [email protected]



12 Electronic Journal of Theoretical Physics 4, No. 16(II) (2007) 11–96

Firstly, many complex physical and chemical phenomena can mimic prints of life so closelythat special methods are required to make the distinction. Secondly, extraterrestrial life,in principle, can be composed of components which are fundamentally different from

those known on Earth. Therefore, the main objective of this paper is to formulate someinvariants of life in terms of phenomenology of behavior.

Modeling of life can be performed on many different levels of description. While thereis no universal agreement on the denition of life, scientists generally accept that thebiological manifestation of life exhibits the following phenomena (Wikipedia): Orga-nization - Living things are composed of one or more cells, which are the basic unitsof life. Metabolism - Metabolism produces energy by converting nonliving materialinto cellular components (synthesis) and decomposing organic matter (catalysis). Livingthings require energy to maintain internal organization (homeostasis) and to produce the

other phenomena associated with life. Growth - Growth results from a higher rate of synthesis than catalysis. A growing organism increases in size in all of its parts, ratherthan simply accumulating matter. The particular species begins to multiply and expandas the evolution continues to ourish. Adaptation - Adaptation is the accommodationof a living organism to its environment. It is fundamental to the process of evolution andis determined by the organism’s heredity as well as the composition of metabolized sub-stances, and external factors present. Response to stimuli - A response can take manyforms, from the contraction of a unicellular organism when touched to complex reactionsinvolving all the senses of higher animals. A response is often expressed by motion: theleaves of a plant turning toward the sun or an animal chasing its prey. Reproduction- The division of one cell to form two new cells is reproduction. Usually the term isapplied to the production of a new individual (asexually, from a single parent organism,or sexually, from at least two differing parent organisms), although strictly speaking italso describes the production of new cells in the process of growth.

In this paper, we will address only one aspect of Life: a biosignature , i.e. mechan-ical invariants of Life, and in particular, the geometry and kinematics of behavior of Livings disregarding other aspects of Life. By narrowing the problem in this way, wewill be able to extend the mathematical formalism of physics’ First Principles to includedescription of behavior of Livings. In order to illustrate the last statement, consider thefollowing situation. Suppose that we are observing trajectories of several particles: someor them physical (for instance, performing a Brownian motion), and others are biological(for instance, bacteria), Figure 1. Is it possible, based only upon the kinematics of theobserved trajectories, to nd out which particle is alive? The test for the proposed modelis to produce the correct answer.

Thus, the objective of this paper is to introduce a dynamical formalism describingthe behavior of Livings. All the previous attempts to develop models for so called ac-tive systems (i.e., systems that possess certain degree of autonomy from the environmentthat allows them to perform motions that are not directly controlled from outside) have

been based upon the principles of Newtonian and statistical mechanics, (A. S. Mikhailov,1990). These models appear to be so general that they predict not only physical, but also



Electronic Journal of Theoretical Physics 4, No. 16(II) (2007) 11–96 13

some biological and economical, as well as social patterns of behavior exploiting such fun-damental properties of nonlinear dynamics as attractors. Not withstanding indisputablesuccesses of that approach (neural networks, distributed active systems, etc.) there is still

a fundamental limitation that characterizes these models on a dynamical level of descrip-tion: they propose no difference between a solar system, a swarm of insects, and a stockmarket. Such a phenomenological reductionism is incompatible with the rst principleof progressive biological evolution associated with Darwin (I. Prigogine, 1980, H. Haken,1988). According to this principle, the evolution of living systems is directed toward thehighest levels of complexity if the complexity is measured by an irreducible number of dif-ferent parts which interact in a well-regulated fashion (although in some particular casesdeviations from this general tendency are possible). At the same time, the solutions tothe models based upon dissipative Newtonian dynamics eventually approach attractors

where the evolution stops while these attractors dwell on the subspaces of lower dimen-sionality, and therefore, of the lower complexity (until a “master” reprograms the model).Therefore, such models fail to provide an autonomous progressive evolution of living sys-tems (i.e. evolution leading to increase of complexity), Figure 2. Let us now extendthe dynamical picture to include thermal forces. That will correspond to the stochasticextension of Newtonian models, while the Liouville equation will extend to the so calledFokker-Planck equation that includes thermal force effects through the diffusion term.Actually, it is a well-established fact that evolution of life has a diffusion-based stochasticnature as a result of the multi-choice character of behavior of living systems. Such anextended thermodynamics-based approach is more relevant to model of living systems,and therefore, the simplest living species must obey the second law of thermodynamicsas physical particles do. However, then the evolution of living systems (during periodsof their isolation) will be regressive since their entropy will increase (I. Prigogine, 1961),Figure 3. As pointed out by R. Gordon (1999), a stochastic motion describing physicalsystems does not have a sense of direction, and therefore, it cannot describe a progressiveevolution. As an escape from this paradox, Gordon proposed a concept of differentiatingwaves (represented by traveling waves of chemical concentration or mechanical defor-mation) which are asymmetric by their nature, and this asymmetry creates a sense of direction toward progressive evolution. Although the concept of differentiating wavesitself seems convincing, it raises several questions to be answered: Who or what arrangesthe asymmetry of the differentiating waves in the “right” direction? How to incorporatetheir formalism into statistical mechanics providing progressive evolution without a vio-lation of the second law of thermodynamics? Thus, although the stochastic extension of Newtonian models can be arranged in many different ways (for instance, via relaxationof the Lipschitz conditions, (M. Zak, 1992), or by means of opening escape-routes fromthe attractors), the progressive evolution of living systems cannot be provided.

The limitations discussed above have been addressed in several publications in whichthe authors were seeking a “border line” between living and non-living systems. It is

worth noticing that one of the “most obvious” distinctive properties of the living systems,namely, their intentionality, can be formally disqualied by simple counter-examples; in-




deed, any mechanical (non-living) system has an “objective” to minimize action (theHamilton principle) as well as any isolated diffusion-based stochastic (non-living) sys-tem has an “objective” to maximize the entropy production (“The Jaynes Principle,”

H. Haken, 1988). The departure from Newtonian models via introduction of dynamicswith expectations and feedback from future has been proposed by B. Huberman and hisassociates (B. Huberman, 1988). However, despite the fact that the non-Newtonian na-ture of living systems in these works was captured correctly, there is no global analyticalmodel that would unify the evolution of the agent’s state variables and their probabilisticcharacteristics such as expectations, self-images etc.

Remaining within the framework of dynamical formalism, and based only upon kine-matics of the particle, we will associate life with the inequality that holds during, at least,some time interval

dH dt < 0 (1)

whereH (t) = −

V

ρ(V, t) ln ρ(V, t)dV (1a)

Here H is entropy of the particle, V is the particle velocity, and ρ is the probability densitycharacterizing the velocity distribution. Obviously, the condition (1) is only sufficient,but not necessary since even a living particle may choose not to exercise its privilege todecrease disorder.

It seems unreasonable to introduce completely new principles for Living’s behavior

since Livings belong to the Newtonian world: they obey the First Principles of Newto-nian mechanics, although these Principles are necessary, but not sufficient: they shouldbe complemented by additional statements linked to the Second Law of thermodynam-ics and enforcing Eq. (1). One of the main objectives of this paper is to extend theFirst Principles of classical physics to include phenomenological behavior on living sys-tems, i.e. to develop a new mathematical formalism within the framework of classicaldynamics that would allow one to capture the specic properties of natural or articialliving systems such as formation of the collective mind based upon abstract images of the selves and non-selves, exploitation of this collective mind for communications and

predictions of future expected characteristics of evolution, as well as for making decisionsand implementing the corresponding corrections if the expected scenario is different fromthe originally planned one. The approach is based upon our previous publications (M.Zak, 1999a, 2003, 2004, 2005a, 2006a, 2007a, 2007b and 2007c) that postulate that evena primitive living species possesses additional non-Newtonian properties which are notincluded in the laws of Newtonian or statistical mechanics. These properties follow froma privileged ability of living systems to possess a self-image (a concept introduced inpsychology) and to interact with it. The proposed mathematical formalism is quantum-inspired: it is based upon coupling the classical dynamical system representing the motordynamics with the corresponding Liouville equation describing the evolution of initialuncertainties in terms of the probability density and representing the mental dynamics.(Compare with the Madelung equation that couples the Hamilton-Jacobi and Liouville




equations via the quantum potential.)The coupling is implemented by the information-based supervising forces that can be associated with the self-awareness. These forcesfundamentally change the pattern of the probability evolution, and therefore, leading to

a major departure of the behavior of living systems from the patterns of both Newtonianand statistical mechanics. Further extension, analysis, interpretation, and application of this approach to complexity in Livings and emergent intelligence will be addressed in thispaper. It should be stressed that the proposed model is supposed to capture the signa-ture of life on the phenomenological level, i.e., based only upon the observable behavior,and therefore, it will not include a bio-chemical machinery of metabolism. Such a limi-tation will not prevent one from using this model for developing articial living systemsas well as for studying some general properties of behavior of natural living systems.Although the proposed model is supposed to be applicable to both open and isolated

autonomous systems, the attention will be concentrated upon the latter since such prop-erties of Livings as free will, prediction of future, decision making abilities, and especially,the phenomenology of mind, become more transparent there. It should be emphasizedthat the objective of the proposed approach is not to overperform alternative approachesto each particular problem (such approaches not only exist, but they may be even moreadvanced and efficient), but rather to develop a general strategy (by extending the FirstPrinciples of physics) that would be the starting point for any particular problem. Theimpotence of the general strategy can be illustrated by the following example-puzzle:Suppose that a picture is broken into many small pieces that are being mixed up; in or-der to efficiently reconstruct this picture, one has to know how this picture should look;otherwise the problem becomes combinatorial, and practically, unsolvable. This puzzleis directly related to the top-down approach to Physics of Life.

The paper presents a review of the author’s publications (M. Zak, 1999a, 2003, 2004,2005a, 2006a, 2006b, and 2006c).

2. Dynamics with Liouville Feedback

2.1 Destabilizing Effect of Liouville Feedback

We will start with derivation of an auxiliary result that illuminates departure fromNewtonian dynamics. For mathematical clarity, we will consider here a one-dimensionalmotion of a unit mass under action of a force f depending upon the velocity v and time t

v = f (v, t ), (2)

If initial conditions are not deterministic, and their probability density is given in theform

ρ0 = ρ0(V ), where ρ ≥0, and∞

−∞

ρdV = 1 (3)

while ρ is a single- valued function, then the evolution of this density is expressed by thecorresponding Liouville equation




∂ρ∂t

+∂

∂v(ρf ) = 0 (4)

The solution of this equation subject to initial conditions and normalization constraints(3) determines probability density as a function of V and t : ρ = ρ(V, t).

In order to deal with the constraint (3), let us integrate Eq. (4) over the whole spaceassuming that ρ →0 at |V | → ∞and |f | < ∞ . Then

∂ ∂t

∞

−∞

ρdV = 0 ,∞

−∞

ρdV = const, (5)

Hence, the constraint (3) is satised for t > 0 if it is satised for t = 0 .Let us now specify the force f as a feedback from the Liouville equation

f (v, t ) = ϕ[ρ(v, t )] (6)

and analyze the motion after substituting the force (6) into Eq.(2)

v = ϕ[ρ(v, t )], (7)

This is a fundamental step in our approach. Although the theory of ODE does not imposeany restrictions upon the force as a function of space coordinates, the Newtonian physicsdoes: equations of motion are never coupled with the corresponding Liouville equation.Moreover, it can be shown that such a coupling leads to non-Newtonian properties of theunderlying model. Indeed, substituting the force f from Eq. (6) into Eq. (5), one arrivesat the nonlinear equation for evolution of the probability density

∂ρ∂t

+∂

∂V {ρϕ[ρ(V, t)]}= 0 (8)

Let us now demonstrate the destabilizing effect of the feedback (6). For that purpose, itshould be noted that the derivative ∂ρ/∂v must change its sign, at least once, within theinterval −∞< v < ∞, in order to satisfy the normalization constraint (3).

But since

Sign∂ v∂v

= Signdϕdρ

Sign∂ρ∂v

(9)

there will be regions of v where the motion is unstable, and this instability generatesrandomness with the probability distribution guided by the Liouville equation (8). Itshould be noticed that the condition (9) may lead to exponential or polynomial growthof v (in the last case the motion is called neutrally stable, however, as will be shownbelow, it causes the emergence of randomness as well if prior to the polynomial growth,the Lipcshitz condition is violated).




2.2 Emergence of Randomness

In order to illustrate mathematical aspects of the concepts of Liouville feedback, as well

as associated with it instability and randomness let us take the feedback (6) in the form

f = −σ2 ∂ ∂v

ln ρ (10)

to obtain the following equation of motion

v = −σ2 ∂ ∂v

ln ρ, (11)

This equation should be complemented by the corresponding Liouville equation (in thisparticular case, the Liouville equation takes the form of the Fokker-Planck equation)

∂ρ∂t

= σ2 ∂ 2ρ∂V 2

(12)

Here v stands for a particle velocity, and σ2is the constant diffusion coefficient.The solution of Eq. (12) subject to the sharp initial condition is

ρ =1

2σ√πtexp(−

V 2

4σ2t) (13)

Substituting this solution into Eq. (11) at V = v one arrives at the differential equation

with respect to v(t) v = v2t

(14)

and therefore,v = C √t (15)

where C is an arbitrary constant. Since v = 0 at t = 0 for any value of C , the solution (15)is consistent with the sharp initial condition for the solution (13) of the correspondingLiouvile equation (12). The solution (15) describes the simplest irreversible motion: it ischaracterized by the “beginning of time” where all the trajectories intersect (that resultsfrom the violation of Lipschitz condition at t = 0 , Fig.4), while the backward motionobtained by replacement of t with (−t) leads to imaginary values of velocities. One cannotice that the probability density (13) possesses the same properties.

For a xed C , the solution (15) is unstable since

dvdv

=12t

> 0 (16)

and therefore, an initial error always grows generating randomness. Initially, at t = 0,this growth is of innite rate since the Lipschitz condition at this point is violated

dvdv → ∞at t →0 (17)

This type of instability has been introduced and analyzed by (Zak, M., 1992).




Considering rst Eq. (15) at xed C as a sample of the underlying stochastic process(13), and then varying C, one arrives at the whole ensemble characterizing that process,(see Fig. 4). One can verify that, as follows from Eq. (13), (Risken., 1989) that the

expectation and the variance of this process are, respectively

MV = 0 , DV = 2 σ2t (18)

The same results follow from the ensemble (15) at −∞ ≤C ≤ ∞. Indeed, the rstequality in (18) results from symmetry of the ensemble with respect to v = 0; the secondone follows from the fact that

DV ∝v2∝t (19)

It is interesting to notice that the stochastic process (15) is an alternative to the followingLangevin equation, (Risken., 1989)

v = Γ( t), M Γ = 0, DΓ = σ (20)

that corresponds to the same Fokker-Planck equation (12). Here Γ( t) is the Langevin(random) force with zero mean and constant variance σ.

The results described in this sub-section can be generalized to n-dimensional case,(Zak, M, 2007b)

2.3 Emergence of Entanglement

In order to introduce and illuminate a fundamentally new non-Newtonian phenomenonsimilar to quantum entanglement, let us assume that the function ϕ(ρ) in Eq. (6) isinvertible, i.e. ρ = ϕ−1(f ). Then Eqs. (7) and (8) take the form, respectively

v = ρ, (7a)

∂ρ∂t

+∂

∂V (ρ2) = 0 (7b)

As follows from Eq. (7) with reference to the normalization constraint (3)

∞

−∞ϕ−1[v(V, t)]dV = 1 (21)

It should be mentioned that non-Newtonian properties of solution to Eq. (8a) such asshock waves in probability space have been studied in (Zak, M., 2004, 2006c), Fig. 24.

Thus, the motions of the particles emerged from instability of Eq. (7a) are entangled:they must satisfy the global kinematical constraint (20). It is instructive to notice thatquantum entanglement was discovered in a special class of quantum states that becomeentangled in the course of their simultaneous creation .

Similar result can be obtained for the feedback (10). Indeed, let us rewrite Eq. (11)

in the formv(t, C ) = −σ2 ∂

∂C ln ρ, (22)




whereρ =

12σ√πt

exp(−C 2

4σ2t) (23)

Integrating Eq. (21) over the whole region −∞ ≤C ≤ ∞, one arrives at the followingglobal constraint in the form

∞

−∞

v(t, C )dC = −σ2[ln ρ(C = ∞) −ln ρ(C = −∞)] = 0 (24)

that entangles accelerations of different samples of the same stochastic process. The sameresult follows from symmetry of the acceleration eld plotted in Fig. 5. It is importantto notice that the Fokker-Planck equation (12) does not impose any constraints upon

the corresponding Langevin equation (20), i.e. upon the accelerations v. In order todemonstrate it, consider two particles driven by the same Langevin force Γ( t)

v1 = Γ( t), v2 = Γ( t), M Γ = 0, DΓ = σ (25)

Obviously, the difference between the particles accelerations, in general, is not zero: itrepresents a stationary stochastic process with the zero mean and the variance 2 σ.Thatconrms the fact that entanglement is a fundamental non-Newtonian effect: it requiresa feedback from the Liouville equation to the equations of motion. Indeed, unlike theLangevin equation, the solution to Eq. (11) has a well-organized structure : as follows

from Eq. (15) and Figs. 4 and 5, the initial “explosion” of instability driven by theviolation of the Lipschitz condition at t = 0 distributes the motion over the family of smooth trajectories with the probability expressed by Eq. (22). Therefore, as followsfrom Eq.(23), the entanglement effect correlates different samples of the same stochas-tic process. As a result of that, each entangled particle can predict motion of anotherentangled particle in a fully deterministic manner as soon as it detects the velocity oracceleration of that particle at least at one point; moreover, since Eqs. (11) and (12) areinvariant with respect to position of the particles, the distance between these particleswill not play any role in their entanglement.

It should be emphasized that the concept of a global constraint is one of the mainattribute of Newtonian mechanics. It includes such idealizations as a rigid body, anincompressible uid, an inextensible string and a membrane, a non-slip rolling of a rigidball over a rigid body, etc. All of those idealizations introduce geometrical or kinematicalrestrictions to positions or velocities of particles and provides “instantaneous” speed of propagation of disturbances. However, the global constraint

∞

−∞

ρdV = 1 (26)

is fundamentally different from those listed above for two reasons. Firstly, this constraintis not an idealization, and therefore, it cannot be removed by taking into account more




subtle properties of matter such as elasticity, compressibility, etc. Secondly, it imposesrestrictions not upon positions or velocities of particles, but upon the probabilities of theirpositions or velocities, and that is where the entanglement comes from.

Continuing this brief review of global constraints, let us discuss the role of the reac-tions to these constraints, and in particular, let us nd the analog of reactions of globalconstraints in quantum mechanics. One should recall that in an incompressible uid,the reaction to the global constraint ∇ ·v ≥ 0 (expressing non-negative divergence of the velocity v) is a non-negative pressure p ≥ 0; in inextensible exible (one- or two-dimensional) bodies, the reaction of the global constraint gij ≤ g0

ij , i , j =1,2 (expressingthat the components of the metric tensor cannot exceed their initial values) is a non-negative stress tensor σij ≥0, i, j =1,2. Turning to quantum mechanics and consideringthe uncertainty inequality

Δ xΔ p ≥ 2 (27)in which Δ x and Δ p are the standard deviation of coordinate and impulse, respectively

as a global constraint, one arrives at the quantum potential2∇

2√ρ2m√ρ as a “reaction” of

this constraint in the Madelung equations

∂ρ∂t

+ ∇ •(ρm∇

S ) = 0 (28)

∂S ∂t

+ (∇S )2 + V −2∇

2√ρ2m√ρ

= 0 (29)

Here ρ and S are the components of the wave function ϕ = √ρeiS/h , and is the Planckconstant divided by 2 π. But since Eq. (27) is actually the Liouville equation, the quantumpotential represents a Liouville feedback similar to those introduced above via Eqs. (6)and (10. Due to this topological similarity with quantum mechanics, the models thatbelong to the same class as those represented by the system (7), (8) are expected todisplay properties that are closer to quantum rather than to classical ones.

2.4 Summary

A new kind of dynamics that displays non-Newtonian properties such as self-generatedrandomness and entanglement of different samples of the same stochastic process has beenintroduced. These novel phenomena result from a feedback from the Liouville equation tothe equation of motion that is similar (but not identical) to those in quantum mechanics.

3. From Disorder to Order

3.1 Information Potential

Before introducing the model of Livings, we have to discuss another non-trivial prop-erty of systems with the Liouville feedback. For that purpose, consider a Newtonian




particle of mass comparable to the mass of the molecules. Such a particle is subjected toa uctuating (thermal) force qL(t) called the Langevin force. The Brownian motion (interms of the velocity v) of that particle is described by a stochastic differential equation

v = qL(t), < L (t) > = 0 , < L (t)L(t ) > = 2 δ(t −t ). (30)

Here q = const is the strength of uctuations. The probability density ρ(V,t) of thevelocity distribution is described by the corresponding Fokker-Planck equation

∂ρ∂t

= q2 ∂ 2ρ∂V 2

(31)

The solution to this equation that starts with the sharp initial value at V = 0

ρ = 12q√πt

exp(−V 2

4q2t) (32)

demonstrates monotonous increase of entropy as a measure of disorder. Another propertyof this solution is its irreversibility: replacement of t with (-t) leads to imaginary valuesof density ρ.

It should be emphasized that Eqs. (29) and (30) are not coupled; in other words, theparticle “does not know” about its own probability distribution. This fact represents ageneral property of Newtonian dynamics: the equations of motion (for instance, the

Hamilton-Jacobi equation) are not coupled with the corresponding Liouville equationas well as the Langevin equations are not coupled with the corresponding Fokker-Planckequation. However, in quantum mechanics, the Hamilton-Jacobi equation is coupled withthe Liouville equation via the quantum potential, and that constitutes the fundamentaldifference between the Newtonian and quantum worlds.

Following quantum formalism, let us couple Eqs. (29) and (30). For that purpose,introduce a function

Π = −α ln ρ(v, t ) (33)

This function can be called “the information potential” since its expected value [-M ln ρ ] is equal to the Shannon information capacity H . (However, this potential cannotbe identied with a potential energy since it depends upon velocities rather than uponpositions). The gradient of this potential taken with the opposite sign can represent aninformation-based force F = -grad Π per unit mass.

v = qL(t) +αρ

∂ρ∂v

. (34)

3.2 Negative Diffusion

Let us apply this force to the particle in Eq. (29) and present this equation in a dimen-sionless form assuming that the parameter α absorbs the characteristic velocity and thecharacteristic time




If one chooses α = q 2, then the corresponding Liouville equation (that takes the formof the Fokker-Planck equation) will change from (30) to the following

∂ρ∂t = q2 ∂ 2

ρ∂V 2 − ∂ ∂V [ρq2

ρ ∂ρ∂V ] = 0, ρ = ρ0(V ) = const (35)

Thus, the information force stops the diffusion. However, the information force can beeven more effective: it can reverse the diffusion process and push the probability densityback to the sharp value in nite time. Indeed, suppose that in the information potential

α = q2 exp √D., where D(t) =∞

−∞

ρV 2dV. (36)

Then the Fokker-Planck equation takes the form

∂ρ∂t

= [q2(1 −exp √D)]∂ 2ρ∂V 2

. (37)

Multiplying Eq.(36) by V 2 , then integrating it with respect to V over the whole space,one arrives at ODE for the variance D(t)

D = 2 q2(1 −exp √D), i.e. D ≤0 if D ≥0 (38)

Thus, as a result of negative diffusion, the variance D monotonously vanishes regardlessof the initial value D(0). It is interesting to note that the time T of approaching D = 0is nite

T =1q2

0

D (0)

dD1 −exp √D ≤

1q2

∞

0

dDexp √D −1

=π

3q2 (39)

This terminal effect is due to violation of the Lipchitz condition, (Zak, M.,1992) at D = 0

dDdD

= −q2

√Dexp √D → ∞at D →0 (40)

Let us turn to a linear version of Eq. (36)

∂ρ∂t

= −q2 ∂ 2ρ∂V 2

. (41)

and discuss a negative diffusion in more details. As follows from the linear equivalent of Eq. ((39)

dDdD

= −q2, i.e. D = D0 −q2t < 0 at t > D 0/q 2 (42)

Thus, eventually the variance becomes negative, and that disqualies Eq. (40) from beingmeaningful. It has been shown (Zak, M., 2005) that the initial value problem for thisequation is ill-posed: its solution is not differentiable at any point. (Such an ill-posednessexpresses the Hadamard instability studied in (Zak, M., 1994)). Therefore, a negativediffusion must be nonlinear in order to protect the variance from becoming negative, (see




Fig. 6.). One of possible realization of this condition is placing a terminal attractor (Zak,M., 1992) at D = 0 as it was done in Eq. (36).

It should be emphasized that negative diffusion represents a major departure from

both Newtonian mechanics and classical thermodynamics by providing a progressive evo-lution of complexity against the Second Law of thermodynamics, Fig. 7.

3.3 Drift

One notes that Eq. (36) is driftless, i.e. its solution preserves the initial mean value of the state variable. Obviously, the drift can be easily introduced via both Newtonian orinformation forces. In our approach we will follow the “Law of Parsimony”, or “Occam’sRazor”: Pluritas non est ponenda sine necessitate, i.e. if a drift can be generated by

classical forces, it is not necessary to duplicate it with the information forces since thatwould divert attention from a unique capability of information forces, namely, from theirrole in progressive evolution of complexity.

3.4 Summary

A progressive evolution of complexity has been achieved via information potential that im-plements the Liouville feedback and leads to a fundamentally new nonlinear phenomenon-negative diffusion- that evolves “against the Second Law of thermodynamics”.

4. Model of Livings

4.1 General Model

Consider a system of n particles as intelligent agents that interact via the informationpotential in the fashion described above. Then, as a direct generalization of Eqs. (33)and (36), one obtains

vi = −ζ n

j =1α ij ∂ ∂v j

ln ρ(v1,...vn , t ), i = 1 , 2, ...n. (43)

where α ij are function of the correlation moments D ks

α ij = α ij (D11 , ...D ks , ...D nn ),

Dks = ∞ −∞

∞ −∞

(vk −ξk)(vs −ξs )ρdvkdvs , ξ j = ∞ −∞

v j ρdv j(44)

and∂ρ∂t

= ζ n

j =1

α ij∂ 2ρ∂V 2 j

(45)




Here ζ is a positive constant that relates Newtonian and information forces. It is intro-duced in order to keep the functions α ij dimensionless.

The solution to Eqs. (42) and (45)

vi = vi(t), ρ = ρ(V 1...V n , t ), i = 1 , 2, ...n (46)

must satisfy the constraint that is an n-dimensional generalization of the constraint D ≥0, namely, a non-negativity of the matrix |D ij |, i.e. a non-negativity of all the left-cornerdeterminants

Det |D ij | ≤0, i, j = 1 , i, j = 1 , 2; ...i, l = 1 , 2, ...n. (47)

4.2 Simplied Model

Since enforcement of the constraints (47) is impractical, we will propose a simpler modelon the expense of its generality assuming that the tensors α ij and D ij are co-axial, andtherefore, the functions (43) can be reduced to the following

α ii = α ii (D11 , ...D nn ), i = 1 , 2, ...n, (48)

where α ii and D ii are the eigenvalues of the corresponding tensors.Referring Eqs. (42) and (45) to the principal coordinates, one obtains

vi = −ζα ii

∂ ∂vi ln ρ(v1,...vn , t ), i = 1 , 2, ...n. (49)

∂ρ∂t

= ζα ii∂ 2ρ∂V 2i

(50)

Since Egs. (49) and (50) have tensor structure, they can be rewritten in arbitrary systemof coordinates using standard rules of tensor transformations.

Let us express Eq. (50) in terms of the correlation moments: multiplying it by V 2i ,then using partial integration, one arrives at an n-dimensional analog of Eq. (37)

D ii = 2 ζα ii (D11 ,...D nn ), n = 1 , 2, ...n, (51)

The last step is to choose such a structure of the functions (48) that would enforce theconstraints (47), i.e.

D ii ≥0, i = 1 , 2, ...n, (52)

The simplest (but still sufficiently general) form of the functions (48) is a neural networkwith terminal attractors

α ii =12

(wij tanh D jj −ci D ii ), i = 1 , 2, ...n, D ii =D ii

D 0 (53)

that reduces Eqs.(51) to the following system (Zak, M., 2006a)

˙D ii = ζ (wij tanh D jj −ci D ii ), i = 1 , 2, ...n, (54)




Here D0 is a constant scaling coefficient of the same dimensionality as the correlationcoefficientsD ii , and wij are dimensionless constants representing the identity of the sys-tem.

Let us now analyze the effect of terminal attractor and, turning to Eq.(54), start withthe matrix |∂ D ii

∂D ii |. Its diagonal elements, i.e. eigenvalues, become innitely negative whenthe variances vanish since

∂ √D ii

∂D ii=

12√D ii → ∞when D ii →0 (55)

while the rest terms are bounded. Therefore, due to the terminal attractor, Eq. ((54)linearized with respect to zero variances has innitely negative characteristic roots, i.e.it is innitely stable regardless of the parameters wij . Therefore the principal variances

cannot overcome zero if their initial values are positive. This provides the well-posednessof the initial value problem.

4.3 Invariant Formulation

Eqs. (49) and (50) can be presented in the following invariant form

v = −ζα •∇v ln ρ, (56)

ρ = ζ ∇2V ρ • •α, (57)

4.4 Variation Principle

Let us concentrate upon the capability of Livings to decrease entropy (see Eq. (1)) with-out external forces. This privileged motion is carried out through negative diffusion thatrequires negativity of the eigenvalues of the tensor α in Eqs. (56) and (57). Redeningthe concept of information potential for n-dimensional case (see Eq. (32))

Π = ζ ln ρ (32a)

one can rewrite Eq. (56) asv = −α •∇Π (56a)

Then, along the trajectory dened by Eq. (56a) the following inequality holds

Π = ∇Π • v = −∇Π • (α •∇Π) > 0 (58)

Thus, the privileged motion of Livings monotonously maximizes the information potentialalong a chosen trajectory. This means that if the initial density ρ0 < 1, and therefore, Π 0 <0 at the chosen trajectory, the density would monotonously decrease along this trajectory,i.e. ρ < 0. Conversely, if ρ0 > 1, and therefore,Π 0 > 0, the density would monotonouslyincrease, i.e. ρ > 0. Hence, the privileged motion of Livings driven by negative diffusionmonotonously decreases the atness of the initial density distribution. Obviously, the




initial choice of the trajectory is determined by Eq. (57). But the most likely trajectoryis those that pass through maxima of density distributions. Since that trajectory has alsomaxima of the information potential (32a) (compare to other trajectories), one arrive at

the following principle: The most likely privileged motion of Livings delivers the globalmaximum to the information potential (32a), (M. Zak, 2005a)

4.5 Remark

Strictly speaking, the formulated variation principle is necessary, but not sufficient forderivation of the governing equations (56) and (57): this principle should be consideredonly as a complement to the Hamiltonian principle taken in its non-variation formulation(since the information “potential” depends upon velocities rather than upon coordinates):

δ R = δS +t1

t0

δ Πdt = 0 . (58a)

Here S - is action, and δ Π is an elementary work of the non-conservative information force

∇•Π. It is obvious that the Hamiltonian principle in the form (58a) leads to equation of motion (56), while the inequality (58) (along with the continuity equation (57)) speciesthe information force.

4.6 SummaryA closed system of dynamical equations governing behavior of Livings has been introducedand discussed.

5. Interpretation of the Model

5.1 Mathematical Viewpoint

The model is represented by a system of nonlinear ODE (56) and a nonlinearparabolic PDE (57) coupled in a master-slave fashion: Eq. (57) is to be solved inde-pendently, prior to solving Eq. (56). The coupling is implemented by a feedback thatincludes the rst gradient of the probability density, and that converts the rst orderPDE (the Liouville equation) to the second order PDE (the Fokker-Planck equation).Its solution, in addition to positive diffusion, can display negative diffusion as well, andthat is the major departure from the classical Fokker-Planck equation. Fig. 7. It hasbeen demonstrated that negative diffusion must be nonlinear with an attractor at zerovariances to guarantee well-posedness of the initial value problem, and that imposes addi-tional constraints upon the mathematical structure of the model, (see Eqs. (47) and (52)).The nonlinearity is generated by a feedback from the PDE (57) to the ODE (56), (thatis the same feedback that is responsible for parabolicity of the PDE (57)). As a result of




the nonlinearity, the solutions to PDE can have attractors (static, periodic, or chaotic)in probability space(see Eqs. (51). The multi-attractor limit sets allow one to introducean extension of neural nets that can converge to a prescribed type of a stochastic process

in the same way in which a regular neural net converges to a prescribed deterministicattractor. The solution to ODE (56) represents another major departure from classicalODE: due to violation of Lipchitz conditions at states where the probability density hasa sharp value, the solution loses its uniqueness and becomes random. However, this ran-domness is controlled by the PDE (57) in such a way that each random sample occurswith the corresponding probability, (see Fig.4).

5.2 Physical Viewpoint

The model represents a fundamental departure from both Newtonian and statistical me-chanics. In particular, negative diffusion cannot occur in isolated systems without helpof the Maxwell sorting demon that is strictly forbidden in statistical mechanics. The onlyconclusion to be made is that the model is non-Newtonian, although it is fully consistentwith the theory of differential equations and stochastic processes. Strictly speaking, it is amatter of denition weather the model represents an isolated or an open system since theadditional energy applied via the information potential is generated by the system “it-self” out of components of the probability density. In terms of a topology of its dynamicalstructure, the proposed model links to quantum mechanics: if the information potential

is replaced by the quantum potential, the model turns into the Madelung equations thatare equivalent to the Schr¨odinger equation (Takabayasi, T., 1953), Fig. 8.It should benoticed that the information potential is introduced via the dimensionless parameter αthat is equal to the rate of decrease of the disorder (-dD/dt), (see Eq. (51)) and a newphysical parameter ζ of dimension [ζ ] = m2/sec 3 describing a new physical quantity thatrelates the rate of decrease of disorder to the specic (information-based) force. For-mally the parameter ζ introduces the information potential in the same way in which thePlanck constant introduces quantum potential. The system of ODE (56) characterizedby velocities vi describes a mechanical motion of the system driven by information forces.

Due to specic properties of these forces discussed above, this motion acquires propertiessimilar to those of quantum mechanics. These properties are discussed below.α .Superposition . In quantum mechanics, any observable quantity corresponds to an

eigenstate of a Hermitian linear operator. The linear combination of two or more eigen-states results in quantum superposition of two or more values of the quantity. If thequantity is measured, the projection postulate states that the state will be randomlycollapsed onto one of the values in the superposition (with a probability proportionalto the square of the amplitude of that eigenstate in the linear combination). Let uscompare the behavior of the model of Livings from that viewpoint. As follows from Eq.(15), all the particular solutions intersect at the same point x = 0 at t = 0 ,and thatleads to non-uniqueness of the solution due to violation of the Lipcshitz condition (seeEq. (17)). Therefore, the same initial condition x = 0 at t = 0yields innite number




of different solutions forming a family (15); each solution of this family appears with acertain probability guided by the corresponding Fokker-Planck equation. For instance,in case of Eq. (15), the “winner” solution is x ≡ 0since it passes through the maxima

of the probability density (13). However, with lower probabilities, other solutions of thefamily (15) can appear as well. Obviously, this is a non-classical effect. Qualitatively,this property is similar to those of quantum mechanics: the system keeps all the solutionssimultaneously and displays each of them “by a chance”, while that chance is controlledby the evolution of probability density (12). It should be emphasized that the choice of displaying a certain solution is made by the Livings model only once, at t = 0, i.e. whenit departs from the deterministic to a random state; since than, it stays with this solutionas long as the Liouville feedback is present.

β . Entanglement . Quantum entanglement is a phenomenon in which the quantum

states of two or more objects have to be described with reference to each other, eventhough the individual objects may be spatially separated. This leads to correlationsbetween observable physical properties of the systems. For example, it is possible toprepare two particles in a single quantum state such that when one is observed to bespin-up, the other one will always be observed to be spin-down and vice versa, thisdespite the fact that it is impossible to predict, according to quantum mechanics, whichset of measurements will be observed. As a result, measurements performed on onesystem seem to be instantaneously inuencing other systems entangled with it.

Qualitatively similar effect has been found in the proposed model of Livings (seeEqs. (21)-(23)) that demonstrate that different realizations of motion emerged frominstability of Eq. (15) are entangled: they must satisfy the global kinematical constraint(23). Therefore, as follows from Eq. (23), the entanglement effect correlates different samples of the same stochastic process. It is instructive to notice again that quantumentanglement was discovered in a special class of quantum states that become entangledin the course of their simultaneous creation.

γ . Decoherence . In quantum mechanics, decoherence is the process by which quantumsystems in complex environments exhibit classical behavior. It occurs when a systeminteracts with its environment in such a way that different portions of its wavefunctioncan no longer interfere with each other.

Qualitatively similar effects are displayed by the proposed model of Livings. In orderto illustrate that, let us turn to Eqs. (11), (12), and notice that this system makes achoice of the particular solution only once i.e. when it departs from the deterministic toa random state; since then, it stays with this solution as long as the Liouville feedbackis present,( σ = 0). However, as soon as this feedback disappears,( σ = 0), the system be-comes classical, i.e. fully deterministic, while the deterministic solution is a continuationof the corresponding “chosen” random solution, (see Fig.9).

δ. Uncertainty Principle . In quantum physics, the Heisenberg uncertainty principlestates that one cannot measure values (with arbitrary precision) of certain conjugate

quantities which are pairs of observables of a single elementary particle. These pairsinclude the position and momentum. Similar (but not identical) relationship follows




from Eq. (15): vv = C 2/ 2, i.e. the product of the velocity and the acceleration isconstant along a xed trajectory. In particular, at t = 0, v and v can not be denedseparately.

5.3 Biological Viewpoint

The proposed model illuminates the “border line” between living and non-living systems.The model introduces a biological particle that, in addition to Newtonian properties,possesses the ability to process information. The probability density can be associatedwith the self-image of the biological particle as a member of the class to which this par-ticle belongs, while its ability to convert the density into the information force - withthe self-awareness (both these concepts are adopted from psychology). Continuing thisline of associations, the equation of motion (such as Eqs (56)) can be identied with amotor dynamics, while the evolution of density (see Eqs. (57) –with a mental dynamics.Actually the mental dynamics plays the role of the Maxwell sorting demon: it rearrangesthe probability distribution by creating the information potential and converting it intoa force that is applied to the particle. One should notice that mental dynamics describesevolution of the whole class of state variables (differed from each other only by initialconditions), and that can be associated with the ability to generalize that is a privilege of living systems. Continuing our biologically inspired interpretation, it should be recalledthat the second law of thermodynamics states that the entropy of an isolated system can

only increase, Eig.3. This law has a clear probabilistic interpretation: increase of entropycorresponds to the passage of the system from less probable to more probable states, whilethe highest probability of the most disordered state (that is the state with the highestentropy) follows from a simple combinatorial analysis. However, this statement is correctonly if there is no Maxwell’ sorting demon, i.e., nobody inside the system is rearrangingthe probability distributions. But this is precisely what the Liouville feedback is doing:it takes the probability density ρ from Equation (57), creates functionals and functionsof this density, converts them into a force and applies this force to the equation of motion(56). As already mentioned above, because of that property of the model, the evolution

of the probability density becomes nonlinear, and the entropy may decrease “against thesecond law of thermodynamics”, Fig.7. Obviously the last statement should not be takenliterary; indeed, the proposed model captures only those aspects of the living systems thatare associated with their behavior, and in particular, with their motor-mental dynamics,since other properties are beyond the dynamical formalism. Therefore, such physiologi-cal processes that are needed for the metabolism are not included into the model. Thatis why this model is in a formal disagreement with the second law of thermodynamicswhile the living systems are not. In order to further illustrate the connection betweenthe life-nonlife discrimination and the second law of thermodynamics, consider a smallphysical particle in a state of random migration due to thermal energy, and compareits diffusion i.e. physical random walk, with a biological random walk performed by abacterium. The fundamental difference between these two types of motions (that may be




indistinguishable in physical space) can be detected in probability space: the probabilitydensity evolution of the physical particle is always linear and it has only one attractor:a stationary stochastic process where the motion is trapped. On the contrary, a typical

probability density evolution of a biological particle is nonlinear: it can have many dif-ferent attractors, but eventually each attractor can be departed from without any “help”from outside.

That is how H. Berg, 1983, describes the random walk of an E. coli bacterium:” If a cellcan diffuse this well by working at the limit imposed by rotational Brownian movement,why does it bother to tumble? The answer is that the tumble provides the cell with amechanism for biasing its random walk . When it swims in a spatial gradient of a chemicalattractant or repellent and it happens to run in a favorable direction, the probability of tumbling is reduced. As a result, favorable runs are extended, and the cell diffuses with

drift”. Berg argues that the cell analyzes its sensory cue and generates the bias internally ,by changing the way in which it rotates its agella. This description demonstrates thatactually a bacterium interacts with the medium, i.e., it is not isolated, and that reconcilesits behavior with the second law of thermodynamics. However, since these interactions arebeyond the dynamical world, they are incorporated into the proposed model via the self-supervised forces that result from the interactions of a biological particle with “itself,”and that formally “violates” the second law of thermodynamics. Thus, the proposedmodel offers a unied description of the progressive evolution of living systems. Basedupon this model, one can formulate and implement the principle of maximum increaseof complexity that governs the large-time-scale evolution of living systems. It shouldbe noticed that at this stage, our interpretation is based upon logical extension of theproposed mathematical formalism, and is not yet corroborated by experiments.

5.4 Psychological Viewpoint

The proposed model can be interpreted as representing interactions of the agent withthe self-image and the images of other agents via the mechanisms of self-awareness. Inorder to associate these basic concepts of psychology with our mathematical formalism,

we have to recall that living systems can be studied in many different spaces such asphysical (or geographical) space as well as abstract (or conceptual) spaces. The lattercategory includes, for instance, social class space, sociometric space, social distance space,semantic space e.t.c.Turning to our model, one can identify two spaces: the physical spacex, t in which the agent state variables vi = x i evolve,(see Eqs.(56)), and an abstract spacein which the probability density of the agent’ state variables evolve (see Eq.(57)).Theconnection with these spaces have been already described earlier: if Eqs. (56) are runmany times starting with the same initial conditions, one will arrive at an ensemble of different random solutions, while Eq. (57) will show what is the probability for each of these solutions to appear. Thus, Eq. (57) describes the general picture of evolution of thecommunicating agents that does not depend upon particular initial conditions. Therefore,the solution to this equation can be interpreted as the evolution of the self- and non-self




images of the agents that jointly constitutes the collective mind in the probability space,Fig. (10). Based upon that, one can propose the following interpretation of the modelof communicating agents: considering the agents as intelligent subjects, one can identify

Eqs. (56) as a model simulating their motor dynamics, i.e. actual motions in physicalspace, while Eq.(57) as the collective mind composed of mental dynamics of the agents.Such an interpretation is evoked by the concept of reection in psychology, (V. Lefebvre,2001). Reection is traditionally understood as the human ability to take the positionof an observer in relation to one’s own thoughts. In other words, the reection is theself-awareness via the interaction with the image of the self. Hence, in terms of thephenomenological formalism proposed above, a non-living system may possess the self-image, but it is not equipped with the self-awareness, and therefore, this self-image isnot in use. On the contrary, in living systems the self-awareness is represented by the

information forces that send information from the self-image (57) to the motor dynamics(56). Due to this property that is well-pronounced in the proposed model, an intelligentagent can run its mental dynamics ahead of real time, (since the mental dynamics is fullydeterministic, and it does not depend explicitly upon the motor dynamics) and thereby, itcan predict future expected values of its state variables; then, by interacting with the self-image via the information forces, it can change the expectations if they are not consistentwith the objective. Such a self-supervised dynamics provides a major advantage forthe corresponding intelligent agents, and especially, for biological species: due to theability to predict future, they are better equipped for dealing with uncertainties, andthat improves their survivability. It should be emphasized that the proposed model,strictly speaking, does not discriminate living systems of different kind in a sense that allof them are characterized by a self-awareness-based feedback from mental (57) to motor(56) dynamics. However, in primitive living systems (such as bacteria or viruses) the self-awareness is reduced to the simplest form that is the self-nonself discrimination; in otherwords, the difference between the living systems is represented by the level of complexityof that feedback.

5.5 Neuro-science Viewpoint

The proposed model represents a special type of neural net. Indeed, turning to Eqs.(54) and reinterpreting the principal correlation moments D ii as neurons’ mean somapotentials, one arrives at a conventional neural net formalism.

α . Classical neural nets . We will start with a brief review of the classical version of this kind of dynamical models in order to outline some of its advantages and limitations.The standard form of recurrent neural networks (NN) is

xi = wij tanh x j −cix i (59)

where x i are state variables, wij are synaptic interconnection, or weights (associated withthe NN topology). The system (59) is nonlinear and dissipative due to the sigmoid func-tion tanh . The nonlinearity and dispassivity are necessary (but not sufficient) conditions




that the system (59) has attractors. The locations of the attractors and their basins inphase (or conguration) space are prescribed by an appropriate choice of the synapticinterconnections wij which can be found by solving the inverse problem (followed by

the stability analysis), or by learning (that is a dynamical relaxation procedure basedupon iterative adjustments of wij as a result of comparison of the net output with knowncorrect answers). In both cases, wij are constants, and that is the rst limitation of recurrent NN. Indeed, although the NN architecture (59) is perfectly suitable for suchtasks as optimization, pattern recognition, associative memory, i.e., when xed topologyis an advantage, it cannot be exploited for simulation of a complex dynamical behaviorthat is presumably comprised of a chain of self-organizing patterns (like, for instance, ingenome) since for that kind of tasks, variable topology is essential. However, there is nogeneral analytical approach to the synthesis of such NN. And now we are coming to the

second limitation of NN (1): their architecture does not have a tensor structure. Indeed,the state variables and the interconnections wij cannot be considered as a vector and atensor, respectively, since their invariants are not preserved under linear transformationsof the state variables. Obviously, the cause of that is the nonlinearity in the form of the sigmoid function. That is why the dynamical system (59) (even with a xed topol-ogy) cannot be decoupled and written in a canonical form; as a result of that, the mainmathematical tools for NN synthesis are based upon numerical runs.

β . Mystery of mirror neuron . The proposed model represents a special type of neuralnet. Indeed, turning to Eqs. (54) and reinterpreting the principal correlation momentsD ii as neurons’ mean soma potentials, one arrives at a conventional neural net formalism.The analysis of Eq. (59) can be linked to the concept of a mirror neuron. The discoveryof mirror neurons in the frontal lobes of macaques and their implications for human brainevolution is one of the most important ndings of neuroscience in the last decade. Mirrorneurons are active when the monkeys perform certain tasks, but they also re when themonkeys watch someone else perform the same specic task. There is evidence that asimilar observation/action matching system exists in humans. In the case of humans, thisphenomenon represents the concept of imitation learning, and this faculty is at the basisof human culture. Hence, a mirror neuron representing an agent A can be activated byan expected (or observed) action of an agent B which may not be in a direct contact withthe agent A at all. Quoting the discoverer of the mirror neuron, Dr. Giacomo Rizzolatti,“the fundamental mechanism that allows us a direct grasp of the mind of others is notconceptual reasoning, but direct simulation of the observed event through the mirrormechanism.” In other words, we do not need to think and analyze, we know immediatelythe intensions of other people. In terms of the mathematical formalism, such a directgrasp of the mind of others represents a typical non-locality, i.e. an “invisible” inuenceon a distance that is similar to quantum entanglement.

γ . Mirror neural nets. In order to capture this fundamentally new phenomenon, wewill apply the approximated version of dynamical formalism developed in the Section

3, and modify Eq. (49) and (53). In the simplest case when α ij = const., the solutiondescribing the transition from initial (sharp) to current density in Eq. (50) is given by




the Gaussian distribution, (Risken,H., 1989),

ρ({V }, t|{V }, t ) = (2 π)n/ 2[Det D (t −t )]−1/ 2 exp{−12 [D−1(t −t )]ij [V i −Gik (t −t )V k ]

×[V j −G jl (t −t)V l ]}(60)

Here the Green function in matrix notation is

G(t) = exp( −at ) = I −at +12

a2t2 ±... (61)

where the matrix elements of a and G are given by a ij and Gij , respectively.If α ij are not constants, for instance, if they are dened by Eqs. (53), the solution

(60) are valid only for small times. (Actually such an approximation follows from ex-

pansion of a probability density in Gram-Charlier series with keeping only the rst termof that series.) Nevertheless, for better physical interpretation, we will stay with thisapproximation in our further discussions. Substituting the solution (60) into Eq.(49),one obtains

vi = ζ {(wij tanh D jj −cii D ii )}{[D−1]ij (v j −G jl vl)} (62)

Eqs. (62) are to be complemented with Eq. (54). (In both equations, the summationis with respect to the index j.) Formally Eq.(54) represents a classical neural net withterminal attractors where the principle variances D ii play the role of state variables.Depending upon the chosen values of constant synaptic interconnections wij , the variances

D ii can converge to static, periodic or chaotic attractors. In particular, if wij = w ji , thenet has only static attractors. However, regardless of synaptic interconnections, the stateD ii = 0 (i = 1,2,...n), is always a terminal attractor that protects the variances fromcrossing zeros. Eqs.(62) represent a stochastic neural net driven by its own principlevariances that, in turn, is governed by the neural net (54). The connection between thenets (54) and (62) is the following: if Eqs (62) are run many times, one will arrive at anensemble of different random solutions, while Eq. (54) will show what is the probabilityfor each of these solutions to appear. Thus, Eq. (54) describes the general picture of evolution that does not depend upon particular initial conditions. Therefore, the solution

to this equation can be interpreted as the evolution of the self- and non-self images of theneurons that jointly constitutes the collective mind in the probability space. Let us recallthat in classical neural nets (59), each neuron receives fully deterministic informationabout itself and other neurons, and as a result of that, the whole net eventually approachesan attractor. In contradistinction to that, in the neural net (62), each neuron receives theimage of the self and the images of other neurons that are stored in the joint probabilitydistribution of the collective mind, Fig. 10. These images are not deterministic: theyare characterized by uncertainties described by the probability distribution, while thatdistribution is skewed toward the expected value of the soma potential, with variancescontrolled by Eq. (54). In our view, such a performance can be associated with themirror neurons. Indeed, as mentioned above, a mirror neuron res both when performingan action and when observing the same action performed by another subject. The way




in which this neuron works is the following. It is assumed that all the communicatingneurons belong to the same class in a sense that they share the same general propertiesand habits. It means that although each neuron may not know the exact values of soma

potentials of the rest of the neurons, it, nevertheless, knows at least such characteristicsas their initial values (to accuracy of initial joint probability density, or, at least, initialexpected values and initial variances). This preliminary experience allows a neuron toreconstruct the evolution of expected values of the rest of the neurons using the collectivemind as a knowledge base. Hence, the neuron representing an agent A can be activated byan expected action of an agent B that may not be in a direct contact with the agent A atall, and that can be associated with the mirror properties of the neuron. The qualitativebehavior of the solution to the mirror-neuron-based net (62) and (54) is illustrated inFig. 11. The collective properties of mirror neurons, i.e. the mirror neural nets, have

a signicant advantage over the regular neural nets: they possess a fundamentally newtype of attractor –the stochastic attractor that is a very powerful generalization tool.Indeed, it includes a much broader class of motions than static or periodic attractors. Inother words, it provides the highest level of abstraction. In addition to that, a stochasticattractor represents the most complex patterns of behavior if the mirror net describes aset of interacting agents. Indeed, consider a swarm of insects approaching some attractingpattern. If this pattern is represented by a static or periodic attractor, the motion of theswarm is locked up in a rigid pattern of behavior that may decrease its survivability.On the contrary, if that pattern is represented by a stochastic attractor, the swarm stillhas a lot of freedom, and only the statistics invariants of the swarm motion is lockedup in a certain pattern of behavior. Fig. 12. It should be emphasized that, due tothe multi-attractor structure, the proposed model provides the following property: if thesystem starts from different initial conditions, it may be trapped in a different stochasticpattern. Such a property, in principle, cannot be provided by regular neural nets orcellular automata since they can have only one stochastic attractor, Fig. 13.

δ. Link to quantum entanglement. Continuing the discussion about a formal similar-ity between the concept of mirror neuron and quantum entanglement, we will emphasizeagain that both of these phenomena result from the global constraint originated fromthe Liouville feedback, namely, the quantum potential (see Eqs. ( 27) and (28)), and theinformation potential (see Eqs. (56) and (57)). In the both cases, the probability densityρ enters the equation of motion imposing the global constraint upon the state variablesvia the normalization condition (see Eq. (23)). However, it should be emphasized thedifference between the status of these models. The model of quantum entanglement iswell-established (at least, within the Schr¨odinger formalism), being corroborated by enor-mous number of experiments. On the other hand, the model of mirror neuron nets hasbeen proposed in this paper. It is based upon two principles that complement classicalmechanics. First principle requires the system capability to decrease entropy by internaleffort to be “alive”; the second principle is supposed to provide global constraint upon

the motions of mirror neurons. Both of these principles are implemented via a speciallyselected feedback from the Liouville equation. This feedback is different from those in




quantum mechanics; however the topologies of quantum and mirror neuron systems aresimilar. Now we are ready to address the fundamental question: how Nature implementsglobal constraints via probability? Does it mean that there exists some sort of “univer-

sal mind”? The point is that the concept of probability does not belong to objectivephysical world; it rather belongs to a human interpretation of this world. Indeed, we canobserve and measure a random process, but we cannot measure directly the associatedprobabilities: that would require a special data processing, i.e. a “human touch”. Asfar as quantum mechanics is concerned, this question is still unanswered . However, forLivings, the global constraint via probability can be associated with the concept of col-lective mind. Indeed, based upon the assumption that all Livings which belong to thesame class possess the identical abstract image of the external world, and recalling that,in terms of the proposed formalism, such an image is represented by the joint probability,

one concludes that in Livings, the global constraint is implemented by the mechanism of “general knowledge” stored in the collective mind and delivered to the mirror neuronsvia the information forces. Several paradigms of self-organization (such as transmissionof conditional information, decentralized coordination, cooperative computing, and com-petitive games in active systems) based upon entanglement effects, have been proposedin (Zak, M., 2002a).

5.6 Social and Economic Viewpoint

One of the basic problem of social theory is to understand “how, with the richness of language and the diversity of artifacts, people can create a dazzlingly rich variety of new yet relatively stable social structures”, (M.Arbib, 1986). Within the framework of the dynamical formalism, the proposed model provides some explanations to this puz-zle. Indeed, social events are driven by two factors: the individual objectives and socialconstraints. The rst factor is captured by the motor dynamics (56), while the social con-straint is created by the collective mind (57). A balance between these factors (expressedby stochastic attractors) leads to stable social structures, while a misbalance (expressedby stochastic repellers) causes sharp transitions from one social structure to another (rev-olutions) or to wandering between different repellers (chaos, anarchy). For an articial“society” of communicating agents, one can assign individual objectives for each agent aswell as the collective constrains imposed upon them and study the corresponding socialevents by analyzing the governing equations (56) and (57). However, the same strategyis too naıve to be applied to a human society. Indeed, most human as members of asociety, do not have rational objectives: they are driven by emotions, inated ambitions,envy, distorted self- and nonself images, etc. At least some of these concepts can beformalized and incorporated into the model. For instance, one can consider emotions tobe proportional to the differences between the state variables v and their expectations χ

E m = c(v −v). (63)




Eq.(63) easily discriminates positive and negative emotions. Many associated concepts(anger, depression, happiness, indifference, aggressiveness. and ambitions) can be derivedfrom this denition (possibly, in combination with distorted self- and nonself images).

But the most accurate characteristic of the human nature was captured by cellular au-tomata where each agent copies the behaviors of his closest neighbors (which in turn,copy their neighbors, etc.). As a result, the whole “society” spontaneously moves towardan unknown emerging“objective”. Although this global objective is uniquely dened bya local operator that determines how an agent processes the data coming from his neigh-bors, there is not known any explicit connection between this local operator and thecorresponding global objective: only actual numerical runs can detect such a connection.Notwithstanding the ingenuity of his model, one can see its major limitation: the modelis not equipped with a collective mind (or by any other type of a knowledge base), and

therefore, its usefulness is signicantly diminished in case of incompleteness of informa-tion. At the same time, the proposed model (56) and (57) can be easily transformedinto cellular automata with the collective mind. In order to do that one has to turn toEqs.(56), replace the sigmoid function by a local operator, and the time derivative -bythe time difference. Then the corresponding Fokker-Planck equation (57) reduces to itsdiscrete version that is Markov chains,( Zak, M., 2000). On the conceptual level, themodel remains the same as discussed in the previous sections. The Fig.14 illustrates apossible approach to the social dynamics based upon the proposed model.

5.7 Language Communications Viewpoint

Language represents the best example of a communication tool with incomplete informa-tion since any message, in general, can be interpreted in many different ways dependingupon the context i.e. upon the global information sheared by the sender and the receiver.Therefore, the proposed model is supposed to be relevant for some language-orientedinterpretations. Indeed, turning to Eqs.(56), one can associate the weighted sum of thestate variables with the individual interpretations of the collective message made by theagents. The sigmoid functions of these sums form the individual responses of the agents

to this message. These responses are completed by the information forces that compen-sate the lack of information in the message by exploiting the global sheared informationstored in the collective mind, (see Eq. (57)). The agent’s responses converted into thenew values of their state variables are transformed into the next message using the samerules, etc. These rules determined by the structure of Eqs. (56) and (57) can be associ-ated with the grammar of the underlying language. In particular, they are responsible forthe convergence to- or the divergence from the expected objective. It should be noticedthat the language structure of the proposed model is invariant with respect to semantics.Hence, in terms of the linguistics terminology that considers three universal structurallevels: sound, meaning and grammatical arrangement, (Yaguello, M.,1998), we are deal-ing here with the last one. To our opinion, the independence of the proposed model uponthe semantics is an advantage rather than a limitation: it allows one to study invariant




properties of the language evolution in the same way in which the Shannon information(that represents rather an information capacity) allows one to study the evolution of information regardless of a particular meaning of the transmitted messages.

Let us now try to predict the evolution of language communications based upon theproposed model. As mentioned earlier, the evolution of the living systems is alwaysdirected toward the increase of their complexity. In a human society, such a progres-sive evolution is effectively implemented by increase or the number of reections in achain”What do you think I think you think, etc.” The society may be stratied intoseveral levels or “clubs” so that inside each club the people will shear more and moreglobal information. This means that the language communications between the membersof the same club will be characterized by the increased capacity of the collective mind(see Eq.(57)), and decreased information transmitted by the messages (see Eqs.(56)). In

the theoretical limit, these messages will degenerate into a string of symbols, that canbe easily decoded by the enormously large collective mind The language communicationsacross the stratied levels will evolve in a different way: as long as the different clubsare drifting apart, the collective mind capacity will be decreasing while the messages willbecome longer and longer. However, the process of diffusion between these two streams(not included in our model) is very likely, see Fig.14.

5.8 Summary

Interpretation of the proposed dynamical model of Livings has been introduced anddiscussed from viewpoints of mathematics, physics, biology, neurosciences, etc.

6. Complexity for Survival of Livings

6.1 Measure of Survivability

In this sub-section, we will apply the proposed model to establish a connectionbetween complexity and survivability of Livings. We will introduce, as a measure of survivability, the strength of the random force that, being applied to a particle, nulliesthe inequality (1). For better physical interpretation, it will be more convenient torepresent the inequality (1) in terms of the variance D

D < 0 (64)

remembering that for normal probability density distribution

H = log 2√2πeD 2 (65)

while the normal density is the rst term in the Gram-Charlier series for representationof an arbitrary probability distribution.

Thus, the ability to survive (in terms of preserving the property (1) under action of arandom force) can be achieved only with help of increased complexity. However, physical




complexity is irrelevant: no matter how complex is Newtonian or Langevin dynamics, thesecond law of thermodynamics will convert the inequality (1) into the opposite one. Theonly complexity that counts is those associated with mental dynamics. Consequently,

increase of complexity of mental dynamics, and therefore, complexity of the informationpotential, is the only way to maximize the survivability of Livings. This conclusion willbe reinforced by further evidence to be discussed in the following section.

6.2 Mental Complexity via Reection of Information

In this sub-section, we will show that communication between living particles via infor-mation potential increases their survivability. For that purpose, consider a system of twoliving particles (or intelligent agents) that interact via the simplest linear information

potential in the fashion described above. We will start with the case when each particleinteracts only with its own image. The motor dynamics of the system is the following

v1 = −ζ (q2 −β 1D1)∂

∂v1ln ρ1(v1, t ), (66)

v2 = −ζ (q2 −β 2D2)∂

∂v2ln ρ2(v2, t ), (67)

Then the mental dynamics in terms of the variance is described by two uncoupled equa-tions

D1 = ζ (q2

−β 1D1) (68)D2 = ζ (q2 −β 2D2) (69)

The solutions to these equations subject to zero initial conditions asymptotically approachthe stationary values q2/β 1 and q2/β 2, respectively, while

D1 > 0, and D2 > 0 (70)

Thus, non-communicating agents (66) and (67) do not expose the property of livingsystems (64) and behave as a physical non-living system.

Let us now increase the complexity by allowing the agents to communicate:

v1 = −ζ (q2 −βD 2)∂

∂v1ln ρ1(v1, t ), (71)

v2 = −ζ (q2 + βD 1)∂

∂v2ln ρ2(v2, t ), (72)

In this case, each agent interacts with the image of another agent. Then the mentaldynamics is represented by two coupled equations

˙D1 = ζ (q

2

−βD 2) (73)D2 = ζ (q2 + βD 1) (74)




This system describes harmonic oscillations induced by a step-function-force originatedfrom noise of the strength q 2, (Zak, M., 2007a).

Obviously, there are periods when

D1 < 0, or D2 < 0 (75)

i.e. the communicating agents expose the property of life (64). It should be emphasizedthat no isolated physical system can oscillate in probability space since that is forbiddenby the second law of thermodynamics, Figure 13.

Oscillation of variances described by Eqs. (73) and (74), represents an exchange of information between the agents, and it can be interpreted as a special type of communica-tions: conversation. During such a conversation, information is reected from one agentto another, back and forth. Obviously, we are dealing here with Shannon information

capacity that does not include semantics.This paradigm can be generalized to the case of n communicating agents

vi = −ζ (q2 −β ij D j )∂

∂viln ρi(vi , t ), β ij = −β ji (76)

D i = ζ (q2 −β ij D j ) (77)

Since the matrix of the coefficients β ij is skew-symmetric, its characteristic roots are pureimaginary, and therefore, the solution to Eq. (77) is a linear combination of weightedharmonic oscillations of different frequencies. The interpretation of this solution is similarto those of the case of two agents: it describes a conversation between n agents. Indeed,the information from the ith agent is distributed among all the rest agents, then reectedand distributed again, etc. Obviously, that solution possesses the property of life in thesense of the inequalities (75), Fig. 16.

6.3 Image Dynamics: What do you think I think you think

In this sub-section we will discuss a special case when additional information is neededto compensate incomplete information of agents about each other. We will start with

the simplest model of two interacting agents assuming that each agent is representedby an inertionless classical point evolving in physical space. That allows us to considerpositions x (instead of velocities v) as state variables. We will also assume that thenext future position of each agent depends only upon its own present position and thepresent position of his opponent. Then their evolutionary model can be represented bythe following system of differential equations

x1 = f 1(x1, x2) (78)

x2 = f 2(x1, x2) (79)

We will start with the assumption that these agents belong to the same class, and there-fore, they know the structure of the whole system (78), (79). However, each of the agents




may not know the initial condition of the other one, and therefore, he cannot calculatethe current value of his opponent’s state variable. As a result of that, the agents try toreconstruct these values using the images of their opponents This process can be associ-

ated with the concept of reection; in psychology reection is dened as the ability of aperson to create a self-nonself images and interact with them.

Let us turn rst to the agent 1. In his view, the system (78), (79) looks as following

x11 = f 1(x11 , x21) (80)

x21 = f 2(x21 , x121 ) (81)

where x11 is the self-image of the agent 1, x21 is the agent’s 1 image of the agent 2, andx121 is the agent’s 1 image of the agent’s 2 image of the agent 1. This system is not closedsince it includes an additional 3-index variable x 121 . In order to nd the correspondingequation for this variable, one has to rewrite Eqs.(30),(31) in the 3-index form. However,it is easily veriable that such form will include 4-index variables, etc., i.e. this chainof equations will never be closed. By interchanging the indices 1 and 2 in Eqs.(78) and(79), one arrives at the system describing the view of the agent 2. The situation can begeneralized from two- to n – dimensional systems. It is easy to calculate that the totalnumber of equations for the m-th level of reection, i.e. for the m-index variables, is

N m = nm . (82)

The chain of reections illustrating the paradigm: “what do you think I think youthink. . . ” is presented in Figure 17.

Thus, as follows from Eq. (82), the number of equations grows exponentially withthe number of the levels of reections, and it grows linearly with the dimensionality nof the original system. It should be noticed that for each m-th level of reection, the

corresponding system of equations always includes ( m +1)-index variables, and therefore,it is always open. Hence, for any quantitative results, this system must be supplementedby a closure, i.e. by additional equations with respect to extra-variables. In order toillustrate how it can be done, let us rst reduce Eqs.(78) and (79) to the linear form

x1 = a11x1 + a12x2, (83)

x2 = a21x1 + a22x2 (84)

Taking a position of the agent 1, we can rewrite Eqs.(83) and (84) in the form of a chain




of reections describing interactions between his images:

x1 = a11x1 + a12x12

x12 = a22x12 + a21x121

x121 = a11x121 + a12x1212

x1212 = a22x1212 + a21x12121

......................................................................................

x1212... 121 = a11x1212 ... 121 + a12x1212 ... 1212

x1212... 1212 = a22x1212... 1212 + a21x1212 ... 12121

(85)

Here the agent 1 starts with Eq. (83) and transforms it into the rst equation in thesystem (85) by replacing the unknown agent’s 2 state variable x2 with the predictedvalue x12 that describes the agent 2 in view of the agent 1. In order to get the governingequation for the new variable x2, the agent 1 transforms Eq. (84) into the second equationin the system (85) by replacing x2 with x12 and x1 with x121 that describes the agent1 view on the agent 2 view on the agent 1. This process is endless: each new equationincludes a new variable that requires another governing equation, etc. Hence, for anyquantitative results, this system must be supplemented by a closure, i.e. by additionalequations with respect to extra-variables. However, before discussing the closure, we will

simplify the indexing of the variables in Eqs. (85) in the following way:

x1 = y1, x12 = y2, x121 = y3, x1212 = y4, etc. (86)

Obviously, the variable yidescribes prediction after i number of reections. Here we as-sumed that the agent 1 has complete information about himself, i.e. x11 = x1. Therefore,in our future discussions, any repeated indices (following one another) will be compressedinto one index.

In the new notations, Eqs. (85) can be presented in a more compact form

y1 = a11y1 + a12y2

y2 = a22y2 + a21y3

y3 = a11y3 + a12y4

y4 = a22y4 + a21y5

......................................................................................

yn−1 = a11yn−1 + a12yn

yn = a22yn + a21yn +1

(87)




It should be noticed that a simple two-dimensional system (83), (84) gave rise to ann-dimensional system of reection chain (87). Although the system (87) is also linear,the structure of its coefficients is different; for instance, if the matrix of coefficients in

Eqs. (83), (84) is symmetric, i.e. a12 = a21 , the matrix of the coefficients in Eqs. (87) isnot symmetric, and therefore, the corresponding properties of symmetric systems, suchas real characteristic roots, are not preserved.

For the purpose of closure, it is reasonable to assume that after certain level of reec-tion, the image does not have signicant change; for instance

x12121212 ≈x121212 , i.e yn ≈yn−2 (88)

This equation complements the system (87).So far, we were dealing with a physical (non-living) system. Let us apply the same

strategy to mental dynamics (73) (74) presenting this system in the form of interactionof the rst agent with its own images

D1 = ζ (q2 −βD 12)D12 = ζ (q2 −βD 121)

D121 = ζ (q2 −βD 1212 ). . . . . . . . . . . . . . . . . . . .etc (89)

or, adopting the notations (86)

y1 = ζ (q2 −βy2)

y2 = ζ (q2 + βy30

.............................

yn−1 = ζ (q2 + βyn )

yn = yn−2

(90)

The system (90) is closed, and the matrix of its coefficients has the following characteristicrootsλ1 = λ2 = ...λ n−2 = 0 , λn−1,n = +

− iβ (91)

Hence, the general solution of Eqs. (90) has the following structure

y1 = C 0q2 + C 1t + C 2t2 + ...C n−2tn−2 + C n−1 sin βt + C n cos βt (92)

The arbitrary constants C 1, C2. . . Cn are supposed to be found from initial conditions,and actually they represent the degree of incompleteness of information that distinguishesthe images from reality. These constants can be specied if actual values of y2 are knownat least at n different instants of time to be compared with the corresponding images of y12.




Obviously, the solution (92) satises the condition (64), since, due to the presence of harmonic oscillations, there are periods when

y1

= D1

< 0 (93)

By interchanging indices 1 and 2 in Eqs. (89)- (93), one arrives at dynamics of interactionof the agent 2 with its images.

It is worth emphasizing that in this section we are dealing with complex mentaldynamics created by a single agent that communicates with its own images, images of images, etc.

Thus, we came to the following conclusion: the survivability of Livings, i.e. theirability to preserve the inequality (2), is proportional to reection-based complexity of mental dynamics that is, to complexity of information potential .

6.4 Chain of Abstractions

In view of importance of mental complexity for survival of Livings, we will take a closerlook into cognitive aspects of information potential. It should be recalled that classicalmethods of information processing are effective in a deterministic and repetative world,but faced with the uncertainties and unpredictabilities, they fail. At the same time, manynatural and social phenomena exhibit some degree of regularity only on a higher level of abstraction, i.e.in terms of some invariants. Indeed, it is easier to predict the state of the

solar system in a billion years ahead than to predict a price of a stock of a single companytomorrow. In this sub-section we will discuss a new type of attractors and associated withthem a new chain of abstraction that is provided by complexity of mental dynamics.

α. Attractors in motor dynamics . We will start with neural nets that, in our terminol-ogy, represent motor dynamics without mental dynamics. The standard form of classicalneural nets is is given by Eqs. (59). Due to the sigmoid functiontanh xi , the system (59) isnonlinear and dissipative, and these two conditions are necessary for existence of severalattractors that can be static, periodic, or chaotic; their type, location, and basins dependupon the choice of the synaptic weights. For instance, if these weights are symmetric

wij = w ji , (94)

then the solution of Eqs. (59) can have only static attractors. As illustrated in Figure18, static attractors perform generalization: they draw a general rule via abstraction, i.e.via removal of insignicant details.

β . Attractors in mental dynamics . Signicant expansion of the concept of an attractoras well as associated with it generalization via abstraction is provided by mental dynamics.We will start with mental neural nets based upon mirror neurons being discussed inSection 5. First, we will reformulate the motor dynamics (49) by changing notations of the state variables from v to x to be consistent with the notations of this section

xi = ( −x1/ 2ii + wij tanh x jj )

∂ ∂x j

ln ρ(x1, ...x n ) (95)




Here, for further convenience, we have introduced new compressed notations

x i = ∞

−∞

xiρdx i , xii = D ii = ∞

−∞

(xi −xi)2ρdx i ,

x iiii = ∞ −∞

(xii −xii )2ρdx ii ...etc (96)

The corresponding mental dynamics in the new notations follows from Eq. (50)

∂ρ∂t

= ( x1/ 2ii −wij tanh x jj )

∂ 2ρ∂X 2 j

(97)

In the same way, the mental (mirror) neural nets can be obtained from Eqs. (54)

xii = (

−x1/ 2

ii + wij tanh x jj ) (98)

where the state variables xii represent variances of ρ.Obviously, the discussion of the performance of the neural net (54) can be repeated

for the performance of the neural net (98).

6.5 Hierarchy of Higher Mental Abstractions

Following the same pattern as those discussed in the previous sub-section, and keepingthe same notations, one can introduce the next generation of mental neural nets starting

with the motor dynamicsx i = [(−x1/ 2

iiii + wij tanh x jjjj )∂

∂x iiln ρ (x11 ,...x nn )]

∂ ∂x i

ln ρ(x1, ...x n ) (99)

Here, in addition to the original random state variables x i , new random variables xii areincluded into the structure of information potential. They represent invariants (variances)of the original variables that are assumed to be random too, while their randomness isdescribed by the secondary joint probability density ρ (x11 , ...x nn ). The correspondingFokker-Planck equation governing the mental part of the neural net is

∂ρ∂t = [( x

1/ 2 jjjj −wij tanh x jjjj )

∂ ∂x ii ln ρ (x11 ,...x nn )]

∂ 2ρ∂X 2 jj (100)

Then, following the same pattern as in Eqs. (95), (97), and (98), one obtains

x ii = ( −x1/ 2 jjjj + wij tanh x jjjj )

∂ ∂x ii

ln ρ (x11 , ...x nn ) (101)

∂ρ∂t

= [(x1/ 2 jjjj −wij tanh x jjjj )

∂ 2ρ∂X 2 jj

(102)

x iiii = (

−x1/ 2

jjjj + wij tanh x jjjj ). (103)

Here Eqs. (101) and (103) describe dynamics of the variances xii and variances of vari-ances xiiii respectively, while Eq. (102) governs the evolution of the secondary joint




probability density ρ (x11 ,..x nn ). As follows from Eqs. (99)- (103), the only variablesthat have attractors are the variances of variances; these attractors are controlled by Eq.(103) that has the same structure as Eq. (98). The stationary values of these variables

do not depend upon the initial conditions: they depend only upon the basins where theinitial conditions belong, and that species a particular attractor out of the whole set of possible attractors. On the contrary, no other variables have attractors, and their valuesdepend upon the initial conditions. Thus, the attractors have broad membership in termsof the variables x iiii , and that represents a high level of generalization. At the same time,such “details” as values of x i and x ii at the attractors are not dened being omitted asinsignicant, and that represent a high level of abstraction.

It should be noticed that the chain of abstractions was built upon only principal vari-ances, while co-variances were not included. There are no obstacles to such an inclusion;

however, the conditions for preserving the positivity of the tensors xij and x ijkq are toocumbersome while they do not bring any signicant novelty into cognitive aspects of theproblem other than increase of the number of attractors, (see Eqs. (42), (43), and (45)).

It is interesting to note that Eqs.(101) and (102) have the same structure as Eqs. (11)and (12), and therefore, the velocities of variances xii are entangled in the same way as theaccelerations v are, (see Eq. (23) ). That means that the chain of abstractions consideredabove, gives rise to an associated chain of entanglements of the variables xii , xiiii , ...etc.

6.6 Abstraction and Survivability

In this sub-section, we will demonstrate that each new level of abstraction in mentaldynamics increases survivability of the system in terms of its capability to increase order(see the condition (64)) regardless of action of random force. For easier physical inter-pretation, we will investigate a one-dimensional linear case by starting with Eq. (66).This equation describes an agent that interacts with its own image in the simple linearform. As shown in the previous section, such an agent does not expose the property of living systems (64) and behaves as a physical non-living particle. Let us introduce nowthe second level of mental complexity when the same agent interacts with its image andthe image of this image. The dynamical model describing such an agent follows from aone-dimensional version of Eqs. (99)- (103) in which the neural net structure is replacedby a linear term and to which noise of the strength q 2 is added

x1 = qL(t) + x1111∂

∂x 11ln ρ (x11 )

∂ ∂x i

ln ρ(x1) (104)

∂ρ∂t

= [q2 −x1111∂

∂x 11ln ρ (x11 )]

∂ 2ρ∂X 211

(105)

x11 = (2 q2 −2x1111 )∂

∂x 11ln ρ (x11 ) (106)

∂ρ∂t

= ( −2q2 + 2 x1111 )∂ 2ρ∂X 211

(107)




x1111 = 4 q2 −4x1111 (108)

In order to prove our point, we have to show that there exists such a set of initialconditions that provides the inequality (64)

x11 < 0 (109)

under the action of random force of a strength q 2. Let us concentrate on Eqs. (106) and(108) and choose the following initial conditions

x1111 = 0 , ρ =2

σ√2πexp(−

X 211

2σ2 ) at t = 0 (110)

where

σ = const , and 0 ≤X 11 < ∞ (111)Now as follows from Eqs. (106) and (108)

0 < x 1111 < q 2 at t > 0,and∂

∂x 11ln ρ (x11) < 0 at x11 > 0, t > 0

(112)

Thereforex11 < 0 at t > 0 (113)

Thus, an additional layer of mental complexity that allows an agent to interact not onlywith its image, but also with the image of that image, makes the agent capable to increaseorder under action of a random force, i.e. increase its survivability.

6.7 Activation of New Levels of Abstractions

A slight modication of the model of motor-mental dynamics discussed above leads to anew phenomenon: the capability to activate new levels of abstraction needed to preservethe inequality (64). The activation is triggered by the growth of variance caused byapplied random force. In order to demonstrate this, let us turn to Eq. (33) and rewriteit in the following form

x = qL(t) + λαρ

∂ρ∂x

(114)

Then Eq. (36), and (37) are modied to

∂ρ∂t

= [q2(−λ exp √D)]∂ 2ρ∂X 2

(115)

D = [2q2(−λ exp √D)] (116)

respectively. Here λ is a new variable dened by the following differential equationλ = λ(1 −λ)D (117)




One can verify that Eq. (117) implements the following logic:

λ = 0 if D ≤0, and λ = 1 if D > 0, (118)

Indeed, Eq. (117) has two static attractors: λ = 1 and λ = 0; when D > 0, the rstattractor is stable; when D < 0 , it becomes unstable, and the solution switches tothe second one that becomes stable. The transition time is nite since the Lipchitzcondition at the attractors does not hold, and therefore, the attractors are terminal,(Zak, M., 2005a). Hence, when there is no random force applied, i.e. q=0, the rstlevel of abstraction does not need to be activated, since then D = 0, and therefore. λ iszero. However, when random force is applied, i.e. q = 0 , the variance D starts growing,i.e. D > 0. Then the rst level of abstraction becomes activated, λ switches to 1, and,according to Eq. (116), the growth of dispersion is eliminated. If the rst level of

abstraction is not sufficient, the next levels of abstractions considered in the previoussub-sections, can be activated in a similar way.

6.8 Summary

A connection between survivability of Livings and complexity of their behavior is es-tablished. New physical paradigms – exchange of information via reections, and chainof abstractions- explaining and describing progressive evolution of complexity in living(active) systems are introduced. A biological origin of these paradigms is associated with

a recently discovered mirror neuron that is able to learn by imitation. As a result, an ac-tive element possesses the self-nonself images and interacts with them creating the worldof mental dynamics. Three fundamental types of complexity of mental dynamics thatcontribute to survivability are identied.

7. Intelligence in Livings

7.1 Denition and General Remarks

The concept of intelligence has many different denitions depending upon the contextin which it is used. In this paper we will link intelligence to the proposed model of Livings, namely, associating it with a capability of Livings to increase their survivabilityvia successful interaction with their collective mind, i.e. with self- image and images of others. For an illustration of this denition, one can turn to the sub-section g of thesection 5, where the preservation of survival (in terms of inequality (64)) is achieved dueto activation of new level of abstraction, (see Eqs. (114)- (118)).

7.2 Intelligent Control in Livings

In this sub-section we will discuss the ability of Livings to controls their own behaviorusing information force as an actuator. These dynamical paradigm links to reective




control being performed by living systems for changing their behavior by “internal effort”with the purpose to increase survivability. In this sub-section we will modify the modelof Livings’ formalism subject to intelligent control. We will start with the linear version

of the Langevin equations known as the Ornstein-Uhlenbeck system

vi = −a ij v j + Li(t),

< L i(t) > = 0 , < L i(t)L j (t ) > = qij δ(t −t ), qij = q ji , i = 1 , 2,...n(119)

subject to initial conditions presented in the form

vi(0) = v0i , (120)

Err (V 0

i ) = ρ(V 0

1 ,...V 0

n ) = ρ0. (121)

v0i = < V 0i > =

∞

−∞

V iρ0dV i ...dV n , i = 1 , 2,...n (122)

Our attention will be concentrated on control of uncertainties represented by non-sharpinitial conditions, Langevin forces Li(t) and errors in values of parameters a i . All theseuncertainties will be measured by expected deviations of the state variables from theirmean values (i.e. variances)

D ij = < (vi

−< v i > )(v j

−< v j > ) > (123)

Now we will introduce the following information-based control

vi = −a ij v j + Li(t) + ζα ij∂

∂v jln ρ (124)

The associated Liouville-Fokker-Planck equation describing evolution of the probabilitydensity ρ reads

∂ρ∂t

= a ij∂

∂V i(V j ρ) + ( qij −ζα ij )

∂ 2ρ∂V 2 j

(125)

whileρ(t = 0) = ρ0 (126)

(Multiplying Eq.(125) by V i , then using partial integration, one obtains for expecta-tions

ddt

< v i > = −a ij < v j >, (127)

Similarly one obtains for variances

D ij = −a il D lj −a jl D li + 2 qij −2ζα ij (128)

Thus, as follows from Eqs.(125) and (128), the control force ζα ij does not affect theexpected values of the state variables ¡ v¿: it affects only the variances.




Before formulating the control strategy, we will discuss the solution to Eq.(125). Inthe simplest case when α ij = const., the solution describing the transition from initialdensity (126) to current density is given by the following equation (compare with Eq.

(60))

ρ({V }, t|{V }, t ) = (2 π)n/ 2[DetD (t −t )]−1/ 2 exp{−12 [D−1(t −t )]ij [V i −Gik (t −t )V k ]

×[V j −G jl (t −t)V l ]}(129)

Here the Green function in matrix notation is expressed by Eq. (61). Substituting thesolution (129) into Eq.(124), one obtains the Langevin equation in the form

vi = −a ij v j + L i(t) + ζ [ασ −1]ij (v j −G jl vl) (130)

If the control forces α ij are not constants, for instance, if they depend upon variances

α ij = α ij ({Dkl}) (131)

the solution (129) as well as the simplied form (130) of the original Langevin equationare valid only for small times. Nevertheless, for better physical interpretation, we will staywith this approximation in our future discussions. First we have to justify the nonlinearform for the control forces (131). For that purpose, let us turn to Eq. (128). Asfollows from this equation, the variances D ij are not “protected” from crossing zeros andbecoming negative, and that would not have any physical meaning. To be more precise,the non-negativity of all the variances, regardless of possible coordinate transformations,can be guaranteed only by non-negativity of the matrix |D | that, in turn, requires non-negativity of all its left-corner determinants

Det |D ij | ≥0, i, j = 1; i, j = 1 , 2; ...i, j = 1 , 2, ...n (132)

In terms of the Fokker-Planck equation (125), negative variances would lead to negativediffusion, and that is associated with ill-posedness of initial-value problems for parabolicPDE. Mathematical aspects of that phenomenon are discussed in (Zak, M.,2005a).

In order to enforce the inequalities (132), we will specify the control forces (131) as

linear functions of variances with one nonlinear term as terminal attractor

α ij = bil D lj + b jl D li + [cD1/ 2]ij , bij = 0 , cij > 0 (133)

Then, as follows from Eq.(128), for small variances

D ij ∼= qij t at D ij −→0 (134)

But the strength-of-noise matrix qij is always non-negative, and therefore, the constraints(132) are satised. The role of the terminal attractor will be discussed below. Now theoriginal dynamical system (124) with intelligent control (133) reads

vi = −a ij v j + L i(t) + ζ (bil D lj + b jl D li + [cD1/ 2]ij )∂

∂v jln ρ (135)




or, applying the approximation (129)

vi = −a ij v j + Li(t) + ζ [αD −1]ij (v j −G jl vl) (136)

where α ij is expressed by Eq. (133).Eqs. (135) and (136) must be complemented by the additional controller that imple-

ments the dynamics of variances

D ij = −a il D lj −a jl D li + 2 qij −2ζ (bil D lj + b jl D li + [cD1/ 2]ij ) (137)

As follows from Eq. (137), the dynamics of the controller depends only upon the designparameters {a }, but not on the state variables {v }of the underlying dynamical system,and therefore, the control strategy can be developed in advance . This strategy mayinclude classical lead-lag compensation for optimization of transient response or largesteady-state errors, while the role of the classical compensation system is played by thecontrolling dynamics (137) in which the design parameters {b}can be modied for appro-priate change of root locus, noise suppression, etc. Fig.19. However, in contradistinctionto classical control, here the compensation system is composed of the statistical invariantsproduced by the underlying dynamical system “itself” via the corresponding Liouville or Fokker-Planck equations.

Let us now analyze the effect of terminal attractor and, turning to Eq.(133), startwith the matrix [ ∂ D ij /∂D lk ]. Its diagonal elements become innitely negative when thevariances vanish

∂ D ij

∂D ij= −2a ii −ζ (4bii −

13

cij σ−1/ 2ij ) → −∞at D ij →0 (138)

while the rest elements ∂ D ij /∂D ij atij = lk are bounded, (compare with Eqs. (54)and (55)). Therefore, due to the terminal attractor, the controller (53) linearized withrespect to zero variances has innitely negative characteristic roots, i.e. it is innitelystable regardless of the parameters {a }of the original dynamical system as well as of thechosen parameters {b}of the controller. This effect has been exploited in (Zak, M.,2006a)for suppression of chaos. Another effect of the terminal attractor is in minimization of

residual noise. A simplied example of such a minimization has been discussed in theSection 2, (see Eqs. (33-37)) where, for the purpose of illustration, it was assumed that thenoise strength q in known in advance and that allowed us to eliminate noise completely.In reality, the noise strength is not known exactly, and it can only be minimized usingthe proposed controller. Let us turn to Eq.(137) at equilibrium, i.e. when D ij = 0, andnd the ratio of residual noise and variance

∂qij

∂D ij= 2 a ii + ζ (4bii +

13

cij D 1/ 2ij →

13

cij D−1/ 2ij ) → ∞at D ij →0 (139)

As follows from Eq. (137), without the terminal attractor, i.e. when cij = 0, this ratiois nite; but with the terminal attractor it is unbounded. This means that the terminalattractor almost completely suppresses noise, (see Fig. 20).




pose a physical model of common sense in connection with the decision making process.This model grew out of the model of Livings proposed and discussed in the previoussections. Previously similar model in quantum implementation has been introduced and

discussed in (Zak, M., 2000b) We will start with the one-dimensional case based uponthe augmented version of Eq. (11)

v = −yζσ2 ∂ ∂v

ln ρ, (140)

∂ρ∂t

= yζσ2 ∂ 2ρ∂V 2

(141)

wherey = −a y(1 −y), a = const > 0 (142)

Eq. (142) has two equilibrium points: y = 0 and y = 1. At both these points, theLipschitz condition does not hold since

∂ y∂y

= a1 −2y

2 y(1 −y)(143)

and therefore∂ y/∂y → ∞if y →0,

∂ y/∂ y →0 if y →1(144)

Hence, y = 1 is a terminal attractor, and y = 0 is a terminal repeller, (Zak,M.,1989). Re-gardless of these “abnormality”, the closed form solution is easily obtained by separationof variables

t =2a

arctan 1 −yy

+ C, C = 0 if y = 0 at t = 0 (145)

However, this “abnormality” becomes crucial for providing a nite time T of transitionfrom the repeller y = 0 to the attractor y = 1

T =πa

, (146)

It should be recalled that in classical theory of ODE, when the Lipschitz condition ispreserved, an attractor is always approached asymptotically, and the transition period istheoretically unbounded. Hence, this limitation is removed due to a special form of thegoverning equation (142). Qualitatively, the result (146) holds even if a = a (t). Thenthe solution to Eq.(142) is

t

0

a(t)dt =2 arctan 1 −yy

, andT

0

a(t)dt = π, i.e. T < ∞, (147)

Similar results can be obtained for a < 0: then the attractor becomes a repeller andwise versa. When the function a(t) changes its sign sequentially, the solution switches




from 0 to 1 and back, respectively, while the transition time is always nite. Selecting,for instance,

a = cos ωt (148)

one arrives at periodical switches (with the frequency ω) from a decision mode to a passive(pre-decision) mode and back, since, in the context of this sub-section, the state y=0 andy=1 can be identied with the passive mode and the decision mode, respectively. Asshown in Fig. 21, after each switch to the decision mode, the system may select differentsolutions from the same family (15), i.e. a different decision, so that the entire solutionwill include jumps from one branch of the family (15) to another. In order to preservethe autonomy of the dynamical model (141), (142), one can present Eq. (148) as alimit-cycle-solution to the following autonomous non-linear ODE

a −ξ1a + ξ2a3

+ ξ3a = 0 (149)The periodic solution (148) to this oscillator is usually associated with modeling brainrhythms, while the frequency ω is found to be a monotonic function of a stimulus intensity.

The model we propose for common sense simulation is based upon augmented n-dimensional version of Eqs.(140) and (141)

vi = ζ [(y −1)f i + y(β i + α ii

∂ ∂V i

ln ρ)], i = 1 , 2, ...n, (150)

∂ρ

∂t=

−

nζ Σi=1 {

[(y

−1)f i + yβ i]

∂ρ

∂V i −yα ii

∂ 2ρ

∂V 2

i }(151)

Heref i = f i(v1,... vn ), α ii = α ii (ρ), β i = β i(ρ) (152)

where vi is expected value of vi ,yis dened by Eqs. (142) and (149), while the functionsf and β provide non-zero mean and non-zero drift of the underlying stochastic process.The solution to Eq. (151) has the same fundamental properties as the solution to itssimplied one-dimensional version (12). Indeed, starting with the initial conditions

vi = 0 , ρ = δ(V i →0) at t = 0 (153)

and looking rst for the solution within an innitesimal initial time interval ε, one reducesEq. (151) to the form

∂ρ∂t

= −nΣ

i=1α ii

∂ 2ρ∂V 2i

(154)

since∂ρ∂V i

= 0 , at t = 0 (155)

and therefore, all the drift terms can be ignored within the interval ε.The solution to Eq. (154) is a multi-dimensional version of Eq. (13). Substitution of

this solution to Eqs. (150) yields

vi = ζ [f i + yβ i +vi

2t] →ζ [

vi

2t] at t →0 (156)




∂ρ∂t

= −n

ζ Σi=1

f i∂ρ∂V i

(161)

At this mode, the motor dynamics (150), and the mental dynamics (151) are not coupled.

The performance starts with mental dynamics that runs ahead of actual time, calculatesexpected values of the state variables as well as the difference Δ at the end of a selectedlead time T, (see Eq. (158), and sends this information to the sensor device (159). Assoon as Δbecomes large, it changes the sign of the control parameter a (see Eqs. (142)and (159)), the value of y changes from zero to one, and the system (160), (161) switchesto the decision mode taking the form (150), (151). After this switch, a deterministictrajectory dened by the solution to Eq. (160) splits into a family of the solution to thecoupled system (150), (151) (see Eq. (157), and Fig. 21). But, as discussed above, thesystem selects only one trajectory of this family at random with the probability dened

by the information forces α i(ρ) and β i(ρ). Thus, the topology of the proposed modeldemonstrates how a decision is made, while the structure of the information forces deneswhat decision is made. This structure will be discussed in the following sub-section.

δ. Decision via choice of attractors. Let us return to the system (150), (151) in thedecision mode

vi = ζ (β i + α ii∂

∂V iln ρ), i = 1 , 2, ...n, (162)

∂ρ∂t

= −n

ζ Σi=1‘

(β i∂ρ∂V i

+ α ii∂ 2ρ∂V 2i

) (163)

and assume that α ii and β i in Eq. (152) depend only upon two rst invariants of theprobability density ρ, namely, upon the means vi and principle variances D ii

α ii = α ii (v1,... vn , D 11,...D n n), β i = β i(v1,... v1, D 11, ...D n n) (164)

As follows from Eqs. (150) and (151), the means vi and the variances D ii must satisfythe following ODE, (Compare with Eqs. (127) and (128))

˙vi = ζβ i(v1,... vn , D 11, ...D n n), (165)

D i i = −ζα i(v1,... v1, D 11, ...D n n) (166)

Thus, although the state variables vi are statistically independent, i.e. D ij

≡0if i = j,

the time evolution of their statistical invariants is coupled via Eqs. (165) and (166).It should be noticed that Eqs. (165) and (166) can be considered as an ODE-based

approximation to the mental dynamics (163).Next we will introduce the following structure into the functions (164)

β i = (vi −γ i1)( vi −γ i2)...(vi −γ in )(Σ wij Δ j ), Δ j = ( vi −vi)2, (167)

α ii = (D ii −ηi1)(D ii −ηi2)...(D ii −ηin )(Σ vij Δ j ), i = 1 , 2, ...n, (168)

where γ ij , ηij , wij , vij are constants, and rewrite the system (165), (166) as follows

˙vi = ζ [n

j =1

(vi −γ ij )](Σ wij Δ j ), i = 1 , 2, ...n, (169)




D ii = −ζ n j =1

(D ii −ηij )(Σ u ij Δ j ), i = 1 , 2, ...n, (170)

We will start our analysis with Eqs. (169). First of all, in that particular setting, theseequations do not depend upon Eqs. (170). Moreover, each equation in the system (169) iscoupled to other equations of the same systems only via the signs of the sums Σ wij Δ j thatinclude contribution from all the equations. Indeed, if Σ wij Δ j > 0, the ith equation hasthe terminal attractors,

v0i = γ i2, γ i4, ...γ im , i = 1 , 2, ...n (171)

and terminal repellersv∗i = γ i1, γ i3, ...γ ik , i = 1 , 2, ...n (172)

where m = n , k = n −1 if n is even, and m = n-1, k =n if n is odd.The n2/ 2 attractors (171) can be pre-stored using n2 weights wij . These attractors

are the limit values of the means of the stochastic process that occurs as a result of switchfrom the pre-decision to the decision making mode, and these attractors are approachedin nite time, (see eq. (146).

Turning to Eqs. (170) and noticing that they have the same structure as Eqs. (169),one concludes that the principle variances D ii of the same stochastic process approachtheir limiting values

D 0

ii= η

i2, η

i4, ...η

im, i = 1 , 2, ...n, (173)

that can be pre-stored using weights uij . Thus, when the system switches to the decisionmaking mode, it chooses a sample of the corresponding stochastic process, and this samplerepresents the decision. The choice of the sample is controlled by the normal probabilitydensity with means and principle variances approaching their limit values (171) and (173).The choice of a particular attractor out of the set of (171) and (173) depends upon theinitial values of means and variances at the end of the pre-decision period: if these valuesfall into the basin of attraction of a certain attractor, they will eventually approach thatattractor, Fig. 22.

ε. Decision via phase transition. In the previous sub-section, we consider the casewhen pre- and post-decision states are not totally disconnected: they belong to the samebasin of attraction. However, in general, decision may require fundamental change of thedynamical structure, and that can be achieved via phase transition. The distinguishingcharacteristic of a phase transition is an abrupt sudden change in one or more physicalproperties of a dynamical system. In physics, a phase transition is the transformationof a thermodynamic system from one phase to another (evaporation, boiling, melting,freezing, sublimation, etc). In engineering, such transitions occur because of change of system conguration (new load, new open valve, new switch or a logic device in controlsystems with variable structure, etc.). In living systems, dynamics includes the conceptof “discrete events”, i.e. special critical states that give rise to branching solutions, ortransitions from deterministic to random states and vice versa (for instant, life-non-life




transitions). First mathematical indication of a phase transition is discontinuities in someof the state variables as functions of time. In classical approach, these discontinuities areincorporated through a set of inequalities imposing unilateral constraints upon the state

variables and the corresponding parameters of the governing differential equations. Themost severe changes in dynamics are caused by vanishing coefficients at the highest spaceor time derivatives. The best example to that is the transition from the Navier-Stokesto the Euler equations in uid dynamics where due to vanishing viscosity, the tangentialvelocity on the rigid boundary jumps from zero (the non-slip condition) to a nal value(the slip condition). In this sub-section, we will modify the proposed model by introducingphase transitions that are accompanied by nite jumps of the state variables.

Let us modify Eq. (150), in the following way

(1 −yi)vi = ζ [f i + ( β i + α i

∂ ∂V i ln ρ)], i = 1 , 2, ...n, (174)

where yi is dened as following

yi = −a i yi(1 −yi),a i = Ai −Δ i , Δ i = ( vi −vi)2 (175)

and assume that Δ n becomes large; that changes the sign of the control parameter an ,the value of yn changes from zero to one, and the system (160), (161) switches to thedecision mode in the form

(1 −yi)vi = ζ [f i + ( β i + α i

∂

∂V i ln ρ)], i = 1 , 2,...n −1, (176)

0 = f n + ( β n + α .n∂

∂V nln ρ), (177)

∂ρ∂t

= −n

ζ Σi=1

[(f i + β i)∂ρ∂V i −α i

∂ 2ρ∂V 2i

], i = 1 , 2,...n −1, (178)

Since Eq. (177) is stationary , one of the state variables, for instance, xn can be expressedvia the rest of the variables. Hence, formally the decision making mode is described byEqs, (162)-(170), but the dimensionality of the system drops from n to n−1; obviously, the

functions f i(v1, ...vn−1), α i(D11 , ...D n−1,n −1), β i(vi ,... vn−1), become different from their n-dimensional origins, and therefore, the location of the attractors (171) and (173) becomesdifferent as well , and this difference may be signicant. The number of decision makingmodes associated with phase transitions is equal to the number of possible sub-spaces,and it grows exponentially with a linear growth of the original dimensionality of themodel.

7.4 Emergent Intelligence

In many cases, in order to simulate the ability of living systems to make decisions, it ismore convenient to introduced Boolean model of motion, or Boolean net. Boolean netconsists of N vertices that are characterized by Boolean variables taking values 0 or 1.




Each vertex receives input from other vertices and updates its value according to a certainrule. Such a model can be associated via a master-slave relationship with the continuousbehavior of the model of Livings discussed above. Indeed, let us turn to Eqs. (56),(57)

and introduce the following Boolean functions:

Y i(t) = 0 if vi(t) < 0, otherwise Y i = 1 (179)

P (Y 1,...Y n ) = 0 if ρ(X 1,..X n ) <12n , and P (Y 1,...Y n ) = 1 otherwise (180)

The conditions (179), (180) extract topological invariants of the solution to Eqs. (56), (57)disregarding all the metrical properties. In terms of logic, Y i can be identied with thestatements variable; then P describes the Boolean truth functions as a set of compoundstatements in these variables, and it can be represented in disjunctive normal form aswell as in the form of the corresponding logic circuit. It should be noticed that changes instatements as well as in the structure of the truth functions are driven by the solution tothe original model (56), (57). Therefore, for non-living systems when there is no feedbackinformation potential Π is applied, in accordance with the second law of thermodynamics,the truth function P (as well as its master counterpart ρ) will eventually approach zero,i.e. the lowest level of logical complexity. On the contrary, as demonstrated above, livingsystems can depart from any initial distribution toward decrease of entropy; it meansthat the corresponding truth function P can depart from zero toward the higher logical complexity without external interactions .

The logics constraints Eqs.(179), (180) can be represented by terminal dynamics(Zak,M., 1992) via relaxing the Lipschitz conditions:

Y = k Y (1 −Y )v(t), (181)

P = m P (1 −P )[ρ(t) −12

] (182)

wherek=const, m=const. sign[v(t)]=const, sign[ ρ (t)]=const.Indeed, Eq. (181) has two static attractors: Y = 0 and Y = 1; when v >0, the rstattractor is stable; when v < 0, it becomes unstable, and the solution switches to the

second one that becomes stable. The transition time is nite since the Lipschitz conditionat the attractors does not hold, and therefore, the attractors are terminal. The same istrue for Eq. (182). In both cases the transition time can be controlled by the constantsk and m to guarantee that the signs of v(t) and ρ(t) do not change during the transitionperiod.

Thus, any dynamical process described by the system (56), (57) can be “translated”into a temporal sequence of logical statements via Boolean dynamics (181), (182) whilethe complexity of the truth function can increase spontaneously, and that representsemergent intelligence. One can introduce two (or several) systems of the type (56), (57)with respect to variables vi and v

∗

i which are coupled only via Boolean dynamics, forinstance

Y i = k Y i(1 −Y i)(vi −v∗i), Y i∗= k∗ Y i (1 −Y i)(v∗i −vi) (183)




In this case, the dynamical systems will interact through a logic-based “conversation”.It should be noticed that, in general, decisions made by a living system may affect

its behavior, i.e. the Boolean functions Y and P can enter Eqs. (56), (57) as feedback-

forces. That would couple Eqs. (56), (57) and Eqs. (181), (182) thereby introducing thenext level of complexity into the living system behavior. However, that case deserves aspecial analysis, here we will conne ourselves only by a trivial example. For that purposeconsider the simplest case of Eqs. (56), (57), (181), and (182) and couple these equationsas following

v = 2( Y −12

)D∂

∂vln ρ(v, t ), (184)

∂ρ∂t

= −2(Y −12

)D∂ 2ρ∂V 2

(185)

whereβ = 2( Y −1

2) (186)

Y = k Y (1 −Y )[v(t −τ ) −1] (187)

P = m P (1 −P )[ρ(t) −12

] (188)

Here τ is a constant time delay. For the initial conditions

v = 0 at τ ≤ t ≤0, ρ = δ(t →0), Y (0) = 0 , P (0) = 1 ,

the solution is approximated by the following expressions

v = v0e1−exp( −βt ) (189)

During the initial period v(t −τ ) > 1 : β = −1, Y = 0 , P = 1 . After that, the solutionswitches to: β = 1, Y=1 , and P = 0. According to the new solution, x(t) starts decreas-ing, and ρ(t) starts increasing. When v(t-τ ) < 1, and ρ(t) > 1/2 , the solution switchesback to the rst pattern etc. Thus, the solution is represented by periodical switchesfrom the rst pattern to the second one and back, Figure 23. From the logical viewpoint,the dynamics simulates the operation of negation: P(0) =1, P(1) =0.

7.5 Summary

Intelligence in Livings is linked to the proposed model by associating it with a capability of Livings to increase their survivability via successful interaction with their collective mind,i.e. with self- image and images of others. The concept is illustrated by intelligent control,common-sense-based decision making, and Boolean-net-based emergent intelligence.

8. Data-driven Model Discovery

The models of Livings introduced and discussed above are based upon extension of the First Principles of Newtonian mechanics that dene the main dynamical topology of




these models. However, the First Principles are insufficient for detecting specic prop-erties of a particular living system represented by the parameters wij as well as by theuniversal constant ζ : these parameters must be found from experiments. Hence, we arrive

at the following inverse problem: given experimental data describing the performance of a living system discover the underlying dynamical model, i.e. nd its parameters. Themodel will be sought in the form of Eqs. (42) and (45)

vi = −ζ n

j =1

α ij∂

∂v jln ρ(v1,...vn , t ), i = 1 , 2, ...n. (190)

∂ρ∂t

= ζ n

j =1

α ij∂ 2ρ∂V 2 j

(191)

where α ij are function of the correlation moments Dks

α ij = −12 (wijks tanh Dks −cij D ij ), i = 1 , 2, ...n,

D ij = D ij

D 0 , c11 = 1 ,(192)

With reference to Eq. (192), Eq. (191) can be replaced by its simplied version (compareto Eq. (51))

D ij = −12 ζ (wijks tanh Dks −cij D ij ), i = 1 , 2, ...n, (193)

Then the inverse problem is reduced to nding the best-t-weights wijks , cij and ζ . Itshould be noticed that all the sought parameters enter Eq. (193) linearly.

We will assume that the experimental data are available in the form of time series forthe state variables in the form

vi = vi(t, C i), C i = 1 , 2,...m i , (194)

Here each function at a xed C i describes a sample of the stochastic process associatedwith the variable vi , while the family of these curves at C i = 1 , 2, ...m i , approximates thewhole ith ensemble, (see Figs. 4 and 9). Omitting details of extracting the correlationmoments

D ij = D ij (t) (195)

from the functions (194), we assume that these moments as well as their time derivativesare reconstructed in the form of time series. Then, substituting D ij , and D ij into Eq.(193) for times t1, ...t q, one arrives at a linear system of algebraic equation with respectto the constant parameters wijks , cij and ζ that, for compression, can be denoted andenumerated as Wi m

i=1

AiW i = B i = 1 , 2, ...m = 2 2n + 2 n (196)

where m is the number of the parameters dening the model, and

Ai = Ai(Dks , Dks ), B = B(Dks , Dks ), (197)




are the coefficients at the parameters W i and the free term, respectively. Introducingvalues of Ai and B at the points t = t j , j = 1 , ...q, and denoting them as Aij , and B j ,one obtains a linear system of 2n q algebraic equations

m

i=1

Aij W i = B j , j = 1 , 2, ...q, (198)

with respect to m unknown parameters. It is reasonable to assume that

2n q ≥m, i.e. q ≥2n + 1 (199)

so the system becomes overdetermined. The best-t solution is found via pseudo-inverseof the matrix

A = {Aij }, i.e. W = A∗B (200)

HereA∗= ( AT A)−1AT , and B = {B j } (201)

As soon as the parameters W are found, the model is fully reconstructed.Remark. It is assumed that the correlation moments found from the experiment must

automatically satisfy the inequalities (47) (by the denition of variances), and therefore,the enforcement of these inequalities is not needed.

9. Discussion and Conclusion

9.1 General Remarks

We will start this section with the discussion of the test-question posed in the Intro-duction: Suppose that we are observing trajectories of several particles: some or themphysical (for instance, performing a Brownian motion), and others are biological (forinstance, bacteria), Figure 1. Is it possible, based only upon the kinematics of the ob-served trajectories, to nd out which particle is alive? Now we are in a better positionto answer this question. First of all, represent the observed trajectories in the form of time series. Then, using the proposed methodology for data-driven model reconstruction

introduced in Section 7 nd the mental dynamics for both particles in the form (193).Finally, calculate the entropy evolution

H (t) = − V

ρ(V, t) ln ρ(V, t)dV (202)

Now the sufficient condition for the particle to be “alive” is the inequality

dH dt

< 0 (203)

that holds during, at least, some time interval, Figure 3. (Although this condition issufficient, it is not necessary since even a living particle may choose not to exercise itsprivilege to decrease disorder).




It should be noticed that the condition (203) is applicable not only to natural, butto articial living systems as well. Indeed, the system (190)-(191) can be simulatedby analog devices (such as VLSI chips, (Mead,C,1989)) or quantum neural nets, (Zak,

M.,1999)) that will capture the property (203), and that justies the phenomenologicalapproach to the modeling of living systems.

The condition (203) needs to be claried from the viewpoint of the second law of thermodynamics. Formally it contradicts this law; however, this contradiction is appar-ent. Indeed, we are dealing here with an idealized phenomenological model which doesnot include such bio-chemical processes as metabolism, breezing, food consumption, etc.Therefore, the concept of an open or an isolated system becomes a subject to interpre-tation: even if the phenomenological model of a living system is isolated, the underlyingliving system is open.

From biological viewpoint, the existence of the information potential that couplesmental and motor dynamics is based upon the assumption that a living system possessesself-image and self-awareness. (Indeed, even such a primitive living system as a viruscan discriminate the self from non-selves). The concepts of self-image and self-awarenesscan be linked to a recently discovered “mirror” properties of neurons according to whicha neuron representing an agent A can be activated by observing action of an agent Bthat may not be in a direct contact with the agent A at all. Due to these privilegedproperties, living systems are better equipped for dealing with future uncertainties sincetheir present motion is “correlated with future” in terms of the probability invariants.Such a remarkable property that increases survivability could be acquired accidentallyand then be strengthening in the process of natural selection.

The ability of living systems to decrease their entropy by internal effort allows oneto make connection between survivability and complexity and to introduce a model of emergent intelligence that is represented by intelligent control and common sense decisionmaking. In this connection, it is interesting to pose the following problem. What isa more effective way for Livings to promote Life: through a simple multiplication, i.e.through increase of the number of “primitives” n, or through individual self-perfection, i.e.through increase of the number m of the levels of reections (“What do you think I thinkyou think. . . ”)? The solution to this problem may have fundamental social, economicaland geo-political interpretations. But the answer immediately follows from Eq. (82)demonstrating that the complexity grows exponentially with the number of the levels of reections m, but it grows only linearly with the dimensionality n of the original system.Thus, in contradistinction to Darwinism, a more effective way for Livings to promote Lifeis through higher individual complexity (due to mutually benecial interactions) rather than trough a simple multiplication of “primitives”. This statement can be associatedwith recent consensus among biologists that the symbiosis, or collaboration of Livings, iseven more powerful factor in their progressive evolution than a natural selection.

Before summarizing the results of our discussion, we would like to bring up the concept

of time perception. Time perception by a living system is not necessarily objective: it candiffer from actual physical time depending upon the total life time of a particular living




system, as well as upon the complexity and novelty of an observed event. In general, onecan approximate the subjective time as

T = T (t,v,H,˙

H,... ) (204)In the simplest case, Eq. (204) can be reduced to a Logtime hypothesis used in psychology(Weber-Fechner law)

T = ln t + Const (205)

However, our point here is not to discuss different modications of Eq. (204), but ratherto emphasize that any time perceptions that are different from physical time lead todifferent motions of the living systems in actual physical space due to coupling of motorand mental dynamics through the information potential

Π = ζ ln ρ(v, t ) (206)

9.2 Model Extension

The proposed model of Livings is based upon a particular type of the Liouville feedbackimplemented via the information potential Eq.(206). However, as shown in (Zak, M.,2004, 2005b, 2006a, 2006c, 2007), there are other types of feedbacks that lead to differentmodels useful for information processing. For instance, if the information force is chosenas

F = α 1ρ + α2 ∂ ∂v ln v + α3ρ ∂

2

ρ∂v2 + α 4ρ ∂

3

ρ∂v3 (207)

(that includes the information force in the second term), then the corresponding Liouvilleequation can describe shock waves, solitons and chaos in probability space, Fig. 24,(Zak,M., 2004). Let us concentrate upon a particular case of Eq. (207) considered inthe Section 2 (see Eqs. (7a) and (8a). The solution of Eq. (8a) subject to the initialconditions and the normalization constraint

ρ0 = ρ0(V ), where ρ ≥0, and∞

−∞

ρdV = 1 (208)

is given in the following implicit form (Whitham, G., 1974)

ρ = ρ0(ξ), V = ξ + ρ0(ξ)t (209)

This solution describes propagation of initial distribution of the density ρ0(V ) with thespeed V that is proportional to the values of this density, i.e. the higher values of ρpropagate faster than lower ones. As a result, any “compressive” part of the wave,where the propagation velocity is a decreasing function of V , ultimately “breaks” to givea triple-valued (but still continuous) solution for ρ(V, t). Eventually, this process leadsto the formation of strong discontinuities that are related to propagating jumps of theprobability density. In the theory of nonlinear waves, this phenomenon is known as the




formation of a shock wave, Fig. 24. Thus, as follows from the solution (209), a single-valued continuous probability density spontaneously transforms into a triple-valued, andthen, into discontinuous distribution. In aerodynamical application of Eq. (8a), when

ρstands for the gas density, these phenomena are eliminated through the model correction:at the small neighborhood of shocks, the gas viscosity ν cannot be ignored, and the modelmust include the term describing dissipation of mechanical energy. The corrected modelis represented by the Burgers’ equation

∂ρ∂t

+∂

∂V (ρ2) = ν

∂ 2ρ∂V 2

(210)

This equation has a continuous single-valued solution (no matter how small is the viscosityν ), and that provides a perfect explanation of abnormal behavior of the solution to Eq.

(8a). Similar correction can be applied to the case when ρ stands for the probabilitydensity if one includes Langevin forces L(t)) into Eq. (7a)

v = ρ + √νL(t), < L (t) > = 0 , < L (t)L(t ) > = 2 δ(t −t ) (211)

Then the corresponding Fokker-Planck equation takes the form (210).It is reasonable to assume that small random forces of strength √ν << 1 are always

present, and that protects the mathematical model Eqs. (7a), and (8a) from singularitiesand multi-valuedness in the same way as it does in the case of aerodynamics. It isinteresting to notice that Eq. (210) can be obtained from Eq. (211) in which random

force is replaced by additional Liouville feedback

v = ρ −ν ∂

∂V ln ρ, ν > 0, (211a)

As noticed above, the phenomenological criterion of life is the ability to decrease entropyby internal effort. This ability is provided by the feedback implemented in Eqs. (7a) and(8a). Indeed, starting with Eq. (1a) and invoking Eq. (8a), one obtains

∂H ∂t = −∂

∂t∞

−∞ρ ln ρdV = −

∞ −∞

ρ(ln ρ + 1) dV = ∞ −∞∂

∂V (ρ2)ln( ρ + 1) dV

= ∞|−∞ρ2(ln ρ + 1) − ∞ −∞

ρdV = −1 < 0(212)

Obviously, presence of small diffusion, when ν << 1, does not change the inequality (212)during certain period of time. (However, eventually, for large times, diffusion takes over,and the inequality (212) is reversed). As shown in (Zak, M., 2006c) the model Eqs. (210),and (211a) exhibits the same mechanism of emergence of randomness as those describedby Eqs. (14) and (15).

If the information force has the following integral form

F = γ ρ(v, t )

v

−∞

[ρ(ζ, t ) −ρ∗(ζ )]dζ (213)




leading to the governing equations

v =1

ρ(v, t )

x

−∞

[ρ(ζ, t )

−ρ∗(ζ )]dζ (214)

∂ρ∂t

+ ρ(t) −ρ∗= 0 (215)

then the solution to the corresponding Liouville equation (215) converges to an arbitrarily prescribed probability density ρ∗(V ). This result can be applied to one of the oldest(and still unsolved) problems in optimization theory: nd a global maximum of a multi-dimensional function. Almost all the optimization problems, one way or another, can bereduced to this particular one. However, even under severe restrictions imposed upon

the function to be maximized (such as existence of the second derivatives), the classicalmethods cannot overcome the problem of local maxima. The situation becomes evenworse if the function is not differentiable since then the concept of gradient ascend cannotbe applied. The idea of the quantum-inspired algorithm based upon the Liouville feedbackEq. (213) is very simple: represent a positive function ψ(v1, v2, ...vn ) to be maximizedas the probability density ρ∗(v1, v2, ...vn ) to which the solution of the Liouville equationis attracted. Then the larger value of this function will have the higher probability toappear as a result of running Eq. (214). If an arbitrary prescribed probability density ischosen as a power law

ρ∗(V ) = Γ(ν +1

2 )√νπΓ( ν 2 )

(1 + V 2

ν )−(ν +1) / 2,

then the system (214), (215) simulates the underlying dynamics that leads to the cor-responding power low statistics ρ∗(V ). Such a system can be applied to analysis andpredictions of physical and social catastrophes, (Zak, M., 2007).

Finally, introduction of the terminal Liouville feedback (Zak, M.,2004, 2005b, 2006a)

F i = γ i(vi −vi)1/ 3 (216)

applied to a system of ODE vi = f i({v}, t ) (217)

vi = f i({v}, t ) + γ i(vi −vi)1/ 3, i = 1 , 2, ...n. (218)

and leading to the governing equations

∂ρ∂t

= −n

i=1

∂ ∂V i {ρ[f i + γ i(vi −V i)1/ 3]}. (219)

has been motivated by the fundamental limitation of the theory of ODE that does not dis-criminate between stable and unstable motions in advance, and therefore, an additionalstability analysis is required for that. However, such an analysis is not constructive: incase of instability, it does not suggest any model modications to efficiently describe




postinstability motions. The most important type of instability, that necessitates postin-stability description, is associated with positive Lyapunov exponents leading to expo-nential growth of small errors in initial conditions (chaos, turbulence). The approach

proposed in (Zak,M., 2005b) is based upon the removal of positive Lyapunov exponentsby introducing special Liouville feedback represented by terminal attractors and imple-mented by the force Eq. (216) . The role of this feedback is to suppress the divergence of the trajectories corresponding to initial conditions that are different from the prescribedones without affecting the “target” trajectory that starts with the prescribed initial con-ditions. Since the terminal attractors include expected values of the state variables asnew unknowns (see Eqs. (218)), the corresponding Liouville equation should be invokedfor the closure (see Eq. (219)). This equation is different from its classical version by ad-ditional nonlinear sinks of the probability represented by terminal attractors. The forces

(216) possess the following properties: rstly, they vanish at vi →vi , and therefore,they do not affect the target trajectory vi = vi(t); secondly, their derivatives becomeunbounded at the target trajectory:

|∂F i∂vi | = |

γ 3

(vi −vi)−2/ 3| → ∞at vi →vi . (220)

and that makes the target trajectory innitely stable thereby suppressing chaos. Thesame property has been exploited for representation of chaotic systems via stochasticinvariants, (Zak, M., 2005b), Fig. 25. Such a representation was linked to the stabilizationprinciple formulated in (Zak, M., 1994) for the closure in turbulence.

So far the extension of the model of Livings discussed above was implemented throughdifferent Liouville feedbacks. Now we will return to the original feedback, i.e. to thegradient of the information potential Eq. (32), while extending the model through adeparture from its autonomy . The justication for considering non-autonomous livingsystems is based upon the following argument: in many cases, a performance of Livingsinclude so called “discrete events”, i.e. special critical states that give rise to branchingsolutions, or to bifurcations. To attain this property, such system must contain a “clock”-a dynamical device that generates a global rhythm. During the rst part of the “clock’s”period, a critical point is stable, and therefore, it attracts the solution; during the second

half of this period, the “clock” destabilizes the critical point, and the solution escapes it inone of possible several directions. Thus, driven by alternating stabilities and instabilities,such a system performs a random walk-like behavior. In order to illustrate it, start withthe following system (compare to Eq. (11))

v = α∂

∂vln ρ(v, t ), α = −2γ √D sin1/ 3 √ω

β √D sin ωt, (221)

where β , γ , and ω are constants. Then one arrives at the following ODE with respectto the square root of the variance D (compare to Eq. (37))

d√Ddt

= γ sin1/ 3 √ωβ

√D sin ωt. (222)




This equation was studied in (Zak, M., 1990, 2005a). It has been shown that at theequilibrium points

√D = πmβ/ √ω, m = ...

−2,

−1, 0, 1, 2,etc (223)

the Lipschitz condition is violated, and the solution represents a symmetric unrestrictedrandom walk on the points Eq. (223). The probability ρ∗that the solution approaches apoint y after n steps is

ρ∗(y, n ) = C mn 2−n , m = 0 .5(n + y), y = D,n = ingr (2ωt/π ) (224)

Here the binomial coefficient should be interpreted as 0 whenever m is not an integer inthe interval [0 , n], and n is the total number of steps. Obviously, the variance D = y2

performs asymmetric random walk restricted by the condition that D is not-negative,while Eq. (224) is to be replaced by the following

ρ∗= C mn 21−n . m = 0 .5(n + D), D ≥0. (225)

Thus, the probability density of the velocity v described by the ODE (221) jumps ran-domly from atter to sharper distributions, and vice versa, so that increase and decreaseof the entropy randomly alternate, and that makes the system “alive”.

Finally, the model (56), (57) can be further generalized by introduction of delay oradvanced time

v = −ζα •∇v ln ρ(t + τ ), (226)

ρ = ζ ∇2V ρ(t + τ ) • •α, (227)

When τ < 0, the dynamics is driven by a feedback from the past (memories); when τ > 0,the dynamics is driven by a feedback from the future (predictions). Although little isknown about the structure of the solution to Eqs. (226), (227), (including existence,uniqueness, stability, etc), the usefulness of such a model extension is obvious for inverseproblems when the solution is given, and the model parameters are to be determined,(see Section 8, Eqs. (190)- (201)).

9.3 Summary

Thus, we have introduced the First Principle for modeling behavior of living systems.The structure of the model is quantum-inspired: it is obtained from quantum mechanicsby replacing the quantum potential with the information potential, Figure 8. As a result,the model captures the most fundamental property of life: the progressive evolution,i.e. the ability to evolve from disorder to order without any external interference. Themathematical structure of the model can be obtained from the Newtonian equations of motion coupled with the corresponding Liouville equation via information forces. Theunlimited capacity for increase of complexity is provided by interaction of the systemwith it’s images via chains of reections: what do you think I think you think. . . .Allthese specic non-Newtonian properties equip the model with the levels of complexitythat match the complexity of life, and that makes the model applicable for description of behaviors of ecological, social and economics systems.




Acknowledgment

The research described in this paper was performed at Jet Propulsion Laboratory

California Institute of Technology under contract with National Aeronautics and SpaceAdministration.




References

[1]Arbib,M., and Hesse, M.,1986,The Construction of Reality, Cambridge Universitypress.

[2]Berg,H., 1983, Random walks in biology, Princeton Univercity Press, New Jersey.

[3]Gordon,R., 1999, The Hierarchical Genome and Differential Waves, World Scientic,

[4]Haken,H.,1988, Information and Self-organization, Springer, N.Y.

[5]Huberman,B., 1988, The ecology of Computation, North-Holland, Amsterdam.

[6]Lefebvre,V.,2001, Algebra of Conscience, Kluwer Acad. Publ.

[7]Mead,C,1989, Analog VLSI and Neural Systems,Addison Wesley.

[8]Mikhailov,A., 1990, Foundations of Synergetics, Springer, N.Y.

[9]Risken,H., 1989, The Fokker-Planck Equation, Springer, N.Y.[10]Perl, J., Probabilistic reasoning in intelligent systems, Morgan Kaufmann Inc. San

Mateo, 1986

[11] Prigogine,I., 1961, Thermodynamics of Irreversible Processes, John Wiley,N.Y.

[12]Prigogine,I., 1980, From being to becoming, Freeman and co., San Francisco.

[13]Whitham, G., Linear and nonlinear waves, Wily-Interscience Publ.;1974.

[14]Yaguello,M,.1998, Language through the looking glass, Oxf. Univ. Press.

[15]Zak, M., ”Terminal Attractors for Associative Memory in Neural Networks,” PhysicsLetters A, Vol. 133, No. 1-2, pp. 18-22. 1989

[16]Zak, M., 1990, Creative dynamics approach to neural intelligence, BiologicalCybernetics, 64, 15-23.

[17]Zak,M.,1992, Terminal Model of Newtonian dynamics, Int. J. of Theor. Phys. 32,159-190.

[18]Zak, M., 1994, Postinstability models in Dynamics, Int. J. of Theor. Phys.vol. 33,No. 11, 2215-2280.

[19]Zak,M., 1999a, Physical invariants of biosignature, Phys. Letters A, 255, 110-118.

[20]Zak, M.,1999b, Quantum analog computing, Chaos ,Solitons &fractals, Vol.10, No.10, 1583-1620.

[21]Zak,M., 2000a, Dynamics of intelligent systems, Int. J. of Theor. Phys.vol. 39, No.8.2107-2140.

[22]Zak,M., 2000b, Quantum decision-maker, Information Sciences, 128, 199-215.

[23]Zak, M.,2002a) Entanglement-based self-organization, Chaos ,Solitons &fractals, 14,745-758.

[24]Zak,M.,2002b, Quantum evolution as a nonlinear Markov Process, Foundations of Physics L,vol.15, No.3, 229-243.

[25]Zak, M., 2003, From collective mind to communications, Complex Systems, 14, 335-361,

[26]Zak,M.,2004, Self-supervised dynamical systems, Chaos ,Solitons &fractals, 19, 645-666,




[27]Zak, M. 2005a, From Reversible Thermodynamics to Life. Chaos, Solitons &Fractals,1019-1033,

[28]Zak, M., 2005b, Stochastic representation of chaos using terminal attractors, Chaos,

Solitons &Fractals, 24, 863-868[29]Zak, M., 2006a Expectation-based intelligent control, Chaos, Solitons&Fractals, 28,

5. 616-626,

[30]Zak, M., 2007a, Complexity for Survival of Livings, Chaos, Solitons &Fractals, 32,3, 1154-1167

[31]Zak, M., 2007b, From quantum entanglement to mirror neuron, Chaos, Solitons&Fractals, 34, 344-359.

[32]Zak, M., 2007c, “Quantum-Classical Hybrid for Computing and Simulations”,Mathematical Methods, Physical Models and Simulation in Science & Technology (inpress).




Figure 17. Chain of reections




Figure 18. Two levels of abstruction.




Figura 19. Block-diagram of intelligent control.




Figure 24. Shock wave and soliton in probability space

10.1.1.87.6917physics of life from first principles.pdf

Documents