_{1}

^{*}

This paper proposes a mathematical formalism for describing the machinery of human mind creativity. Since some functions of mind support discreteness of cognition, but others support its continuity, while time in physics is always continuous, an attempt to overcome the continuous/discrete duality of mind performance is implemented via utilization of special critical points: terminal attractors and repellers that act as autonomous biological clock. In this paper, attention is concentrated upon the discrete mode of the mind activity since the continuous mode was introduced and discussed in details in our previous publications. The fundamental novelty of the model is in its capability to move from disorder to order without external inputs in violation of the second law of thermodynamics; that suggests that this kind of dynamics requires extension of modern physics to include physics of life. Conceptually the discrete model links to its continuous version represented by a hypothetical particle of life that is briefly discussed prior to derivation of the discrete model. However, the discrete model should be considered as the next step in study of mind dynamics since it provides a bridge to mathematical origin of self-generated novelties in such brunches as mathematical logic and linguistic, i.e. the highest level activities of human mind. The proposed model deals with rules of mind activity rather than with its content.

This paper continues an effort to model life based upon the first principles, with emphasis to capture the mind creativity mystery. The effort is based upon a hypothetical particle of life introduced in [

This paper proposes a mathematical formalism for describing the machinery of human mind creativity based upon the hypothetical model of particle of life. A mind is understood as the set of cognitive faculties that enables consciousness, perception, thinking, judgment, and memory―a characteristic of humans. Adopting terminology of computer science, we will associate brain with a computer, and mind―with a set of algorithms implemented by this computer. This allows us to distance ourselves from biological aspects of brain and concentrate on in- trinsic mathematical properties of mind. Similar abstraction―ignorance of metabolism and reproduction―made the particle of life an isolated system. Since some functions of mind support discreteness of cognition, but others support its continuity, while time in physics is always continuous, an attempt to overcome the continuous/dis- crete duality of mind performance is implemented via utilization of special critical points: terminal attractors and repellers, [

It should be emphasized that the proposed model of mind as a part of the hypothetical particle of life is based only upon one assumption: The ability to move from disorder to order without external sources, i.e. in violation of the second law of thermodynamics. Starting with that postulate, mathematical formalism takes over and leads to a quantum-inspired model of mind.

In this section, we briefly discuss a hypothetical model of L-particle introduced in [

We will start with derivation of a one-dimensional case that illuminates departure from Newtonian dynamics. For mathematical clarity, we will consider here a motion of a unit mass under action of a force f depending upon the velocity v and time t and present it in a dimensionless form

referring all the variables to their representative values

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

while

The solution of this equation subject to initial conditions and normalization constraints (2) determines proba- bility density as a function of V and t:

Remark. Here and below we make distinction between the random variable v(t) and its values V in probability space.

In order to deal with the constraint (2), let us integrate Equation (3) over the whole space assuming that

Hence, the constraint (2) is satisfied for

Let us now specify the force

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

This is a fundamental step in our approach. Although the theory of ODE does not impose any restrictions upon the force as a function of space coordinates, the Newtonian physics does: equations of motion are never coupled with the corresponding Liouville equation. Nevertheless such a Liouville feedback does not lead to any physical inconsistence.

Substituting the force

In order to illustrate mathematical aspects of the concepts of Liouville feedback in systems under considera- tion as well as associated with its instability and randomness, let us take the feedback (3) in the form

and therefore, Equations (7) and (8) form the following system

We will start analysis of this system with the simple case when

to obtain the following equation of motion

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

Here v stands for a particle velocity, and

Since

the solution of Equation (14) subject to the sharp initial condition

describes diffusion of the probability density.

Substituting this solution into Equation (13) at V = v, one arrives at the differential equation with respect to v(t)

and therefore,

where C is an arbitrary constant. Since v = 0 at t = 0 for any value of C, the solution (18) is consistent with the sharp initial condition for the solution (16) of the corresponding Liouvile Equation (14). The solution (18) de- scribes the simplest irreversible motion: it is characterized by the “beginning of time” where all the trajectories intersect (that results follows from the violation of Lipchitz condition at t = 0,

It is easily verifiable that the solution (18) has the same structure as the solution of the Madelung equation [

Further analysis of the solution (18) demonstrates that this solution is unstable since

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

This type of instability has been introduced and analyzed in [

Considering first Equation (18) at fixed C as a sample of the underlying stochastic process (16), and then va- rying C, one arrives at the whole ensemble characterizing that process (see

The same results follow from the ensemble (18) at

It is interesting to notice that the stochastic process (16) is an alternative to the following Langevin equation, [

that corresponds to the same Fokker-Planck Equation (14). Here

Thus, the emergence of self-generated stochasticity is the first basic non-Newtonian property of the dynamics with the Liouville feedback.

We will continue with another extreme case of the feedback (9) assuming that

obtaining

and therefore, the equation of motion and the Liouville equation are

The solution of Equation (27) subject to the initial conditions

This solution subject to the initial conditions and the normalization constraint, describes propagation of initial distribution of the density

In aerodynamical application of Equation (27), when

As shown in [

Then the corresponding Fokker-Planck equation takes the form (29). It is reasonable to assume that small random forces of strength

t is interesting to notice that Equation (29) can be obtained from Equation (26) in which random force is re- placed by an additional Liouville feedback

and that is exactly Equation (11).

Equation (29) is known as the Burgers equation, and it has the exact analytical solution [

where

Although this solution is well known, it should be emphasized that in our case it occurs in the probability space. This means that if one runs Equation (31) independently many times starting with different initial condi- tions and compute the statistical characteristics of the family of these solutions, he will arrive at the evolution of the probability density described by Equation (33).

As a particular case of the solution (32), consider a long-term behavior of an initial step

that diffuses into the steady profile [

moving with the constant speed

when v/t is fixed, and

In this case, Equation (33) describes a stationary stochastic process to be applied to the proposed model of mind.

One can verify that the additional (normalization) constraint imposed upon the probability density that is a state variable of the Burgers Equation (29) is satisfied

Indeed, as follows from (34), if the normalization constraint is satisfied at the initial condition, it will be satis- fies for all times.

An important non-classical property of the solution of this equation is a decrease of entropy. Indeed,

Obviously, presence of small diffusion, when

The inequality (35) is of fundamental importance: it confirms existence of such an isolated dynamical system that violates the second law of thermodynamics by decreasing the entropy during some period of time while ob- eying the rules of mathematics. Therefore if the corresponding hypothetical particle exists, it cannot belong to the modern physics, as we know it; actually it should belong to an extended physical world in the form of a quantum-classical hybrid. However, the assumption about existence of such a particle is justified by similarity between its dynamical properties and those of living systems. The analysis of these similarities is performed in [

The proposed model illuminates the “border line” between living and non-living systems. The model intro- duces a biological particle that, in addition to Newtonian properties, possesses the ability to process information. The probability density can be associated with the self-image of the biological particle as a member of the class to which this particle belongs, while its ability to convert the density into the information force―with the self-awareness (both these concepts are adopted from psychology). Continuing this line of associations, the equ- ation of motion, Equation (10) can be identified with a motor dynamics, while the evolution of density (see Eq- uation (11))―with a mental dynamics. Actually the mental dynamics play the role of the Maxwell sorting de- mon: it rearranges the probability distribution by creating the information potential and converting it into a force that is applied to the particle. One should notice that mental dynamics describes evolution of the whole class of state variables (differed from each other only by initial conditions), 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 recalled that the second law of thermodynamics states that the entropy of an isolated system can only increase. This law has a clear probabilistic interpretation: increase of entropy corresponds to the passage of the system from less probable to more probable states, while the highest probability of the most disordered state (that is the state with the highest entropy) follows from a simple combinatorial analysis. However, this statement is correct only if there is no Maxwell’s sorting demon, i.e., nobody inside the system is rearranging the probability distri- butions. But this is precisely what the Liouville feedback is doing: it takes the probability density

The proposed model can be interpreted as representing interactions of the agent with the self-image and the images of other agents via the mechanisms of self-awareness. In order to associate these basic concepts of psy- chology with our mathematical formalism, we have to recall that living systems can be studied in many different spaces such as physical (or geographical) space as well as abstract (or conceptual) spaces. The latter category includes, for instance, social class space, sociometric space, social distance space, semantic space etc. Turning to our model, one can identify two spaces: the physical space x, t in which the agent state variables

Since we are planning to concentrate upon discrete aspect of cognitive performance of mind, the most transpa- rent mathematical tool for that are time-delay equations. However, the problem with these equations is in their nonlocality. Indeed because the known laws of physics are all local, and because nonlocal interactions combined with relativity lead to causal paradoxes, many physicists find nonlocal models unacceptable. On the first sight, in our case this argument can be dismissed on the ground that the model of particle of life (that is planned to be the base of the model of mind) is quantum-inspired and it manifests entanglement that is a fundamentally non- local effect [

The governing equations of classical dynamics are based upon Newton’s laws

where L is the Lagrangian and q and

The structure of the function (37) does not prescribed by the Newton’s laws, and, strictly speaking, some ad- ditional assumptions should be made in order to define them. The natural assumption (that has never been chal- lenged) is that these functions can be expanded in the Taylor series with respect to an equilibrium state

Obviously this requires the existence of the derivatives

i.e., F_{i} must satisfy the Lipchitz condition. This condition allows one to describe the Newtonian dynamics within the mathematical framework of classical theory of differential equation. However, there is a certain price for such mathematical “convenience”: The Newtonian dynamics with dissipative forces remain fully reversible in the sense that its governing equations are invariant with respect to time inversion

In order to trivialize the mathematical part of our argument, let us consider the one-dimensional motion of a particle decelerated by a friction force

in which m is mass and w is velocity. Invoking the assumption (39) one can linearize the force F with respect to the equilibrium v = 0:

and the solution of Equation (40) for

As follows from Equation (42), the equilibrium w = 0 cannot be approached in finite time. A usual explana- tion of this paradox is that to accuracy of our limited scale of observation, the particle “actually” approaches the equilibrium in finite time. In other words, eventually the trajectories (42) and w = 0 become so close that we cannot distinguish them. The same type of explanation is used for emergence of chaos: if two trajectories origi- nally are “very close” and then they diverge exponentially, the same initial conditions can be applied to either of them, and therefore, the motion cannot be traced.

Turning to the example (40), let us assume that

in which N is an odd number.

By selecting a large N, one can make k close to 1, so that Equations (41) and (43) will be almost identical every- where excluding a small neighborhood of the equilibrium point w = 0 , while, as follows from Equation (43), at this point

Hence the condition (39) is violated; the friction force grows sharply at the equilibrium point, and then it gradually approaches the straight line (41). This effect can be interpreted as a jump from static to kinetic friction. It appears that this small difference between the friction forces (41) and (43) leads to fundamental change in Newtonian dynamics: the time of approaching the equilibrium w = 0 becomes finite. Indeed, as follows from (40) and (43)

Obviously this integral diverge in the classical case when

As shown in [

Therefore the uniqueness of the solution at w = 0 is violated, and the motion for t < t_{1} or t < t_{2} is totally for- gotten. This is a mathematical implication of irreversibility in dynamics (43).

So far we were concerned with the stabilizing effects of dissipative forces. However, as is well known from dynamics of non-conservative systems, these forces can destabilize the motion when they feed external energy into the system (the transmission of energy from laminar to turbulent flow in fluid dynamics, or from rotation to oscillations in dynamics of flexible systems). In order to capture the fundamental properties of these effects in the case of non-Lipchitz, or “terminal” dissipative forces (43) by using the simplest mathematical model, let us turn to Equation (43) and assume that now the friction force feeds energy into the system

One can verify that that for Equation (46), the equilibrium point w = 0 becomes a terminal repeller, and since

it is infinitely unstable. If the initial condition is infinitely close to this repeller, the transient solution will escape it during a finite time period

while for a regular repeller the time period would be infinite. But in addition to that, terminal repellers possess even more surprising characteristics: the solution of Equation (46) becomes totally unpredictable.

Indeed, two different motions described by Equation (49) are possible for “almost the same” initial conditions

The most essential property of this result in that the divergence of these two solutions is characterized by an unbounded rate

In contrast to the classical case where_{0} since during this interval the initial infinitesimal distance between the solutions becomes finite. Thus a terminal repeller represents a vanishingly small, but infinitely powerful “pulse of unpredictability” that is pump- ed into the system via terminal dissipative forces. Obviously, failure of the uniqueness of the solution here re- sults from the violation of the Lipchitz condition (39) at w = 0.

As known in theory of chaos and turbulence, a combination of stabilizing and destabilizing effects can lead to a new phenomenon: intermittency. In order to describe similar effects in dynamics with terminal dissipating forces, let us slightly modify Equation (46)

Here stabilization and destabilization effects alternate. With the initial conditions

the exact solution of Equation (52) consists of a regular solution

and a singular solution

During the first period

It is important to notice that each time the system escapes the terminal repeller, the solution splits into two symmetric branches, so that there are 2^{n} possible scenarios of the oscillations with respect to the center w = 0, while each scenario has the probability 2^{−n} to occur, where n is the number of cycles. Hence the motion (54) re- sembles chaotic oscillations known from classical dynamics: it combines random characteristics with the attrac- tion to the center. However in classical case, chaos is caused by a supersensitivity to the initial conditions, while the uniqueness of the solution for the unique initial conditions is guaranteed. In contrast to that, the chaos in the oscillations (54) is caused by the failure of the uniqueness of the solutions at the equilibrium points, and it has a

well-organized probabilistic structure. Since the time of approaching the equilibrium point w = 0 by the solution (54) is finite, this type of chaos can be called terminal.

The model proposed in this section is based upon dynamical simulation of terminal oscillations introduced above. We start with a rectilinear motion of a particle of unit mass (m = 1) described by the following differen- tial equations:

where x and w are the position and velocity, respectively, and

Let us start with Equation (56). Actually this is a particular case of Equation (52) at k = 1/3. The solution of this equation subject to the initial conditions (53) is

and a singular solution

These two solutions coexist at t = 0, and that is possible because at this point the Lipchitz condition fails

As shown in

It should be emphasized again that the noise

Turning to Equation (57), one obtains the distance between two adjacent equilibrium positions of the particle

where

Thus the equilibrium positions of the particle are

while the positive and negative signs randomly alternate with the probability p, where p is the probability that

Thus, the solution of Equations (56) and (57) combines acting (the transition from one equilibrium point to another), and “thinking” (the decision making process based upon

Since the solutions of Equations (56) and (57) include random components, it is useful to describe them in terms of probabilities. For that purpose, let us introduce a function

supplemented by the normalization constraint

In case of unrestricted random walk,

Equation (63) can be considered as a terminal version of the Liouville equation generated by the system (56), (57). Indeed, as the classical Liouville equation (see Equation (3)), it describes evolution of an initial error dis- tribution. However, in terminal version of dynamics, this error occurs only in discrete points representing ter- minal repellers, and that makes the Liouville equation discrete as well.

One can notice that in the limit case

In this sub-section we will analyze possible structures of the noise that is responsible for triggering the transition from one equilibrium to another in the direction depending upon

Regardless of the type of the corresponding terminal repeller, the noise can be derived from a sample of an underlying stationary stochastic process characterized by some probability density

where

and

Then

while

Obviously any stationary stochastic process uniquely defines the function

In general,

the variance

or the Shannon uncertainty (entropy)

i.e.

Actually, all the variety and complexity of the particle behavior described by Equations (56) and (57) is de- fined by the structure of the function (73). A variety of structures of this function as well as specific characteris- tics of the corresponding behaviors was analyzed in [

with application of the terminal model of punctuated evolution to Boolean dynamics.

In this section, we start with a critical review of the dynamics of a set of n variables

Any model of a Boolean net starts from n binary elements

In the time stepping, each of these Boolean elements is defined by a function of all the elements (75), or at least, some of them. More precisely, the value of

A typical Boolean network consists of a set of n nodes, and each of these nodes receives inputs from K other nodes. For K = n, the dynamics can be presented in the following form

where the synapse weights

In [

The interest to Boolean dynamics as a candidate for the model of mind is due to its capability to process dis- crete-events with application to logic and linguistic that is the fundamental privilege of human mind. However, the existing models of Boolean dynamics described above have two limitations that disqualify them from that purpose.

The first limitation is their non-locality since they are presented by difference rather than differential equa- tions. However as mentioned in Section 3, since the known laws of physics are all local, and because nonlocal interactions combined with relativity lead to causal paradoxes, many physicists find nonlocal models unacceptable.

The second limitation is that these models are not isolated systems. As emphasized in the Introduction, the proposed model of mind as a part of the hypothetical particle of life is based only upon one assumption: the abil- ity to move from disorder to order without external sources, i.e. in violation of the second law of thermodynam- ics. Starting with that postulate, mathematical formalism is supposed to take over and to lead to a quantum-in- spired model of mind.

In order to remove these limitations, we apply the terminal model of punctuated evolution presented in the previous section. We will concentrate upon the case of Equation (73) when

and consider the equation

subject to the initial conditions

The solution consists of a regular solution

and a singular solution

These two solutions coexist at t = 0, and that is possible because at this point the Lipchitz condition fails

During the first period

with an unstable transitional point

Let us concentrate on the variable w. Turning to Equation (80) and selecting the constants

one arrives at a Boolean variable (75). Unlike a “classical” Boolean variable that takes values 0 and 1, this vari- able takes values (−1) and 1, while zero plays the role of a transitional point that, as a result of its instability, provides a passage to the stable points (−1) or 1. This transitional point plays another important role: it separates future from past in such a way that at this point, past is completely forgotten, and the motion starts afresh. The last property is a fundamental characteristic of terminal attractors/repellers.

Applying the formula for the binary entropy function

where

one calculates the change of the entropy: it starts with zero value at t = 0

Then it jumps to its maximum value at

Finally it returns to its initial value (88) at

On the first sight, it seems surprising that entropy may decrease (see the green line). However, turning to the governing Equation (78), one concludes that such an “inverse” evolution results from the fact that this system is not isolated, although it is closed, (there is no mass exchange). Indeed, it consumes energy via the harmonic os- cillations

Prior to generalizing Equation (78) to the n-dimensional case, we will emphasize the role of the term (77). When

and the term (77) provide stability of Equation (90). As a result of that, the particle returns to its initial position w = 0 thereby performing restricted one-step random walk. Obviously without this term, the random walk would be unrestricted, and the variable w would not be a Boolean one.

In order to introduce not only the Boolean variables, but the Boolean functions as well, we will recall that

Therefore Equation (90) can be approximated as

Let us now move to the n-dimensional case

where the synapse weights

However, in order to prevent the system (93) from an unrestricted walk, the synapse weights should be chosen such that that the system

is asymptotically stable. The sufficient conditions for that are the following

i.e.

Let us turn to the analysis of the system (93). Comparing this system with the one-dimensional case (78), one notice that each Boolean variable w_{i} is not affected by the rest of the variables, and the role of the Boolean func- tions

is to provide the stability of the system (93) at _{i} are changed with changes of the corresponding Boolean variables.

In this section, we discuss the system (93) as a candidate for the model of the human mind. Unlike the Boolean dynamics model (76) that is nonlocal, the model (93) is described by the system of differential (but not differ- ence!) equations, and therefore, it is local. Hence the first limitation of the model (76) formulated in the previous Section is removed in the system (93). However the second limitation is still there: the system (93), as well as the system (76), is not isolated. Indeed the system (93) consumes energy via the harmonic oscillations

In order to achieve that, we combine the model of mind with the corresponding model of the particle of life (L-particle) allowing the mind to “borrow” energy from the L-particle. Such a symbiosis is natural from the physiological viewpoint. Thereby we introduce a master-slave system that is isolated from the external world. Such an idealization allows us to capture distinguished properties of livings that possess intelligence.

Let us turn to the system (13), (14) which describes the dynamics of L-particle. From its solution given by Equation (18), one can express

Turning to the stationary stochastic process generated by L-particle and described by Equation (33), one can adapt from it random values of noise (66)

In order to obtain random samples

Now with help of Equations (97) and (98), the system (93) can be presented in the following form

The system is still not closed because of additional variables v and

Now the system (99), (100) and (101) is closed and isolated. Because of the master-slave coupling we can as- sume that the functions (97) and (98) are preset, and analyze Equations (99) separately from Equations (100) and (101).

As follows from Equation (97), the system (99) was set up based upon preserving the time scale of the L-par- ticle. However, in general, the mind may have its intrinsic time scale. Indeed, substituting

Now we can rewrite Equations (99) in the following form

However, there is still a limitation to be removed: the Boolean functions

do not change their structure in the course of dynamics evolution. In order to make their structure plastic while preserving the stability constraints (95) we add a free random term

that can be associated with the activation threshold, (compare with Equation (76)).

The governing equations for these terms will be presented in the form of unrestricted random walk, (see Equ- ation (58))

with equilibrium values

These free terms randomly change the structure of the Boolean functions in course of their dynamics evolu- tion.

Now we can present the proposed model of human mind in the final form

The system (107)-(110) is closed and isolated.

Prior to analysis of this model, we have to make a comment concerning the Boolean variables w and the Boo- lean functions B. Strictly speaking, they are not Boolean since they can take three values: −1, 0, 1; however zero values are transitional, and they appear with periodic regularity after each of the values −1 and 1. So if we asso- ciate the proposed dynamics with information processing, the zero values can be ignored since they are fully predictable. The formal difference between the standard Boolean functions taking values 0 and 1 can be removes by a trivial variable transformation.

Starting with Equation (107) and compare it with Equation (93), one concludes that each Boolean variable w_{i} is not affected by the rest of the variables, and the role of the Boolean functions (104) is to provide the stability of the system (107) at

The Hamming distance between two state configurations of equal length is the number of positions at which the corresponding symbols are different. In another way, it measures the minimum number of substitutions required to change one state configuration in into the other. Due to the property of the Boolean dynamics pointed out above, any two strings of configurations at different times are separated by transitional zero points, and therefore, are uncorrelated.

Let us calculate the probability that two neighboring configuration sequences have zero Humming distance. This event can happen if all n Boolean variables stay at the same places during the next cycle. Since each variable can stay or change with the same probability 1/2, the probability of the event is

where

Then

The minimum probability that two configuration strings have the Humming distance

As follows from Equation (104), the values of the Boolean functions B_{i} change with changes of the corres- ponding Boolean variables. But in addition to that, they also change because the Boolean functions are changed themselves due to additional random inputs

The system (107), (108) can be expanded to introduce a new generation of Boolean functions

In the same way, next generations of Boolean functions

The objective of this work is to extend the model of a hypothetical particle of life [

world does not provide sufficient information, a human turns for an “advise” to his experience, and that is asso- ciated with a common sense.

Therefore the first challenge of our objective is to preserve isolation of the model of mind from the external world. Such idealization simplifies mathematical formalism without diminishing the proof of concept.

The second challenge is to preserve locality of the model of mind as a part of L-particle: since the known laws of physics are all local, and because nonlocal interactions combined with relativity lead to causal paradoxes, many physicists find nonlocal models unacceptable. However to present a discrete-event dynamics in a local way is a formidable problem: all existing approaches to discrete event dynamics are nonlocal.

Adopting terminology of computer science, we will associate brain with a computer, and mind―with a set of algorithms implemented by the brain. This allows us to distance ourselves from biological aspects of brain and concentrate on intrinsic mathematical properties of mind. And this leads us to the third challenge: to derive the model of human mind based only upon one assumption―the ability to move from disorder to order without ex- ternal sources, i.e. in violation of the second law of thermodynamics. Starting with that postulate, mathematical formalism is supposed to take over and lead to a quantum-inspired model of mind.

Based upon these objective and challenges, this work proposes a mathematical formalism for describing the machinery of human mind creativity. Since some functions of mind support discreteness of cognition, but others support its continuity, while time in physics is always continuous, an attempt to overcome the continuous/dis- crete duality of mind performance is implemented via utilization of special critical points: terminal attractors and repellers that act as autonomous biological clock. In this paper, attention is concentrated upon the discrete mode of the mind activity since the continuous mode was introduced and discussed in details in our previous publica- tions.

The fundamental novelty of the model is in its capability to move from disorder to order without external in- puts in violation of the second law of thermodynamics; that suggests that this kind of dynamics requires exten- sion of modern physics to include physics of life. Conceptually the discrete model links to its continuous version represented by a hypothetical particle of life that is briefly discussed prior to derivation of the discrete model. However, the discrete model should be considered as the next step in study of mind dynamics since it provides a bridge to mathematical origin of self-generated novelties in such brunches as mathematical logic and linguistic, i.e. the highest level activities of human mind.

The proposed model deals with rules of mind activity rather than with its content, i.e. with grammar, but not semantic.

The model is represented by a system of differential equations with terminal attractors and repellers that pro- vide the capability to process discrete-event-based flow of information. This system has master-slave architec- ture: the master equations (see Equations (109) and (110)) describe dynamics of the corresponding L-particle, while the slave equations (see Equations (107), (108), (113) and (114)) present dynamics of Boolean functions that capture the mind activity such as mathematical logic and linguistic. The Boolean dynamics consists of a se- quence of two-step cycles during which a new values of the uncorrelated Boolean functions are exposed. It should be recalled that in terms of mathematical logic, each Boolean function is characterized by the truth table, [

The chain of slave equations can be extended thereby generating new logical statements.

From the thermodynamics viewpoint, during each cycle, entropy of the slave system first grows from zero, and then drops back to zero. Such behavior does not violate the second law of thermodynamics since the slave system is not isolated, although it is closed: It exchanges energy with the master system describing L-particle. However the whole system that includes the master and slave equations is isolated, and nevertheless it generates new logical statements, new patterns of behavior and new information, and that violates the second law of ther- modynamics. It should be emphasized that such a unique property of L-particle is due to the feedback from the Liouville Equation (110) representing the mental dynamics, to Equation (109) representing the motor dynamics. This feedback implements a new type of force―the information force that does not exist neither in Newtonian nor in quantum mechanics (