Despite the mass of empirical data in neuroscience and plenty of interdisciplinary approaches in cognitive science, there are relatively few applicable theories of how the brain as a coherent system functions in terms of energy and entropy processes. Recently, a free energy principle has been portrayed as a possible way towards a unified brain theory. However, its capacity, using free energy and entropy, to unify different perspectives on brain function dynamics is yet to be established. This multidisciplinary study attempts to make sense of the free energy and entropy not only from the perspective of Helmholtz thermodynamic basic principles but also from the information theory framework. Based on the proposed conceptual framework, we constructed (i) four basic brain states (deep sleep, resting, active wakeful and thinking) as dynamic entropy and free energy processes and (ii) stylized a self-organizing mechanism of transitions between the basic brain states during a day period. Adaptive transitions between brain states represent homeostatic rhythms, which produce complex daily brain states dynamics. As a result, the proposed simulation model produces different self-organized circadian dynamics of brain states for different types of chronotypes, which corresponds with the empirical observations.

Though the presented here interpretation of free energy and entropy stems mainly from the thermodynamic principles, at the same time we imply that it has a strong relation with the information theory as well, which is crucial to understand the meaning of the thermodynamic free energy and entropy employment for modelling the brain state dynamics.

Despite the wealth of empirical data in neuroscience, there are relatively few global theories about how the brain works. A recently proposed free energy principle for adaptive systems tries to provide a unified account of action, perception and learning. Although this principle has been portrayed as a unified brain theory, its capacity to unify different perspectives on the brain function has yet to be established (Friston,

Historically, Hinton realized first that some tough problems can be solved in machine learning by treating a prediction error of neural networks as free energy, and then minimizing it (Hinton and Terrence,

However, in more general terms, the free energy principle is essentially a mathematical formulation of how adaptive systems resist a natural tendency of disorderliness. We can see that although the motivation is quite straightforward, the implications are complicated and diverse. This diversity allows the principle to account for many aspects of brain structure and function and lends it the potential to unify different perspectives on how the brain works. Admittedly, the number of physiological and sensory states in which an organism can be is limited, and these states define the organism’s phenotype. Mathematically, this means that the probability of these sensory states must have low entropy. In other words, there is a high probability that a system will be in any of a small number of states, and a low probability that it will be in the remaining states. Biological agents must therefore minimize the long-term average of entropy to ensure that their sensory errors remain low. In other words, biological systems somehow manage to violate the fluctuation theorem, which generalizes the second law of thermodynamics (Friston,

Following Friston

Before delving deeper into the proposed model details, let us recall that the research on the human brain state dynamics was basically concerned with impaired alertness and cognition, which was mathematically associated with the fundamental circadian sleep-wake cycle of restful and active brain states (Medeiros

However, such two-state approaches reduce the brain state space to the minimum. Admittedly, there are well known sleep states such as NREM (deep sleep) and REM (dreaming), which we think are imperative to employ for modelling too. In a similar way, there are quite a few “wake” states too. For instance, we have chosen active wakefulness and thinking states for transitions modelling. The former represents physical activity without mental concentration (dominated by alpha and beta EEG signals), whereas the latter represents a concentrated mental activity (dominated by beta and gamma EEG signals) (Kezys and Plikynas,

Let us elucidate a bit more about the currently dominating two-state modelling approach. The field of brain states modelling has a strong history of using mathematical models to illustrate an understanding of sleep-wake cycling and circadian rhythms in general (Refinetti

Recent advances in the clarification of the neural anatomy and physiology involved in the regulation of sleep and circadian rhythms have motivated the development of more detailed mathematical models that extend the approach introduced by the classical reciprocal-interaction model (Booth and Diniz Behn,

The fact that a physiological rhythm could oscillate not only in the absence of periodic changes in the environment, but also at a period different from that of behavioural cyclicity, established the endogenous and physiologic nature of human circadian rhythms for the first time by Kleitman (

The endogenous factors are being investigated in various ways, mainly using neurophysiological data of neuronal activities, for instance, one can be referred to the sleep-wake studies (Borbely,

Admittedly, both the ‘sleep pressure’ relaxation (homeostatic) oscillator and circadian oscillator are of intrinsic endogenous nature. The latter oscillations have been genetically encoded following daily natural rhythms of the sun’s activity during a day. However, despite recent discovery of above mentioned sleep-regulating cerebrospinal fluid substances or the role of adenosine, there is a lack of conceptual understanding of fundamental processes, which produce such endogenous homeostatic ‘sleep pressure’ regulators. Although there are some biological markers, the literature does not propose comprehensive explanations pertaining to the underlying fundamental mechanism.

In this regard, we were not so much concerned about finding a feasible empirically based physiological model for circadian or homeostatic rhythms modelling. Instead, we were concerned about finding a conceptual way for modelling and understanding of the fundamental principles of the so-called “homeostatic pressure” mechanism. The underlying fundamental reasoning of the homeostatic drive has yet to be understood.

Thus, the authors treat the brain as an organ with an intrinsic adaptive behaviour, constrained by fundamental physical and biological laws. It is assumed that self-organized oscillatory dynamics of free energy and entropy serves to keep brain homeostasis within certain confines. In this way, the brain as a complex biological organ, presumably maintains oscillatory self-organization that evokes brain states dynamics during a day.

In this regard, the main conceptual idea of this paper concerns deductive assumption, that the homeostatic relaxation oscillator can be nailed down to the fundamental free energy and entropy rhythmic processes taking place in the brain during a day period. Based on this assumption, we constructed a new way of modelling oscillations in the brain system that exhibit homeostatic adaptive behaviour. We argue that proposed fundamental endogenous model can simulate homeostatic rhythmic dynamics of brain states in a more meaningful way compared with the classical ‘sleep pressure’ modelling approach.

The free energy term, according to Helmholtz, is a quantity defined as the amount of useful work that is obtainable from a system while keeping its volume and temperature constant. Like the total internal energy, the free energy is a thermodynamic state function.

The term ‘stylized’, used for the free energy and entropy modelling, indicates that we are using only metaphoric estimates for the conceptual modelling purposes. We clearly admit that stylized estimates of free energy and entropy do not represent real physical values. However, stylized estimates represent similar mutual relations and constraints as real ones. Hence, they do fit for the conceptual modelling purposes.

The term ‘chronotype’ is used as the internal circadian rhythm or body clock of an individual that influences the cycle of sleep and activity in a 24-hour period. A nice overview of probing the mechanisms of chronotype using quantitative modelling can be found in Phillips

The term ‘open system’ is used as in thermodynamics and physics – a system where matter and energy can enter or leave, in contrast to a closed system where energy can enter or leave but matter cannot.

The term negentropy, according to Willard Gibbs, is the amount of entropy that may be increased without changing the internal energy or increasing its volume. In other words, it is the difference between the maximum possible entropy, under assumed conditions, and its actual entropy. Usually, negentropy is denoted as negative; therefore, we use the modulus. Negentropy for the dynamically ordered sub-system can be redefined as the specific entropy deficit relative to the surrounding chaos.

REM – Rapid Eye Movement during a dreaming state while sleeping. NREM – Non Rapid Eye Movement during a deep sleep state.

In the next section, the conceptual model is presented in terms of entropy and free energy. The third section describes the stochastic modelling of marginal and in-between transitions among BBS. The fourth section presents simulation results. The fifth section gives a brief discussion. The last section makes conclusions.

In this section, a specific purely conceptual research question is posed: whether there is a way to model the basic brain states (BBS), using free energy principles. After a discussion in terms of information theory below, this section describes how stylized Helmholtz free energy principles can be applied. Meanwhile, BBS characterization is presented in the third section, where each state is parameterized and a set of equations for the transition probabilities between states is set out. There we describe a discrete non-Markov stochastic process over four states, where transition probabilities depend basically on the (i) entropy and corresponding energy level of the current brain state, (ii) time of day and (iii) chronotype.

Hence, before delving deeper into the modelling subject, let us remind the context of this endeavour. In fact, the physiological mechanisms underlying interindividual differences in the chronotype are yet to be established, despite that both the circadian and homeostatic processes, proposed in the mainstream research, are involved. Admittedly, physiologically-based models are developed by combining the models of the sleep-wake switch and circadian pacemaker, providing a means for examining how interactions between these systems affect the chronotype (Phillips

Following the traditional approach, some circadian (e.g. period and amplitude) or homeostatic (e.g. clearance and production rates) parameters should be adjusted in order to obtain different behaviour of chronotypes. However, in our model, different chronotypes are obtained not by substantial tailoring of circadian and homeostatic terms, but by introduction of chronotype dependent probabilities of transitions between brain states, which in turn are related to the specific dynamic patterns of entropy and energy change.

Coming back to the main issue, while searching for the most basic generalization of the neural metabolic processes, our attention was attracted to the universal thermodynamic laws. Admittedly, the brain as a complex biological system has to function according to these laws, where free energy as a primal energy source and entropy as a primal measure of order, play a key role. Following this line of thought, we admit that agent’s brain states are essentially dependent on the means of using available free energy, which can be exploited for every kind of cellular and consequently neural metabolic activities. It implies that neural metabolic activities, being specific for each BBS, can be recognized in terms of the specific dynamic patterns of entropy and energy change.

Admittedly, there are a few free energy interpretations. We adapted the Helmholtz free energy principle, which is commonly used for systems held at constant volume and temperature (as it is in the brain). For a system at constant temperature and volume, the Helmholtz energy is minimized at equilibrium. In fact, the equilibrium is a major condition. It applies to gases, liquids, solid matter and even living cells and organs. But how is this equilibrium achieved in the brain? Before delving into explanations, we do acknowledge that, strictly speaking, blood circulation makes the brain function as an open system. Blood stream, pressure and temperature are kept constant over time. The blood provides nutrients (energy source) and keeps away waste products while fuelling the whole brain system.

ATP (nucleoside also called a nucleoside triphosphate) transports chemical energy within cells for metabolism (synthesis of proteins, synthesis of membranes, movement of the cell, cellular division, transport of various solutes, etc.). Hence, ATP is the molecule that carries both types of energy – potential and kinetic too. However, our paper does not elaborate on the forms of nucleoside (ATP, ADP. AMP, etc.) energy. It is out of our research scope. We simply call it a biological form of energy.

In this sense, the brain can be understood as a biological engine, which keeps equilibrium (minimum of free energy), constant volume and temperature while performing a useful work, i.e. maintaining basic brain states, which are specific concerted neural processes involved in the consumption of comparatively large amounts of free energy (usually, even at rest, the brain consumes energy 10 times more than the rest of the body per gram of tissue, which indicates very intensive and dynamic neural processes).Similarly, like inanimate matter in idealized conditions, the brain operates in highly idealistic and stable conditions regulated by the homeostatic physiological mechanisms. Homeostasis refers to stability, balance, or equilibrium within a cell, organ or the body. Hence, through the homeostasis it is the ability of an organism to keep a constant internal environment in the brain as an organ. In this sense, homeostasis is an important characteristic of equilibrium observed in the living forms. Keeping a stable internal environment requires constant adjustments as conditions change inside and outside the brain (e.g. osmoregulation, thermoregulation, chemical and endocrine metabolic regulation, etc.). In fact, all vertebrates have a blood-brain barrier that allows metabolism inside the brain to operate differently from metabolism in other parts of the body. Glial cells play a major role in brain metabolism by controlling the chemical composition of the fluid that surrounds neurons, including levels of ions and nutrients.

In our generalized model (see below), we take the above mentioned considerations, assuming that free energy is constantly minimized, i.e. is time constant. Hence, we do not solve here the problem of free energy minimization while doing some cognitive tasks. Instead, we are focusing on (i) the more general representation of basic brain states as entropy and energy processes, and (ii) modelling of stochastic transitions between basic brain states during a day period.

Now let us examine some basic principles of thermodynamic (Helmholtz) free energy

Hence, the product of

In our model, we also propose two fundamental ingredients for the total internal energy

Of course, an EEG only measures a tiny part of the total emitted bioelectromagnetic energy in the entire brain system. However, following recent neuroscience developments, we assume that analyses of this tiny EEG registered part of the entire bioelectromagnetic field in the brain is capable of differentiating basic brain states (Buzsaki,

Hence, according to the proposed approach free energy

Following Eq. (

In order to have

For some people it might be better understood in terms of a field-effect transistor, where the terminals are labelled as gate, source, and drain, and a voltage at the gate (i.e. potential energy in our model) can control the current between the source and drain (i.e. kinetic energy in our model). For others, who are more theory-driven our approach might be better understood in terms of McFadden’s electromagnetic theory of consciousness, Pribram’s holonomic brain theory, Hameroff-Penrose Orchestrated Objective Reduction theory, etc. Such type of the research frontier has made room for field-theoretic modelling of consciousness (Libet,

While looking for basic stylized mathematical functions suitable to represent the above-mentioned energy relationships, we employed a nonlinear logistic function, observed naturally in various biological systems. Admittedly, this function finds many applications throughout a vast range of fields, including biology, neural networks, ecology, biomathematics, chemistry, economics, geosciences, sociology, political sciences, etc.

Here we applied the classical logistic function, widely used for the versatile growth modelling inspired by nature. Hence, we assume that it is able to depict various nonlinear energy dependencies on

The generalized logistic function has plenty of parameters that allow its flexibility and ability for adaptation in many applied cases.

Hence, the logistic function is employed for the mathematical representation of the

In order to plot the stylized

Principal scheme of basic relationships between the entropy

In Fig.

First, we remind that just like the total internal energy, the free energy is a thermodynamic state function. According to Helmholtz, the free energy is a quantity defined as the amount of useful work obtainable from a system while keeping its volume and temperature constant. Second, the brain operates under highly idealistic and stable conditions regulated by the homeostatic physiological mechanisms. Homeostasis refers to stability, balance, or equilibrium within an organ (brain). It is the ability of an organism to keep a constant internal environment in the brain as an organ. In this sense, homeostasis is an important characteristic of equilibrium observed in the living forms, which keeps constant the free energy level at all costs.

When free energy starts to increase, self-organizing brain processes accelerate (transition to the active brain states takes place) in order to reduce the free energy level. The opposite happens when free energy starts to decrease. The brain processes start to slow down (transition to the passive brain states takes place) in order to maintain the free energy level within allotted bounds. Thus, the free energy level fluctuates within narrow bounds. However, we made simplification assuming that free energy is fixed and for modelling purposes equals 1. This chosen number can be different, but it does not make a big difference in terms of the scale-invariant features of the abstract model. Namely, we can use any multiplayer if needed.

In the proposed model, free energy as a constant factor was achieved using (i) aforementioned self-regulating dynamics of brain states, and (ii) interchange mechanism between potential

Certainly, to know empirically and model exact values of the brain entropy and energy is not possible at the current state of neuroscience. Therefore, we have to emphasize again that the differential Helmholtz free energy equation should be interpreted as a means to model prevailing brain states in terms of stylized entropy and energy. Hence, we clearly admit that stylized estimates of free energy and entropy do not represent real physical values. However, the universal thermodynamic free energy equation revealed some important relations and constraints between entropy and energy dynamics, which most probably takes place in the brain.

In the next section, each basic brain state (BBS) as a dynamic process will be parameterized using above described stylized entropy and energy time dependent relations. Besides, we will introduce a stylized self-organizing mechanism of transitions between the basic brain states during a day period, where adaptive transitions between brain states lead to homeostatic rhythms and activity patterns of complex daily rhythmic brain states.

In this section, we first construct analytical representations of four basic brain states (BBS) as processes led by the specific entropy dynamics. The presented numerical modelling is not inductively derived from empirical data, although, inevitably it has been framed by some well-known empirical observations. The presented deductive approach utilizes modelling of brain states as stylized entropy processes, described in Section

The second part of this section concerns analytical modelling of transitions between states as a discrete non-Markov stochastic process, based on the (i) entropy margins and transition probabilities, (ii) entropy and corresponding energy level of the state, (iii) time of day, and (iv) chronotype.

Hence, the presented model aims to simulate dynamics of BBS homeostatic-driven rhythms during a day. That is, we strive to create a self-regulating process of brain states dynamics using the above-described theoretical setup. In this section, we briefly present some practical ideas related to the simulation model design. It mainly concerns description of probabilistic marginal and in-between transitions between BBSs.

In the next section, numerical simulation results are presented, that indicate multiple probabilistically repeating occurrences of each state during a day. That corresponds to the experimental observations of the dynamics of real human states during a daytime and night-time, e.g. multiple repeating cycles of altering duration for the REM (BBSRE as dreaming), NREM (BBSDS – deep sleep) and wakeful (BBSAW) states during the night-time (Dijk,

According to the literature, under normal circumstances, during the night-time there are 1) 3–4 deep sleep cycles (BBSDS) of diminishing duration, 2) 4–5 cycles of the REM dreaming state (BBSRE) of increasing duration, and 3) around 4 cycles of the wakeful state (BBSAW) of increasing duration.

Before delving deeper into the analytical model details, let us remember that historically the mathematical two-process model was introduced by the well-known seminal work of Daan

Hence, traditional circadian models employ time dependent, exponential, two-process growth and decline functions, which are bounded by the circadian harmonic function. Hence, transitions occur when the exponents of the vaguely explained term “sleep pressure” approach the harmonic (circadian) function. In essence, it means that the marginal values for the two-process states are determined by the circadian harmonic function and vary during the day following the harmonic function.

In our model, entropy

It is important to remind that entropy

According to the model setup, during activity (thinking and physically active states) entropy increases as brainwork processes increase the waste of free energy in the form of heat (entropy). Whereas, during resting (NREM and REM sleep) entropy decreases as brainwork processes are eliminating the excess of heat (entropy). Naturally, both processes have some bounds, that direct to the brain’s self-organizing mechanism to stop the current state process and switch to the opposite process (active vs. resting). Hence, entropy bounds act like a thermostat to keep system condition within certain homeostatic confines.

However, there is no fundamental difference between imposing a threshold that is oscillating and then adding some stochasticity to this, as was done for some of the earlier two-process model simulations, and introducing a probabilistic transition function that is oscillating in the presented model approach. In both cases, the rhythmicity is essentially imposed as part of the modelling assumptions. However, the main conceptual difference lies in the interpretation of the underlying oscillatory mechanisms. That is, the classical two-process model does not define a mechanism for occurrence of “sleep pressure” (a relaxation oscillator), although, it uses some biological markers intrinsic to this mechanism. Whereas, our model defines a possible fundamental mechanism for occurrence of a relaxation oscillator, which consequently eliminates the need to employ the “sleep pressure” term.

In our model, the underlying oscillatory mechanism is not solely based on the oscillating probabilistic transition function. It is important to note that the main novelty is in the simulation of the basic four brain states as self-organized energy and entropy processes, bounded by the daytime dependent entropy floor

In the proposed model, transitions between BBS are visualized in the form of directed graphs, see Fig.

Probabilistic transitions invoked by the marginal entropy limits are indicated using solid lines, and that, invoked by the in-between probabilities, are indicated using dashed lines. In this way, the basic brain state related processes (free energy and entropy) are governed by the thermodynamic processes (see Eqs. (

As can be inferred from Fig.

Directed graphs for the depiction of nodes (BBS) and the transitions between them (directed lines). The more frequent states (shaded nodes) indicate the average time spent in the corresponding BBS and the thickness of the directed lines indicates the probability level of the transitions. Part (a) indicates marginal transitions when the

It is important to emphasize that according to the model setup at each time moment an agent’s brain state is basically described by the entropy

Thus, each 10 min time period during a day is represented by 4 specific

In order to obtain a set of entropy curves, a new homeostatic-circadian function

Entropy space network

Hence, a new set of proposed mathematical

homeostatic individual curves

circadian movements of the homeostatic curves

In fact, the variable

Let us explain that the states as time varying entropy processes were defined analytically in Eqs. (

Thus, each state is progressing in the entropy space, see Fig.

We have arbitrary chosen a discrete set of corresponding curves (the same number of curves for a day and night

The proposed entropy space network of the

Admittedly, the law of entropy conservation should hold during transitions between states. Hence, there should be no gaps between the entropy levels when the transitions between states take place. That is, the continuity of entropy level holds during transitions and the next state proceeds from where the last state ends. The proposed entropy-based BBS modelling approach follows the conservation law in a mathematical sense; however, due to some programming approximations an attentive reader can notice slight shifts between

Let us see an example for the BBSDS: during the night-time, the occasionally repeating deep sleep state gradually becomes shorter as the brain alternately shifts from the NREM to REM and awaking states (this is a well-known fact, well reported and documented in the mainstream literature (Booth and Diniz Behn,

Depending on the obtained curves position (see Fig.

Let us discuss more about newly introduced in-between transitions, that were additionally incorporated into the model in order to add some stochasticity. According to the chosen model setup, they occasionally happen in all BBS processes before they reach the marginal entropy conditions (

In the proposed model, the general expression for the in-between probabilistic transitions, depending on whether it is daytime or night-time, is expressed using a simple harmonic function

Daytime and night-time functions of transition probabilities

In the proposed model, we apply the sinusoidal function (see Eq. (

We do not assume that, for instance, evening types (owls) for some reason have a longer intrinsic period then morning types (early birds) (Phillips

Hence, the model simulates the behavioural manifestation of the underlying circadian rhythms in terms of people’s chronotypes (night owl or early bird), see Fig.

In this section, we aim to (i) show simulation results and (ii) examine the influence of some basic and optional model parameters.

In fact, there are various methods for the meta level analysis of parameter dependent simulation of input–output relations (Kamiński,

An example of the corresponding simulation results of the basic brain state (BBS) dynamics for the two chronotypes is provided in Fig.

Admittedly, night owl chronotype people tend to feel most energetic just before they go to sleep at night. Early bird chronotype people, as opposed to night owls, feel more energetic early in the daytime (Roenneberg

The simulation results show some behaviour, close to the real life, at night-time too. For instance, we observe an increase of DSRE(), REAW() and decrease of THDS(), REDS(), AWDS() of transitions at the end of the night period, see Fig.

BBS dynamics of two chronotypes: early bird and night owl. The graphs were obtained using default basic and optional parameters, except the checking time period (

Statistical averages for the reiterated estimates of the number of BBS transitions during the daytime and night-time periods are provided in Table

The analysis of transitions has revealed slight differences between chronotypes in terms of the total number of transitions during the daytime and night-time. That is, for the early birds more frequent transitions between states are during the night. An opposite tendency is observed for the night owls. It can be explained, keeping in mind that active states of brain have to be consciously controlled in order to prolong their duration. Otherwise, brain is controlled by involuntary (subconscious) transitions, which happen more often due to the intrinsic wandering nature of the mind. Therefore, early birds, being more consciously active in the daytime, tend to have less wandering transitions during the daytime, while night owls, being more consciously active at night-time, tend to have less wandering transitions during the night period.

In Table

The number of transitions among BBSs. The results were averaged for 20 agents. Time period

Number of transitions | Early bird chronotype | Night owl chronotype | ||||

Total number of transitions | 102 | 89 | 45 | 121 | 79 | 45 |

– daytime | 44 | 41 | 24 | 65 | 42 | 23 |

– night-time | 58 | 48 | 21 | 56 | 37 | 22 |

Total number of marginal transitions | 41 | 46 | 28 | 50 | 44 | 25 |

– daytime | 19 | 19 | 16 | 27 | 23 | 12 |

– night-time | 22 | 27 | 12 | 23 | 21 | 13 |

Total number of in-between transitions | 61 | 43 | 17 | 71 | 35 | 20 |

– daytime | 25 | 22 | 8 | 38 | 19 | 11 |

– night-time | 36 | 21 | 9 | 33 | 16 | 9 |

Duration of staying in BBSs (reiterated for 20 agents). The total time spent in each BBS is denoted as

Duration, |
Daytime/night-time | Early bird chronotype | Night owl chronotype | ||||

358 | 295 | 228 | 225 | 235 | 352 | ||

358 | 531 | 638 | 246 | 435 | 471 | ||

358 | 138 | 127 | 429 | 162 | 155 | ||

366 | 476 | 447 | 541 | 607 | 462 | ||

Daytime | 44 (1 tr.) | 0 | 91 (1 tr.) | 16 (2 tr.) | 32 (2 tr.) | 62 (2 tr.) | |

Night-time | 12 (27 tr.) | 15 | 20 (7 tr.) | 10 (19 tr.) | 19 (9 tr.) | 28 (8 tr.) | |

Daytime | 13 | 16 | 37 | 5 | 14 | 27 | |

Night-time | 16 | 18 | 44 | 14 | 17 | 31 | |

Daytime | 15 | 14 | 14 | 10 | 14 | 25 | |

Night-time | 16 | 14 | 86 | 14 | 11 | 30 | |

Daytime | 18 | 19 | 23 | 16 | 20 | 31 | |

Night-time | 6 | 13 | 32 | 15 | 26 | 45 |

The standard deviation of the total time spent in BBS (

Following the statistics in Table

We also have to discuss those cases where the DS state occasionally occurs in the daytime. As we can see in Table

Additionally, the concerted circadian model is also capable of simulating the empirically observed sleep-wake cycles during the night-time. Depending on the model parameters, we can obtain a different number of REM (i.e. RE state that can be associated with the dreaming state), NREM (DS state can be associated with the deep sleep state) and awakening states (in our model it can be associated with active wakefulness (AW) and thinking (TH) states) cycles during the night-time. The results obtained are relatively close to the numerous neurophysiological research results for sleeping stages during the night-time (Booth and Diniz Behn,

Next, in Fig.

The number of transitions and the average time (

The authors argue that the model simulation, based on the conceptual framework, can be interpreted using entropy

Hence, according to the model, the nature of underlying fundamental brain processes for each BBS can be directly revealed by the temporal dynamics of stylized entropy as well as stylized kinetic and potential energy patterns, see Fig.

According to thermodynamics and statistical mechanics, the most general interpretation of entropy results in a measure of uncertainty about a system or, in other words, disorder. In fact, such a measure has the opposite meaning to the order observed in the inner structures and the behaviour of living systems. This provides clear incentives for living systems to employ another measure, which is called negentropy. This term was first used by Schrödinger. He introduced the concept of negative entropy for a living system as entropy that it exports to keep its own entropy low. Negentropy for the dynamically ordered sub-system can be redefined as the specific entropy deficit relative to the surrounding chaos. In this way, negentropy can be understood as a measure of the distance

This makes perfect sense as a random variable with a Gaussian white noise distribution would need the maximum length of data to be accurately described. If

Gaussian white noise refers to the probability distribution with respect to the average value, in this context, the probability of the signal reaching certain amplitude, while the term white refers to the flat power spectral density distribution. In general, Gaussian noise is not necessarily white noise, yet neither property implies the other.

In this way, negentropy serves as a measure of order while entropy, as a measure of disorder. Hence, it is apparent that unlike engineering (where negentropy takes the form of digital information and is quantized in bits), social and biological systems are much more complex self-organizing processes and require a more sophisticated approach (Plikynas,Similarly, we assume that another directly related variable, i.e. kinetic energy

It is important to note that the BBS dynamics (see Fig.

In fact, turning points in the diagram indicate transitions between BBS, which occur because the simulated brain system either reaches marginal limit

After the marginal point has been reached, following the scheme of transitions, an agent is probabilistically redirected into another BBS process (see Fig.

Dynamics of entropy and stylized energy (kinetic and potential) for both chronotypes. Periodicity of the applied probabilistic in-between transitions

Thus, entropy

Comparison of entropy

Below we examine the average levels of fundamental factors, i.e. entropy

In the deep relaxation DS (deep sleep) state, differently from all other states, for both chronotypes the average (i) entropy

In the relaxation (RE) state, we observe a unique behaviour for both chronotypes: entropy

Energy and entropy distribution in the active AW and TH states is more even.

Now let us examine entropy

Hence, in each simulation, the probabilistic nature of the marginal and in-between transitions between the BBS creates a unique

Samples of the BBS entropy space charts depicted to illustrate the daily

In the charts, each curve represents a corresponding BBS process in terms of the entropy

Hence, it is not clear whether the simulations give results that are different or better than previous models. However, we had a great difficulty to find any similar model and simulation results to compare. Sure, there are some related studies like (Borbely,

the unique free energy and entropy model setup to describe the brain states as entropy and energy dependent processes,

four-process states (not two!),

entropy, chronotype and day time dependent stochastic transitions between states setup.

Hence, our model is different in kind. It models brain states dynamics in a different framework. Therefore, it is not an easy task to make a comparative analysis of our results with other approaches.

In sum, a thorough additional research needs to be done to examine, in detail, the issues and criteria that will help identify the validity of the proposed model by comparison with the established phenomenological two-process models, coupled oscillator models and the reciprocal interaction models (Booth and Diniz Behn,

Concerning further research directions, the presented model needs thorough parameter estimation, sensitivity analysis, and verification. Surely, it will be a main direction of the further research. In the prospective research, additional empirically-based investigations are needed to calibrate and validate the presented conceptual model as well.

Next, the presented model can also be expanded for simulating transitions between other types of mental states. Namely, it can be adapted for modelling dynamics of emotional states. Let us recall that psychologists map emotional states in two major axes – arousal (high to low) and valence (pleasure to displeasure). Such two-dimensional map has been theorized by Russell and Barrett (

Due to the involved complexity of interstate transitions, such modelling could start from Paul Ekman’s six basic emotional states (happiness, hate, grief, hope, fear, desire) model (Ekman and Cordaro,

In future, we also foresee that this kind of research framework of brain states dynamics can also take a different application sphere. That is, it may be applied in the domain of cognitive agents and multi-agent systems research. It is an emerging multidisciplinary research trend in the domains of artificial intelligence and multi-agent systems. Simulation of agent state dynamics for a single agent leads to the simulation of collective state dynamics for groups or societies of agents. It concerns the simulation and prediction of individual and collective behavioural phenomena such as dynamics of emotional states, political moods, fashion trends, social capital distributions, cultural traits, etc. However, in order to get there, we first have to find a universal way to simulate human-like brain states and transitions between them in the most abstract and fundamental way for a single agent. We suppose that the intrinsic and universal nature of free energy and entropy terms serves well for this purpose.

In short, this paper provides a deductive conceptual framework, using universal free energy and entropy terms, that can provide a better understanding of brain states dynamics as self-organized energy and entropy processes. Based on the proposed modelling framework, we also presented a pilot simulation model to showcase dynamical transitions between the basic brain states during the day.

It is important to note that the presented conceptual model has been constructed on the deductive theoretical assumptions and does not directly stem from the empirical data modelling. The main purpose was to find out what inner fundamental processes in the brain can cause the main experimentally observable brain state dynamics.

According to the literature review, for the first time, stylized thermodynamic Helmholtz free energy and entropy terms were used to differentiate brain states and describe stochastic dynamics of transitions between states. We shared ideas how universal and intrinsic free energy and entropy principles can be employed for a better understanding and simulation of brain states as self-organized energy and entropy processes. Even a conceptual possibility to model such dynamics, using entropy terms, gives a very substantial new knowledge about the implicit self-organizing nature of entropy and energy processes taking place in the brain.

It is important to note that in our model, entropy constraints trigger motion (transitions between brain states). That is, marginal entropy constraints lead to the transition processes between attractor states so that brain functioning can be optimized during the day. In this way, the brain behaves as self-regulating and adaptive behavioural mechanism. Thus, we proposed each basic brain state to model as specific endogenous (entropy and energy) processes that, change over time following intrinsic patterns. Brain states are interpreted as specific entropy and energy-related thermodynamic processes, that follow characteristic pattern changes over time. Each process leads to the marginal entropy and energy boundaries, where stochastic transitions take place. Due to the complexity involved, we have approached such a process in a reductionist way, using stylized entropy and energy evaluations.

Our model is based on the deductive assumption that the homeostatic relaxation oscillator can be nailed down to the fundamental rhythmic processes of free energy and entropy, taking place in the brain during the day. Based on this assumption, we have constructed a new way of modelling the homeostatic relaxation oscillator. We argue that the proposed endogenous model can simulate homeostatic rhythmic dynamics of brain states in a more meaningful way as compared to the classical ‘sleep pressure’ approach.

The homeostasis simulation for two chronotypes was able meaningfully to differentiate the behaviour of both chronotypes. We obtained statistically significant differences of brain state dynamics for two chronotypes during the day. For instance, our simulations have revealed experimentally observed chronotype dependent features as follows: owls tend to feel most energetic just before sleep at night, while early birds feel more energetic early in the day.

The simulation setup has revealed how two major simulated chronotypes (early birds and night owls) behave with respect to the basic brain states, entropy and energy dynamics, depending on a few basic parameters such as the recalculation time period

In sum, the simulation results show that after additional theoretical and empirical studies, the proposed conceptual research framework has a potential to (i) describe BBS using entropy and energy terms, (ii) generate homeostatic rhythms for different chronotypes, (iii) provide empirical predictions of brain state dynamics, (iv) be used for the studies of societies composed of such agents, and (v) be applied in the artificial intelligence domain, machine learning, and robotics while mimicking human-like robot state dynamics. In the prospective research, the chronotype behaviour modelling can be related with the social jetlag (misalignment of the biological and social time), cognitive abilities, depressive mood, insomnia, daytime sleepiness, etc.