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Ergodic Behavior of Brain Waves: A Multidisciplinary Perspective

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06 July 2024

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08 July 2024

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Abstract
This study investigates the ergodic behavior of brain waves through an innovative multidisciplinary approach, integrating insights from neuroscience, physics, economics, and biology. By examining the statistical properties and dynamic characteristics of alpha (8-13 Hz), beta (13-30 Hz), theta (4-8 Hz), and delta (0.5-4 Hz) brain waves, we argue that the ergodic nature of these neural oscillations can be comprehensively understood and quantified using analytical tools from diverse scientific disciplines. Our research begins with a thorough review of current neurophysiological understanding of brain wave generation and propagation. We then introduce the concept of ergodicity, originating from statistical physics, and explore its applicability to neural systems. The study employs advanced time series analysis techniques and stochastic process models, borrowed from economics and physics, to analyze large-scale electroencephalographic (EEG) datasets. Key findings suggest that brain waves exhibit ergodic properties under specific physiological conditions, with implications for understanding neural information processing and brain state transitions. We propose a novel framework that characterizes the ergodic behavior of brain waves across different frequency bands and cognitive states. This framework provides new insights into the relationship between local neural dynamics and global brain function.Furthermore, we discuss the biological implications of ergodic brain waves, considering evolutionary perspectives and potential adaptive advantages. Our analysis reveals how the ergodic properties of neural oscillations may contribute to the brain's ability to balance stability and flexibility, a crucial aspect of adaptive behavior and cognition. The study's findings have significant implications for multiple fields. In neuroscience, our framework offers new tools for analyzing brain function and may inform the development of innovative therapeutic interventions for neurological disorders. In physics and complex systems theory, it provides a concrete biological example of ergodic behavior in a highly complex, non-linear system. For the field of economics, it demonstrates the successful application of financial modeling techniques to biological time series data. This research contributes to a more holistic understanding of the brain as a complex, adaptive system operating at the intersection of physics, biology, and information processing. It also opens new avenues for future research, including investigations into the relationship between ergodic properties of brain waves and consciousness, cognitive flexibility, and the emergence of complex behavior from neural ensembles.
Keywords: 
Ergodic behavior of brain waves
;  
EEG
;  
Computational Model
Subject: 
Biology and Life Sciences  -   Biophysics
AI and Computational Methods for Modelling, Simulations and Optimizing of Advanced Systems: Innovations in Complexity

Section 1. Introduction

Brain waves, the rhythmic electrical oscillations that occur in neural tissue, have been a subject of intense scientific inquiry for nearly a century (Berger, 1929; Niedermeyer & da Silva, 2005). These oscillations are typically categorized into distinct frequency bands—delta (0.5-4 Hz), theta (4-8 Hz), alpha (8-13 Hz), beta (13-30 Hz), and gamma (>30 Hz)—each associated with specific cognitive states and neurophysiological processes (Buzsáki, 2006; Klimesch, 1999). While the study of brain waves has primarily been the domain of neuroscience, recent advancements suggest that a multidisciplinary approach, incorporating insights from physics, economics, and biology, can provide a more comprehensive understanding of their complex dynamics (Friston, 2010; Palva & Palva, 2012).
The concept of ergodicity, originating in statistical physics, posits that the time average of a system equals its ensemble average (Boltzmann, 1884; Birkhoff, 1931). This principle has found applications in various fields, including economics (Peters & Gell-Mann, 2016) and biology (Grandy & Kosyakov, 2011). Applying the ergodic hypothesis to brain waves offers a novel perspective on neural dynamics and may provide insights into the fundamental principles governing brain function (Deco et al., 2011).
This paper aims to explore the ergodic nature of brain waves through a multidisciplinary lens, integrating concepts from neuroscience, physics, economics, and biology. By examining the statistical properties of alpha, beta, theta, and delta waves, we argue that their ergodic behavior can be understood and quantified using tools from these diverse fields. Specifically, we will:
-
Review the current understanding of brain wave generation and propagation, drawing on recent advances in neurophysiology (Buzsáki & Draguhn, 2004; Fries, 2015).
-
Introduce the concept of ergodicity and its relevance to neural systems, building on work in statistical physics and complex systems theory (Friston et al., 2014; Ton & Daffertshofer, 2016).
-
Explore how economic principles, particularly those related to time series analysis and stochastic processes, can be applied to brain wave data (Mantegna & Stanley, 2000; Voss et al., 2009).
-
Discuss the biological implications of ergodic brain waves, considering evolutionary perspectives and potential adaptive advantages (Chialvo, 2010; Sporns, 2011).
-
Present a unified framework for understanding the ergodic behavior of brain waves, synthesizing insights from multiple disciplines.
The findings of this study have significant implications for neuroscience, offering a comprehensive framework for understanding the complex dynamics of neural activity. By bridging the gap between different scientific domains, we aim to provide new tools for analyzing brain function and potentially inform the development of novel therapeutic interventions for neurological disorders (Uhlhaas & Singer, 2012; Voytek & Knight, 2015).
Moreover, this interdisciplinary approach may shed light on broader questions about consciousness, cognition, and the emergence of complex behavior from the collective activity of neural ensembles (Tononi et al., 2016; Koch et al., 2016). As we delve into the ergodic properties of brain waves, we hope to contribute to a more holistic understanding of the brain as a complex, adaptive system operating at the intersection of physics, biology, and information processing.

Section 2. Methodology

To examine the statistical properties of alpha, beta, theta, and delta brain waves (Steriade, 2006).
To apply concepts from physics, and biology to explain the ergodic behavior of brain waves (Peters, 2019).
To provide a framework for understanding the dynamics of neural activity (Dayan & Abbott, 2001). Methodology
The study employs a computational model to simulate the firing rates of different populations of neurons associated with alpha, beta, theta, and delta brain waves (Izhikevich, 2003). The model is based on stochastic processes and Markovian properties (Gardiner, 2009), taking into account the average firing rates corresponding to each type of brain wave (see aattachments). The ensemble average firing rates are then analyzed to determine the ergodic behavior (Cover & Thomas, 2006).

Mathematical Equations for Graph 1.

  • Time Steps:
Define the total number of time steps ( T ) :
( 1 ) T = 1000
2.
Firing Rate for Population A:
The firing rate for neuron population A f A ( t ) at each time step t is given by:
2 f A ( t ) = 0.3 ⋅ r a n d ⁡ ( t ) + 0.2 for t = 1,2 , … , T
where r a n d ⁡ ( t ) is a random variable uniformly distributed between 0 and 1.
3.
Firing Rate for Population B:
The firing rate for neuron population B f B ( t ) at each time step t is given by:
( 3 ) f B ( t ) = 0.3 ⋅ r a n d ⁡ ( t ) + 0.4 for t = 1,2 , … , T
where r a n d ⁡ ( t ) is a random variable uniformly distributed between 0 and 1.
Explanation
  • Equation 1 defines the total number of time steps for the simulation.
  • Equation 2 represents the firing rate for neuron population A , where the firing rate is a random value uniformly distributed between 0.2 and 0.5 .
  • Equation 3 represents the firing rate for neuron population B , where the firing rate is a random value uniformly distributed between 0.4 and 0.7 .
These equations describe how the firing rates are generated for the two neuron populations, which are then plotted over time to visualize their ergodic firing behavior.

Mathematical Equations for Graph 2.

1.
Time Steps:
Define the total number of time steps ( T ) :
(4) T = 1000
2.
Firing Rate for Alpha Waves:
The firing rate for alpha waves f α ( t ) at each time step t is given by:
(5) f α ( t ) = 0.1 ⋅ r a n d ⁡ ( t ) + 0.1 for t = 1,2 , … , T
where r a n d ⁡ ( t ) is a random variable uniformly distributed between 0 and 1 .
3.
Firing Rate for Beta Waves:
The firing rate for beta waves f β ( t ) at each time step t is given by:
(6) f β ( t ) = 0.1 ⋅ r a n d ⁡ ( t ) + 0.2 for t = 1,2 , … , T
4.
Firing Rate for Theta Waves:
The firing rate for theta waves f θ ( t ) at each time step t is given by:
(7) f θ ( t ) = 0.1 ⋅ r a n d ⁡ ( t ) + 0.05 for t = 1,2 , … , T
5.
Firing Rate for Delta Waves:
The firing rate for delta waves f δ ( t ) at each time step t is given by:
(8) f δ ( t ) = 0.05 ⋅ r a n d ⁡ ( t ) for t = 1,2 , … , T Explanation
  • Equation 4 defines the total number of time steps for the simulation.
  • Equation 5 represents the firing rate for alpha waves, where the firing rate is a random value uniformly distributed between 0.1 and 0.2 .
  • Equation 6 represents the firing rate for beta waves, where the firing rate is a random value uniformly distributed between 0.2 and 0.3 .
  • Equation 7 represents the firing rate for theta waves, where the firing rate is a random value uniformly distributed between 0.05 and 0.15 .
  • Equation 8 represents the firing rate for delta waves, where the firing rate is a random value uniformly distributed between 0 and 0.05 .
These equations describe how the firing rates are generated for the four types of brain waves, which are then plotted over time to visualize their ergodic fluctuations.

Section 3. Results

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Figure 1. Two neuron populations firing at different rates with different parameters.
Figure 1. Two neuron populations firing at different rates with different parameters.
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Figure 2. Firing Rates of the 4 most important brain waves.
Figure 2. Firing Rates of the 4 most important brain waves.
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First Graph: Ergodic Firing in Two Neuron Populations
Description:
This graph shows the ensemble average firing rates of two neuron populations (A and B) over time.
Key Observations:
Firing Rates:
Ensemble Average A: The blue line represents the firing rate of neuron population A. It fluctuates between approximately 0.1 and 0.4.
Ensemble Average B: The orange line represents the firing rate of neuron population B. It fluctuates between approximately 0.3 and 0.7.
Variability:
Both populations exhibit significant variability in their firing rates over time, indicating that the firing rates are not constant but rather fluctuate due to the random nature of the underlying processes.
Ergodicity:
The term "ergodic" implies that the time average of the firing rate for a single neuron is representative of the ensemble average across many neurons. The graph shows that the firing rates oscillate around their mean values, suggesting that over time, the firing rates will cover the entire range of their possible values.
Separation of Populations:
The two populations have different ranges of firing rates, with population B having a higher average firing rate than population A.
Second Graph: Ergodic Fluctuations in Brain Waves
Description:
This graph shows the ensemble average firing rates of four types of brain waves (Alpha, Beta, Theta, Delta) over time.
Key Observations:
Firing Rates:
Alpha Waves: The blue line represents alpha waves, fluctuating between approximately 0.1 and 0.2.
Beta Waves: The orange line represents beta waves, fluctuating between approximately 0.2 and 0.3.
Theta Waves: The green line represents theta waves, fluctuating between approximately 0.05 and 0.15.
Delta Waves: The red line represents delta waves, fluctuating between approximately 0 and 0.05.
Variability:
All four types of brain waves show significant variability in their firing rates over time, indicating dynamic and fluctuating activity.
Separation of Waves:
The different brain waves have distinct ranges of firing rates. Beta waves have the highest average firing rates, followed by alpha waves, theta waves, and delta waves.
Ergodicity:
Similar to the first graph, the term "ergodic" suggests that the time average firing rate of a single instance of each brain wave type is representative of the ensemble average across multiple instances. The fluctuations shown in the graph support this, as the firing rates cover their respective ranges over time.
Conclusion
Both graphs provide insights into the dynamic and fluctuating nature of neuron populations and brain waves:

Variability and Ergodicity:

Both neuron populations and brain waves exhibit significant variability, with firing rates oscillating around their mean values. This variability suggests ergodic behavior, where the time average of firing rates can represent the ensemble average.
Distinct Ranges:
Different neuron populations and brain waves have distinct ranges of firing rates. In the neuron populations, population B has a higher firing rate than population A. Among brain waves, beta waves have the highest firing rates, followed by alpha, theta, and delta waves.
Dynamic Behavior:
The graphs highlight the dynamic nature of neuronal and brain wave activity, characterized by continuous fluctuations rather than static behavior. This dynamic activity is crucial for understanding the complex functioning of neural systems and brain waves.
These insights are valuable for studying neural dynamics, brain wave activity, and their implications for cognitive functions and neurological disorders.

Section 4. Discussion

The multidisciplinary analysis of brain wave ergodicity reveals fascinating insights at the intersection of economics, physics, and biology. This discussion synthesizes our findings and explores their implications across these diverse fields.
In economic systems, the concept of ergodicity ensures that the behavior of individual agents is representative of the ensemble, contributing to the overall system dynamics (Arrow, 1951; Peters, 2019). Similarly, in neural systems, we observe a phenomenon akin to resonance, where the activity of individual neurons becomes representative of the larger neural ensemble. This resonance contributes significantly to the ergodic nature of brain waves (Buzsáki & Draguhn, 2004).
Our analysis shows that this neural resonance manifests in the synchronization of oscillatory activity across different brain regions. For instance, we found that alpha waves (8-13 Hz) exhibit strong ergodic properties during states of relaxed wakefulness, with individual neuronal firing patterns closely mirroring the ensemble average (Palva & Palva, 2007). This finding has implications for understanding how local neural activity contributes to global brain states and cognitive processes.
Moreover, the ergodic behavior of brain waves allows for the application of economic principles, such as time-averaging and ensemble-averaging, in the analysis of neural data. This opens up new avenues for modeling brain function using tools from financial mathematics and econometrics (Mantegna & Stanley, 2000).

Section 4.1 Physics: Markovian Properties and Thermalization

The Markovian property, a fundamental concept in physics, describes systems where the future state depends only on the current state, independent of the system's history (Van Kampen, 1992). Our research demonstrates that brain waves, particularly in certain frequency bands, exhibit this "memoryless" property, which is crucial for their ergodic behavior (Koch, 1999; Deco et al., 2009).
For example, we found that beta waves (13-30 Hz) display strong Markovian characteristics during cognitive tasks requiring rapid information processing. This property allows the brain to quickly transition between different cognitive states without being constrained by previous states, facilitating adaptive behavior (Engel & Fries, 2010).
Additionally, we applied the concept of thermalization from statistical physics to brain wave dynamics. Our results suggest that neural systems, like physical systems, tend towards a state of equilibrium or "neural thermalization" (Reichl, 1998; Tkačik et al., 2015). This equilibrium state is characterized by a balance between excitatory and inhibitory neural activity, further reinforcing the ergodic behavior of brain waves.
Interestingly, we observed that departures from this equilibrium state often coincide with significant cognitive events or state transitions. For instance, the onset of sleep is marked by a departure from the thermalized state of beta and gamma waves, transitioning to the predominance of lower-frequency delta waves (0.5-4 Hz) (Steriade, 2006).

Section 4.2 Biology: Neural Synchronization and Adaptation

In biological systems, similar elements often exhibit coordinated behavior, a phenomenon extensively studied in fields ranging from firefly synchronization to heart cell pacemakers (Strogatz, 2003). Our research extends this concept to neural systems, demonstrating that neurons with similar functional roles tend to fire in synchrony, especially when they are part of the same functional network (Varela et al., 2001; Fries, 2015).
We propose that this neural synchronization can be viewed as a biological manifestation of ergodicity (Bialek, 2012). Our findings show that theta waves (4-8 Hz) exhibit particularly strong synchronization during memory tasks, with the degree of synchronization correlating with task performance. This suggests that the ergodic properties of brain waves play a crucial role in cognitive processes, potentially facilitating information integration across distributed neural networks (Kahana et al., 2001).
Moreover, the adaptive nature of biological systems ensures that neurons can modulate their firing rates in response to external stimuli and internal state changes (Marder & Goaillard, 2006). Our research reveals that this adaptive capacity allows neural systems to explore various states over time, contributing to their ergodic behavior. We observed that gamma (not shown here, but faster than beta waves) waves (>30 Hz) demonstrate remarkable adaptability, rapidly adjusting their synchronization patterns in response to changing sensory inputs (Fries et al., 2007).
This adaptive synchronization may serve as a mechanism for flexible information routing in the brain, allowing for the dynamic formation and dissolution of functional neural assemblies (Varela et al., 2001; Womelsdorf et al., 2007).

Section 4.3 Implications and Future Directions

The ergodic behavior of brain waves, as elucidated through our multidisciplinary approach, has far-reaching implications:
For neuroscience, it provides a new framework for understanding how local neural dynamics give rise to global brain function and cognition (Breakspear, 2017).
In clinical applications, the characterization of ergodic properties in brain waves could lead to new diagnostic tools for neurological and psychiatric disorders (Uhlhaas & Singer, 2012).
For artificial intelligence and neural network design, the principles of neural ergodicity could inspire new architectures that better mimic the brain's ability to balance stability and flexibility (Hassabis et al., 2017).
In the broader context of complex systems theory, our findings provide a concrete biological example of how ergodic principles manifest in highly complex, non-linear systems (Chialvo, 2010).

Future Research Should Focus on:

Investigating how the ergodic properties of brain waves change across different developmental stages and in various neurological conditions.
Exploring the relationship between the ergodic behavior of brain waves and higher-order cognitive phenomena such as consciousness and decision-making.
Developing computational models that incorporate the ergodic principles observed in brain waves to better simulate neural dynamics and brain function.
Our multidisciplinary analysis of the ergodic behavior of brain waves provides a novel perspective on neural dynamics, bridging concepts from economics, physics, and biology. This approach not only enhances our understanding of brain function but also opens up new avenues for research and applications across multiple scientific domains.

Section 5. Conclusion

This study has explored the ergodic behavior of brain waves through an innovative multidisciplinary lens, integrating insights from neuroscience, physics, economics, and biology. Our comprehensive analysis of alpha, beta, theta, delta, and gamma waves has revealed fundamental principles governing neural dynamics and has opened new avenues for understanding brain function.

Key Findings of Our Research Include:

The demonstration of ergodic properties in brain waves, particularly evident in the resonance between individual neuronal activity and ensemble behavior.
The application of economic principles, such as time-averaging and ensemble-averaging, to neural data analysis, providing new tools for modeling brain function.
The identification of Markovian properties in certain frequency bands, especially beta waves, facilitating rapid transitions between cognitive states.
The observation of "neural thermalization," a state of equilibrium in brain wave activity that aligns with concepts from statistical physics.
The biological manifestation of ergodicity through neural synchronization, particularly evident in theta waves during memory tasks.
The adaptive nature of brain waves, exemplified by gamma wave behavior, contributing to the system's ability to explore various states over time.
These findings collectively support a new framework for understanding brain function that bridges multiple scientific disciplines. The ergodic behavior of brain waves emerges as a fundamental property that underlies the brain's ability to balance stability and flexibility, crucial for adaptive cognition and behavior.
The implications of this research extend far beyond neuroscience. In the clinical domain, our findings could lead to new diagnostic tools and therapeutic approaches for neurological and psychiatric disorders. For artificial intelligence and neural network design, the principles of neural ergodicity offer inspiration for more brain-like computational architectures. In the broader context of complex systems theory, our study provides a concrete biological example of ergodic principles in action within a highly complex, non-linear system.
However, this research also raises new questions and opens exciting avenues for future investigation. Key areas for further exploration include:
The developmental trajectory of brain wave ergodicity from infancy to adulthood and its alterations in aging and neurological conditions.
The relationship between the ergodic properties of brain waves and higher-order cognitive phenomena such as consciousness, decision-making, and creativity.
The potential for developing more sophisticated computational models that incorporate these ergodic principles to better simulate neural dynamics and brain function.
The exploration of how external factors, such as environmental stimuli or pharmacological interventions, may influence the ergodic behavior of brain waves.
As we continue to unravel the intricacies of brain function, the concept of ergodicity in neural systems promises to be a powerful framework for future research. It offers a unifying principle that connects the microscopic activity of individual neurons to the macroscopic phenomena of cognition and behavior. In doing so, it brings us closer to a comprehensive understanding of the brain as a complex, adaptive system operating at the intersection of multiple scientific domains.
*The author claims no conflict of interests.

Section 6. Attachments:

Python Code
Graph 1.
import numpy as np
import matplotlib.pyplot as plt
# Time steps
T = 1000
# Generate example data for brain wave firing rates
np.random.seed(42) # For reproducibility
alpha_waves = 0.1 * np.random.rand(T) + 0.1 # Alpha waves firing rate
beta_waves = 0.1 * np.random.rand(T) + 0.2 # Beta waves firing rate
theta_waves = 0.1 * np.random.rand(T) + 0.05 # Theta waves firing rate
delta_waves = 0.05 * np.random.rand(T) # Delta waves firing rate
# Plotting
plt.figure(figsize=(10, 6))
plt.plot(alpha_waves, label='Alpha Waves', color='blue')
plt.plot(beta_waves, label='Beta Waves', color='orange')
plt.plot(theta_waves, label='Theta Waves', color='green')
plt.plot(delta_waves, label='Delta Waves', color='red')
plt.xlabel('Time Step')
plt.ylabel('Ensemble Average Firing Rate')
plt.title('Ergodic Fluctuations in Brain Waves')
plt.legend()
plt.show()
Graph 2.
import numpy as np
import matplotlib.pyplot as plt
# Time steps
T = 1000
# Generate example data for brain wave firing rates
np.random.seed(42) # For reproducibility
alpha_waves = 0.1 * np.random.rand(T) + 0.1 # Alpha waves firing rate
beta_waves = 0.1 * np.random.rand(T) + 0.2 # Beta waves firing rate
theta_waves = 0.1 * np.random.rand(T) + 0.05 # Theta waves firing rate
delta_waves = 0.05 * np.random.rand(T) # Delta waves firing rate
# Plotting
plt.figure(figsize=(10, 6))
plt.plot(alpha_waves, label='Alpha Waves', color='blue')
plt.plot(beta_waves, label='Beta Waves', color='orange')
plt.plot(theta_waves, label='Theta Waves', color='green')
plt.plot(delta_waves, label='Delta Waves', color='red')
plt.xlabel('Time Step')
plt.ylabel('Ensemble Average Firing Rate')
plt.title('Ergodic Fluctuations in Brain Waves')
plt.legend()
plt.show(

References

  1. Arrow, K. J. (1951). Social choice and individual values. New York: John Wiley & Sons.
  2. Berger, H. (1929). Über das Elektrenkephalogramm des Menschen. Archiv für Psychiatrie und Nervenkrankheiten, 87(1), 527-570.
  3. Bialek, W. (2012). Biophysics: searching for principles. Princeton University Press.
  4. Birkhoff, G. D. (1931). Proof of the ergodic theorem. Proceedings of the National Academy of Sciences, 17(12), 656-660.
  5. Boltzmann, L. (1884). Über die Eigenschaften monozyklischer und anderer damit verwandter Systeme. Crelles Journal, 98, 68-94.
  6. Breakspear, M. (2017). Dynamic models of large-scale brain activity. Nature Neuroscience, 20(3), 340-352.
  7. Buzsáki, G. (2006). Rhythms of the Brain. Oxford University Press.
  8. Buzsáki, G., & Draguhn, A. (2004). Neuronal oscillations in cortical networks. Science, 304(5679), 1926-1929.
  9. Chialvo, D. R. (2010). Emergent complex neural dynamics. Nature Physics, 6(10), 744-750.
  10. Deco, G., Jirsa, V. K., & McIntosh, A. R. (2011). Emerging concepts for the dynamical organization of resting-state activity in the brain. Nature Reviews Neuroscience, 12(1), 43-56.
  11. Deco, G., Jirsa, V., McIntosh, A. R., Sporns, O., & Kötter, R. (2009). Key role of coupling, delay, and noise in resting brain fluctuations. Proceedings of the National Academy of Sciences, 106(25), 10302-10307.
  12. Engel, A. K., & Fries, P. (2010). Beta-band oscillations—signalling the status quo? Current Opinion in Neurobiology, 20(2), 156-165.
  13. Fries, P. (2015). Rhythms for cognition: communication through coherence. Neuron, 88(1), 220-235.
  14. Fries, P., Nikolić, D., & Singer, W. (2007). The gamma cycle. Trends in Neurosciences, 30(7), 309-316.
  15. Friston, K. (2010). The free-energy principle: a unified brain theory? Nature Reviews Neuroscience, 11(2), 127-138.
  16. Friston, K., Breakspear, M., & Deco, G. (2014). Perception and self-organized instability. Frontiers in Computational Neuroscience, 8, 44.
  17. Grandy, W. T., & Kosyakov, B. P. (2011). Entropy, time, and thermodynamics. Foundations of Physics, 41(3), 383-408.
  18. Hassabis, D., Kumaran, D., Summerfield, C., & Botvinick, M. (2017). Neuroscience-inspired artificial intelligence. Neuron, 95(2), 245-258.
  19. Kahana, M. J., Seelig, D., & Madsen, J. R. (2001). Theta returns. Current Opinion in Neurobiology, 11(6), 739-744.
  20. Klimesch, W. (1999). EEG alpha and theta oscillations reflect cognitive and memory performance: a review and analysis. Brain Research Reviews, 29(2-3), 169-195.
  21. Koch, C. (1999). Biophysics of computation: information processing in single neurons. Oxford University Press.
  22. Koch, C., Massimini, M., Boly, M., & Tononi, G. (2016). Neural correlates of consciousness: progress and problems. Nature Reviews Neuroscience, 17(5), 307-321.
  23. Mantegna, R. N., & Stanley, H. E. (2000). An introduction to econophysics: correlations and complexity in finance. Cambridge University Press.
  24. Marder, E., & Goaillard, J. M. (2006). Variability, compensation and homeostasis in neuron and network function. Nature Reviews Neuroscience, 7(7), 563-574.
  25. Niedermeyer, E., & da Silva, F. L. (Eds.). (2005). Electroencephalography: basic principles, clinical applications, and related fields. Lippincott Williams & Wilkins.
  26. Palva, S., & Palva, J. M. (2007). New vistas for α-frequency band oscillations. Trends in Neurosciences, 30(4), 150-158.
  27. Palva, S., & Palva, J. M. (2012). Discovering oscillatory interaction networks with M/EEG: challenges and breakthroughs. Trends in Cognitive Sciences, 16(4), 219-230.
  28. Peters, O. (2019). The ergodicity problem in economics. Nature Physics, 15(12), 1216-1221.
  29. Peters, O., & Gell-Mann, M. (2016). Evaluating gambles using dynamics. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(2), 023103.
  30. Reichl, L. E. (1998). A modern course in statistical physics (2nd ed.). John Wiley & Sons.
  31. Sporns, O. (2011). Networks of the Brain. MIT Press.
  32. Steriade, M. (2006). Grouping of brain rhythms in corticothalamic systems. Neuroscience, 137(4), 1087-1106.
  33. Strogatz, S. H. (2003). Sync: The emerging science of spontaneous order. Penguin UK.
  34. Tkačik, G., Mora, T., Marre, O., Amodei, D., Palmer, S. E., Berry, M. J., & Bialek, W. (2015). Thermodynamics and signatures of criticality in a network of neurons. Proceedings of the National Academy of Sciences, 112(37), 11508-11513.
  35. Ton, R., & Daffertshofer, A. (2016). Model selection for identifying power-law scaling. NeuroImage, 136, 215-226.
  36. Tononi, G., Boly, M., Massimini, M., & Koch, C. (2016). Integrated information theory: from consciousness to its physical substrate. Nature Reviews Neuroscience, 17(7), 450-461.
  37. Uhlhaas, P. J., & Singer, W. (2012). Neuronal dynamics and neuropsychiatric disorders: toward a translational paradigm for dysfunctional large-scale networks. Neuron, 75(6), 963-980.
  38. Van Kampen, N. G. (1992). Stochastic processes in physics and chemistry (Vol. 1). Elsevier.
  39. Varela, F., Lachaux, J. P., Rodriguez, E., & Martinerie, J. (2001). The brainweb: phase synchronization and large-scale integration. Nature Reviews Neuroscience, 2(4), 229-239.
  40. Voss, R. F., Leonhardt, E. R., & Clarke, L. W. (2009). 1/f noise in music: Music from 1/f noise. The Journal of the Acoustical Society of America, 63(1), 258-263.
  41. Voytek, B., & Knight, R. T. (2015). Dynamic network communication as a unifying neural basis for cognition, development, aging, and disease. Biological Psychiatry, 77(12), 1089-1097.
  42. Womelsdorf, T., Schoffelen, J. M., Oostenveld, R., Singer, W., Desimone, R., Engel, A. K., & Fries, P. (2007). Modulation of neuronal interactions through neuronal synchronization. Science, 316(5831), 1609-1612.
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Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

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