176
T. Stankovski
exist between the observed data. Finally, one could study the causal relations between
dynamical models and observe the effective connectivity. In this way, the interactions
can be reconstructed in terms of coupling functions which define the underlaying
interaction mechanism.
With their ability to describe the interactions in detail, coupling functions have
received a significant attention in the scientific community recently [69, 70]. Three
crucial aspects of coupling functions were studied: the theory, methods and applications. Various methods have been designed for reconstruction of coupling functions
from data [15, 33, 35, 39, 66, 80]. These have enabled applications in different
scientific fields including chemistry [32], climate [41], secure communications [44,
68], mechanics [34], social sciences [56], and oscillatory interaction in physiology
for cardiorespiratory and cardiovascular interactions [26, 35, 40, 59, 79].
Arguably, the greatest current interest for coupling functions is coming from neuroscience. This is probably because the brain is a highly-connected complex system
[51], with connections on different levels and dimensions, many of them carrying
important implications for characteristic neural states and diseases. Coupling functions are particularly appealing here because they can characterize the particular
neural mechanisms behind these connections. Recent works have encompassed the
theory and inference of a diversity of neural phenomena, levels, physical regions,
and physiological conditions [4, 13, 47, 49, 61, 71, 73, 76–78, 81, 86].
The chapter gives an overview of the topic of coupling function, with particular
focus on their use and suitability to neuroscience. This will be explained through
observations on two levels of brain connectivity—the neurons and the brainwaves
level. The relationship between the appropriate theory and methods will be also
given. On systemic level, the focus will be on neuronal oscillations, thus positioning
around and complementing the main topic of the book—biological oscillators. The
chapter will finish by outlook and some thoughts on the future developments and
uses of coupling function in neuroscience. However, before going into greater detail,
first the basics of what coupling functions are discussed briefly bellow.
11.1.1 Coupling Function Basics
The system setup to be studied is one of an interacting dynamical systems, with
the focus of coupled oscillators. Then, coupling functions describe the physical rule
specifying how the interactions occur and manifest. Because they are directly connected with the functional dependencies, coupling functions focus not only on if the
interactions exist, but more on how they appear and develop. For example, when
studying phase dynamics of coupled oscillators the magnitude of the phase coupling
function affects directly the oscillatory frequency and will describe how the oscillations are being accelerated or decelerated by the influence of the other oscillator.
Similarly, if one considers the amplitude dynamics of interacting dynamical systems,
the magnitude of coupling function will prescribe how the amplitude is increased or
decreased due to the interaction.
T. Stankovski
exist between the observed data. Finally, one could study the causal relations between
dynamical models and observe the effective connectivity. In this way, the interactions
can be reconstructed in terms of coupling functions which define the underlaying
interaction mechanism.
With their ability to describe the interactions in detail, coupling functions have
received a significant attention in the scientific community recently [69, 70]. Three
crucial aspects of coupling functions were studied: the theory, methods and applications. Various methods have been designed for reconstruction of coupling functions
from data [15, 33, 35, 39, 66, 80]. These have enabled applications in different
scientific fields including chemistry [32], climate [41], secure communications [44,
68], mechanics [34], social sciences [56], and oscillatory interaction in physiology
for cardiorespiratory and cardiovascular interactions [26, 35, 40, 59, 79].
Arguably, the greatest current interest for coupling functions is coming from neuroscience. This is probably because the brain is a highly-connected complex system
[51], with connections on different levels and dimensions, many of them carrying
important implications for characteristic neural states and diseases. Coupling functions are particularly appealing here because they can characterize the particular
neural mechanisms behind these connections. Recent works have encompassed the
theory and inference of a diversity of neural phenomena, levels, physical regions,
and physiological conditions [4, 13, 47, 49, 61, 71, 73, 76–78, 81, 86].
The chapter gives an overview of the topic of coupling function, with particular
focus on their use and suitability to neuroscience. This will be explained through
observations on two levels of brain connectivity—the neurons and the brainwaves
level. The relationship between the appropriate theory and methods will be also
given. On systemic level, the focus will be on neuronal oscillations, thus positioning
around and complementing the main topic of the book—biological oscillators. The
chapter will finish by outlook and some thoughts on the future developments and
uses of coupling function in neuroscience. However, before going into greater detail,
first the basics of what coupling functions are discussed briefly bellow.
11.1.1 Coupling Function Basics
The system setup to be studied is one of an interacting dynamical systems, with
the focus of coupled oscillators. Then, coupling functions describe the physical rule
specifying how the interactions occur and manifest. Because they are directly connected with the functional dependencies, coupling functions focus not only on if the
interactions exist, but more on how they appear and develop. For example, when
studying phase dynamics of coupled oscillators the magnitude of the phase coupling
function affects directly the oscillatory frequency and will describe how the oscillations are being accelerated or decelerated by the influence of the other oscillator.
Similarly, if one considers the amplitude dynamics of interacting dynamical systems,
the magnitude of coupling function will prescribe how the amplitude is increased or
decreased due to the interaction.