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T. Stankovski
Importantly, the neurons are highly interconnected forming a complex brain network. Their interactions give rise to different neural states and functions. In terms
of system interactions, such brain interactions could lead to qualitative transitions
like synchronization and clustering, on the whole or part of the brain network. When
observing the neuronal models as dynamical systems, the mechanisms of the interactions are defined by the neuronal coupling functions. On this level, coupling functions
have been studied extensively, although more in an indirect way through the neuronal phase response curve (PRC) [2, 10]. Namely, coupling function is a convolution
between two functions, the phase response curve and the perturbation function [37]
i.e. one function of how an oscillator responds to perturbations and the second function defining the perturbations from the second oscillator, respectively. There are
generally two types of such response curves, type I with all positive, and type II with
positive and negative values. Different types of phase response curves were studied
(especially theoretically) forming different types of neuronal models [6, 11, 48]. The
phase response curves are typically defined for weakly coupled units [45, 53].
An important feature of the neuronal oscillations are that they are excitable and
have non-smooth spike-like trajectories. Such dynamics of the neuronal oscillations
are highly nonlinear. For many applications, the neuronal activity is studied completely through the timing of the spike events [17]. In general, such spike-like oscillations act similar as a delta function, hence the phase response curves will have a
similar delta function-like form [12]. This can have direct effect when observing the
coupling function which can be a convolution between the a delta-like functions.
In terms of methods for neuronal coupling functions, a number of methods exist
for reconstructing the neuronal phase response curves and the associate coupling
functions [16, 77]. However, there are many open problems on this task and many
applications on different types of signals from interacting neurons are yet to be
resolved.
11.2.2 Coupling Functions on Brainwave Level
Studying some kind of property of a large number of neurons at once, as a whole
or region of the brain, scales up the observation on higher level. In this way the
resultant measurement of the brain, or region of the brain, is in a way some kind
of mean field, a sum of all the functional activities of the individual neurons in a
group, ensemble or network. For example such measurements include the neural
EEG, iEEG, NIRS, MRI, CT and PET, which measure different characteristics like
the electrical activity, the hemodynamic activity, the perfusion etc. of the whole brain
or on specific spatially localized brain regions.
Arguably, the most used high level observable is the EEG. Electroencephalography (EEG) is a noninvasive electrophysiological monitoring method to record
electrical activity of the brain. EEG measures voltage fluctuations resulting from
ionic current within the neurons of the brain [46]. EEG measures electrical activity
over a period of time, usually recorded from multiple electrodes placed on the scalp
T. Stankovski
Importantly, the neurons are highly interconnected forming a complex brain network. Their interactions give rise to different neural states and functions. In terms
of system interactions, such brain interactions could lead to qualitative transitions
like synchronization and clustering, on the whole or part of the brain network. When
observing the neuronal models as dynamical systems, the mechanisms of the interactions are defined by the neuronal coupling functions. On this level, coupling functions
have been studied extensively, although more in an indirect way through the neuronal phase response curve (PRC) [2, 10]. Namely, coupling function is a convolution
between two functions, the phase response curve and the perturbation function [37]
i.e. one function of how an oscillator responds to perturbations and the second function defining the perturbations from the second oscillator, respectively. There are
generally two types of such response curves, type I with all positive, and type II with
positive and negative values. Different types of phase response curves were studied
(especially theoretically) forming different types of neuronal models [6, 11, 48]. The
phase response curves are typically defined for weakly coupled units [45, 53].
An important feature of the neuronal oscillations are that they are excitable and
have non-smooth spike-like trajectories. Such dynamics of the neuronal oscillations
are highly nonlinear. For many applications, the neuronal activity is studied completely through the timing of the spike events [17]. In general, such spike-like oscillations act similar as a delta function, hence the phase response curves will have a
similar delta function-like form [12]. This can have direct effect when observing the
coupling function which can be a convolution between the a delta-like functions.
In terms of methods for neuronal coupling functions, a number of methods exist
for reconstructing the neuronal phase response curves and the associate coupling
functions [16, 77]. However, there are many open problems on this task and many
applications on different types of signals from interacting neurons are yet to be
resolved.
11.2.2 Coupling Functions on Brainwave Level
Studying some kind of property of a large number of neurons at once, as a whole
or region of the brain, scales up the observation on higher level. In this way the
resultant measurement of the brain, or region of the brain, is in a way some kind
of mean field, a sum of all the functional activities of the individual neurons in a
group, ensemble or network. For example such measurements include the neural
EEG, iEEG, NIRS, MRI, CT and PET, which measure different characteristics like
the electrical activity, the hemodynamic activity, the perfusion etc. of the whole brain
or on specific spatially localized brain regions.
Arguably, the most used high level observable is the EEG. Electroencephalography (EEG) is a noninvasive electrophysiological monitoring method to record
electrical activity of the brain. EEG measures voltage fluctuations resulting from
ionic current within the neurons of the brain [46]. EEG measures electrical activity
over a period of time, usually recorded from multiple electrodes placed on the scalp