184
T. Stankovski
The large populations of interacting neurons, in form of ensembles and networks,
have been studied extensively in theory. The celebrated Kuramoto model [36, 37]
has been exploited in particular. It is a model of large population of phase oscillators, one which has an exact analytic solution for the synchronization state of the
whole ensemble. The coupling functions is a simple sine function of the phase difference. Kuramoto discussed that this coupling function is not very physical, however
his interest was in finding an analytically solvable model. The Kuramoto model
has been particularly popular in neuroscience with its ability to describe analytically the synchronous states of large populations of neurons [1, 5, 52]. Other two
recently introduced approaches, known as the Ott and Antonsen [50] and Watanabe
and Strogatz [83] reductions, provide reduced model equations that exactly describe
the collective dynamics for each subpopulation in the neural oscillator network via
few collective variables only. A recent review provides a comprehensive and updated
overview on the topic [4]. The theoretical studies on the large-scale brainwave interactions are often performed through the common framework of two or few coupled
oscillatory systems [55].
To infer coupling functions from data one needs to employ methods based on
dynamical inference. These are class of methods which can reconstruct a model of
ordinary or stochastic differential equations from data. The coupling functions are
integral part of such models. In this chapter examples were shown from the use of
a specific method based on dynamical Bayesian inference [64, 66, 67]; however
any other method based on dynamical inference (often referred to also as dynamic
modelling or dynamic filtering) can also be used [15, 33, 35, 39, 80]. The differences
between the results of these methods in terms of the coupling functions are minor and
not qualitatively different. Often, there is a need for coupling functions to be inferred
from networks of interacting systems, and several methods have been applied in this
way [39, 54, 72]. In neuroscience, such methods have been used mainly on two
to several brainwave oscillation systems, and it has been argued that the precision
and feasibility are exponentially reduced as the number of systems increases and
it is recommended not to go beyond N > 10 [57]. For this reason and due to the
exponentially increasing demand for larger number of systems, there are not many
effective methods for inference of coupling functions in low-level large populations
of neuronal interactions.
In terms of methodology and analysis, few other aspects are important when
analysing coupling functions. One is that once coupling functions are inferred they
give the qualitative mechanisms but for any quantitative evaluations and comparisons
(for example in a multisubject neuroscience study) one can conduct coupling function
analysis i.e. it can calculate the coupling strength and the similarity of the form of
coupling function [35, 69, 79]. Also, of paramount importance is to validate if the
inferred coupling functions are statistically significant in respect of surrogate time
series [38, 63]. Usually one tests whether the strengths of the coupling functions
are significantly higher than the coupling strengths of a large number of randomized
T. Stankovski
The large populations of interacting neurons, in form of ensembles and networks,
have been studied extensively in theory. The celebrated Kuramoto model [36, 37]
has been exploited in particular. It is a model of large population of phase oscillators, one which has an exact analytic solution for the synchronization state of the
whole ensemble. The coupling functions is a simple sine function of the phase difference. Kuramoto discussed that this coupling function is not very physical, however
his interest was in finding an analytically solvable model. The Kuramoto model
has been particularly popular in neuroscience with its ability to describe analytically the synchronous states of large populations of neurons [1, 5, 52]. Other two
recently introduced approaches, known as the Ott and Antonsen [50] and Watanabe
and Strogatz [83] reductions, provide reduced model equations that exactly describe
the collective dynamics for each subpopulation in the neural oscillator network via
few collective variables only. A recent review provides a comprehensive and updated
overview on the topic [4]. The theoretical studies on the large-scale brainwave interactions are often performed through the common framework of two or few coupled
oscillatory systems [55].
To infer coupling functions from data one needs to employ methods based on
dynamical inference. These are class of methods which can reconstruct a model of
ordinary or stochastic differential equations from data. The coupling functions are
integral part of such models. In this chapter examples were shown from the use of
a specific method based on dynamical Bayesian inference [64, 66, 67]; however
any other method based on dynamical inference (often referred to also as dynamic
modelling or dynamic filtering) can also be used [15, 33, 35, 39, 80]. The differences
between the results of these methods in terms of the coupling functions are minor and
not qualitatively different. Often, there is a need for coupling functions to be inferred
from networks of interacting systems, and several methods have been applied in this
way [39, 54, 72]. In neuroscience, such methods have been used mainly on two
to several brainwave oscillation systems, and it has been argued that the precision
and feasibility are exponentially reduced as the number of systems increases and
it is recommended not to go beyond N > 10 [57]. For this reason and due to the
exponentially increasing demand for larger number of systems, there are not many
effective methods for inference of coupling functions in low-level large populations
of neuronal interactions.
In terms of methodology and analysis, few other aspects are important when
analysing coupling functions. One is that once coupling functions are inferred they
give the qualitative mechanisms but for any quantitative evaluations and comparisons
(for example in a multisubject neuroscience study) one can conduct coupling function
analysis i.e. it can calculate the coupling strength and the similarity of the form of
coupling function [35, 69, 79]. Also, of paramount importance is to validate if the
inferred coupling functions are statistically significant in respect of surrogate time
series [38, 63]. Usually one tests whether the strengths of the coupling functions
are significantly higher than the coupling strengths of a large number of randomized