11 Coupling Functions in Neuroscience
177
First we consider two coupled dynamical systems given in the following general
form:
˙
x = f 1 (x) + g 1 (x, y)
˙
y = f 2 (y) + g 2 (x, y),
(11.1)
where the functions f 1 (x) and f 2 (y) describe the inner dynamics, while g 1 (x, y)
and g 2 (x, y) describe the coupling functions in the state space. Then, given that
the two dynamical systems are oscillators, and under the assumption that they are
weakly nonlinear and weakly coupled, one can apply the phase reduction theory [37,
45, 53]. This yields simplified approximative systems where the full (at least two
dimensional) state space domain is reduced to a one dimensional phase dynamics
domain:
˙
φ 1 = ω 1 + q 1 (φ 2 , φ 1 )
˙
φ 2 = ω 2 + q 2 (φ 1 , φ 2 ),
(11.2)
where φ 1 , φ 2 are the phase variables of the oscillators, ω 1 , ω 2 are their natural frequencies, and q 1 (φ 2 , φ 1 ) and q 2 (φ 1 , φ 2 ) are the coupling functions in phase dynamics
domain. For example, in the Kuramoto model [37] they were prescribed to be sine
functions from the phase differences:
˙
φ 1 = ω 1 + ε 1 sin(φ 2 − φ 1 )
˙
φ 2 = ω 2 + ε 2 sin(φ 1 − φ 2 ),
(11.3)
where ε 1 , ε 2 are the coupling strength parameters. Apart from this example of sinusoidal form, the coupling functions q 1 (φ 2 , φ 1 ) and q 2 (φ 1 , φ 2 ) can have very different
and more general functional form, including a decomposition on a Fourier series.
Given in the phase dynamics like this Eq. 11.2, the coupling functions q 1 (φ 2 , φ 1 ) and
q 2 (φ 1 , φ 2 ) are additive to the frequency parameters ω 1 , ω 2 , meaning that their higher
or lower values will lead to acceleration or deceleration of the affected oscillations,
respectively.
Coupling function can be described in terms of its strength and form. The coupling
strength is a relatively well-studied quantity, and there are many statistical methods
which detect measures proportional to it (e.g., the mutual-information based measures, transfer entropy and Granger causality). It is the functional form of the coupling function, however, that has provided a new dimension and perspective probing
directly the mechanisms of the interactions. Where, the mechanism is defined by
the functional form that gives the rule and process through which the input values
are translated into output values i.e. for the interactions it prescribes how the input
influence from one system is translated into the output effect on the affected or the
coupled system.
In this way a coupling function can describe the qualitative transitions between
distinct states of the systems e.g., routes into and out of synchronization, oscillation
177
First we consider two coupled dynamical systems given in the following general
form:
˙
x = f 1 (x) + g 1 (x, y)
˙
y = f 2 (y) + g 2 (x, y),
(11.1)
where the functions f 1 (x) and f 2 (y) describe the inner dynamics, while g 1 (x, y)
and g 2 (x, y) describe the coupling functions in the state space. Then, given that
the two dynamical systems are oscillators, and under the assumption that they are
weakly nonlinear and weakly coupled, one can apply the phase reduction theory [37,
45, 53]. This yields simplified approximative systems where the full (at least two
dimensional) state space domain is reduced to a one dimensional phase dynamics
domain:
˙
φ 1 = ω 1 + q 1 (φ 2 , φ 1 )
˙
φ 2 = ω 2 + q 2 (φ 1 , φ 2 ),
(11.2)
where φ 1 , φ 2 are the phase variables of the oscillators, ω 1 , ω 2 are their natural frequencies, and q 1 (φ 2 , φ 1 ) and q 2 (φ 1 , φ 2 ) are the coupling functions in phase dynamics
domain. For example, in the Kuramoto model [37] they were prescribed to be sine
functions from the phase differences:
˙
φ 1 = ω 1 + ε 1 sin(φ 2 − φ 1 )
˙
φ 2 = ω 2 + ε 2 sin(φ 1 − φ 2 ),
(11.3)
where ε 1 , ε 2 are the coupling strength parameters. Apart from this example of sinusoidal form, the coupling functions q 1 (φ 2 , φ 1 ) and q 2 (φ 1 , φ 2 ) can have very different
and more general functional form, including a decomposition on a Fourier series.
Given in the phase dynamics like this Eq. 11.2, the coupling functions q 1 (φ 2 , φ 1 ) and
q 2 (φ 1 , φ 2 ) are additive to the frequency parameters ω 1 , ω 2 , meaning that their higher
or lower values will lead to acceleration or deceleration of the affected oscillations,
respectively.
Coupling function can be described in terms of its strength and form. The coupling
strength is a relatively well-studied quantity, and there are many statistical methods
which detect measures proportional to it (e.g., the mutual-information based measures, transfer entropy and Granger causality). It is the functional form of the coupling function, however, that has provided a new dimension and perspective probing
directly the mechanisms of the interactions. Where, the mechanism is defined by
the functional form that gives the rule and process through which the input values
are translated into output values i.e. for the interactions it prescribes how the input
influence from one system is translated into the output effect on the affected or the
coupled system.
In this way a coupling function can describe the qualitative transitions between
distinct states of the systems e.g., routes into and out of synchronization, oscillation