80
R. S. MacKay
˙
θ = μ + 1 − cos θ
(5.6)
˙
μ = 0.
Since μ is constant, one could think of it as an external parameter, but I wish to
consider it as a state variable because coupling from another neuron can make μ
change in time. It is best to think of μ as bounded, so the attracting cylinder can be
considered an invariant annulus.
They arise in modelling of “excitable neurons” whose frequency goes to zero as a
parameter (μ) is varied and then settle at a μ-dependent resting state, or in reverse go
from a resting state to large amplitude periodic spiking. An example is the MorrisLecar model [21], but it was [9] who identified the phenomenon as the unfolding of
a saddle-node on a cycle (I proposed this independently to physiologist H.Barlow in
the same year and then in 1991 proposed to C. Koch the extension to allow crossover
at a “saddle-node loop” [25] to the unfolding of a homoclinic orbit to a saddle). Thus
the non-autonomous version has an attracting NH annulus-cylinder.
I had an undergraduate student study networks of such neurons in 1989/90, with
the state μ of each neuron driven by the spiking of some others (with time-delay
kernels), which produced periodic bursting [19].
Two class I neurons coupled not too strongly have a NH attracting annulus×
annulus-cylinder. Generic bifurcation diagrams in the autonomous case were given
in [3]. The dynamics on it has attracting submanifolds of various types. The nonautonomous case has non-autonomous versions of them.
The theory of this paper applies just as well to class I neurons as to ordinary
oscillators, with the addition of the μ-direction for each class I neuron.
5.9 Extension to Chaotic Oscillators
The approach can also be extended to chaotic oscillators if they have an attracting
NH submanifold containing the attractor. For example, think of a Rössler attractor
[24], which is contained in a solid torus in R
3 . Then the non-autonomous system has
a solid-torus-cylinder. A Rössler attractor can be phase-locked to forcing, meaning
that the dynamics is attracted onto a disk-cylinder (a solid torus is the product of
a disk and a circle). This should be quite easy because the Rössler attractor was
observed to be nearly phase-coherent. I interpret that as meaning that there is a
cross-section with nearly constant return time (equivalently, for a given cross-section
there is a constant c > 0 and a function b : → R such that the return time
τ (x) = c + b( f (x)) − b(x), where f : → is the return map).
Synchronisation of chaotic attractors with NH cylinders of dimensions N 1 +
1, N 2 + 1 means there is a NH cylinder for the coupled system with dimension
less than N 1 + N 2 + 1.
Even better, the theory of NH submanifolds extends to NH laminations [15]. A
lamination is a topological space in which each point has a neighbourhood home-
R. S. MacKay
˙
θ = μ + 1 − cos θ
(5.6)
˙
μ = 0.
Since μ is constant, one could think of it as an external parameter, but I wish to
consider it as a state variable because coupling from another neuron can make μ
change in time. It is best to think of μ as bounded, so the attracting cylinder can be
considered an invariant annulus.
They arise in modelling of “excitable neurons” whose frequency goes to zero as a
parameter (μ) is varied and then settle at a μ-dependent resting state, or in reverse go
from a resting state to large amplitude periodic spiking. An example is the MorrisLecar model [21], but it was [9] who identified the phenomenon as the unfolding of
a saddle-node on a cycle (I proposed this independently to physiologist H.Barlow in
the same year and then in 1991 proposed to C. Koch the extension to allow crossover
at a “saddle-node loop” [25] to the unfolding of a homoclinic orbit to a saddle). Thus
the non-autonomous version has an attracting NH annulus-cylinder.
I had an undergraduate student study networks of such neurons in 1989/90, with
the state μ of each neuron driven by the spiking of some others (with time-delay
kernels), which produced periodic bursting [19].
Two class I neurons coupled not too strongly have a NH attracting annulus×
annulus-cylinder. Generic bifurcation diagrams in the autonomous case were given
in [3]. The dynamics on it has attracting submanifolds of various types. The nonautonomous case has non-autonomous versions of them.
The theory of this paper applies just as well to class I neurons as to ordinary
oscillators, with the addition of the μ-direction for each class I neuron.
5.9 Extension to Chaotic Oscillators
The approach can also be extended to chaotic oscillators if they have an attracting
NH submanifold containing the attractor. For example, think of a Rössler attractor
[24], which is contained in a solid torus in R
3 . Then the non-autonomous system has
a solid-torus-cylinder. A Rössler attractor can be phase-locked to forcing, meaning
that the dynamics is attracted onto a disk-cylinder (a solid torus is the product of
a disk and a circle). This should be quite easy because the Rössler attractor was
observed to be nearly phase-coherent. I interpret that as meaning that there is a
cross-section with nearly constant return time (equivalently, for a given cross-section
there is a constant c > 0 and a function b : → R such that the return time
τ (x) = c + b( f (x)) − b(x), where f : → is the return map).
Synchronisation of chaotic attractors with NH cylinders of dimensions N 1 +
1, N 2 + 1 means there is a NH cylinder for the coupled system with dimension
less than N 1 + N 2 + 1.
Even better, the theory of NH submanifolds extends to NH laminations [15]. A
lamination is a topological space in which each point has a neighbourhood home-