Slow vs Fast Variables in the Brain and "Hyper-Riccati Systems"

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 08/27/04 

Date: Fri, 27 Aug 2004 14:09:36 +0000 (UTC)

 From Osher Doctorow mdoctorow@comcast.net

COPYRIGHT NOTICE
Slow vs Fast Variables in the Brain and "Hyper-Riccati Systems"
Copyright by Owner Osher Doctorow Ph.D.
First Published 2004.

Debin Huang of Shanghai U. in "Stabilizing near-hyperbolic chaotic
systems and its potential applications in neuroscience," arXiv:nlin.
CD/0405014 v2 11 May 2004, addresses the control problem on near-
hyperbolic chaotic systems:

1) du/dt = g_u(u, v), dv/dt = rg_v(u, v)

where u is in R^(n1), v in R^(n2), 0 < r << 1, where the n1 u
components are fast and the n2 v components are slow, which Huang
tells us arise naturally in many scientific disciplines from
astrophysical models to biological cells. In the brain, the slow
components may represent a recorder or container storing acquired
knowledge and may be attached on some neurons, and they may be often
ignored but actually contribute to multipurpose flexibility because
related knowledge must be excited to respond to signals that enter
the brain. Huang points out that cognitive science indicates that
new knowledge displaces some old knowledge in the brain and this
reduces the freedom of brain topological structure. This is
analogous to the failure of stabilization of an inverted triple
pendulum when out-of-planar motions become very large. The brain
may be feeding back to control chaos in this situation as possibly
indicated by onset of regular bursts in groups of irregularly
bursting neurons whose individual properties differ. (1) and its
discrete versions are commonly used to model bursting, spiking,
chaotic phenomena in neuroscience.

Huang gives two examples: the Hindmarsh-Rose model neuron governed by:

2) dx1/dt = x2 - 3x1^2 - x1^3 - x3 + I
   dx2/dt = 1 - 5x1^2 - x2
   dx3/dt = -rx3 + 4r(x1 + 1.6)

with 0 < r << 1 with x1 the membrane potential of the neuron, x3 a
slow adaptation current, x2 a recovery variable, I is external curr-
ent, and this admits a chaotic attractor when r = .0012 and I =
3.281 from previous studies. A second example, the Rossler hyper-
chaos system, has:

3) dx1/dt = -x2 - x3
   dx2/dt = x1 + .25x2 + x4
   dx3/dt = 3 + x1x3
   dx4/dt = -.5x3 + .05x4

Both (2) and (3) contain "Riccati type terms" as well as cross-
linking terms from the other variables (other simultaneous equa-
tions). The x1^3 term in (2) goes beyond the Riccati type situa-
tion, although since x1 is a slow adaptation current it might be
that x1^3 approximately drops out with appropriate scaling or
that it is something specific to chaos.

Huang develops control methods for the above examples and for more
general situations. He points out that many chaotic phenomena seen
in actual systems are nonhyperbolic, so near-nonhyperbolicity is
important. Feedback strengths are adapted or updated according to
change in feedback proportional to xi^2 in R^n (i = 1, 2, ..., n)
where the proportionality constants are negative.

Osher Doctorow