Download (intermediate-range) elements in brain dynamics

Commentary on Nunez
1
Freeman & Kozma
Local-global interactions and the role of mesoscopic
(intermediate-range)
elements
in
brain
dynamics
Walter J. Freeman and Robert Kozma
Department of Molecular and Cell Biology, LSA 129
University of California at Berkeley, Berkeley, CA 94720-3200
Tel:
(510)-642 - 4 2 2 0
Fax: (510)-643-6791
Email: w f r e e m a n @ s o c r a t e s . b e r k e l e y . e d u
Commentary on the target article
“Toward a Qualitative Description of Large Scale Neocortical Dynamic
Function and EEG,” by Paul L. Nunez
to be published in Behavioral and Brain Sciences
10 September 1999.
Abstract 59 words; text 1,000 words.
Abstract
A unifing theory of spatiotemporal
brain dynamics
should
incorporate
multiple spatial and temporal scales. Between the microscopic (local) a n d
macroscopic
(global)
(intermediate-range)
corresponding
dynamics
components
elements
mathematical
proposed
by
Nunez,
mesoscopic
should be integral parts of models. T h e
formalism
requires
tools
and the use of aperiodic (chaotic) attractors.
of
nonlinear
Some relations
between local-mesoscopic and mesoscopic-global components are outlined.
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Commentary on Nunez
Linear
models,
transitions
in
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Freeman & Kozma
amplitude-dependent
neocortical
nonlinearities,
and p h a s e
dynamics
The work by Nunez is a valuable
contribution
to studies
on spatio-
temporal dynamics of brain functions, and it opens a great adventure into
the
yet
largely
undiscovered
electroencephalographic
macroscopic level.
territory
(EEG) measurements
of
interpretation
and brain
of
imaging at t h e
Nunez introduces a local-global model of neocortical
dynamics based on second order partial differential equations (PDEs) w i t h
given boundary conditions, which usually represent
periodic closure.
great advantage is that global and local effects co-exist in this model.
new material is clear and thought-provoking.
A
The
Qualitative considerations
regarding the circular causality between local and global parts of his m o d e l
and the consequences of such interactions at various temporal and spatial
scales,
including
dispersion
relationships,
are
the
highlights
of h i s
approach.
Nunez assumes linearity of the PDEs in his search for solutions. In t h e
Appendix he considers some of the effects of nonlinearities on his model.
These considerations, however, do not develop some important aspects of
nonlinearities that have crucial impact on the properties of brain dynamics
at
various
scales.
In
particular,
the
static
sigmoid
input-output
nonlinearity governed by the thresholds and refractory periods of n e u r o n s
introduces amplitude-dependent
rapid
and repetitive
phase
function (Freeman, 1992).
nonlinearities, which are crucial for t h e
transitions
that
characterize
normal
brain
These transitions lie well beyond the scope of
linear analysis, for which the greatest
2
virtue
is the determination
of
Commentary on Nunez
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Freeman & Kozma
stability by evaluation of Lyapunov exponents, after linearization of t h e
equations at operating points far from the point attractors determined b y
equilibria (Freeman, 1975).
PDEs suffer from the limitation that analytic kernels are usually r e q u i r e d
to get satisfactory solutions, and these are hard to come by in spatial
organizations
of brain
integrodifferential
activity.
equations
For this
or
their
reason
we
prefer
compartmentalized
to u s e
equivalent,
arrays of difference equations solved by numerical integration.
IDEs also
facilitate computations of chaotic attractors.
The role of mesoscopic elements in brain dynamics
For heuristic purposes we define an intermediate level of brain function
between
single neurons
or sparse
networks
of dendritic
bundles
and
cortical columns operating at a microscopic level, and those large b r a i n
parts
whose activities
are
observed
with
scalp EEG, fMRI, PET, a n d
comparable optical imaging techniques in humans.
We find it necessary t o
introduce the mesoscopic level to interpret data taken with 8x8 arrays of
electrodes
over cortical surfaces (Freeman, 1992; Barrie, Freeman
Lenhart, 1996).
and
These domains having diameters of 0.5 to 2 cm are m u c h
larger than columns, barrels and glomeruli, but they are at or below t h e
lower
limits
of spatial
resolution
by
macroscopic
methods.
Their
properties are determined by the self-organizing chaotic dynamics of local
populations of neurons, in which the delays introduced by the conduction
velocities of the axons of participating neurons provide the limitations o n
mesoscopic sizes and durations.
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Commentary on Nunez
Mesoscopic effects
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Freeman & Kozma
operating at spatial and temporal scales of 1 cm a n d
100 ms mediate between the two extremes
major lobes of the forebrain.
and
in duration
of single neurons
and t h e
They correspond in size to Brodmann's a r e a s
to psychophysical
events
that
compose
perceptions.
Mesoscopic effects provide a link between extreme local fragmentation a n d
global unity.
They change continually in space and time, requiring a v e r y
close relationship between dynamic events, e.g., EEG bursts, and the m e d i a
through which the propagation occurs.
This requires a nonlinear a p p r o a c h
(Skarda and Freeman, 1987; Freeman, 1992).
In physics the importance of
intermediate-range effects is well recognized (Kozma, 1998).
We illustrate the problem with Nunez' ocean wave analogy.
Propagation of
such waves leaves largely unchanged the properties of the water t h r o u g h
which transmission
takes place. In mathematics
order PDE formalizes this independence.
the
propagating
nonlinear
inseparable
signal
effects
are
and
the
essential
from the medium.
of a 2 n d
Extensive interactions b e t w e e n
neural
tissue,
in brains,
Neural
the linearity
however,
in which
tissues
are
the
reveal
that
dynamics
not passive
through which effects propagate as waves do in air and water.
is
media
The b r a i n
medium itself has an intimate relationship with the dynamics. There is a
continuous
excitation
in the
neural
regimes. Occasionally due to external
crosses a threshold.
tissue,
usually
in s u b - t h r e s h o l d
stimuli, for example, the activity
At that point the properties of the medium drastically
change in a phase transition to accommodate changed external conditions.
Mesoscopic elements are needed to introduce these nonlinearities, which
are the essence of adaptation through perception and learning.
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Commentary on Nunez
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Freeman & Kozma
Acknowledgment
This work was supported
by ONR grant N14-90-J-4054
and ARO MURI
grant DAAH04-96-1-0341.
References
Barrie,
J.M., Freeman,
W.J. and
Lenhart,
M. (1996)
discriminative training of spatial patterns
and phase in neocortex of rabbits.
Modulation
by
of gamma EEG a m p l t u d e
J. Neurophysiol. 76: 520-539.
Freeman, W.J. (1975) Mass Action in the Nervous System.
New York:
Academic Press.
Freeman, W.J. (1992) Tutorial on neurobiology - from single neuron
to
brain chaos, Int. J. Bifurcation & Chaos 2: 451-482.
Kozma, R. (1998)
Intermediate-range
coupling generates low-dimensional
attractors deeply in the chaotic region of one-dimensional
lattices,
Physics Letters A: 244, 85-91.
Skarda, C.A., Freeman W.J. (1987) How brains make chaos in order to m a k e
sense of the world, Behavioral & Brain Sciences, 10:161-195.
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