Below is the Abstract and Conclusion of a 100+ page paper submitted to
Physical Review. Comments are welcome, and I will fulfill reprint requests
after publication of this paper.
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Statistical mechanics of neocortical interactions:
A scaling paradigm applied to electroencephalography
Lester Ingber
Science Transfer Corporation, P.O. Box 857, McLean, VA 22101
A series of papers has developed a statistical mechanics of
neocortical interactions (SMNI), deriving aggregate behavior of
experimentally observed columns of neurons from statistical
electrical-chemical properties of synaptic interactions. While
not useful to yield insights at the single neuron level, SMNI has
demonstrated its capability in describing large-scale properties
of short-term memory and electroencephalographic (EEG) systemat-
ics. The necessity of including nonlinear and stochastic struc-
tures in this development has been stressed. In this paper, a
more stringent test is placed on SMNI: The algebraic and numeri-
cal algorithms previously developed in this and similar systems
are brought to bear to fit large sets of EEG/evoked potential
data being collected to investigate genetic predispositions to
alcoholism and to extract brain "signatures" of short-term
memory. It is demonstrated that SMNI can indeed fit this data
within experimentally observed ranges of its underlying
neuronal-synaptic parameters, and use the quantitative modeling
results to examine physical neocortical mechanisms to discrim-
inate between high-risk and low-risk populations genetically
predisposed to alcoholism. Since this first study is a control
to include relatively long time epochs, similar to earlier
attempts to establish such correlations, this discrimination is
inconclusive. However, the model is shown to be consistent with
EEG data and with neocortical mechanisms previously published
using this approach. This paper explicitly identifies similar
nonlinear stochastic mechanisms of interaction at the
microscopic-neuronal, mesoscopic-columnar and macroscopic-
regional scales of neocortical interactions. These results give
strong quantitative support for an accurate intuitive picture,
portraying neocortical interactions as having common
algebraic/physics mechanisms that scale across quite disparate
spatial scales and functional/behavioral phenomena, i.e.,
describing interactions among neurons, columns of neurons, and
regional masses of neurons.
PACS Nos.: 87.10.+e, 05.40.+j, 02.50.+s, 02.70.+d
VI. CONCLUSION
We have outlined in some detail a reasonable approach to
extract more ``signal'' out of the ``noise'' in EEG data, in
terms of physical dynamical variables, than by merely performing
regression statistical analyses on collateral variables. To
learn more about complex systems, we inevitably must form func-
tional models to represent huge sets of data. Indeed, modeling
phenomena is as much a cornerstone of 20th century science as is
collection of empirical data.
We have been able to fit these sets of EEG data, using
parameters either set to experimentally observed values, or being
fitted within experimentally observed values. The ranges of
columnar firings are consistent with a centering mechanism
derived in earlier papers.
The ability to fit data to these particular SMNI functional
forms goes beyond nonlinear statistical modeling of data. The
plausibility of the SMNI model, as emphasized in this and previ-
ous SMNI papers, as spanning several important neuroscientific
phenomena, suggests that the fitted functional forms may yet help
to explicate some underlying biophysical mechanisms responsible
for the normal and abnormal behavioral states being investigated,
e.g., excitatory and/or inhibitory influences, background elec-
tromagnetic influences from nearby firing states (by using SMNI
synaptic conductivity parameter in the fits).
There is much more work to be done. We have not yet
addressed the "renormalization" issues raised, based on the
nature of EEG data collection, and which are amenable to this
framework. While the fitting of these distributions certainly
compacts the experimental data onto a reasonable algebraic model,
a prime task of most physical theory, in order to be useful to
clinicians (and therefore to give more feedback to theory) more
data reduction must be performed. We are experimenting with
path-integral calculations and some methods of "scientific visu-
alization" to determine what minimal, or at least small, set of
"signatures" might suffice to be faithful to the data yet useful
to clinicians. We also are examining the gains that might be
made by putting these codes onto a parallel processor, which
might enable real-time diagnoses based on non-invasive EEG
recordings.
In order to detail such a model of EEG phenomena we found it
useful to seek guidance from ``top-down'' models, e.g., the non-
linear string model representing nonlinear dipoles of neuronal
columnar activity. In order to construct a more detailed
``bottom-up'' model that could give us reasonable algebraic func-
tions with physical parameters to be fit by data, we then needed
to bring together a wealth of empirical data and modern tech-
niques of mathematical physics across multiple scales of neocort-
ical activity. At each of these scales, we had to derive and
establish reasonable procedures and sub-models for climbing from
scale to scale. Each of these sub-models could then be tested
against some experimental data to see if we were on the right
track. For example, at the mesoscopic scale we checked con-
sistency of SMNI with known aspects of visual and auditory
short-term memory; at the macroscopic scale we checked con-
sistency with known aspects of EEG and propagation of information
across neocortex. Here, we have demonstrated that the currently
accepted dipole EEG model can be derived as the Euler-Lagrange
equations of an electric-potential Lagrangian.
The theoretical and experimental importance of specific
scaling of interactions in neocortex has been quantitatively
demonstrated: We have shown that the explicit algebraic form of
the probability distribution for mesoscopic columnar interactions
is driven by a nonlinear threshold factor of the same form taken
to describe microscopic neuronal interactions, in terms of
electrical-chemical synaptic and neuronal parameters all lying
within their experimentally observed ranges; these threshold fac-
tors largely determine the nature of the drifts and diffusions of
the system. This mesoscopic probability distribution has suc-
cessfully described STM phenomena and, when used as a basis to
derive most likely trajectories using the Euler-Lagrange varia-
tional equations, it also has described the systematics of EEG
phenomena. In this paper, we have taken the mesoscopic form of
the full probability distribution more seriously for macroscopic
interactions, deriving macroscopic drifts and diffusions linearly
related to sums of their (nonlinear) mesoscopic counterparts,
scaling its variables to describe interactions among regional
interactions correlated with observed electrical activities meas-
ured by electrode recordings of scalp EEG, with apparent success.
These results give strong quantitative support for an accurate
intuitive picture, portraying neocortical interactions as having
common algebraic/physics mechanisms that scale across quite
disparate spatial scales and functional/behavioral phenomena,
i.e., describing interactions among neurons, columns of neurons,
and regional masses of neurons.
It seems reasonable to speculate on the evolutionary des
