Statistical Mechanics of Neocortical Interactions. EEG Dispersion Relations
LESTER INGBER
Abstract—An approach is explicitly formulated to blend a local with a global theory to
investigate oscillatory neocortical firings, to determine the source and the information-
processing nature of the alpha rhythm. The basis of this optimism is founded on a statistical
mechanical theory of neocortical interactions which has had success in numerically detailing
properties of short-term-memory (STM) capacity at the mesoscopic scales of columnar
interactions, and which is consistent with other theory deriving similar dispersion relations at
the macroscopic scales of electroencephalographic (EEG) and magnetoencephalographic
(MEG) activity.
Manuscript received13March 1984. This project has been supported entirely by personal contributions
to Physical Studies Institute and to the University of California at San Diego Physical Studies Institute
agencyaccount through the Institute for Pure and Applied Physical Sciences.
The author is with Physical Studies Institute, Drawer W,Solana Beach, California 92075, and the Insti-
tute for Pure and Applied Physical Sciences, University of California at San Diego, La Jolla, California
92093.
Statistical Mechanics of Neocortical ... -2- Lester Ingber
I. OBJECTIVES
If the development of artificial intelligence systems is to benefit from knowledge of howneocortex
processes macroscopic patterned information at multiple spatial-temporal scales, then this knowledge
must be gained by at least testing viable theoretical formulations based on neocortical properties against
empirical data plausibly related to such processing. To w ards this end, in addition to the neuroscientific
relevance of this work, an approach is formulated to determine just what proportion of local and global
cortical circuitry givesrise to the alpha frequency. This has strong implications for other behavioral
studies which seek correlations between macroscopic EEG-MEG data and their underlying neuronal
mechanisms. This calculation is an essential bridge to understand howneuronal specificity provides
mechanisms underlying neuropsychological states, e.g., selective and global ‘‘attention.’’ The statistical
mechanical techniques employed are quite general [1]. E.g., theyhav e been applied to study nucleon-
nucleon velocity-dependent [2] Riemannian contributions to the binding energy of nuclear matter [3,4],
and to study the nonlinear dynamics of financial markets [5]. The former application of these
mathematical techniques yields insights into representing mesoscopic firing patterns by eigenfunctions of
aLagrangian; the latter application is particularly interesting in the context of describing neocortical
interactions more as a ‘‘neural throng’’reminiscent of social interactions [6], than as ‘‘hard-wired’’simple
local circuits.
II. BACKGROUND
Statistical Mechanics of Neocortical Interactions.Aseries of published studies, have demonstrated
that several scales of neocortical interactions can be consistently analyzed with the use of methods of
modern nonlinear nonequilibrium statistical mechanics [7-10]. A more extensive background for these
studies, with a fairly comprehensive set of references to other approaches, is detailed in these papers, but
Appendix A givesanoutline of these calculations. The formation, stability,and interaction of spatial-
temporal patterns of columnar firings nowcan be explicitly calculated, to test hypothesized mechanisms
Statistical Mechanics of Neocortical ... -3- Lester Ingber
relating to information processing. Adetailed scenario has been calculated of columnar coding of
external stimuli, short-term storage via hysteresis, and long-term storage via synaptic modification [8].
This development supports the possibility of parallel processing of local information via microscopic
circuits and of global patterned information via mesoscopic columnar mechanisms [11-13].
One of the most dramatic successes of this theory has been to produce a nonphenomenological
calculation of a macroscopic ‘‘observable’’from microscopic synaptic dynamics: the derivation of STM
capacity [9,10], i.e., the ‘‘rule of 7 ± 2.’’ [14-16] This theory demonstrates that empirical values of
chemical and electrical parameters of synaptic interactions establish several minima of the path-integral
Lagrangian as a function of excitatory and inhibitory columnar firings. The number of possible minima,
their time scales of hysteresis and probable reverberations, and their nearest-neighbor (NN) columnar
interactions are all consistent with well-established empirical rules of human STM capacity.Both the
nonlinear and statistical natures of the interactions developed by this theory are tested by the derivation of
STM capacity.Thus, aspects of conscious experience are derivedfrom neuronal firing patterns, using
modern methods of nonlinear nonequilibrium statistical mechanics to develop realistic explicit synaptic
interactions. This result at least partially justifies the process by which microscopic activity has been
statistically developed to describe mesoscopic and macroscopic activity.For example, although future
refinements can takeinto consideration state-dependent complexinteractions among intra-neuronal
components [17,18], the assumptions outlined in Appendix A appear to suffice to detail STM capacity.
In the wakeofthis interesting result obtained for STM capacity,this paper speculates that this
theory also be applied to macroscopic EEG-MEG phenomena. As outlined in Appendix A, microscopic
neuronal synaptic interactions, consistent with anatomical observations, are first spatially averaged over
minicolumnar afferent and macrocolumnar efferent domains, defining a physiological ‘‘mesocolumn.’’
These spatially ordered domains, ∼10
−2
cm, retain intimate contact with the original physical synaptic
parameters, are consistent with observed columnar physiology,and are a suitable substrate for
macroscopic spatial-temporal regions, ∼ tens of cm, described by a path-integral Lagrangian formalism of
coupled excitatory-inhibitory spatial-temporal firing states. NN interactions among mesocolumns support
regions of alternating columnar structures. Long-ranged influences from extrinsic and inter-regional
Statistical Mechanics of Neocortical ... -4- Lester Ingber
afferents drive these short-ranged interactions, giving rise to columnar mechanisms affecting macroscopic
activity.Within neighborhoods of established most-probable stationary firing minima determined by the
Euler-Lagrange equations, the linearized field equations give rise to a dispersion relation relating firing
frequencies and spatial wav e ve ctors, i.e., exhibiting properties of classical wav e phenomena.
Origins of Time Dependencies of Scalp EEG.Other researchers have dev e loped quite different
approaches to investigating macroscopic neocortical activity,e.g., stressing that systematics of alpha
rhythm of EEG can be modeled by resonant modes of macroscopic dipole-layered firing patterns of
neocortex[19-22]. These resonances, in linearized coupled excitatory-inhibitory spatial-temporal integral
equations describing dipole-layered sources, give rise to a macroscopic dispersion relation relating firing
frequencies to spatial wav e ve ctors, consistent with empirical observations. As demonstrated in Appendix
A, typical synaptic parameters result in mesoscopic dispersion relations consistent with these macroscopic
dispersion relations.
While manyother investigators also accept dipole layers to model EEG activity,atleast to the
extent of recognizing activity perpendicular to laminae, theyalso demonstrate that there are respectable
candidates for mechanisms that might fundamentally be responsible for macroscopic activity,other than
those proposed here which detail synaptic dynamics of mesocolumnar interactions [23-30]. For example,
giventhe present lack of empirical knowledge, it is possible to formulate macroscopic neocortical activity
in terms of statistics of either membrane or synaptic microscopic neuronal activities, albeit that these two
are obviously empirically dependent on each other [31]. Therefore, the results derivedinAppendix A
might be interpreted either as suggesting that mesocolumnar activity instigates macroscopic activity,or
rather as suggesting that mesocolumnar activity strongly interacts with ongoing macroscopic activity
which is instigated or sustained by other mechanisms.
III. PRESENT ISSUES
The twoapproaches outlined above inSection II, i.e., local mesocolumnar versus global non-
mesocolumnar,giv e rise to important alternative conjectures: (1) Is the alpha rhythm a global resonance
Statistical Mechanics of Neocortical ... -5- Lester Ingber
of primarily long-ranged cortical interactions? If so, can relatively short-ranged local firing patterns
effectively modulate this frequencyand its harmonics, to enhance their information processing across
macroscopic regions? (2) Or,does global circuitry imply boundary conditions on collective mesoscopic
states of local firing patterns, and is the alpha rhythm a manifestation of these collective local firings? (3)
Or,isthe truth some combination of (1) and (2) above?
Using this mesocolumnar approach, within their empirical ranges, sets of synaptic parameters can
be examined to determine local dispersion relations [8] similar to those obtained from the global
dispersion relations [19]. The results of such a calculation, outlined in Appendix A, clearly demonstrate
that this local theory predicts alpha frequencies and spatial wav e numbers compatible with those predicted
by the global resonance model. Forexample, the possibility of generating alpha rhythm from multiple
mechanisms at multiple scales of interactions, e.g., as discussed above,may account for its presence
under manyphysiological conditions [19]. Note that these results, similar to results derivedfor STM
capacity [9,10], are not obtained by ‘‘fitting’’theoretical parameters mocking neuronal mechanisms to
empirical data. Rather,these results are obtained by taking reasonable synaptic parameters, developing
the statistical mechanics of neocortical interactions, and then discovering that indeed theyare consistent
with the empirical macroscopic data. Furthermore, this theory allows the local and global approaches to
complement each other at a common levelofformal analysis — i.e., the ‘‘equations of motion’’
analogous to
Σ
(forces) = d(momentum) /dt describing mechanical systems: A more detailed calculation
will include contributions from most probable states of the stochastically averaged microscopic system in
the local approach, i.e., the linearized Euler-Lagrange equations, and will include contributions from
normal modes of the linearized macroscopic system in the global approach, i.e., resonances of the dipole
field equations.
It is plausible that studies of the source of the alpha rhythm will give direct insight into related
mechanisms underlying evokedpotentials. In anycase, initially a study the Euler-Lagrange variational
equations can determine just what kinds of spatial-temporal structures can be supported by the
mesocolumnar system, giveninitial driving forces that match/mismatch firing eigenfunctions (patterns of
columnar firings) currently possessed by a givenset of synaptic parameters, and under conditions of