Freeman's mass action

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Author: Dr. Walter J. Freeman, University of California, Berkeley, California
Author: Dr. Robert Kozma, Computational NeuroDynamics Lab, University of Memphis, TN

The phrase Freeman's Mass Action (FMA) designates the neural forces exerted by self-organized neural populations while constructing intentional action and perception. FMA explains how the force exerted by a brain population within itself condenses its spontaneous background noise into signals, which are patterns that are carried by condensates called wave packets. The patterns guide the operations that recall memories and execute the action-perception cycle.

1 Introduction
- 1.1 Box: Neural Networks versus Neurodynamics: A paradigm shift
2 Mesoscopic state variables: The wave packet
3 Mesoscopic phases, dissipative structures, and order parameters
4 See also: Advances in FMA theory, experimental analysis and applications
5 Recommended Reading
6 Scholarpedia References
7 References

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Introduction

Brains sustain spontaneous neural activity at and near Self-organized Criticality at extraordinary metabolic cost. The energy transformed and dissipated appears in diverse forms. These include electric and magnetic fields of potential, and electrochemical gradients that exert forces at synapses and trigger zones. Neural forces act over distributed distances by transmission and reception of spatiotemporal patterns of action potentials. The motor systems embedded through much of the central nervous system transduce the patterns of the neural forces into controlled release of muscular energy in patterns of behavior, which are fed back through sensory systems by exteroception, proprioception and interoception. Thus FMA is energy in many forms, all derived from metabolism. It is exerted by neural populations that act in the brain on neurons in other populations, in the body through the sensorimotor systems, and in the environment through the body.

FMA theory treats the brain as an open thermodynamic system, not as a logical neural network, information processor, or chemical system. The brain is tightly constrained in temperature, pressure, volume and mass, but its arteriovenous circulation supplies metabolic energy and dissipates waste heat and entropy. The brain requires these energy sources and sinks to construct complex, evolving, spatiotemporal patterns of neural activity within itself and, through the body, of behavior. Neurons, brains and bodies conform to the first and second Laws of Thermodynamics in energy transformations, while the creation of information and knowledge by the organization of dissipative energy patterns in brains transcends these constraints. This transcendence poses the central challenge for the theory of FMA [Freeman, 1975, Ch. 1].

The term FMA has a triple meaning, referring historically to the 19th century Law of Mass Action in chemistry; to Lashley'spluripotentiality of cortical function revealed by stimulation and ablation (see Pribram Holonomic brain theory); and to the capacity of neural populations to synchronize their oscillations during perception transiently and repeatedly in continuous fields [Freeman, 1975]. Phenomenologists [Dreyfus, 2008] have provided the philosophical context in which to describe how neurons by collective action execute intentional behaviors through the action-perception cycle (Intentionality). Ilya Prigogine in studies of nonequilibrium thermodynamics introduced the term dissipative structure to denote a large-scale spatiotemporal pattern of energy in matter, which self-organizes by nonlinear interactions with diffusion-dependent delays among molecules, and which feeds on energy. Hermann Haken in studies of lasers introduced the concept of the order parameter [Sethna, 2008, Sect. 9.2] by which to evaluate the intensity of the collective force among particles (here neurons), which interact at intensities above a threshold of the rate of energy dissipation, and which organize a field of action that constrains the particles sustaining it in circular causality. The use of these basic concepts broadly grounds FMA theory in neuroscience, psychology, physics, engineering, and philosophy.

Though physically confined in the skull, the brain is an infinite dimensional dynamical system. Neuroscientists measure its anatomical structures and physiological operations and construct models that represent their knowledge of brain dynamics in finite state spaces. The dimensions of functional measurement spaces are constrained by the kinds and numbers of their sensors giving brain signals as multiple time series. The techniques for structural and functional measurements impose several levels for describing the signals as state variables. The three levels relevant here are micro-, meso-, macro. Histology using stains that reveal cellular architectures combined with microelectrode recordings give microscopic state variables that represent dendritic currents and axonal pulse trains of neurons singly and in discrete networks. Brain structural imaging in comparative gross anatomy, when combined with EEG, MEG, functional MRI, BOLD, and related techniques for estimating the metabolic energy that sustains brain activity, give macroscopic state variables. Measures of cortical and subcortical multicellular architectures combined with recordings of brain fields of electric potentials at cortical surfaces (electrocorticograms [ECoG] (Fig. 1) and local field potential [LFP]) in cortical depths give mesoscopic state variables Mesoscopic Brain Dynamics.

The aim of FMA theory is correlation of all types of neural state variables with measurements of intentional behavior. The correlation requires reducing the infinite complexity of behavior to a finite state space and then defining the system state variables that co-vary with neural measurements at each level. FMA theory is a framework in which relations among state variables across levels and disciplines can be described, modeled, and explained using the concepts of dissipative structures (the forms of FMA) and order parameters (measures and indices for the intensities of collective, condensing pressures of FMA).

At the microscopic level each action potential releases a neurochemical transmitter substance that diffuses down a concentration gradient across a synaptic cleft and exerts a microscopic force that is directed: topologically by the synapse from one neuron to another, and geometrically by the locations and orientations of the axon and dendrite, e.g., parallel or perpendicular to the cortical surface. The neurochemical substance activates an electromotive force that drives ionic current from synapse to trigger zone and return in a closed loop. The electrochemical force is also directed: topologically from synaptic input to axonal output of each neuron, and geometrically in accord with the orientation of a current dipole set by the locations of the sites of entry and exit of ionic current (sources and sinks) across the neural membrane. The source-sink geometry determines the dendritic potential field. At the mesoscopic level the aggregate force of billions of diverse synapses and trigger zones loses microscopic geometry over the multiple types of actions, so it lacks well defined direction and acts as an isotropic pressure. The cumulative effect exerted by all ions and chemicals at each point in the two surface dimensions of cortex is reflected in the gradient of the electric potential recorded in the electrocorticogram (ECoG). The variations of potential reveal mesoscopic spatial and temporal gradients in FMA. These variations are the targets for finding neural correlates of behavior. The identities of the microscopic substances and ions that exert the pressures are irrelevant at the mesoscopic level; only the net concentrations and charge densities matter, as they appear in the ECoG patterns.

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Box: Neural Networks versus Neurodynamics: A paradigm shift

A paradigm shift is more than a new theory; it encompasses new models, techniques, classic experiments, and rules of evidence. A comparable shift occurred late in the 19th century, when most physicists conceived electricity and magnetism in Newtonian terms as forces exerted by point charges acting at a distance instantaneously on other point charges. Michael Faraday reconceived the forces as fields, and James Clark Maxwell devised new mathematics to describe the fields, which led to discovery of the electromagnetic spectrum [Arianrhod, 2003; Barrett, 2008]. Neural networks are Newtonian models, because they treat microscopic neural pulses as point processes at trigger zones and synapses, often instantaneous [Engel et al., 1991] with zero lag in transmission. FMA is Maxwellian in conceiving mesoscopic neural activity as a continuum of energy density with finite transmission velocities. The wave packet defined here is conceived as a vector field that is comparable to Maxwell’s A field. Just as Maxwell's equations subsume the Newtonian laws of Faraday, Ampére, Coulomb and Oersted, neurodynamics [Freeman, 2001] in FMA theory subsumes the point processes of neural operations in neural networks on pulse trains and postsynaptic potentials at microscopic entry into and exit from the mesoscopic neural operations on dendritic wave densities and axonal pulse densities.

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Mesoscopic state variables: The wave packet

The synaptic interactions that support FMA at the underlying microscopic level operate in the cortical “neuropil” (the dense carpet of axons, dendrites, capillaries and cell bodies forming the outer shell of the cerebrum). The packing density of neurons ( $10^5/mm^3$ ) and synapses ( $10^9/mm^3$ ) can only be roughly estimated; each neuron synapses with ~ $10^4$ others, yet sparsely with only $1\%$ of the neurons within its dendritic arbor [Braitenberg and Schüz, 1998]. It is the processes of spatial and temporal summation and smoothing of microscopic activities of neurons and glia that create the mesoscopic FMA and provide the topological foundation for defining the state variables and order parameters. FMA theory holds that smoothing by dense serial synaptic interaction and transmission by divergent-convergent pathways delete extraneous microscopic information and noise while preserving that fraction of transmission which constitutes mesoscopic signals. The alignment of the cortical neurons in palisades and in layers facilitates the summing of the synchronized extracellular dendritic currents of the interactive neurons in many-to-one transmissions. The small size and immense number of synapses and the long distances of axonal transmission provide the topological conditions for sustaining the mesoscopic continuum.

Figure 1: ECoG recordings from olfactory cortex. Above. A food-deprived cat at rest is aroused by an odor of fish and searches for it by sniffing. Below. After feeding to satiety there is no arousal. The upper and lower signals were recorded from the surface and depth of the olfactory cortex; the middle signal is the difference across the dipole field of the cortical pyramidal cells with optimal common mode rejection [Freeman, 1975, Fig. 7.17 p. 442].

Figure 2: Representations of (A) gamma band pass filtered ECoG, (B) analytic power, (C) $log_{10}$ analytic power, and (D) instantaneous frequency in rad/s from the analytic phase, $\Phi_{x,y}(t)$ , in rad divided by the digitizing step in s. The downward spikes in (C) mark the onets of phase transition from a receiving phase to a transmitting phase.

The ECoG signals are discrete samples of potential differences with respect to a common distant point, which are provided by an array of closely spaced surface electrodes. Proper positioning of the array and reference emphasizes the contributions of each area of cortex to the ECoG. The potential differences (volt/cm) accompany the extracellular dendritic current density ( $amp/cm^2$ ) as ions flow across the fixed extracellular tissue resistance ( $ohm.cm/cm^2$ ). The extracellularly measured ECoG power ( $volt.amp$ ) is much too weak to rapidly synchronize the contributory neurons [Freeman and Baird, 1989]; even if it did, it would degrade the spatial patterns by cross-talk. No, the agency that imposes coordination is the inferred dense synaptic pressure ( $dynes/cm^2$ ) of FMA. The ECoG amplitude, which is digitized at each point in time, t, and cortical location, x,y, is an indirect index of FMA intensity through the causal sequence: dendritic current – transmembrane potential – pulse frequency – presynaptic transmitter release – postsynaptic transmitter action.

Better temporal resolution of local FMA intensity is given by band pass filtering and applying the Hilbert transform for brain waves, which separates the analytic amplitude, $A_{x,y}(t)$ , and analytic phase, $\Phi_{x,y}(t)$ . The local temporal rate of change in energy dissipation (watts, $dyne.cm/s$ ), i. e., power, is estimated from the square of the analytic amplitude. The local temporal rate of change in analytic phase is estimated by dividing successive temporal differences in rad, $\Phi_{x,y}(t)$ - $\Phi_{x,y}(t-1)$ , by the digitizing step in s to get the analytic frequency $\omega(t)$ . From these data the local temporal and spatial wavelengths of FMA are calculated for every point in observed space and time [Freeman, 2004]. The global spatial rates of change in the aggregate are estimated from the spatial standard deviations of $A_{x,y}(t)$ and $\omega(t)$ over each time frame between null spikes.

The form of FMA is a locally continuous distribution of power called a wave packet [Freeman, 1975] that is represented by its activity density function, $V_{x,y}(t)$ [Freeman, 1975]. The state variables of each wave packet are bounded in spectral, temporal, and spatial state spaces. The location and width of a peak in spectral power gives the spectral boundaries, which set the pass band width (Fig. 2, A) of neuropil operating as a band pass filter. The maxima of fluctuations in analytic power (B) locate successive wave packets. The logarithm of analytic power (C) reveals downward spikes that often drop below the peaks by 4 to 6 orders of magnitude. These are called null spikes. The rates of change in the analytic frequency, $\omega(t)$ , show spikes in mean and variance that coincide with the downward spikes (D), at which the analytic phase is undefined. Plateaus with low temporal and spatial standard deviations ( $SD_T$ and $SD_X$ ) of analytic frequency $\omega(t)$ manifesting near-stationarity coincide with peaks of analytic power and serve to locate the start and end of wave packets in time. The spikes are beats from mixing distributed frequencies in the pass band [Rice, 1945; Freeman, 2009]. The spikes reveal endogenous phase discontinuities (phase slip) in the filtered ECoG, which accompany the onsets of wave packets; the discontinuities appear as sudden phase re-setting in polar plots [Freeman, 2004, a, b].

Spatial patterns of amplitude modulation (AM) of the carrier wave are revealed by the analytic amplitude, $A_{x,y}(t)$ or the root mean square of the band pass filtered ECoG, $v_{x,y}(t)$ , in the pass band centered at $\omega_{0}(t)$ . The patterns are observed through the window of $n$ electrodes [Freeman. 2004a,b] (Fig. 3). The $n$ values of analytic power at each time step, $A_{x,y}^{2}(t)$ , when divided by the mean power, ${\hat{A}}^{2}(t)$ , yield a normalized $n \times 1$ feature vector, ${\mathbb{A}}^{2}(t)$ , that represents the ECoG pattern (the dissipative structure) by the location at time, t, of a point in $n$ -space (a finite projection from infinite brain state space, not 3-D brain anatomy or 2-D cortical surface). A time series of the tip of the feature vector gives a trajectory in the $n$ -space across a landscape of basins of attraction shown as dwell times creating clusters of points. The entry of the trajectory into a basin is shown by a discontinuity in the analytic phase, $\Phi_{x,y}(t)$ , and a spike in $\omega(t)$ . On exit there is no apparent discontinuity until the next spike in $\omega(t)$ demarcates the transit to a new basin. The center of gravity of a cluster (shown by examples as contour plots in Fig. 3) represents the AM pattern of a wave packet. Two or more centers of gravity in cortical state space in trained subjects provide the basis for classification of AM patterns with respect to conditioned stimuli [Freeman, 2005].

Each AM pattern is accompanied by a phase modulation (PM) pattern of the carrier in the form of a phase cone [Freeman and Baird, 1987; Freeman and Barrie, 2000] (Fig. 4). The location of the apices and the sign (lead - or lag +) are fixed in each wave packet but vary randomly from each wave packet to the next. The phase gradient in rad/mm (slope of the cone) divided by the carrier frequency in rad/s determines the phase velocity in m/s, which is correlated with the conduction velocity of axons running parallel to the surface. The soft spatial boundary of the wave packet is estimated from the reciprocal of the gradient, when the lag equals $\pm \pi/4$ , which occurs at the half power radius ( $cos 45^{\circ} \approx .7$ ). Serial spatial plots of the band pass filtered ECoG reveal vortices [Kozma and Freeman, 2008; Freeman and Vitiello, 2009] with clockwise, counterclockwise or no rotation (pulsation), indicating that the wave packet is a vector field.

Figure 3: Spatial patterns of wave packets in ECoG. Left: Example of AM carrier wave, $\omega(t)$ , in one frame. Center: Contour plots of AM patterns from 8x8 electrodes (4x4 mm) in frames showing differences correlated with stimuli. Right: Demonstration of lack of invariance of AM patterns with respect to stimuli, showing that AM patterns are not representations of stimuli; they are operators expressing activated memories about stimuli that are continually updated with new experience. The vertical differences reflect short-term memory changes on formation of an assembly. The horizontal differences reflect long-term memory changes in consolidation. [Freeman, 1999].

Figure 4: Spatial patterns of temporal analytic phase differences in successive time steps show six conic patterns within the soft spatial boundary determined by the phase gradient and carrier frequency. The direction of the gradient is defined everywhere except at the apex of the cone.

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Mesoscopic phases, dissipative structures, and order parameters

Cortex as a thermodynamic system has multiple states that resemble various states of matter referred to as phases [Sethna, 2008], among them awake, asleep and seizure. Two phases in sensory cortices are essential for the action-perception cycle: a receiving phase during behavioral search and expectancy that is organized by ‘’preafference’’ under limbic control, and a transmitting phase that is triggered by reception of an expected conditional stimulus [Freeman, 2008]. The phase transition from receiving to transmitting occurs when a null spike supervenes upon the activity in a Hebbian nerve cell assembly, which has been excited by a conditioned stimulus, and which provides activation energy in a brief burst of narrow band oscillation.

The receiving phase is characterized by frequency tuning at the microscopic level [Traub et al., 1996; Whittington et al., 2000]. The ECoG shows high spatial and temporal variances around the shared instantaneous frequency, which suggest a search trajectory in a high dimensional state space. Each sensory cortex contains synaptic webs of connectivity that have been structured by reinforcement learning, and that embody the knowledge that a subject already has about expected stimuli. The connectivity supports a latent landscape of nonconvergent ("chaotic") attractors corresponding to an array of sensory outcomes of an intended act of observation. Each attractor is accessed through a Hebbian assembly that is presumed to be sensitized by preafference. The transmitting phase is characterized by dramatic reduction in spatiotemporal variance in amplitude and instantaneous frequency, as the sensory cortex converges to an attractor. The resulting AM pattern that is classifiable with respect to the selecting conditioned stimulus is an expression of stored memories that are relevant to the subject in the context set by preafference. The accompanying phase cone has no classificatory value. It serves to define the boundaries of the wave packet carrying the AM pattern, and to reveal some aspects of the mesoscopic dynamics by which the phase transition and AM pattern formation occur. The phase cone and vortex are attributed to a phase transition [Sethna, 2008, Sect. 12; K] Neuropercolation in the distributed medium of the neuropil, for which the phase velocity is determined by the conduction velocity of the action potentials sustaining the FMA.

The normalized feature vector, ${\mathbb{A}}^{2}(t)$ , is chosen as the vectorial order parameter for FMA for two reasons. First, it is the most direct measure of the order in the wave packets that is relevant to behavior [Freeman, 2005]. Second, the disappearance of the order parameter in the null spikes (Fig. 2, C) coincides with the disappearance of mesoscopic order. It marks the initiation of the phase transition to the transmitting phase. By inference the relaxation of the order parameter frees the microscopic neurons from existing mesoscopic constraint, so that they are available for capture by a different constraint, which is provided through the impact of a conditioned stimulus that excites a Hebbian nerve cell assembly, which gives access to an attractor with its associated AM pattern [Freeman, 2009]. FMA provides the energy needed to read out a memory from its globally distributed synaptic network, shape the coordinated firing of millions of neurons in the actualized recollection, broadly disseminate the retrieved knowledge through the forebrain, and quickly terminate the pattern in anticipation of the next sensory sample.

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Scholarpedia References

Freeman and Cacchione, Intentionality
Freeman Hilbert transform for brain waves
Freeman and Erwin, Freeman K-set
Freeman Scale-free neocortical dynamics
Freund and Kali Interneurons
Kozma Neuropercolation
Liljenstrom Mesoscopic Brain Dynamics
Pribram Holonomic brain theory
Tang and Wiesenfeld Self-organized Criticality

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References

Arianrhod R (2003) Einsteins�s Heroes. Oxford UK: Oxford UP.
Barrett TW (2008) Topological Foundations of Electromagnetism. Hackensack, NJ: World Scientific.
Barrie JM. Freeman WJ and Lenhart M (1996) Modulation by discriminative training of spatial patterns of gamma EEG amplitude and phase in neocortex of rabbits. J Neurophysiol 76: 520-539.
Berthoz A, Petit J-L [2008] The Physiology and Phenomenology of Action (Macann C, trans). New York: Oxford UP.
Braitenberg V, Schüz A (1998) Cortex: Statistics and Geometry of Neuronal Connectivity, 2nd ed. Berlin: Springer-Verlag.
Dreyfus H [2007] Why Heideggerian AI failed and how fixing it would require making it more Heideggerian. Philosophical Psychology 20[2]: 247-268.
Engel AK, Kreiter AK, König P and Singer W [1991] Synchronization of oscillatory neuronal responses between striate and extrastriate visual cortical areas of the cat. Proc Nat Acad Sci USA 88: 6048-6052.
Freeman WJ [1986] Petit mal seizure spikes in olfactory bulb and cortex caused by runaway inhibition after exhaustion of excitation. Brain Res Rev 11: 259-284.
Freeman WJ [2003] Neurodynamic models of brain in psychiatry. Neuropsychopharmacology 28, S1: 54-63. http://repositories.cdlib.org/postprints/990
Freeman W.J. [2004a] Origin, structure, and role of background EEG activity. Part 1. Analytic amplitude. Clin Neurophysiol 115: 2077-2088. http://repositories.cdlib.org/postprints/1006
Freeman W.J. [2004b] Origin, structure, and role of background EEG activity. Part 2. Analytic phase. Clin Neurophysiol 115: 2089-2107. http://repositories.cdlib.org/postprints/987.
Freeman W.J. [2005] Origin, structure, and role of background EEG activity. Part 3. Neural frame classification. Clin Neurophysiol 116 (5): 1118-1129. http://repositories.cdlib.org/postprints/2134/
Freeman WJ [2006] Origin, structure, and role of background EEG activity. Part 4. Neural frame simulation. Clin Neurophysiol 117/3: 572-589. http://repositories.cdlib.org/postprints/1480/
Freeman WJ [2008] A pseudo-equilibrium thermodynamic model of information processing in nonlinear brain dynamics. Neural Networks 21: 257-265. http://repositories.cdlib.org/postprints/2781
Freeman WJ [2009] Deep analysis of perception through dynamic structures that emerge in cortical activity from self-regulated noise. Cognitive Neurodynamics 3(1): 105-116. http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s11571-009-9075-3
Freeman WJ, Baird B [1987] Relation of olfactory EEG to behavior: Spatial analysis: Behav Neurosci 101:393-408.
Freeman WJ, Baird B. [1989] Effects of applied electric current fields on cortical neural activity. Chapter in: Schwartz E (ed.) Computational Neuroscience. New York: Plenum Press. pp. 274-287.
Freeman WJ, Barrie JM [2000] Analysis of spatial patterns of phase in neocortical gamma EEGs in rabbit. J Neurophysiol 84: 1266-1278.
Freeman WJ, Burke BC, Holmes MD [2003] Aperiodic phase re-setting in scalp EEG of beta-gamma oscillations by state transitions at alpha-theta rates. Human Brain Mapping 19(4): 248-272.
Freeman WJ, Cao Y [2008] Proposed renormalization group analysis of nonlinear brain dynamics at criticality. Ch. 27 in: Advances in Cognitive Neurodynamics ICCN 2007 (R. Wang et al. eds.). Heidelberg: Springer, pp. 147-158.
Freeman WJ, Holmes MD, West GA, Vanhatalo S [2006] Dynamics of human neocortex that optimizes its stability and flexibility. Intern J Intelligent Syst 21: 1-21. http://repositories.cdlib.org/postprints/2385
Freeman WJ, Holmes MD, West GA, Vanhatalo S [2006] Fine spatiotemporal structure of phase in human intracranial EEG. Clin Neurophysiol 117(6): 1228-1243.
Freeman WJ, Kozma R, Bollobás B, Ballister P [2009] Chapter 7. Scale-free cortical planar networks. Handbook of Large-Scale Random Networks. Series: Bolyai Mathematical Studies, Vol. 18. Bollobás B, Kozma R, Miklos D (Eds.) New York: Springer. http://www.springer.com/math/numbers/book/978-3-540-69394-9
Freeman WJ, Rogers LJ, Holmes MD, Silbergeld DL [2000] Spatial spectral analysis of human electrocorticograms including the alpha and gamma bands. J Neurosci Methods 95: 111-121.
Freeman WJ, Vitiello G [2006] Nonlinear brain dynamics as macroscopic manifestation of underlying many-body field dynamics. Physics of Life Reviews 3: 93-118. http://repositories.cdlib.org/postprints/1515
Freeman WJ, Vitiello G [2009] Dissipative neurodynamics in perception forms cortical patterns that are stabilized by vortices. J. Physics Conf Series, in press.
Freeman WJ, Zhai J [2009] Simulated power spectral density (PSD) of background electrocorticogram (ECoG). Cognitive Neurodynamics 3(1): 97-103. http://dx.doi.org/10.1007/s11571-008-9064-y.
Haken H [1983] Synergetics: An Introduction. Berlin: Springer-Verlag.
Köhler W [1940] Dynamics in Psychology. New York NY: Grove Press
Kozma R, Freeman WJ [2001] Chaotic resonance: Methods and applications for robust classification of noisy and variable patterns. Intern J Bifurc Chaos 10: 2307-2322.
Kozma R, Freeman WJ [2002] classification of EEG patterns using nonlinear dynamics and identifying chaotic phase transitions. Neurocomputing 44: 1107-1112.
Kozma R, Huntsberger T, Aghazarian H, Tunstel E, Ilin R, Freeman WJ [2008] Implementing intentional robotics principles using SSR2K platform. Advanced Robotics 22(12): 1309-1327.
Kozma R, Freeman WJ (2008) Intermittent spatio-temporal de-synchronization and sequenced synchrony in ECoG signals. Special Issue: Synchronization in Complex Networks, Suykens J, Osipov G (eds). Chaos 18, 037131. http://link.aip.org/link/?CHA/18/037131DOI: 10.1063/1.2979694
Li X, Li G, Wang L, Freeman WJ [2006] A study on a bionic pattern classifier based on olfactory neural system. Intern J Bifurc Chaos 16(8): 2425-2434.
Merleau-Ponty M [1945/1962] Phenomenology of Perception. (Smith C, trans.). New York: Humanities Press.
Ohl FW, Scheich H, Freeman WJ [2001] Change in pattern of ongoing cortical activity with auditory category learning. Nature 412: 733-736.
Orsucci F [2009] Mind Force on Human Attraction. London: World Scientific.
Pincus D, Freeman WJ, Modell A [2007] A neurobiological model of perception: Considerations for tranference. J Psychoanalytic Psych 24(4): 623-640.
Pockett S, Bold GEJ, Freeman WJ [2009] EEG synchrony during a perceptual-cognitive task: Widespread phase synchrony at all frequencies. Clin Neurophysiol, in press. doi:10.1016/j.clinph.2008.12.044
Prigogine I [1980] From Being to Becoming: Time and Complexity in the Physical Sciences. San Francisco: W H Freeman.
Principe, J.C., Tavares, V.G., Harris, J.G. and Freeman, W.J. [2001] Design and implementation of a biologically realistic olfactory cortex in analog VLSI. Proc IEEE 89: 1030-1051.
Rice SO [1945] Mathematical Analysis of Random Noise. Bell System Technical J. 24: 46-156, Section 3.10, pp. 98-100.
Ruiz Y, Li G, Freeman WF, Gonzalez E. [2009] Detecting stable phase structures on EEG signals to classify brain activity amplitude patterns. J Zhejiang Univ, in press.
Sethna JP [2008] Statistical Mechanics: Entropy, Order Parameters, and Complexity. Clarendon, Oxford UP. http://pages.physics.cornell.edu/sethna/StatMech/EntropyOrderParametersComplexity.pdf
Traub RD, Whittington MA, Stanford IM, Jefferys JGR. [1996] A mechanism for generation of long-range synchronous fast oscillations in the cortex. Nature 383: 421-424. doi:10.1038/383621a0
Whittington MA, Faulkner HJ, Doheny HC, Traub RD [2000] Neuronal fast oscillations as a target site for psychoactive drugs. Pharmacol Therap 86: 171-190.
Wright JJ [2009] Cortical phase transitions: Properties demonstrated in continuum simulations at mesoscopic and macroscopic scales. New Math Nat Comp 5(1): 159-184

Dr. Robert Kozma, Computational NeuroDynamics Lab, University of Memphis, TN, was invited on 27 January 2009.

Suggested by:	Mr. Daniel Rojas-Libano, PhD student, University of Chicago, Chicago, IL, USA
Invited by:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Action editor:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

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Categories: Neuroscience | Dynamical systems

Freeman's mass action

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Contents

Introduction

Box: Neural Networks versus Neurodynamics: A paradigm shift

Mesoscopic state variables: The wave packet

Mesoscopic phases, dissipative structures, and order parameters

See also: Advances in FMA theory, experimental analysis and applications

Advances in theory:

Advances in experimental neuroscience:

Advances in applications:

Recommended Reading

Scholarpedia References

References

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