Qualia as the Atoms of Consciousness

from an Estate in the beautiful Napa Valley of California Wine Country. Now moved to Telegraph Hill where Don Johnson is shooting Nash Bridges outside my window. My Green Explorer is visible in the shot where the guy in the Angel Suit sitting in the back seat of the yellow convertible asks for a popsicle.

Jack Sarfatti's notes and heuristic musings on the quantum physics of mind/brain in the context of

The genome of a higher metazoan cell has 10^4 to 10^5 structural and regulatory genes with joint orchestrated activity in development from fertilized egg.

Human immune system has 10^8 different kinds of antibody molecules again working together all over the body in a synchronized way.

More than 10^9 neurons work together.

There are two mechanisms: self-organization and natural selection. Henry Stapp only recognizes natural selection in his orthodox quantum theory of the mind/brain. Back-action beyond orthodox quantum theory is required for the natural order of self-organization.

"Much of the order we see in organisms may be the direct result not of natural selection but of the natural order selection was privileged to act on." p. 173 Kauffman

Complex systems able to adapt "achieve a 'poised state' near the boundary between order and chaos..."

Classical dynamical systems have attractors which can be simple or chaotic.

Attractors confine the system to small parts of its state space of possible behaviors. This is essential for self-organization.

So far, there is no need for quantum back-action which is a peculiar property of consciousness in my theory.

Random classical Boolean networks of 2-state switches have three phases, ordered, complex and chaotic.

The ordered phase has a static non-active frozen connected component which is a large connected cluster that percolates across the system leaving behind isolated islands of unfrozen dynamically fluctuating activity.

The transition from order to chaos is a role exchange between unfrozen and frozen, between "Fire" and "Ice" as in the Nordic Sagas. The frozen connected component disappears in the chaotic phase and is replaced by an unfrozen connected component.

"Instead, a connected cluster of unfrozen elements, free to fluctuate in activities, percolates across the system, leaving behinds isolated frozen islands. ... small changes in initial conditions unleash avalanches of changes which propagate to many other unfrozen components. .... The transition region, on the edge between order and chaos, is the complex regime. ... In this transition region, altering the activity of single unfrozen elements unleashes avalanches of change with a characteristic size distribution having many small and few large avalanches." p.174 Kauffman

The transition for chaos to order can occur over large regions with only local couplings between the cellular automata precluding the necessity for quantum nonlocality at first sight in such things as ontogeny, immune response and coherent neuronal activity which are "zombie" behavior. The specifically quantum nonlocality is needed for subjective felt-consciousness.

Fitness landscapes of Boolean switching networks are smooth in the ordered phase like ancient mountain ranges, and rugged in the chaotic. Only on the complexity edge between order and chaos are the networks optimized to evolve by accumulating useful variations from natural selection.

What happens for quantum non-Boolean networks when each switch is in a coherent superposition of the two classically mutually exclusive states? Does this really happen in the human microtubule? The position of the thermally shielded single control electron at the alpha-beta boundary of the protein dimer is EPR entangled with the dimer conformations of "open" and "closed". Suppose we have N dimers. Let 1 be open and 0 be closed. We then have 2^N possible strings of N bits. The quantum wavefunction is then a coherent superposition of these 2^N strings. This ignores the magnetic spin correlations of the spatially separated electrons.

Newtonian dynamical systems and their attractors. In Newtonian dynamics, a system of ordinary differential equations for the rates of change of a variable in terms of the present values of all the variables that affect it is constructed. Bohm's hidden variable formulation of nonrelativistic quantum mechanics is able to use these equations with an added nonlocal quantum force. There is no memory of the past and no precognition of the future. Absolute simultaneity is also supposed. Therefore, the speeds of things must be small compared to the speed of light for this approximation to be accurate. A more realistic model of the brain-mind will put in both memory and precognition making the system nondeterministic. Determinism depends on the locality of the differential equations. As soon as you put in finite timelags and spatial actions at a distance, it's a whole new ball game even on the classical level.

Each relevant variable forms a dimension of the state space. The system point has a path in this classical state space. The material brain is represented by such a system point. The nonmaterial, but still physical, quantum mind exerts a nonlocal and contextual force on this classical system point representing the material brain.

In classical Newtonian mechanics, only one path passes through each point in state space. Different paths can converge, in an infinite time, to a zero-dimensional point-attractor in state space. All the points of state space that lie on paths that limit to this point-attractor form its basin of attraction. Each possible felt-experience in the stream of consciousness corresponds to the system point of the material brain being in a basin of attraction partially determined by the quantum mind, and partially determined by the external forces of the environment that is the mechanism of Darwinian natural selection in the evolution of all the biospheres in the universe.

"The existence of attractors in physical systems often requires some form of driving and friction which prevents conservation of energy within the system itself. ... Just as a mountainous region may have many lakes and drainage basins, so may a dynamical system have many attractors, each draining its own basin. Therefore, it is natural to conceive of the state space as being partitioned into disjoint basins of attraction. When released from an initial state, the dynamical system is on a trajectory lying in only one basin, and the system flows to that basin's attractor. ... the different attractors constitute the total number of long term alternative behaviors of the system. ....attractors are typically much smaller than the volume of states in their basins, the system becomes boxed into an attractor unless perturbed by an outside force. " pp.176-7, Kauffman.

Attractors have stability properties. They need not be zero-dimensional points. For example, saddle-point attractors are stable against small external forces acting along some directions in state space, but not in others. Basins of attraction are separated by regions called "separatrixes" of one less dimension than the number of variables.

The stability of point-attractors is found by linearizing the system of ordinary differential equations around that point attractor. Look at the eigenvalues of the matrix M of the RHS linearized system

where X is the column vector representing a point in classical state space. Stability requires that all eigenvalues are negative. Positive eigenvalues indicate that the attractor is unstable. Complex eigenvalues indicate a frictional spiraling.

Non-point attractors include the one-dimensional limit cycles i.e., a closed loop of states for the long-term behavior. There are toroidal attractors. The simplest is the two-dimensional toroidal attractor made from two limit-cycles. If the ratio of the periods is irrational the system point covers all of the two dimensional toroid. Similarly with higher dimensional toroids made from more limit cycles. Finally there are the strange chaotic attractors. Two points very close together on such a strange attractor have paths that widely diverge from each other. The dimension of a strange attractor can be a fraction. This is because one must take the statistical average of the dimensions from all the points on the strange attractor. The measure of dimension from each point on the strange attractor "depends on how the density of points on the attractor changes with radius in all... dimensions away from any arbitrary point on the attractor" (p. 178) This is why they are called "fractals". This is extreme sensitivity to small changes in initial conditions. This "Butterfly effect" is why a single quantum event in the brain, whose final cause is in the timelike future of that event, can influence the felt-experience of that future cause before it actually happens. Fred Hoyle writes about this in his book, The Intelligent Universe (1986).

In addition to classical state space, there is parameter space as well as quantum Hilbert space, as well as the external Darwinian forces of natural selection, all affecting the motion of the brain system point in classical state space.

Structural Stability

"Any point in parameter space corresponds to ... a fixed set of basins of attraction and attractors in the corresponding state space of the dynamical system. The set of basins of attraction is often called the basin-portrait of the dynamical system. ... For particular changes of the parameters.... sudden dramatic changes in trajectories and attractors can occur. Such sudden changes are called bifurcations.... For example, a basin might contract to nothing, or a new basin appear... The values of parameters at which bifurcations occur divide parameter space into disjoint volumes. ... volumes in parameter space defined by bifurcation surfaces are like soap bubbles. The volumes are reasonably large relative to the bifurcation surfaces which divide them. Thus for most changes in the parameters, the system remains within one volume in parameter space and the dynamical behavior does not change dramatically. Dynamical systems having this property are said to be structurally stable. Their dynamics typically changes only slightly as parameters change but does jump across bifurcation surfaces..."

Fractal strange attractors in state space destroy structural stability.

" ... in many systems with strange attractors, tuning the parameters leads to a succession of bifurcations at successively smaller intervals in parameter space. ... such is seen in the famous period-doubling bifurcations ... which may underlie the onset of turbulence in fluid flow ... In these cases, the volumes separating qualitatively different behaviors become sinuous, intertwined labyrinths in parameter space. Here, tiny changes in parameters in almost any direction can lead to successive dramatic changes in the dynamical behavior of the correlated system. ... structurally stable systems adapt on correlated landscapes. By contrast, chaotic systems, which are not structurally stable, adapt on uncorrelated landscapes. Very small changes in the parameters pass through many interlaced bifurcation surfaces and so change the behavior of the system dramatically... alterations that cause the systems to pass from ordered to chaotic behavior will inevitably alter the statistical structure of their fitness landscapes from smooth to rugged... an intermediate ruggedness optimizes evolvability... the optimum may lie near the boundary between order and chaos." pp. 180-1, Kauffman

The fitness landscape for the brain material system point in 3n-configuration state space is determined by the Bohm quantum force in addition to the external sensory forces. What is the parameter space of the brain material system point? In my back-action theory, unlike Stapp's theory which has no back-action, the path of the brain system point directly modifies the mental quantum force acting on that same brain system point. This is the process of "self-measurement" which is not the same as the von Neumann measurement of orthodox quantum theory where the observed is different from the observer. In self-measurement, the observer and the observed are the same system. This is the "strange loop" of Godel's "self-reference" that Doug Hofstadter, in Godel, Escher, Bach, conjectured was required for the generation of felt-consciousness.

The actual path space of the brain system point is a subspace of the parameter space. That is, in a living system, the state space is its own parameter space. To repeat, this is the creative Godelian strange feedback loop of self-determination required for free will. This is how the conscious living system breaks free of simple-minded classical Newtonian determinism.

Closed systems conserve energy and hyper-volume in 6-n phase space (i.e., Liouville's theorem). Living matter is open dissipating energy. Living system points do not wander randomly and ergodically in their state space like their dead closed inanimate cousins. The inanimate ergodic flow is replaced by an adaptive flow in state space.

Now let's look at Bader's breakthrough in quantum chemistry which should seamlessly connect to complex adaptive systems. Bader is mainly concerned with visualizable 3-space for the single electron, but his methods should generalize to 3-n configuration space for many-particle problems.

Bader's Topological Quantum Chemistry

Bader starts with the 3D-spatial topology of the single-electron charge density rho(x,y,z). The critical points are the maxima, minima and saddle points where grad rho = 0.

The e's are orthonormal Cartesian unit vectors. There are 9 second derivatives of rho for its curvature that form the real symmetric Hessian 3x3 matrix. In 3n-space for n-electrons this is a 3nx3n matrix. Diagonalize the Hessian. This is the simplest eigenvalue problem in which we rotate to a new frame in which all of the off-diagonal mixed second derivatives of rho vanish.

"The new coordinate axes are called the principal axes of curvature because the magnitudes of the three second derivatives of rho with respect to these axes is extremized. The principal axes will correspond to symmetry axes if the critical point is at the origin... the corresponding curvatures are equal. ... The trace of the Hessian matrix, the sum of its diagonal elements, is invariant to a rotation of the coordinate system." p. 18 Bader

This 3-D rotation invariant trace of the Hessian matrix is the Laplacian of rho, i.e.,

The principal axes are the eigenvectors, and the principal curvatures are the eigenvalues lambda(i), i = 1,2,3, of the Hessian matrix operator, Hess. The eigenvalues are the roots of the polynomial f(lambda) formed from

where I is the unit 3x3 matrix and det is the determinant of high school algebra.

Let's compare Bader's approach with Bohm's pilot-wave/hidden-variable (i.e., beable) quantum theory of the single electron in an atom or molecule. The fundamental quantity is the quantum potential Q rather than rho. At the single-electron mean-field level,

Using the chain rule of Calculus 101

Therefore, the Bohm quantum potential is

The part of Q that depends on the square of grad rho is intrinsically negative. The part that depends on Lap rho can be of either sign. The nonlocal context-dependent Bohm quantum force field on the electron is - grad Q which will involve third space derivatives of the mean field charge density. Note that Bader's critical points where grad rho = 0, leave a finite Q (1/2) rho^-1 Lap rho.

Now I return to Bader's most interesting and technologically practical theory.

The eigenvalues of the 3x3 single-electron Hessian matrix of the critical points are real. They can equal zero. The rank of a critical point is equal to the number of non-zero eigenvalues. The signature of the critical point is the algebraic sum of the signs of the eigenvlaues. However, I would guess that bifurcations correspond to eigenvalues going complex. The imaginary part signals an instability. "With relatively few exceptions, the critical points of charge distributions for molecules at or in the neighborhood of energetically stable geometrical configurations of the nuclei are all of rank three. ... It is in terms of the properties of critical points with rank 3 that the elements of molecular structure are defined. A critical point with rank less than 3, i.e., with at least one zero curvature, is said to be degenerate. Such a critical point is unstable in the sense that a small change in the charge density, as caused by a displacement of the nuclei, causes it either to vanish or to bifurcate into a number of non-degenerate or stable rank 3 critical points." p. 18, Bader.

Denoting critical points by (rank, signature), there are four types of rank 3. Bader shows:

(3,-3) is a local maximum of the electron charge density at the critical point. All the nuclei in a molecule lie at these critical points.

(3,-1) are the all-important sub-system boundary (i.e., interatomic surface) saddle points in which rho is a maximum at the critical point in the plane of negative curvatures and is a minimum along the line through the critical point perpendicular to that plane. This (3,-1) type critical point is found between every pair of nuclei which are considered to be linked by a chemical bond in conventional empirical chemistry.

(3,+1) ring critical points are also saddles with the opposite max/min properties relative to the (3,-1). Thus, rho is a minimum in the plane of positive curvatures and a maximum along the line through the critical point perpendicular to that plane. The minimum planar region of the (3,+1) critical point is generally bounded by a ring of (3,-1) type critical points.

(3,+3) are cage critical points that have all positive curvatures so that rho is a local minimum at the critical point. For example, C4H4 has a (3,+3) critical point in rho in the center of the cage structure.

Such cages are important in the microtubules which form the Eccles Gates at the boundary between quantum mind and classical brain. The Bohm quantum force -grad Q is literally the force of mind on the material brain. Back-action of brain on mind is how the brain literally changes its mind allowing us to have felt-consciousness. The motion of the brain system point in 3n-space into a basin of attraction is the felt-experience of the information content encoded in that attractor on the fitness landscape. Bifurcations of the attractor structure changing the fitness landscape is how the mind-brain adapts to external Darwinian environmental selection pressures. The back-action provides the intrinsic self-organization that can resist natural selection pressures to some extent. The mind is the macroscopically coherent quantum pilot-wave of the brain material substrate at the nanoscale microtubule level which is the hidden-variable or "beable" of the mind.

".. the principal topological features of a charge distribution can be summarized using the rank and signature classification scheme of its critical points. It has further demonstrated the existence of a connection between the number and kind of critical points appearing in a charge distribution and its conventional chemical structure." p. 22 Bader.

Bader then studies the gradient vector field grad rho. The gradient vector of the scalar field points in the direction of greatest spatial rate of change of that scalar. Therefore, it is perpendicular to contour surfaces of the scalar field of constant charge density in this particular case. The critical points grad rho = 0 are sources where lines of grad rho start, or else, sinks where they end. Two gradient trajectories cannot cross.

Consider a single eigenvalue and its attached eigenvector of Hess at a critical point. If the eigenvalue is negative, i.e., a (1,-1) 1-D critical point, then we have a sink where lines of grad rho end. Similarly, (1,+1) is a source in 1-D, e.g., nanowires. Let's go to 2D, e.g. anyons, high Tc superconductors.... (2,-2) is a sink in the plane. (2,+2) is a source. The saddle point is a sink along the line in which the curvature eigenvalue is negative, and it is a source along the perpendicular line of positive eigenvalue. Paths of grad rho that miss the critical point look like hyperbolas.

Now we come to the all important case of (3,-1). Look at all the paths that start or end at this critical point. Consider the tangent plane through this (3,-1) critical point that contains the two negative eigenvalues. All of those grad rho paths end at that critical point. Two paths start in opposite directions along the direction of positive eigenvalue perpendicular to the negative eigenvalue tangent plane through the (3,-1) critical point. The set of paths that end at the (3, -1) critical point sweep out the interatomic surfaces that define the atoms inside molecules. This is possible because the chemical binding energies of a few electron volts are small compared to the rest mass of the nuclei of billions of electron volts. These interatomic surfaces are generally nonplanar. They are planar only if the (3,-1) critical point lies in a symmetry plane of the molecule.

"The set of trajectories defined by linear combinations of the pair of eigenvectors associated with the two negative eigenvalues terminate at the critical point and define a surface. The charge density is a maximum in the surface at the critical point and a minimum at this same point along the perpendicular axis." p. 27 Bader

Definition of the classical basin of attraction.

Given a critical point attractor, "there exists an open neighborhood B of the attractor that is invariant to the flow grad rho such that any trajectory originating in B terminates at the attractor. The largest neighborhood satisfying these conditions is called the basin of the attractor." (p. 28, Bader). Note, that there is no obvious apriori 1-1 connection between these critical point basins of attraction in 3D space and the single-electron eigenfunctions of a quantum measurement observable.

The positions of the nuclei are (3,-3) critical points in the rho scalar field in 3D space.

"... real space, is partitioned into disjoint regions, the basins of which contain one point attractor or nucleus. ... An atom, free or bound, is defined as the union of an attractor and its associated basin." Bader, p.28

"The result is a single variational principle which defines the observables, their equations of motion, and their average values for the total system or for an atom within the system. The generalization of the action principle to a subsystem of some total system is unique, as it applies only to a region that satisfies a particular constraint on the variation of its action integral. The constraint requires that the subsystem be bounded by a surface of zero flux in the gradient vectors of the charge density, i.e., grad rho.n = 0 for all points on the surface S. ... it is necessary that the atomic surface not be crossed by any trajectories ... the imposition of the quantum boundary condition of zero flux leads directly to the topological definition of an atom. ... all trajectories in the vicinity of a given nucleus terminate at that nucleus and no trajectories cross from one basin to another." Bader pp.29-31

Note that these Bader trajectories are not the actual paths of the hidden-variable electron system point in Bohm's theory.

"Atomic surfaces undergo continuous deformations as atoms move relative to one another. They are, however, not destroyed as atoms separate." Bader p. 31

Zero flux surfaces also define linked groupings of atoms as well as individual atoms. They also define Wigner-Seitz cells in solids and solutes in solutions.

".. the average properties of a subsystem and their equations of motion are determined by the same quantum mechanical expressions as apply to an 'isolated' system. The major difference between the properties of a subsystem and those of a system considered to be isolated is to be found in the fluctuation of their average values. Such fluctuations vanish for an isolated system ..." Bader p. 32

"Atomic interaction lines" are the pairs of outgoing gradient paths leaving the (3,-1) critical point along the direction of the eigenvector with positive eigenvalue. Both of these outgoing paths end at the (3, -3) critical points where nuclei of the molecule are located. That is, the (3,-1) critical points are between (3,-3) nuclei which form chemical bonds in the molecule. The (3, -1)critical point is where the atomic interaction line intersects the interatomic boundary surface "and charge is so accumulated between the nuclei along the length of this line".

"These two unique gradient paths define a line through the charge distribution linking the neighboring nuclei along which rho is a maximum with respect to any neighboring line. Such a line is found between every pair of nuclei whose atomic basins share a common interatomic surface... The existence of a (3, -1) critical point and its associated atomic interaction line indicates that electronic charge density is accumulated between the nuclei that are so linked. ... Both theory and observation concur that the accumulation of electronic charge between a pair of nuclei is a necessary condition if two atoms are to be bonded to one another. This accumulation of charge is also a sufficient condition when the forces on the nuclei are balanced and the system possesses a minimum energy equilibrium internuclear separation. ... The line of maximum charge density linking the nuclei is called a 'bond path' and the (3, -1) critical point referred to as a 'bond critical point'. ... a 'molecular graph is defined as the union of ...the bond paths or atomic interaction lines... the (3, -1) critical points are found to link certain, but not all pairs of nuclei in a molecule." Bader pp. 32-3

Rings and cages.

"If the bond paths are linked so as to form a ring of bonded atoms... then a (3, +1) critical point is found in the interior of the ring. ... the eigenvectors associated with the two positive eigenvalues of the Hessian matrix of rho at this critical point generate an infinite set of gradient paths which originate at the critical point and define ... the ring surface. ... All the trajectories which originate at ... the (3, +1) or ring critical point, terminate at the (3, -3) ring nuclei, but for the set of single trajectories, each of which terminates at one of the bond (3, -1) critical points whose bond paths form the perimeter of the ring. ... The remaining eigenvector of a (3, +1) ring critical point, its single negative eigenvector, generates a pair of gradient paths which terminate at the (3, +1) critical point, and define a unique axis perpendicular to the ring surface at the critical point ... A ring ... is defined as a part of the molecular graph which bounds a ring surface.

If the bond paths are so arranged as to enclose the interior of a molecule with ring surfaces, then a (3, +3) cage critical point is found in the interior of the resulting cage. The charge density is a local minimum at a cage critical point. ... Trajectories only originate at such a (3, +3) critical point and terminate at (3, -3) nuclei, and at (3, -1) bond and (3, +1) ring critical points, thereby defining a bounded region of space. A cage ... is a part of a molecular graph which contains at least two rings, such that the union of the ring surfaces bounds a region of R3 which contains a (3, +3) critical point. " Bader pp. 35 - 7

To review

(3,-3) nucleus

(3,-1) bond

(3,+1) ring

(3,+3) cage

to be continued