COHERENCE, SELF-SIMILARITY AND BRAIN ACTIVITY

COHERENCE, SELF-SIMILARITY AND BRAIN
ACTIVITY ∗

Giuseppe Vitiello

Salerno University, Italy

∗ G. Vitiello, My Double Unveiled. Amsterdam: John Benjamins, 2001

W. J. Freeman and G. Vitiello, Physics of Life Reviews 3, 93 (2006)
q-bio.OT/0511037
W. J. Freeman and G. Vitiello, J. Phys. A: Math. Theor. 48, 304042 (2008)
arXiv:q-bio/0701053
W. J. Freeman and G. Vitiello, Vortices in brain waves, arXiv:0802.3854

1

The mesoscopic activity of neocortex:

dynamical formation of spatially extended domains of amplitude mod-
ulated (AM) synchronized oscillations with near zero phase dispersion.

These “packets of waves” form in few ms, have properties of location,
size, duration (80−120 ms) and carrier frequencies in the beta-gamma
range (12 − 80 Hz),

re-synchronize in frames at frame rates in the theta-alpha range (3 −
12 Hz) through a sequence of repeated collective phase transitions.

Such patterns of oscillations cover much of the hemisphere in rabbits
and cats and over domains of linear size of 19 cm in humans

2

The patterns of phase-locked oscillations are intermittently present in
resting, awake subjects as well as in the same subject actively engaged
in cognitive tasks requiring interaction with environment,

so they are best described as properties of the background activity of
brains that is modulated upon engagement with the surround∗ .

Neither the electric field of the extracellular dendritic current nor
the extracellular magnetic field from the high-density electric current
inside the dendritic shafts, which are much too weak, nor the chemical
diffusion, which is much too slow, appear to be able to fully account
for the observed cortical collective activity.

∗ W.
J. Freeman, Clin. Neurophysiol. 116 (5), 1118 (2005); 117 (3), 572 (2006)
W. J. Freeman, et al., Clin. Neurophysiol. 114, 1055 (2003)

3

It turns out that the many-body dissipative model∗ is able to account
for the dynamical formation of synchronized neuronal oscillations † :

each AM pattern is described to be consequent to spontaneous break-
down of symmetry triggered by external stimulus and is associated
with one of the quantum ﬁeld theory (QFT) unitarily inequivalent
ground states.

Their sequencing is associated to the non-unitary time evolution im-
plied by dissipation.

∗ G. Vitiello, My Double Unveiled. Amsterdam: John Benjamins, 2001.
† W. J. Freeman and G. Vitiello, Physics of Life Reviews 3, 93 (2006)
q-bio.OT/0511037
W. J. Freeman and G. Vitiello, J. Phys. A: Math. Theor. 48, 304042 (2008)
arXiv:q-bio/0701053
W. J. Freeman and G. Vitiello, Vortices in brain waves, arXiv:0802.3854

4

Lashley dilemma

Concept of “mass action” in the storage and retrieval of memories in
the brain:

“...Here is the dilemma. Nerve impulses are transmitted ...form cell
to cell through definite intercellular connections. Yet, all behavior
seems to be determined by masses of excitation...within general fields
of activity, without regard to particular nerve cells... What sort of
nervous organization might be capable of responding to a pattern
of excitation without limited specialized path of conduction? The
problem is almost universal in the activity of the nervous system.” ∗

Pribram: analogy between the fields of distributed neural activity in
the brain and the wave patterns in holograms † .
∗ K.
Lashley, The Mechanism of Vision, Journal Press, Provincetown MA, 1948, pp.
302-306
† K. H. Pribram, Languages of the Brain. Engelwood Cliffs NJ: Prentice-Hall, 1971
5

• resorting to classical nonlinear dynamics

⇒ synchrony, very good!

But not enough: only kinematics, it says “how” not “why”. We want
the dynamics (the forces), not just the “description”.

The problem we have to solve is: How can we “obtain” synchrony and
coherence as the output (the eﬀects) of the dynamics, not putting
them (by hands) in writing down the model equations.

6

An alternative approach is therefore necessary.

• The dissipative quantum model of brain has been then proposed ∗

∗ G.
Vitiello, Int. J. Mod. Phys. B 9, 973 (1995)
G. Vitiello, My Double Unveiled. Amsterdam: John Benjamins, 2001

7

The dissipative quantum model of brain

is the extension to the dissipative dynamics of the many-body model
proposed in 1967 by Ricciardi and Umezawa ∗

the extended patterns of neuronal excitations may be described by
the spontaneous breakdown of symmetry (SBS) formalism in QFT.
Umezawa †: “In any material in condensed matter physics any par-
ticular information is carried by certain ordered pattern maintained
by certain long range correlation mediated by massless quanta. It
looked to me that this is the only way to memorize some informa-
tion; memory is a printed pattern of order supported by long range
correlations...”

∗ L.M. Ricciardi and H. Umezawa, Kibernetik 4, 44 (1967)
C. I. J. Stuart, Y. Takahashi and H. Umezawa, J.Theor. Biol. 71, 605 (1978);
Found. Phys. 9, 301 (1979)
† H.Umezawa, Math. Japonica 41, 109 (1995)
8

In QFT long range correlations are indeed dynamically generated
through the mechanism of SBS.

These correlations manifest themselves as the Nambu-Goldstone (NG)
boson particles or modes,

which have zero mass and therefore are able to span the whole system.

The NG bosons are coherently condensed in the system lowest energy
state, the vacuum or ground state (Bose-Einstein condensation).

Due to such correlations the system appears in an ordered state.

The vacuum density of the NG bosons provides a measure of the
degree of ordering or coherence: the order parameter, a classical
ﬁeld specifying (labeling) the observed ordered pattern.
9

Example of NG modes: phonons, magnons, Cooper pairs.

The water matrix is more than the 80% of brain mass and it is there-
fore expected to be a major facilitator or constraint on brain dynamics.

⇔ the quantum variables are the electrical dipole vibrational ﬁeld of
the water molecules and of other biomolecules ∗.

The spontaneous breakdown of the rotational symmetry of the elec-
trical dipole vibrational ﬁeld dynamically generates the NG quanta,
named the dipole wave quanta (DWQ).

The neuron and the glia cells and other physiological units are NOT
quantum objects in the many-body model of brain.

∗ E.Del Giudice, S. Doglia, M. Milani and G. Vitiello, Nucl. Phys. B 251 (FS 13),
375 (1985); Nucl. Phys. B 275 (FS 17), 185 (1986).
M. Jibu and K. Yasue, Quantum brain dynamics and consciousness. Amsterdam:
John Benjamins, 1995.
M. Jibu, K. H. Pribram and K. Yasue, Int. J. Mod. Phys. B 10, 1735 (1996)
10

The recall of recorded information occurs under a stimulus able to
excite DWQ out of the corresponding ground state.

Such a stimulus is called “similar” to the one responsible for the
memory recording.

Similarity between stimuli thus refers not to their intrinsic features,
but to the reaction of the brain to them; to the possibility that under
their action DWQ are condensed into, or excited from the ground
state carrying the same label.

11

• The many-body model fails in explaining the observed coexistence
of AM patterns and also their irreversible time evolution.

• One shortcoming of the model is that any subsequent stimulus
would cancel the previously recorded memory by renewing the SBS
process, thus overprinting the ’new’ memory over the previous one
(’memory capacity problem’).

• The fact that the brain is an open system in permanent interaction
with the environment was not considered in the many-body model.

⇒ Include Dissipation!!

12

In the QFT formalism for dissipative systems the environment is de-
scribed as the time-reversal image of the system ∗ .

This is realized by doubling the system degrees of freedom:

external stimulus ⇒ SBS ⇒ dynamical generation of DWQ Aκ

˜
dissipation ⇒ doubling: Aκ → (Aκ , Aκ)

˜
Aκ ≡ “time-reversed mirror image” or “doubled modes”

energy ﬂux balance ⇔ E0 = ESyst − EEnv = κ ¯ Ωκ(NAκ − NAκ ) = 0
h ˜

∗ E. Celeghini, M. Rasetti and G. Vitiello, Annals Phys. 215, 156 (1992)

13

The canonical commutation relations are the usual ones and
˜† ˜
[ Aκ, Aλ ] = 0 = [ Aκ, Aλ ] etc.. (1)
The Hamiltonian for the infinite collection of damped harmonic os-
˜
cillators Aκ (a simple prototype of a dissipative system) and the Aκ
is
H = H 0 + HI

¯ Ω κ (A † A κ − A † A κ ) ,
˜κ ˜
H0 = h κ
κ
¯ Γ κ (A † A † − A κ A κ ) ,
κ ˜κ ˜
HI = i (2)
h
κ
Ωκ is the frequency, Γκ the damping constant.

κ generically labels degrees of freedom such as, e.g., spatial momen-
tum, etc..

˜
The Aκ and Aκ modes are actually quasi-massless, i.e. they have a
non-zero effective mass, due to finite volume effects.
14

† ˜† ˜
- {|NAκ , NAκ } ≡ simultaneous eigenvectors of AκAκ and AκAκ,
˜

- NAκ and NAκ non-negative integers
˜

˜
- |0 0 ≡ |NAκ = 0, NAκ = 0 the vacuum state for Aκ and Aκ: Aκ|0 0 =
˜
˜
0 = Aκ|0 0 for any κ.

The balanced nonequilibrium state is a ground state.

At some initial time t0 = 0, we deﬁne it to be a zero energy eigenstate
of H0 and denote it by |0 N
⇒ the ”memory state” |0 N is a condensate of equal number of Aκ and
˜
mirror Aκ for any κ: NAκ − NAκ = 0.
˜

label N ≡ {NAκ = NAκ , ∀ κ, at t0 = 0} ≡ order parameter identifying
˜
the vacuum |0 N (the ”memory state”) associated to the information
recorded at time t0 = 0.
15

Balancing E0 to be zero, does not fix the value of either EAκ or EAκ
˜
for any κ. It only fixes, for any κ, their difference.

⇒ at t0 we may have infinitely many perceptual states, each one in
one-to-one correspondence to a given N set: a huge memory capacity.

The important point is:

{|0 N } and {|0 N }, N = N , are ui in the infinite volume limit:

N 0|0 N −→ 0 ∀ N, N , N =N . (3)
V →∞

In contrast with the non-dissipative model, a huge number of sequen-
tially recorded memories may coexist without destructive interference
since infinitely many vacua |0 N , ∀ N , are independently accessible.

16

The commutativity of H0 with HI ([H0 , HI ] = 0) ensures that the num-
ber (NAκ −NAκ ) is a constant of motion for any κ, which also guaranties
˜
that H0 remains bounded from below if this has been assumed to hold
at t0 .

The state |0 N is given, at ﬁnite volume V , by |0 N = exp (−iG(θ))|0 0,
with generator
θκ ( A † A † − A κ A κ ) .
κ ˜κ ˜
G(θ) = −i (4)
κ
and N 0|0 N = 1 ∀ N .

The average number NAκ is given by

NAκ = N 0|A† Aκ|0 N = sinh2 θκ , (5)
κ
which relates the N -set, N ≡ {NAκ = NAκ , ∀κ, at t0 = 0} to the θ-set,
˜
θ ≡ {θκ, ∀κ, at t0 = 0}.

17

We also use the notation NAκ (θ) ≡ NAκ and |0(θ) ≡ |0 N .

˜
The θ-set is conditioned by the requirement that A and A modes
satisfy the Bose distribution at time t0 = 0:
1
NAκ (θ) = sinh2 θκ = βE (6)
,
e κ−1
β≡ k 1T is the inverse temperature at t0 = 0. {|0 N } is recognized to be
B
a representation of the CCR’s at finite temperature: |0 N is a thermo
field dynamics (TFD) state in the real time formalism and can be
shown to be an SU (1, 1) squeezed coherent state.

˜
The mirror A modes actually account for the quantum noise of Brow-
nian nature in the fluctuating random force in the system-environment
coupling (entanglement).

18

Stability of order parameter N against quantum fluctuations is a man-
ifestation of the coherence of boson condensation.

⇒ memory N not affected by quantum fluctuations. In this sense, it
is a macroscopic observable. |0 N is a “macroscopic quantum state”.

⇒ “change of scale” (from microscopic to macroscopic scale) dynam-
ically achieved through the coherent boson condensation mechanism.

19

At finite volume V , time evolution of |0 N is formally given by
H
|0(t) N = exp −it I |0 N (7)
¯
h
1
exp tanh (Γκ t − θκ )A†A† |0 0 ,
˜
=
κ cosh (Γκt − θκ )
obtained by using the commutativity between HI and G(θ).

|0(t) N is an SU (1, 1) generalized coherent state, it is specified by the
initial value N , at t0 = 0, and N 0(t)|0(t) N = 1, ∀t.

Provided κ Γκ > 0,

lim N 0(t)|0 N ∝ lim exp −t Γκ = 0 . (8)
t→∞ t→∞ κ
In the infinite volume limit we have (for d3 κ Γκ finite and positive)

N 0(t)|0 N −→ 0 ∀t = 0 ,
V →∞
N 0(t)|0(t ) N −→ 0 ∀t,t , t=t . (9)
V →∞
20

In agreement with observations:

the QFT dissipative dynamics ⇒

∗ (quasi-)non-interfering degenerate vacua (AM pattern textures)

∗ (phase) transitions among them (AM patterns sequencing)

∗ huge memory capacity

The original many-body model could not describe these features.

22

In the “memory space”, or the brain state space (the space of uir),
{|0 N }, for each N -set, describes a physical phase of the system and
may be thought as a “point” identified by that specific N .

The system may shift, under the influence of one or more stimuli act-
ing as a control parameter, from vacuum to vacuum in the collection
of brain-environment equilibrium vacua (E0 = 0), i.e. from phase to
phase,

⇒ the system undergoes an extremely rich sequence of phase transi-
tions, leading to the actualization of a sequence of dissipative struc-
tures formed by AM patterns, as indeed experimentally observed.

23

Let |0(t) N ≡ |0 N at t, speciﬁed by the initial value N at t0 = 0.

Time evolution of |0(t) N = trajectory of ”initial condition” speciﬁed
by the N -set in the space of the representations {|0(t) N } .

Provided changes in the inverse temperature β are slow, the changes
in the energy Ea ≡ k Ek Nak and in the entropy Sa are related by
1
˙
Ek Nak dt =
dEa = dSa = dQ (10)
,
β
k
i.e. the minimization of the free energy dFa = 0 holds at any t

⇒ change in time of the condensate, i.e. of the order parameter,
turns into heat dissipation dQ.

24

Dissipation ⇒ time-evolution of |0(t) N at ﬁnite volume V controlled
by the entropy variations ⇒ irreversibility of time evolution (breakdown
of time-reversal symmetry) ⇒ arrow of time (a privileged direction in
time evolution)

25

Mesoscopic background activity conforms to scale-free power law
noise.

Figure 11. Evidence is summarized showing that the mesoscopic background activity conforms to
scale-free, low-dimensional noise [Freeman et al., 2008]. Engagement of the brain in perception and other
goal-directed behaviors is accompanied by departures from randomness upon the emergence of order (A),
as shown by comparing PSD in sleep, which conforms to black noise, vs. PSD in an aroused state showing
excess power in the theta (3 − 7 Hz) and gamma (25 − 100 Hz) ranges. B. The distributions of time
intervals between null spikes of brown noise and sleep ECoG are superimposed. C,D. The distributions
are compared of log10 analytic power from noise and ECoG. Hypothetically the threshold for triggering
a phase transition is 10−4 down from modal analytic power. From [Freeman, O’Nuillain and Rodriguez,
2008 and Freeman and Zhai, 2009]

Fig. 1. The ﬁrst ﬁve stages of Koch curve.

Let H(L0 ) denote lengths, surfaces, volumes.

Scale trasformation: L0 → λL0 .

H(λL0 ) = λd H(L0 )

1 λ=1 .
The square S of side L0 scales as 22 S, 2

1
The cube V scales as 23 V .

Thus in λd , d = 2 and d = 3 for surfaces and volumes.

S( 1 L0) 1
1 and V ( 2 L0) = p = 1 .
2
Note: S(L ) = p = 4 8
V (L0)
0

In both cases p = λd .

For lengths L0, p = 1 ; 1 = λd and p = λd and thus d = 1.
2 2d

For the Koch curve: the relation p = λd gives:

q = 1d
qα = 1, where α = 4,
3

i.e.

d = ln 4 ≈ 1.2619.
ln 3

The non-integer d is called

fractal dimension , or self-similarity dimension .

u1,q (α) ≡ q α u0, q = 1d ,
Stage n = 1: α=4
3
d = 1 to be determined.

u2,q (α) ≡ q α u1,q (α) = (q α)2 u0.
Stage n = 2:

By iteration:

un,q (α) ≡ (q α) un−1,q (α), n = 1, 2, 3, ...

i.e., for any n

un,q (α) = (q α)n u0.

which is the “self-similarity” relation characterizing fractals.

Notice! The fractal is mathematically deﬁned only in the limit of
inﬁnite number of iterations (n → ∞).

1
√ (q α)n
Notice!
n!

is the basis in the space of entire analytical functions, where coherent
states are represented.

|qα|2 n
∞ (qα) |n
|qα = exp(− 2 ) √
n=0 n!

a |qα = qα |qα ,

this establish a link between fractals and coherent states and conﬁrms
the role of coherence in brain activity.

the operator (a)n acts as a “magnifying” lens: the nth iteration of
the fractal can be “seen” by applying (a)n to |qα (and restricting to
real qα):

qα|(a)n |qα = (qα)n = un,q (α), qα → Re(qα).

Other predictions in agreement with experiments :

• very low energy required to excite correlated neuronal patterns,

• AM patterns have large diameters, with respect to the small sizes
of the component neurons,

• duration, size and power of AM patterns are decreasing functions
of their carrier wave number k,

• there is lack of invariance of AM patterns with invariant stimuli,

• heat dissipation at (almost) constant in time temperature,

26

• the occurrence of spikes (vortices) in the process of phase transi-
tions,

• the whole phenomenology of phase gradients and phase singularities
in the vortices formation,

• the constancy of the phase ﬁeld within the frames,

• the insurgence of a phase singularity associated with the abrupt
decrease of the order parameter and the concomitant increase of
spatial variance of the phase ﬁeld,

27

• the onsets of vortices between frames, not within them,

• the occurrence of phase cones (spatial phase gradients) and random
variation of sign (implosive and explosive) at the apex,

• that the phase cone apices occur at random spatial locations,

• that the apex is never initiated within frames, but between frames
(during phase transitions).

• The model leads to the classicality (not derived as the classical
limit, but as a dynamical output) of functionally self-regulated and
self-organized background activity of the brain.

28

A crucial neural mechanism:

the event that initiates the transition to a perceptual state is an
abrupt decrease in the analytic power of the background activity to
near zero (a null spike), associated with the concomitant increase of
spatial variance of analytic phase.

The null spikes recur aperiodically at rates in the theta (3 − 7 Hz) and
alpha (8 − 12 Hz) ranges,

it has rotational energy at the geometric mean frequency of the pass
band, so it is called a vortex.

The vortex occupies the whole area of the phase-locked neural activity
of the cortex for a point in time.

Between the null spikes the cortical dynamics is (nearly) stationary
for ∼ 60 − 160 ms. This is called a frame.
29

During periods of high amplitude the spatial deviation of phase (SDX )
is low,

the phase spatial mean tends to be constant within frames

and to change suddenly between frames,

The reduction in the amplitude of the spontaneous background ac-
tivity induces a brief state of indeterminacy in which the signiﬁcant
pass band of the electrocorticogram (ECoG) is near to zero and the
phase of ECoG is undeﬁned.

Each null spike initiates a spatial phase cone.

The phase cone is a spatial phase gradient that is imposed on the
carrier wave of the wave packet in a frame by the propagation velocity
of the largest axons having the highest velocity in a distribution.
30

The arriving stimulus can drive the cortex across a phase transition
process to a new AM pattern.

The observed velocity of spread of phase transition is ﬁnite, i.e. there
is no “instantaneous” phase transition.

These features have been documented as markers of the interface
between microscopic and mesoscopic phenomena.

31

Figure 10. Null spikes are observed by band pass filtering the EEG (A), applying the Hilbert
transform [Freeman, 2007b] to get the analytic power (B), and taking the logarithm (C). On each channel
the downward spikes coincide with spikes in analytic frequency (D) reflecting increased analytic phase
variance. The flat segment between spikes reflects the stability of the carrier frequency of AM patterns.
The spikes form clusters in time but are not precisely synchronized. One or more of these null spikes
coincides with phase transitions leading to emergence of AM patterns. The modal repetition rate of the
null spikes in Hz is predicted to be 0.641 times the pass band width in Hz [Rice, 1950, p. 90, Equation
3.8-15].

The possibility of deriving from the microscopic (quantum) dynamics
the classicality of trajectories in the memory space is one of the merits
of the dissipative many-body ﬁeld model.

These trajectories are found to be classical deterministic chaotic tra-
jectories ∗

The manifold on which the attractor landscapes sit covers as a “clas-
sical blanket” the quantum dynamics going on in each of the repre-
sentations of the CCR’s (the AM patterns recurring at rates in the
theta range (3 − 8 Hz)).

∗ E.Pessa and G. Vitiello, Mind and Matter 1 59 (2003)
E. Pessa and G. Vitiello, Intern. J. Modern Physics B 18, 841, (2004)
G. Vitiello, Int. J. Mod. Phys. B 18, 785 (2004)

37

The emerging picture is that a stimulus selects a basin of attraction
in the primary sensory cortex to which it converges, often with very
little information as in weak scents, faint clicks, and weak ﬂashes.

The convergence constitutes the process of abstraction.

Each attractor can be selected by a stimulus that is an instance of
the category (generalization) that the attractor implements by its AM
pattern:

⇒ the waking state consists of a collection of potential states, any
one of which but only one at a time can be realized through a phase
transition.

38

The speciﬁc ordered pattern generated through SBS by an external
input does not depend on the stimulus features. It depends on the
system internal dynamics.

⇒ The stored memory is not a representation of the stimulus.

The model accounts for the laboratory observation of lack of invari-
ance of the AM neuronal oscillation patterns with invariant stimuli

The engagement of the subject with the environment in the action-
perception cycle is the essential basis for the emergence and main-
tenance of meaning through successful interaction and its knowledge
base within the brain.

It is an active mirror, because the environment impacts onto the self
independently as well as reactively.

The brain-environment “inter-action” is ruled by the free energy min-
imization processes.
39

The continual balancing of the energy ﬂuxes at the brain–environment
interface amounts to the continual updating of the meanings of the
ﬂows of information exchanged in the brain behavioral relation with
the environment.

By repeated trial-and-error each brain constructs within itself an un-
derstanding of its surround, which constitutes its knowledge of its
own world that we describe as its Double ∗.

∗ G.
Vitiello, Int. J. Mod. Phys. B 9, 973 (1995)
G. Vitiello, My Double Unveiled. Amsterdam: John Benjamins, 2001

40

Giuseppe Vitiello
My Double Unveiled
Advances inin Consciousness Research
Advances Consciousness Research

“The other one, the one called Borges, is the one things happen
to....It would be an exaggeration to say that ours is a hostile rela-
tionship; I live, let myself go on living, so that Borges may contrive
his literature, and this literature justifies me....Besides, I am destined
to perish, definitively, and only some instant of myself can survive
him....Spinoza knew that all things long to persist in their being; the
stone eternally wants to be a stone and a tiger a tiger. I shall remain
in Borges, not in myself (if it is true that I am someone)....Years ago
I tried to free myself from him and went from the mythologies of the
suburbs to the games with time and infinity, but those games belong
to Borges now and I shall have to imagine other things. Thus my life
is a flight and I lose everything and everything belongs to oblivion, or
to him.
I do not know which of us has written this page.”∗

∗ JorgeLouis Borges, “Borges and I”, in El hacedor, Biblioteca Borges, Alianza
Editorial, 1960.
41

In conclusion,

John von Neumann noted that

“...the mathematical or logical language truly used by the central
nervous system is characterized by less logical and arithmetical depth
than what we are normally used to. ...We require exquisite numerical
precision over many logical steps to achieve what brains accomplish
in very few short steps” ∗.

The observation of textured AM patterns and sequential phase tran-
sitions in brain functioning and the dissipative quantum model de-
scribing them perhaps provide a way to the understanding of such a
view.
∗ J.
von Neumann, The Computer and the Brain. New Haven: Yale University Press,
1958, pp.80-81
42

Much work remains to be done in many research directions,

such as the analysis of the interaction between the boson condensate
and the details of the electrochemical neural activity,

or the problems of extending the dissipative many-body model to
account for higher cognitive functions of the brain.

At the present status of our research, the study of the dissipative
many-body dynamics underlying the richness of the laboratory obser-
vations seems to be promising.

43

COHERENCE, SELF-SIMILARITY AND BRAIN ACTIVITY

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