The quantum model of the brain proposed by Ricciardi and Umezawa is extended to dissipative dynamics in order to study the problem of memory capacity. It is shown that infinitely many vacua are accessible to memory printing in a way that in sequential information recording the storage of a new information does not destroy the previously stored ones, thus allowing a huge memory capacity. The mechanism of information printing is shown to induce breakdown of time-reversal symmetry. Thermal properties of the memory states as well as their relation with squeezed coherent states are discussed.

Note comments enclosed in [square brackets] in the quotes are by me not by Dr. Vitiello.

1. Introduction

The purpose of this paper is the study of the problem of memory capacity in the Ricciardi-Umezawa quantum model of brain[1] by resorting to recent results on dissipative systems in quantum field theory (QFT)[2].

Coupling coefficients and activity thresholds of artificial neuron units are central ingredients in neural network machines simulating the brain functions. Ricciardi and Umezawa[1] have observed that in the study of the natural brain it is pure optimism to hope to determine the values of the coupling coefficients and the activity thresholds of all neurons by means of anatomical or physiological methods.

On the other hand, special activities of the natural brain persist in spite of changes in the number of alive neurons. In other words, the functioning of the whole brain appears not significantly affected by the functioning of the single neuron, neither do physiological observations show the existence of special long-lived neurons or the existence of a large redundancy in specialized neuronal circuits.

Besides the neurons, many other thousands of elements, such as glia cells, play a role in the brain activity, which, again, is not critically dependent on the single cell functioning.

A characterizing feature of the brain activity is instead related with nonlocality, namely with the existence of simultaneous responses in several regions of the brain to some external stimuli. This suggests that the brain may be in states characterized by the existence of long-range correlations among its elementary constituents; such long-range correlations seem to play a more fundamental role than the functioning of the single cell in the brain activity.

Storing and recalling information appear as a diffuse activity of the brain not lost even after destructive action of local parts of the brain or after treatments with electric shock or with drugs. This suggests to model these memory activities as coding of the brain states whose stability is to be derived as a dynamical feature rather than as a property of special neural nets which would be critically damaged by the above destructive actions. Stable long-range correlations and diffuse, nonlocal properties related with a code specifying the system state are dynamical features of quantum origin.

Ricciardi and Umezawa[1] have thus proposed a quantum model where the elementary constituents of the brain exhibit coherent behaviour and macroscopic observables are derived as dynamical output from their interaction.

Pioneering proposals relating advanced results in quantum optics, such as holography, with brain models were put forward by Pribram[3]. In more recent years, an analysis of non-algorithmic and non-computational character of brain functions has been made by Penrose[4], who has also proposed the quantum framework as the proper one too bridge microscopic dynamics with macroscopic functional activity of the brain.

For a general account of the application of modern statistical mechanics and spin glass theory to brain system see refs. [5] and [6]. In the quantum model of Ricciardi and Umezawa the elementary constituents are not the neurons and the other cells and physiological units, which cannot be considered as quantum objects, but some dynamical variables, called corticons, able to describe stationary or quasi-stationary states of the brain.

A crucial assumption, based on the fact that the brain is an open system in interaction with the external world, is that information printing is achieved under the action of external stimuli producing breakdown of the continuous symmetry associated to corticons.

As well known, in spontaneously broken symmetry theories the Lagrangian is invariant under some group, say G , of continuous transformations; however, the minimum energy state, i.e. the ground state or vacuum, of the system is not invariant under the full group G , but under one of its subgroups. In this case, general theorems of QFT[7] show that the vacuum is an ordered state and collective modes (called Nambu-Goldstone bosons) propagating over the whole system are dynamically generated and are the carriers of the ordering information (long-range correlations).

 In other words, order manifests itself as a global property dynamically generated and the quantum numbers characteristic of the collective mode act as coding for the ground state.

Ordering and coding are thus achieved by the condensation of collective modes in the vacuum.

One important point, is that the collective mode is a gapless mode and therefore its condensation in the vacuum does not add energy to it. As a consequence, the stability of the ordering and of the coding is ensured.

Another consequence is that infinitely many vacua with different degrees of order may exist, corresponding to different densities of the condensate. In the infinite volume limit these vacua are each unitarily inequivalent and thus represent different possible physical phases of the system, which then appears as a complex system with many macroscopic configurations.

The actual phase is determined once one among the many degenerate vacua is selected as an effect of some external action. Transitions among these vacua are in general not implementable (i.e., the non-existence of unitary transformations relating different vacua) in the infinite volume limit; however, in the case of open systems these transitions may occur (phase transitions), for large but finite volume, due to coupling with external environment.

The inclusion of dissipation leads thus to a picture of the system "living over many ground states" (i.e., continuously undergoing phase transitions)[8]. It is interesting to observe that even very weak (although above a certain threshold) perturbations may drive the system through its macroscopic configurations[8]. In this way, occasional (random) weak perturbations are recognized to play an important role in the complex behavior of living systems. The observable specifying the ordered state is called the order parameter and acts as a macroscopic variable since the collective modes present coherent dynamical behavior.

The order parameter is specific to the kind of symmetry brought into play and may thus be considered as a code specifying the vacuum. The value of the order parameter is related to the density of [Bose-Einstein] condensed Goldstone bosons in the vacuum and specifies the [structural domain] phase of the system in relation to the considered symmetry.

Since physical properties are different for different phases, the value of the order parameter may be considered as a code number specifying the system state.

In conclusion, code numbers specifying the phases may be organized in classes corresponding to different kinds of dynamical symmetry.

A typical example of spontaneous breakdown of symmetry is provided by the ferromagnet where the Lagrangian is invariant under the spin rotation group, but the ground state is invariant only under rotations around the direction of the magnetization. The collective modes are the spin-wave quanta or magnons and the system phases are indeed macroscopically characterized (i.e., coded) by the value of the magnetization, which is the order parameter. The magnetic order is thus a diffused, i.e., macroscopic, feature of the system.

The collective mode of the Ricciardi-Umezawa brain model has been called the symmetron[9] and the information storage function is represented by the coding of the ground state through symmetron condensation. The corticon has been assumed[9] to be a two-state system and the associated symmetry is a phase symmetry.

By following Frohlich[10], Del Giudice et al.[11-18] who have assumed that the symmetry to be spontaneously broken in living matter is the rotational symmetry for electrical dipoles. Such an assumption is phenomenologically based on the fact that living matter is made up of water and other biomolecules equipped with electric dipoles. The (electric) polarization density thus plays the role of an order parameter and the associated Goldstone modes have been named dipole wave quanta (dwq).

In the QFT approach to living matter the dynamical generation of collective modes thus sheds some light on the problem of change of scale in biological systems, namely the problem of the transition from the microscopic scenario to the many macroscopic functional properties (many macroscopic configurations) of the living systems.

Del Giudice et al. have shown the super-radiant or "lasering" behavior of water electrical dipoles[15] and the self-focusing propagation of the electrical field in ordered water[13], thus providing a conjecture for the formation of microtubules[13,18].

It has been shown[15] that the coherent interaction of water molecules with the quantized radiation field leads to a time scale for the coherent long-range interaction much shorter (10-14 sec ) than the one of short range interactions. Water coherent domains are therefore protected from thermalization.

Solitary wave propagation on biomolecular chains, as proposed by Davydov[19], has also been studied[12] and related with the triggering of the breakdown of symmetry.

Spontaneous breakdown of electric dipole rotational symmetry has turned out to be useful in further developments of the quantum brain model (referred to as quantum brain dynamics ( QBD )) worked out by Jibu and Yasue[20-24] who have identifed the Ricciardi-Umezawa symmetron modes with dwq and the corticon with the electric dipole field. They have obtained the first understanding of anesthesia and have elaborated a formalism for the super-radiant propagation of the electromagnetic field in cytoskeleton microtubules, also in relation to computational functions possibly associated with them [25].

Summing up, in the quantum brain model external stimuli are information printed by triggering the spontaneous breakdown of symmetry.

The stability of the memory is ensured by the fact that coding occurs in the lowest energy state and the memory's nonlocal character is guaranteed by the coherence of the dwq (or symmetron) condensate.

The recall process is described as the excitation of dwq modes under external stimuli of a nature similar to the ones producing the memory printing process.

When the dwq modes are excited the brain "consciously feels"[9] the pre-existing ordered pattern in the ground state.

Short-term memory is associated with metastable excited states of dwq condensate[1]. For a discussion on this poin t see also ref.[26].

The electrochemical activity observed by neurophysiology provides, according to Stuart et al.[9], a first response to external stimuli which, through some intermediate interaction, has to be coupled with the dipole field (or corticon) dynamics so as to allow the coding of the ground state.

One possibility, according to QFT approach to living matter[11-13], is that electrochemical activity may trigger, e.g. through ATP reaction, solitary dipolar waves on biomolecular chains. These solitary waves may in turn produce domains of nonzero polarization in the surrounding water molecules and the associated dwq condensation.

In the original brain model it is conjectured that the formation of ordered local domains may play a relevant role in this intermediate coupling[9].

In the quantum brain model only one kind of symmetry is assumed (the dipole rotational symmetry). Thus there is only one class of code numbers. Suppose a vacuum with a specific code number has been selected by the printing of a specific information. The brain then sets in that state and no other vacuum state is successively accessible for recording other information, unless a phase transition to the vacuum specified by the new code number is produced under the external stimulus carrying the new information. This will destroy the previously stored information (i.e., overprinting. Vacua labelled by different code numbers are accessible only through a sequence of phase transitions from one to another one of them.

Such a problem of memory capacity was already mentioned by Stuart et al.[9], who realized that the model was too simple to allow the recording of a huge number of information printings. Stuart et-al. then proposed that the model could be extended in such a way to present a huge number of symmetries (i.e., a huge number of code classes) and "a realistic model would therefore require a vector space of extremely high dimensions"[9], which however would introduce serious difficulties and spoil its practical use.

The purpose of the present paper is to show that, by taking into account the fact that the brain is an open system with dissipative dynamics, one may reach a solution to the problem of memory capacity which does not require the introduction of a huge number of symmetries.

It will be shown that, even by limiting the analysis to one kind of symmetry, infinitely many vacua are accessible to memory printing in a way that in a sequential information recording, the successive printing of information does not destroy the previous ones, thus allowing a huge memory capacity . Taking into account dissipation is crucial in reaching such a result.

Section 2 is devoted to the presentation of the dissipative quantum brain dynamics (DQBD). Its connection with thermal field theory and squeezed coherent states is discussed in sections 3 and 4, respectively.

2. Dissipative quantum dynamics of the brain: In this section the quantum model of the brain proposed by Ricciardi and Umezawa is extended to dissipative dynamics by resorting to some results on dissipative systems in QFT[2,27,28].

It will be shown that the problem of memory capacity may have a solution in the framework of dissipative quantum brain dynamics.

Let us start by the (trivial) observation that "only the past can be recalled".

This means that memory printing breaks the time-reversal symmetry of the brain dynamics and is another way to express the (obvious) fact that brain is an open, dissipative system coupled with external world.

As a matter of fact, in the quantum brain model, spontaneous breakdown of dipole rotational symmetry is triggered by the coupling of the brain with external stimuli. Here, however, our attention is focused on the fact that once the dipole rotational symmetry has been broken (and information has thus been recorded), then, as a consequence time-reversal symmetry is also broken.

Before the information recording process, the brain can, in principle, be in any one of the infinitely many (unitarily inequivalent) vacua. After information has been recorded, the brain state is completely determined and the brain cannot be brought to the state configuration in which it was before the information printing occurred (NOW you know it!).

Thus, information printing introduces the arrow of time in to brain dynamics.

Due to this memory printing process, the time evolution of the brain states is intrinsically irreversible. Ricciardi and Umezawa[1] have studied the brain's non-stationary or quasi- stationary states in the stationary approximation, thus avoiding damped oscillations.

In the following discussion, on the contrary , we will consider non-stationary states without using the stationary approximation.

A central feature of the quantum dissipation formalism[2,27,28]is the duplication of the field describing the dissipative system.

Let a and a' denote the gapless dwq mode [i.e., second quantized annihilation operators of the Frohlich dipole wave quanta which are Goldstone modes that Bose-Einstein condense in a pumped dissipative open system analogous to a laser] and the doubled [ time-reversed advanced mirror] mode [from the future] required by the canonical quantization of damped systems[2,27].

The a' mode is the "time-reversed mirror image"[2] of the a mode and represents the environment mode.

We have thus at to = 0 the splitting, or foliation , of the space of states in to infinitely many unitarily in-equivalent represen tations of the canonical commutation rules.

The freedom thus introduced by the degeneracy among the vacua ... , plays a crucial role in solving the problem of memory capacity . A huge number of sequentially recorded information [patterns] may coexist without destructive interference because the infinitely many vacua ... are independently accessible.

Recording the information in the code ... does not necessarily pro duce destruction of previously printed information [encodings] ...contrary to the nondissipative case, where differently coded vacua are accessible only through a sequence of phase transitions from one to another one of them.

In the present dissipative case the "brain (ground) state" may be represented as the collection (or the superposition) of the full set of memory states....

Alternatively, one may also think of the brain as a complex system with a huge number of macroscopic states (the memory states). In order to better clarify this point, it is useful to consider the dynamical group structure associated with our system.

...... the SU(1;1) structure of the dissipative dynamics introduces m -coded "replicas" of the system (foliation of the state space) and information printing can be performed in each replica without destructive interference with recorded informations in the other replicas.

In the nondissipative case the ... consecutive information printing produces overprinting.

The non-existence in the infinite volume limit of unitary transformations which may map one representation of code to another one code ... guaranties that the corresponding printed informations are indeed different or distinguishable informations (i.e., N is a good code) and that each information printing is also protected against interference from other information printings (i.e., absence of confusion among information [patterns]).

The effect of the finite (realistic) size of the system may however spoil unitary inequivalence and may lead to "association" of memories.

We therefore realize that in the dissipative dynamics ruled by the Hamiltonian (3a), the difference in [the number of modes a] number of their mirror time-reversed modes a'] is kept constantly zero during time evolution. Formally, at finite volume V, the time evolution of the memory state ... is given by ... a generalized coherent state.

In the infinite volume limit, time evolution of [the selected vacuum] would be rigorously frozen ... the associated Hilbert spaces [for each distinct degenerate vacuum state] are unitarily inequivalent to each other ...however, in realistic situations, a finite lifetime may be possible due to effects of the system boundaries (cf. Eq.(12)).

Time evolution of the memory state ... is thus represented as the trajectory of the "initial condition" specified by the [code] in the space of the representations ... of the canonical commutation relations.

The non-unitary character of time-evolution implied by damping is consistently recovered in the unitary inequivalence among represen tations at different times in the infinite-volume limit.

....we thus conclude that the memory state and its time-evolved state are squeezed coherent states. The degeneracy of the memory states is labelled by different codes). In conclusion, the memory code ...is nothing but the squeezing parameter classifying the squeezed coherent states ...Note that to different squeezed states are associated with unitarily inequivalent representations of the canonical commutation relations in the [unphysical] infinite volume limit: In dissipative quantum brain dynamics the huge (infinite) number of squeezed states, labelled by the squeezing parameter constitute the memory capacity.

Let us now summarize the main points of the discussion presented in this paper:

1. In dissipative quantum brain dynamics infinitely many vacua coexist and a huge number of informations may be sequentially recorded without destructive interference.

2. The problem of memory capacity in the quantum brain model, arising from the fact that vacua labelled by different code numbers belonging to the same class are accessible only by a sequence of phase transitions, however, a solution is in the intrinsic dissipative character of brain dynamics.

3. As we have indeed stressed, the process of information printing by itself produces the breakdown of time-reversal symmetry and thus introduces the arrow of time into brain dynamics.

4. The key point is that the resulting dissipative dynamics cannot be worked out without the the introduction of the time-reversed image (the tilde-system) of the original system.

5. As a consequence, energy degeneracy is introduced and the brain ground state may be represented as a collection (or superposition) of infinitely many degenerate vacua or memory states, each of them labelled by a different code number and each of them independently accessible to information printing (without reciprocal interference). Many information storage levels may then coexist thus allowing a huge memory capacity.

6. Differently stated, the brain system may be viewed as a complex system with (infinitely) many macroscopic configurations (the memory states). Dissipation, which is intrinsic to the brain dynamics, is recognized to be the root of such a complexity, namely of the huge memory capacity.

7. Time evolution of the coded memory state is represented as a trajectory of the initial conditions ... running over the states each one minimizing the free energy functional.

8. Memory states have also been shown to be squeezed coherent states.

Let us close the paper with few more comments. The quantum field theory approach to living matter does not require the introduction of other symmetries than the dipole rotational symmetry (and the electromagnetic gauge symmetry , see ref. [13]).

In the case other symmetries could be required in future developments of such an approach, to each broken symmetry will be associated a code class. Then the memory state will carry as many labels (codes) as there are dynamical symmetries broken. In such a case, the Goldstone modes associated with a special label may interfere with the Goldstone modes associated to some other label of the same state. This may produce fluctuations in their [Bose-Einstein] condensation and thus originates the mechanism of "association" of memories, by which some information is recorded with some "confusion" due to the presence of elements belonging to a different information; or also, the recalling of some information may trigger the recalling of some other information.

As already observed, association of memories may also occur when, as in the present paper, only one kind of symmetry is considered. In such a case, "interferences" are due to the realistic (finite) size of the system (boundary effects) making the memory states not exactly orthogonal (unitary nonequivalence is spoiled).

...In addition to breakdown of time-reversal (discrete) symmetry , already mentioned in the previous sections, we also have spontaneous breakdown of time-translation (continuous) symmetry . Dissipation (i.e. energy non-conservation) has been thus described in this paper (see also ref. [2]) as an effect of the breakdown of time translation and time-reversal symmetry.

Finally, according to the original quantum brain model, the recall process is described as the excitation of dipole wave quanta modes under an external stimulus which is "essentially a replication signal"[9] of the one responsible for memory printing.

When Frohlich dipole wave quanta are excited, the brain "consciously feels"[9] the presence of the condensate pattern in the corresponding coded vacuum.

The replication signal thus acts as a probe by which the brain "reads" the printed information.

In this connection we observe that the dipole wave quanta may acquire an effective nonzero mass due to the effects of the system finite size[12]. Such an effective mass will then introduce a threshold in the excitation energy of dipole wave quanta so that, in order to trigger the recall process an energy supply equal or greater than such a threshold is required. Nonsufficient energy supply may be experienced as a "difficulty in recalling".

At the same time, however, the threshold may positively act as a "protection" against unwanted perturbations (including thermalization) and contribute to the memory state stability.

In the opposite case of zero threshold, any replication signal could excite the recalling and the brain would fall into a [manic?] state of "continuous flow of memories".

The [time-reversed advanced tilde] system is indeed a "replication" of the [retarded] system and plays in fact a central role in the recalling process.

Eqs.(16) show that the creation (excitation) of the [retarded] mode is equivalent, up to a factor, to the destruction (from the memory state) of the [tilde advanced] mode. In this sense the coupling term of the [tilde advanced] mode with the [nondissipative retarded] mode in the Hamiltonian can be seen as a self-interaction term of the [retarded] system, thus confirming the role of [tilde advanced] system in "self-recognition" processes.

Remarkably, the [advanced] tilde-system also represents the environment effects and cannot be neglected since the brain is an open system. Therefore, the [advanced precognitive remote-viewing ?] tilde-modes can never be eliminated from the brain dynamics. The tilde-modes thus might play a role as well in the unconscious brain activity.

...Moreover, we have seen that the [tilde] ~ A system is the time-reversed image of the [retarded] A system. Thus the [tilde]~ A system is the "mirror in time" system. This fact, together with the role of the [tilde] ~ A modes in the self-recognition processes, leads us to conjecture (also accepting the literary image of consciousness as a "mirror") that the [advanced from the future] tilde-system is actually responsible for consciousness mechanisms.

Consciousness emerges as a manifestation of the dissipative quantum dynamics of the brain.

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Acknowledgments: I would like to warmly thank H. Umezawa for many stimulating discussions on the fascinating subject of brain dynamics. I am also glad to thank E. Del Giudice, M. Jibu, R. Penrose, K.H. Pribram, M. Rasetti and K.Yasue for the stimulating inputs I received from them. Finally , I am grateful to M. Rasetti for his encouragement in writing this paper.

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