Quantum-like Networks

The following sections are included:

• Dedication
• Introduction
• About the Author
• Contents
• Notes to the Reader

There are few entries in the lexicon more abused than the word “quantum.” It is one of the aims of this foundational chapter to precisely elucidate why and how we came to use this term in the context of a study of very complex systems such as those found in living organisms. This we attempt here by rigorously rehearsing the logic that arises when we try to cope with complexity by blocking or ignoring the differences between states of such systems which would produce negligibly discernible differences of effect. Roughly speaking, we regard such states as being confusable if their effects are indistinguishable. (We call this hiding the variables, which may be a misnomer since it is the differences in the variables that we are hiding.) We explain in this chapter how this leads to a quantum-like logic, quite independently of any considerations of quantum physics itself, except for the purposes of post facto comparisons and interpretations. This ultimately reduces to a consideration of subsets of Euclidean spaces from this point of view which will form the basis of the neuronal network model introduced in subsequent chapters…

In this chapter we introduce our network models and lay the groundwork for what might be described as their physics-like aspects. These notions will be applied and developed further in later chapters in concert with the other face of our model, namely the computational structure, which will be the subject of the next chapter…

We have so far considered the generalization of Boolean logic called orthologic (OL) as the appropriate logic under our doctrine of hidden variables. This was used to justify the move to full finite dimensional real Hilbert spaces as appropriate to a logical treatment of the state spaces of our network models. Now OL is modeled by single ortholattices. That is to say, in our cases, by the subspaces of a single finite dimensional Hilbert space. However, we have already seen how different networks may be combined and connected to each other and how this entails combinations of, and maps between, their respective state spaces. OL does not seem capable of reflecting such interactions. In fact there is an overarching or external logic that can deal with the case of many spaces, relative to which OL may be regarded as the logic internal to a single space. In this chapter we introduce this logic and a slight variant of it…

The coins of our realm emerged in Chapter 2 as the so-called firing patterns. These are the states of possible subsets of co-operating or co-spiking b-neurons of a given b-network, along with their possible superpositions. The eigenstates of the p-Hamiltonian of the b-network were invested with cognitive significance since they are more stable than other states. Of course, a judgement as to what exactly this cognitive significance is, its cognitive semantics as it were, is beyond the remit of our model. Perhaps they are (possibly short-lived) memoranda. In Chapter 3 we were pleasantly surprised to find that these firing patterns are storage capable commodities in the sense of our sequent calculus GN. We pointed out in that chapter that these two attributes are independent of each other, eigenspectra being aspects of the internal dynamics of individual b-networks, while the sequent calculus refers to structural issues: how these networks combine and interact. In what follows we will exploit both the internal dynamics and the external structural logic, separately and in tandem, to derive results for our networks that seem to be analogous to actual memory processes in brains…

J. Z. Tsien formulated his “theory of connectivity” to explain certain cognitive processes which were revealed in a long series of careful in vivo recording experiments upon rodent brains [60, 96, 97, 101]. Tsien’s “power-of-two” law has also been found serendipitously (for the case of three inputs) in a more recent study of macaque brains [67]. There is evidence for it in humans as well [34]: cf. section 5.5. Roughly speaking, the idea is that certain assemblages of neurons become specialized over the course of natural selection to respond to stimuli of particular significance to the survival of the organism. Moreover, the memories induced upon the introduction of these stimuli follow a particular pattern: namely, as new types of stimuli are introduced, all possible combinations of them are registered and stored for future use. This immediately implies a pre-existing hierarchy in which all combinations, from the most general in a category to the individual stimuli in the category, may be quickly apprehended and processed. The primitive selective advantage of such a system is apparent: a general DANGER! response to a frightening experience of some kind is clearly more survivable than a time-consuming pattern-matching search through a set of memories of other frightening experiences to decide the lethality of the one we are confronting. (Cf. the pattern completion mechanism discussed in section 4.2.) As a side effect, such an arrangement can contribute to higher cognitive functions, as seems to be the case in humans…

Combinators with rules of use generate certain patterns. These are generally deemed well-formed and the study of well-formedness and its absence in the context of linguistics, formal logic, and other disciplines, has been hotly pursued for decades if not centuries. Such studies generally come under the rubric syntax…

The following sections are included:

• Appendix A: Appendix to Chapter 1
  • Some logical and mathematical results
    • Modal identities
    • Modal propositions
    • Ortholattices
    • Subsets of ℝⁿ\{0}
    • Orthologic: Models, completeness and the Modal Embedding Theorem
• Appendix B: A Mathematics Primer
  • Some multilinear algebra
    • Tensor products
    • Exterior products
    • Exterior algebras
    • The Plücker Embedding
  • A note on convolution

The following sections are included:

• Bibliography
• Index