Generation of the Stream of Consciousness by Neuronal Synaptic Interaction

In previous work, a quantum mathematical formalism associated an element of experience with a single sensory neuron, as a local reduction of a global mental state. In contrast to the binary objective states of neuronal polarisation/depolarisation, neuronal experience was modeled as a continuous variable, the instantaneous value of which could only be estimated statistically from an ensemble of evoked responses to stereotyped stimulus presentation. In the present work, the quantum operations formalism of energy dissipation through amplitude damping is adopted to explain how smooth evolution of conscious experience might arise from discrete spikes and discontinuous synaptic transmission between neurons.


Conceptual Background
At the click of a Geiger counter, an observer infers that a nearby Americium-74 nucleus has decayed to Neptunium-72, emitting an alpha particle. Yet before and after this nuclear transmutation, the universe is in a coherent superposition of states of parent and of daughter radionuclide. Moreover, the state vector of the universe moves smoothly and continuously through Hilbert space over the lifetime of the radionuclide from the parent to daughter eigenspace, a decay that had appeared spontaneous and random generated by the time-independent interaction between nucleus and its immediate environment. Even considering only the reduced state of this one nucleus, the probability weighting of the parent in an incoherent mixture with daughter radionuclide must smoothly decay over time. These dynamics only appear as an abrupt transition because out of the mixture of parent and daughter radionuclides only one or other can be manifest in the world of the observer at a point in time.
This account raises the question whether other apparently discrete spontaneous events arising in the natural world might be generated similarly. This would be particularly plausible where transition events in subsystems were known to correlate with the state and smooth trajectory of a spatially distributed system of which subsystems were all part. The brain is such a multipartite and anatomically distributed system, in which the stream of unified consciousness is driven by transition events in neuronal components. Specifically, we consider whether a single spike, and the chemical synaptic transmission that results, might be modelled as a continuous interaction between an efferent and afferent neuron.

Existing Model: axioms and quantum foundations
In the author's previous work, two 'bridging principles' for the mind-brain relation were presented as axiomatic foundations of a neuronal model of sensory perception. The first postulate was that the "contribution of a single sensory neuron to the neural code for the sensory environment parallels the relationship between a neuronal element of experience and the overall sensory percept" [1]. As an example, if as suggested by the experiment of Britten [2] the organism's behavioural response to direction of visual motion, within a limited region of the visual field, can be predicted almost entirely by the response of a single, carefully selected neuron in the motionsensitive cortex V5, then the neuron singlehandedly encodes this aspect of the sensory environment.
The first bridging principle goes further to say that such a V5 neuron also affords, within a limited region, this same limited aspect of motion experience. The second postulate extended the mindbrain parallel to neuronal relations: "however the similarity between environmental features specified by a pair of neurons is encoded in their joint activity, this objective association parallels the qualitative proximity of respective neuronal elements of experience" [3]. Previous work emphasised perceptual rather than conceptual experience, but in what follows it will be assumed that these bridging principles can be extended to a more general correspondence between objective and subjective representation of information. A third and final bridging principle was suggested in section 1.1 of the current paper, but is here stated again formally: neuronal synaptic transmission generates not only the evolution of an objective neural representation of information but also the dynamics of conscious experience (those elements of experience and their relations corresponding to that objective representation by the first and second bridging principles).
Notwithstanding the close association, implied by these bridging principles, between elements of experience and their neural correlates, a purely physicalist approach, it was argued, would fail to capture two essential features of consciousness, subjectivity and unity. A starting point to the development of a model to describe these features was the principle of evoked responses in neurophysiology. Neural responses are quantised as spikes. Over an ensemble of stereotyped stimulus presentations, the best description of a sensory neuron's response to the stimulus at a point in time is its instantaneous firing rate, averaged across trials, not the number of spikes that have already occurred on an individual trial, or the objective state of the neuron that pertains at that moment. The quantum mathematical formalism is consistent with an interpretation that the state of a system really exists and has statistical effects over an ensemble of trials with identical starting conditions, even though the state itself is not directly measurable. In the quantum mechanical mathematical formalism of neural signaling that was developed, this feature of the quantum state was identified with the subjectivity of a neural element of experience. Spikes were considered to be objective transition events between definite levels of neuronal potential, even as an element of experience, inaccessible to direct measurement, smoothly decayed (like the quantum conception of alpha decay mentioned above). When the model was applied to the neurophysiologic scenario of repeated stereotyped stimulus presentations, the expectation value of spike potential across trials 3 of 25 was not only proportional to the instantaneous firing rate, but also to a scalar quantity of experience inherent in the inaccessible neural state (a 'perception value', see below).
If only this part of the quantum formalism had been appropriated, consciousness would have been neuro-anatomically locally explicable, consistent with the 'microconsciousness' theory of Zeki: "Activity in each separate processing node generates a microconsciousness for the attribute for which that node is specialised. Consequently, there are several microconsciousnesses, corresponding to the activity of cells at different nodes within different processing systems" [4]. It seemed to the author that such "summative atomism" was a poor description of the phenomenon: "a sub-mind is an atrocious monstrosity, just as is a plural-mind -neither having any counterpart in anybody's experience, neither being in any way imaginable" [5]. Indeed, the unity of consciousness seemed to be a fundamental feature. A unified consciousness was considered to be singular and not plural; complete in itself, not a fraction of a greater whole. In contrast, sensation, cognition, emotion and volition were considered not complete experiences but aspects of the subject's consciousness.
Whatever level of perceptual or conceptual experience is introspected upon, it always seemed possible to consider a higher level at which the relation between simultaneous percepts or concepts is experienced. The unified mental state was thus proposed to exist at the apex of a hierarchy of bound percepts and concepts. The author shared with Nagel the view that "It seems inevitable that psychophysical explanation will apply first at the level of some kind of elements of experience; but if these elements come together in a single consciousness, they must also be components of a single point of view" [6]. This suggested that, just as the bound percept might be resolved along multiple perceptual dimensions, perhaps a unified consciousness might usefully be represented as a mental state vector in a subjective space spanned by all of the dimensions of perceptual experience.
With these ideas in mind, two other features of the quantum mathematical formalism, superposition of states and tensor product combination of state spaces, were applied to a neuronal model that had been extended to the whole brain. This facilitated description of the mental state as a pure, inseparable state (a vector or uni-dimensional projector) on this combined system of vast dimension. There was proposed a direct correspondence between the unity of experience, a property of the system as a whole, and such coherence of the state of that system. The reduced state of any subsystem of one or many neurons would be a mixed-state density operator. Were this not so, the completeness privilege of consciousness described above would be lost. Such a mental state would be not only unified but also emergent: it would reduce to, but be not entirely constituted in the semi-classical phenomenal correlates of firing rate at the single neuron level, mentioned above.
The objective manifestation of an inseparable pure mental state would be firing correlations between neurons in anatomically remote regions that have no direct synaptic connection with each other, perhaps correlations that could not have been established classically by synaptic input from a common source.
In a decoherence theoretic interpretation of quantum mechanics, the emergence of a probability-weighted mixture of discrete objective states is the local manifestation of a smooth, unitary evolution generated by the interaction between a system and its environment [7]. In the further development of the neuronal model of consciousness to be presented below, it is proposed 4 of 25 that such a decoherence theoretic framework can reconcile discontinuous synaptic transmission with the stream of consciousness. Specifically, complex neuronal vector spaces, which might have seemed an unnecessary extravagance when modelling subjectivity and unity, will facilitate a description of unitary evolution of the mental state 1 . Of course, if this were merely the application of quantum physics to the brain, considered as an open quantum system, its quantum state would no longer remain pure after a vanishingly small decoherence time, although the state of a very large composite system including the brain and some part of its physical environment would remain pure. In order to maintain the picture of a pure mental state, the only interaction that will be considered in this neuronal model is that occurring at the synapse. Interactions that occur between sub-neural structures and their physical environment will be neglected.
Of course, classical neural dynamics, generated by synaptic transmission, are already well described. At a synapse on the dendrite of a recipient neuron, excitatory neuro-transmitters, released as a spike depolarises the pre-synaptic terminal, increase sodium permeability by opening chemically-gated channels and partially depolarise the post-synaptic membrane. Partial depolarisation at the trigger zone, by passive (or 'electrotonic') conduction of this excitatory postsynaptic potential, may reach a threshold at which voltage-gated sodium channels open, making the transmembrane potential abruptly positive. This opens similar channels in the adjacent neural membrane, so that complete depolarisation extends rapidly throughout the neuron as a spike. What is to be presented here will challenge none of this. Partly this is because the model is a coarsegrained description, neglecting all this subcellular activity to represent a spike as nothing more than a Dirac delta function. More importantly though, the contention is that the continuous evolution of the mental state reduces to the discrete synaptic transmission of information of the classical neuron doctrine at a local level of explanation, in the same way that the click of the Geiger counter is a local manifestation of the unitary evolution of the state vector of the universe.

Existing Model: mathematical formalism
In previous work, a unified consciousness, or mental state, was modelled as a vector on a composite tensor product of single neuronal complex vector spaces [3]. Each of these neuronal spaces was spanned by an orthonormal basis of integer 'action potential' states | ⟩. These were eigenvectors of a hermitian number operator : states of inevitability of corresponding integer spike counts , but in contradistinction to conventional neuroscience not the depolarisation events themselves. A superposition principle was introduced in which pure states | ⟩, sums of integer action potential states weighted by complex amplitudes ⟨ | ⟩, were all valid single neuronal states.
If a state | ⟩ were normalised, the squared modulus |⟨ | ⟩| 2 would be the prior probability that a neuron in that state would ultimately spike n times, and the expectation value of , 〈 〉, a sum of eigenvalues weighted by these respective probabilities, would be the expected spike count. The proposed equivalence of using the uni-dimensional projector | ⟩⟨ | or the vector | ⟩ for the description of a pure state of a neuron implied that a global phase across amplitudes ⟨ | ⟩ 1 Unitary evolution implies that the evolution operator when multiplied by its adjoint yields the identity (see section 3.1 Unitary dynamics of isolated neurons). It is a concept unrelated to the unity of consciousness, modelled in this formulation as an inseparable, pure mental state of the composite neuronal system.

Materials and Methods
Here we argue reductio ad absurdum that without synaptic transmission, action potentials would remain sequestered in neurons, perception values would be unchanging and consciousness static. We extend the quantum mechanical formalism summarised above to show that a single neuron isolated in this way could nevertheless undergo a unitary evolution generated by in which integer action potential eigenstates remain unchanged but for acquisition of a relative phase − . We extrapolate that in any neural system in which, by virtue of isolation, there is conservation of action potential, the subjective state of that system could nevertheless undergo a unitary evolution . But synaptic interaction within that system would require that the eigenvectors of were no longer single neuron integer action potential eigenstates. Applying this principle to a hypothetical isolated 2-neuron system, we model the oscillatory flow of a single action potential between component neurons and by synaptic transmission as a Jaynes-Cummings interaction [8]. We use an operator sum formalism to reduce this unitary evolution to a description of the continuous evolution of the single neuron state . We qualify the perception value concept, replacing the general constant with 'nervous energy': a neuron-specific experience-per-spike dependent on . We show how, in a realistic brain of polysynaptic neuronal interaction, this allows reconciliation of oscillatory flow of action potential between a neuronal pair with exponential decay in firing of the efferent neuron, of rate constant , predicted by the memoryless firing assumption [1]. 6 of 25

Unitary dynamics of isolated neurons
If we were able to set = 0, and in doing so abruptly impose upon the brain complete neuronal isolation, then a neuron's action potential could no longer be realised as spikes. If a single neuron were in a pure state | ⟩ with probability |⟨ | ⟩| 2 of eventual spikes prior to this imposed isolation, it would still have the same probability of spikes after such isolation were relieved.
While synaptic transmission to or from the neuron remained impossible, 〈 〉 and perception value would remain constant. In the proposed mathematical formalism the evolution of single neuronal elements of experience is entirely attributable to spikes and synaptic transmission.
Neuronal isolation would nevertheless be compatible with an evolution in which, over time , incremental integer action potential states acquire a relative phase . This implies an association between relative phase and experience that will be justified below. An operator on the single neuron space to effect such an evolution would be unitary, meaning that † is the identity for example the operator This particular is generated by , and leaves the phase of the ground state |0⟩ unchanged. Of course any operator that imposed the same relative phase but a different global phase on the neural state would describe an identical evolution. The evolution of an isolated single neuron in a superposition of 0-and 1-action potentials is depicted in Figure 1. It will be mathematically convenient, when we consider the contribution of such single-neuron evolution to a multi-neuron system, to use an equivalent , generated by an operator sharing with eigenvectors belonging to eigenvalues of unit separation (so that they again acquire relative phase − ), presented here in

Unitary dynamics of synaptic interaction
More generally of course, a multi-neuron system might be isolated, in the sense that there could be no synaptic transmission to or from the wider neural environment, even as such interaction was ongoing between component neurons. Action potential would be conserved within the system, even as it flowed between components. Within a 2-neuron system (Figure 2), an interaction between neurons might generate a unitary evolution that would not preserve |0 ⟩|1 ⟩ or|1 ⟩|0 ⟩ eigenvectors, but would nevertheless conserve one action potential. If neuron is initially in the 1-action potential state, while is initially in the ground state, then with prior probability ( ), after some brief time has elapsed, a single spike may have occurred in .
The reduced state of neuron will then be in a mixture of the 0-action potential state weighted by probability ( ) and the 1-action potential state weighted by the probability 1 − ( ): This is consistent with an evolution that takes the pure initial state |0 ⟩|1 ⟩ to a weighted superposition of |0 ⟩|1 ⟩ and |1 ⟩|0 ⟩, where the squared modulus of the amplitude of the second term is the firing probability: As in the non-interacting neuron model above, we choose an evolution in which |0 ⟩|0 ⟩ and |1 ⟩|1 ⟩ acquire a positive and negative phase respectively, determined by single neuron evolution, but the phase of the |0 ⟩|1 ⟩ term in the superposition does not change. The phase of the |1 ⟩|0 ⟩ term is chosen to simplify the mathematical form of the operator that generates this evolution.
The columns of the operator that achieves these transformations, ( ) , are orthonormal (since is unitary) and reflect the symmetry of the system:

Evolution of the reduced neural state: spike elements
More generally still, if neuron is initially in a superposition of 0-and 1-action potential states weighted by 0 and 1 respectively (and neuron is again assumed to start in the ground state) then a matrix equation for the unitary evolution of the initial product state is  Figure 3, and in matrix form:

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The action potential then 'sloshes' back and forth between neurons at a frequency of . 23 24 25

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The perception value was supposed to be a general description of the subjective  value of , so it would seem reasonable to allow the level of percep tual experience that 75 could be attributed to a neuron at a given instantaneous firing rate to be determined by a 76 neuron-specific , if that would lead to a resolution of the paradox.

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To achieve this, we will need a 'nervous energy' operator on the neuron al space  There is still an oscillatory flow of action potential, but as the discrepancy in between efferent and afferent neuron increases, the weighting of the 1 -action potential state in ( ) 103 dips to a lesser extent ( 2 2 + 2 ) and is more promptly restored (period Ω ).

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These dynamics describe the interaction between a single source and target neuron.

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Only when we discard the fanciful model of two isolated but interacting neurons, and will ultimately lead to a spike in any of one of these targets will be extremely small. We 123 will assume a constant weak coupling between a single source neuron and any one of 124 a large number of targets .

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If each target neuron were initially in the ground state then collectively they would     The definition of consciousness adopted in this paper is a general one provided by  196 quantum consciousness, whose ideas on conscious evolution will be discussed below.

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One might also question the validity of one or all of the three bridging principles. In

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There is one other reason why quantum mechanics has been attractive for theorists of

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So what then is this function of consciousness? We will consider this first from a

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A caveat to this discussion is that these putative benef its to brain function are non-

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For the current computational model of brain to be implemented in the future, a 325 potential objective neural property constraining will need to be explored. The finger, its fundamental frequency or pitch rises, perhaps to B. Considering the A and B 329 fundamentals to be analogous to the 1-action potential of a larger or smaller neuron 330 respectively, natural/artificial harmonics created by lesser finger pressure at 1 2 , 1 3 , 1 4 of 331 the length of the string could be considered analogues of 2 -, 3-, 4-action potential states.

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When this model is extended to the level of the complex neural system that is the 353 whole brain, a multitude of synaptic interactions between comp onent neurons generates a

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There has been telling criticism of previous quantum models of neural signalling.

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Quite apart from these big questions of context of the current theory in contemporary 438 science, there remains, of course, much to be don e to extend the model to ever more 439 realistic neural systems. Firstly we need to acknowledge that spikes rarely occur as 440 isolated events. Future work will need to relate the decay of superposition states of 441 multiple action potentials to the probability dist ribution of multiple-spike trains.

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Secondly, there is a need to explore how a directional flow of action potential through 444 a neuron can be modelled. All that has been described in the current paper is the continuous 445 evolution of a neural element of experience that parallels objective discontinuous spike 446 formation and synaptic transmission. Considering the subjective aspect, the specific 447 starting condition considered in the paper, one action potential in the source neuron and 448 no action potentials in a reservoir of target neurons, is a highly structured state. In an 449 information theoretic sense, the entropy of the combined subjective state increases through 450 synaptic transmission. The dynamics that ensue from this specific starting condition are that will need to be told in future work. The limitation of the current model that it seemed 453 to allow bidirectional or reversible flow of action potential through the neural circu it: not state still cannot be time-independent. As we extend our definition of brain to include 495 structures that seem ever less likely to support consciousness, such as thalamic relay 496 nuclei, spinal cord and dorsal root ganglia, external influences will still modulate the 497 potential of component neurons to fire. According to the model we have developed, there 498 is still a flow of action potential to and from the composite neural system.

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In our description of source and target neurons detailed above, we could have

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Conflicts of Interest: The author declares no conflict of interest.