Thomas L. Clarke

University of Central Florida/Institute for Simulation and Training

3280 Progress Drive

Orlando, FL 32826

(407)658-5030, FAX: (407)658-5059

tclarke@ist.ucf.edu

There is a little evidence and much speculation that the highest brain functions such as intelligence and computation are fundamentally quantum mechanical.

This paper presents a simple superregenerative mechanism for coupling quantum phenomena to neural phenomena. This mechanism should not only be useful in explaining brain function, but also suggests mechanisms for engineering quantum effects into neural and other computations.

The brain can be thought of as an outer layer of cortex and deeper thalamic structures. The prominence of the cortex in man suggests that it has much to do with intelligence. Areas of cortex which perform distinct data processing functions have been identified, and associated signal pathways to thalamic structures have been traced.

While the location of the fundamental process of attention or consciousness remains elusive, there is strong evidence that the deeper thalamic structures are vital to attention. Edelman (1989) has sketched how reentrant loops between thalamic structures and the cortex may underly consciousness. In a similar way Cauller and Kulics (1991) invoke back projections within the cortex that involve thalamic regions. Crick and Kock (1990) and also Llinas and coworkers (1994) have discovered coherent oscillations at 40 Hz between the cortex and the thalamus that may be involved in consciousness.

Beginning with Hoffman (1985) who suggested similarities between neural maps in the brain and category theory, Clarke and Ronayne (1991) investigated the application of differential geometric concepts in intelligence. These cortical neural maps have a natural implementation in artificial neural networks. More recently (Pratt, 1992; Clarke, 1998) connection has been made between Hoffman’s ideas and quantum mechanics.

From this background the recent surge of interest in quantum computation and its possible connection to consciousness has been of great interest. Hameroff (1994) has been an advocate of the view that quantum events are significant in the brain at the level of microtubule activity within neurons. Within the field of pure computation Shor (1994) and others have explored algorithms and implementations of quantum computation to real problems such as composite number factoring.

Hameroff’s ideas for providing a role for quantum mechanics in the brain lack an explicitly detailed mechanism for connecting quantum events with neural dynamics. The 40 Hz oscillation provides a clue as to how this connection can be made.

One of the oldest radio receivers is the superregenerator (Lawson and Uhlenbeck, 1950). This receiver uses a single vacuum tube (or other non-linear amplifying element) to detect week signals by mounting the tube in an oscillating circuit. The circuit oscillations are periodically stopped or quenched either by additional circuitry or by self-limitations designed into the basic oscillator circuit. The result is that the oscillation amplitude grows exponentially until it is quenched.

If externally quenched, the oscillator begins oscillation with amplitude determined by the signal level and then grows exponentially possibly reaching saturation level. As a result the average oscillation level depends on the initial signal level. To quantify this, the signal, s, will be assumed to have the form where e is the slowly varying envelope function which modulates the carrier wave at angular frequency . Since the envelope is band limited to, the signal will be narrow-band in the frequency domain so that the rapid oscillation at the carrier frequencycan be suppressed, either by mathematical convenience or physically by heterodyning all signals to base-band. The quenching operation occurs at the discrete times . In the time interval the oscillator amplitude will begin at the value and will grow to where is the time constant of oscillator amplitude growth. In a successful superregenerative design so that the exponential amplification factor is large, but the quenching time is shorter than the Nyquist sampling interval . The situation is as shown in Figure 1.

If self quenched the oscillator signal grows until it reaches the quenching level at which time it is reset. The osciallation thus grows more rapidly when a signal is present so that the average oscillation level is a function of the signal level as shown in Figure 2. If is the oscillator amplitude at which the oscillator self quenches, then the time before quenching is and the average oscillator output level is . Proper self-quenched design locates the logarithmic transfer function so that non-linearity is minimized as the same time keeping the maximum (time of growth from noise initiated oscillation to ) below .

By this simple expedient a single vacuum tube or other non-linear active element is able to amplify signals at the level of thermal or quantum noise. It is natural to hypothesize that the 40 Hz oscillation in the brain represents a quenching signal for a superregenerator-like process within the brain. Superregeneration thus provides a mechanism for the amplification of quantum events such as those postulated within microtubules by Hameroff.

It is very easy to implement superregeneration within a neural network, natural or artificial. The output of two neurons cross connected as a flip flop grows exponentially from a metastable state to the final state depending on initial noise or signal (Mead and Conway, 1980). As shown in Figure 3, the flip-flop output voltage obeys where is the initial voltage of the flip-flop, is the metastable equilibrium voltage and is the circuit time constant. If these neurons are reset by an external quenching signal, the result is that the average output level is proportional to the initial imbalance of the neural flip-flop. The flip-flop acts as a superregenerative receiver. Circuitry of this nature is easily integrated; in fact the sense amplifiers of dynamic RAM chips can be regarded as superregenerative detectors.

This superregenerative mechanism should provide a convenient means to implement quantum computation in a neural network. Recent work shows that the radio frequency signals from nuclear magnetic resonance (NMR) in a liquid may provide the most convenient and robust implementation of quantum computation (Gershenfeld and Chuang, 1998). Simply connecting driving and sensing coils to a neural network thus adds a basic a quantum computational capability to the network

The neurons connected to the sensing coil can be configured, or may automatically configure themselves when subject to backpropagation or genetic algorithms to take advantage of the quantum possibilities. With suitable adjustment of network weights the resulting superregenerative elements make the spin echo data available for processing by the network. Since neural networks can be viewed as processing complex-valued data (Clarke, 1990) the NMR signals integrate naturally into the neural computation.

Biologically, NMR based neural quantum computation is perhaps more plausible than Hameroff’s microtubule approach since the brain is full of "wet" NMR active nuclei. Further NMR quantum computation mediated via superregeneration is a simple process that biological evolution could be expected to hit upon it by variation and selection.

The mathematical details of superegenerator based neural quantum computation are currently being investigated; particularly is being given to developing extensions of training algorithms such as backpropagation. One natural application, for superregenerative neural quantum computation is speech recognition using a complex Markov model. A wide variety of algorithms such as Shor's algorithm make use of the ability of a quantum computer to rapidly evaluate DFTs and the applications of these algorithms to superregenerative neural quantum computation are being explored. Simple apparatus to physically test superregenerative neural quantum computation are also being explored.

Cauller and Kulics, 1991, The neural basis of the behaviorally relevant N1 component of the somatosensory evoked potential in awake monkeys: Evidence that backward cortical projections signal conscious touch sensation, Experimental Brain Research, 84:607-619.

Clarke, T. L., 1998, A comparison of linear logic with wave logic"; Fifth International Symposium on Artificial Intelligence And Applied Mathematics.

Clarke, T.L. and T. Ronayne, 1991, Differential geometric approach to learning in machine vision, Proceedings 1991 Florida Artifical Intelligence Research Symposium (FLAIRS).

Clarke, T. L., Generalization of neural networks to the complex Plane, Proceedings 1990 International Joint Conference on Neural Networks (IJCNN).

Crick, F. and C. Loch, 1990, Towards a neurobiological theory of consciousness, Seminars in the Neurosciences 2:263-275.

Edelman, G. M., 1989, The Remembered Present: A Biological Theory of Consciousness: New York, Basic Books.

Girard, J.Y., 1987. "Linear logic", Theoretical Computer Science, 50:1-102.

Gershenfeld, N. and Isaac L. Chuang, I. L., 1998, Quantum computing with molecules, Scientific American, June.

Hameroff, S. R. 1994, Quantum coherence in microtubules: A neural basis for emergent consciousness? Journal of Consciousness Studies, 1: 91-118.

Hoffman, W. C, 1985, Some reasons why algebraic topology is important in neurophysiology: perceptual and cognitive systems as fibrations, Int. J. Man-Machine Studies, 22: 613-650.

Lawson, James L., and George E. Uhlenbeck, 1950. Threshold Signals: New York, McGraw Hill.

Llinas, R. R, U. Ribary, M. Joliot, and X.J. Wang, 1994, Content and context in temporal thalamocortical binding. In G. Buzsaki, R. R. Llina, and W. Singer eds, Temporal Coding in the Brain: Berlin, Springer-Verlag.

Mead, Carver, and Lynn Conway, 1980. Introduction to VLSI Systems: Reading, Mass, Addison-Wesley.

Pratt, Vaughan R, 1992. Linear logic for generalized quantum mechanics. In Proceedings of the Workshop on Physics and Computation, pages 166-180, Dallas, Texas, IEEE Press.

Shor, Peter W, 1994, "Algorithms for Quantum Computation: Discrete Logarithms and Factoring"; 35th Annual Symposium on Foundations of Computer Science, IEEE; pp 124-134.