On single neurons

Please note that you must be a member by then, or you will no longer be able to access the material. Members who submit an essay for the assessment at the end of the course will be given a grade that will correspond to where they would have finished in the corresponding Stanford class. While not a formal Stanford qualification, it will be a realistic appraisal of their level of knowledge, and I will be happy to write a reference based on it.

The first  lecture was self-explanatory

We now begin the first technical lecture;

 p/word as before


We will stay with this material until everyone is comfortable with it.

This is why we're doing this work;

Current theme; Single neurons -  classical and quantum

Ours (2004) was the first work to show how single neurons could realistically perform processing of sensory data expressed simply as spectral such data. This work has since been corroborated by, for example, Tiago Branco et al. (2010). Essentially, we argued that subthreshold oscillations of the neuron allowed groups of neurons to “own” part of the spectrum. That can be conceived of using only classical physics..

Since our original work, quantum coherence at physiological temperatures has been demonstrated for biological systems in photosynthesis at the 3nm level characteristic of gap junctions in neurons (Hoyer et al, 2011). This finding converges with a controversy about quantum effects in neurons related to consciousness. While, in related work, we question the assumption in the later that “phase coherence” has in fact been demonstrated in the brain, there is a long-attested corpus of observations suggestive of entropically minimal states several times a second there.

We therefore speculate that gap junctions might allow a quantum superposition of states of the membrane potential of each neuron to be communicated to thousands of others. This will lead to entanglement of a scale that would allow the Fourier decomposition we envisage for the classical case be extended to a quantum description. This is the only currently physiologically plausible story about Quantum effects in the brain

In fact, we have  data to indicate that much of the statistical inferences in classical EEE/ECOG evince premature closure, and that this approach is certainly not ready – pace, the ORCH OR proponents – for the non-classical world.

The existence of phase coherence in gamma waves in the brain, and the relation of this phenomenon to consciousness, is a point of much consensus, with only the recent work of Pockett and her colleagues contradicting it. It has been further argued that the entropically minimal state resulting from this phase coherence might yield an environment conducive to quantum coherence.

While we believe that the work of Walter Freeman indeed is indicative of entropically minimal states in th brain occurring several times a second, we believe that the EEG/ECOG signal is too coarse to indicate synchrony. Indeed, we have findings from PCA , among other methods, indicating that a 64-electrode grid produces at most two signals. As for phase coherence, the stated electronic specifications of the equipment use expressly prohibit any such inference, as the error range of the equipment is too large. So this study of single neurons over the next few classes is REALLY important

O Nuallain, S CSLI, Stanford and T. Doris(2004) http://bcats.stanford.edu/previous_bcats/bcats04/html/nuallain.html

Hoyer et al, (2011) http://arxiv.org/pdf/1106.2911.pdf

See also  Branco et al (2010) http://www.sciencemag.org/content/329/5999/1671


The critical  paper on neural resonance is appended below 

WHAT IS NEURAL RESONANCE FOR?

SEAN O NUALLAIN, University of Ireland USA AND TOM DORIS

ABSTRACT. Vast amounts of research, both theoretical and experimental, are being carried out about neural resonance, subthreshold oscillations, and stochastic resonance in the nervous system. In this paper, we first offer a radically different computational model of neural functioning, in which integrate and fire behaviour is seen as a special case of the more encompassing resonate and fire processes. After commenting on the explanatory vista opened up by this model, we speculate on its utility for signal processing.

   KEYWORDS: Subthreshold oscillations; neural resonance; signal processing; resonate and fire.

1.  Introduction

While neural resonance can exist without subthreshold oscillations, a vast literature connects the two. For Wu et al (2001), the oscillations emerge from membrane resonance. The resonant current is steady-state potassium current, amplified by a sodium current. Izhikevich (2002) most explicitly drew consequences from the fact that the Hodgkin-Huxley model is a resonator. His point that a neuron's firing may depend on the timing of its afferent impulses is one that we believe to be well-taken. We have been careful to ensure that our model caters to all the possible scenarios (in-phase doublets, and so on) that he envisages. Like Wu et al (op. cit.) he interrelates subthreshold oscillations and bursts, coming to the conclusion that the intervals in bursts may be significant for communication. This is one line of reasoning that emerges, transformed and extended, in our work.

System level phenomena are also increasingly beginning to attract attention. Wu et al (ibid.) comment that a single excitatory stimulus to a mesencephalic V neuron can result in high-frequency spiking in a whole network under certain circumstances. Even more interestingly, the phenomenon of stochastic resonance (SR) has come into focus in neuroscience. SR is essentially a non-linear systems phenomenon through which, apparently paradoxically, a noisy environment can be exploited to amplify a weak signal. Reinker et al (2004) integrate the two resonance phenomena by asserting that subthreshold neural resonance manifests itself when thalamocortical neurons are stimulated with sine waves of varying frequency, and stochastic resonance emerges when noise is added to these stimuli.

The possibility that these phenomena have computational utility has not been lost on these and other researchers. However, we believe that ours is the first work credibly to interrelate the signal-processing task faced millisecond to millisecond by the brain with the phenomena in question. In their review article, Hutcheon et al (2000) comment that resonance and oscillation may have a role in such phenomena as gamma waves. Rudolph et al (2001) venture a more specific conjecture; responsiveness of neo-cortical pyramidal neurons to subthreshold stimuli can indeed be enhanced by SR, and under certain conditions the statistics of this background activity, as distinct from its intensity, could become salient. Obviously, such forms could have computational consequences.

For Freeman et al (2003), the conversion of sensory data into meaning is mediated by those gamma wave processes. The distinction between ours and Freeman's approach, which we are admirers of, is that we are looking for the resonant frequencies at the microscopic level in single neurons using novel solutions to the 4th order Hodgkin-Huxley equation, whereas Freeman finds them at the mesoscopic level in the characteristic frequencies of populations. Nevertheless, the thrust of the two approaches, and the critique of the integrate-an-fire model, is similar.

Yet the integrate -and-fire (INF) neuron has emerged largely intact, even if supplemented with resonant abilities (Reinker et al, op cit.). In this paper, our first goal is to call the integrity of the INF paradigm into question in a novel way. In particular , we wish to show that INF behaviour can be viewed as a specific phase in the cycle of a different neural model, the resonate-and-fire model (RNF). Our model caters to all the bursting situations-doublet, triplet etc -identified by Izhikevich (2002). However, our background as computer scientists impels us on another previously unexplored path at this stage. What actually are the sensory data that the brain is operating on? Intriguingly, a decomposition of such stimuli into their constituent power spectra affords a vista in which each resonating neuron may accomplish a part of a Fourier transform. These digital analog signalling processing (DASP) concerns form the next part of the paper. We recognise that, since the frequencies involved are changing, a more complex function approximation method like the Hilbert transform may be closer to neuroscientific reality; however, the ethos whereby individual neurons or groups thereof have the roles proposed remains the same.

Yet the way ahead may be more fascinating still. While quantum computing, as distinct from quantum cryptography, may still be a generation away, computational tasks such as data base search have already been achieved by exploiting the phenomenon of classical wave interference. In the most speculative part of the paper, we propose that dendro-dendritic connections may be complicit in this. Particularly in neocortex the dendrodendritic connections have only recently been recognized, since they are comparatively uncommon, in contrast to the axosynaptic connections among pyramidal cells, accounting for maybe 85 Finally, we allude to further work that we have done in which the RNF paradigm is applied to some classical problems with artificial neural nets (ANNS).

2. The Resonate and Fire Model

The Hodgkin-Huxley system exhibits a stable low amplitude oscillation which can be considered in isolation to the production of action potentials. Izhikevich has done preliminary work on the possibility that neurons may exhibit either integrative or resonance properties. He posits that the neuron experiences a bifurcation of the rest state and depending on the outcome subsequently behaves as either an integrator or a resonator.

If the rest state disappears via fold or saddle-node on invariant circle bifurcations, then the neuron acts as an integrator; the higher the frequency of the input, the sooner it fires. If the rest state disappears via an Andronov-Hopf bifurcation, then the neuron acts as a resonator; it prefers a certain (resonant) frequency of the input spike train that is equal to a low-order multiple of its eigenfrequency. Increasing the frequency of the input may delay or even terminate its response.

Integrators have a well-defined threshold manifold, while resonators usually do not. Integrators distinguish between weak excitatory and inhibitory inputs, while resonators do not, since an inhibitory pulse can make a resonator fire.

Izhikevich  points out that the Hodgkin-Huxley model exhibits behaviors which are a superset of the standard IFN model. The low amplitude oscillation of the membrane potential can be sustained for long periods without the need for an action potential to result. Only when the amplitude of oscillation reaches a threshold value does depolarisation and action potential generation ensue. The resonance phase of the process is non-trivial. Complex waveforms are permissible, and would suggest that this phase of neuronal behaviour is of some importance to the behaviour of the cognitive apparatus. The oscillations are directly related to the action potential, since the same parameter, membrane potential, is central to both phases. Since the action potential is of undoubted importance to the activity of the brain, it would appear that an intimately related phenomenon should be given thorough consideration. The IFN model is the result of a view of the neuron which only considers a brief period prior to the generation of the action potential. As such, we will show that the resonate and fire model is a superset of the IFN, that it is capable of capturing all of the properties of the IFN in addition to new and interesting capabilities with strong evidence supporting the idea that such properties are critical to the transduction of sensory data.

The physical basis for the resonate and fire model lies in the fact that every object has a frequency or a set of frequencies at which they naturally vibrate when struck, strummed or somehow distorted. Each of the natural frequencies at which an object vibrates is associated with a standing wave pattern. Standing waves are formed when oscillations are confined to a volume, and the incident waveform from the source interferes with the reflected waveform in such a way that certain points along the medium appear to be standing still. Such patterns of interference are produced in a medium only at specific frequencies referred to as harmonics. At frequencies other than the set of harmonic frequencies, the pattern of oscillation is irregular and non-repeating. While there are an infinite number of ways in which an object can oscillate, objects prefer only a specific set of modes of vibration. These preferred modes are those which result in the highest amplitude of vibration with the least input energy. Objects are most easily forced into these modes of vibration when disturbed at frequencies associated with their natural frequencies.

The model described here seeks to compromise between plausibility in the biological domain, and efficiency in the computational domain. The level of granularity of the model is an essential factor in this compromise. In order to model systems with many interacting neurons, it was necessary to avoid the computational overhead of compartmental models. The current model provides no spatial extent for its neurons. The mathematical physics governing the harmonic oscillator is used as a basis for the development of the resonate and fire model. The entity that actually oscillates is the membrane potential. The driving forces are the input spikes received on the neuron's dendritic field. The neuron's oscillations are lightly damped under normal conditions. For a brief period after firing, the oscillation is heavily damped, reflecting the quiescence period found in biological neurons, typically referred to as the absolute refractory period. The fundamental frequency of the neuron is a tunable parameter, in our consideration; the details which would determine this quantity in the biological instance are omitted. We treat it simply as a single parameter that may be set arbitrarily.

The oscillation of the membrane potential can alternatively be viewed as the oscillation of the threshold at which the action potential is generated. The arrival of an excitatory pulse to a dendrite will result in the summation of the current membrane potential with the new input. If the current membrane potential is high, smaller input will result in the threshold being reached and an action potential being generated. Similarly, if the current potential is low, a larger input will be required to force the resultant potential across the threshold. From this viewpoint, the resonate and fire model can be seen to be a superset of the IFN model. The behaviour of the IFN model can be simulated with a resonate and fire neuron with a low resonant frequency (long period). Input spikes are then summed in the usual manner with negligible influence from the oscillation of the membrane potential.

The IFN model, in which two neurons that innervate a third node with excitatory connection are always considered to cooperate, does not apply here. Such an event sequence also illustrates the other side of selective innervation, when the post-synaptic neuron is not selected by the pre-synaptic neuron, by virtue of the fact that its resonant frequency means that the interspike delay is not an integral multiple of the period of oscillation.

Such properties have obvious applications, one can envision an array of neurons forming a ``spectrographic map''; each neuron in the array is attuned to a different resonant frequency. Two input neurons innervate every neuron in the map, so that when the two input neurons fire, the time between their firing (inter-spike delay) will cause a single neuron in the map to react most positively. The neuron that reacts with an action potential is the neuron whose resonant period (the inverse of the frequency) most closely matches the inter-spike delay. Such an arrangement can be generalized to implement a pseudo-Fourier transform of an input channel. Each neuron in the spectrographic map will ``own'' a particular narrow frequency band. The input channel is a signal containing multiple frequencies superimposed upon one another. The input innervates all neurons in the map, which produce action potentials if their particular resonant frequency is present in the original signal.

The implementation details of the resonate and fire model are straightforward. We consider an idealized harmonic oscillator, similar to a mass on a spring. There is a single point of equilibrium in such a system, where the position of the mass is at the point where the spring is neither compressed nor stretched. The mass is assumed to be floating in free space outside the influence of the gravitational force, while the other end of the spring is bound to an idealized fixed point. The mass is displaced from the equilibrium point by the arrival of an impulse (push) of negligible duration. The displacement of the mass then oscillates back and forth past the equilibrium position. The spring exerts a ``return force'' proportional to the magnitude of the displacement. The frequency of oscillation is determined by both the size of the mass and the magnitude of the return force exerted by the spring. In the real world, all such oscillations gradually die off (though remain at the same frequency), due to the damping effects of friction.

A more familiar analogy would be that of a playground swing. Here the equilibrium position of the swing seat is directly below the supporting bar, i.e. hanging straight down. When we push the swing, it begins to swing to and fro (oscillate) past the equilibrium point. If we want to make the swings ``higher'' (increase the amplitude of oscillation) we must push the swing ``in phase'' with the basic oscillation. This simply means that we must push it as it is at the top of the back swing, or heading away from us. If we push it as it is coming toward us, we are pushing ``out of phase'' with the basic oscillation, and the amplitude thereby is decreased.

The mathematical details of the model follow directly from the math used to describe harmonic oscillation in bodies such as the mass on a spring, pendulums and playground swings. The task here is to translate the basic ideas into a form applicable to the resonate and fire neuron. Additionally we must formulate this in a manner that is amenable to computational implementation.

The starting point for analysis is to consider the mass on a spring arrangement. Here we have a mass that is displaced from the equilibrium point by  at any given moment; this displacement may be positive or negative. Due to the physical form of the spring, the mass always experiences a return force in the opposite direction to the current displacement:

                                                                                                                                

 where  is a positive constant referred to as the spring constant. This equation captures the fact that the return force is proportional to the current displacement. This is a key fact in that such systems are characterized among  Harmonic Oscillators. The basic behaviour of Harmonic Oscillators is captured by the differential equation:

                                                                                                                                 

By Newton's second law, we can relate the mass, return force and acceleration thus:

                                                                                                                                 

Substituting we arrive at

                                                                                                                             

The above equation is simply shorthand for that which we know intuitively. It states that the current acceleration is proportional to the current displacement, and in the opposite direction. For the purposes of simulation, we rewrite the equation in its more common form, replacing  and  with the term , defined below.

                                                                                                                             

The term  is defined as

                                                                                                                                 

 This result allows us to re-express the acceleration term in terms of :

                                                                                                                               

 A particular example of an equation which represents a solution to the general differential relation described above is written

                                                                                                                                                 

 where  is any constant length and  is any constant angle. The parameters which give an oscillator its unique properties are ,  and . The value of  determines the amplitude of oscillation, that is how far the maximum displacement from equilibrium will be. The  term determines the strength of the returning force. This in turn determines how quickly the mass returns to the equilibrium point (and indeed the velocity at which the equilibrium is passed). This equates to the more familiar concept of the frequency of oscillation. The frequency of oscillation is the number of complete cycles performed per second, and is the inverse of the period, the length of time required to complete a single cycle.

The period of oscillation of such a system is denoted  and related to the other terms as follows:

                                                                                                                                   

 In a fashion similar to the delta functions used to describe the IFN, we now demonstrate the operation of the resonate and fire model in mathematical terms. First, we must define some variables unique to the model:

                                                                                                                               

 where  is the resonant frequency of node , and  is the frequency of the global clock. The global clock frequency determines the granularity of simulation and may be set to any value, the default used to produce the graphs discussed previously is 1000. The term  is referred to as the counter multiplier for node . This term is introduced since it may be calculated once the resonant frequency is specified, and thus does not need to be calculated in subsequently.

                                                                                                   

The rate of change of the membrane potential  of neuron , or its velocity, is denoted by . The change in the velocity for the current time step is calculated first. The contribution from input pulses from all pre-synaptic neurons is calculated by the sum of products term , where  is the weight of the connection from neuron  to neuron , and  is the current (axonal) output of neuron . The current axonal output is always either a  or a , since action potentials are all or none events. The return force's contribution to the velocity calculation is expressed as , which is the expression we arrived at for  previously, divided by . We divide by  because we are performing a  time slice calculation; in each step of the calculation we are simulating a period of time that is the inverse of the global clock frequency. The final term is the damping factor. The damping constant,  ranges from  to , and is typically assigned a value of around . The effect of this parameter is to cause the oscillation to gradually die off, slowly reducing the amplitude, as seen previously in the graphs.

                                                                                                                                 

The calculation of the new membrane potential, , is straightforward once we have calculated the new velocity. In a single period of the global clock,  will change by the product of the current velocity and the time that we are simulating. Since period is the inverse of the frequency, this sum can be expressed as shown above. At this point we have calculated the new membrane potential. All that remains is to handle the production of action potentials.

                                                                                                                                                 

The above equation is the mathematical characterization of the model's method for deciding the output of neuron , denoted . The result is simply that if  is greater than , which denotes the threshold, then  is set to , otherwise it is set to . There are a number of actual mathematical functions that provide suitable implementations of , however in the computational implementation a single ``if'' statement suffices.

The mathematical structures described thus far handle axonal inputs from pre-synaptic neurons. Another major feature of the model is direct dendro-dendritic connections. This aspect is accommodated through a simple extension to the delta rule.

                                                                         

The new sum of products term  is the sum across all neurons providing dendritic inputs to neuron , of the products of the current membrane potential of neuron , , minus the current membrane potential of neuron ,  and the weight of the dendritic connection from neuron  to neuron , denoted . This factor is the key element in the creation of the dendritic field, through which waveforms may propagate. The difference between the axonal inputs and the dendritic connections in this model is that axonal inputs permit the transmission of single impulses, . The term  is non-zero only when neuron  has generated an action potential, while the term  is almost always non-zero, hence the difference between the two sum-of-product terms. The dendritic connections transmit electrical ``pressures'' which cause recipient neurons' membrane potentials to become closer to their own.

It is easy to extend this model to provide for propagation delays. Each neuron is modeled as a set of parameters, including the current value of  and . We extend this to provide a history of the values of these parameters. As each time step of the simulation passes, the new value calculated for  and  becomes the ``current'' value, while the old current value is stored in the history record. Axonal and dendritic connections are then augmented to specify which element of the history array they refer to, so that instead of using the current value of  or  in the delta rule, we may use the value as it was  time steps ago. For convenience of implementation, the current value is stored in the history array as element , element  is the value as it was during the last time slice, and so on. The terms  and  which represent the parameters of the connection, are augmented to account for this, with a superscript,  indicating the element of the history array that they refer to. This additional parameter is a fundamental property of the network topology of a resonate and fire network. So the final delta rule, which encapsulates resonance, axonal inputs, the dendritic field, and propagation delays, becomes

                                                                         

The model described above has been implemented using the C programming language.

3. Signal processing and RNF.

Sherrington (1906) first suggested the concept of the integrate-and-fire neuron. Under this scheme, the higher the frequency of the input spike trains, the larger the input activity is considered to be. The neuron is then assumed to respond with a firing rate that is a function of the input firing rates. McCulloch and Pitts (1943) formalised the model and showed how to encode any logical proposition in a network of their neurons. Similarly, any network could be shown to encode a logical proposition.

Eccles (1957) used spinal cord recordings to correlate the spike frequency with the intensity of the applied stimulus as well as the intensity of the perceived sensation. Under the frequency-coding scheme, neurons encode information by frequency modulation of action potentials output on the axon. Increased firing rates in the presence of certain stimuli were taken to indicate that the neuron under observation was reacting directly to the presence of the feature which it was tuned to react to.

An alternate view of neuronal signaling which uses frequency coding as its basic component is that of ``population coding'' (Georgopoulos et al., 1982). Under this scheme, the intensity or salience of the content is conveyed using frequency modulation, but the content itself is represented by a distributed combination of spike trains across a population of neurons.

In terms of visual processing, the assumption of feature detection follows an Euclidean geometry hierarchy. First, there are point and line detectors. These feed into edge and boundary detectors, and so on up the scale. Barlow (1972) suggested the possibility of such hierarchies when he made the claim that aspects of perceptual awareness are related to the activity of specific neurons. The ``grandmother'' cell hypothesis follows logically from this sequence. This concludes that there can be a single cell in the brain which is best attuned for a single recognition task, such as the recognition of a single human face (Grandma's). There has been some experimental evidence for such ``fine tuning'' of individual neurons, such as the demonstration of Tanaka (1993) of ``grandmother'' style cells in monkeys which respond to moderately complex figures.

There are numerous problems with such specific specialization of function at the cellular level. From a redundancy viewpoint, it is simply bad design to have a single point of failure of the recognition process as would be the case were a single cell assigned to a single pattern. A key feature distinguishing neural networks from other computational devices is the property of graceful degradation - meaning that a large part of the system can be destroyed without completely annihilating the behaviour of the system.

Hubel and Weisel's work on the receptive fields of individual neurons in the cat's striate cortex was taken by many as proof positive that visual perception followed the Euclidean hierarchy of points, lines and contours, shapes and forms (1959). Each stage was seen to be built on the previous. The basic assumption underlying this scheme is that the visual processing operation begins with a two dimensional retinal image formed by the eye. As  observes, the situation is much more complex than that. The optical image is a flow in at least three dimensions, the retinal image is curved, not flat, and the perceptual system has evolved to operate under conditions where the subject is moving. As experiments (Rock, 1983) show, the primitives of perception are ``relations between changes in oculocentric and egocentric direction. Lines and edges are not the primitives that configure the perceptual process; lines and edges result from the perceptual process, they do not determine it.'' .

This is not to say that the whole paradigm of viewing neural perceptual stages as feature extraction exercises is wrong. Rather that it is time to examine carefully the assumptions underlying the choice of features that we think are being extracted. Ultimately, sensory data come in the form of a power spectrum, a continuous stream of intensity values. The fact that hair cells in the ear are tuned to specific frequencies, and the existence of neurons in the inferior colliculus specifically oriented to pitch extraction is now commonplace in the literature (see, for example, Braun 2000). We make the following suggestions: 

• The stimulus for processing of sensory data is ultimately a power spectrum
• Conventional neural net systems have great difficulty in handling phenomena like rotational invariance, scaling, and so on
• These problems can be avoided by considering the action of RNF neurons that own a part of the frequency spectrum

We now wish to open out the discussion to talk about the specific role of dendro-dendritic connections, and the possibility that the well-examined phenomenon of stochastic resonance may point to a general process, ubiquitous in the brain, of computing with wave interference.

4. Computing with wave interference; the role of dendro-dendritic connections.

While quantum computing has become bogged down by the decoherence phenomenon, Ian Walmsley and his associates, inter alia, have demonstrated the possibility of computing by wave interference alone. Their celebrated demonstration is effectively an interference-based optical computer. Our work described above exemplifies the possibility of neurons implementing Fourier transforms by stealth, as it were, by a single neuron "owning" a particular bandwidth. In this brief section, we wish to suggest the possibility that the structure of dendro-dendritic connections affords a more flexible and potentially computationally powerful means of signal processing. In general, we are suggesting, such mechanisms perform the work of computation in the brain; INF effectively handles computation.

Pribram sees the dendritic microprocess as a central location of computational activity in the brain. Spike trains, action potentials are seen more as communicative devices than as the essence of the computational process. Izhikevich's resonate and fire neuron and the neural model described later place greater emphasis on the dendritic microprocess than conventional neural network models.

An important departure in Pribram's (1991) work is the emphasis on the role of dendro-dendritic connections. Such connections are similar to normal axonal-dendritic synaptic connections; however, the entity being transmitted is not an action potential, instead it is the current internal state of the neuron. In this way, Pribram proposes that computations can occur which involve multiple neurons, but which do not utilise axonal action potentials. This is not to say that action potentials are relegated to insignificance in the model; rather dendritic processes have been promoted to a level on a par with action potentials and conventional axonal transmission.

Recent evidence from experimental studies have confirmed that subthreshold dendritic dynamics are complex and would appear to have an important role to play in the computational activity of the brain. Particularly, calcium channels (Schutter ,1993) react strongly to subthreshold inputs. Callewaert , Eilers and Konnerth (1996) express the case for the dendritic process thus:

Recent results obtained by using high resolution imaging techniques provide clear evidence for new forms of neuronal signal integration. In contrast to the quickly spreading electrical potentials, slower intracellular signals were found that are restricted to defined dendritic compartments. Of special significance seem to be highly-localized, short-lasting changes in calcium concentration within fine branches of the neuronal dendritic tree. These calcium signals are evoked by synaptic excitation and provide the basis for a dendritic form of signal integration that is independent of the conventional electrical summation in the soma. There is experimental evidence that dendritic integration is critically involved in synaptic plasticity.

The general feature whereby neurons can ``tune in'' to a particular frequency component of the aggregate oscillation in the dendritic field provides an important computational asset to the model as a whole. It is also a phenomenon predicted to exist in biological neurons by Llinas (1988). The fact that the dendritic field supports such interference effects has deep ramifications; the modes by which the brain performs computation may be very different to the current action-potential centric paradigm.

Readers familiar with Young's slit experiment may find the analogy useful. In this case, the light source is the input neuron, while the slits correspond to the two output neurons. The screen on which the interference pattern appears is the entire set of possible values of the delay constants; for a particular pair of values we are measuring the interference at a single point on the screen. So, for each experimental simulation of the network, we select a value of the delay constants. As for Young's slit experiment, if the distance from the slits to the point on the screen is exactly the same, then waves from each slit arrive in-phase and constructively interfere. If, however the distance differs by exactly half a wavelength, then destructive interference occurs and the waves cancel each other out. In addition to standard inputs coming from the axons of presynaptic neurons, the RFN model implements inputs from the dendrites of other neurons, transmitting the current activation of the pre-synaptic node. This feature is directly inspired by Pribram (1991), who emphasizes the role of such channels in the computational process in the brain.

Here we have modeled the feature in a manner similar to the standard axonal input -- the sum of the products of connection weight and pre-synaptic output is augmented with the sum of product of dendritic connection weight and the current activation of the pre-synaptic neuron. Therefore, the only difference is that the current activation is used instead of the current output.

On its own, this mechanism would not be very useful. The contribution from dendro-dendritic connections to a post-synaptic neuron's activation would simply be the linear sum of the current activations of its pre-synaptic neurons. This situation is corrected by the addition of the delay mechanism discussed previously. Each dendro-dendritic connection has an associated weight, and delay. The delay corresponds to a propagation delay in the biological case. As the diagrams illustrate this mechanism permits an innervated neuron to position itself in any position in the interference field of a set of neurons, by tuning the delay parameters of its dendritic connections.

We have also implemented a neural net architecture using this basic idea. Neurons are to learn which frequencies to respond to.

5. Conclusions

This paper makes a set of claims ranging in strength from categorical to extremely tentative. The fact that after a century of modern neuroscience we have yet to establish the neural basis for a single symbolic cognitive act must surely give pause. Elsewhere (O Nualláin, (2003) ) we speculate that entirely different formalisms like Lie groups may be appropriate for analysis of brain function in addition to the Hilbert and other transforms hinted at here. It is uncontroversial at this stage to contend that old-fashioned INF needs greatly to be augmented. We contend that RNF may offer a superset formalism. We go on to posit that dendro-denritic connections may yield a fundamental set of new insights, which we look forward to pursuing.

References

 Barlow H. B. (1972) Single Neurons and Sensation: A neuron doctrine for perceptual psychology. Perception. Perception 1, 371-394.

 Biebel, U.W., Langner, G., 1997. Evidence for "pitch neurons" in the auditory midbrain of chinchillas. In: Syka, J. (Ed.), Acoustic Signal Processing in the Central Auditory System. Plenum Press, New York

 Braun, M., 2000. Inferior colliculus as candidate for pitch extraction: multiple support from statistics of bilateral spontaneous otoacoustic emissions. Hear. Res. 145, 130-140.

 Braun, M., 1999. Auditory midbrain laminar structure appears adapted to f 0 extraction: further evidence and implications of the double critical bandwidth. Hear. Res. 129, 71-82.

 J. C. Eccles (1957). The Physiology of Nerve Cells. Academic Press, New York, 1957

 G Callewaert, J Eilers, and A Konnerth Axonal calcium entry during fast 'sodium' action potentials in rat cerebellar Purkinje neurones J Physiol (Lond) 1996 495: 641-647

 Georgopoulos, A., Kalaska, J., Caminiti, R.,   Massey, J. (1982). On the relations between the directionof two-dimensional arm movements and cell discharge in primate motor cortex. Journal ofNeuroscience, 2(11), 1527-1537.

 Hubel and Wiesel (1959) Receptive fields of single neurons in the cat's striate cortex

 Hutcheon, B. and Yarom, Y. "Resonance, oscillation, and the intrinsic frequency preferences of neurons" Trends Neurosci. 2000 May; 23(5): 216-22

 Izhikevich (2002) "Resonance and selective communication via bursts in neurons having subthreshold oscillations" Biosystems 67(2002) 95-102

 Izhikevich (2000) "Resonate and fire neurons" The neurosciences institute

 Llinas, R. (1988) The intrinsic electrophysiological properties of mammalian meomeric insights into electrophysiological function

 Langner, G., Schreiner, C.E., Biebel, U.W., 1998. Functional implications of frequency and periodicity coding in auditory midbrain. In: Palmer, A.R., Rees, A., Summerfield, A.Q., Meddis, R. (Eds.), Psychophysical and Physiological Advances in Hearing. Whurr, London, pp. 277-285.

 Langner, G., Schreiner, C.E. and Merzenich, M.M. (1987) Covariation of latency and temporal resolution in the inferior colliculus of the cat. Hear. Res. 31, 197-201

 McCulloch, W. and Pitts, W. (1943). A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics, 7:115 - 133.

 Rees, A. and Sarbaz, A. (1997) The influence of intrinsic oscillations on the encoding of amplitude modulation by neurons in the inferior colliculus. In: J. Syka (Ed.), Acoustic Signal Processing in the Central Auditory System, Plenum Press, New York, pp. 239-252

 O Nualláin, Seán (2003) The Search for Mind; third edition. Exeter: England

 Pribram, K. (1991) Brain and Perception: holonomy and structure in figural processing. N.J. : Lawrence Erlbaum

 Reinker, S, E. Puil, and R.M. Miura (2004) "Membrane Resonance and Stochastic resonance modulate firing patterns of Thalamocortical neurons: Journal of computational Neuroscience 16 (1): 15-25, January-February, 2004

 Rock, I. (1983) The logic of perception. Cambridge, Mass: MIT Press

 Rudolph, M. and A. Destexhe (2001) "Do neocortical pyramidal neurons display stochastic resonance?" Journal of computational neuroscience 11,19-42

DeSchutter, E. and Bower, J.M. (1993) Parallel fiber inputs gate the Purkinje cell response to ascending branch synaptic inputs. Soc. Neurosci. Abst. 19:1588.

 Sherrington CS. 1906. Integrated Action of the Nervous System. Cambridge University Press: Cambridge, UK

 Wu, M, C-F Hsiao, and S.C. Chandler (2001) "Membrane reonance and subthreshold membrane oscillations in Mesencephalic V Neurons: Participants in Burst Generation The Journal of Neuroscience, June 1, 2001, 21(11):3729-3739