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A Simplified Global Competitive Model

The features of any model of inhibitory cells with lateral connections which support global competition would seem to need global connection across the whole net. However, NRT cells have not been observed to have connections across the whole NRT sheet, in spite of their extensive dendrite trees (especially in the rostral part of NRT). Thus the range over which any competition between the NRT cells can effectively take place will be limited. The localised properties of such competition have already been analysed over a decade ago [17], in terms of the even more simplified model of a single sheet with excitatory input and a Mexican hat style of lateral coupling. The range over which the competition can occur is of order that of the range of connections on the sheet. This feature of the sheet is not satisfactory for global control of the form of global guidance as in attention and consciousness [45]. It would seem that without considerable extensions of the lateral connections on NRT (or a small enough sheet, as might be the case in the rat) global guidance could not arise, and attentional control would be weak. A number of localised activities could then be supported on cortex, in disagreement to the ability to attend only to a single object at a time.

Figure 1: The structure of the first model of the thalamus-NRT-cortex complex, in which the cortex is dropped, and competition occurs only between the inputs to the thalamiccompetition occurs only between the inputs to the thalamic relay cell ; the strengths of the outputs indicate the winners and losers of the competition carried out between the corresponding NRT neurons by means of their inhibitory lateral connections.

The important feature of NRT, noted at least for more advanced mammals, above the rat, is the presence of dendro-dendritic synapses. These latter allow the NRT to be considered as a totally connected net even without the presence of axon collaterals of the form considered in Figure 1. Such dendro-dendritic synapses arise between horizontal cells in the outer plexiform layer of the retina [15], in the form of electrical gap junctions. They were modelled as linear resistors in the mathematical model of the retina in Taylor [43]. On NRT, the dendro-dendritic synapses appear to be chemical ones (as high-magnification electromicrographs show vesicles on or on both sides of the synapses). Such synapses need to be modelled in a non-linear fashion. In general, the dendro-dendritic synaptic contributions to the membrane potential at a particular cell at the point on the NRT sheet might be approximated as a sum of contributions from the nearby synapsing cells, each depending on the membrane potential differences between the cells. Thus, a typical form would be


in terms of some non-linear function F. Working with only small changes of potentials, F can be linearised, to give the contributions


where is positive in the case of inhibitory action in the dendro-dendritic synapse. For values of close to , the continuum limit can be taken of the NRT system, and the expression 2 reduces to [43]:


where is the two-dimensional Laplacian operator in Cartesian coordinates and a is a real constant determined by the average spacing between the neurons of the net.



Upon neglect of the lateral connection matrix , and with and diagonal, we obtain the simpler equation


where , , and we assume AB<1 to prevent infinite gain in the thalamus-NRT feedback loop system. The expression (6) is the basis of the simple version of the global gating model.

What can we deduce about the dependence of the response of the NRT cells' potentials (and hence that of the thalamic cells) from (6). We claim that (6) instantiates a form of competition in the spatial wavelength parameters of incoming inputs .

Physical systems with this underlying description have been investigated in a number of cases: spatially inhomogeneous superconducting states on tunnel injection of quasi-particles [28]; a Peierls insulator under strong dynamic excitation of electron-hole pairs or in the presence of electromagnetic radiation [13]. These models, and related ones for growth and dispersal in a population [38].

We may see precise forms of competition arising from the NRT modelled by (6) by looking at these equations for inputs made of sums of plane waves. Thus, if is composed of a set of separate waves of wave-numbers , so , then for suitable coefficients , from (6),


Thus, NRT activity will also display the same spatial oscillation as the input, but now with amplification of those waves with


Such augmentation corresponds to a process of competition on the space of the Fourier transform of inputs, where the Fourier transform for an input involves the global recombination


It is in this manner that we can see how NRT can exercise global control, by singling out those components of by (9) for which (8) is true. Other values of do not have such amplification. In other words, it would appear that the NRT would oscillate spatially with wavelength , with net strength given by the component of the input with the same wavelength.

The way in which global control arises now becomes clear. Only those inputs which have special spatial wavelength oscillations are allowed through to the cortex, or are allowed to persist in those regions of cortex strongly connected to the NRT: the thalamus-NRT system acts as a spatial Fourier filter. There is evidence for this in that feature detectors occur in a regular manner across striate cortex [27] as well as face-coding appears to have a spatial lattice structure [23]. Other explanations of the spatial periodicity of striate cortex feature detectors have been proposed [16] but these are consistent with our present proposal which may only add a further spatial instability to that explored in those references by non-NRT processes.

next up previous
Next: Modelling the Global Up: A Basis for Long-Range Previous: Introduction