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Scaling Issues

 

  Having adopted essentially two levels of analysis, we are in the position of using biological mechanisms operating at the level of individual neurons to explain even relatively complex, high-level cognitive phenomena. This raises the question as to why these basic neural mechanisms should have any relevance to understanding something that is undoubtedly the product of millions or even billions of neurons -- certainly we do not include anywhere near that many neurons in our simulations! This scaling issue relates to the way in which we construct a scaled-down model of the real brain. It is important to emphasize that the need for scaling is at least partially a pragmatic issue having to do with the limitations of currently available computational resources. Thus, it should be possible to put the following arguments to the test in the future as larger, more complex models can be constructed. However, scaled-down models are also easier to understand, and are a good place to begin the computational cognitive neuroscience enterprise.

  We approach the scaling problem in the following ways.

  The first argument amounts to the idea that our neural network models are performing essentially the same type of processing as a human in a particular task, but on a reduced problem that either lacks the detailed information content of the human equivalent or represents a subset of these details. Of course, many phenomena can become qualitatively different as they get scaled up or down along this content dimension, but it seems reasonable to allow that some important properties might be relatively scale invariant. For example, one could plausibly argue that each major area of the human cortex could be reduced to handle only a small portion of the content that it actually does (e.g., by the use of a 16x16 pixel retina instead of 16 million x 16 million pixels), but that some important aspects of the essential computation on any piece of that information are preserved in the reduced model. If several such reduced cortical areas were connected, one could imagine having a useful but simplified model of some reasonably complex psychological phenomena.

  figure507
Figure: Illustration of scaling as performed on an image -- the original image in (a) was scaled down by a factor of 8, retaining only 1/8th of the original information, and then scaled back up to the same size and averaged (blurred) to produce (b), which captures many of the general characteristics of the original, but not the fine details. Our models give us something like this scaled-down, averaged image of how the brain works.

  The second argument can perhaps be stated most clearly by imagining that an individual unit in the model approximates the behavior of a population of essentially identical neurons. Thus, whereas actual neurons are discretely spiking, our model units typically (but not exclusively) use a continuous, graded activation signal. We will see in chapter 2 that this graded signal provides a very good approximation to the average number of spikes per unit time produced by a population of spiking neurons. Of course, we don't imagine that the brain is constructed from populations of identical neurons, but we do think that the brain employs overlapping distributed representations, so that an individual model unit can represent the centroid of a set of such representations. Thus, the population can encode much more information (e.g., many finer shades of meaning), and is probably different in other important ways (e.g., it might be more robust to the effects of noise). A visual analogy for this kind of scaling is shown in figure 1.3, where the sharp, high-resolution detail of the original (panel a) is lost in the scaled-down version (panel b), but the basic overall structure is preserved.

  Finally, we believe that the brain has a fractal character for two reasons: First, it is likely that, at least in the cortex, the effective properties of long-range connectivity are similar to that of local, short-range connectivity. For example, both short and long-range connectivity produce a balance between excitation and inhibition by virtue of connecting to both excitatory and inhibitory neurons (more on this in chapter 3). Thus, a model based on the properties of short-range connectivity within a localized cortical area could also describe a larger-scale model containing many such cortical areas simulated at a coarser level. The second reason is basically the same as the one given earlier about averaging over populations of neurons: if on average the population behaves roughly the same as the individual neuron, then the two levels of description are self-similar, which is what it means to be fractal.

In short, these arguments provide a basis for optimism that models based on neurobiological data can provide useful accounts of cognitive phenomena, even those that involve large, widely distributed areas of the brain. The models described in this book substantiate some of this optimism, but certainly this issue remains an open and important question for the computational cognitive neuroscience enterprise. The following historical perspective on this enterprise provides an overview of some of the other important issues that have shaped the field.

 


next up previous contents
Next: Historical Context Up: Basic Motivations for Computational Previous: Levels of Analysis

Randall C. O'Reilly
Fri Apr 28 14:15:16 MDT 2000