near the edge of the retina. But, even if the language and some of the tools of probability theory cannot be introduced, I believe a designer can take advantage of the ability to make finite samples from the family of all possible locations and safely estimate the angle being displayed.

Second, suppose we take for the mobile figure a bounded region of the plane bounded by a finite number of simple closed curves. Consider a fixed region of the same degree of generality; we study the area of overlap of the fixed figure with the mobile one. This was the case which struck me as relevant in my initial concern over the alleged description of the Perceptron. The integral of this over all positions of the mobile figure (I would call it the expected value of this random variable if I had a probability measure to work) turns out to be 277 times the product of the areas of the fixed and mobile figures. Corollary: nobody can successfully tell a circle from a square of the same area by examining the "average" amount of overlap of either with a mobile area of any shape whatsoever the result only depends on areas, not shape.

Third, if we compute the perimeter of overlap rather than area, and again "average" this, that is integrate this with respect to the kinematic measure for all locations, the result is 277 (AoL1 + A1 Lo) where the subscript O refers to the fixed figure and the subscript 1 refers to the mobile one (the result is symmetric anyway).

If we compute the total curvature of overlap for each position of the mobile figure, and again integrate this over all positions using kinematic measure, we get 2 TT (AoK1 + A1 Ko + LoL1). In particular, if both fixed and mobile figures are convex domains, their total curvatures Ko and K1 will be 2TT, as will the total curvature of the overlap in every position in which there is overlap. Since the integrand is then constant, we can factor 2 out of the entire identity (as we did with the constant 2 when considering intersections of lines with a convex curve) and we obtain the kinematic measure of all locations of a convex mobile figure which overlap with a convex fixed figure to be 2 TT (Ao + A1) + LOL1. This again permits a sort of retinal version of locations of an arbitrary convex figure "within" a convex retina where unfortunately the word "within" misleadingly means "intersecting" or equivalently, "overlapping." As before, knowing the kinematic measure of all the retinal locations, we may divide the kinematic measure by this constant, producing a true probability measure on the submanifold of retinal locations.

Let us consider a doubly infinite array of congruent figures or cells filling the plane without overlap, as with parallelograms or hexagons; in each consider a congruently placed replica of a curve, of length Lo. Now consider a mobile curve, of length L1, not necessarily small enough to be contained in any one cell., The kinematic measure for this mobile figure may be taken to be dx dy do where (x, y) are the coordinates of a point fixed in the mobile figure. Let's call this point the base point of the mobile figure. If we integrate the number of intersections of the infinitely repeated fixed figure with the mobile figure, over the submanifold of all positions of the mobile figure having the base point within any one cell,

the result is 4LoL1 where Lo is the length of the fixed figure and Ly is the length of the mobile one. This is again a theorem with a retinal or probability version, since the kinematic measure of all locations and orientations of the mobile figure within any one cell of area A is

If we divide the integral (or equivalently, the kinematic measure with respect to which it was performed) of the number of intersections by this normalizing constant we get the average number of intersections, for a random position of the mobile figure with base point in the cell A. This then is ratio of the last two, namely 2LOL1/TT A. The special case in which the cells are rectangles of sides a and b, and the fixed curve is a pair of adjacent sides, while the mobile curve is of length, shows that the average number of intersections is 2(a + b) /π ab. If we let a tend to infinity we get (2/77)(b) for the average number of crossings of the mobile curve with an infinite grid of parallel lines of distance b apart. This is the general form of the Buffon needle problem referred to at the beginning. As you see, it isn't the simplest theorem in the subject, viewed in this framework.

My final remark is this. There has been some progress made as to the question of the number of tosses that are required in this kind of random process to get specified accuracy. Back in 1841 Cauchy proved a theorem in what would now be called Integral Geometry: if you project a closed convex curve on the direction making an angle with the x-axis, and average the resulting projections, regarding → as chosen at random between 0 and 77, the result is 2/7 times the length of the curve.

Now this remark has been used by the mathematician Steinhaus in 1930 to design a machine very similar in idea to the ideas that I have now. (I didn't know of Steinhaus' work when I first suggested that Integral Geometry would be a good trick for pattern recognition.) Steinhaus was interested in designing a so-called longimeter, a means for measuring lengths in the field of a microscope by using this trick of projections. He knew that Cauchy's result called for the use of the average over all angles between 0 and TT. He asked what was the error caused by using a numerical integration, replacing the integral by a sum involving six equally spaced terms. Now that's an extremely small sample, only six, and its not a sample of six independent observations from among all angles. The six are dependent: having chosen the first, all the others are found by advancing thirty degrees at a time. He was able to show (and it is really a trivial fact having to do with the accuracy of numerical integration for the cosine function, which the general case reduces to immediately) that by using such a sample you are always within 2.26% of underestimate and 1.15% of overestimate of length. So there is at least some evidence that some finite samples are very good approximations to these averages.

MATHEMATICAL MODELS IN SENSORY PERCEPTION

Wilson P. Tanner, Jr.
University of Michigan

I would like to congratulate those who have encouraged and sponsored the development of Bionics as a new applied science. It is very likely to lead to a boon in both the engineering and biological sciences. After all, many living organisms have the property of adaptiveness incorporated in their design. An understanding of this property should permit the design of systems which are not obsolete by the time they are brought into being. From the biologists' side, it is a healthy exercise to attempt to design mechanisms which perform some of the functions of the organisms he observes. These attempts frequently force him to a more precise definition of terms than he might otherwise attempt.

As a psychologist who has worked closely with engineers in the formulative stages of the design of countermeasure systems over a period of years, I have been interested in the design of systems which "learn." The task of a countermeasure is to render a measure obsolete. In thinking of this problem, one soon adopts the philosophy that for every measure there is a countermeasure, and that for every countermeasure there is a countercountermeasure, etc. The obsolescence problem is inevitable. The problem is not to eliminate obsolescence; it is to make obsolescence less damaging. In order to get some insight into the problem, I attempted to design a model which would "learn" a very trivial task. I defined as "learning" the convergence upon an optimum performance over a sequence of acts. as the task to study one in which I could define an optimum performance, namely, that of testing statistical hypotheses. The problem is illustrated in Fig. 1. There is an observation space which can be defined on a linear axis, x. There are statistical hypotheses, H and H, defined on the axis, Ho and it is the observer's task to state whether each input x, arose from H or H. The optimum behavior in this case is to accept the hypothesis H whenever

I took

and otherwise to accept the hypothesis H. In the equation (x) is the likelihood ratio: f (x) and f (x) are probability densities conditional upon the truth of the hypotheses, and H respectively; P(H) and P(H) are a priori probabilities of the truth of the hypotheses; and R is a risk function.

The likelihood ratio to be associated with any value of x is dependent upon the separation between the hypotheses, which we will call d'. In other words, in order to behave optimally, the observer needs to know both d' and the a priori probabilities.

Suppose that the observer knows that the hypotheses are normal with equal variance, that he knows that the hypothesis H has mean O on the x axis, and that he knows R. Suppose also that he knows that d' and P(Ã) are fixed values through the experiment, with P(H) = 1 - P(H). However,

he does not know what these values are. ment, he can now compute estimates of d' prior to each stage of the experiment. lowing equations:

As he proceeds through the experiand P(H) based on the observations These estimates depend on the fol

There are thus two measures on the sequence of inputs both of which are functions of P() and d'. The decision rule now is that if

(2)

(3)

(4)

then the hypothesis is accepted as true; otherwise, H is accepted as true. In the equation, x is the current input, and d' and P(H) are estimates based on the sequence of inputs.

If one observes a system such as this in operation and considers the performance as a function of the current input, he will note a system which converges on an optimum performance, or "learns." On the other hand, if one considers the sequence as the input, at any stage the performance bears the same relation to optimum at all times. This depends only on how nearly optimum the procedure for estimating d' and P(H) is. Looked at in this

way,

the performance does not converge on optimum performance and the concept of "learning" disappears. There is a suggestion that the degree of adaptiveness depends on the level of generality of the point at which the system is fixed. If the fixed rule is general, then some time is lost before an optimum for a specific situation is achieved.

This discussion is highly speculative, and one might question whether or not it is clearly bionics. What can we expect of bionics in the future? I do not intend to consider the possible contributions of micro-biology, but rather the contributions of what I shall refer to as functional biology. It is my opinion that there is not very much in the latter field which has been established with sufficient confidence to be accepted as "fact.

I base this opinion on a number of facts. In the first place, there are of the order of 1010 neurons in the human nervous system. This constitutes a system capable of tremendously complex behavior. Even so, the flexibility exhibited by human behavior suggests that single functions cannot be assigned neuron by neuron. It seems more likely that the neural organization exhibits the same flexibility as the behavior, and that under different conditions the functional organization assumes different forms.

The notion that neurons might serve different functions under different conditions can be supported by a number of observations. I am told, for example, that under conditions of excitement cerebral hemorages leading to the destruction of 50,000 or so neurons are frequent occurrences. The loss of these leads to no observable changes in behavior. Either what they did was unimportant, an unlikely notion; or their function is performed by elements which are still active.

Wasteful redundancy likewise seems unlikely. Whatever the scheme, it must approximate some sort of optimum. For the present, at least, we are a long way from an understanding of the principles of the over-all organization; we will have to be satisfied with small theories. These theories should be constructed to permit flexibility. They should be constructed with the idea that they are specific to the particular situations, and that under different conditions different theories may apply. Two theories which appear to be contradictory may both be valid. The test may be that the consequences of both theories cannot be expected to exist simultaneously.

All of this sounds like the understanding of functional biology is an insurmountable task. It does, however, indicate that there is a tremendous field of basic research ahead of us. The development of this field can contribute to the science of bionics having a future development of a highly exciting nature.

The work that I am going to describe might be looked at in a number of ways. It might be considered as a contribution to a theory of hearing. It might be looked at as a contribution to the development of scientific techniques with experiments in hearing designed to test the feasibility of these techniques. I hope also that it might serve as an outline of one type of program which might lead to the development of the basic scientific knowledge essential to the growth of bionics.

The four phases of this work that I will cover are: 1) the development of mathematical models specifying optimum performance; 2) the design of experiments and the interpretation of data within this framework; 3) a review of the data so far collected; and 4) the development of a descriptive model of the hearing process, without reference to specific neural mechanisms, but keeping in mind that eventually the model should be expected to agree with an exceedingly flexible mechanism optimized over a large set of experiments rather than one to agree with a collection of mechanisms each optimized for a single experiment.

As I proceed, I will attempt to make explicit the philosophical foundations upon which the program is based. Unless one understands clearly what we are trying to do and why we are doing the things we are, they are likely to miscontrue our goals and bring up irrelevant arguments.

We adhere to the basic assumption that the hearing system is a subsystem of an intelligent system. Consequently, we wanted a theory which describes a system capable of assuming many forms, i.e., a flexible system. The particular form it takes at any particular time depends on instructions from the intelligent system. Only at a very general level is there a fixed decision rule, perhaps, homeostasis. At all other levels, the decision rule at any time is the one which best contributes to the general purpose at the highest level. For any particular system, the rule which contributes best at any time depends on the conditions at the particular time. This applies to the auditory system, and the theory must lead to methods and techniques for dealing with the different decision rules.