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Implementing neural firing: Towards a new technology
Mathl. Comput. Modelling Vol. 26, No. 4, pp. 113-124, 1997 Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177197 $17.00 + 0.00
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s0895-7177(97)00149-0
Implementing Neural Firing: Towards a New Technology T. L. SAATY AND L. G. VARGAS University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. (Received February 1996; accepted May 1996) Abstract-Following earlier work on neural firing and synthesis, we use the expressions we derived elsewhere in the literature for that purpose, to show how we can represent images and sounds with the aid of CLOS (Common Lisp Object System) computer program of the LISP computer language.
Keywords-Diitributions,
Neurons,Density,Images,Sounds.
1.
INTRODUCTION
The biophysical mechanisms for the generation of action potentials was first modeled by Hodgkin and Huxley in 1952 [l]. Since then many researchers have used differential equations to model the behavior of neurons and other cells which exhibit bursting activity in response to a stimulus, e.g., Both et al. [2], Chay and Keizer [3], Plant and Kim [4], and Rinzel [5]. However, no one until now developed a representation using Dirac distributions. Two questions are: how to represent neural firing as a distribution, and what is the fundamental characteristic of the firing of many neurons simultaneously that makes it possible to create optic images, sounds, tastes, and other sensations. We illustrate here how we think vision and hearing occur. In this process, we rely on existing electronic technology to approximate and represent the firing of neurons that genuinely create particular images recognized by people. We are able to do that because vision and hearing are vibratory phenomena that transmit their signals from a distance through a medium. Taste, smell, and touch require direct contact with the sense organs by acting on particular chemicals and materials with texture in such specialized ways, so far not well understood. Further developments in technology such as those discussed at the end of the paper are needed to enable us to extend our work to represent sensation of that kind in a way that is similar to what we have done for vision and sound. All our perceptions occur in what appears to us as patches of a continuum. When a neuron fires, it contributes a “point” to this continuum. Since the number of neurons is finite, the total number of diverse points arising from firings at any time is finite and we can only obtain a discrete approximation to the continuum. This approximation is good if one can show that the totality of firings comes “close” to each point. Closeness is mathematically represented by the density of firings. Our goal is to show how the synthesis of impulses in neurons connected by chemical synapses can be interpreted as a meaningful representation of external stimuli. There are two basic assumptions in our model: (1) only those stimuli or inputs that maximize the overall response are selectively accumulated in the synthesis process, and
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(2) the response of networks of neurons have the property of being dense in the space of all responses. This implies that the output of a finite, but large, number of neurons is sufficient to approximate stimuli. The approximation would be our perception of the stimuli. In [6], we gave a global view of how neurons work as a group to create images and impressions. We synthesized the solutions derived from the response of individual neurons to obtain the general form of the response of many neurons. To describe how a diversity of stimuli may be grouped abstractly into visual, auditory, sensual, and other kinds of response, the outputs of such synthesis are interpreted as points of an inner direct sum of a space of functions of the form {t”e-flt, cr, p _>0}, or more generally, as distributions (generalized functions in the sense of Schwartz) that can be used to represent images, sound, and perhaps other sensations.
2. A NEURAL
NETWORK
To use the functions {toe-@, a,@ 2 0) to represent images and sounds, we created a 2-dimensional network of neurons consisting of layers. For illustrative purposes, we assume that there is one layer of neurons corresponding to each of the stimulus values. Thus, if the list of stimuli consists of n numerical values, we created n layers with a specific number of neurons in each layer. Under the assumption that each numerical stimulus is represented by the firing of one and only one neuron, each layer of the network must also consist of n neurons with thresholds varying between the largest and the smallest values of the list of stimuli. We also assumed that the firing threshold of each neuron had the same width. Thus, if the perceptual range of a stimulus varies between two values 81 and tiz, and each layer of the network has n neurons, then a neuron in the i th position of the layer will fire if the stimulus value falls between 8i+ (i - 1) (8s - 01/n - 1) and 8i + i (0, - @i/n - 1). This task is computationally demanding even for such simple geometric figures as the bird and the flower shown in Figures 1 and 2. For example, for the bird picture, the stimuli list consists of 124 values, and we would need 1242 = 15376 neurons, arranged in 124 layers of 124 neurons each. To implement this network, we define the neumn class in the object oriented programming language called LISP using CLOS (Common Lisp Object System). Each neuron is now an object and its characteristics are represented as class-slots (see Table 1). Finally, given this class, we defined instances of neurons. For example, the neuron in the i th position of a layer is obtained as follows: sstq
neuron-i
make-instance : threshold-lower : threshold-upper
‘neuron
e2 - e1
& f (i - 1) n-l 62 -
81 + i p n-l
oc 2
4
6
8
Figure 1. Sample picture of a bird.
01
>>
Implementing
115
Neural Firing
9
6
‘c
5 4,
. . .:-
.:*..
. .. -.111
.#
.. ..:..
3
.. ::
2 0
2
4
6
Figure 2. Sample picture of a rose.
Before we describe in detail how the network can be used to represent images and sound, we summarize the mathematical model on which the neural density representation is based.
3. SEQUENTIAL
FIRINGS
OF A NEURON
The functions {tQe-flt , a, p 1 0) result from modeling the neural firing as a pairwise comparison process in time. It is assumed that a neuron compares neurotransmitter-generated charges in increments of time. This leads to the continuous counterpart of a reciprocal matrix known as a reciprocal kernel. A reciprocal kernel K is an integral operator that satisfies the condition K(s, t)K(t, s) = 1, for all s and t. The response function w(s) of the neuron in spontaneous activity results from solving the homogeneous integral equation b w(s)
=
x0
s0
K(s,
t)w(t)
&
(1)
where K is a compact integral operator defined on the space Ls (0, b] of Lebesgue square integrable functions. If the reciprocal kernel K(s, t) > 0, on o 5 s, t _
s,” taegct) dt
satisfies (1) for some choice of g(t). Because finite linear combinations of the functions {Fe-@, a, p 2 0) are dense in the space of bounded continuous functions C[O, b] we can approximate taeg(t) by linear combinations of t Q e -@, and hence, we substitute g(t) = -& ,B 2 0, in the eigenfunction w(t). The density of neural firing is not completely analogous to the density of the rational numbers in the real number system. The rationals are countably infinite, the number of neurons is finite but large. In speaking of density here, we may think of making a sufficiently close approximation (within some predescribed bound rather than arbitrarily close). We also showed that neural responses are impulsive, and hence, the brain is a discrete firing system. It follows that the spontaneous activity of a neuron during a very short period of time in which the neuron fires is given by w(t) = 5
Tk(t
-
7k)Qe-P(t-Tk1,
(2)
k=l
if the neuron Mountcastle However, as are constant
fires at the random times, rk, k = 1,2,. . . , R. The empirical findings of Poggio and [7] support the assumption that R and the times, rk, Ic = 1,2,. . . , R are probabilistic. observed by Brinley [8], the parameters (Yand ,0 vary from neuron to neuron, but for the firings of each neuron. Nonspontaneous activity can be characterized as a
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Table 1. The Neuron Class. (def class neuron 0 ((list-of-neurons
:initarg
:list-of-neurons
: initf orm ’ 0 :accessor list-of-neurons :allocation :class) (neuron-name :initarg :neuron-name :initform ” ” :accessor neuron-name> (property :initarg :property :initform jJ ” :accessor property) (threshold-upper :initarg :threshold-upper :initform 0 :accessor threshold-upper) (threshold-lower :initarg :threshold-lower :initform 0 :accessor threshold-lower) (beginning-of-excitation :initarg :beginning-of-excitation :initform 0 :accessor beginning-of-excitation) (time-to-maximum-excitation : initarg:time-to-maximum-excitation :initform 0 :accessortime-to-maximum-excitation) (impulse-value :initarg :impulse-value :initform 0 :accessor impulse-value) (refractary-period :initarg :refractary-period :initform 0 :accessor refractary-period : allocation : class) (tau : initarg :tau :initform 0 : accessor tau> (alpha :initarg :alpha :initform 0 :accessor alpha) (beta : initarg :beta :initform 0 : accessor beta) :gamma-c (gamma-c : initarg :initform 0 :accessor gamma-c) (impulse :initarg :impulse :initform ‘0 :accessor impulse) (impulsive-sequence :initarg :impulsive-sequence :initform ‘0 :accessor impulsive-sequence))) perturbation
of background
activity.
To derive the response function when neurons are stimulated
from external sources, we consider an inhomogeneous neuron in addition equation
to existing spontaneous
activity.
equation
to represent stimuli acting on the
Thus, we solve the inhomogeneous
F’redholm
of the 2”d kind given by b w(s)
This
equation
has a solution
-
x0
s0
K(s,
in the Sobolev
t)w(t) space
dt = f(s). W:(Q)
of distributions
(3) (in the sense of
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Figure 3. First five seconds of Haydn’s Symphony No. 102.
Figure 4. First five seconds of Mozart’s Symphony No. 40. Table 2. Coordinates
of random sample of original bird contour.
(5.27.0.97) (5.61.6.34) (5.83.6.69) (5.99,0.87) @X05,6.22)
(6.08,6.25) (6.42.2.06) (6.7OJ.72)
(6.08.5.53) (6.42,0.81)
(6.11,5.66)
(6.36.4.66)
(8.23.6.03) (8.30.6.41) (8.86J.84) (9.09,0.65) (9.17.1.56)
Schwartz) in L,,(O) whose derivatives of order k also belong to the space LP(0), where R is an open subset of R”.
4.
DATA
COLLECTION
We now make the neurons of the network fire in response to a stimulus such as an image or sound. To do so, we sampled pictures and sounds to create a set of numerical values that would then be approximated by the firing of neurons. In the case of pictures, we sampled the contour of a picture (see Figures 1 and 2). (See [6], for a previous approximation as linear combinations.)
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118
Table 3. Random sample of coordinates
of original rose contour.
143350 162376 ~)(2.03415)(217448) t5$6.,8)) b:82;7:00] @:83;1:81) (0.87j:Ol) (0:9ili21) (1.01,247) (1.11.2.78) 1.23.3.07)(1.39.3.33)(153,350) (1.723.76) (1.98.4.69)(2.13.4.25)(227.4.48] g45.4.76) $584.93) (2.77.5.09) (3.04,5.31)(3.37565) I3.94.6.03)(4.66,6.41)(5426.78) (5.92,7.00)(2.66,5.11) 69.5.4S)(2675.62 69.5.13) .725.40) (272,562) (2.72,5.62)(MOJ.34) (3.48.5.80)(3.99.6.10)(4.45.6.33)(4.85654) (5.30,6.77)F5.y6.79) (5.62.7.14)(5.56,7.22)(548,720) (5.38.7.13) (5.26,7.05)(5.32,720) (5.38,7.28)(5.46.7.35)(5.56,7.36)(5.20,7.11)(5227.26) (5.28,7.40)(5.38,751) (5.53.7.60)(5.36,6.72)(555,677) (5.73.6.83)(5.70.6.70)(5.63.6.63)(5.48,653) (560,653) (5546.52) (561,652) (5.75.6.57)(X86,6.63 (5.75.6.48) 5.65.6.42)(553,635) (5.66.6.33)(5.76,6.33)(5.88,6.38)(5.9x.6.43)(6.10,6.55)(629,668) (626,6&i) (6.1L6.88)(5.96.6.93I (5.83.6.98)!5.73,7.03)(5.65.7.16) (5.51.7.61)(5.56,7.73)(5.6Q7.85)(5.65,7.90)(5.80,8.@3)(5.95,8.08)(6.11,8.11)(6.30.8.13) 6.41.8.23) (6.91.793) (6.98,7.65)(6.99.7.60)(6.99,7.55)(X06,7.45 (-/.14,7.35)(7.2Q7.25)(7.16.7.15)E7 JL7.0 (6.60,6.87)(6.4L6.87)(6.X,6.88) (6.30,8.16)(6.43,8.13 (6.70.7.73) I (6.56,8.06)(6.63,7.93)(6.58B7.83) (6.71.7.56)(6.76,750) (6X.7.40) (6.93,7.30)(6.98,7.23)(6.93,7.05)(6.85,6.95)(5.78.8.10)(5.85,8.26)(5.98,8.49)(6.13,8.73)(6.25,8.91) (6.26.9.03 (6.289.13) (6.33.9.19)(6.469.19) (6.50.9.19)(661.9.10) (6.66.9.04)(6.67,8.94 6.828.83) (6M8.75) (6.96.8.73)(-X&8.75) (7.25.8.69j (7.36,8.61)(7.41.8.59)(744,851) (7.46.8.43)(7.47,8.30)(7.528.19) (7.57,8.11 lb 64,s .02) (-7.67.7.90)(7.65.7.81)c/.62,7.71) (-l.S4,7.60)(7.59.7.51)(7.64.7.43)(7.68,7.43)(7.75,7.36)(7.78,7.30)(7.76,7.19)(7.68.7.12)(7.64.7.12)(X51,7.12) (7.38,7.07)(7.22,7.09) 12,7.12)(7X19,7.5 (7.20.7.54)(7.28,7.46)(l&,7.39) (7.60.7.43)(624,6.85) (6.40.6.77)(6.56,6.72)(6.74,6.67)(6.9L6.71)(7.096.79) (6.18,8.78)(6.77.8.65)(6.88.8.57)(7.03,8.56)(7.14.8.48)(7.22.8.37)c7.28,8.26)(7.32.8.11)(7.36,8.02) &:30,688) (746,6983 (7.599,7.07) (7.43,7.94)(7.54.7.86)(X59.7.74)(7.57.7.63) 6.63,8.51)&X69,8.56) (6.84.8.59)(6.88,8.45)(6.96.8.30)(7.04.8.19)(X14,8.05)(7.24,7.94) (7.32.7.89 (7.28.7.73)(7247.63) (7.19,7.54)t6.34,8.96)(6.37,8.80)(6.43.8.64 6.49.8.48)(6.51,8.36) 6.57 8.22) 6.49,8.14)(6.44.8.41) (6.41.8.59j (6.35.8.73)(6.28.8.85)(6.26.8.94)(5.42,6.83)(5.45,6.92)(554.7.01G 5.547.08) (5.63J.15) b*.10,5.38)b .24,5.36)(X38,5.36) (X47.5.36) (3.56.5.36)(X08.5.32)(3.225.32) (X36.5.32)(3.47,5.32)(X54,5.31)(3.56,5.25) Table 4. Sample of Haydn’s Symphony No. 102. +0.40351086987844076 +0.08536055177194633 -0.18623578609774188 -0.37247157219548377 -0.2793536791466128 -0.13191453200875253 +0.007757341276924608 -0.05432125408898935 -0.16295382969170952 -0.3181503181064944 -0.2793536791466128 -0.100875234325795SS +0.29487829427572054 +0.4966287629273 117 +0.5043861042042362 +0.17072110354389267 -0.OS43212S408898935 -0.2871110204235374 -0.2793536791466128 -0.131914S3200875253 +0.02328195640603237 -0.038796638959881585
+0.27159633786968823+0.02328195640603237 -0.3026356355526452 -0.3569568896416345 -0.23279969090980662 -0.07759327791976317 +o.o -0.08536055 177194633 -0.22503242505762347 -0.3181503181064944 -0.25607172274058043 +O.O3879663895988fS8S iO.37247157219548377 +0.558707358293225 +0.387986254749333 +0.1086425081779787 -0.1551964884147849 -0.29487829427572054 -0.24831438146365584 -0.06983593664283856 +O.O155146825S3849217 -0.06983593664283856 Table 5. Sample of Mozart’s
0 08594 f&J7031 +O.U4688 +0.01562 +0.03906 +0.03125 +0.01%2 -0.01562 -0.02344 -0.04688 -0.03125 -0.01562 -0.03906 -0.03906 -0.06250 -0.05469 -0.11719 -0.10156 -0.11719
UU/812 :0:07031 +0.04688 +0.02344 %E +0:01562 -0.01562 -0.03906 -0.04688 -0.02344 -0.00781 -0.04688 -0.04688 -0.06250
+0.03906 +0.03125 +0.03125 w:: -0:01562 -0.03906 -0.04688 -0.02344 -0.00781 -0.04688 -0.01469 -0.07031 -0.062SO -0.11719 -0.10156 -0.11719 -0.12500 -0.12500 -0.11719 .nnclv7~
007812 fo:07031 +0.03906 +0.03906 +0.03125 %",% -0:01562 -0.03906 -0.04688 -0.02344 -0.00781 -0.05469 -0.062SO -0.062SO -0.06250 -0.11719 -0.10156 -0.11719 -0.11719 -0.12500 :iE:~
Symphony No. 40.
+0.03125 +0.01562 10.00781 -0.01562 -0.03906 -0.04688 -0.01562 -0.00781 -0.04688 -0.06250 -0.06250 -0.07031 :;:;g -0:11719 -0.11719 -0.12500 :%%
~~~+o.l6295382969t70952 -0.06983593664283856 -0.36471423091855915 -0.31g1SO3181064944 -0.18623578609774188 -0.02328195640603237 -0.02328195640603237 -0.1163998494549033 I -0.2871110204235374 -0.3026356355526452 -0.2017504686515911 +0.18623578609774188 +0.426792826284473 1 +0.5664646995701502 +0.26382906401750505 +0.038796638959881585 -0.24055704Of8673f2 -0.29487829427572054 -0.1939931273746665 -0.015514682553849217 -0.007757341276924608 -0.09311789304887094
OUIU31 :0:05469 +0.0312s +0.03906 +0.03125 +0.00781 +0.00781 -0.01562 -0.03906 -0.03906 -0.01562 -0.00781 ::g:;; -0:05469 -0.07812 -0.11719 -0.10938 -0.11719 -0.11719 _ __ _. -0.11719 :K',~
+0.02344 +0.03906 +0.03125 +o.ooooo -0.00781 -0.02344 -0.03906 -0.03906 -0.01562 -0.01562 -0.03125 -0.06250 -0.06250 -0.08594 -0.10938 -0.10938 -0.11719
-0.10938 -0.10938 -0.10938 .ntn,rl;
+0.02344 +0.03906 +0.03125 +0.00781 -0.01562 -0.02344 -0.03906 -0.03906 -0.01562 -0.02344 -0.03906 -0.07031 -0.07031 -0.09375 -0.10938 -0.10938 -0.11719 -n.1n930 _ ____. -0.10938 -0.10156 A IlllCI.
+0.04688 +0.03125 +0.01562 -0.01562 -0.02344 -0.03906 -0.03906 -0.02344 -0.03906 :g:gg -0:06250 -0.10938 -0.10938 -0.11719 -0.12soo -0.10938 .._._.. -0.11719 -0.10156 ." rn,rr.
a digitized version of sound using a subroutine in Mathematical built for that purpose (see Figures 3 and 4). The graphical approximation of pictures required less computer resources than sound, partly because discrete pictures as an instantaneous whole evoke recognition in the mind of the observer while even the simplest sound is perceived sequentially and requires a large number of data points for the representation to have greater cohesion.
In the case of sound, we sampled
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Implementing
Neural Firing
119
.. ..
124
. :
N e
.
.
u r 0 n
.
:
. __ ._ : :
._
.
..
._ 1.
.-. . . _ ... . _-. --.
S . -- .
0
0
.
._ .
_ -.
:
.
..
. 124
Layers Figure 5. A 124 x 124 neural network.
0.6 .
0.00o1c).00040.00060.0008o.oo1o.0010.00120.0014 Figure 6. Firing of neuron-o. 12
‘. 1
2.5
5
7.5
Figure 7. Neural approximation
10
12.5
15
of Figure 1.
Picture Experiment
In the graphics experiment, the bird and rose pictures required 124 and 248 data points, respectively, whereas the sound experiment required 1000 times more data points. Once the (x, y)-coordinates of the points were obtained, the x-coordinate was used to represent time and the y-coordinate to represent response to a stimulus. The numerical values associated with the
T. L. SAATY et al.
120
9 8 7 6
.
r; 5 :
1
*.-r#.h#
m.
.. .: -
4:
.:’
..
3 . 21 0
2’ .. .. .. *. .. ..
2
4
Figure 8. Neural approximation
6 of Figure 2.
Figure 9. Neural approximation
of the first five seconds of Haydn’s Symphony No.
Figure 10. Neural approximation 40 in G minor.
of the first five seconds of Mozart’s Symphony No.
drawings in Figures 1 and 2 are given in Tables 2 and 3. These numbers the neurons in the networks built to represent the bird and the rose.
provided
the input
to
Sound Experiment In the sound experiment, we first recorded with the aid of Mathematics the first few seconds of Haydn’s Symphony No. 102 in B-flat major and Mozart’s Symphony No. 40 in G minor. The result is a set of numerical amplitudes between -1 and 1 shown in Figures 3 and 4. Each of these amplitudes was used to make neurons fire when the amplitude falls within a prescribed threshold range. Under the assumption that each neuron fires in response to one stimulus, we would need the same number of neurons as the sample size, i.e., 117,247 in Haydn’s symphony and 144,532 in Mozart’s symphony. Our objective was to approximate the amplitude using one neuron for
Implementing Neural Firing
121
Table 6. Numerical data for Figure 7. ( 1.67 3.53500) ( 1.92 3.44571) ( 2.23 2.92935) ( 2.45 3.34641) ( 2.89 3.09816) ( 3.42 1.63845) ( 3.70 3.03858 ) ( 3.86 4.34934 ) ( 4.45 4.93521 ) ( 4.86 5.27283I ( 5.36 4.13088 j ( 5.64 6.61338) ( 5.83 6.64317j ( 5.95 6.33534) ( 5.99 5.95800) ( 6.08 5.49129) ( 6.40 4.93521 ) ( 6.45 2.42292) ( 6.80 2.82012) ( 7.14 5.21325) ( 7.67 0.61566 ) ( 7.89 3.91242 ) ( 7.99 6.08709) ( 8.14 5.12388 ) ( 8.20 5.46150) ( 8.27 1.92642) ( 8.45 2.01579) ( 8.86 1.82712) ( 9.08 0.55608) ( 9.17 0.74475) ( 9.23 1.08237) each amplitude
( 1.69 3.49536) ( 1.96 3.19746)
( 1.80 ( 2.06 ( 2.30 ( 2.61 ( 3.20 ( 3.67 ( 3.74 ( 4.08 ( 4.61 t 5.24 ( 5.45 ( 5.70 i 5.89
( 2.27 3.37620) ( 2.61 2.45271) ( 3.02 2.01579) ( 3.52 2.87970) ( 3.7 3.346410) ( 4.02 459759 ) ( 4.49 1.11216) t 5.05 5.431711 ( 5.36 6.08709j ( 5.64 6.92120) i 5.86 4.68696j ( 5.99 0.86391) ( 6.05 534234 ) ( 6.08 6.20625) ( 6.42 0.80433) ( 6.52 1.82712) ( 7.02 1.63845) ( 7.36 1.70796) ( 7.67 1.79733) ( 7.92 4.19046) ( 8.05 4.65717) ( 8.17 4.93521) ( a.2 6.484290) ( 8.27 5.76933) ( 8.67 1.92642) ( 8.98 1.76754) ( 9.09 0.64545) ( 9.17 1.14195) ( 9.23 1.36041)
( 5.99
( 6.05 ( 6.11 ( 6.42 ( 6.61 ( 7.08 ( 7.37 ( 7.67
( 7.99 ( 8.08 ( 8.17 ( a.23 ( 8.30 ( 8.70 ( 9.02 ( 9.11 ( 9.17 ( 9.30
3.47550) 3.12795) 2.82012) 3.25704) 1.79733) 3.62445) 1.41999) 1.27104) 5.12388) 5.76933 1 6.45450j 4.37913) 4.34934j 5.05437) 5.89842) 5.62038) 2.04558) 2.60166) 3.03858) 3.28683) 3.53508) 1.88670) 4.40892) 5.52108) 5.98779) 5.92821) 1.82712) 1.60866) 0.86391) 1.54908) 1.27104)
( 3.24 ( 3.67 ( 3.74 ( 4.20 ( 4.86 ( 5.27 i 552 ( 5.74 i 5.89 ( 5.99 ( 6.05 ( 6.36 ( 6.42 ( 6.70 ( 7.11 ( 7.49 ( 7.80 ( 7.99 ( a.11 ( 8.20 ( 8.23 ( 830 ( 8.77 ( 9.05 ( 9.14 ( 9.20 ( 5.61
data displayed
5. NEURAL The network described
3.29676) 337620 ) 2.60166) 2.23425) 2.91942) 3.84291) 4.09116) 4.74654) 1.02279) 0.96321) 6.45459j 6.83184) 4.90542j 5.24304) 6.17646) 4.62738) 2.23425) 1.70796) 0.71496) 5.49129) 5.76933) 4.13088) 455787 ) 0.58587) 651408 ) 6.36513) 0.52629) 0.89370) 139020 ) 0.96321) 6.29562)
to play back the music.
value, and then use the resulting values in Mathematics
A small sample of the numerical
in Figures 3 and 4 is given in Tables 4 and 5.
REPRESENTATION
in Section 2 and the data sampled to form the picture given in Figure 1,
were used to create a 124 x 124 network of neurons consisting each layer.
( 1.83 ( 2.11 ( 2.45 ( 2.83
of 124 layers with 124 neurons in
The result of the firing of the network is given in Figure 5. Each dot is generated
by the firing of a neuron in response
to a stimulus
falling within the neuron’s
lower and upper
thresholds. To speed the computations, mated
we made a simplifying
by the firings of one neuron on adjusting
rather than creating value by estimating
a neuron for each value approximated. the parameters
neuron-0
the value in question,
(make-instance
: citation
Dirac distribution in Figure 6.
excitation
With these parameters,
where (Y = 2n(2000)2(0.0007)2,
p = o/2000,
in the neighborhood
of the stimulus
instead of approximating {Pe-@,
a
Q, p 2 0) such that
we created
0.0007 2000
: ref ractary-period . * . >I. of one millisecond.
stimuli can be approxi-
‘neuron
:impulse-value
This neuron reaches its maximum
Thus,
CY,p, and y for the functions
the firing of the neuron reproduces (setq
assumption-all
those firings to the intensity
= 0.001 in 0.7 of a millisecond
a single firing generated
and y = P”+‘/(2000T’((r
and has a refractory
period
by the neuron is given by
+ l)),
which behaves
of the origin, and its graphical representation
as a
is illustrated
122
T. L. SAATY et al. Table 7. Numerical data for Figure 8. ( i I i ( f
i ( ( ( ( ( i ( I i I i ( ( ( I i (
i ( ( ( ( ( ( f i ( i ( ( i ( i ( ( i ( ( ( ( ( ( (
0.73 1.13 2.03 2.94 5.82 1.1 I 1.98 2.77 5.42 2.69 3.48 5.44 5.26 5.20 5.36 5.48 5.86 5.76 6.26 5.65 5.56 6.11 6.83 7.06 6.98 6.26 6.58 6.93 5.85 6.28 6.66 7.12 7.46 7.67 7.64 7.68 1.12 7.60 6.91 6.18 7.22 7.54 6.84 7.24 6.34 6.57 6.28 5.56 3.47 3.47
1.19733) 3.o485lj 4.22025) 5.27283) 6.95100) 2.76054) 4.06137) 5.05437) 6.73254) 5.09409) 5.75940) 6.14241) 7.OOO65) 7.06023) 6.67296) 6.48429) 6.58359) 6.28569) 6.80205) 7.10988) 7.61589) 8.05323) 8.00358) 7.39785) 6.98079) 6.83184) 7.77519) 7.24890) 8.2O218) 9.06609) 8.97672) 8.66890) 8.37099) 7.84470) 7.37799) 7.O7016) 7.07016) 7.37799) 6.66303) 8.71854) 8.31141) 7.80498) 8.52987) 7.88442) 8.89728) 8.16246) 8.78805) 7.03044) 5.32248) 5.28276)
I i ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
0.77 1.99593) 1.29 3.30669) 2.17 4.44864) 3.27 5.61045) 0.83 1.79733) 1.23 3.04851) 2.13 4.22025) 3.04 5.27283) 5.92 6.95100) 2.72 5.36220) 3.99 6.05730) 5.62 7.09002) 5.32 7.14960) 5.22 7.20918) 5.55 6.72261) 5.6 6.484290) 5.75 6.43464) 5.88 6.33534) 6.11 6.83184) 5.56 7.26876) 5.60 7.79505) 6.30 8.07309) 6.91 7.87449) 7.14 7.29855) 6.86 6.93114) 6.30 S.lO288) 6.70 7.67589) 6.98 7.17939) 5.98 8.43057) 6.33 9.12567) 6.67 8.87742) 7.25 8.62917) 7.47 8.24190) 7.65 7.75533) 7.68 7.37799) 7.64 7.07016) 7.09 7.49715) 6.24 6.80205) 7.09 6.74247) 6.77 8.58945) 7.28 8.20218) 7.59 7.68582) 6.88 8.39085) 7.32 7.83477) 6.37 8.73840) 6.49 8.08302) 6.26 8.87742) 5.63 7.09995) 3.56 5.32248) ( 3.54 5.27283)
( i ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
(
0.82 1.43 2.35 3.84 0.87 1.39 2.27 3.37 2.66 2.72 4.45 5.56 5.38 5.28 5.73 5.50 5.65 5.98 5.96 5.53 5.65 6.41 6.98 7.20 6.73 6.43 6.71 6.93 6.13 6.46 6.82 7.36 7.52 7.62 7.75 7.51 7.20 6.40 7.30 6.88 7.32 7.57 6.96 7.28 6.43 6.44 5.42 3.10 3.08 3.56
2.19453) 3.47550) 4.72668) 5.98779) 1.99593) 3.30669) 4.44864) 5.61045) 5.07423) 5.58066) 6.28569) 7.16946) 7.22904) 7.34820) 6.78219) 6.47436) 6.37506) 6.38499) 6.88149) 7.37799) 7.84470) 8.17239) 7.59645) 7.19925) 6.86163) 8.07309) 7.50708) 7.00065) 8.66889) 9.12567) 8.76819) 8.54973) 8.13267) 7.65603) 7.30848) 7.07016) 7.48722) 6.72261) 6.83184) 8.51001) 8.05323) 7.57659) 8.24190) 7.67589) 8.57952) 8.35113) 6.78219) 5.34234) 5.28276) 5.21325)
t 0.91 2.452711 i 1.62 3.73368) ( 2.48 4.89549) ( 4.56 6.36513) ( 0.92 2.19453) ( 1.53 3.47550) ( 2.45 4.72668) ( 3.94 5.98779) ( 2.69 5.41185) ( 2.72 5.58066) ( 4.82 6.49422) ( 5.48 7.14960) ( 5.46 7.29855) ( 5.38 7.45743) (5.70 6.65310) ( 5.61 6.47436) ( 5.53 6.30555) ( 6.10 6.50415) ( 5.83 6.93114) ( 5.51 7.47729) ( 5.80 7.97379) ( 6.58 8.15253) ( 6.99 7.54680) ( 7.16 7.09995) ( 6.60 6.82191) ( 6.56 8.00358) ( 6.76 7.44750) ( 6.85 6.90135) ( 6.25 8.84763) ( 6.50 9.12567) ( 6.84 8.68875) ( 7.41 8.52987) ( 7.57 8.05323) ( 7.54 7.54680) ( 7.78 7.24890) ( 7.38 7.02051) ( 7.28 7.40778) ( 6.56 6.67296) ( 7.46 6.93114) ( 7.03 8.5ooo8) ( 7.36 7.96386) ( 6.63 8.45043) ( 7.04 8.13267) ( 1.24 757659) ( 6.49 8.42064) ( 6.41 852987) ( 5.45 6.87156) ( 3.24 5.32248) ( 3.22 5.28276)
( 1.01 2.76054) i 1.88 4.06137) ( 2.67 5.05437) f 5.32 6.732541 i 1.01 2.45271) ( 1.72 3.13368) i 2.58 4.895493 ( 4.66 6.36513) ( 2.67 5.58066) ( 3.00 5.30262) f 5.30 6.72261) i 5.38 7.08OO9j ( 5.56 7.30848) ( 5.53 7.54680) ( 5.63 6.58359) ( 5.75 6.52401 j ( 5.66 6.28569) ( 6.20 6.63324) i 5.73 6.980793 I 5.51 7.55673) i 5.95 8.02344) ( 6.71 8.09295) ( 6.99 7.49715) ( 7.11 7.03044) f 6.41 6.821911 i 6.63 7.87446) ( 6.85 7.34820) i 5.78 8.04330) ( 6.26 8.96679) ( 6.61 9.03630) ( 6.96 8.66889) ( 7.44 8.45043) ( 7.64 7.96386) ( 7.59 7.45743) ( 7.16 7.13967) ( 7.22 7.04037) ( 7.46 7.33821) ( 6.74 6.62331) ( 7.59 7.02051) ( 7.14 8.42064) i 7.43 7.88442) ( 6.69 S.SOOO8) ( 7.14 7.99365) ( 7.19 7.48722) ( 6.51 8.30148) ( 6.35 8.66889) ( 5.54 6.96093) ( 3.38 5.32248) ( 3.36 5.28276)
The firing of neuron-O reaches its maximum at t = 0.0007 and is approximately equal to 0.9933. Using the maximum value as the approximation factor, we are implicitly setting the error of our approximation equal to 0.0067. Thus, if the value of the impulse is measured at 100, then the approximation given would be lOO(O.9933) M 99.33. The resulting approximations are exhibited in Figures 7 and 8.
6. SUMMARY
AND
CONCLUSIONS
The basic assumption we made to represent the response to a sequence of individual stimuli by the network of Section 2 is that all the layers are identical, and each stimulus value is represented by the firing of a neuron in each layer. A shortcoming of this representation is that it is not invariant with respect to the order in which the stimuli are fed into the network. It is known in the case of vision that the eyes do not scan pictures symmetrically if they are not symmetric, and hence, our representation must satisfy some order invariance principle. Taking into account this principle would allow us to represent images independently of the form in which stimuli are input into the network. For example, we recognize an image even if it is subjected to a rotation,
Implementing
Neural Firing
Table 8. Sample of neural approximation +0.55469 -0.31250 -0.17188 -0.00781 -0.33594 +0.28906 +0.67969 -0.13281 -0.17969 +0.00781 -0.29688 +0.07812 +0.64844 -0.06250 +0.0aaoo +0.03906 -0.4453 1 +0.15625 +0.39062 -0.12500 +0.19531 +a.42969 -0.33594 +0.03906 +0.26562 -0.26562 +0.0703 1 i-O.46094 -0.21875 +0.203 12 +0.07812 -0.54688 +0.11719 +0.29688 -0.18750 +0.34375
+0.39062 -0.39062 -0.10156 -0.04688 -0.31250 +0.42188 +0.53906 -0.24219 -0.10156 -0.03125 -0.32812 +0.23438 +0.56250 -0.16406 +0.07812 -0.03125 -0.43750 +0.26562 iO.30469 -0.17188 +0.35156 +0.33594 -0.35938 +0.20312 +0.15625 -0.33594 +0.23438 +0.43750 -0.22656 +a.29688 -0.10156 -0.55469 +0.28906 +0.26562 -0.17188 +0.46875
+a.2578 1 -0.39844 -0.03906 -0.08594 -0.28906 +0.51562 +0.38281 -0.29688 -0.03125 -0.05469 -0.35156 0.36719 +0.46094 -0.21094 +O. 16406 -0.11719 -0.41406 +0.32812 +0.21875 -0.17969 +0.50781 +0.21875 -0.37500 +0.28906 0.04688 -0.35938 +0.38281 0.36719 -0.22656 +0.3203 1 -0.25781 -0.50781 +0.42969 +0.19531 -0.14062 iO.49219
0 164OG :0:14oa2 CO.10938 tO.06250 +0.09375 +0.08594 tO.06250 ~0.01562 +0.00781 -0.02344 +o.ooooo to.01562 -0.01562 -0.01562 -0.04688 -0.03906 -0.12500 -0.10156 -0.12500 -0.14062 -0.11719 -0.12500
+0.15625 +0.14062 +0.10938 +0.07812 +0.09375 +0.08594 +0.06250 +0.01562 -0.01562 -0.02344 +0.00781 +0.03125 -0.02344 -0.02344 -0.04688
+0.15625 +0.14062 +0.09375 +0.08594 +0.08594 +O.O8S94 +0.05469 +0.01562 -0.01562 -0.02344 +0.00781 +0.03125 -0.02344 -0.03906 -0.06250
:g;;;: -0:10156 -0.12500 -0.14062 -0.12500 .0.12500 .0.09375 :o".:o"::: -0.10156 -0.14062 $;;W& -0.11719 -oh719 -0.10156
-0.04688 -0.12500 -0.10156 -0.12500 -0.14062 -0.14062 -0.12500 -0.09375 -0.0937s -0.14844 -0.11719 -0.10156
:oo.g;:;: -0:08594 -pm;;
-0.07031 -0.08594 -0.08594 -0.09375 -0.07031
-0.08594 -0.09315 -0.09375 -0.08594 -0.08594
:ooy;:;;t
-0.09375 -0.11719
:p; -0:lllSS -0.11719
-0.11719 -0.12500 -0.16406 -0.11719 -0.07031 -0.10156 -0.10156 -0.04688 -0.01562 -0.02344 -0.03906 +0.00781
.0:08594 -0.0937s -0.12500 -0.12500 -0.17188 -0.11719
_ _ _. _ ::::;:;
-0:03906 +o.oOooo -0.02344 -0.04688 +0.00781
0 15625 :0:14062 +0.09375 +0.09375 +0.08594 +0.07812 +0.05469 +0.01562 -0.01562 -0.02344 +0.00781 +0.03125 -0.03906 -0.04688 -0.04688
to Haydn’s Symphony
+0.16406 -0.38281 +0.03125 -0.14062 -0.25781 +0.58594 +0.26562 -0.30469 +0.03906 -0.08594 -0.37500 +0.46875 +0.36719 -0.21875 +0.23438 -0.203 12 -0.35938 +0.36719 +0.15625 -0.14844 +0.61719 +0.0703 1 -0.35938 +0.32812 -0.00781 -0.34375 +0.49219 0.23438 -0.21094 +.29688 -0.34375 -0.41406 +0.49219 +0.078 12 -0.12500 +0.45312
Table 9. Sample of neural approximation
+. +0.132L?1 +0.09375 io.09375 +0.08594 +0.06250 +0.05469 +0.01562 -0.01562 -0.02344 +0.01562 +0.03125 -0.02344 -0.04688 -0.04688 -0.06250 -0.12500 -0.12500 -0.04688 -0.10156 -0.10156 -0.12500 -0.12500 -0.12500 -0.12soo -0.14062 -0.14062 -0.12500 -0.12500 -0.10156 -0.10156 -0.10156 -0.11719 -0.14844 -0.14062 -0.11719 -0.11719 -0.08594 -0.09375 -0.07031 :oo~~;::t: -0.08594 -0.09375 -0:08594 -0.08594 -0.07031 -0.08594 :;$:g: -0:12.500 -0.10156 -0.11719 -0.10156 -0.14062 -0.10156 -0.14062 -0.16406 -0.16406 -0.11719 -0.11719 -0.07031 :g::g -0.09375 -0.10156 -0:08594 -0.06250 -0.06250 -0.03906 -0.03906 -0.02344 -0.03906 -0.02344 -0.01562 +0.00781 +0.00781
123
+0.08594 -0.33594 +0.0703 1 -0.21875 -0.18750 +0.67188 +0.19531 -0.30469 +0.08594 -0.14844 -0.34375 +0.57812 +0.26562 -0.18750 +0.25000 -0.28906 -0.29688 +0.40625 +0.09375 -0.09375 +0.64062 -0.07812 -0.35156 +0.33594 -0.03906 -0.27344 +0.50781 0.0703 1 -0.17969 +0.26562 -0.39062 -0.30469 +0.46094 -0.04688 -0.08594 +0.41406
to Mozart’s
+U.l4062 +0.11719 +0.08594 +a.09375 +0.08594 +0.05469 +0.05469 +0.01562 -0.OlS62 -0.01562 +0.01562 +0.03125 -0.01562 -0.03906 -0.03906 -0.07031 -0.12500 -0.11719 -0.12500 -0.12500 -0.12500 -0.12500 -0.10156 -0.11719 -0.14062 -0.12500 -0.08594 -0.07031 -0.09375 -0.09375 -0.08594 -0.08594 -0.10156 -0.12500 .0.10156 -0.14062 -0.16406 -0.11719 -0.08594 -0.09375 -0.07031 -0.04688 -0.03906 -0.04688 -0.01562 +0.00781
No. 102. -0.03 125 -0.28906 +0.05469 -0.29688 -0.06250 +0.75000 +0.10938 -0.28906 +0.07812 -0.21875 -0.25000 +0.66406 +0.16406 -0.14062 +0.203 12 -0.37500 -0.16406 +a.43750 +0.03125 -0.01562 a.59375 -0.203 12 -0.28906 +0.34375 -0.07812 -0.17969 to.46875 -0.078 12 -0.09375 +0.25000 -0.43750 -0.20312 +0.39062 -0.13281 +o.ooooo +0.40625
-0.17188 -0.22656 +0.03 125 -0.33594 +0.10938 +0.75781 -0.00781 -0.25000 +0.04688 -0.25781 -0.09375 +0.68750 +0.05469 -0.07812 +0.12500 -0.42969 -0.00781 0.45312 -0.04688 +0.07812 +0.51562 -0.28906 -0.14062 +0.33594 -0.16406 -0.06250 +0.453 12 -0.17188 +0.04688 +0.19531 -0.48438 -0.05469 +0.33594 -0.17969 +0.16406 +0.38281
Symphony No. 40.
0 14062 :0:117L9 +0.07812 +0.09375 +0.08594 +0.03906 +O.O312S +0.00781 -0.01562 -0.01562 +0.01562 +0.01562 +o.ooooo -0.04688 -0.04688 -0.08594 -0.11719 -0.11719 -0.12500 -0.11719 -0.11719 -0.11719 -0.10156 -0.11719 -0.14062 -0.12.500 -0.08594 -0.07031 :oo~~~::~ -0:08594 -0.08594 -0.09375 -0.12500 -0.11719 -0.14844 -0.14844 -0.lOlS6 -0.09375 -0.09375 -0.04688 -0.03906 -0.03906 -0.04688 +o.ooooo +0.00781
0 15615 :0:132*1 +0.07812 +0.09375 +0.08594 +0.05469 +O.OISbZ +0.00781 -0.01562 -0.01562 +0.01562 r0.00781 -0.01562 -0.06250 -0.06250 -0.09375 -0.11719 -0.11719 -0.12500 -0.11719 -0.11719 -0.10156 -O.lOISb -0.12500 -0.14062 -0.l2SOO -0.07031 -0.07031 -0.09375 -0.09375 -0.08594 -0.08594 -0.08594 -0.12500 -0.11719 -0.16406 -0.14062 -0.10156 -0.10156 -0.09375 -0.03906 -0.02344 -0.03906 -0.04688 lo.ooooo +0.01562
+O.l562S +0.11719 to.07812 +0.10938 +0.08594 +0.06250 +0.01562 +0.00781 -0.01562 -0.01562 +0.00781 -0.01562 -0.01562 -0.04688 -0.04688 -0.11719 -0.11719 -0.12500 -0.14062 -0.11719 -0.12500 -0.10156 -0.10156 -0.12500 -0.14062 -0.12500 -0.08594 -0.08594 -0.09375 -0.09375 -0.08594 -0.08594 -0.08594 -0.12500 -0.12500 -0.17188 -0.12500 -0.09375 -0.10156 -0.09375 -0.03906 -0.01562 -0.03906 -0.04688 +o.ooooo +0.01562
1
124
T. L. SAATY et al.
or to some sort of deformation. Thus, the invariance principle must include afhne and similarity transformations. This invariance would allow the network to recognize images even when they are not identical to the ones from which it recorded a given concept, e.g., a bird. The next step would be to use the network representation given here with additional conditions to uniquely represent patterns from images, sounds, and perhaps other sources of stimuli such as smell.
REFERENCES 1. A.L. Hodgkin and A.F. Huxley, A quantitative description of membrane current and its applications to conduction and excitation in nerve, Journal of Physiology 117, 509-544, (1952). 2. Ft. Both, W. Finger and R.A. Chaplain, Model predictions of the ionic mechanisms underlying the beating and bursting pacemaker characteristics of molluscan neurons, Biological Cybernetics 23, l-11, (1976). 3. T.R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic P-cell, Biophysical Journal 42, 181-190, (1983). 4. R.E. Plant and M. Kim, Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations, Biophysical Journal 16, 227-244, (1976). 5. J. Rinzel, Bursting oscillations in an excitable membrane model, In Proceedings of the gh Dundee Conference on the Theory of Ordinary and Partial Differenctial Equations, (Edited by S.D. Sleeman, R.J. Jarvis and D.S. Jones), Springer-Verlag, New York, (1984). 6. T.L. Saaty and L.G. Vargas, A model of neural impulse firing and synthesis, Journal of Mathematical Psychology 37 (2), 200-219, (1993). 7. G.F. Poggio and V.B. Mountcastle, Functional organization of thalamus and cortex, In Medical Physiology, (Edited by V.B. Mountcastle), Chapter 9, C.V. Mosby Co., St. Louis, MO, (1980). 8. F.J. Brinley, Jr., Excitation and conduction in nerve fibers, In Medical Physiology, (Edited by V.B. Mountcastle), Chapter 2, C.V. Mosby Co., St. Louis, MO, (1980). 9. R.E. Plant, Bifurcation and resonance in a model for bursting nerve cells, Journal of Mathematical Biology 11, 15-32, (1981).
Pergamon PII:
s0895-7177(97)00149-0
Implementing Neural Firing: Towards a New Technology T. L. SAATY AND L. G. VARGAS University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. (Received February 1996; accepted May 1996) Abstract-Following earlier work on neural firing and synthesis, we use the expressions we derived elsewhere in the literature for that purpose, to show how we can represent images and sounds with the aid of CLOS (Common Lisp Object System) computer program of the LISP computer language.
Keywords-Diitributions,
Neurons,Density,Images,Sounds.
1.
INTRODUCTION
The biophysical mechanisms for the generation of action potentials was first modeled by Hodgkin and Huxley in 1952 [l]. Since then many researchers have used differential equations to model the behavior of neurons and other cells which exhibit bursting activity in response to a stimulus, e.g., Both et al. [2], Chay and Keizer [3], Plant and Kim [4], and Rinzel [5]. However, no one until now developed a representation using Dirac distributions. Two questions are: how to represent neural firing as a distribution, and what is the fundamental characteristic of the firing of many neurons simultaneously that makes it possible to create optic images, sounds, tastes, and other sensations. We illustrate here how we think vision and hearing occur. In this process, we rely on existing electronic technology to approximate and represent the firing of neurons that genuinely create particular images recognized by people. We are able to do that because vision and hearing are vibratory phenomena that transmit their signals from a distance through a medium. Taste, smell, and touch require direct contact with the sense organs by acting on particular chemicals and materials with texture in such specialized ways, so far not well understood. Further developments in technology such as those discussed at the end of the paper are needed to enable us to extend our work to represent sensation of that kind in a way that is similar to what we have done for vision and sound. All our perceptions occur in what appears to us as patches of a continuum. When a neuron fires, it contributes a “point” to this continuum. Since the number of neurons is finite, the total number of diverse points arising from firings at any time is finite and we can only obtain a discrete approximation to the continuum. This approximation is good if one can show that the totality of firings comes “close” to each point. Closeness is mathematically represented by the density of firings. Our goal is to show how the synthesis of impulses in neurons connected by chemical synapses can be interpreted as a meaningful representation of external stimuli. There are two basic assumptions in our model: (1) only those stimuli or inputs that maximize the overall response are selectively accumulated in the synthesis process, and
113
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T. L. SAATY et al.
(2) the response of networks of neurons have the property of being dense in the space of all responses. This implies that the output of a finite, but large, number of neurons is sufficient to approximate stimuli. The approximation would be our perception of the stimuli. In [6], we gave a global view of how neurons work as a group to create images and impressions. We synthesized the solutions derived from the response of individual neurons to obtain the general form of the response of many neurons. To describe how a diversity of stimuli may be grouped abstractly into visual, auditory, sensual, and other kinds of response, the outputs of such synthesis are interpreted as points of an inner direct sum of a space of functions of the form {t”e-flt, cr, p _>0}, or more generally, as distributions (generalized functions in the sense of Schwartz) that can be used to represent images, sound, and perhaps other sensations.
2. A NEURAL
NETWORK
To use the functions {toe-@, a,@ 2 0) to represent images and sounds, we created a 2-dimensional network of neurons consisting of layers. For illustrative purposes, we assume that there is one layer of neurons corresponding to each of the stimulus values. Thus, if the list of stimuli consists of n numerical values, we created n layers with a specific number of neurons in each layer. Under the assumption that each numerical stimulus is represented by the firing of one and only one neuron, each layer of the network must also consist of n neurons with thresholds varying between the largest and the smallest values of the list of stimuli. We also assumed that the firing threshold of each neuron had the same width. Thus, if the perceptual range of a stimulus varies between two values 81 and tiz, and each layer of the network has n neurons, then a neuron in the i th position of the layer will fire if the stimulus value falls between 8i+ (i - 1) (8s - 01/n - 1) and 8i + i (0, - @i/n - 1). This task is computationally demanding even for such simple geometric figures as the bird and the flower shown in Figures 1 and 2. For example, for the bird picture, the stimuli list consists of 124 values, and we would need 1242 = 15376 neurons, arranged in 124 layers of 124 neurons each. To implement this network, we define the neumn class in the object oriented programming language called LISP using CLOS (Common Lisp Object System). Each neuron is now an object and its characteristics are represented as class-slots (see Table 1). Finally, given this class, we defined instances of neurons. For example, the neuron in the i th position of a layer is obtained as follows: sstq
neuron-i
make-instance : threshold-lower : threshold-upper
‘neuron
e2 - e1
& f (i - 1) n-l 62 -
81 + i p n-l
oc 2
4
6
8
Figure 1. Sample picture of a bird.
01
>>
Implementing
115
Neural Firing
9
6
‘c
5 4,
. . .:-
.:*..
. .. -.111
.#
.. ..:..
3
.. ::
2 0
2
4
6
Figure 2. Sample picture of a rose.
Before we describe in detail how the network can be used to represent images and sound, we summarize the mathematical model on which the neural density representation is based.
3. SEQUENTIAL
FIRINGS
OF A NEURON
The functions {tQe-flt , a, p 1 0) result from modeling the neural firing as a pairwise comparison process in time. It is assumed that a neuron compares neurotransmitter-generated charges in increments of time. This leads to the continuous counterpart of a reciprocal matrix known as a reciprocal kernel. A reciprocal kernel K is an integral operator that satisfies the condition K(s, t)K(t, s) = 1, for all s and t. The response function w(s) of the neuron in spontaneous activity results from solving the homogeneous integral equation b w(s)
=
x0
s0
K(s,
t)w(t)
&
(1)
where K is a compact integral operator defined on the space Ls (0, b] of Lebesgue square integrable functions. If the reciprocal kernel K(s, t) > 0, on o 5 s, t _
s,” taegct) dt
satisfies (1) for some choice of g(t). Because finite linear combinations of the functions {Fe-@, a, p 2 0) are dense in the space of bounded continuous functions C[O, b] we can approximate taeg(t) by linear combinations of t Q e -@, and hence, we substitute g(t) = -& ,B 2 0, in the eigenfunction w(t). The density of neural firing is not completely analogous to the density of the rational numbers in the real number system. The rationals are countably infinite, the number of neurons is finite but large. In speaking of density here, we may think of making a sufficiently close approximation (within some predescribed bound rather than arbitrarily close). We also showed that neural responses are impulsive, and hence, the brain is a discrete firing system. It follows that the spontaneous activity of a neuron during a very short period of time in which the neuron fires is given by w(t) = 5
Tk(t
-
7k)Qe-P(t-Tk1,
(2)
k=l
if the neuron Mountcastle However, as are constant
fires at the random times, rk, k = 1,2,. . . , R. The empirical findings of Poggio and [7] support the assumption that R and the times, rk, Ic = 1,2,. . . , R are probabilistic. observed by Brinley [8], the parameters (Yand ,0 vary from neuron to neuron, but for the firings of each neuron. Nonspontaneous activity can be characterized as a
T. L. SAATY et al.
116
Table 1. The Neuron Class. (def class neuron 0 ((list-of-neurons
:initarg
:list-of-neurons
: initf orm ’ 0 :accessor list-of-neurons :allocation :class) (neuron-name :initarg :neuron-name :initform ” ” :accessor neuron-name> (property :initarg :property :initform jJ ” :accessor property) (threshold-upper :initarg :threshold-upper :initform 0 :accessor threshold-upper) (threshold-lower :initarg :threshold-lower :initform 0 :accessor threshold-lower) (beginning-of-excitation :initarg :beginning-of-excitation :initform 0 :accessor beginning-of-excitation) (time-to-maximum-excitation : initarg:time-to-maximum-excitation :initform 0 :accessortime-to-maximum-excitation) (impulse-value :initarg :impulse-value :initform 0 :accessor impulse-value) (refractary-period :initarg :refractary-period :initform 0 :accessor refractary-period : allocation : class) (tau : initarg :tau :initform 0 : accessor tau> (alpha :initarg :alpha :initform 0 :accessor alpha) (beta : initarg :beta :initform 0 : accessor beta) :gamma-c (gamma-c : initarg :initform 0 :accessor gamma-c) (impulse :initarg :impulse :initform ‘0 :accessor impulse) (impulsive-sequence :initarg :impulsive-sequence :initform ‘0 :accessor impulsive-sequence))) perturbation
of background
activity.
To derive the response function when neurons are stimulated
from external sources, we consider an inhomogeneous neuron in addition equation
to existing spontaneous
activity.
equation
to represent stimuli acting on the
Thus, we solve the inhomogeneous
F’redholm
of the 2”d kind given by b w(s)
This
equation
has a solution
-
x0
s0
K(s,
in the Sobolev
t)w(t) space
dt = f(s). W:(Q)
of distributions
(3) (in the sense of
Implementing
Neural Firing
117
Figure 3. First five seconds of Haydn’s Symphony No. 102.
Figure 4. First five seconds of Mozart’s Symphony No. 40. Table 2. Coordinates
of random sample of original bird contour.
(5.27.0.97) (5.61.6.34) (5.83.6.69) (5.99,0.87) @X05,6.22)
(6.08,6.25) (6.42.2.06) (6.7OJ.72)
(6.08.5.53) (6.42,0.81)
(6.11,5.66)
(6.36.4.66)
(8.23.6.03) (8.30.6.41) (8.86J.84) (9.09,0.65) (9.17.1.56)
Schwartz) in L,,(O) whose derivatives of order k also belong to the space LP(0), where R is an open subset of R”.
4.
DATA
COLLECTION
We now make the neurons of the network fire in response to a stimulus such as an image or sound. To do so, we sampled pictures and sounds to create a set of numerical values that would then be approximated by the firing of neurons. In the case of pictures, we sampled the contour of a picture (see Figures 1 and 2). (See [6], for a previous approximation as linear combinations.)
T. L. SAATY et al.
118
Table 3. Random sample of coordinates
of original rose contour.
143350 162376 ~)(2.03415)(217448) t5$6.,8)) b:82;7:00] @:83;1:81) (0.87j:Ol) (0:9ili21) (1.01,247) (1.11.2.78) 1.23.3.07)(1.39.3.33)(153,350) (1.723.76) (1.98.4.69)(2.13.4.25)(227.4.48] g45.4.76) $584.93) (2.77.5.09) (3.04,5.31)(3.37565) I3.94.6.03)(4.66,6.41)(5426.78) (5.92,7.00)(2.66,5.11) 69.5.4S)(2675.62 69.5.13) .725.40) (272,562) (2.72,5.62)(MOJ.34) (3.48.5.80)(3.99.6.10)(4.45.6.33)(4.85654) (5.30,6.77)F5.y6.79) (5.62.7.14)(5.56,7.22)(548,720) (5.38.7.13) (5.26,7.05)(5.32,720) (5.38,7.28)(5.46.7.35)(5.56,7.36)(5.20,7.11)(5227.26) (5.28,7.40)(5.38,751) (5.53.7.60)(5.36,6.72)(555,677) (5.73.6.83)(5.70.6.70)(5.63.6.63)(5.48,653) (560,653) (5546.52) (561,652) (5.75.6.57)(X86,6.63 (5.75.6.48) 5.65.6.42)(553,635) (5.66.6.33)(5.76,6.33)(5.88,6.38)(5.9x.6.43)(6.10,6.55)(629,668) (626,6&i) (6.1L6.88)(5.96.6.93I (5.83.6.98)!5.73,7.03)(5.65.7.16) (5.51.7.61)(5.56,7.73)(5.6Q7.85)(5.65,7.90)(5.80,8.@3)(5.95,8.08)(6.11,8.11)(6.30.8.13) 6.41.8.23) (6.91.793) (6.98,7.65)(6.99.7.60)(6.99,7.55)(X06,7.45 (-/.14,7.35)(7.2Q7.25)(7.16.7.15)E7 JL7.0 (6.60,6.87)(6.4L6.87)(6.X,6.88) (6.30,8.16)(6.43,8.13 (6.70.7.73) I (6.56,8.06)(6.63,7.93)(6.58B7.83) (6.71.7.56)(6.76,750) (6X.7.40) (6.93,7.30)(6.98,7.23)(6.93,7.05)(6.85,6.95)(5.78.8.10)(5.85,8.26)(5.98,8.49)(6.13,8.73)(6.25,8.91) (6.26.9.03 (6.289.13) (6.33.9.19)(6.469.19) (6.50.9.19)(661.9.10) (6.66.9.04)(6.67,8.94 6.828.83) (6M8.75) (6.96.8.73)(-X&8.75) (7.25.8.69j (7.36,8.61)(7.41.8.59)(744,851) (7.46.8.43)(7.47,8.30)(7.528.19) (7.57,8.11 lb 64,s .02) (-7.67.7.90)(7.65.7.81)c/.62,7.71) (-l.S4,7.60)(7.59.7.51)(7.64.7.43)(7.68,7.43)(7.75,7.36)(7.78,7.30)(7.76,7.19)(7.68.7.12)(7.64.7.12)(X51,7.12) (7.38,7.07)(7.22,7.09) 12,7.12)(7X19,7.5 (7.20.7.54)(7.28,7.46)(l&,7.39) (7.60.7.43)(624,6.85) (6.40.6.77)(6.56,6.72)(6.74,6.67)(6.9L6.71)(7.096.79) (6.18,8.78)(6.77.8.65)(6.88.8.57)(7.03,8.56)(7.14.8.48)(7.22.8.37)c7.28,8.26)(7.32.8.11)(7.36,8.02) &:30,688) (746,6983 (7.599,7.07) (7.43,7.94)(7.54.7.86)(X59.7.74)(7.57.7.63) 6.63,8.51)&X69,8.56) (6.84.8.59)(6.88,8.45)(6.96.8.30)(7.04.8.19)(X14,8.05)(7.24,7.94) (7.32.7.89 (7.28.7.73)(7247.63) (7.19,7.54)t6.34,8.96)(6.37,8.80)(6.43.8.64 6.49.8.48)(6.51,8.36) 6.57 8.22) 6.49,8.14)(6.44.8.41) (6.41.8.59j (6.35.8.73)(6.28.8.85)(6.26.8.94)(5.42,6.83)(5.45,6.92)(554.7.01G 5.547.08) (5.63J.15) b*.10,5.38)b .24,5.36)(X38,5.36) (X47.5.36) (3.56.5.36)(X08.5.32)(3.225.32) (X36.5.32)(3.47,5.32)(X54,5.31)(3.56,5.25) Table 4. Sample of Haydn’s Symphony No. 102. +0.40351086987844076 +0.08536055177194633 -0.18623578609774188 -0.37247157219548377 -0.2793536791466128 -0.13191453200875253 +0.007757341276924608 -0.05432125408898935 -0.16295382969170952 -0.3181503181064944 -0.2793536791466128 -0.100875234325795SS +0.29487829427572054 +0.4966287629273 117 +0.5043861042042362 +0.17072110354389267 -0.OS43212S408898935 -0.2871110204235374 -0.2793536791466128 -0.131914S3200875253 +0.02328195640603237 -0.038796638959881585
+0.27159633786968823+0.02328195640603237 -0.3026356355526452 -0.3569568896416345 -0.23279969090980662 -0.07759327791976317 +o.o -0.08536055 177194633 -0.22503242505762347 -0.3181503181064944 -0.25607172274058043 +O.O3879663895988fS8S iO.37247157219548377 +0.558707358293225 +0.387986254749333 +0.1086425081779787 -0.1551964884147849 -0.29487829427572054 -0.24831438146365584 -0.06983593664283856 +O.O155146825S3849217 -0.06983593664283856 Table 5. Sample of Mozart’s
0 08594 f&J7031 +O.U4688 +0.01562 +0.03906 +0.03125 +0.01%2 -0.01562 -0.02344 -0.04688 -0.03125 -0.01562 -0.03906 -0.03906 -0.06250 -0.05469 -0.11719 -0.10156 -0.11719
UU/812 :0:07031 +0.04688 +0.02344 %E +0:01562 -0.01562 -0.03906 -0.04688 -0.02344 -0.00781 -0.04688 -0.04688 -0.06250
+0.03906 +0.03125 +0.03125 w:: -0:01562 -0.03906 -0.04688 -0.02344 -0.00781 -0.04688 -0.01469 -0.07031 -0.062SO -0.11719 -0.10156 -0.11719 -0.12500 -0.12500 -0.11719 .nnclv7~
007812 fo:07031 +0.03906 +0.03906 +0.03125 %",% -0:01562 -0.03906 -0.04688 -0.02344 -0.00781 -0.05469 -0.062SO -0.062SO -0.06250 -0.11719 -0.10156 -0.11719 -0.11719 -0.12500 :iE:~
Symphony No. 40.
+0.03125 +0.01562 10.00781 -0.01562 -0.03906 -0.04688 -0.01562 -0.00781 -0.04688 -0.06250 -0.06250 -0.07031 :;:;g -0:11719 -0.11719 -0.12500 :%%
~~~+o.l6295382969t70952 -0.06983593664283856 -0.36471423091855915 -0.31g1SO3181064944 -0.18623578609774188 -0.02328195640603237 -0.02328195640603237 -0.1163998494549033 I -0.2871110204235374 -0.3026356355526452 -0.2017504686515911 +0.18623578609774188 +0.426792826284473 1 +0.5664646995701502 +0.26382906401750505 +0.038796638959881585 -0.24055704Of8673f2 -0.29487829427572054 -0.1939931273746665 -0.015514682553849217 -0.007757341276924608 -0.09311789304887094
OUIU31 :0:05469 +0.0312s +0.03906 +0.03125 +0.00781 +0.00781 -0.01562 -0.03906 -0.03906 -0.01562 -0.00781 ::g:;; -0:05469 -0.07812 -0.11719 -0.10938 -0.11719 -0.11719 _ __ _. -0.11719 :K',~
+0.02344 +0.03906 +0.03125 +o.ooooo -0.00781 -0.02344 -0.03906 -0.03906 -0.01562 -0.01562 -0.03125 -0.06250 -0.06250 -0.08594 -0.10938 -0.10938 -0.11719
-0.10938 -0.10938 -0.10938 .ntn,rl;
+0.02344 +0.03906 +0.03125 +0.00781 -0.01562 -0.02344 -0.03906 -0.03906 -0.01562 -0.02344 -0.03906 -0.07031 -0.07031 -0.09375 -0.10938 -0.10938 -0.11719 -n.1n930 _ ____. -0.10938 -0.10156 A IlllCI.
+0.04688 +0.03125 +0.01562 -0.01562 -0.02344 -0.03906 -0.03906 -0.02344 -0.03906 :g:gg -0:06250 -0.10938 -0.10938 -0.11719 -0.12soo -0.10938 .._._.. -0.11719 -0.10156 ." rn,rr.
a digitized version of sound using a subroutine in Mathematical built for that purpose (see Figures 3 and 4). The graphical approximation of pictures required less computer resources than sound, partly because discrete pictures as an instantaneous whole evoke recognition in the mind of the observer while even the simplest sound is perceived sequentially and requires a large number of data points for the representation to have greater cohesion.
In the case of sound, we sampled
lMathematica
is a trademark of Wolfram Research Inc.
Implementing
Neural Firing
119
.. ..
124
. :
N e
.
.
u r 0 n
.
:
. __ ._ : :
._
.
..
._ 1.
.-. . . _ ... . _-. --.
S . -- .
0
0
.
._ .
_ -.
:
.
..
. 124
Layers Figure 5. A 124 x 124 neural network.
0.6 .
0.00o1c).00040.00060.0008o.oo1o.0010.00120.0014 Figure 6. Firing of neuron-o. 12
‘. 1
2.5
5
7.5
Figure 7. Neural approximation
10
12.5
15
of Figure 1.
Picture Experiment
In the graphics experiment, the bird and rose pictures required 124 and 248 data points, respectively, whereas the sound experiment required 1000 times more data points. Once the (x, y)-coordinates of the points were obtained, the x-coordinate was used to represent time and the y-coordinate to represent response to a stimulus. The numerical values associated with the
T. L. SAATY et al.
120
9 8 7 6
.
r; 5 :
1
*.-r#.h#
m.
.. .: -
4:
.:’
..
3 . 21 0
2’ .. .. .. *. .. ..
2
4
Figure 8. Neural approximation
6 of Figure 2.
Figure 9. Neural approximation
of the first five seconds of Haydn’s Symphony No.
Figure 10. Neural approximation 40 in G minor.
of the first five seconds of Mozart’s Symphony No.
drawings in Figures 1 and 2 are given in Tables 2 and 3. These numbers the neurons in the networks built to represent the bird and the rose.
provided
the input
to
Sound Experiment In the sound experiment, we first recorded with the aid of Mathematics the first few seconds of Haydn’s Symphony No. 102 in B-flat major and Mozart’s Symphony No. 40 in G minor. The result is a set of numerical amplitudes between -1 and 1 shown in Figures 3 and 4. Each of these amplitudes was used to make neurons fire when the amplitude falls within a prescribed threshold range. Under the assumption that each neuron fires in response to one stimulus, we would need the same number of neurons as the sample size, i.e., 117,247 in Haydn’s symphony and 144,532 in Mozart’s symphony. Our objective was to approximate the amplitude using one neuron for
Implementing Neural Firing
121
Table 6. Numerical data for Figure 7. ( 1.67 3.53500) ( 1.92 3.44571) ( 2.23 2.92935) ( 2.45 3.34641) ( 2.89 3.09816) ( 3.42 1.63845) ( 3.70 3.03858 ) ( 3.86 4.34934 ) ( 4.45 4.93521 ) ( 4.86 5.27283I ( 5.36 4.13088 j ( 5.64 6.61338) ( 5.83 6.64317j ( 5.95 6.33534) ( 5.99 5.95800) ( 6.08 5.49129) ( 6.40 4.93521 ) ( 6.45 2.42292) ( 6.80 2.82012) ( 7.14 5.21325) ( 7.67 0.61566 ) ( 7.89 3.91242 ) ( 7.99 6.08709) ( 8.14 5.12388 ) ( 8.20 5.46150) ( 8.27 1.92642) ( 8.45 2.01579) ( 8.86 1.82712) ( 9.08 0.55608) ( 9.17 0.74475) ( 9.23 1.08237) each amplitude
( 1.69 3.49536) ( 1.96 3.19746)
( 1.80 ( 2.06 ( 2.30 ( 2.61 ( 3.20 ( 3.67 ( 3.74 ( 4.08 ( 4.61 t 5.24 ( 5.45 ( 5.70 i 5.89
( 2.27 3.37620) ( 2.61 2.45271) ( 3.02 2.01579) ( 3.52 2.87970) ( 3.7 3.346410) ( 4.02 459759 ) ( 4.49 1.11216) t 5.05 5.431711 ( 5.36 6.08709j ( 5.64 6.92120) i 5.86 4.68696j ( 5.99 0.86391) ( 6.05 534234 ) ( 6.08 6.20625) ( 6.42 0.80433) ( 6.52 1.82712) ( 7.02 1.63845) ( 7.36 1.70796) ( 7.67 1.79733) ( 7.92 4.19046) ( 8.05 4.65717) ( 8.17 4.93521) ( a.2 6.484290) ( 8.27 5.76933) ( 8.67 1.92642) ( 8.98 1.76754) ( 9.09 0.64545) ( 9.17 1.14195) ( 9.23 1.36041)
( 5.99
( 6.05 ( 6.11 ( 6.42 ( 6.61 ( 7.08 ( 7.37 ( 7.67
( 7.99 ( 8.08 ( 8.17 ( a.23 ( 8.30 ( 8.70 ( 9.02 ( 9.11 ( 9.17 ( 9.30
3.47550) 3.12795) 2.82012) 3.25704) 1.79733) 3.62445) 1.41999) 1.27104) 5.12388) 5.76933 1 6.45450j 4.37913) 4.34934j 5.05437) 5.89842) 5.62038) 2.04558) 2.60166) 3.03858) 3.28683) 3.53508) 1.88670) 4.40892) 5.52108) 5.98779) 5.92821) 1.82712) 1.60866) 0.86391) 1.54908) 1.27104)
( 3.24 ( 3.67 ( 3.74 ( 4.20 ( 4.86 ( 5.27 i 552 ( 5.74 i 5.89 ( 5.99 ( 6.05 ( 6.36 ( 6.42 ( 6.70 ( 7.11 ( 7.49 ( 7.80 ( 7.99 ( a.11 ( 8.20 ( 8.23 ( 830 ( 8.77 ( 9.05 ( 9.14 ( 9.20 ( 5.61
data displayed
5. NEURAL The network described
3.29676) 337620 ) 2.60166) 2.23425) 2.91942) 3.84291) 4.09116) 4.74654) 1.02279) 0.96321) 6.45459j 6.83184) 4.90542j 5.24304) 6.17646) 4.62738) 2.23425) 1.70796) 0.71496) 5.49129) 5.76933) 4.13088) 455787 ) 0.58587) 651408 ) 6.36513) 0.52629) 0.89370) 139020 ) 0.96321) 6.29562)
to play back the music.
value, and then use the resulting values in Mathematics
A small sample of the numerical
in Figures 3 and 4 is given in Tables 4 and 5.
REPRESENTATION
in Section 2 and the data sampled to form the picture given in Figure 1,
were used to create a 124 x 124 network of neurons consisting each layer.
( 1.83 ( 2.11 ( 2.45 ( 2.83
of 124 layers with 124 neurons in
The result of the firing of the network is given in Figure 5. Each dot is generated
by the firing of a neuron in response
to a stimulus
falling within the neuron’s
lower and upper
thresholds. To speed the computations, mated
we made a simplifying
by the firings of one neuron on adjusting
rather than creating value by estimating
a neuron for each value approximated. the parameters
neuron-0
the value in question,
(make-instance
: citation
Dirac distribution in Figure 6.
excitation
With these parameters,
where (Y = 2n(2000)2(0.0007)2,
p = o/2000,
in the neighborhood
of the stimulus
instead of approximating {Pe-@,
a
Q, p 2 0) such that
we created
0.0007 2000
: ref ractary-period . * . >I. of one millisecond.
stimuli can be approxi-
‘neuron
:impulse-value
This neuron reaches its maximum
Thus,
CY,p, and y for the functions
the firing of the neuron reproduces (setq
assumption-all
those firings to the intensity
= 0.001 in 0.7 of a millisecond
a single firing generated
and y = P”+‘/(2000T’((r
and has a refractory
period
by the neuron is given by
+ l)),
which behaves
of the origin, and its graphical representation
as a
is illustrated
122
T. L. SAATY et al. Table 7. Numerical data for Figure 8. ( i I i ( f
i ( ( ( ( ( i ( I i I i ( ( ( I i (
i ( ( ( ( ( ( f i ( i ( ( i ( i ( ( i ( ( ( ( ( ( (
0.73 1.13 2.03 2.94 5.82 1.1 I 1.98 2.77 5.42 2.69 3.48 5.44 5.26 5.20 5.36 5.48 5.86 5.76 6.26 5.65 5.56 6.11 6.83 7.06 6.98 6.26 6.58 6.93 5.85 6.28 6.66 7.12 7.46 7.67 7.64 7.68 1.12 7.60 6.91 6.18 7.22 7.54 6.84 7.24 6.34 6.57 6.28 5.56 3.47 3.47
1.19733) 3.o485lj 4.22025) 5.27283) 6.95100) 2.76054) 4.06137) 5.05437) 6.73254) 5.09409) 5.75940) 6.14241) 7.OOO65) 7.06023) 6.67296) 6.48429) 6.58359) 6.28569) 6.80205) 7.10988) 7.61589) 8.05323) 8.00358) 7.39785) 6.98079) 6.83184) 7.77519) 7.24890) 8.2O218) 9.06609) 8.97672) 8.66890) 8.37099) 7.84470) 7.37799) 7.O7016) 7.07016) 7.37799) 6.66303) 8.71854) 8.31141) 7.80498) 8.52987) 7.88442) 8.89728) 8.16246) 8.78805) 7.03044) 5.32248) 5.28276)
I i ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
0.77 1.99593) 1.29 3.30669) 2.17 4.44864) 3.27 5.61045) 0.83 1.79733) 1.23 3.04851) 2.13 4.22025) 3.04 5.27283) 5.92 6.95100) 2.72 5.36220) 3.99 6.05730) 5.62 7.09002) 5.32 7.14960) 5.22 7.20918) 5.55 6.72261) 5.6 6.484290) 5.75 6.43464) 5.88 6.33534) 6.11 6.83184) 5.56 7.26876) 5.60 7.79505) 6.30 8.07309) 6.91 7.87449) 7.14 7.29855) 6.86 6.93114) 6.30 S.lO288) 6.70 7.67589) 6.98 7.17939) 5.98 8.43057) 6.33 9.12567) 6.67 8.87742) 7.25 8.62917) 7.47 8.24190) 7.65 7.75533) 7.68 7.37799) 7.64 7.07016) 7.09 7.49715) 6.24 6.80205) 7.09 6.74247) 6.77 8.58945) 7.28 8.20218) 7.59 7.68582) 6.88 8.39085) 7.32 7.83477) 6.37 8.73840) 6.49 8.08302) 6.26 8.87742) 5.63 7.09995) 3.56 5.32248) ( 3.54 5.27283)
( i ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
(
0.82 1.43 2.35 3.84 0.87 1.39 2.27 3.37 2.66 2.72 4.45 5.56 5.38 5.28 5.73 5.50 5.65 5.98 5.96 5.53 5.65 6.41 6.98 7.20 6.73 6.43 6.71 6.93 6.13 6.46 6.82 7.36 7.52 7.62 7.75 7.51 7.20 6.40 7.30 6.88 7.32 7.57 6.96 7.28 6.43 6.44 5.42 3.10 3.08 3.56
2.19453) 3.47550) 4.72668) 5.98779) 1.99593) 3.30669) 4.44864) 5.61045) 5.07423) 5.58066) 6.28569) 7.16946) 7.22904) 7.34820) 6.78219) 6.47436) 6.37506) 6.38499) 6.88149) 7.37799) 7.84470) 8.17239) 7.59645) 7.19925) 6.86163) 8.07309) 7.50708) 7.00065) 8.66889) 9.12567) 8.76819) 8.54973) 8.13267) 7.65603) 7.30848) 7.07016) 7.48722) 6.72261) 6.83184) 8.51001) 8.05323) 7.57659) 8.24190) 7.67589) 8.57952) 8.35113) 6.78219) 5.34234) 5.28276) 5.21325)
t 0.91 2.452711 i 1.62 3.73368) ( 2.48 4.89549) ( 4.56 6.36513) ( 0.92 2.19453) ( 1.53 3.47550) ( 2.45 4.72668) ( 3.94 5.98779) ( 2.69 5.41185) ( 2.72 5.58066) ( 4.82 6.49422) ( 5.48 7.14960) ( 5.46 7.29855) ( 5.38 7.45743) (5.70 6.65310) ( 5.61 6.47436) ( 5.53 6.30555) ( 6.10 6.50415) ( 5.83 6.93114) ( 5.51 7.47729) ( 5.80 7.97379) ( 6.58 8.15253) ( 6.99 7.54680) ( 7.16 7.09995) ( 6.60 6.82191) ( 6.56 8.00358) ( 6.76 7.44750) ( 6.85 6.90135) ( 6.25 8.84763) ( 6.50 9.12567) ( 6.84 8.68875) ( 7.41 8.52987) ( 7.57 8.05323) ( 7.54 7.54680) ( 7.78 7.24890) ( 7.38 7.02051) ( 7.28 7.40778) ( 6.56 6.67296) ( 7.46 6.93114) ( 7.03 8.5ooo8) ( 7.36 7.96386) ( 6.63 8.45043) ( 7.04 8.13267) ( 1.24 757659) ( 6.49 8.42064) ( 6.41 852987) ( 5.45 6.87156) ( 3.24 5.32248) ( 3.22 5.28276)
( 1.01 2.76054) i 1.88 4.06137) ( 2.67 5.05437) f 5.32 6.732541 i 1.01 2.45271) ( 1.72 3.13368) i 2.58 4.895493 ( 4.66 6.36513) ( 2.67 5.58066) ( 3.00 5.30262) f 5.30 6.72261) i 5.38 7.08OO9j ( 5.56 7.30848) ( 5.53 7.54680) ( 5.63 6.58359) ( 5.75 6.52401 j ( 5.66 6.28569) ( 6.20 6.63324) i 5.73 6.980793 I 5.51 7.55673) i 5.95 8.02344) ( 6.71 8.09295) ( 6.99 7.49715) ( 7.11 7.03044) f 6.41 6.821911 i 6.63 7.87446) ( 6.85 7.34820) i 5.78 8.04330) ( 6.26 8.96679) ( 6.61 9.03630) ( 6.96 8.66889) ( 7.44 8.45043) ( 7.64 7.96386) ( 7.59 7.45743) ( 7.16 7.13967) ( 7.22 7.04037) ( 7.46 7.33821) ( 6.74 6.62331) ( 7.59 7.02051) ( 7.14 8.42064) i 7.43 7.88442) ( 6.69 S.SOOO8) ( 7.14 7.99365) ( 7.19 7.48722) ( 6.51 8.30148) ( 6.35 8.66889) ( 5.54 6.96093) ( 3.38 5.32248) ( 3.36 5.28276)
The firing of neuron-O reaches its maximum at t = 0.0007 and is approximately equal to 0.9933. Using the maximum value as the approximation factor, we are implicitly setting the error of our approximation equal to 0.0067. Thus, if the value of the impulse is measured at 100, then the approximation given would be lOO(O.9933) M 99.33. The resulting approximations are exhibited in Figures 7 and 8.
6. SUMMARY
AND
CONCLUSIONS
The basic assumption we made to represent the response to a sequence of individual stimuli by the network of Section 2 is that all the layers are identical, and each stimulus value is represented by the firing of a neuron in each layer. A shortcoming of this representation is that it is not invariant with respect to the order in which the stimuli are fed into the network. It is known in the case of vision that the eyes do not scan pictures symmetrically if they are not symmetric, and hence, our representation must satisfy some order invariance principle. Taking into account this principle would allow us to represent images independently of the form in which stimuli are input into the network. For example, we recognize an image even if it is subjected to a rotation,
Implementing
Neural Firing
Table 8. Sample of neural approximation +0.55469 -0.31250 -0.17188 -0.00781 -0.33594 +0.28906 +0.67969 -0.13281 -0.17969 +0.00781 -0.29688 +0.07812 +0.64844 -0.06250 +0.0aaoo +0.03906 -0.4453 1 +0.15625 +0.39062 -0.12500 +0.19531 +a.42969 -0.33594 +0.03906 +0.26562 -0.26562 +0.0703 1 i-O.46094 -0.21875 +0.203 12 +0.07812 -0.54688 +0.11719 +0.29688 -0.18750 +0.34375
+0.39062 -0.39062 -0.10156 -0.04688 -0.31250 +0.42188 +0.53906 -0.24219 -0.10156 -0.03125 -0.32812 +0.23438 +0.56250 -0.16406 +0.07812 -0.03125 -0.43750 +0.26562 iO.30469 -0.17188 +0.35156 +0.33594 -0.35938 +0.20312 +0.15625 -0.33594 +0.23438 +0.43750 -0.22656 +a.29688 -0.10156 -0.55469 +0.28906 +0.26562 -0.17188 +0.46875
+a.2578 1 -0.39844 -0.03906 -0.08594 -0.28906 +0.51562 +0.38281 -0.29688 -0.03125 -0.05469 -0.35156 0.36719 +0.46094 -0.21094 +O. 16406 -0.11719 -0.41406 +0.32812 +0.21875 -0.17969 +0.50781 +0.21875 -0.37500 +0.28906 0.04688 -0.35938 +0.38281 0.36719 -0.22656 +0.3203 1 -0.25781 -0.50781 +0.42969 +0.19531 -0.14062 iO.49219
0 164OG :0:14oa2 CO.10938 tO.06250 +0.09375 +0.08594 tO.06250 ~0.01562 +0.00781 -0.02344 +o.ooooo to.01562 -0.01562 -0.01562 -0.04688 -0.03906 -0.12500 -0.10156 -0.12500 -0.14062 -0.11719 -0.12500
+0.15625 +0.14062 +0.10938 +0.07812 +0.09375 +0.08594 +0.06250 +0.01562 -0.01562 -0.02344 +0.00781 +0.03125 -0.02344 -0.02344 -0.04688
+0.15625 +0.14062 +0.09375 +0.08594 +0.08594 +O.O8S94 +0.05469 +0.01562 -0.01562 -0.02344 +0.00781 +0.03125 -0.02344 -0.03906 -0.06250
:g;;;: -0:10156 -0.12500 -0.14062 -0.12500 .0.12500 .0.09375 :o".:o"::: -0.10156 -0.14062 $;;W& -0.11719 -oh719 -0.10156
-0.04688 -0.12500 -0.10156 -0.12500 -0.14062 -0.14062 -0.12500 -0.09375 -0.0937s -0.14844 -0.11719 -0.10156
:oo.g;:;: -0:08594 -pm;;
-0.07031 -0.08594 -0.08594 -0.09375 -0.07031
-0.08594 -0.09315 -0.09375 -0.08594 -0.08594
:ooy;:;;t
-0.09375 -0.11719
:p; -0:lllSS -0.11719
-0.11719 -0.12500 -0.16406 -0.11719 -0.07031 -0.10156 -0.10156 -0.04688 -0.01562 -0.02344 -0.03906 +0.00781
.0:08594 -0.0937s -0.12500 -0.12500 -0.17188 -0.11719
_ _ _. _ ::::;:;
-0:03906 +o.oOooo -0.02344 -0.04688 +0.00781
0 15625 :0:14062 +0.09375 +0.09375 +0.08594 +0.07812 +0.05469 +0.01562 -0.01562 -0.02344 +0.00781 +0.03125 -0.03906 -0.04688 -0.04688
to Haydn’s Symphony
+0.16406 -0.38281 +0.03125 -0.14062 -0.25781 +0.58594 +0.26562 -0.30469 +0.03906 -0.08594 -0.37500 +0.46875 +0.36719 -0.21875 +0.23438 -0.203 12 -0.35938 +0.36719 +0.15625 -0.14844 +0.61719 +0.0703 1 -0.35938 +0.32812 -0.00781 -0.34375 +0.49219 0.23438 -0.21094 +.29688 -0.34375 -0.41406 +0.49219 +0.078 12 -0.12500 +0.45312
Table 9. Sample of neural approximation
+. +0.132L?1 +0.09375 io.09375 +0.08594 +0.06250 +0.05469 +0.01562 -0.01562 -0.02344 +0.01562 +0.03125 -0.02344 -0.04688 -0.04688 -0.06250 -0.12500 -0.12500 -0.04688 -0.10156 -0.10156 -0.12500 -0.12500 -0.12500 -0.12soo -0.14062 -0.14062 -0.12500 -0.12500 -0.10156 -0.10156 -0.10156 -0.11719 -0.14844 -0.14062 -0.11719 -0.11719 -0.08594 -0.09375 -0.07031 :oo~~;::t: -0.08594 -0.09375 -0:08594 -0.08594 -0.07031 -0.08594 :;$:g: -0:12.500 -0.10156 -0.11719 -0.10156 -0.14062 -0.10156 -0.14062 -0.16406 -0.16406 -0.11719 -0.11719 -0.07031 :g::g -0.09375 -0.10156 -0:08594 -0.06250 -0.06250 -0.03906 -0.03906 -0.02344 -0.03906 -0.02344 -0.01562 +0.00781 +0.00781
123
+0.08594 -0.33594 +0.0703 1 -0.21875 -0.18750 +0.67188 +0.19531 -0.30469 +0.08594 -0.14844 -0.34375 +0.57812 +0.26562 -0.18750 +0.25000 -0.28906 -0.29688 +0.40625 +0.09375 -0.09375 +0.64062 -0.07812 -0.35156 +0.33594 -0.03906 -0.27344 +0.50781 0.0703 1 -0.17969 +0.26562 -0.39062 -0.30469 +0.46094 -0.04688 -0.08594 +0.41406
to Mozart’s
+U.l4062 +0.11719 +0.08594 +a.09375 +0.08594 +0.05469 +0.05469 +0.01562 -0.OlS62 -0.01562 +0.01562 +0.03125 -0.01562 -0.03906 -0.03906 -0.07031 -0.12500 -0.11719 -0.12500 -0.12500 -0.12500 -0.12500 -0.10156 -0.11719 -0.14062 -0.12500 -0.08594 -0.07031 -0.09375 -0.09375 -0.08594 -0.08594 -0.10156 -0.12500 .0.10156 -0.14062 -0.16406 -0.11719 -0.08594 -0.09375 -0.07031 -0.04688 -0.03906 -0.04688 -0.01562 +0.00781
No. 102. -0.03 125 -0.28906 +0.05469 -0.29688 -0.06250 +0.75000 +0.10938 -0.28906 +0.07812 -0.21875 -0.25000 +0.66406 +0.16406 -0.14062 +0.203 12 -0.37500 -0.16406 +a.43750 +0.03125 -0.01562 a.59375 -0.203 12 -0.28906 +0.34375 -0.07812 -0.17969 to.46875 -0.078 12 -0.09375 +0.25000 -0.43750 -0.20312 +0.39062 -0.13281 +o.ooooo +0.40625
-0.17188 -0.22656 +0.03 125 -0.33594 +0.10938 +0.75781 -0.00781 -0.25000 +0.04688 -0.25781 -0.09375 +0.68750 +0.05469 -0.07812 +0.12500 -0.42969 -0.00781 0.45312 -0.04688 +0.07812 +0.51562 -0.28906 -0.14062 +0.33594 -0.16406 -0.06250 +0.453 12 -0.17188 +0.04688 +0.19531 -0.48438 -0.05469 +0.33594 -0.17969 +0.16406 +0.38281
Symphony No. 40.
0 14062 :0:117L9 +0.07812 +0.09375 +0.08594 +0.03906 +O.O312S +0.00781 -0.01562 -0.01562 +0.01562 +0.01562 +o.ooooo -0.04688 -0.04688 -0.08594 -0.11719 -0.11719 -0.12500 -0.11719 -0.11719 -0.11719 -0.10156 -0.11719 -0.14062 -0.12.500 -0.08594 -0.07031 :oo~~~::~ -0:08594 -0.08594 -0.09375 -0.12500 -0.11719 -0.14844 -0.14844 -0.lOlS6 -0.09375 -0.09375 -0.04688 -0.03906 -0.03906 -0.04688 +o.ooooo +0.00781
0 15615 :0:132*1 +0.07812 +0.09375 +0.08594 +0.05469 +O.OISbZ +0.00781 -0.01562 -0.01562 +0.01562 r0.00781 -0.01562 -0.06250 -0.06250 -0.09375 -0.11719 -0.11719 -0.12500 -0.11719 -0.11719 -0.10156 -O.lOISb -0.12500 -0.14062 -0.l2SOO -0.07031 -0.07031 -0.09375 -0.09375 -0.08594 -0.08594 -0.08594 -0.12500 -0.11719 -0.16406 -0.14062 -0.10156 -0.10156 -0.09375 -0.03906 -0.02344 -0.03906 -0.04688 lo.ooooo +0.01562
+O.l562S +0.11719 to.07812 +0.10938 +0.08594 +0.06250 +0.01562 +0.00781 -0.01562 -0.01562 +0.00781 -0.01562 -0.01562 -0.04688 -0.04688 -0.11719 -0.11719 -0.12500 -0.14062 -0.11719 -0.12500 -0.10156 -0.10156 -0.12500 -0.14062 -0.12500 -0.08594 -0.08594 -0.09375 -0.09375 -0.08594 -0.08594 -0.08594 -0.12500 -0.12500 -0.17188 -0.12500 -0.09375 -0.10156 -0.09375 -0.03906 -0.01562 -0.03906 -0.04688 +o.ooooo +0.01562
1
124
T. L. SAATY et al.
or to some sort of deformation. Thus, the invariance principle must include afhne and similarity transformations. This invariance would allow the network to recognize images even when they are not identical to the ones from which it recorded a given concept, e.g., a bird. The next step would be to use the network representation given here with additional conditions to uniquely represent patterns from images, sounds, and perhaps other sources of stimuli such as smell.
REFERENCES 1. A.L. Hodgkin and A.F. Huxley, A quantitative description of membrane current and its applications to conduction and excitation in nerve, Journal of Physiology 117, 509-544, (1952). 2. Ft. Both, W. Finger and R.A. Chaplain, Model predictions of the ionic mechanisms underlying the beating and bursting pacemaker characteristics of molluscan neurons, Biological Cybernetics 23, l-11, (1976). 3. T.R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic P-cell, Biophysical Journal 42, 181-190, (1983). 4. R.E. Plant and M. Kim, Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations, Biophysical Journal 16, 227-244, (1976). 5. J. Rinzel, Bursting oscillations in an excitable membrane model, In Proceedings of the gh Dundee Conference on the Theory of Ordinary and Partial Differenctial Equations, (Edited by S.D. Sleeman, R.J. Jarvis and D.S. Jones), Springer-Verlag, New York, (1984). 6. T.L. Saaty and L.G. Vargas, A model of neural impulse firing and synthesis, Journal of Mathematical Psychology 37 (2), 200-219, (1993). 7. G.F. Poggio and V.B. Mountcastle, Functional organization of thalamus and cortex, In Medical Physiology, (Edited by V.B. Mountcastle), Chapter 9, C.V. Mosby Co., St. Louis, MO, (1980). 8. F.J. Brinley, Jr., Excitation and conduction in nerve fibers, In Medical Physiology, (Edited by V.B. Mountcastle), Chapter 2, C.V. Mosby Co., St. Louis, MO, (1980). 9. R.E. Plant, Bifurcation and resonance in a model for bursting nerve cells, Journal of Mathematical Biology 11, 15-32, (1981).