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A THEORY or NEUROMIME NETS CONTAIMNGL ': ‘ " '

RECURRENT *NHIBITION, WITH AN ANALYSis :_

OF A HIPPQCAMPUS MODEL

Thesis far the Degree of PhD. " . 7 .,

.mcmem STATE umvensm .
DUANE‘G. LEET '
1971

 

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This is to certify that the

thesis entitled

A THEORY OF NEUROMIME NETS CONTAINING
RECURRENT INHIBITION, WITH AN ANALYSIS
OF A HIPPOCAMPUS MODEL.

presented by
Duane G. Leet

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Systems Science

WWW ‘Eva/vvvf‘

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Date April 30, 1971

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ABSTRACT

A THEORY OF NEUROMIME NETS CONTAINING
RECURRENT INHIBITION, WITH AN ANALYSIS
OF A HIPPOCAMPUS MODEL

BY

Duane G. Leet

A novel system component called the functal can be set to
realize any one of many different functions. A functal net is an
interconnected array of functals, function generators, and delays.
Some fundamental time-domain properties of these nets are developed.

A functal net model of recurrent inhibition as found in the
CA3 sector of mammalian hippocampus is presented. The model
contains a rank of functals, which are somewhat like adaptive threshold
logic units, and a rank of function generators, which are threshold
logic units. The two ranks are interconnected through delays, and the
function generators inhibit the functals. The only assumption on the
connectivity between ranks is that, for each element in the first rank,
there exists at least one direct circuit path from that element through
some element of the second rank and then back to itself.

The most important characteristic of the model's input-output
transformation is that a single input can be transformed into a
sequence of outputs. This sequence terminates, for a given input,
with the continuous repetition of either a single output or a sequence
of outputs. Some properties of the model's output sequences are
derived, and an algorithm is deve10ped for generating, for any
particular N-functal net, all output sequences which that net could
possibly produce.

A trainable functal net is one in which the functions realized by
its functals are under the control of an external structure called the
trainer, which Operates according to a specified algorithm. Both
the trainer and the trainable functal net are part of a new canonical
system called a functal system, some fundamental prOperties of which

are discussed.

 

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model of
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derive:

Duane G. Leet

The CA3 model is incorporated into an automate. theoretic
model of the hippocampus that is designed to take advantage of
certain of the CA3 model's properties. There does not exist a
training algorithm for this model that can always change the function
realized by one of its functals to any other arbitrarily specified
function. But an algorithm is given that can produce defined changes
as long as the parameters of,the CA3 model meet certain speci-
fications.

The function realized by a functal system model whenever it
is placed in a new environment is called the initial function. The
selection of initial functions is discussed, and an algorithm is

derived to select them automatically.

A THEORY OF NEUROMIME NETS CONTAINING
RECURRENT INHIBITION, WITH AN ANALYSIS
OF A HIPPOCAMPUS MODEL
BY

(9

Duane C? Le et

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCT OR OF PHILOSOPHY

Department of Electrical Engineering and Systems Science

1971

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of'

ACKNOWLEDGEMENTS

The author gratefully acknowledges the aid and encourage-
ment given by his academic advisor, Dr. William Kilmer, not
only on matters related to the research project, but on enurnerable
other matters as well.

The author's father, Gerald Leet, his grandparents, James
and Ruby Leet, his inlaws, Floyd and Irene Layman, and other
members of his wife's family have contributed financial and spiri-
tual support during his graduate studies. The author is also deeply
endebted to the late Dr. Leroy Augenstein for his support and
encouragement.

Finally, this thesis could not have been completed without
the inexhaustible patience, cheerfulness, and sustaining support

of the author's wife, Chris.

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TABLE OF CONTEN TS

PTER 1. INTRODUCTION
What is the Function of Recurrent Inhibition?
Hippocampus Morphology
The Expositional Problem
What is the Function of the Hippocampus ?

HHHHO

HA
.1.
..2
..3
..4
HAPTER 2. FUNCTAL SYSTEMS
1. An Informal Description of the Functal System
. 2. The Functal
3 The Functal Net
4 The Concepts Of State and Output Foundations
5. Properties of Output and State Levels
5. 1. Properties of State Levels
5. 2. Properties of Output Levels
6 The Target Table Generator
7 The Target Table
8 A General Training Structure and Algorithm
8. 1. Movement Within Foundations Under Input and
Function Change
8. 2. The Operation of the Trainer Under Input or
Function Change
2. 9. The Trainer, Target Table Relationship
2. 10. Measures of Functal Net Performance
2. 11. The Functal System Analysis Problem

PPPPPPPPPNPO

5"

CHAPTER 3. A FUNCTAL SYSTEM MODEL OF THE

HIPPOCAMPUS. PART 1

3. 1. Introduction

3. 2. The Input and Output Sets of the Hippocampus System
Model

3. 3 The Input Buffer
3. 4 The CA3 Sector Net
3. 4. 1. Introduction
3. 4. 2. The Pyramidal Cell Model: The Pyramidal
Cell Logic Unit
3. 4. 3. The Basket Cell Model: The Basket Cell Logic Unit
3. 4. 4. The Connectivity and Delays
3. 4. 5. The Operational Algorithm
3. 5 The Output Buffer
CHAPTER 4. PROPERTIES OF THE CA3 SECTOR NET
4.1. Introduction
4. 2. The Output Sequence Set of a PCLU

iv

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20

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24
25

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27
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29
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4. 3. The Output Sequence Set of the Hippocampus Net
4. 4. Rules for Successful CA3 Sector Net Training
Using Algorithm 4. l. 2

CHAPTER 5. A FUNCTAL SYSTEM MODEL OF THE

HIPPOCAMPUS. PART 2

5. l. The Target Table

5. 2. The Trainer

5. 3. The Error Correction Information for the Change of

Function Controller

5. 4. The Read/Write Head and Its Controller
HAPTER 6. THE INITIAL CONDITIONS PROBLEM
I. The Phase Concept

2. Phase 1: Target Table Training

1.

2.

3.

C
6.
6.
CHAPTER 7. DISCUSSION
7. Summary

7. Comments on the Neuroscientific Aspects Of this Study
7. Comments on the Engineering Aspects of the Study

LIST OF REFERENCES

APPENDIX A. BACKGROUND ON THE DEVELOPMENT OF
THE HIPPOCAMPUS NET

A. l. The Pyramidal Cell Logic Unit

A. 2. The Basket Cell Logic Unit

A. 3. The Connectivity

APPENDIX B. A COMPUTER PROGRAM BASED ON
ALGORITHM 4. 3. l

45
46

53
53

53
57

59

61
61
62

75
75
76
78
81

83
83
86
88

89

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Table 3. 3. 1

Table 4. 2. 1

Table 5. 2. 1

Table 6. 2. 1

LIST OF TABLES
The Truth Table for the Input Buffer of the
Functal Systems Model of the Hippocampus

The Computations, Over Several Periods,
of the CA3 Sector Net

The Truth Table for the Change-of- State
Controller

The Function Assignment for Example 6. 2. 1

vi

30

44

56

65

Figure . l. 1
Figure . 2. 1
Figure . 3. 1
Figure 1. 1
Figure . 5. 1
Figure . 8. 1
Figure l. 1
Figure . 4. 1
Figure . 4. 2
Figure . 4. 3
Figure l. 1
Figure . 2. 1
Figure . 4. 1
Figure A. 1

Figure A. 2

Figure B. 1

LIST OF FIGURES
A schematic of a section of the CA3 sector
of the hippocampus.
Dorsal hippocampus and connections.

A phase diagram for the example given in
Section 1. 3.

The functal system surrounded by an environment.

A typical state level structure.
The basic trainer structure of a functal system.

The overall structure of the hippocampus
system model

The pyramidal cell logic unit.
The basket cell logic unit (BCLU).

A general form of connectivity for the CA3
sector net.

A PCLU and its special BCLU.

The training structure of the functal system
model of the hippocampus.

Read-head controller, target table relationship
for a target table sequence.

The pyramidal cell firing rate equations.

The basket cell firing rate equations.

FORTRAN listing for TTABLE.

vii

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CHAPTER 1

INTRODUCTION

1. l . What is the Function of Recurrent Inhibition?

Recurrent inhibition can be described in terms of components
and connectivity and interneuronal relationships. The components,
which are neurons, are arranged in two ranks. The first rank re-
ceives inputs from elsewhere in the nervous system and from
neurons in the second rank; it sends outputs elsewhere and to
neurons in the second rank (Figure 1.1. 1). The second rank
receives inputs only from the first rank and sends outputs only to
the first rank. The interneuronal relationship, termed interneuronal
inhibition, holds when a neuron in the second rank decrements the
impulse frequencies of those neurons in the first rank to which it is
connected.

Recurrent inhibition is found in many regions of vertebrate
nervous systems: sensory systems [1 ], the cerebellum [2], the
hippocampus [3], and perhaps the Spinal cord [4, 5 ]. For this
reason, understanding its function should be of interest to neuro-
scientists.

When a neuroscientist speaks of the function of a structure,
he is usually referring to its specialized actions or purposes.
Within this context, a number of functions have been attributed to
structures containing recurrent inhibition, or its close relative,
lateral inhibition:

1. enhancement of contrast [1 ] and the detection of edges

[6 ].
2. blockage of low-level inputs [1],
3. amplification of time -varying signals of certain

frequencies [7 ],

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4. selective response to signal patterns flowing in one
direction in a two-dimensional space [8] ,

5. the generation of two periodic signals approximately 180
degrees out of phase with each other from a single input
[9].

6. production of quasi-impulse responses to step inputs [10],

and

7. preferential response to stimuli having certain orientations

[11].

Mathematical readers usually interpret the function of a
structure to be the list of input-output correspondences produced by
it, where the word "list" presupposes only an algorithm (essentially
an ordered set of instructions) that can generate any input-output
pair of the list. With this interpretation, the neuroscientist's
"functions" can be regarded as a list of vaguely defined algorithms,
each of which indicates how a certain subset of the set of all
possible inputs is related to the set of outputs.

In order to avoid confusion over these two meanings of
function, the following convention will be adopted: if "function" is
meant in the neurOphysiological sense, the word ”task" will be used
in its place; if "function" is meant in the mathematical sense, the
word ”function" will be used. This thesis represents the first
attempt known to the author to investigate the function of recurrent

inhibition.

1. Z. Hippocampus Morphology

In general, in order to determine the function of any opera-
tional unit, 'it is necessary to measure its inputs and outputs
simultaneously. The vertebrate central nervous system does not
lend itself to this approach because the inputs and outputs of its
various subunits are for the most part inaccessable, undecipherable,
and apparently highly variable in the frequency domain. Furthermore,
the subunits themselves change rapidly with time.

An alternative approach to the determination of the function

of the operational unit is to model it mathematically or by computer

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simulation (or some blend of both). The hippocampus is well suited to
this approach. In particular, a wealth of both morphological and neuro-
physiological data exists on it (see Kilmer [12], References and
Appendix B), which makes component modeling comparatively easy.
The hippocampus also has a highly stylized connectivity and the CA3
sector clearly exhibits all of the known indicators of recurrent inhibi-
tion (see Figure 1.1.1); thus its circuit organization is easily carica-
tured. Two kinds of inputs (ignoring the commissural fibers) and their
origins, plus two kinds of outputs and their destinations are known to
exist (see Figure 1.2.1). Thus, the inputs and outputs of any model are
defined and their characteristics can be compared with the available
hippocampal electrophysiological data. In summation, the hippocampus
is the neural structure of choice for an investigation by mathematical

model and computer simulation of the function of recurrent inhibition.

1. 3. The Expositional Problem

The following example points up the difficulty of communicating
the principles of circuit actions for a neural net of the complexity
found in Figure 1. l. l and of concisely describing the net's function.

Consider the neuron net shown in Figure l. l. l, ignoring all of
the direct pyramid-to-pyramid connections. Assume all activity in the
net is allowed to die out, and then apply an input to the net sufficient to
cause P3 to produce a moderate number of pulses per second (fire at
a moderate rate) and to cause P5 to produce a large number of pulses
per second (fire at a high rate). If this occurs at time to (Figure
l. 3. l), and the leading edges of both trains of pulses require the same
time to reach the basket cell rank, then both B3 and B4 will be
affected at time t1; suppose B3 responds by firing at a moderate rate
and B4 responds by firing at a high rate. Assuming these pulse trains
require the same time to travel to the pyramidal cell rank, both P3
and P5 will be affected at the time t2; suppose P3 reacts by completely
turning off and P5 reacts by decreasing its output to a moderate rate.
At some later time t3 these changes will be felt by the basket cells;
as a result suppose B3 turns off and B4 decreases its output to a
moderate rate. At time t4 these changes will be felt by the pyramids;

as a result suppose P3 begins firing at a moderate rate again and P5

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In order to circumvent these nasty expositional problems, this
paper has developed a formal language, called functal system theory,
for systems of the kind exemplified by the hippocampus. The reader
is urged not to become discouraged by what may seem to him to be
excessive formalism in the following chapters; the formalism is
justified by the compactness with which it expresses functions involving
recurrent inhibition.

In addition to modeling the hippocampus as a functal system,
some of the operational principles of the class of neuromime nets to
which the model belongs are also given, along with characteristics of

the output sequences of such nets.

1. 4. What is the Function of the Hippocampus ?

An hypothesis of the primary task of the mammalian hippo-
campus has been proposed by W. Kilmer and T. McLardy [12].
Previously, Kilmer and W. McCulloch [13] proposed that the task of
the mammalian reticular formation is to decide the basic mode of
behavior of an animal. A mode might be to fight, take flight, groom,
mate, or eat. It is plausible to suggest that another structure exists
which takes modal and current sensory information and generates
commands for acts within modes. For instance, if the mode decision
is to fight, another structure may select the tactics or style to be
used. Kilmer and McLardy believe that the hippocampus is part of
this structure, at least during the animal's behavior-formative period.

Functal system theory is used in this paper to provide an
interpretation of the hippocampus's function which supports this
hypothesis. In a few words, the interpreted function is trainable

re cur rent inhibition.

CHAPTER 2

FUNCTAL SYSTEMS

2. 1. An Informal Description of the Functal System

The hippocampus and its associated structures appear to be
related to a theoretical structure called a functal system. Informal
definitions of each of the components and of the overall operation of
the system are as follows (see Figure 2. l. l).

l. The INPUT GENERATOR interprets the present environ-
ment according to its built-in predisposition and produces an input
from a finite set of possible inputs.

2. The INPUT BUFFER, which is under the control of the
TRAINER, performs a combinational circuit transformation on the
input and produces the input to the functal net.

3. The TRAINABLE FUNCTAL NET has these
characteristics:

a. It is an array of three types of elements: functals,
function generators, and delays.

b. Each functal is capable of realizing any one of many
functions. Each function being realized is under the control
of the trainer.

c. A fixed connectivity exists between the elements of the
net (the rules defining the connectivity may involve probability
density functions).

The functal net generates a finite sequence of outputs for a given single
input.

4. The TARGET TABLE GENERATOR has observed the
environment by this time and has constructed a target table.

5. The TARGET TABLE contains the functions that each of
the functals is required to generate. All communication with the
target table is controlled by the READ/WRITE HEAD AND CONTROLLER.

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required output.

7. The OUTPUT BUFFER is combinational circuitry under
the control of the trainer.

8. The EFFECTOR DEVICES use the output from the output
buffer to allow the entire system to interact with the environment.
It may also be true that this output affects the environment directly.

A formal discussion of functal system theory is presented in
the remainder of this chapter. Throughout the discussion it will be
assumed that the functal net and all its associated structures and

algorithms operate synchronously in discrete time.

2. 2. The Functal

The intuitive concept of a functal is that it is a mechanism
(that is, an algorithm or physical device) which can realize any one
of a finite number (greater than 1) of different functions. If the
domains and ranges of the functions are assumed to be finite sets, and
if time is assumed discrete, then:
Definition 2. 2. l

 

A functal can be represented over all time by

(H. v.3? . >3)
and at any time t by

om = F1 (Mm. zm. t)
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a(t)eZ‘., Fi(.)€T , Z(t)ey, and M(t)ep. .
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Definition 2. 2. 2

p. , a finite set, is the controlled mpg; set of the functal. The

 

elements of IJ. are called controlled inputs. (u is under direct

 

external control. )
Definition 2. 2. 3
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internal input set}

 

 

is the internal input sequence set. The elements of y are called

internal
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Definiti

is the i
can re:

Delinit

is the

Defini

Defini

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empj
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11

internal inputs. (y takes into account possible inputs to the functal
that cannot be directly controlled. )
Definition 2. 2. 4

F1
‘3" = {Fi: pr-* 73}
is the function set. (The function set contains the functions the functal
can realize. )

Definition 2. 2. 5

Z = {O'(t) = H1(t) Hz(t) . . . :Hl(t) is an element of the
finite output setJ'C }

 

is the set of output seflences.
Definition 2. 2. 6

 

Any member of 2, 0(t), is called an output sequence.
Definition 2. 2. 7

 

 

Any vector element _
i
h1(t)

. i
H1(t) = h2(t)

 

 

i
Lhn‘tL
of an output sequence is called an ou_tput sequence element (or simply

an output).

Definition 2. 2. 8

 

 

A component h;(t) of an output vector is called an output
element.

Definition 2. 2. 9

Thesequence
l 2
0'.t = h.t h.t
J() J() J() ,

is called an output sequence component.

The specification of both controlled and internal inputs
emphasizes a basic prOperty of functals: only controlled inputs to a
functal are provided by the input generator; internal inputs are
generated within the functal net. In particular, feedback is one kind
of internal input. If it is present the functal can generate a sequence,

even when there is only a single input from the input generator.

‘ 'I J'“. ‘P

.. ..f
55;).

2.3. 'I

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the out‘
atthe <

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Z ist

even 1
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under

ASSUr

12

2. 3. The Functal Net
Definition 2. 3. l

A functal net consists of functals plus unit delay elements at

 

the output of each functal, function generators plus unit delay elements
at the output of each generator, and a connectivity scheme relating
these.

The delay elements are included for two reasons. First, the
outputs of some of the elements in a functal net will be in a feedback
configuration. The standard way to analyze such nets is to insert
unit delays. Second, all physically realizable functal nets will have
delays in lines and elements.

The concept of state plays a fundamental role in understanding
the behavior of functals:

Definition 2. 3. 2

 

The outputs of the delay elements can be ordered in a vector

called the state vector 06 Q, the state vector set. The ordering will

 

 

be Q = (X, Z), where X is the output of the delay elements associated
with the functals and Z is the output of the delay elements associated

with the function generators. X is called the functal state vector and

 

Z is called the generator state vector.

 

If the functal net is considered to be a single functal of an
even larger net, then the elements H of the output sequence set will,
are those

f
components of X considered outputs and Hg are those components

by convention, have the form H = (Hf, Hg), where H

of Z considered outputs.

A special kind of functal net is the trainable functal net:
Definition 2. 3. 3

A trainable functal net is a functal net whose function is under
the control of a defined structure called the trainer.

The trainer will be discussed in more detail after the character

of the input-output relationship of a general functal net is revealed.

2. 4. The Concepts of State and Output Foundations
There is a graphic viewpoint which can promote some initial
understanding into the design .and analysis problems for functal nets.

Assume that at some initial time to the vector of functions currently

.4‘. 'l'n—V'fl' _

IM”\ 1 t"-

realized
the statl

Definiti

is calle

Definfi'

combii
For e2
new st
these

Defini
N

trary

Defin
\

funct

l3

realized by the functals is F(to), the controlled input vector is 1(t0),
the state vector is Q(to), and the output vector is H(to).
Definition 2. 4. l

 

The quadruple
Lu) = < Fm, om, 1m. Hm >
is called the_12_c_:ls_ of the functal net at time t.
Definition 2. 4. 2

 

 

The locus L(to) is called the initial locus.

Consider each and every combination of F and 1. Within each
combination, place the net in every possible state in the state set.
For each state allow the net to compute for one period and record the
new state. Construct a standard state table or state diagram from
these data. The resulting representation is given a special name.
Definition 2. 4. 3

 

A state level is the state structure associated with any arbi-

 

trary but specified combination of F and I. The notation is 1(F, 1).
Definition 2. 4. 4

The set of all state levels is called the state foundation of the

 

functal net. .

The fact that the output vector H is a subvector of the state
vector Q can be used to construct an equivalent set of definitions for
the output.

Definition 2. 4. 5

 

An output level lo(F, I) is the output structure associated with

 

a combination of any F and any 1.
Definition 2. 4. 6

 

An ouflaut foundation is the set of all possible output levels for

 

a given functal net.

2. 5. Properties of Output and State Levels
2. 5. 1. Properties of State Levels

Kauffman [14] has demonstrated the following property for
nets of arbitrarily connected switching elements having fixed inputs:
Definition 2. 5. l

A state with the prOperty that the net remains in the state once

it is entered is called an gquilibrium state.

 

‘u' - V—annr.u.

I.) \ 2.; n. 3.34 :7-

Definith
uni—'—

cafled 2

leads h

Proper

pr0per

14

Definition 2. 5. 2

 

A subsequence of states that is continuously repeated is
called a state cycle.
Definition 2. 5. 3

A subsequence of states with the property that it eventually

 

leads to a state cycle is called a state run-in.

 

PrOperty 2. 5. l

 

Each state of a level has one and only one of the following
properties:

1. It is in a state run-in.

2. It is an equilibrium state.

3. It is in a state cycle.

It is clear that this prOperty is true for function generators
of arbitrary but finite domains and ranges.

A useful relation between state run-ins and state cycles is the
following.
Definition 2. 5. 4

Within a state level, all states belonging to run-ins to the

 

same cycle plus all the states belonging to the cycle form a set of

states called the state cycle complex.

 

A typical state level is shown in Figure 2. 5. 1.

One state in each state cycle complex will assume particular
importance:
Definition 2. 5. 5

 

Any state in a state cycle complex may be designated as a

start state for the cycle.

 

Property 2. 5. 2

 

There can be only one start state per state cycle complex.
Proper_ty 2. 5. 3

A given start state will lead to a unique state cycle.
Definition 2. 5. 6

 

A sequence of states (run-in plus cycle) in state level 1(F, I)
with start state qo will be denoted by C(qo, F, I), and will be

called a state sequence.

 

An important property of any state sequence is:

i it it \L
r ii: t -

The 31

value c

15

State Space

start state

 

run-in

 

start state

 

 

 

/ cycle

 

 

 

g \- equilibrium 8 tate
2

Figure 2. 5. 1. A typical state level structure.

The state space is three-dimensional, with each state being binary
valued.

 

Prooerty
'1

indicate;

second I
2.5. 2.

for eve:

Definiti
called :

state 1'

{s e
DEADII
is can
D . .
4%

is:

139$

Same

Outpu

D .
Q

two

16

Propertj 2. 5. 4
The second occurrence of any state in the sequence C(q, F, 1)
indicates the completion of the state cycle and the beginning of a

second pass through the cycle.

2. 5. 2. PrOperties of Output Levels

Since the output vector H is subvector of the state vector Q,
for every state cycle there is a corresponding output cycle:
Definition 2. 5. 7

A sequence of outputs which continuously repeats itself is
called an outgut cycle.

There will also be sequences of outputs corresponding to the
state run-ins:
Definition 2. 5. 8

 

A sequence of outputs which eventually leads to an output cycle

is called an output run-in.

 

Corresponding to the equilibrium state:
Definition 2. 5. 9
An output which continuously repeats itself is called an

equilibrium ou_tput.

 

Finally, the definition corresponding to the state cycle complex
is:
Definition 2. 5. 10

Within an output level, all outputs belonging to run-ins to the
same cycle plus all the outputs belonging to the cycle form a set of
outputs called the ou_tput cycle cormlex.
Definition 2. 5. ll

 

 

An output cycle plus output run-in in level lo(F, I) with initial
output H1 is an cumut seqmgnce of the net and is signified by
«(HR F, 1).

Of course the similarity between the output sequence of the
functal definition and the above definition is no accident. Indeed,

6 (H1, F, I) = HIHZHB... = F(I, 2).

Now, consider a specific state level l(F, I) and the following

two state cycle 8:

17

(The cycles are listed in the form
ql(tl) q1(tl+d) ql(tl+2d)

q2(t1) q2(tl+d) q2(tl+ 2d)

qndl) qn(tl+d) qn(tl+2d) )

 

 

cycle numberl cycle number 2
00000 0000
10011 1011
11001 1101
00000 1111

If q1 and q2 are defined as the outputs, then the sequences
0 0 0 0 0 and 0 0 0 0 0 0
l 0 0 l l 1

outputs are in more than one output cycle and the second occurrence

1 0 l 1 are the output cycles. Note that the 0 and

of g in cycle number 1 did not signal the end of the cycle. These
observations can be generalized in the following properties:
PrOperty 2. 5. 5

Any single output may be in more than one output cycle complex.
Property 2. 5. 6

It is not possible to determine the end of an output cycle
by comparing the current output with previous outputs.

These two properties play a significant role in determining

the complexity of the functal system trainer.

2. 6. The Target Table Generator

A target table can be generated whenever it is both advan-
tageous to do so and conditions permit. This generally requires the
target table generator to know what functions can be trained from
each function the net can realize. In other words, the target table
generator should have available as a reference the following class of
sets:
Definition 2. 6. l

 

6’ = {1.0! i: 21 i is a convergence set}

is the convergence class.

is the

table

requii
that tl
just c

each t

On the
ofthe

in par
2.7.

Sequel

Called

Sequer.
target

seqUEn
D '-
~£igg£

sequent

18

Definition 2. 6. 2
2! i = {er T : the functal realizing the

 

function FiGT can be trained to realize
the function Fk}

is the converflnce set of the function F,.

 

Implicit in these definitions is the requirement that the target
table generator must also have knowledge of the set T . This
requirement should not be taken lightly. In the real system it implies
that the target table generator and the functal net must be more than
just casually related: they must have evolved in a way that allows
each to know what it can expect from the other.

This kind of relationship could come about very naturally
in a neural system if the target table generator structure
grew the functal net to perform a deligated task.

On the other hand, in the design of the artificial system, the design
of the target table generator and the functal net will have to proceed

in parallel.

2. 7. The Target Table

The target table contains a list of output sequences, one
sequence for every possible input to the functal net.
Definition 2. 7. 1

When contained in a target table, an output sequence will be

 

called a target sequence, with the notation 0*(t).

 

As with the output sequence, the target sequence is a vector
sequence. The following definitions locate the various parts of the
target sequence.

Definition 2. 7. 2

A target sequence element HJ*(t) corresponds to the output

 

sequence element of Definition 2. 2. 7.
Definition 2. 7. 3

 

A taget sequence commonent element, or M: hg*(t),
corresponds to the output element of Definition 2. 2. 8.
Definition fi2_. 7. 4

A target sequence component «1*(t) corresponds to the output

sequence component of Definition 2. 2. 9.

Defh

term

finct

1eng1
conn
Inod:
ataz

Supp
targ

2. a.
2. s.

a1go:

desc

eithe

19

Definition 2. 7. 5

 

The set of target sequence components for a single output
terminal and over all possible input values to the net is called a

functal section of the target table.

 

In order to keep the target table as compact as possible, the
length of a target sequence is limited to the maximum length of any
component's run-in plus cycle. Along with this convention, a
modified regular expression "notation is used when explicitly listing
a target sequence. This notation is best defined by example.
Suppose the functal net has four outputs and the components of the

target sequence for some input I are:

0' 1*”) = 0123456456456 .....
0' 24(1) = 0000000 .....

63*(1) = 234523452345 .....
64*(1) = 8722222222222 .....

Then the notation for these is:

61*(I) = 0123456(456)*
_ (72*(1) : 00*

63*(1) = 2345(2345)*

04*(1) = 8722*

As one target sequence, the notation is:

0123 456456456456 456456456456 *
0*(1) = 0000 000000000000 000000000000

2345 2345 2345 2345 2345 2345 2345

8722 222222222222 222222222222

2. 8. A General Training Structure and Algorithm

2. 8. 1. Movement Within Foundations Under Input and Function Change

In this section a general form for a trainer structure and
algorithm is proposed. First, though, it will be necessary to
describe what happens in the foundations when there are changes
either in the inputs to a net or in the functions of a net.

Assume that the initial locus of the net is

20

Leo) = < Ftto), mo). one). Hue) > .

Therefore, the net is in:
a. state level l(F(tO), I(to) ).
b. state sequence C(Q(to), F(to). 100)).
c. output level lo(F(to), l(to) ),
output sequence 0' (H(to), F(to), l(to) ).

Assuming l(to) and F(to) are not changed, (b) and ((1) define the
future of the net.

Now suppose the input is changed and is effective at time t1.
At this time the net is in state Q(tl) and this becomes a new start
state. This means that the net is in:

a. state level l(F(to), I(tl) ),

b. state sequence C(Q(t1), F(to), l(tl) ),

c. output level lo(F(to), l(tl) ),

d. output sequence 0' (H(tl),F(to), l(tl) ).

Finally, suppose the function that the net is realizing is

changed and becomes effective at time t At this time the net is in

state Q(t2) and this becomes a new stari state. Therefore, the net
18 m:

a. state level l(F(t2). l(tll ).

b. state sequence C(Q(tz). F<t2)’ l(tl) ).

c. output level lo(F(tZ), I(tl) ),

output sequence 61(H(t2), F(t2), l(tl) ).

2. 8. 2. The Operation of the Trainer Under Input or Function Change
Suppose the target table has entries s*(l(to) ) and cr*(I(tl) ).

The ideal situation would be that when the input changes,
a: = ,
6 (ml) ) a (Hal). Ftto). 1(t1) )

Of course, in general there is no assurance that this will happen.
The structure in Figure 2. 8. l is proposed to insure that the ideal
situation will occur, at the cost of some "dead time" when the input
changes. The function of the components of the structure are:
INPUT-CHANGE DETECTOR: Detects changes in the input.

In]

Det

 

Gen

21

From Read/Write
Head Controller

From f \
Input Comparator

Buffer \ From
Input
Generator

 

 

 

Change -of - Start-State
Function Decision Table
Controlle r Gene rator

 

 

 

 

 

 

I
To Net 1

Start-
State
Table

 

 

 

 

1

 

Input -

Change F——¢ ‘ Change-of-

Detector State

 

 

 

 

 

 

 

F
i To Input 1:12:11 __ Controller
and Output

Generator
Buffer s

 

1 To Net

From
Input
Generator From
Output

Buffer

 

Figure 2. 8. l The basic trainer structure of a functal system.

STAE
Cthk

COhi

DEX:

CPLA

STAN

Algo

beco

Sigr
Out;
nect
Out;
Pal?

8tat

by t}
diffe
deCi

nett

refil

22

START-STATE TABLE: Contains a list of start states, one for each
possible input.

CHANGE-OF-STATE CONTROLLER: Retrieves a start state from
the start-state table and places the net in that state.

COMPARATOR: Uses the inputs it receives to compare the output

it receives from the net with the appropriate entry in the

 

target table. It may also be necessary to signal the comparator
to suspend operations until the output associated with a new
input has made its way to the comparator.

DECISION CONTROLLER: When an error has been detected, this
component decides whether to change the start state or the
function.

CHANGE-OF-FUNCTION CONTROLLER: Changes the function being
realized by the net.

START-STATE TABLE GENERATOR: Changes the start-state table.

The algorithm the structure follows is:
Algorithm 2. 8. 1

1. If the input changes, the input change detector output
becomes one.

2. This activates the change-of-state controller, which
signals the input buffer to withhold the new input from the net and the
output buffer to generate some ”neutral'l output. (It may also be
necessary to signal the comparator to suspend operations until the
output associated with the new input has found its way to the com-
parator. ) Then the change-of-state controller consults the start-
state table and imposes a new start state on the net.

3. The comparator is continually comparing outputs generated
by the net with the desired outputs in the target table. If there is a
difference between any two of them, the comparator notifies the
decision controller. This controller, in turn, activates either the
change-of-function controller or the start-state table generator.

4. The change-of-function controller attempts to train the
net to the sequence in the target table by changing the function being
realized by the net.

5. The start-state table generator attempts to train the net

23

to the sequence in the target table by changing the start state of the
input in question.

The detailed manner in which the decision controller, the
change-of-function controller, and the start-state table generator
operate is, of course, of utmost interest. However, very little more
can be said about these devices without having a particular situation
in mind, such as the hippocampus and its environment.

Two kinds of time intervals will be of interest to the trainer.

Definition 2. 8. l

 

The time period is the time during which one output is com-

 

puted by the net.

As mentioned previously, when there is an output error, the
generation of outputs is suspended and training takes place. During
training the passage of time is indicated by the number of time pulses
that occur.

Definition 2. 8. 2

The time pulse is the fundamental time quantum of the net.

 

 

2. 9. The Trainer, Target Table Relationship

There are two methods the trainer might use to compare an
output sequence with a target sequence. Either it can wait until the
entire sequence has been generated and then compare it with the target
sequence, producing a single judgment on the similarity of the two
sequences; or it can compare the outputs one at a time, producing
judgments after each output. The first alternative is undesirable
because it implies that a sequence is equivalent to a single net output.
Consequently:

Assumption 2. 9. l

 

Each output of the functal net is assumed to have an effect on
the environment and/or the effector devices. Furthermore the
trainer has the capability to compare any single output with the
corresponding entry in the target table.

Since the target sequence consists of a run-in plus a cycle,
it might be efficient to compare the output sequence with the target

sequence only until the completion of the first output cycle.

24

After that, and for as long as the input does not change, the output
and corresponding target sequence element would no longer be
compared. There are two reasons, however, that this approach is
undesirable. First, from Property 2. 5. 6 the generated sequence
may be a subsequence of a longer sequence. Second, no restrictions
are placed on when the target table can be changed. Therefore,
Assumption 2. 9. 2

The trainer has a mechanism for detecting the end of an
output cycle and a mechanism for finding the beginning of the cycle
in the target sequence-representation whenever the end is
detected.
Example 2. 9. l

 

Each target sequence 0*(t) can be visualized as being on an
erasable tape. A mark on a companion tape can be used to indicate
the end of the run-in and the beginning of the cycle. The read/write
head and controller can be assumed to consist of two parts: a read-
head and its controller, and the write-head and its controller. The
read-head controller could initialize the read-head at the first element
of the tape whenever a new input is detected and it could move the
read-head to the right one element every time period. If the final
element in the representation is read and the input has not changed,
then the controller could move the read-head back (to the left) in the
tape until it found the beginning-of-cycle indicator on the companion

tape.

2. 10. Measures of Functal Net Performance

It is important to have measures by which nets, natural or
man-made, can be compared. Some of the obvious measures are the
number of functals, the number of function generators, the similarity
of connectivity, and the similarity of the training algorithm. Others
are:

l. The convergence set. One measure of the versatility of a
functal net is the number of net functions that the net can be trained
to realize from a specified function. This measure is the cardi-

nality of the convergence set of the specified function.

25

2. The statistics of output sequence lengths. Another class of
measures of interest to some studies is the statistics of the output
sequence lengths. For instance, it might be of importance to look at
the distribution of run-in, cycle, or equilibrium lengths or the
percentage of sequences in the three categories.

3. The statistics of convergence times. It will not be of
much value to be able to train a functal net if a prohibitively long time
is required for training. Hence, the following measure.

Definition 2. 10. l

The convergence time T = T(0'(I), 6*(1) ) is the number of

 

 

time pulses required to train a functal net to produce 6*(1) when it
was originally producing (7(1).
Some of the convergence time statistics are:
a. The longest training period for a convergence set.
b. The shortest training period for a convergence set.
c. The longest training period for the entire convergence class.
The shortest training period for the entire convergence
class.

e. The average convergence times for any of the above.

2. 11. The Functal System Analysis Problem
The following procedure has been found useful in modeling and
analyzing a natural system (the hippocampus) by a functal system.
Step 1. In the natural system identify:
1 The input and output spaces.
The functals.
The function generators.
The rules for connectivity.
The delay times between and within elements.

The trainer and its algorithm.

"31‘”9999'?’

The target table generator and its algorithm.

Step 2. Specify the model-~equations and parameters--of each of the
above.

Step 3. If possible, isolate the physical elements and establish that
the element models are satisfactory; that is, the input-output trans-

formations are within an acceptable range of those found in the

26

physical system. Go back to Step 2 or perhaps even Step 1 if changes
need to be made in either the model or the natural system concepts.
Step 4. Verify that the model behaves in the way the modeler (L93)
expects it to. In some cases this may lead to a jump to either Step 2
or Step 7.

Step 5. Record properties and measure the characteristics of the
model.

Step 6. Design experiments or interpret existing data to obtain
comparable properties and characteristics of the physical system.
Step 7. Compare the results of Steps 5 and 6. If the comparison is
poor, then make a judgment as to the cause and return to the appro-
priate previous step to revise' either the physical system concepts or
the model.

It should be emphasized that Step 7 may lead to entirely new
concepts of the physical system. This is one of the two main contri-
butions functal net modeling (or any modeling, for that matter) can
make. The other contribution is that the functal net concepts are so
designed that computer component analogs can be constructed of the
physical system; this, especially when the physical system is neural

in nature, would lead to entirely new kinds of machines.

CHAPTER 3

A FUNCTAL SYSTEM MODEL OF THE HIPPOCAMPUS. PART 1

3. 1. Introduction \

The hippocampus system model presented in this chapter and
Chapter 5 (see Figure 3. l. l) is the compromise design which resulted
from intense negotiations to simultaneously satisfy four competing
interests: The anatomical and neurOphysiological data on neurons and
nervous systems in general and the hippocampus in particular, the
ability to meaningfully simulate the model either in hardware or soft-
ware, the experimenter's intuition, and the functal system theory.

The discussion of the model has been divided into two parts.
The first part, this chapter, discusses the model of the hippocampus
CA3 sector and those other elements of the canonical functal system
which do not rely on certain theoretical properties of the CA3 model.
These include the input and output sets and buffers. The next chapter
develops some of the computation theory of the CA3 model and then
Chapter 5 completes the specification of the hippocampus system

model.

3. 2. The Input and Output Sets of the Hippocampus System Model
There is insufficient evidence from the firing rate data of the
mossy fiber input or the pyramidal cell output of the CA1 sector to
determine the significant ranges for the input and output variables of
any hippocampus system model. It seems reasonable, therefore, to
choose the simplest range possible, the binary set (low firing rate,

high firing rate). The corresponding model values will be (0, l).

3. 3. The Input Buffer
The input buffer defines the connectivity of each component Ij

of the system input vector I to each pyramidal cell analog k in the

27

28

   
    
   

 
   
 

Read / Write
Head and
Controller

 
  
 

Target Table
Generator

Trainer

  

CA3
Sector
Net

 
   
       
 

Input
Gene rator

Figure 3. 1.1. The overall structure of the hippocampus system
model.

29

CA3 model. It also acts as a combinational switching circuit with
Table 3. 3. l as the truth table (where M is the input matrix to the
CA3 sector model). From the table it is clear that SD is used to do
one of three things: block the input (SD = 0), allow the input to pass
unchanged (SD = 1), or amplify the input (SD = 2).
In the hippocampus itself, it is possible that the dentate per-

forms the input buffer function, with SD corresponding to the septal
input, I corresponding to the perforant path fibers, and M corres-

ponding to the mossy fiber output of the dentate.

3. 4. The CA3 Sector Net
3. 4. 1. Introduction

The CA3 sector net is intended to be a model of the CA3
sector of the hippocampus. The major elements in the net are the
pyramidal cell logic units (PCLUs), which are functals representing
the pyramidal cells; and the basket cell logic units (BCLUs), which
are function generators representing the basket cells. The inputs to
the net are the matrix M and the vector S. The latter is assumed
to correspond to the septal input. The output of the net, the vector
H, represents the Schaffer and fimbrial fornix collaterals. The
values of the elements of the output are taken from the set {0, l, 2},
where 0 is assumed to represent a low firing rate, 1 a medium firing
rate, and 2 a high firing rate. The design rationale for the net is

given in Appendix A.

3. 4. 2. The Pyramidal Cell Model: The Pyramidal Cell Logic Unit

This functal, which is a close kin to the adaptive threshold
logic unit so prevalent in the literature of neural modeling, is designed
to generate a digital approximation to the firing rate of a pyramidal
cell.

The primary difference between the adaptive threshold

logic unit and the PCLU is that an individual weight of

a PCLU can only be adjusted in one direction.

There are two major parts to the PCLU, the discriminant

function and the training algorithm.

30

Table 3. 3. l.

The Truth Table for the Input Buffer of the
Functal Systems Model of the Hippocampus

 

SD 1. Does a connection mk. eM
J exist between input J
I. and k?
J
0 O 0
0 1 . 0
1 0 . 0
l 1 yes 1
1 1 no 0
2 0 . 0
2 1 yes 2
2 1 no 0

31

Definition 3. 4. l
The discriminant function of PCLU i, denoted by

 

[‘Aim - Min) - Bim . zim‘lT = yitt). is
12
T2 > Ai(t) - Mi(t) - Bi(t) ° Zi(t) 2 T1 iff yi(t) : l
Ai(t) - Mi(t) - Bi(t) - Zi(t) 2 T2 iff yi(t) = 2
Ai(t) - Mi(t) - Bi(t) ° Zi(t) < Tl iff yi(t) : 0
where
mij(t) : 1 or 2 implies aij(t) mij(t) : aij(t).

The vectors and constants in the discriminant function have been
given names:
Definition 3. 4. 2

 

The vector A16 dam is the vector of mossy fiber (mf) weijlgs
for PCLU i.
Definition 3. 4. 3

The vector Bie 0n is the vector of feedback weights for
PCLU i.
Definition 3. 4. 4

The vector Wi : (Ai’ Bi) is the weight vector for PCLU i.
Definition 3. 4. 5

The vector Mic (Tm is the set of mossy fiber (meinputs to
PCLU i. (Mi is a row of the matrix M. )
Definition 3. 4. 6

The vector Zie an is the set of feedback inputs to PCLU i.
Definition 3. 4. 7

The constants T and T e 691 are the 1957131; and pm

1 2
thre 8 holds re spe ctively.

 

 

 

 

 

 

 

 

 

 

 

According to the discriminant function, each mf input is
multiplied by a corresponding mf weight and each feedback input is
multiplied by a corresponding feedback weight. (From Figure 3. 4. 1
note that a two is equivalent to a one in this multiplication. ) The total
PCLU contribution is subtracted from the total mossy fiber contribu-

tion (the inhibition effect) and the result is compared with the two

32

Sim

Min) Q vim

Zi(t)

(a) Schematic

yi(t) = FAN) - Mia) - Bin) . ziufl
T
12
where
mij(t) = l or 2 implies
aij(t) mij(t) : aijm.

(b) Discriminant function

 

Si(t) mij(t) e Ai(t+d) Bi(t+d)
‘ 0 {0,1} Ai(t) Bim
0 {0. 2} Ai(t) Bi(t)
1 {0,1} Ai(t) Bi(t) + ézi(t)
1 {0,2} Aim + AMim Bim

 

 

(c) Weight adjustment table

Figure 3. 4. 1. The pyramidal cell logic unit.

 

33

thresholds.

The other major part of the PCLU is the training algorithm,
which adjusts the weight vector if necessary. The mode of this
adjustment is determined by the septal fiber input.

Definition 3. 4. 8
The scalar function si(t)€(0, 1) is the septal inliut to PCLU i.

 

 

Figure 3. 4. 1 summarizes the algorithm. Expressed verbally:
a. If si(t) = l and the mf input has components from the
set (0, 2), then every component of the mf weight vector A having a
nonzero mf input is increased by some fixed amount A .
b. If si(t) : 0, then no change is made in any weight vector.
c. If s,(t) : l and the mf input has components from the
set i 0, 1}, then every component of the feedback vector Bi having
a nonzero feedback input is increased by some fixed amount 6.
It is important to note that the connectivity of the net requires

the feedback inputs Zi to be internal inputs (see Definition 2. 2. 3).

3. 4. 3. The Basket Cell Model: The Basket Cell Logic Unit

The well-known function generator called the threshold logic
unit is used as the basket cell model in the net. Renamed the
basket cell logic unit, or BCLU, the representation is shown in
Figure 3. 4. 2. The definitions of interest are:

Definition 3. 4. 9
[Vim - Him—IT = zim

is the discriminant function of BCLU i, where

 

I
. 2 ° :
Vi Hi(t) T iff zi(t) l
I
s < . :
Vi Hi(t) T lff zi(t) 0
and where
v..h..(t) : v.. iff h..(t) : 1 or 2
1J 1J 1J 1J

l
O

(t) iff hij(t) = 0 or vij(t) = o.

vijhij

Definition 3. 4. 10

The vector Vic 01 is the vector of weights for BCLU i.
Definition 3. 4. 11

The positive integer T is the BCLU threshold.

 

 

 

34

Hi(t) e 211 (t)

(a) Schematic

T
where
v..h..(t) ._ v.. iff h..(t) : l or 2
ij 1) 13 1]
v..h..(t) = 0 otherwise
1.1 1J

(b) Dis criminant function

Figure 3. 4. 2. The basket cell logic unit (BCLU).

‘4... Iurualliafirw. ...,...nuzwlbj
a... . ...V
T .

35

Definition 3. 4. 12
The vector Hi(t)’ with components from the set {0, 1, 2},
is the vector of inputs to BCLU i.

 

3. 4. 4. The Connectivity and Delays

The pattern of connectivity in the CA3 sector net (Figure 3. 4. 3)
is an extreme simplification of the connection scheme of the natural
system. The mossy fiber input feeds a rank of PCLUs. At the
output of each PCLU there is a unit delay; the output of these delays
is used as the input to a rank of BCLUs and also as the output of the
net. The output of each of the BCLUs first passes through a unit
delay and then feeds the PCLU rank.

There are two rules that might be used when defining a
specific connectivity. The first is suggested by the CA3 sector
morphology: a PCLU should feed the BCLUs in only a limited surround
of the PCLU, and a BCLU should feed PCLUs over an area several
times as large. The second rule is suggested by the behavior of the
model (as developed in the next chapter): a direct path should exist
from each PCLU i to at least one BCLU and back to PCLU i. If it
is assumed that only one BCLU per PCLU is connected in this
fashion, then:
Definition 3. 4. 13

The BCLU in the direct PCLU i - BCLU - PCLU i path is
called the special BCLU of PCLU i.

As will be seen in subsequent chapters, the extent of both the

 

trainer and the target table generator's knowledge of a functal net's
connection scheme plays an important part in determining the
operating characteristics of those structures (for example, their
versatility when changing the net's function). In order to emphasize
this point, the connectivity of the CA3 sector net is specified only to
the extent of its trainer and target table generator's knowledge. That
is, it is assumed reasonable for both structures to know about the
Special BCLUs; it is assumed unreasonable to suppose that they know
the first connectivity rule. Therefore, the special BCLU connectivity

rule is the only one assumed for the CA3 sector net.

36

 

551(t)

Y’ (t) (t- l) = 11 (hi
1v11(t) ‘ l -J.‘\\e 3,1 l e-— l11(t)

 

 

 

 

 

532(t)
Mz(t)
O
531(t)
ylu) yltt-l) = hlmi
Iv11(t 'T, j—L,/(' ,. ==. 111(t)
2' (t)

21“) I
zI(t) ‘ 9 HI“)

Figure 3. 4. 3. A general form of connectivity
for the CA3 sector net.

37

A more complex and seemingly more realistic connection
scheme for a CA3 sector model is presented in the Appendix. It is
suggested that part of the reason for the complexity of the connec-
tivity in the natural hippocampal system is to overcome the restraints
placed on the natural trainer's activities because of its lack of
knowledge of the hippocampal structure. This observation appears to
present a paradox, but perhaps the explanation is that, after a certain
critical level of connection complexity, more complexity tends to elim-
inate the need for detailed knowledge on the part of the trainer and
target table generator; they can deal instead with generalities.

Finally, a comment on the delays. It may be that there has
been a significant oversimplification in the placement and magnitudes
of the model's delays. Unfortunately, a more complex arrangement
would remove the behavior of the model from the realm of the author's

existing intuition.

3. 4. 5. The Operational Algorithm

In order to discuss computational prOperties of the net it is
necessary to be specific about the order in which the computations
occur. This order is:
Advance the state.
Compute the new outputs of the PCLUs.
Compute the new weight vectors of the PCLU.

4:.pr

Compute the new outputs of the BCLUs.

3. 5. The Output Buffer
The computation of the output buffer obeys the following truth

 

table:
him ¢(t)
0 0
1 0
2 1

In addition, if the output buffer input CFO = 0, then all output buffer
outputs are zero.
Presumably the natural structure which performs this function

is the CA1 sector. This should not, however, be taken as the full

extent of the functional SOphistication of this area.

CHAPTER 4

PROPERTIES OF THE CA3 SECTOR NET

4. 1. Introduction

The properties of the CA3 sector net presented in this chapter
are important in two ways- First, the trainer and target table
generator designs depend, to a large extent, on the computational
properties of the functal net they control. Second, the properties
constitute an analysis of the function of recurrent inhibition as it
occurs in the net.

A necessary preliminary assumption concerns the major role
played by the special BCLU in net computation.
Assumption 4. l. l.

The output of the special BCLU of PCLU i is assumed
to be nonzero whenever the input to the BCLU from PCLU i is

 

nonzero.
The most basic CA3 sector net property is the following.
Property 4. l. l.

 

Two M = 0 inputs place the hippocampus net in the zero state.
Proof: Suppose the net is in some state Q = (H, Z) and has a PCLU
output Y and a BCLU output 2;. Furthermore, assume that the
mossy fiber input matrix M = 0. The ope rational algorithm of the
net outlined in Section 3. 4. 5 says that the next time period will see
the state of the net become Q = (Y, Z'a) , the PCLU output become 0
and the BCLU output become 2%). If the mf input remains zero for
the next time period, then the state of the net, the PCLU output, and
the BCLU output will beecme Q = (0, 2.3), 0, and 0 respectively.
The state of the net for the next time period will be Q = (O, 0). QED
Therefore, any time a zero state is desired it is only necessary to
apply a zero input for at least two time periods.

The structure of the hippocampus and intuition made it difficult

to justify the existence of the start state table, the start state table

38

39

generator, and the decision components of the general functal system.
By giving the change of state controller the capability to apply a zero
input to the net (through the input buffer) and making the following
assumption, it was possible to entirely eliminate these troublesome
components from the hippocampus system model.

Assumption 4. l. 2.

 

The only start state of a target sequence will be the zero state.
Based on this assumption, the following orthodox trainer and
target table generator operating algorithm was defined.

Algorithm 4. 1. l.

 

1. Assume the function realized by the net is also contained
in the target table. Change the table to a new function which is
contained in the convergence set of the original function.

2. Place the net in a zero state by applying two successive
zero inputs.

3. When a conflict between the computed and the desired output
of any one PCLU is detected, modify the net by increasing the mf
weights if the gene rated output is lower in magnitude than the desired
output, or the feedback weights if the gene rated output is higher in
magnitude than the desired output of the PC LU (using the Weight
Adjustment Table of Figure 3. 4. l).

4. Reset the entire net to a zero state. Recompute the output
sequence and go to Step 3 if an error is detected.

5. Training is successful when this output sequence and all
others are generated correctly.

In addition, the original method of evaluating and improving the
performance of this algorithm was defined to be: Maximize the inter—
section between each convergence set of the system and the function
set of the net.

As the following example illustrates, both of these definitions
proved to be unworkable because the convergence class of the net
cannot be defined.

Example 4. l. l.

 

Let “k = 0 2 2 l 1 (l l)* be an output sequence component
of the function being realized by the PCLU of Figure 4. 1. 1. Suppose

the corresponding target sequence component is changed to

4O

 

 

 

H.
XJ

Figure 4. l. 1. A PCLU and its special BCLU.

M is the mossy fiber input vector, Z is the feedback input vector
from BCLUs other than the special one, and ij is the input to the
BCLU from PCLUs other than PCLU j.

41

O'k* = 0 2 2 0 0 (0 0)*. According to Algorithm 4. 1. l the feedback
weights will be increased when the error at the h2 pair position is

detected. (The general sequence notation is a'k* = 0 h1 h1 h2 h2 h3

h3 . . . ) Since the special BCLU existence is assured by Definition
3. 4. 13 and its output is assured of being nonzero whenever the input
from its PCLU is nonzero, the training will succeed at least for the
h2 pair. Successful training for the next pair, h3, cannot be
guaranteed, however,” since there is no assurance that either one

of the following conditions is true: _ , 5 ,

l. The Zxk feedback input from BC LUs other than the special
BCLU are nonzero.

2. The input to the special BCLU from other PCLUs is sufficient
to cause the special BC LU output to be nonzero, even though the input
from PC LU k is zero.

In conclusion, it is not possible to say whether or not any function
containing (rk’l‘ is in the convergence set of any function containing a'k.

The definition of a new algorithm and method of evaluation was
based on two prOperties of the net discovered while evaluating
Algorithm 4. l. l. The first is implied by the previous example: If
the atom of a target table function is changed, then the atoms and
elements following it in the sequence cannot be predicted. (It is
important to note that this statement does not imply anything about the
atoms preceding the altered atom. )

The second property involved a consideration of whether or not
successful training can be guaranteed if only one atom of a target table
function is allowed to change. The following example demonstrates
that in some cases it would be necessary to make multiple atom
changes in order to insure successful training as defined in Algorithm
4. l. 1, Step 5.

Example 4. l. 2.

Consider a net in state Q = (H, Z) = 0. The disc. fcn. of a
silngle PCLU k is Ak - Mk - Bk- 2;, which reduces to Ak - Mk for
h . A well-known property of disc. fcns. of this form is: If
M113 M17; and hl(M11() = a, then hl(M]:) 2 a. This implies that if

1
h *(Mllc) = a , then hl*(M12() Z a. Therefore, changing one atom in the
first element of a target sequence will generally require changing atoms

of several other target sequences.

42

The problem of defining multiple atom changes is equivalent to
the function set definition problem and the latter can be solved in two
steps:

1. Develop an algorithm for generating target sequences.

2. Develop an algorithm for constructing the entire target
table from target sequences.

It has not been possible to develop the second algorithm. Even if one
was developed, however, it is unlikely that an algorithm of such
apparent complexity could be imitated by a nervous system. This is
especially true in light of the reasonableness of the following algorithm
and method of evaluation:

Algorithm 4. l. 2.

1. Assume the function realized by the net is also contained

 

in the target table. Change one atom pair hkhk in the table.

2. Place the net in a zero state by applying two successive
zero inputs.

3 When a conflict between the computed and the desired output
is detected, modify the offending PCLU's disc. fcn. by either (a)
increasing the mf weights if the generated output is lower in magnitude
than the desired output or (b) increasing the feedback weights if the
generated output is greater in magnitude than the desired output.

4. Reset the net to a zero state. Recompute the output sequence
and go to step 3 if an error is detected in any target sequence element
through the element containing the original alteration. Otherwise,
training is considered successful.

The method of evaluation and improvement was to determine the atom
alteration rules which, when used during step 1, would make it possible

for this algorithm to succeed.

4. 2. The Output Sequence Set of a PCLU

The output sequence set of an arbitrary PCLU in the CA3 sector
net has some important properties which will ultimately allow the set
of output sequences of an N PC LU net to be completely generated. The

properties are also interesting in their own right.

43

Property 4. 2. 1.

 

If a net is initially in state Q = 0 , then the output sequence

of any PCLU i for any input Mi will be of the form

Ui(Mi) = o h1 h1 h‘2 h2 h3 h3

Proof: The proof can be summarized by Table 4. 2. l,which traces the
state of the net, the feedback input Zi , and the computed output bl;
through several time periods.-

Picking up the action at tl , the input Mi extracts an output
of yl from the PC LU. Since the input to all the BCLUs (H) is still 0,
their output is collectively 0. Therefore, the new state formed for the
t2 computations will be as shown.

For t2 the input to the PCLUs in general and for the PCLU i
in particular is no different than it was for tl . Therefore, the output
will not change. The input to the BCLUs has changed, however, and a
new collective output of Zl' should be expected. The state shift leaves

(H1, Zl) as the state for the t computations.

During t3 the feedback3input to the PCLUs can be nonzero for
the first time. This is reflected in the change in the PCLU i output.
The BCLU inputs have not changed, so their output remains the same
and the output of the BC LUs is different.

It is clear that such a pattern will continue as long as the input
to the net or the functions computed by the functals do not change. QED

Property 4. 2. 2.

 

The output sequence component a'k(Mk) = 0 h1 h1 h2 h2 . . .

generated by PCLU k must satisfy the following set of inequalities:
l 2 3 4 5

(1) h2h,h,h,h,.
(2) h2 s h3, h4, h5, 116, .
(3) h3 2 h4, h5, h6,

(4) h4 s h5, h6, h7, .

(5) h5 2 h6, h7, h8, .

(Note that the k subscript has been suppressed. )
Proof: The predicate for h1 is [Ak ' M13 . The predicate for any
other j > 1 is [Ak ° Mk - Bk ' ZiJ . Clearly (I) is true. Therefore,

44

Table 4. 2. l.

The Computations, Over Several Periods, of
the CA3 Sector Net

 

period state of net feedback input output delayed output

Q = (H, Z) Zi Yik hi

to (0, O) O 0 0

t1 (0, 0) o yl 0

t2 (H1, 0) o yl hl

t3 (H1, Z2) Zi2 y2 hl

t4 (H2, Z2) Z12 y2 hZ

t5 (H2, Z3) 2.: 3'3 h2

t6 (H3, Z3) 2: y3 h3

45

HW(H1) 2 HW(Hj) for all j and Hj 3 H1. Since the BCLUs are
threshold functions, HW(Z2) Z HW(Zj) and Zj D Z2. As a result
B - 25 s B . z2 and h‘2 s hj for all j)! 2. Therefore, HW(HZ) s
HW(Hj), j )6 2. Any PCLUs which fire for h2 will certainly fire for
hj, so H2 D Hj . This completes the proof of (2).

Continuing, H'W(Z3) s HW(Zj) and Z3 D 23, j2 4. Therefore,

B - 235 B - zJ and h32 hJ. HW(H3)2 HW(HJ) and so on. QED

k k
4. 3. The Output Sequence Set of the Hippocampus Net
The following property was implied in the proof of Property
4. 2. 1.
Property 4. 3. l.

 

The CA3 sector net has output sequences of the form

0'(M) = 0H1H1H2H2H3H3...

Recall that in general the repetition of a subsequence in an out-
put sequence does not imply that the subsequence is a cycle. The CA3
sector net is nearly an exception, but the argument for it being an ex-

ception is purely academic, as can be seen by the following property.

Property 4. 3. Z.

 

If 0'(M)=0H1Hl...H1H1... HJHJ...

and H1 = HJ (j > i), then H1 H1 H1+1 H1+1
is a cycle.

Proof: Assume a state (Hi-l

, 21) produced an output Yi. This will
become Hi during the next period and the new state will be (Hi. Zi).
The next state will be (Hi, Zi+l). Similarly, assume a state
(de, Zj) produces an output Yj. This will become Hj during the
next period and the new state will be (Hj, Zj). The next state will be
(Hi, 25“). If Hi: H5, the 21“: 23+1 and (Hi, 2””) = (Hj, 23'“). The
two equal states mark the boundaries of a cycle. QED
Properties 4. 2. 2, 4. 3. l, and 4. 3. 2 can be combined in an
algorithm which exhaustively lists all possible output sequences of a
CA3 sector net containing N PC LUs .

Algorithm 4. 3. l.

 

1. Generate an output H1 from the set of 3N possible outputs.

2. Generate an output H2 from the same set.

46

3a. If the sequence HlHlHZH2 satisfies Property 4. 2. 2, go

to 4.

3b. Otherwise, go to 2 until every possible output candidate
for H2 has been tested. Then go to l and repeat until every possible
output candidate for H1 has been tested.

4. If the sequence HlHlHZHZ satisfies Property 4. 3. 2, add
the sequence to the list of output sequences and go to 3b. Otherwise
K = 3 and continue.

5a. Generate an output for HK from the set of possible outputs.
If the sequence HlHlHZH2 . . . HKHK satisfies Property 4. 2. 2, go to
6.

5b. Otherwise, generate another output for I-l'K and test again
until the set of outputs for the K-th element in the sequence has been
exhausted. Then gene rate a new output for HK-l and reinitialize the
set of outputs to be tested for HK. The algorithm terminates when
all possible outputs for H1 have been tested.

6. If the sequence HlHlHZH2 . . . HKHK satisfies Property

4. 3. 2, add the sequence to the output sequence set and go to 5b.
A version of this algorithm with the ability to generate all

possible target sequences for a net with N PCLUs was programmed
on the CDC 6500 computer (see Appendix B). Since it would be pro-
hibitively expensive to allow the program to generate all possible
target sequences, a representative sample was taken for several
values of N and for target sequences with the first element containing
all twos and the second element containing all zeros (to give target
sequences of maximum length). Sixteen was the longest output run-
in length found (for N=5), with the length increasing slowly with in-
creasing N. Only output cycles of length 4 and equilibria were found;

there were approximately equal numbers of each.

4. 4. Rules for Successful CA3 Sector Net Training Using Algorithm
4. l. 2.

The results presented in the previous two sections, along with
those below, are sufficient to deve10p the rules which assure success-
ful training using Algorithm 4. l. 2. The key word in the following

property is "guaranteed. "

47

Properpy 4. 4. 1.

 

Using Algorithm 4. l. 2 and its associated success criterion,
training of the hippocampus net is guaranteed to be successful if and
only if the following rules are obeyed. (The subscripts have been
omitted for simplicity. )

Rule 1: If 0'(I) = 0 h1 h1 0"(1) , then changes in h1 are made

according to the following table.

1 l
h h * T2 _ T1
1 provided A < —N—- , N the number of
0 2 mf inputs to
PCLUj
l 2

Rule 2: If a'(I) = 0 h1 hlhzhzo'U). then changes in h2 are

made according to the following table

 

h1 h2 h2*
T2 ' T1
2 0 1 provided A < —-——1—\I——— , N the number of
2 0 2 mf inputs to
PCLUj
2 1 2
2 2 0 T2 _ T1
2 2 1 provided 6 < T— , L the number of
2 1 0 feedback inputs
to PCLU i.
1 l 0

Rule 3: If 0'(I) = 0(2200)(2200)>=< 1th1 0"(1) and h1 = 1, then
h1* = 0 or 2.

Rule 4: If «(1) = 022(0022)* hlhl 0"(1) and h1 = 1, then
h‘*

= 0 or 2.
Proof:
"Rule 1: Assume the net is realizing the function in the target
table; change the atoms h”. Clearly, if h” is increased to 2, then

h1 can be increased to 2 by increasing the mf weights and training
*
will be successful. If h1 is increased to 1, it is necessary that an

increase in the mf weights not force an output of bl: 2. The condition
12' T1
A < T

of the disc. fen. will be less than the T2- T1 gap. Note that h1 can

will prevent this from happening, since any one increment

never be decreased, since the feedback input for h1 will always be

zero vector. "

48

Rule 2: The table associated with Rule 2 defines the changes
that can be made in the second pair of atoms of a target sequence com-
ponent with guaranteed success. The changes in but are dependent on
h , since this output element defines the upper bound on any change.

If h'2 must be increased from 0 to 1, then the same A limit must be
observed as was defined in the proof of Rule 1. There is, of course,
no problem if h2 is increased to 2 (assuming h1 = 2). But note that
the alternative h1 = l and h2 is increased from 0 to 1 has been
omitted from the table. Any attempt to increase the disc. fcn. to
produce h2 = 1 under these conditions may inadvertently produce

h1 = 2. Since hl cannot be decreased, the training would have to be
considered a failure.

In general, H2 is the first output element associated with a
nonzero feedback input. The existence of the special BCLU guarantees
that if h1 is nonzero, the feedback input vector is nonzero. This in
turn guarantees that the disc. fcn. of the PCLU j can be decreased by
increasing the feedback weights. This is the justification for the
inclusion of the last four entries in the table under Rule 2. Note that
a change in h2 from 2 to 1 requires a condition on 6 . This condition
prevents the disc. fcn. from dropping from a value above T2 to a
value T1 or lower with a single increment of the feedback weights.

Rule 3: This rule summarizes the changes that can occur with

guaranteed success when the atom altered is h1* , i Z 3 and odd.

Suppose h1* is increased. From Property 4. 2. 2, the bound on this

 

increase is determined by hi-Z. The possible changes are:
hi-Z hi hi*
(a) l 0 1
(b) 2 O l
(C) 2 0 2
(d) 2 l 2

For alternatives (a), (b), and (c) h1 = 0 implies hk = 0, all
even k < i (using Property 4. 2. 2). If any of these changes are made,
then, when the error in the output is detected, the reaction of the

trainer is to increasethe mf weights. In doing so, it is entirely

49

possible that some of the disc. fcns. of the even elements will be
inadvertently increased over the T1 threshold. When the PC LU
generates the incorrect output sequence element upon reinitialization,
the response of the trainer is to increase the feedback weights. The
possibility exists that this will force the disc. fcn. of hi to fall below
the desired threshold. To correct this, the mf weights are increased
again, creating the situation where the even elements may again become
incorrect. The trend is clear and the conclusion is that success cannot
be guaranteed if any of changes (a), (b), or (c) are made.

From the information given and Property 4. 2. 2, the output

sequence component associated with alternative ((1) is of the form

«(1): o (2 2 h‘z‘2 h )(2 2 hk hk)* 1 1 o"(i)
\_V_J
h‘ hi
where k < i and even, ha and hke (0,1) and Property 4. 2. 2 holds.
If hk = 1 for any even k < i , then the situation is the same as in the

other three alternatives: there is the possibility of unstable training.
Therefore, all sequence components are eliminated except those of the
form suggested by the rule itself. The crucial step in the proof is to
demonstrate that the fatal trainer instability of the other alternatives

. does not occur.

Let the j-th PCLU generate a'j(I) and change hi* to 2. When
the change is first detected by the trainer, the mf weights are increased
to produce the correct value of the disc. fcn. for h, D1(tl ) 2 T2.
However, as in the previous alternatives, D k1(t )> — Tl may be true for
some even k < i. In order to compensate for this error, the trainer
increases the feedback weights, thus decreasing the disc. fcns. until,
in particular, Dk(t2) < T] . So far the script .is the. same as in all of
the other alternatives. Note, however, that originally hk< hi, k < i
and even. Since 2; 3 Z? k < i and even, hi> hk implies HW(Z}) <
HW(Z;<). The important prOperty is the strictly less than of the
Hamming weight relation. This implies that the change in the disc.
fcn. for hi is strictly less than the change in the disc. fcn. for hk:

Di(tl) - Di(t2) < Dk(tl) - Dk(t2)

50

If Di(t2) < TZ , the trainer will attempt to compensate by increasing
the mf weights again. This time, if conditions are right, i D k3(t )
will be less than Dk (tl ). If Dk (t3 ) is still greater than Tl , the
compensation in the feedback weights need be no greater than the
compensation for Dk (t2 ), and it can be less. If D (t4 ) is still less
than T2 , the mf weights will be increased less than the increase that
occurred during the computation of D 1(t3). Eventually Di (tn ) 2 T2
wlhile at the same time Dk(tn) is not increased enough to force

h to be incorrect.

To complete the proof, note that any changes in the k-th
component, k S i, are corrected before the change can affect the other
PCLUs. Therefore, the other sequence components through target
sequence element i do not change during the training for the k-th
component.

Now suppose hi, i Z 3 and odd, is decreased. The bound on
the decrease is determined by hi-1 and the possible changes are
(again from Property 4. 2. 2):

i-l i

 

(a)
(b)
(C)
(d)

Hooos‘
NNNt—D‘
HOHOD‘

Alternatives (b), (c), and (d) can be eliminated in short order as

successful training candidates. In all cases hk = 2 , k < i and odd,

 

2i: Since the mf weights increase in "quantum jumps, " it would, in
general, not be possible to recompute D1(t1) exactly; the value actually
computed may range from a quantum higher to a quantum lower. If it
is the former, then it is possible that the difference D1(t3 ) - Di(t2) Z
Di(t1) - D1(tz). In this case, the feedback weights would be required
to increase the same amount as before to correct h However, the
next time hthe mf weights are increased, the increment required for a
correct will be even less than before. Eventually this extra
negative weight will be great enough that the contribution of the mf
weights will be less than the contribution of the feedback weights, no
matter what the magnitude of the quantums, and the proof will proceed
as outlined.

51

and it is entirely possible that Z? = Z} for at least one of those k's.

If this is so, then any attempt to decrease Di by increasing the feed-
back weights decreases Dk by the same amount. Therefore, hk
will become incorrect at the same time h1 becomes correct. The
trainer will respond by increasing the mf weights, but the effect is
felt equally by both hk and hi. The result is training instability.

Alternative (a) implies output sequence components of the form

0'(I) = 0(hlh100)(hk hkO O)* l l 0"(I)

h1 hi
where k < i and odd and h1 , hke (1, 2) , along with Property 4. 2. Z.
If any of the hk or h1 is one, then training instability may occur.
This leaves only output sequences of the form given in the rule state-
ment. In order to reduce hi, the feedback weights are increased, and
all of the disc. fcns. Dk, k < iand odd, are reduced. Perhaps some
will be reduced to below T2. Consequently, the mf.weights will be
increased to compensate, with the possibility that D1 is forced to a
value above Tl . Fortunately, a property of the same nature as
described in alternative (d) of the previous set exists to prevent
training instability: Since Z}; 3 25.1 if h1< hk for all k originally, then

HW(Z. ) > HW(Zj ). Therefore, the Dk will not be decreased as much

as D , and eveJntually D1< T1, while DkZ T2 for all k.

Rule 4: The final rule summarizes the changes that can occur
with guaranteed success when the atom altered is h1 , i 2 2 and even.
If h1 is increased, then the upper bound is determined by hlfll and

the possible changes are:

3 O

hi-l 1 hi*

 

(a)
(b)
(c)
(d)

NNNH
HOOD
NND—‘I—l

Successful training for the (a), (b), and (c) alternative cannot

be guaranteed, since h1 = 0 implies that hk = 0, k < iand even.

52

The output sequence components accompanying alternative ((1)
are of the form:

a-(I) = 022(hk hk22)>-'< 1 1 «'(1)
“W"

hi h1

where k is even, hk, his (0, l). The subset of sequence components
where bk 2 l for any k can be immediately eliminated, leaving
sequences of the form given in the rule. Successful training is
guaranteed for these by the same argument as was used for (d) in the
first set of alternatives in Rule 3.

If hi is decreased, then the lower bound will be determined by

hl-2 and the possible changes are:

i-Z hi

 

h h *
(a) 0 l 0
(b) O 2 l
(c) 0 2 0
(d) l 2 1

Successful training for alternative (b), (c), and (d) cannot be
guaranteed since h1 = 2 implies hk = 2, k< i and odd.
The output sequence components accompanying alternative (a)

are of the form:

0'(I) = o 2 2 (hk bk 2 2)* 1 1 0"(1)
L—V'J

hi hi

where k is even and hk

, his (1, 2). Again the subset of sequence
components where hk = l for any k can be eliminated, leaving
sequences of the form given in the rule. Successful training is
guaranteed for the remainder by the same argument as was used for

(a) of the second set of alternatives in Rule 3. QED

.3». .4m

. , .. flurrsfluflu“

(Alt. . 19V

CHAPTER 5

A FUNCTAL SYSTEM MODEL OF THE HIPPOCAMPUS. PART 2..

S. l. The Target Table

The previous chapter noted that if a net is realizing the
function in the target table and then one pair of atoms is changed,
the output function of the net after training could then differ greatly
from the function in the table. Consequently, if orthodox functal
system training techniques were used, that is, if the entire new target
table had to be realized by the net, training would be unstable and the
net would be essentially useless. The following assumption summarizes
a target table form different from the one originally defined in
Section 2. 7 which helps to circumvent this problem.

Assumption 5. l. l.-

 

The target table can contain a set of target sequences for
each input. The interpretation given to each set of target sequences
is: Any output sequence not contained in a set for a particular input
is considered to be harmful to the entity of which the hippocampus or
its model is a part. Those output sequences which are target
sequences are either neutral or beneficial to the entity.

The target table is a conceptual device which makes explicit
the relationship between the natural system and its environment. It
is not intended that a physical structure exist to hold the table. All
neuroscientific interpretations of target tables must comply with this
fact.

5. Z. The Trainer

Algorithm 4. l. 2. has been modified to be compatible with the
new target table concept. The new trainer Operating algorithm for
one time period is outlined in Algorithm 5. 2. l and the trainer structure
associated with it is given in Figure 5. Z. l. The following is a
description of the algorithm.

53

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Input-
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etector

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5. 2. l.

 

 

 

 

 

 

 

 

 

 

 

M
RHCF
Read- CFRHS
10 Head 1 CFRHB
ontroller CFRHR
RHC
H
Comparator from
(C) net
CCF
’ CFRHR
CFRHB
CFRHS
IC Change- RHCF
of— CFcb
Function —
Control S to net To
L.— H Output
from Buffer
net
AMP
Change-of-
IC State
Controller
SD

To Input Buffe r

The training structure of the functal system model
of the hippocampus.

56

At the beginning of the time period, the input-change detector
examines the input for any change since the last time period. If a
change occurred, then IC = l and the trainer does the following:

1. Its change-of—state controller sets SD = O for two time

pulses. This holds the mossy fiber inputs at zero for
the same length of time. (See Table 5. Z. l. )

Table 5. 2. l.

The Truth Table for the Change-of-State Controller

 

IC AMP SD
0 0 l
0 l 2
l O 0
l l 0

2. Its read-head controller positions the read head at the
beginning of the appr0priate target sequence tape.
3. Its change-of-function controller ignores the comparator

output.

If IC = 0, then the change-of—function controller interrogates
the flag ERROR to determine if an uncorrected error has occurred
since the current input was applied.

If ERROR = 0, then an error has not occurred and the
comparator compares the current output with the target sequence
element information on line RHC (see Section 5. 4. ). If there is a
match, then the output of the comparator, CCF = l. The change -of-
function controller senses this and sends the read-head controller a
signal CFRHR = l for one time pulse in order to advance the read-
head. If there is no match, then the change-of-function controller
outputs CFRHS = l for a time pulse. This causes the read-head
controller to search through the other target sequences in the set
assigned to the current input for another target sequence candidate.
If one is not found, then the change-of-function controller is notified

by RHCF: 1. In response, it sets the flag ERROR = l and stores the

57

current input, the previous output, and the error correction informa-
tioni in a memory area called STORE; if STORE is full, the error
is ignored.

If ERROR = 1, then the current input and output are compared
with each of the sets of data in STORE. If an identical pair is not
found, then CFRHR = 1 for one time pulse to advance the read-head.
If an identical pair is found, then the change-of-function controller
sets the output of the output buffer to a neutral value (CFO = l) for
the next time period:): :I: , clears all the stored valuesi, sets ERROR =
O, and uses the error correction information to set AMP or SD. If
the AMP is set, indicating an mf weight adjustment, then the change-
of-state controller computes SD according to Table 5. 2. 1. Finally,
the read-head is advanced with a CFRHR = 1 signal.

5. 3. The Error Correction Information for the Change of Function
Controller

As described in the, previous section, when a bona fide error
is detected by the change-of—function controller, error correction
information is stored as well as the current input and the output

preceeding the erroneous one. This information consists of a PCLU

 

2): Any error procedure will require that the net be in the state that

led to the error. From the results of Chapter 4, the output just prior
to the erroneous output is uniquely related to that state. Therefore,
if the weights are activated exactly when this output is detected, the
state for the next period will be the one desired. The rules for deter—
mining which weights are adjusted, i. e. , the error correction infor-
mation, are discussed in the next section.

i 4: Note that the weights are adjusted after the output is computed.
Therefore, the output will be the same erroneous value it was before,
even though the weights are changed. The CFO = 1 signal prevents
this output from forcing the same erroneous, and presumed dangerous,
response from the entity containing the model, thus increasing the
chances for survival.

$ If the stored values are not cleared, then the problem of training
interference exists. This condition occurs when the training for one
target sequence unintentionally changes the state sequence of another
erroneous sequence. If this happens, then the output stored for that
sequence may never occur, or it might occur in the wrong place in the
sequence. This is another technique to increase the chances for
survival.

58

number and the weight vector, mf or feedback, to be adjusted. These
two quantities can be determined by a number of different methods.
Among them:

Method 1: Randomly select both the PCLU and the weight

vector to be modified.

Method 2: Select the PCLU whose disc. fcn. is closest to

one of the thresholds. In a sense this PCLU is
the most uncertain of its output components. A
weight vector is chosen which will force the disc.
fcn. toward the nearest threshold.

Method 3: Compare the computed output with the target-

sequence element and determine the erroneous
PCLU. A weight vector to be modified is chosen
which will adjust the disc. fcn. in the right
direction.

There are advantages and disadvantages to each of these methods.
Method 1 should be the slowest to arrive at an acceptable output, a
possible disadvantage, but the change-of-function controller does not
need to know the CA3 sector net output, a possible advantage. Method
2 is perhaps the most "neurophysiological" method. The change -of—
function controller would require the values of the disc. fcns. of each
of the PCLUs. This might be a disadvantage in any model, but such
information could be easily provided in the natural system if the firing
rate were proportional to the disc. fcn. over the range of interest.
The change -of-function controller would also require some idea of the
average threshold values of the PCLUs. In the natural system this
information could be genetically provided.

Method 3 is the most direct method, and the one be st adapted
to simulation. However, the change-of-function controller would
require the target sequence elements and the hippocampus output as
inputs. It can be reasonably argued that if the target sequence
elements are actually stored somewhere else, as this method implies,
then the existence of the hippocampus can not be justified, since the

storage of sequences is its proposed main function.

_ 1...: dfiJ.) j 1.13.“...9 Lflrfi.~

‘blIuLTV

59

5. 4. The Read/Write Head and Its Controller
The read/write head and its controller are organized in a
manner similar to that described in Example 2. 9. l. The target
table is assumed to consist of a number of multidimensional tapes,
one tape for each target sequence (see Figure 5. 4. l). The dimension
of the tapes is exactly N + l, where N is the number of PCLUs in‘
the net: The extra dimension is for the beginning-of—cycle indicator.
The tapes are read by a read -head, which is under the control
of the read-head controller. The exact position of the read-head is
located by the triple (M, T, E), where M is the mf input to the net, T
is a number indicating the target sequence within the set for the input,
and E is the number of an element within the sequence. The various
inputs to the read-head controller from the other elements of the
hippocampus system model, and their effect on this triple, are:
CFRHB: An input from the change-of-function controller
of the trainer. Sets E = l.
CFRHR: Also an input from the change -of-function controller.
This sets E = E + 1 unless E is equal to the far right element
of the sequence. If this is true, the controller uses the
beginning of cycle dimension on the tape to reposition the
read-head to the correct element in the sequence.
IC: An input from the input change detector. Sets E = l and
T = l.
CFRHS: An input from the change—of-function controller.
If this input is one, T = T + 1, unless every target sequence
in the set for this input has been inspected, in which case T =
l and RHCF = 1. After T is increased, the new tape is
inspected to determine if all elements prior to the element
indicated by E are equal to the corresponding elements of the
old tape. If they are not, then T is increased again.
The outputs of the read-head controller are RHCF and RHC, the latter
being the contents of the element currently under the read-head.
The target table generator modifies the target table through
the write —head controller. The details of operation of this controller

do not concern us here.

60

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TA
Read Head
1 l 2 Z 3 3 4 4
O h1 h1 h1 h.l h1 h1 h1 h PCLU #1
0 PCLU #2
O
0
I I
I
I I
I I
I l
O
O PCLU #N
Beginning of cycle
Read-Head
Controller

 

 

Figure 5. 4. l. Read-head controller, target table relationship for
a target table sequence.

CHAPTER 6

THE INITIAL CONDITIONS PROBLEM

6. 1. The Phase Concept

The behavior of the entity of which the hippocampus system
model is a part is assumed to be influenced by the output of the CA3
sector net. Furthermore, in any steady state situation, the function
generated by the net is the same as the function in the target table.
Any changes in the target table are made, of course, by the target
table generator. Presumably, it receives a signal from a perfor-
mance evaluator whenever the response of the environment to past
behavior is considered harmful. Upon receipt of such a signal, the
generator makes a change in an atom of the target table. In doing
so, it consults the rules developed in Chapter 4 and perhaps other rules
on how to make the reSponse more palatable to the judgment device.

A tacit assumption in this description is that the target table
contains a function at the beginning of the period of interest. The
problem of the initial assignment of the functions to the net and target
table is not trivial: the assignment should be realistic in terms of
hippocampus data and it should insure maximum survivability to the
entity of which it is a part. Another problem that cannot be overlooked
is the placement of the initial function in the net.

Fortunately, a bit of hippocampus data offers a plausible solu-
tion to both problems. Purpura [15] has observed that basket cells
do not make connections to the pyramidal cell somata until about the
third week after birth in kittens. This suggests that at birth the
output of the hippocampus is generated by pyramidal cells that are not
yet communicating to each other through basket cells. Furthermore,
it must be assumed that the function in the hippocampus at birth is
something other than the trivial function, and that the function, in part,

controls the newborn kitten's instinctive behavior. It is also

61

62

reasonable to extrapolate on these data and to assume that when the
basket cell axons do begin to make connections, they are at first
limited to the pyramids immediately surrounding the basket cell.

In applying these ideas to the solution of the two aforementioned
problems, it seemed reasonable that if nature felt maximum survi-
vability was insured by a hippocampus initially containing no recurrent
inhibition, then the initial function of the CA3 sector net need not be
any more sOphisticated than what can be generated by a CA3 sector
net with no recurrent inhibition. This and another idea on the
subsequent growth of the basket cells were formalized by assuming
that the lifetime of the net is divided into the following three phases:

Phase 1 -- The functions to be generated by the net are from
the set of functions that can be generated by nets not having recurrent
inhibition.

Phase 2 -- The functions to be generated by the net are from
the set of functions which can be generated by a net with only
PCLU-to-special BCLU-to-PCLU connections.

Phase 3 -- No restrictions.

6. 2. Phase 1: Target Table Training

It should be clear that the target sequences generated during
Phase 1 will be equilibria with components taken from the set
{00*, 01*, 02*}. To illuminate some secondary principles, assume
that the only target sequence components allowed during Phase 1 areIII
{ 00*, 02*}. Then the initial function assignment is reduced to one
of defining the relationship between the entity's behavior and all of
the net's ZN output vectors with components from the set { 0, 2}.
This problem, although considerably simpler than the general initial
function assignment problem, is still formidable. When a function

of the right form is found, however, the results developed in this

 

2‘: Note that this is comparable to saying that the pyramidal cell output
is either firing at a very high rate or a very low rate, but not at a
moderate, or "fine-tuned" rate.

63

section permit it to be placed in the target table and trained into the
net prior to the net's placement in its environment.

The first result is an algorithm for generating any function
that can be generated by the net while it is in Phase 1. It is based
on a property presented informally in the second example of Section
4. l and more formally here:

PrOperty 6. 2. 1
:et M]: and M:

 

be two different mf inputs to a PCLU k.

1 lMllcCM: and .h11((M:) = a, .
then h'k(M'k) 2 a. The algor1thm itself simply assigns function values
arbitrarily to all inputs with Hamming weight of one, and then uses
the above property to generate restrictions on the assignment of
function values for all inputs with Hamming weight of two. Successive
application of the property for inputs with successively larger
Hamming weights will result in a complete function;
Algorithm 6. Z. 1

Do Steps 1-5 for each PCLU in the net. I).

l. Initialize by assigning values to h1 for all M s. t.
HW(M) = l.

2. For each hl(M) = Z assigned, replace all zeros in M

 

with x. Call the resulting set (L, with elements M*.

RULE: It must be true that h1(M*) : Z.

3. Consider all inputs M s. t. HW(M) is one greater than
the last Hamming weight considered. Assign values to the h1 for
these M within the restrictions set by RULE.

4. If the current Hamming weight is N, the dimension of
the mf input vector, then go to 5; otherwise go to 2.

5. The output sequence components are of the form
0’(M) = 0 111*.

Example 6. 2. 1

Let N = 5 for the PCLU under consideration.

 

 

* The ”k" subscript (for PCLU k) has been omitted from the description.

 

64

Step 1. Assume it is desirable to assign values from the
range set (0, Z) to the mf inputs with a Hamming weight of 1 in the
manner indicated by column A of Table 6. Z. 1.

Step 2. The set 11* is (xxxxl, xxxlx). This set automatically
assigns a range value of 2 as shown in column B of Table 6. 2. 1.

Step 3. The inputs with a Hamming weight of 2 not yet
assigned range values are 01100, 10100, and 11000. The values
chosen are as indicated in column C of Table 6. Z. 1.

Step 4. N = 2. Therefore, go to 2.

Step 2. The set 11* contains the additional elements lxlxx,
llxxx, and xllxx. An assignment of 2 is forced on the only input
not yet paired with a range value, as is shown in column D. The
algorithm terminates at this point because the function is complete.

The following example demonstrates a disturbing prOperty of
the hippocampus system model: It is not possible to train the net to
realize every function the net is theoretically capable of generating
during Phase 1.

Example 6. Z. 2
Suppose a portion of the target table for one PCLU k is:

 

flux/111‘: 00011) = 2
Max/1: = 00101) = 0
1110.4: = 01001) = 2
1110.4: = 10001) = 2
hl(M: = 10100) = 2

Assume W = 0 initially and the inputs are presented in the order of

their superscripts. At the completion of training for M116
A = (0, 0, 0, aA, aA),

where a is an integer, A is the mf weight increment, and ZaA 2 T2.

If a and A are in the right proportion, then at the completion of
3

training for Mk’

I
0: la 3' 3) 3A.

65

Table 6. 2. l.

The Function Assignment for Example 6. 2. l.

mf input

 

2 222 222 222 222 222 222 222

00010101010101010101010101010101
00110011001100110011001100110011
0000.11.11000011110000111100001111
00000000.111111110000000011111111
00000000000000001111111111111111

66

At the completion of training for M4

k!
_ .1. .1. Z
A ‘ (4! Z! 09 l: 4) 3A °
Finally, when the training for M: is complete,
_ 5 .1. 2. Z
A -’ ( 8! z) 8, l, 4) 3A.

Note that h1(M12<) = 2, which is incorrect. It is not possible for the
trainer to correct this error. Therefore, the function in the target
table will not be realized by the net.

Fortunately, an algorithm has been developed which generates
trainable Phase 1 functions. The algorithm generates the function
as signed to only one PCLU, but since the PCLUs remain independent
throughout Phase 1 (the feedback weight vector remains 0), it can be
applied to each PCLU to generate the entire function. A required
definition is that the set of mf inputs in the domain of h1 which have
not yet been given values in the range is called the Eo_o_l. The initial
pool, containing ZN-l elements, is denoted by (Pl.
Algorithm 6. Z. Z

1. Let K=l and szo.

2. 6k = {M:HW(M) = K and Meth}. Assign

h1(M), Mask, as desired except that the following rule must be

obeyed. l
RULE: If h (M) = 2, MeSk, then Mka.
3. K = K+ 1. (Pk = k_l -3k_l —mk_l, where
(RJ = {M:HW(M)>J, MC M', M'e (SJ. and

hIIM') = 2} .

If 0k: 4), continue. Otherwise, select Xk from the set:

{M: HW(M) = K-l, hl(M) = o, and MC xk_ }

l

and go to Z.
4. For each M, h1(M) = hl(M) i> 1. STOP.

Example 6. Z. 3
Let N = 6.

 

67

Step 2. .31 = {000001, 000010, 000100, 001000, 010000,
100000} .
RULE is inconsequential for this set. Suppose all members of 51
are assigned a 0 output.

Step 3. K = Z, 092 : (Pl - 51, m1 = 4). 02 is not empty.
Let X2 = 000001.

Step 2. :32 has 15 elements. Only the elements in the set
{100001, 010001, 001001, 000101, 000011} satisfy RULE.
Suppose hl(000011) = 2; the remainder are assigned 0 outputs.

Step3. K: 3. (PB: 6’2- 82- (RZ. 02
X3 = 000101.

Step 2. Only the elements in the set {100101, 010101, 001101}
satisfy RULE. Suppose hl(001101) : 2; the remainder are assigned

is not empty. Let

0 outputs.

Step 3. K = 4. 6’4 = (P3 - <93 - 633. 03 is not empty. Let
X4 = 010101.

Step 2: 84 = { 110101 }. This is also the only element which
satisfied RULE. Suppose hl(110101) : 2.

Step 3. K: 5. 0’5: 6’4 - 84 - (R4: s. STOP.

The ability of the system to train the net to realize an
”Algorithm 6. Z. 2" function depends on the following conditions being
satisfied.

Conditions 6. 2. l

1. The net is initially generating the trivial function, with
all W = 0.

2. The function generated by Algorithm 6. Z. 2 is in the target
table.

3. The mf input vectors are presented to the net in order of
increasing Hamming weight.

4. Each input is held for as long as is required to train the
net to generate the correct output.

In addition, there is a fifth condition consisting of two relations
between the values assigned to A, TI' and T2, that requires a more
lengthy discussion. One of these, relating A and TI’ is particularly

complex, and the following property is presented in an attempt to ease

68

the shock of the more general result. Note that the sets 5k defined
in Algorithm 6. Z. Z are required, but since they must be computed
anyway, this is not an inconvenience. Also, once training is complete

for the 1nputs 1n j' no other 1nputs W111 requ1re a tra1n1ng sessmn.

PrOperty 6. 2. 2
For every PCLU in the net, if

 

(a) Conditions 6. 2. 2 are satisfied,

(b) JN - J+1A Z T where J is the lowest K for which

Z,
317;”).
$312< = {M:M€SK and hl(M)=2},

and N is the dimension of the mf input to the PCLU.

N-J+l .
(c) (J-1)JN"J z 11" < Tl/A.
i=1
(d) |5§| = N- J+l,
then Algorithm 5. 2. 1 will successfully train the net to realize the

function in the target table.

Proof:
Let
,1; = {M:HW(M)=K and hl(M)=2},
.1; = {M:HW(M)=K and h1(M)=0},
0 N 0
H = U H1
i=1
N
H2 = U H32
. 1
121

A necessary and sufficient condition for a function generated by a
PCLU is:
A ° M < Tl, Meuo (1)

A'M 2 T Metz (2)

Z,
The proof consists of developing an expression for the largest

A ° M, Mep. 0 over all functions generated by Algorithm 6. Z. 2

obeying (d). It will be used to construct relations between T1’ T2,

69

and A such that the satisfaction of relations (1) and (2) is insured.

Example
Let N = 5, J = Z, and X2 = 00001. Then the function has
5: = p: = {00011, 00101, 01001. 10001}.

After training for the first vector in p. g, the weight compo-
nent(s) ax corresponding to the nonzero components of XJ will have

avalue CA, C an integer, where

J CA 2 T2. (3)

C represents the number of training trials required to drive the
discriminant above T2.
Example

After training is complete for 00011, the mf weight vector
A will be A = (0, 0, 0, 1, l)CA.

Training for the second vector in p. g benefits from the
previous training, since the discriminant at the start of training will
already have a value (J-l)CA. Therefore, the increment required
of the apprOpriate weight components is l/J(CA ). Of course C
must contain J as a factor.

Example
After training is complete for 00101, A = (0, 0, %, 1, %)CA.
At the completion of all training, the components ax have

a magnitude:

N-J+1 1 .
a : CA 2 J '1. (4)
X .
1:1
Example
After training is complete, A = (é, %. i" 1, -1—85- )CA.

In order to add A an integral number of times to a weight
component, it is necessary that

c = JN'J.

Furthermore, in order for the training to be successful, it is

(5)

necessary that both X and the input of highest Hamming weight

J

assigned a zero output, which will always be l-X produce dis-

JD
criminants less than Tl' But since the weight components ax are

incremented every time any weight component is incremented, and

70

the number of components ax is at least equal to the number of
other components incremented during any one training session, the

discriminant for XJ will be at least as large as the discriminant

for l-XJ. Therefore, it is necessary that

(J-l)ax < T (6)

l

or, N-J+l l-i
(J-l) CA 2 J < Tl' (7)
i=1

Example

With c: 25‘2 = 8, A = (1, 2, 4, 8, 15)A.

Note that A - (l -XJ) = A ~XJ = 15A.

Relations (3), (5), and (7) are enough to insure that training
will be successful if the functions are of the kind discussed so far.

They can be used to compute the values to be assigned to A, T

1’
and T2 of the PCLU before training begins. QED
Example

16A 2 T2. (3)
15A < T1. (7)
T2 = 104, T1 = 9.8 x 103, and A = 650 satisfy these inequalities.

If an Algorithm 6. Z. 2 function does not obey (d), then the

expression for the largest A ° M, Map. 0 can be awarded to either

 

A - XJ or A - (l-XJ), as the following examples demonstrate.
Example 6. 2. 1
Let
,5: = {(000011)}
2
e83 = 4»
2
(S4 : 4’
.32 = {(111101)}
5

The mf weight vector after training is: A = (— 1, %)CA.
Therefore, since X2 = (000001),

A- (1-x2)= 9/5 CA > A- x2: 6/5 CA.

71

Examle 6. Z. 2

Let
s; = {(0000111), (0001011)}
6: = {(0110011), (1010011)}
The mf weight vector after training is:
A = (1, 4, 5, 16, 48, 69, 69)C/48 A.

Therefore, since X3 = (0000011),

A- (1-x3) = 74/48 CA < A- x3 = 138/48 CA.
Therefore, for the general Algorithm 6. Z. 2. function,

max { (1-1) ax, A- (1-xJ)} < T1

The following prOperty includes the Specific values for this expression.
PrOperty 6. 2. 3

For every PCLU in the net, if

(a) Conditions 6. Z. 2 are satisfied,

(b) J CA 2 T2, J as defined in Property 6. 2. Z,

 

(c) max{(J l)ax, -.(1xJ)} < T1, where
2A
IeSJ I 1‘1'I82I
ax = CA EJ J' +
i=1 2 "7
457:] ISII _
x Z: IIj J 2: I
l j i=1
see see
C1 C2 -'
Z
8
I JI l-i 1_J|5 I
A- (l-XJ) : CA 2 J + J1
i=1
2
— 57t| '51.) .
x z: (1-k+1) n j J 2: z."
!,k j i=1
see see
C3 C2
C1 -- This sum is over all subscripts I of5f where

1 2 J+1 and 5: 7! .1).
C2 -- This product is over all j, J < j < k such that 5? 7! ()1.

72

C3 -- In addition to the I defined in C1, k is the largest
k' for which i, ,1! 4) and yet k' < 1.

2 2

I5I-1 IS I

J IIK K
K

where the product is over all subscripts of 5; greater than J.

((1) C=J

Proof:

At the completion of training for 5 g:

2
a (5 ) = CA 2: J '1
x J i=1

2 2
A . (l-XJ) : D($J) : ax(8J)'

If [8;] had been one greater, say due to some input Y,
then the amount of increase required in the discriminant A ° XJ
would have been 2

-||
JJ JCA.

This quantity would have been divided among the J - lax weight
components and one other weight component whose value had remained
zero up to that time.

The next input requiring training is in 3 120 K > J. It differs
from Y only in having more than one other weight component which
has remained zero. Therefore, the increase required to attain
J CA is divided among K 2components, and the ax increment is:

IS JI .

CA (J/K) J- (1)

If there is another element in 3 120 then it will differ from the
preceeding vector in only two components in the same way two vectors
are different in 8;. Therefore, the discriminant of the new input
before training is short of J CA by (1). If this value is divided evenly
among the K weight components associated with nonzero input
components, then the increment to any one weight component is:
2 46%|

CA(J/K ) J . (2)

2.

In general, after the completion of training for 5 K

73

Z
a(5) = a(5)+CAJ-J J 2 '1
x K X J ..
1-1
Each of the weight components selected by XK - XJ (there are K-J

of these) are increased by the same amount as the ax
the sum of all other increments to all the remaining weights is equal
to the ax increment. Therefore
2
3 2 Is I
2 2 'l J| K -i
D(8K) = D(,SJ)+ (K-J+l) J-J 2 K CA.

i=1
If another 5 Z, L > K, is not empty, then, by the same
reasoning as for the 8; case, the necessary total increment for

the first input of this set must equal:

-52 -l _52
K[J-KIK| J|J|]CA.

This quantity is divided among L components. Therefore, the ax
increment is 1/L of this. In general, after training is complete for

this set:

2 2
-|8 |—1 1-|5 |
24,551) = ax(612<)+CAK [K K J J
2
ISLI ,
x 2: L'1
i=1
and
2 z 48;) 1463,)
D(SL) : D(5K)+(L-K+1)K J
2
151,) ,
x 2 L“.
i=1

By an extension of this argument, the expressions at the
completion of all training are those given in the statement of the
property.

The property is proved if a technicality involving the integer
C is cleared up. The smallest quantity a weight component can be
increased by is A. In order for C to contain all factors that might
occur during a training session and thereby allow an increase of A

and no more, C should contain all of the factors given in (d). QED.

74

Example
The computation of (c) for Example 6. 2. l.

ax(CA(2O + 20 (no product term) 5'1) = (1+ 1/5)CA.
ax = 6/5 CA.
Therefore,

(J-l) = (2.1)ax : 6/5 CA.

a
x

A - (1-Xj) : CA(1+ 20(5-Z+l) (1/5) ) = 9/5 CA.
Therefore,

max{(J-1) ax, A- (l-XJ)} = 9/5 CA.

Example
The computation of (c) for Example 6. 2. 2.

(D
II

2 l-i 1-2 2 -i
CA (2 3 + 3 (no product term) 2 4 ).

x i=1 i=1
ax = 69/48 CA.
Therefore,
(J-l) ax = 2ax = 138/48 CA.
A- (l-XJ) = CA {1 +1/3(4-3+1)(1/4+1/16) }
a 74/48 CA.
Therefore,

max { (J-l)ax, A- (1-xJ)} = 138/48 CA.

CHAPTER 7

DISCUSSION

7. 1. Summary

An automaton model of the CA3 sector of mammalian hippo-
campus is presented. The connectivity between the PCLU (the py-
ramidal cell model) rank and the BCLU (the basket cell model) rank
is left unspecified except that a direct PCLU-BCLU-PCLU loop is
required for each PCLU. It is assumed that whenever the output of
a PCLU's delay is nonzero, the output of its special BCLU is also
nonzero. The input to each PCLU is a vector Mi with components
having values from the set {0, l} . The output of the model is a
time-sequence of vectors of the form

0’(Mi) = 0H1H1H2H2H3H3. . . ,

with each vector HJ having components hit with values from the set
{0, l, 2} . Assuming each nontrivial input is separated by a zero
input to clear circulating quantities left over from the previous input,

the output sequences are shown to have these properties:

1. Each sequence terminates in either an equilibrium or

a cyclze. 3

4 5

-hk, hk’ , etc.

3. Iin=Hj, j > i, thenHlI-ll....HJ-1HJ-lisacycle.

IV

a. at... sit.

An algorithm is developed to generate all possible output sequences
of any model containing N PCLUs.

The characteristics of a training structure for reshaping the
output sequences of the foregoing model are also presented. This
structure is supported by a target table containing a set of allowed

output sequences for each input to the model. It is assumed that

75

76

when the system is placed in its environment for the first time, the
function realized by the model is contained in the target table. In
order to insure this, a special training session is held before the
model is placed in its environment. An algorithm (Algorithm 6. 2. 2)
is developed to generate the function placed in the target table for
the special session. If certain parameters (the mossy fiber and
feedback weights) are set correctly (to zero) at the beginning of this
session, the function realized by the model at the completion of
training is the function in the target table.

After the system is placed in its environment, desired changes
in the model's function are registered by changing the target table.
The trainer compares the output sequence generated in response to
a net input, M, with the sequences in the target table. If no match
can be found (which implies a change in the target table has been
detected), a marker is set. The next time M occurs as the net input,
the output sequence up to the point of the fault is generated, and then
a training session is triggered. It is proved that the training session
is guaranteed to "succeed" if and only if both the change in the target
table and the selection of some of the model's parameters (A, 5, T1’
and T2) are in accordance with certain rules (PrOperty 4. 4. 1). To
"succeed, " the output sequence must be the same as the target table
sequence only up to and including the element containing the change.
It is understood that the outputs following the subsequence just
described, as well as any other output sequence of the model, may
be altered by this training session. The model's new function may or
may not be the same as the functions in the target table. If it is not
the same, then, more training sessions are required.

A number of other ancillary results on the time-domain
behavior of CA3-like automata were also obtained, both analytically

and by computer simulation.

7. 2. Comments on the Neuroscientific Aspects of this Study

As sume that the hippocampus is a memory bank containing
transformations of single inputs into output sequences, and that its
task is to make act decisions. Furthermore, assume that a trainer

is available for changing the output sequence that any input is trans-

77

formed into and that it operates in the manner described in Chapter 5.
The following observations might now be of speculatory interest to
neuroscientists.

The results of Section 6. l on training phases, together with
Property 4. 4. 1, suggest an increase in both the capability of the
trainer and the complexity of the hippocampus‘s transformations as
it matures. At birth, and during phase 1, a single input is related
to a single output; that is, the relevant output is not sequential.

At this stage, the trainer can increase the firing rate of an output

but not decrease it. As the hippochus matures, and in particular
as the basket cell rank begins to make connection with pyramidal
cells, the outputs of the hippocampus can become sequential in nature,
involving oscillations. The trainer now has the ability to decrease
the output rate, but at the risk of forcing the output into oscillations.
The trainer cannot yet suppress these oscillations. This capability

is achieved only when the basket cells have made connections with a
sufficient number of pyramidal cells.

A second observation is related to the assumed ability of the
natural system to avoid training instabilities. Recall that in the
model, successful training can be guaranteed if and only if certain
rules are followed when altering the target table and certain relation-
ships are obeyed when specifying the CA3 sector model's parameters.
But even then undesirable changes can occur in other output sequences.
In fact, it is possible that: (1) either these changes cannot be
corrected; or (2) as each change is corrected, another Inismatch
occurs. Such training instabilities might be dangerous to an animal.

A third observation involves the problem the trainer has in
selecting the output to be retrained when a mismatch occurs. As
mentioned in Section 5. 3, one approach would be to select the most
“uncertain" PCLU. Another approach, involving training all PCLUS
at once, might also be used.

A related observation involves the knowledge a hypothetical
natural target table generator has of the connectivity of the natural
functal net. From computer simulations, it appears that the more
information the target table generator has about the connectivity, the

more freedom it has in making changes in the target table that are

78

guaranteed realizable by the net. On the other hand, the more
connectivity knowledge the target table generator has, the greater

the information that must be genetically stored and the greater the
chance for a connectivity error to occur during growth. In the
author's opinion, the weight of evidence supports only the most
general kind of connectivity knowledge on the part of the natural
target table generator, and hence supports a limited function changing
capability with safety.

The final observation pertains to the code employed by the
natural system to convey act information. If the hippocampus is
indeed an act computer, there must be a direct relationship between
behavior and the hippocampus's output. Since the behavior of a
mammal often consists of essentially a stimulus-directed Markovian
sequence of actions, each output of the hippocampus might well be
related in a nontrivial way to its preceeding output. In other words,
a hippocampus output associated with a certain behavioral act on one
occasion may be associated with a different behavioral act on another
occasion. The original function of the hippocampus would have to be
compatible with this, as would the hypothetical target table generator

when it decided on changes in the hippoc ampal output function.

7. 3. Comments on the Engineering Aspects of the Study

The functual system theory developed in this report introduces
a new perspective for understanding interconnected arrays of variable
function nonlinear function generators (functals). Useful applications
of this theory may arise in fields other than neurocybernetics.

It is generally ac'cepted that the nervous system combines
memory and logic in the same location in an extremely effective way.
The Kilmer-McCulloch Retic model, the Kilmer-McLardy hypothesis
of the task of the hippocampus, and the hippocampus model presented
in this report suggest a partial organization of a robot controller
which takes advantage of this prOperty. Consider the design of the
controller for a moon rover. The controller can be imagined as a

hierarchy of subcontrollers with the apex occupied by the Retic, which

79

commands the mode of the rover. As an example, suppose one of the
_ modes is "proceed with the search. "

The rover would receive information on its environment
through its sensory transducers. A reasonable choice of transducers
for a moon rover might be a 3-D television camera, temperature and
pressure sensors (for internal state monitoring), and tactile sensors
(on probes, shovels, and bumpers). The data from these would be
fed into processors designed to extract certain kinds of information.
Some of these may be assigned the task of processing data for input
to the hippocampus system.

The hippocampus occupies the next level of the hierarchy; it
computes the acts within a mode. For example, the acts within the
"proceed with the search" mode might define the direction and speed
of the rover and the search mode of its camera system. The acts
associated with an input configuration would have to be programmed
on earth according to the best information available. Once on the
moon, however, if either a Situation occurred which was found to be
harmful to the rover or an unexpected situation occurred, then the
hippocampus would be retrained. From the hippocampus the act
command would be passed on to lower levels where the actual motor
command sequences would be generated.

There are many problems yet to be solved while pursuing the
details of any hippocampus system design for a robot. Most of these
are analogous to problems yet to be solved in the natural system.
Among these are:

(l) the definition of the code assigned to each output;

(2) a determination of whether the code is context-sensitive

or context-free;

(3) the definition of an initial function for a net which affords
the robot maximum protection and versatility;

(4) the specification of the connectivity of the net (Do usable
connectivities exist which increase the freedom of the
trainer? );

(5) the specification of the trainer rules to (a) guarantee

successful training, (b) select the PCLU to be trained,

80

(c) select the direction in which the PCLU is changed,
and (d) allow the new output to fit smoothly into the act

sequence.

10.

ll.

12.

LIST OF REFERENCES

Von Bekesy, G. , Sensory Inhibition (Princeton University Press,

1967).

Eccles, J. C., Ito, M., and Szentagothai, J., The Cerebellum
as a Neuronal Machine (Springer-Verlag, New York, 1967).

Eccles, J. C. , "Postsynaptic inhibition in the central nervous
system, " The Neurosciences (Gardner C. Quarton, Theodore
Melnechuk, and Frances O. Schmitt, eds. , The Rockefeller
University Press, New York, 408-426, 1967).

 

Wilson, V. J. , "Inhibition in the central nervous system, "
Scientific American 214, 102-110 (1966).

Scheibel, M. E. and Scheibel, A. B. , "Spinal motorneurons,
interneurons and Renshaw cells. A Golgi study, " Arch. Ital.
Biol. 104, 328-353 (1966).

Maturana, H. R. , Lettvin, J. Y. , McCulloch, W. S. , and Pitts,
W. H. , "Anatomy and physiology of vision in the frog (Rana
pipiens), J. Gen. Physiol. 4_3 (No. 6, Pt. 2), 129-175 (1960).

Ratliff, F. , "On fields of inhibitory influence in a neural
network, " Neural Networks (E. R. Caianiello, ed. , Springer-
Verlag, New York, 6-23, 1968).

Barlow, R. B. Jr. , and Levick, W. R. , "The mechanism of
directionally selective units in the rabbit's retina, " J. Physiol.
(Lond.) 178, 477-504 (1965).

Wilson, D. M. , and Waldron, I. , "Models for the generation of
the motor output pattern in flying locusts, " Proceedings of the
IEEE pp, 1058-1064 (1968).

Ratliff, F., and Mueller, C. (3., "Synthesis of "On-Off" and
"Off" responses in a visual-neural system, " Science 126, 840-
841 (1957).

Hubel, D. H. , and Wiesel, T. N. , "Receptive fields, binocular
interaction and functional architecture in the cat's visual cortex, "
J. Physiol. (Lond.) 160, 106-154 (1962).

Kilmer, W. L. , "A circuit model of the hippocampus of the brain, "
AFOSR Scientific Report, Division of Engineering Research,

Michigan State University (July 1970).
81

13.

14.

15.

82

Kilmer, W. L. , "The reticular formation: Part 1, Modeling
studies of the reticular formation; Part II, The biology of the
reticular formation, " AFOSR Scientific Report, Division of
Engineering Research, Michigan State University (February 1969).

Kauffman, S. A. , "Metabolic stability and epigenesis in randomly
constructed genetic nets," J. Theoret. Biol. 22, 437-467 (1969).

Purpura, D. B. , in Basic Mechanisms of the Epilepsies

(Jasper, H. H., Ward, A.A., and Pope, A., eds., Little, Brown
and Co., Boston, 1969).

APPENDIX A

BACKGROUND ON THE DEVELOPMENT OF THE HIPPOCAMPUS NET

A. 1. The Pyramidal Cell Logic Unit

The pyramidal cell model as originally conceived was the set
of continuous firing rate equations shown in Figure A. 1. In this
figure, Equation 2 says that the firing rate of model pyramidal cell j
at the axon hillock at time t, yj(t) is a linear function of xJ.(t) only
when xj(t) is in the range from 0 to a .. If xj(t) is less than

mYJ
zero, then .(t) = 0. If x, t is reater than a ., then .(t) is
YJ J( ) g myj YJ

equal to the maximum value of amyjayj°
The function xJ.(.) as defined in Equation 1 consists of Six
terms: I
1. Z) 6.. .. a.. t-T ..) zt-T..
1:1 ji in 31' A31 ( 31)

This term represents the effect of the firing rates of the basket cells
on the firing rate of the pyramidal cell. To explain the concepts which
were used to develop this term, assume synaptic contact is made

between basket cell i and pyramid j. At time t-Tji the basket cell

fired at a rate zi(t-'rji). This signal traveled through various
collaterals to bouton j, i, arriving there at time TAji’ altered by
an amount in' (Note: By convention, if there is no connection

between basket cell i and pyramid j, then in: 0. ) At or near the
bouton, the signal is modified by the memory process ajiu-TAji)’
which is defined in Equation 4. Finally, the Signal passes through
the dendritic arbor and soma of the pyramidal cell as an inhibitory
post-synaptic potential and arrives at the axon hillock at time t,
having been altered on the way by an amount €ji'

K
2. Z}
k=l

This term represents the effect of the septal fiber firing rate on the

O'jk sk(t-'rsjk)

83

84

X(t) = -€yA(t-TA)z(t-'r) + 0’s(t-‘rs) + 0M(t-'TM)
+ 8Y(t-tx) - F(t) (1)
Y t) - t) t) (t) )T (2)
( — (y1( . y2( . y,
where
0 . S 0
xJ(t)
:: < .
yj(t) aijj(t), 0 < xj(t) amyJ
a .0. ., x.(t) Z a .
1711'] VJ J mYJ
I:
l"(t) : \II So exp [-% (t-w)] x(w)dw + F0 (3)
A(t) — 1+ 111 if M” ( )d
_ ( - )exp - T 0 exp[- 'T—flyz w-‘rz w
(4)
Figure A. 1. The pyramidal cell firing rate equations.

The expressions are for J pyramidal cells, I basket cells, K

septal fibers, and N mossy fibers. The dimensions of the vectors

are: X(t), l"(t), \II, 6, l" , o. , c1, and C:Jxl; Z(t):Ix1;
0 mx x

s(t) : le; M(t) : le. The dimensions of the matrices are:

A(t), A, II, TA, 7, e, y:JxI; 0, 7x:JxJ; TM, 9 :JxN;

'r, 0' : JxK.

85

firing rate of the pyramidal cell. The memory effects between the

septum and the cell are assumed to be constant relative to the

basket to pyramidal cell memory. This will also be true of both

the mossy fiber and other pyramidal cell inputs discussed below.
N

. 23 0. - .
3 n=1 jn mn(t TMjn)

This term represents the effect of the mossy fiber input on the firing

rate of the pyramidal cell.

J
4. E (3. y (t-‘r
[:1 j! I

This term represents the input from other pyramids and the possible

xj 1)

feedback from pyramid j itself.
5. I‘. t
J( )

This term is the variable threshold defined by Equation 3. This
expression is an attempt at a simple linear continuous equation for
the kind of firing rate dependence on the input rate threshold above
which nerve spikes are generated: the threshold increases as the
firing rates of the inputs to the neuron increases in the recent past.
The equation is a convolution of the potential function with an
exponential decay. Thus, at some time t the threshold is made up
of a constant term plus an infinite number of terms of the form

f(w) exp{ - 1/7 (t-w)} o s w s t.
Therefore, the value of the potential function which occurred at time
w = 0 will have decayed the most, since it would have the value

f(0) exp (- t/T);
and the value of the potential function occurring at time w = t will
have decayed not at all, since it would have the value

f(t) - 1 = f(t).

Equation 4 is an attempt to give the pyramidal cell model a
memory, where memory can be loosely defined as a device for storing
records of events which have occurred in time previous to the present.

The aji-th entry expresses the concept that the memory
process becomes larger as the basket cell i's firing rate zi(t-'rz) in

the recent past becomes larger and approaches 1 in the limit: that is,

86

88

t
_ , 1221 _ a
A — So exp I: )‘ji in zi(w iji)dw large,

exp [-A/Tji] -> 0 and aji(t) -' 1

If the basket cell i's firing rate zi(t-'rz) has been very small in the
past, then a.i(t) approached some minimum value nji: that is, as

the A expression defined above becomes small,

- .. -’ t -*
exp ( A/TJI) l and aji( ) lIJ1

The PCLU as defined in Chapter 3 is an extreme simplification
of this continuous model. Some more of the more important simplifying
assumptions are:

1. There is a constant threshold.

2. There are no inputs from other PCLUS.

3. The septal input controls the magnitude of A and is not of
primary importance in the determination of the pyramidal cell firing
rate.

4. The memory has no decay.

5. Most time lags are omitted.

A. 2. The Basket Cell Logic Unit

The terms in the basket cell firing rate equations, Figure A. 2,
are analogous to terms in the pyramidal cell firing rate equations.
Z(t), Equation 2, is the basket cell firing rate vector. It is expressed
in the same form as Y(t), with azi being the proportionality con-
stant and amzi being the maximum permissible value of di(t)°

D(t) is the basket cell firing rate potential vector, and it is
analogous to X(t). The first term on the right hand side of Equation 1
is of the same form as Equation 1, Figure A. 1, term 1: ¢ corres-
ponds to 6; A correSponds to y; G(-) correSponds to A(-). The
second term in the expression, (Z(t), is the threshold for the basket
cells. Its Equation 3 is analogous to Equation 3 of Figure A. l. The
last term in the expression, 2;, is the constant firing rate potential
vector for the basket cells, and it is analogous to C of the pyramidal

cell expression. The memory expression, Equation 4, is of the same

form as the memory expression for the pyramidal cells.

87

 

D(t) = <1> A G(t-TQ) Y(t-TR) - G(t) + 1; (1)
Z(t) = (21m. 210) )T. (2)
where
0 di(t) S 0
7‘1") = l “zidi(t" 0 S di't' < Clmzi
a .0. ,, d.(t) Z a Z.
_ mm m 1 m 1
t 1
(Z(t) : {2150 exp [- R (t-w)-] D(w) + 90 (3)
I:
_ l. - .I_)t‘w -
G(t) _ l + (p -1) exp {g 30 expl: v J AY(t 'rv)dw
(4)

Figure A. 2. The basket cell firing rate equations.

The expreséions are for J pyramidal cells and I basket cells. The
dimensions of the vectors are: D(t), Z(t), (Z(t), 91. 90, g, and
p:lxl; Y(t) : Jxl. The dimensions of the matrices are: (b, G(t),

TV: go “'9 V31x-J; A3JXIo

TR, TQ,

88

It is clear from the BCLU model presented in Chapter 3
that some radical simplification of this model has been made. The
major additional assumption for the BCLU over and above those

presented in the previous section is that there is no memory process.

A. 3. The Connectivity

As originally conceived, the connectivity of the hippocampus
model was based on the concept of a card. A card was defined as
(1) all pyramidal cell (PC) models connected to one septal fiber,
plus (2) all basket cell (BC) models which receive inputs from the PCs
of the card (it was assumed that a BC did not receive inputs from two
different cards), plus (3) a cell in CA1 which received inputs from
every PC in the card. The output of the card was the output of this
last cell. The communication between cards was accomplished by
BC collaterals to the PCS of other cards.

This concept was modified to the connectivity described in
Chapter 3, with one septal fiber per PC, because it seemed possible

that a card could be modeled as a single PC.

APPENDIX B
A COMPUTER PROGRAM BASED ON ALGORITHM 4. 3. 1

Figure B. l is a CDC 6500 FORTRAN EXTENDED (MSU)
listing of the program TTABLE and its subroutines. This program
generates all possible output sequences of a CA3 sector net model
containing N PCLUs. It does so according to Algorithm 4. 3. 1.
Note that one data card is required in order to specify the number
N; the format of this card is 10X, 15. The output sequences are
printed in rows of ten; the format for a typical sequence is demon-

strated by the following, which is an actual output sequence generated
by TTABLE with N = 5:

PCLU l- 1 0 l 0 l 0 l (The output sequence component for PCLU l.)
PCLUZ-ZOlOOOO
PCLU3-2021222
PCLU4-2010101
PCLU 5 2 0 0 0 81-9 0.
cycle

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