This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Biol. Cybern. 52, 1417152 (1985) Biological
Cybernetics © SpringerVerlag 1985 “Neural” Computation of Decisions in Optimization Problems J. J. lI’opﬁeldl'2 and D. W. Tank2 1 Divisions of Chemistry and Biology, California Institute of Technology, Pasadena, CA 91125, USA
2 Department of Molecular Biophysics, AT&T Bell Laboratories, Murray Hill, NJ 07974, USA Abstract. Highlyinterconnected networks of non
linear analog neurons are shown to be extremely
effective in computing. The networks can rapidly
provide a collectivelycomputed solution (a digital
output) to a problem on the basis of analog input
information. The problems to be solved must be
formulated in terms of desired optima, often subject to
constraints. The general principles involved in con
structing networks to solve speciﬁc problems are dis
cussed. Results of computer simulations of a network
designed to solve a difﬁcult but welldeﬁned optimiza
tion problem — the TravelingSalesman Problem — are
presented and used to illustrate the computational
power of the networks. Good solutions to this problem
are collectively computed within an elapsed time of
only a few neural time constants. The effectiveness of
the computation involves both the nonlinear analog
response of the neurons and the large connectivity
among them. Dedicated networks of biological or
microelectronic neurons could provide the compu
tational capabilities described for a wide class of
problems having combinatorial complexity. The
power and speed naturally displayed by such collective
networks may contribute to the effectiveness of biolog
ical information processing. _—____——___——————— I Introduction A large class of logical problems arising from real
world situations can be formulated as optimization
problems, and thus qualitatively described as a search
for the best solution. These problems are found in
engineering and commerce, and in perceptual prob
lems which must be rapidly solved by the nervous
systems of animals. Wellstudied problems from com
merce and engineering include: Given a map and the
problem of driving between two points, which is the best route? Given a circuit board on which to put
chips, what is the best way to locate the chips for a good
wiring layout (Kirkpatrick et al., 1983)? Analogous,
but only partially characterized problems in biological
perception and robotics include: 1 Given a monocular
picture, what is the best threedimensional description
of the locations of the objects? Indeed, what are the
“objects”? In each of these optimization problems, an
attempt can be made to quantify the vague criterion
“best” by the use of a speciﬁc mathematical function to
be minimized. While a cost function may be speciﬁed, real world
data used to evaluate it is generally not precise. Also,
complex cost functions usually involve somewhat
arbitrary weightings and forms of the various contri
butions. From an engineering viewpoint, these com—
plications imply that little meaning can be attached
to “best”. Often, what is truly desired is a very good
solution, which will be uniquely best only for simple
tasks. In many situations, a very good answer com
puted on a time scale short enough so that the solution
can be used in the choice of appropriate action is more
important that a nominallybetter “best” solution.
This is especially true in the biological and robotics
tasks of perception and pattern recognition, because
these problems typically have an immense number of
variables and the task of searching for the mathemat
ical optimum of the criterion can often be of consid
erable combinatorial difﬁculty, and hence time
consuming. The computational powers routinely used by ner
vous systems to solve perceptual problems must be
truly immense, given the massive amount of sensory
data continuously being processed, the inherent dif
ﬁculty of the recognition tasks to be solved, and the
short time (msec—secs) in which answers must be found. 1 (Poggio and Torre, 1985; Terzopoulos, 1984; lkeuchi and
Horn, 1981; Julesz, 1971; Marr, 1982; Sebestyn, 1962) 142 Most general purpose digital computers would fail to
provide this combination of power and speed. One of
the central goals of research in neuroscience is to
understand how the biophysical properties of neurons
and neural organization combine to provide such
impressive computing power and speed. An under
standing of biological computation may also lead to
solutions for related problems in robotics and data
processing using nonbiological hardware and soft ware.
It is clear from studies in anatomy, neurophysi— ology, and psychophysics that part of the answer to
how nervous systems provide computational power
and speed is through parallel processing. The mam
malian visual system computes elementary feature
recognition massively in parallel (J ulesz, 1981; Ballard
et al., 1983). At the level of neural architecture,
anatomy and neurophysiology have revealed that the
broad category of parallel organization is manifest in
several different but interrelated forms. Parallel sen
sory input channels, such as the individual rods and
cones in the vertebrate retina, allow rapid remote
sensing of the environment and data transmission to
processing centers. Likewise, parallel output channels,
for example in corticocortical projections in the cortex,
connect different processing modules (see, for example
Goldman—Rakic, 1984). Another manifestation of par—
allelism occurs in the large degree of feedback and in
terconnectivity in the “local circuitry” of speciﬁc pro
cessing areas (see, for example Shepherd, 1978). The
idea that this large degree of local connectivity between
the simple processing units (neurons) in a speciﬁc
processing area of the nervous system is an important
contribution to it’s computational power has led to the
study of the general properties of neural networks 2 and
also several “connectionist” theories in perception
(Ballard, in press; Feldman and Ballard, 1982). The
connectionist theories employ logical networks of two
state neurons in a digital clocked computational
framework to solve model pattern recognition problems.
There is a major feature of neural organization which is not included in connectionist models but
which can act synergistically with parallel feedback
and connectivity to greatly enhance computational
power. This feature is that the biological system
operates in a collective analog mode, with each neuron
summing the inputs of hundreds or thousands of
others in order to determine its graded output. An
analog system is made powerful in computation by its
ability to adjust simultaneously and selfconsistently
many interacting variables (Poggio and Koch, 1984;
Jackson, 1960; Huskey and Korn, 1962). Although 2 (Hopﬁeld, 1984; Gelperin et al., in press; Hopﬁeld, 1982;
Hinton and Sejnowski, I983; Arbib, 1975) very fast, analog summation is inevitably less accurate
than digital summation. This compromise is not
critical, however, in perceptual tasks formulated as
optimization problems. The computational load of
rapidly reducing this sensory input to the desired
“good” solution is already immense; inaccuracies and
uncertainties are already present and the computa
tional load is meaninglessly increased by high digital
accuracy. Parallel analog computation in a network of
neurons is thus a natural way to organize a nervous
system to solve optimization problems. In this paper we quantitatively demonstrate the
computational power and speed of collective analog
networks of neurons in solving optimization problems
rapidly. We demonstrate that even for very difﬁcult
problems, making a collective decision is rapid, requir
ing an elapsed time of only a few characteristic times of
the “neurons” comprising the network. This speed,
needed for real—time processing of sensory information
by a nervous system, can be provided by collective
analog computational circuits because all of the
neurons simultaneously and continuously change their
analog states in parallel. When compared to modern
digital general purpose computers constructed with
conventional silicon integrated circuits (VLSI), the
“neural” computational circuits we describe have
qualitatively different features and organization. In
VLSI the use made of analog calculations in minimal
(Mead and Conway, 1980). Each logic gate will
typically obtain inputs from two or three others, and a
huge number of independent binary decisions are
made in the course of a computation. In contrast, each
nonlinear neural processor (neuron) in a collective
analog computational network gets inputs from tens
or hundreds of others and a collective solution is
computed on the basis of the simultaneous interactions
of hundreds of devices. Recognizing that the nature of perceptual problems
is similar to other optimization problems (Poggio and
Torre, 1985; Hinton and Sejnowski, 1983; Ter
zopoulos, 1984) and that computing power is best
illustrated on a difficult but well understood problem,
we will show here how to organize a computational
network of extensively interconnected nonlinear ana
log neurons so that it will solve a well characterized,
but nonbiological, optimization problem. We have
chosen as an illustration the “TravelingSalesman
Problem” (TSP), for which the computational dif
ﬁculty has been much studied (Lawler et al., in press;
Garey and Johnson, 1979). The solution to the TSP
problem, and indeed, the solution to many optimi
zation problems is a discrete answer. However, the
neurons in the networks we describe operate in
an analog mode. Hence, unlike “connectionist” ap
proaches to solving perceptual problems in networks which use strictly twostate neurons, the formulation
of problems to be solved by an analog computational
network requires embedding what seemed to be dis
crete problems in a continuous decision space in which
the neuronal computation operates. We demonstrate
here how a continuous decision space and continuous
computation can be related to discrete computation
and why a continuous space can improve the quality of
the solutions obtained by highlyinterconnected
neural networks. 11 Analog Computational Networks The general structure of the analog computational
networks which can solve optimization problems is
shown in Fig. lb. These networks have the three major
forms of parallel organization found in neural systems:
parallel input channels, parallel output channels, and a
large amount of interconnectivity between the neural
processing elements. The processing elements, or
“neurons”, are modelled as ampliﬁers in conjunction
with feedback circuits comprised of wires, resistors and
capacitors organized so as to model the most basic (0)
1 Y inverting amplifier V amplifier 0 resnstor in ‘nj network Fig. 1a and b. a The inputoutput relation for the “neurons” or
analog ampliﬁers. b The analog circuit. The output of any neuron
can potentially be connected to the input of any other neuron.
Black circles at intersections represent resistive connections
(TIJ75) between outputs and inputs. Connections between inverted
outputs and inputs represent negative (inhibitory) connections 143 computational features of neurons, namely axons,
dendritic arborization, and synapses connecting the
different neurons. The ampliﬁers have sigmoid mono—
tonic inputoutput relations, as shown in Fig. 1a. The
function Vj=gj(uj) which characterizes this input
output relation describes the output voltage of
ampliﬁer due to an input voltage u,. The time
constants of the ampliﬁers are assumed negligible.
However, like the input impedance caused by the cell
membrane in a biological neuron, each ampliﬁer j has
an input resistor 91 leading to a reference ground and
an input capacitor C j. These components partially
deﬁne (see below) the time constants of the neurons and
provide for integrative analog summation of the
synaptic input currents from other neurons in the
network. In order to provide for both excitatory and
inhibitory synaptic connections between neurons
while using conventional electrical components, each
ampliﬁer is given two outputs, a normal (+) output
and an inverted (—) output. The minimum and
maximum outputs of the normal ampliﬁer are taken as
0 and 1, while the inverted output has corresponding
values of 0 and — 1. A synapse between two neurons is
deﬁned by a conductance TU which connects one of the
two outputs of ampliﬁer j to the input of ampliﬁer 1'.
This connection is made with a resistor of value
RU: l/T,j. If the synapse is excitatory (T,J~>0), this
resistor is connected to the normal (+) output of
ampliﬁer j. For an inhibitory synapse (Til<0), it is
connected to the inverted (—) output of ampliﬁer j. The
matrix Ti] deﬁnes the connectivity among the neurons.
The net input current to any neuron i (and hence the
input voltage u,) is the sum of the currents flowing
through the set of resistors connecting its input to the
outputs of the other neurons. Thus the normal and
inverted output for each neuron allow for the construc—
tion of both excitatory and inhibitory connections
using normal (positive valued) resistors; biological
neurons do not require a normal and inverted output
since excitatory and inhibitory synapses are deﬁned by
use of different receptor/ion channel combinations. As
indicated in Fig. 1b, our circuits include an externally
supplied input current I , for each neuron. These inputs
can be used to set the general level of excitability of the
network through constant biases, which effectively
shift the inputoutput relation along the u, axis, or to
provide direct parallel inputs to drive speciﬁc neurons.
Although this “neural” computational circuit is de
scribed here in terms of ampliﬁers, resistors, capacitors,
etc., we have shown (Hopﬁeld, 1984; Gelperin et al., in
press) that networks of neurons whose output consists
of action potentials and with connections modelled
after biological excitatory and inhibitory synapses
could compute in a similar fashion to this conventional
electronic hardware. 144 The equation of motion describing the time evo
lution of this circuit is: C.(du./dr)= g T. F1 'Vj_ui/Ri+Ii (1) J Vj:gj(uj) R, is a parallel combination of g, and the Rij: 1/R.=1/a.+ UK... (2) For simplicity, in the present work we have always
chosen gi= g, RizR, and Ci=C, independent of i,
though this is not necessary. Dividing by C and
redeﬁning Ej/C and I i/C as TU and 1,, the equations of
motion become: N
j=l
1=RC
Vj=g(“j) For an “initialvalue” problem, in which the input
voltages of the neurons are given values ui at time t = 0,
this equation of motion provides a full description of
the time evolution of the state of the circuit. Integration
of this equation in a digital computer allows any
hypothetical network to be simulated. In earlier work (Hopﬁeld, 1984) it was shown that
the equations of motion for a network with symmetric
connections (Tij = T},) always lead to a convergence to
stable states, in which the outputs of all neurons
remain constant. Also, when the width of the ampliﬁer
gain curve in Fig. 1a is narrow ~ the highgain limit »
the stable states of a network comprised of N neurons
are the local minima of the quantity J
i:1j=1 N N N
E=—1/2 Z Z TiVst— 2 VJ.» (4)
i:1 The state space over which the circuit operates is the
interior of the Ndimensional hypercube deﬁned by
V]. = 0 or 1. However, in the highgain limit, the minima
only occur at corners of this space. Hence the stable
states of the network correspond to those locations in
the discrete space consisting of the 2” corners of this
hypercube which minimize [Eq. (4)]. Networks of neurons with this basic organization
can be used to compute solutions to speciﬁc optimi
zation problems by ﬁrst choosing connectivities (Tlj)
and input bias currents (1,) which appropriately repre
sent the function to be minimized. The methods in
volved in this selection are discussed below. Following
this construction or “programming” of the network, an
initial set of input voltages ui are provided, and the
analog system then converges to a stable state which minimizes the function. We interpret the solution to
the problem from the ﬁnal stable state. For the
problems considered here, the solutions are discrete
(digital) and the gain is chosen high enough so that in
the ﬁnal stable state each neuron has a normal (+)
output V, very near 0 or 1. The set of outputs then
provides a digital answer which represents a solution
the problem. Before we consider the form of a network which
solves the TSP, it is instructive to consider how a
simpler optimization problem can be solved by one of
these computational networks. Although not inter
preted as an optimization problem at that time, an
example was actually provided in earlier work (Hop
ﬁeld, 1984) where it was shown how the same compu
tational circuit described above could, with the proper
choice of connection strengths, operate as a Content
AddressableMemory (CAM). The normal outputs of
the N ampliﬁers comprising the memory circuit 7
which for that application were allowed the range —1
to +1, instead of the 0 to 1 range described above —
were always — 1 or 1 in the highgain limit and the state
of these outputs represented a binary word in memory.
A memory, stored in the network by an appropriate
choice of 7}}. elements, could be “retrieved” by setting
the outputs of the ampliﬁers in the binary pattern of the
recall key and then allowing the system to converge to
a stable state. This stable state was interpreted as the
memory word evoked by the key. Each recall “solved”
the “problem” of which of the stored binary words was
“closest” to the key word. We can understand how to construct an appropri
ate computational circuit for the CAM, considered
now as a simple example of an optimization problem,
by examining the E function. Since E [Eq. (4)] deter
mines the stable states of the network, then if we wish
the stable states to be a particular set of m memory
states V‘s: {1,2, ,m} we must choose the connec
tion strengths (TU) and the input bias currents (1,) of the
network such that Eq. (4) is a local minima when the
system is in each one of the states VS. Since Eq. (4) is
quadratic a guess might be: E=—1/2_§ (VSV)? (5) If the state vector V (with components V;) is a random
vector, then each parenthesized term is very small. But
if Vis one of the memories VS, then one term in the sum
is N2. Hence the network has an energy minima of
depth approximately —1/2N2 at each of the assigned
memories. Equation (5) can be rewritten in the stan
dard form [Eq. (4)] of the energy function if all 1,20
and the 7}]. elements are deﬁned as: 7—1:}: sgl Vists _ This equation for TU is the storage algorithm presented
earlier (Hopﬁeld, 1984) except for an additive constant.
It is derived above by thinking of the CAM as an
optimization problem and then making a judicious
Choice of the representation of the energy function in
terms of the desired memories. In a practical applica—
tion of a CAM, for example a network used to store
telephone numbers, in addition to the storage al
gorithm above, a transformation used to code the real
world information into the binary word memory data
representation is required. Taken together, the data
transformation and the algorithm for the Ej can be
considered the “map” of this problem onto the analog
computational network. The basic property of the analog computational
networks described above is the minimization of E.
The CAM example illustrates that through the con
struction of an appropriate energy function for the
circuit, and a strategy for interpreting the state of the
outputs as a solution, a simple optimization problem
may be “mapped” onto the network. We have recently
found that these circuits can also rapidly solve difﬁcult
optimization problems which have both contraints in
the possible solutions and also combinatorial com
plexity. A network designed to solve the “Traveling
Saleman Problem” illustrates this computational
power. III The TSP Problem The “TravelingSalesman Problem” (TSP) is a classic
of difﬁcult optimization. It is simple to describe,
mathematically well characterized and makes a good
vehicle for describing the used of neural analog compu
tational networks to solve difﬁcult optimization prob
lems. A set of n cities A, B, C,... have (pairwise) dis
tances of separation (1,13,11,10 ...,dBC.... The prob—
lem is to ﬁnd a closed tour which visits each city once,
' returns to the starting city, and has a short (or
minimum) total path length. A tour deﬁnes some
sequence B, F, E, G, ..., W in which the cities are
visited, and the total path length d of this tour is d=dBF+dFE+... +dWB. The actual best solution to a TSP problem is
computationally very hard — the problem is
np—complete (Garey and Johnson, 1979), and the time
required to solve this problem on any given computer
grows exponentially with the number of cities. The solution to the n—city TSP problem consists of
an ordered list of n cities. To “map” this problem onto
the computational network, we require a representa
tion scheme which allows the digital output states of the neurons operating in the highgain limit to be I45 decoded into this list. We have chosen a representation
scheme in which the ﬁnal location of any individual
city is speciﬁed by the output states of a set of n
neurons. For example, for a 10city problem, if city A is
in position 6 of the tour which is the solution to the
problem then, as shown below, this is represented by
the sixth neuron out of a set of ten having an output
V6: 1 with all other outputs at 0: 0000010000. This representation scheme is natural, since any
individual city can be in any one of the n positions in
the tour list. For n cities, a total of n independent sets of
n neurons are needed to represent a complete tour.
This is a total of N =n2 neurons. The output state of
these n2 neurons which we will use in the TSP
computational network is most conveniently dis
played as an nxn square array. Thus, for a 5city
problem using a total of 25 neurons, the neuronal state (7) shown above would represent a tour in which city C is
the ﬁrst city to be visited, A the second, E the third, etc.
[The total length of the 5city path is dCA+dAE+dEB
+dBD + duo] Each such ﬁnal state of the array of
outputs describes a particular tour of the cities. Any
city cannot be in more than one position in a valid tour
(solution) and also there can be only one city at any
position. In the n x n “square” representation this
means that in an output state describing a valid tour
there can be only one “1” output in each row and each
column, all other entries being zero. Likewise, any such
array of output values, called a permutation matrix,
can be decoded to obtain a tour (solution). An example
of the ﬁnal state of a lOcity problem is shown in
Fig. 2d. For an ncity TSP problem, there are n! states of the
general form [Eq. (7)] above. However, a tour de
scribes an order in which cities are visited. For an ncity
problem, there are 2n tours of equal pathlength, for
each path has an nfold degeneracy of the initial City on
a tour and a 2fold degeneracy of the tour sequence
order. There are thus n!/2n distinct paths for closed
TSP routes. Because of our representation of neural outputs of
the TSP computational network in terms of 11 rows of n
neurons, the N symbols V, will be described by double
indicies nyj. The row subscript has the interpretation
of a city name, and the column subscript the position of (a)...III (b) . . . . ..I
I . . . . . ...l . . . . . ..I
 ll . . . . . . ..I.
I . . . . . .. I . . . . . . ..I
....u . ........
III III II .. . II. . . . . . .. I u u    . u. n. . . . . . . . . . . . . . . . . . . . . . ... (c) . . . . (d) . . . . I . . . . . .... . . . . . . . ...B . . . . . . ... I . . . . . . ... . . . . . . . .. D .....TT... .I.ff.'..FCITY ... .II ..... ....G . . . . . . ..        H .l . . . . . .. ... . . . . . . .l . . . . . 1234567391
Fa“ POSITION IN PATH
PATH = DHIFGEAJCB Fig. 2a—d. The convergence of the lOcity analog circuit to a tour.
The linear dimension of each square is proportional to the value
of V)“. a, b, c intermediate times, d the ﬁnal state. The indices in d
illustrate how the ﬁnal state is decoded into a tour (solution of
TSP) that city in a tour [cf (7)]. We will use these two indices
instead of one to label all of the neurons because it
simpliﬁes the interpretation of the equations describ
ing the energy function. Like in the CAM problem, this
E function will aid construction of an appropriate
computational network for the TSP. The TSP Energy Function To enable the N neurons in the TSP network to
compute a solution to the problem, the network must
be described by an energy function in which the lowest
energy state (the most stable state of the network)
corresponds to the best path. This can be separated
into two requirements. First, the energy function must
favor strongly stable states of the form of a permu
tation matrix, such as those shown in Eq. (7) or in
Fig. 2d, rather than more general states. Second, of the
n! such solutions, all of which correspond to valid
tours, it must favor those representing short paths. An
appropriate form for this function can be found by
considering the high gain limit, in which all ﬁnal
normal (+) outputs will be 0 or 1. As before, the space
over which the energy function [Eq. (4)] is minimized
in this limit is the 2” corners of the Ndimensional
hypercube deﬁned by Vi=0 or 1. Consider those
corners of this space which are the local minima (stable
states) of the energy function E=A/2§ z z VX,VX,+B/22 § 2 thVn i j$i 1 X42Y + C/2 VXi—n)2 , (8) where A, B, and C are positive. The ﬁrst triple sum is
zero if and only if each city row X contains no more
than one “1”, (the rest of the entries being zero). The
second triple sum is zero if and only if each “position in
tour” column contains no more than one “1” (the rest
of the entries being zero). The third term is zero if and
only if there are n entries of “1” in the entire matrix.
Thus this energy function evaluated on the domain of
the corners of the hypercube has minima with E = 0 for
all state matrices with one “1” in each row and column.
All other states have higher energy. Hence, including
these terms in an energy function describing a TSP
network strongly favors stable states which are at least
valid tours in the TSP problem, and fulﬁlls the ﬁrst
requirement for E. The second requirement, that E favor valid tours
representing short paths, is fulﬁlled by adding one
additional term to the function. This term contains
information about the length of the path correspond
ing to a given tour, and its form can be chosen as
1/21); Z zderXi(Vr,i+1+Vr,i—1) (9) Y—‘FX i For notational convenience, subscripts are deﬁned
modulo n, in order to express easily “end effects” such
as the fact that the n’th city on a tour is adjacent in the
tour to both city (n— l) and city 1: i.e., VY,,,+J= V”.
Within the domain of states which characterize a valid
tour, the above term [Eq. (9)] is numerically equal to
the length of the path for that tour. An appropriate total energy function for the TSP
network consists of the sum of Eq. (8) and Eq. (9). If
AB, and C are sufficiently large, all the really low
energy states of a network described by this function
will have the form of a valid tour. The total energy of
that state will be the length of the tour, and the states
with the shortest path will be the lowest energy states. Through Eqs. (3) and (4), the quadratic terms in this
energy function deﬁne a connection matrix and the
linear terms deﬁne input bias currents. Using the
row/column neuron labeling scheme described earlier
for each of the two indices, the implicitly deﬁned
connection matrix is (with brief descriptions of the
meanings of the various terms): Tm,“ = —A6Xy(l —5,~j) “inhibitory connections within
each row” ~ B5,,(1 — 6H) “inhibitory connections within
each column” —C “global inhibition”
_Ddxr(5j,i+ 1 + 513— 1) “data term” [5,,=1 if i=j and is 0 otherwise]. (10)
The external input currents are:
1,“: + Cn “excitation bias”. (ll) The “data term” contribution, with coefﬁcient D, to
TX”). is the input which describes which TSP problem
(i.e., where the cities actually are) is to be solved. The
terms with A, B, and C coefﬁcients provide the general
constraints required for any TSP problem. With this E
function guiding the dynamics of the circuit, the
network should compute the solution by choosing a
ﬁnal state which has the form of a permutation matrix
[Eq. (7)] after starting in some initial unbiased state.
The “data term” contribution controls which one of
the n! set of these properly constrained ﬁnal states is
actually chosen as the “best” path. IV TSP Simulation Results A network for a lOcity problem using the connection
matrix deﬁned in Eq. (10) and the input bias terms of
Eq.(ll) was simulated on a digital computer. The
locations of the 10 cities were chosen at random (with
uniform probability density) on the interior of a two
dimensional square of edge length 1. These choices
deﬁned a particular set of d H and hence TX”, through
Eq. (10). The analog network for this problem gen
erated the equations of motion dun/(h: ‘uxl/T—A Z VXj—B Z VYi
j¢i Hx
_C V ,_
<29; X} n)
’ngxﬂVmH‘l'I/mq) (12) Vxl = g(uXi) = %(1 + tanh(uXi/u0)) (for all X, 1'). These equations of motion have the form described in
an earlier section, but show the speciﬁc contributions
made by the TX”, and I X, terms. The parameter “n”
was not ﬁxed as 10, but was used to adjust the neutral
position of the ampliﬁers which would otherwise also
need an adjustable offset parameter in their gain
functions. The offset hyperbolic tangent form of the
gain curve was chosen to resemble real neural input
output relations as well as the characteristics of a
simple transistor ampliﬁer. The set of parameters in
these equations of motion is overcomplete, for the time
it takes to converge is in arbitrary units. Without loss
of generality, r can be set to 1. In our simulations, an appropriate general size of
the parameters was easily found, and an ancedotal
exploration of parameter values was used to find a
good (but not optimized) operating point. Results in
this section refer to parameter sets at or near A=B=500 C=200 0:500 u0=0.02 n=15. 147 Since we have no a priori knowledge of which tours
are best, and the network already has in the “data
term” the necessary input to solve the problem, we
want to pick the initial values of the neural input
voltages (um) without bias in favor of any particular
tour. A sensible choice might seem to be uXizuoo,
where am is a constant which is chosen so that at t=0 ZZVXi=10
x i which is also, approximately, the desired value of this
sum at I: 00. However, this unbiased choice is a
disaster to the computation. Since each path has 2n
equivalent tours describing it, the system has no way to
choose one of them given an unbiased start, and thus
cannot converge to a tour at all. The problem is
equivalent to the fact (in classical physics) that a pencil
poised exactly vertically on its point must not fall over,
since to do so would be to choose a direction in which
to fall. A similar problem of “broken symmetry”
appears in magnetic phase transitions (Anderson,
1984). It is therefore necessary to add some noise (SuX, “Xi = “00 + 514m to the initial values. This has the desired effect of
breaking the symmetry and allowing the system to
choose a tour, but also inserts a small random bias into
the choice of path. Figure 2 shows the results of a simulation which
illustrate the convergence of a typical such starting
state to a ﬁnal path. The symmetrybreaking 614,“ were
each randomly chosen uniformly in the interval: 0.1u0_.<__5qu§0.lu0 . The linear dimensions of the squares in Fig. 2 are
proportional to the outputs of the “neurons” in the
array. Initially they are very nearly uniform and as time
passes (Figs. 2a—c) they converge to a ﬁnal time
independent state (Fig. 2d). The set of V,“ are not a
permutation matrix of form [Eq. (7)] throughout the
computation. This is because the actual domain of
function of the network is not at the corners of the
N dimensional hypercube defined by VXi=0 or 1, but
rather in it’s interior. However, notice that the ﬁnal
outputs (Fig. 2d) produce a permutation array with
one neuron “on” and the rest “off” in each row and
column, and this state thus represents a legitimate tour.
The choice of network parameters which provides
good solutions is a compromise between always ob
taining legitimate tours (D small) and weighting the
distances heavily (D large). Also, as expected, too large
u0 (low gain) results in ﬁnal states in which the values of
V,“ are not near 1 or 0. These states are not permuta
tion matrices and hence represent invalid tours. Too
small u0 yields a poorer selection of good paths. (a) (b) l I l l J J
O 0.2 0.4 06 0.8 1 O 0.2 0.4 0.6 0.8 1 0 0.2 04 06 0.8 1 Fig. 3a—c. a, b Paths found by the analog convergence on 10
random cities. The example in a is also the shortest path. The
city names A J used in Fig. 2 are indicated. c A typical path
found using a twostate network instead of a continuous one (a) (b) (c)
4 0:119 D=426 0:507
0.8
06
0.4
O 2
O O 02 0.406 0.8 1 O 0.2 0.4 0.60.8 1 O 02 0.40608 1 Fig. 4a—c. a A random tour for 30 random cities. b The Lin
Kernighan tour. c A typical tour obtained from the analog
network by slowing increasing the gain (a) (b) 4000
12500
10000 3000
El tr
Lu
g 7500 a}: 2000
D 3
z 5000 Z
1000
2500
o O L. O 5 10 15 TOUR LENGTH TOUR LENGTH Fig. 5a and b. a A histogram of the number of different paths
between length L and L+0.l for the TSP with 10 cities. The *‘s
below the xaxis give the histogram for the number of times a
path between L and L+ 0.1 was found by the analog network in
20 tries (conditions as in text). The region for L< 3.0 has been
magniﬁed by a factor of 100 for clarity. b A histogram of the
number of different paths between length L and L+0.1 for the
TSP problem with 30 cities. The arrow indicates the tour length
for the KernighanLin solution while the asterisk at 5.6 indicates
the path length of a solution obtained by the analog network at
ﬁxed gain width. The asterisk at 5.0 indicates a better solution
obtained by slowly increasing the gain A convergence from a given starting state is
deterministic, but starting states which are different
due to a different choice of 6X, may lead to different
ﬁnal states. Figures 3a and 3b show two typical paths
obtained with different 6,“. Although different, both
are good solutions to the problem. Figure 3a is also the best path, found by exhaustive search of all paths in a
separate calculation. To see how well a selection was being made of good
paths, we compare the paths chosen by the network
with the lengths of all possible paths. There are 10!/20
= 181,440 total distinct paths, and a histogram of their
length distribution is shown in Fig. 5a. The paths
found in 20 convergences from random states are also
shown as the histogram (* symbols) below the xaxis.
(Of these 20 starting states, 16 converged to legitimate
tours.) About 50% of the trials produced one of the 2
shortest paths. Hence the network did an excellent job
of selecting a good path, preferring paths in the best
10‘5 of all paths compared to random paths. Because a typical biological neuron may be con
nected to 1000—10,000 others, it is relevant to investi
gate how the computational power of the network
grows with the number of neurons. We therefore
studied a 900 neuron system describing a TSP on
30 cities. Because the time to simulate the differential
equations in a digital computer scales somewhat worse
than n3, our results are fragmentary. We are not yet
well located in parameter space and parameter choice
seems to be a more delicate issue with 900 neurons than
with 100. The particular set of 30 random cities we
used 3 are believed to have the minimum path length of
4.26 for the path shown in Fig. 4b. The 30city system
converged to paths of length less than 7 commonly,
and less than 6 occasionally. For 30 cities, there are
30!/60=4.4><1030 paths. A direct evaluation of the
length of 105 random paths found an average of 12.5,
and none shorter than 9.5. A path of about average
path length is shown in Fig. 4a. The path length
histogram of the random sampling is shown in Fig. 5b.
From a statistical estimate and the known shortest
path, there should be about 108 paths shorter than
length 7. Thus, in a single convergence, the network
provided a very good solution to the problem, exclud
ing poor paths by a factor of 10—22 to 10—23. V The Computational Process The collective computations we have described using
nonlinear analog circuits have aspects from both
conventional digital and analog computers. In conven
tional analog computation, the differential equation
which is solved by the electrical circuit is generally the
same equation that the programmer wishes to solve in
the real world (Tomovic and Karplus, 1962). The variables in the analog computer are closely related to 3 The list of 30 cities used in these experiments and the solution
shown in Fig. 4b computed using the Lin/Kernighan algorithm
(Lin and Kernighan, 1973) were provided by David Johnson the realworld variables whose behavior is sought. In
the present case, however, the circuit differential
equation to be solved is of no intrinsic interest. This
differential equation is essentially a program by which
an answer to a question can be found. Digital compu
tation conventionally involves ﬁnding a data represen
tation and algorithm for the problem, by which the
ﬁxed hardware will eventually construct the desired
answer. The collective mode we have described com
bines the programming and data representation as
pects characteristic of digital computation, but re
places the usual stereotyped logical behavior of a
digital system by a stereotyped form of analog
computation. Why is the computation so effective? The solution
to a TSP problem is a path and the decoding of the
TSP network’s ﬁnal stable state to obtain this discrete
decision or solution requires having the ﬁnal VXj values
be near 0 or 1. However, the actual analog computa
tion occurs in the continuous domain, 0 g VXJ. g 1. The
decisionmaking process or computation consists of
the smooth motion from an initial state on the interior
of the space (where the notion of a “tour” is not even
deﬁned) to an ultimate stable point near enough to a
corner of the continuous domain to be able to identify
with that corner. It is as though the logical operations
of a calculation could be given continuous values
between “True” and “False”, and evolve toward cer—
tainty only near the end of the calculation. Although there may be no precise “tour” interpre
tation of a state vector which does not have the form
[Eq. (7)], a qualitative interpretation can be made.
Suppose row C has an appreciable value in columns 5
and 6 and nowhere else, and no other row A has much
greater value in these same columns. Then it might be
said that city C was being considered (simultaneously)
for both position 5 and position 6, that other possi
bilities were not as likely, and that later in time this
positional ambiguity should be resolved. Figure 2a is
an illustration of an intermediate time state in a TSP
calculation on 10 cities using random noise initial
conditions. At this stage of the calculation, it appears
that A wants to be in position 6 or 7 in the tour. Cities
B, C, and D want to be in positions 9, 10, or 1, but it is
not at all clear which pairing of B, C, D with 9, 10, 1 will
be present in the ﬁnal state. Similarly, position 5 is
going to be captured by either city F or E, but again the
order is not clear. A decision is already apparent as to
roughly where on the tour various cities will be, and
this is of itself important information to convey to the
other cities: it suggests restrictions on the possibilities
which these others should be considering. The rough
assignments in this example are plausible, as can be
seen from looking at the 10 city map in Fig. 3a. Indeed,
the computation works because the intermediate states 149 so interpreted are reasonable. Though not precisely
deﬁned in terms of a tour, they represent the simulta—
neous consideration of many similar tours. Interpre
ted in this way, during a convergence, the network
moves from states corresponding to very roughly
deﬁned tours to states of higher reﬁnement, until a
single tour is left. This general computational strategy
will work well in optimization problems for which
good solutions cluster, and each excellent solution has
many almost as good which are similar to it. In a direct test of the contribution which
intermediatestate analog processing makes to the
ability of the computational network to solve the TSP
problem, separate simulations of a 10—city problem
were performed using a deterministic network which
minimized E using a decision space which consisted
only of the corners of the 2” dimensional cube. Such a
procedure led to tours little better than random. An
example of a solution for the 10city problem is shown
in Fig. 30. Thus the analog characteristics and inter
mediate state processing are important for good TSP
solutions. Unlike our analog network procedure, Kirkpatrick
has approached constraint satisfaction problems on a
discrete decision space by a Monte Carlo approach
using an effective temperature and an annealing proce
dure (Kirkpatrick et al., 1983). This “simulated anneal
ing” method has several important features. First, it
causes many conﬁgurations to be averaged near a
given one, which has the effect of smoothing the surface
along which a search is being done. This prevents the
system from becoming stuck in minor energy minima,
since these are smoothed out. Second, it gives the
possibility of climbing out of a local minimum into
another one if the annealing goes on long enough. (As
the problem size gets larger, the truly best solution to a
problem using simulated annealing is generally not
found because the annealing procedure would take an
inﬁnite amount of time.) The analog procedure used by
the computational networks also smooths the energy
surface during the search but does not allow recovery
from local minima in the solution space. Through the
“spin representation” we have constructed in earlier
work (Hopﬁeld, 1982), there is a direct means of
showing the smoothing effect. Consider the effective
ﬁeld solution to the expectation value of for a set of
Ising spins, each restricted to a value V,=0 or 1, at
temperature T with an energy E as in Eq. (8) and
Eq. (9). Effective ﬁeld models (see Wannier, 1966)
replace a variable (such as a particular V,) which occurs
in an energy by its expectation value (Vi) when
evaluating the probability distribution of any other
variable. This wellknown approximation allows the
statistical mechanics of complicated systems to be
approximated by a closed set of equations relating the 150 expectation values. For the TSP problem, these equa
tions are (Vi) =e+Hi/"T/(l +e+H'/"T)= % [l +tanh(H,/2kT)]
H.= 272x19» (13) If the analog system, with the gain function used, is
allowed to come to equilibrium, one has tit/T:ZEMﬂ‘lmﬁfZEﬂ/Frhi (14)
j j 01‘ V} = g(rh,~) = %(l + tanhrhi/uo). The solution to our analog system at gain width u0 is
thus equivalent to an effective ﬁeld solution at temper
ature kT=uo/21. This relation between the present
system and effective ﬁeld models of spin systems forms
a basis for understanding why the interior of the
continuous decision space is smoothly related to the
corners. In some spin glass problems, the effective ﬁeld
description followed continuously from high tempera
tures is expected to lead to a state near the thermody
namic ground state (Thouless et al., 1977; Gross and
Mezard, 1984). In the analog network system de
scribed, the circuit does not literally follow the effective
ﬁeld solution from high temperatures, but instead
jumps to the solution at a particular temperature by a
dynamic means. In general this need not be the same
branch of the effective ﬁeld solution that would be
obtained by equilibrium slow cooling. However, in
addition to the smoothing effects which the analog
system has at ﬁxed gain widths, a computation ana
logous to following effective ﬁeld solutions from high
temperatures can also be performed by slowly turning
up the analog gain (decreasing uO and hence decreasing
the effective “temperature”) from an initially low value.
This fonn of “annealing” has been applied to the
30city TSP problem. It results in even better compu
tational performance (Fig. 4c), though it has been only
slightly studied. VI Heuristics and Variants Heuristics are rules of thumb which are not necessarily
true but which are helpful guides to ﬁnding solutions.
Digital computer algorithms which are designed to
solve difﬁcult computational problems frequently
make use of heuristics. They can be added to analog
computational networks by changing the connection
matrix, changing the initial state, and changing steady
state inputs. For example, in the TSP problem in two
spatial dimensions on random cities, inspection shows
that the best pathways essentially always connect a city to one if its four nearest neighbors. A heuristic
embodying this rule can be added by replacing the true
distance between cities which are further apart than
that by augmented distances. This will lessen the
possibility that such an unnecessary link occurs in a
ﬁnal path, while maintaining the appropriate distance
measure for good paths. Another possible heuristic is
that, in the usual TSP problem on a planar two
dimensional surface, if cities A and B are as far apart as
possible, they will tend to occur near opposite sites of
the tour, i.e., position m and near m + n/2. Correlations
in the noise put into the initial state or modiﬁcations of
the connection matrix can add this heuristic to the
system. A person looking at the 10—city TSP problem
quickly ﬁnds a very good path, and one might therefore
feel that it is an easy problem. Our ability to do so is
based on the fact that all the relevant relationships can
be seen in a twodimensional drawing. No such
capacity is available to the analog network. The
problem the network solves has no necessity of being
geometrically “ﬂat” or even of being described by a
spatial geometry. The same numerical set of 45 (in
used in the 10city TSP already described can be
randomly assigned to letter pairs X, Y. If this is done,
no geometric representation of the problem is possible,
and our ability to solve the problem visually com
pletely disappears. In computer simulations of the
lOcity TSP we found that the network ﬁnds this
problem somewhat more difﬁcult, but nevertheless
typically converges to solutions among the best
60 paths. Although we have chosen to illustrate the capa
bilities of analog computational networks using the
TSP, the applicability of the methods to other prob
lems appears broad. A variety of seemingly unrelated
problems can be mapped onto the analog network. A
simple example is the transposition code problem.
Given an alphabet A Z and a message in English
which is written in a transposition code, (i.e., A<—>i,
RHj, CHI, code lij=word CAR), ﬁnd the code. To
solve the problem, let P A: the frequency of letter A in English
P A B2 the frequency of letter sequence AB in English
Pi: the frequency of letter i in the message PU: the frequency of sequence ij in the message. A state matrix of the form of Eq. (7) describes a labeling
of each particular row by the particular column in
which that row contains a 1. If the column numbers are
replaced by the code letters, then each such matrix
describes a l : l correspondance between English let
ters and code word letters. Thus a permutation matrix describes a transposition code. Therefore the appropri
ate energy function needs the same A, B, C terms as
Eq. (8) and in addition DAZB(PAB—Pij)2VAiVBj+E;(PA~Pi)2V/ti (15)
“J This energy function should suffice to ﬁnd the “correct”
codes, or at least one nearly correct. While it would be
nice also to use higher order correlations of letter
frequency, the problem of implementing triples Tm, in
hardware is severe. The transposition code problem
can thus be mapped onto a network using a slightly
modiﬁed TSP E function. Other problems can also be
mapped onto the network, but utilize quite different
speciﬁc E functions. For example, we have found
constructions by which the Vertex Covering Problem
(see for example Garey and Johnson, 1979) and the best
match with gaps between two linear sequences prob
lem can be mapped onto analog computational
circuits. VII Conclusion Both analog networks of biological neurons and
networks of microelectronic neurons could rapidly
solve difﬁcult optimization problems using the
strategies we have presented. “Rapid” is measured on
the time scale of the devices used. Since the conver
gence time of the network will be a few characteristic
times of the devices from which it is built, one may
expect convergence times of 10—100 ms for networks of
biological neurons, while semiconductor circuits
should converge in 10—5 to 10’7s. The time scale
expected for biological systems is consistent with the
known computation times in perception and pattern
recognition problems which organisms solve quickly. The power of the computation carried out is
demonstrated by the selection it makes between the
possible answers it might give. In the TSP network
consisting of 100 neurons, the selectivity was 10‘4 to
10‘ 5. This is the fraction of all possible solutions which
were put forward by the network as putative “best”
answers. Since there were only about 2 x 105 possible
paths, one of the best few was always selected. The computational power of the TSP network
scales favorably with the size of the system. Under the
most favorable circumstances, (and only then) the
computing power, as measured by the selectivity
deﬁned above should be raised to the power of at when
the size of the system is multiplied by a. We might then
have expected a selectivity of (10""5)+9 =1O‘3’9'5 for
the 900 neuron system. The actual selectivity of about
10‘22 corresponds to (10’4'5)+5 or thus corresponds
to the scaling expected for the ideal case and 500 151 neurons. This should be contrasted with the case when
the computation is not truly done in parallel and
collectively, but is instead simply partitioned. In this
case, the selectivity would change by a factor of l/oc,
and hence would only be about 10’ 5'5 for the 30city
problem. The combination of speed and power of the
computational networks is based on the analog char
acter of the devices involved. Real neurons have the
kind of response characteristic used here, and it is to be
expected that biology will make use of that fact.
“Simulated Annealing” (Kirkpatrick et al., 1983) on a
digital computer or in models using twostate neurons
(Hinton and Sejnowski, 1983) is intrinsically slow
when measured in units of the time constants of devices
from which the computer is constructed because of the
long time necessary to calculate conﬁgurational aver
ages and to climb from one valley into another. This
approach ignores the very important use that can be
made of analog variables to represent probabilities,
expectation values, or the superposition of many
possibilities. Making use of the analog variables seems
a key to the combination of high speed and compu—
tational power in real networks. It was not necessary to
plan such a use — the real physical systems naturally
perform in this fashion. The inputs in the TSP problem (the distances
between cities) occur as a modulation of the connec
tions between neurons. This form of input is rather
different in concept from the usual way of Viewing the
inputs as additively driving a processing network. In
real neurons, such a modulation could be done, for
example, by attenuating distal signals in the dendritic
arbor (Koch et al., 1983) by means of a proximal
inhibitory shunting input. This new mechanism of
inputing information is both biophysically reasonable
and computationally effective. The elements of the computational networks we
have described were given properties that biological
neurons are known to possess, particularly the large
connectivity and analog character. It is difﬁcult to
imagine a system which would more efficiently solve
such complex problems using a small number of
“neurons”. Because many recognition tasks and per
ception problems can be set in the form of a con
strained optimum with combinatorial complexity, the
effectiveness of neural computation in these problems
may rely on casting the optimization problem into a
format which can be done collectively by a network. Although we have demonstrated remarkable com
putational power in networks of simple neurons, real
neurons are rather more complex. However, adding additional features to the neurons comprising the network should increase the complexity of a computa
tional task which the network can do. Nevertheless, it 152 should be recognized that the working together of an
entire nervous system involves a host of additional
features, including hierarchy, anatomy, wiring limi
tations, nonreciprocal connections, and propagation
delays. The present work describes only the simulation
of a partial, but powerful, computation which a
module of intensely interconnected very simple
neurons might perform. VI I 1 Acknowledgements. The authors thank A. Gelperin and H.
Berg for critical readings of the manuscript in draﬁ.The work at Cal Tech was supported in part by National Science Foundation
grant PCM8406049. References Anderson, P.W.: Ch. 2. In: Basic notions of condensed matter
physics. Menlo Park: Benjamin Cummings 1984 Arbib, M.A.: Artiﬁcial intelligence and brain theory: unities and
diversities. Ann. Biomed. Eng. 3, 238—274 (1975) Ballard, D.H.: Cortical connections and parallel processing:
structure and function. Behav. Brain Sci. (1985, in press) Ballard, D.H., Hinton, G.E., Sejnowski, T.J.: Parallel visual
computation. Nature 306, 21726 (1983) Feldman, J .A., Ballard, D.H.: Connectionist models and their
properties. Cog. Sci. 6, 2057254 (1982) Garey, M.R., Johnson, D.S.: Computers and intractability. New
York: W.H. Freeman 1979 Gelperin, A.E., Hopﬁeld, J.J., Tank, D.W.: The logic of Limax
learning. In: Model neural networks and behavior. Selver
ston, A., ed.. New York: Plenum Press 1985 GoldmanRakic, P.S.: Modular organization of the prefrontal
cortex. Trends Neurosci. 7, 419—424 (1984) Gross, DJ ., Mezard, M.: The simplest spin glass. Nucl. Phys. FS
3240, 431—452 (1984) Hinton, G.E., Sejnowski, T.J.: Optimal perceptual inference. In:
Proc. IEEE Comp. Soc. Conf. Comp. Vision & Pattern
Recog., pp.448—453, Washington, DC, June 1983 Hopﬁeld, J .J .: Neural networks and physical systems with
emergent collective computational abilities. Proc. Natl.
Acad. Sci. USA 79, 2554—2558 (1982) Hopﬁeld, J .J .: Neurons with graded response have collective
computational properties like those of twostate neurons.
Proc. Natl. Acad. Sci. USA 81, 308873092 (1984) Huskey, H.D., Korn, G.A.: Computer Handbook, pp. 14. New
York: McGraw Hill 1962 Ikeuchi, K., Horn, B.K.P.: Numerical shape from shading and
occluding boundaries. Artif. Intell. 17, 1417184 (1981)
Jackson, W.A.: Analog computation, pp. 578. New York: McGraw Hill 1960 J ulesz, B.: In: Foundations of cyclopean vision. Chicago: Univer
sity of Chicago Press 1971 J ulesz, B.: Textons, the elements of texture perception, and their
interactions. Nature 290, 91—97 (1981) Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by
simulated annealing. Science 220, 671—680 (1983) Koch, C., Poggio, T., Torre, V.: Nonlinear interactions in a
dendritic tree: Localization, timing, and role in information
processing. Proc. Natl. Acad. Sci. USA 80, 2799 (1983) Lawler, E.L., Lenstra, A., Rinnooy Kan, H.G., Eds.: The traveling
salesman problem. New York: John Wiley (in press) Lin, 8., Kernighan, B.W.: An effective heuristic algorithm for the
travelingsalesman problem. Oper. Res. 21, 4987516 (1973) Marr, D.: Vision. New York: W. H. Freeman 1982 Mead, C., Conway, L.: Introduction to VLSI systems, pp. 12, 265.
Reading, MA: AddisonWesley 1980 Poggio, T., Koch, C.: An analog model of computation for the ill
posed problems of early vision. M.I.T. Artif. Intell. Lab. A.I.
Memo No. 783, 1984 Poggio, T., Torre, V.: Ill—posed problems in early vision. Proc.
Roy. B. (1985, in press) Sebestyn, G.S.: Decisionmaking processes in pattern recog
nition. New York: McMillan 1962 Shepherd, G.M.: Microcircuits in the nervous system. Sci. Am.
238, 924103 (1978) Terzopoulos, D.: Multilevel reconstruction of visual surfaces:
variational principles and ﬁnite element representations. In:
Multiresolution image processing and analysis. Rosenfeld,
A., ed.. Berlin, Heidelberg, New York: Springer 1984 Thouless, D.J., Anderson, P.W., Palmer, R.G.: Solution of
“solvable model of a spin glass”. Phil. Mag. 35, 593—601
(1977) Tomovic, R., Karplus, W.J.: High speed analog computers, p. 4.
New York: Wiley 1962 Wannier, G.H.: Statistical physics, p. 3301f. New York: Wiley
1966 Received: February 16, 1985 Prof. J. J. Hopﬁeld Division of Chemistry and Biology
California Institute of Technology
Pasadena, CA 91125 USA ...
View
Full Document
 Fall '09

Click to edit the document details