Proc. NatL Acad. Sci. USA Vol. 79, pp. 2554-2558, April 1982 Biophysics Neural networks and physical systems with emergent collective computational abilities (associative memory/parallel processing/categorization/content-addressable memory/fail-soft devices) J. J. HOPFIELD Division of Chemistry and Biology, California Institute of Technology, Pasadena, California 91125 and Bell Laboratories, Murray Hill, New Jersey 07974 Contributed by John J. Hopfweld, January 15, 1982 ABSTRACT Computational properties of use to biological or- ganisms or to the construction of computers can emerge as col- lective properties of systems -having a large number of simple equivalent components (or neurons). The physical meaning ofcon- tent-addressable memory is described by an appropriate phase space flow of the state of a system. A model of such a system is given, based on aspects of neurobiology but readily adapted to in- tegrated circuits. The collective properties of this model produce a content-addressable memory which correctly yields an entire memory from any subpart of sufficient size. The algorithm for the time evolution of the state of the system is based on asynchronous parallel processing. Additional emergent collective properties in- clude some capacity for generalization, familiarity recognition, categorization, error correction, and time sequence retention. The collective properties are only weakly sensitive to details ofthe modeling or the failure of individual devices. Given the dynamical electrochemical properties ofneurons and their interconnections (synapses), we readily understand schemes that use a few neurons to obtain elementary useful biological behavior (1-3). Our understanding of such simple circuits in electronics allows us to plan larger and more complex circuits which are essential to large computers. Because evolution has no such plan, it becomes relevant to ask whether the ability of large collections of neurons to perform "computational" tasks may in part be a spontaneous collective consequence of having a large number of interacting simple neurons. In physical systems made from a large number of simple ele- ments, interactions among large numbers of elementary com- ponents yield collective phenomena such as the stable magnetic orientations and domains in a magnetic system or the vortex patterns in fluid flow. Do analogous collective phenomena in a system of simple interacting neurons have useful "computa- tional" correlates? For example, are the stability of memories, the construction of categories of generalization, or time-se- quential memory also emergent properties and collective in origin? This paper examines a new modeling ofthis old and fun- damental question (4-8) and shows that important computa- tional properties spontaneously arise. All modeling is based on details, and the details of neuro- anatomy and neural function are both myriad and incompletely known (9). In many physical systems, the nature of the emer- gent collective properties is insensitive to the details inserted in the model (e.g., collisions are essential to generate sound waves, but any reasonable interatomic force law will yield ap- propriate collisions). In the same spirit, I will seek collective properties that are robust against change in the model details. The model could be readily implemented by integrated cir- cuit hardware. The conclusions suggest the design of a delo- calized content-addressable memory or categorizer using ex- tensive asynchronous parallel processing. The general content-addressable memory of a physical system Suppose that an item stored in memory is "H. A. Kramers & G. H. Wannier Phys. Rev. 60, 252 (1941)." A general content- addressable memory would be capable of retrieving this entire memory item on the basis of sufficient partial information. The input "& Wannier, (1941)" might suffice. An ideal memory could deal with errors and retrieve this reference even from the input "Vannier, (1941)". In computers, only relatively simple forms ofcontent-addressable memory have been made in hard- ware (10, 11). Sophisticated ideas like error correction in ac- cessing information are usually introduced as software (10). There are classes of physical systems whose spontaneous be- havior can be used as a form of general (and error-correcting) content-addressable memory. Consider the time evolution of a physical system that can be described by a set of general co- ordinates. A point in state space then represents the instanta- neous condition of the system. This state space may be either continuous or discrete (as in the case of N Ising spins). The equations ofmotion ofthe system describe a flow in state space. Various classes offlow patterns are possible, but the sys- tems of use for memory particularly include those that flow to- ward locally stable points from anywhere within regions around those points. A particle with frictional damping moving in a potential well with two minima exemplifies such a dynamics. If the flow is not completely deterministic, the description is more complicated. In the two-well problems above, if the frictional force is characterized by atemperature, it must also produce a random driving force. The limit points become small limiting regions, and the stability becomes not absolute. But as long as the stochastic effects are small, the essence of local stable points remains. Consider a physical system described by many coordinates X1 XN, the components of a state vector X. Let the system have locally stable limit points Xa, Xb, **. Then, if the system is started sufficiently near any Xa, as at X = Xa + A, it will proceed in time until X Xa. We can regard the information stored in the system as the vectors Xa, Xb, . The starting point X = Xa + A represents a partial knowledge of the item Xa, and the system then generates the total information Xa. Any physical system whose dynamics in phase space is dom- inated by a substantial number of locally stable states to which it is attracted can therefore be regarded as a general content- addressable memory. The physical system will be a potentially useful memory if, in addition, any prescribed set of states can readily be made the stable states of the system. The model system The processing devices will be called neurons. Each neuron i has two states like those of McCullough and Pitts (12): Vi = 0 2554 The publication costs ofthis article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertise- ment" in accordance with 18 U. S. C. ��1734 solely to indicate this fact.

Proc. Natl. Acad. Sci. USA 79 (1982) 2555 ("not firing") and Vi = 1 ("firing at maximum rate"). When neu- ron i has a connection made to it from neuron j, the strength of connection is defined as Tij. (Nonconnected neurons have Tij 0.) The instantaneous state ofthe system is specified by listing the N values of Vi, so it is represented by a binary word of N bits. The state changes in time according to the following algo- rithm. For each neuron i there is a fixed threshold U,. Each neuron i readjusts its state randomly in time but with a mean attempt rate W, setting Vi �� 1 Ui ] Vi0if IT.,V. joi Thus, each neuron randomly and asynchronously evaluates whether it is above or below threshold and readjusts accord- ingly. (Unless otherwise stated, we choose Ui = 0.) Although this model has superficial similarities to the Per- ceptron (13, 14) the essential differences are responsible for the new results. First, Perceptrons were modeled chiefly with neural connections in a "forward" direction A - B -* C -- D. The analysis of networks with strong backward coupling proved intractable. All our interesting results arise as consequences of the strong back-coupling. Second, Percep- tron studies usually made a random net ofneurons deal directly with a real physical world and did not ask the questions essential to finding the more abstract emergent computational proper- ties. Finally, Perceptron modeling required synchronous neu- rons like a conventional digital computer. There is no evidence for such global synchrony and, given the delays of nerve signal propagation, there would be no way to use global synchrony effectively. Chiefly computational properties which can exist in spite of asynchrony have interesting implications in biology. The information storage algorithm Suppose we wish to store the set of states V8, s = 1 n. We use the storage prescription (15, 16) Tij= (2V - 1)(2Vj - 1) [2] S but with Tii = 0. From this definition Tijjs =E (2V,- 1) I VJ(2Vj-1) Hjs. [3] The mean value of the bracketed term in Eq. 3 is 0 unless s - s', for which the mean is N/2. This pseudoorthogonality yields TiVs (Hs') (2Vs' - 1) N/2 i [4] and is positive if VW' = 1 and negative if Vf' = 0. Except for the noise coming from the s # s' terms, the stored state would al- ways be stable under our processing algorithm. Such matrices T,. have been used in theories of linear asso- ciative nets (15-19) to produce an output pattern from a paired input stimulus, S1 -* 01. A second association S2 -�� 02 can be simultaneously stored in the same network. But the confusing simulus 0.6 Si + 0.4 S2 will produce a generally meaningless mixed output 0.6 01 + 0.4 02 Our model, in contrast, will use its strong nonlinearity to make choices, produce categories, and regenerate information and, with high probability, will generate the output 01 from such a confusing mixed stimulus. A linear associative net must be connected in a complex way with an external nonlinear logic processor in order to yield true computation (20, 21). Complex circuitry is easy to plan but more difficult to discuss in evolutionary terms. In contrast, our model obtains its emergent computational properties from simple properties of many cells rather than circuitry. The biological interpretation of the model Most neurons are capable of generating a train of action poten- tials-propagating pulses ofelectrochemical activity-when the average potential across their membrane is held well above its normal resting value. The mean rate at which action potentials are generated is a smooth function of the mean membrane po- tential, having the general form shown in Fig. 1. The biological information sent to other neurons often lies in a short-time average of the firing rate (22). When this is so, one can neglect the details of individual action potentials and regard Fig. 1 as a smooth input-output relationship. [Parallel pathways carrying the same information would enhance the ability of the system to extract a short-term average firing rate (23, 24).] A study of emergent collective effects and spontaneous com- putation must necessarily focus on the nonlinearity of the in- put-output relationship. The essence of computation is nonlin- ear logical operations. The particle interactions that produce true collective effects in particle dynamics come from a nonlin- ear dependence of forces on positions of the particles. Whereas linear associative networks have emphasized the linear central region (14-19) of Fig. 1, we will replace the input-output re- lationship by the dot-dash step. Those neurons whose operation is dominantly linear merely provide a pathway of communica- tion between nonlinear neurons. Thus, we consider a network of "on or off" neurons, granting that some of the interconnec- tions may be by way of neurons operating in the linear regime. Delays in synaptic transmission (of partially stochastic char- acter) and in the transmission of impulses along axons and den- drites produce a delay between the input of a neuron and the generation of an effective output. All such delays have been modeled by a single parameter, the stochastic mean processing time 1/W. The input to a particular neuron arises from the current leaks of the synapses to that neuron, which influence the cell mean potential. The synapses are activated by arriving action poten- tials. The input signal to a cell i can be taken to be [5] I Tijvj where Tij represents the effectiveness of a synapse. Fig. 1 thus / Q ~~~~~~~~~/ 0 , P�� I I' 0 a)-jaz-Present Model W t --Linear Modeling w .'C E -0.1 / 0 Membrane Potential (Volts) or "Input" FIG. 1. Firing rate versus membrane voltage for a typical neuron (solid line), dropping to 0 for large negative potentials and saturating for positive potentials. The broken lines show approximations used in modeling. Biophysics: Hopfield