Misplaced Pages

#473526

69-565: Hebb is a surname. Notable people with the surname include: Hebbian theory in psychology (including Hebb's rule , AKA Hebb's postulate ) Hebbian theory Let us assume that the persistence or repetition of a reverberatory activity (or "trace") tends to induce lasting cellular changes that add to its stability. ... When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A ’s efficiency, as one of

138-488: A Hebbian learning algorithm. One of the key features of Hopfield networks is their ability to recover complete patterns from partial or noisy inputs, making them robust in the face of incomplete or corrupted data. Their connection to statistical mechanics, recurrent networks, and human cognitive psychology has led to their application in various fields, including physics , psychology , neuroscience , and machine learning theory and practice. One origin of associative memory

207-423: A biological basis for errorless learning methods for education and memory rehabilitation. In the study of neural networks in cognitive function, it is often regarded as the neuronal basis of unsupervised learning . Hebbian theory concerns how neurons might connect themselves to become engrams . Hebb's theories on the form and function of cell assemblies can be understood from the following: The general idea

276-406: A certain state V s {\displaystyle V^{s}} and distinct nodes i , j {\displaystyle i,j} w i j = V i s V j s {\displaystyle w_{ij}=V_{i}^{s}V_{j}^{s}} but w i i = 0 {\displaystyle w_{ii}=0} . (Note that

345-592: A certain time, the state of the neural net is described by a vector V {\displaystyle V} , which records which neurons are firing in a binary word of N {\displaystyle N} bits. The interactions w i j {\displaystyle w_{ij}} between neurons have units that usually take on values of 1 or −1, and this convention will be used throughout this article. However, other literature might use units that take values of 0 and 1. These interactions are "learned" via Hebb's law of association , such that, for

414-412: A complete undirected graph G = ⟨ V , f ⟩ {\displaystyle G=\langle V,f\rangle } , where V {\displaystyle V} is a set of McCulloch–Pitts neurons and f : V 2 → R {\displaystyle f:V^{2}\rightarrow \mathbb {R} } is a function that links pairs of units to a real value,

483-407: A content addressable memory system, that is to say, the network will converge to a "remembered" state if it is given only part of the state. The net can be used to recover from a distorted input to the trained state that is most similar to that input. This is called associative memory because it recovers memories on the basis of similarity. For example, if we train a Hopfield net with five units so that

552-404: A correlation matrix is always a positive-definite matrix , the eigenvalues are all positive, and one can easily see how the above solution is always exponentially divergent in time. This is an intrinsic problem due to this version of Hebb's rule being unstable, as in any network with a dominant signal the synaptic weights will increase or decrease exponentially. Intuitively, this is because whenever

621-500: A few. However, while it is possible to convert hard optimization problems to Hopfield energy functions, it does not guarantee convergence to a solution (even in exponential time). Initialization of the Hopfield networks is done by setting the values of the units to the desired start pattern. Repeated updates are then performed until the network converges to an attractor pattern. Convergence is generally assured, as Hopfield proved that

690-421: A huge batch of training data. Hebbian theory was introduced by Donald Hebb in 1949 in order to explain "associative learning", in which simultaneous activation of neuron cells leads to pronounced increases in synaptic strength between those cells. It is often summarized as "Neurons that fire together wire together. Neurons that fire out of sync fail to link". The Hebbian rule is both local and incremental. For

759-481: A new state V s ′ {\displaystyle V^{s'}} is subjected to the interaction matrix, each neuron will change until it matches the original state V s {\displaystyle V^{s}} (see the Updates section below). The connections in a Hopfield net typically have the following restrictions: The constraint that weights are symmetric guarantees that

SECTION 10

#1732791473474

828-424: A new state of neurons V s ′ {\displaystyle V^{s'}} is introduced to the neural network, the net acts on neurons such that where U i {\displaystyle U_{i}} is the threshold value of the i'th neuron (often taken to be 0). In this way, Hopfield networks have the ability to "remember" states stored in the interaction matrix, because if

897-427: A particular action, the individual will see, hear, and feel the performing of the action. These re-afferent sensory signals will trigger activity in neurons responding to the sight, sound, and feel of the action. Because the activity of these sensory neurons will consistently overlap in time with those of the motor neurons that caused the action, Hebbian learning predicts that the synapses connecting neurons responding to

966-483: A pattern. When several training patterns are used the expression becomes an average of individual ones: where w i j {\displaystyle w_{ij}} is the weight of the connection from neuron j {\displaystyle j} to neuron i {\displaystyle i} , p {\displaystyle p} is the number of training patterns and x i k {\displaystyle x_{i}^{k}}

1035-522: A single layer of neurons, where each neuron is connected to every other neuron except itself. These connections are bidirectional and symmetric, meaning the weight of the connection from neuron i to neuron j is the same as the weight from neuron j to neuron i . Patterns are associatively recalled by fixing certain inputs, and dynamically evolve the network to minimize an energy function, towards local energy minimum states that correspond to stored patterns. Patterns are associatively learned (or "stored") by

1104-459: Is a formulaic description of Hebbian learning: (many other descriptions are possible) where w i j {\displaystyle w_{ij}} is the weight of the connection from neuron j {\displaystyle j} to neuron i {\displaystyle i} and x i {\displaystyle x_{i}} the input for neuron i {\displaystyle i} . Note that this

1173-470: Is a zero-centered sigmoid function. The complex Hopfield network, on the other hand, generally tends to minimize the so-called shadow-cut of the complex weight matrix of the net. Hopfield nets have a scalar value associated with each state of the network, referred to as the "energy", E , of the network, where: This quantity is called "energy" because it either decreases or stays the same upon network units being updated. Furthermore, under repeated updating

1242-478: Is also diagonalizable , and the solution can be found, by working in its eigenvectors basis, to be of the form where k i {\displaystyle k_{i}} are arbitrary constants, c i {\displaystyle \mathbf {c} _{i}} are the eigenvectors of C {\displaystyle C} and α i {\displaystyle \alpha _{i}} their corresponding eigen values. Since

1311-483: Is an elementary form of unsupervised learning, in the sense that the network can pick up useful statistical aspects of the input, and "describe" them in a distilled way in its output. Despite the common use of Hebbian models for long-term potentiation, Hebb's principle does not cover all forms of synaptic long-term plasticity. Hebb did not postulate any rules for inhibitory synapses, nor did he make predictions for anti-causal spike sequences (presynaptic neuron fires after

1380-513: Is an old one, that any two cells or systems of cells that are repeatedly active at the same time will tend to become 'associated' so that activity in one facilitates activity in the other. Hebb also wrote: When one cell repeatedly assists in firing another, the axon of the first cell develops synaptic knobs (or enlarges them if they already exist) in contact with the soma of the second cell. [D. Alan Allport] posits additional ideas regarding cell assembly theory and its role in forming engrams, along

1449-471: Is highly likely for the energy function of the SK model to have many local minima. In the 1982 paper, Hopfield applied this recently developed theory to study the Hopfield network with binary activation functions. In a 1984 paper he extended this to continuous activation functions. It became a standard model for the study of neural networks through statistical mechanics. A major advance in memory storage capacity

SECTION 20

#1732791473474

1518-551: Is human cognitive psychology , specifically the associative memory . Frank Rosenblatt studied "close-loop cross-coupled perceptrons", which are 3-layered perceptron networks whose middle layer contains recurrent connections that change by a Hebbian learning rule. Another model of associative memory is where the output does not loop back to the input. W. K. Taylor proposed such a model trained by Hebbian learning in 1956. Karl Steinbuch , who wanted to understand learning, and inspired by watching his children learn, published

1587-436: Is not included in the traditional Hebbian model. Hebbian learning and spike-timing-dependent plasticity have been used in an influential theory of how mirror neurons emerge. Mirror neurons are neurons that fire both when an individual performs an action and when the individual sees or hears another perform a similar action. The discovery of these neurons has been very influential in explaining how individuals make sense of

1656-419: Is pattern learning (weights updated after every training example). In a Hopfield network , connections w i j {\displaystyle w_{ij}} are set to zero if i = j {\displaystyle i=j} (no reflexive connections allowed). With binary neurons (activations either 0 or 1), connections would be set to 1 if the connected neurons have the same activation for

1725-1150: Is said to follow the Storkey learning rule if it obeys: w i j ν = w i j ν − 1 + 1 n ϵ i ν ϵ j ν − 1 n ϵ i ν h j i ν − 1 n ϵ j ν h i j ν {\displaystyle w_{ij}^{\nu }=w_{ij}^{\nu -1}+{\frac {1}{n}}\epsilon _{i}^{\nu }\epsilon _{j}^{\nu }-{\frac {1}{n}}\epsilon _{i}^{\nu }h_{ji}^{\nu }-{\frac {1}{n}}\epsilon _{j}^{\nu }h_{ij}^{\nu }} where h i j ν = ∑ k = 1   :   i ≠ k ≠ j n w i k ν − 1 ϵ k ν {\displaystyle h_{ij}^{\nu }=\sum _{k=1~:~i\neq k\neq j}^{n}w_{ik}^{\nu -1}\epsilon _{k}^{\nu }}

1794-410: Is the correlation matrix of the input under the additional assumption that ⟨ x ⟩ = 0 {\displaystyle \langle \mathbf {x} \rangle =0} (i.e. the average of the inputs is zero). This is a system of N {\displaystyle N} coupled linear differential equations. Since C {\displaystyle C} is symmetric , it

1863-453: Is the largest eigenvalue of C {\displaystyle C} . At this time, the postsynaptic neuron performs the following operation: Because, again, c ∗ {\displaystyle \mathbf {c} ^{*}} is the eigenvector corresponding to the largest eigenvalue of the correlation matrix between the x i {\displaystyle x_{i}} s, this corresponds exactly to computing

1932-409: Is the mathematical model of Harry Klopf . Klopf's model reproduces a great many biological phenomena, and is also simple to implement. Because of the simple nature of Hebbian learning, based only on the coincidence of pre- and post-synaptic activity, it may not be intuitively clear why this form of plasticity leads to meaningful learning. However, it can be shown that Hebbian plasticity does pick up

2001-402: Is unstable. Therefore, network models of neurons usually employ other learning theories such as BCM theory , Oja's rule , or the generalized Hebbian algorithm . Regardless, even for the unstable solution above, one can see that, when sufficient time has passed, one of the terms dominates over the others, and where α ∗ {\displaystyle \alpha ^{*}}

2070-623: The k {\displaystyle k} -th input for neuron i {\displaystyle i} . This is learning by epoch (weights updated after all the training examples are presented), being last term applicable to both discrete and continuous training sets. Again, in a Hopfield network, connections w i j {\displaystyle w_{ij}} are set to zero if i = j {\displaystyle i=j} (no reflexive connections). A variation of Hebbian learning that takes into account phenomena such as blocking and many other neural learning phenomena

2139-525: The Lernmatrix in 1961. It was translated to English in 1963. Similar research was done with the correlogram of D. J. Willshaw et al. in 1969. Teuvo Kohonen trained an associative memory by gradient descent in 1974. Another origin of associative memory was statistical mechanics . The Ising model was published in 1920s as a model of magnetism, however it studied the thermal equilibrium, which does not change with time. Roy J. Glauber in 1963 studied

Hebb - Misplaced Pages Continue

2208-747: The Nobel Prize in Physics for their foundational contributions to machine learning, such as the Hopfield network. The units in Hopfield nets are binary threshold units, i.e. the units only take on two different values for their states, and the value is determined by whether or not the unit's input exceeds its threshold U i {\displaystyle U_{i}} . Discrete Hopfield nets describe relationships between binary (firing or not-firing) neurons 1 , 2 , … , i , j , … , N {\displaystyle 1,2,\ldots ,i,j,\ldots ,N} . At

2277-546: The Hebbian learning rule takes the form w i j = ( 2 V i s − 1 ) ( 2 V j s − 1 ) {\displaystyle w_{ij}=(2V_{i}^{s}-1)(2V_{j}^{s}-1)} when the units assume values in { 0 , 1 } {\displaystyle \{0,1\}} .) Once the network is trained, w i j {\displaystyle w_{ij}} no longer evolve. If

2346-405: The Hopfield network can be performed in two different ways: The weight between two units has a powerful impact upon the values of the neurons. Consider the connection weight w i j {\displaystyle w_{ij}} between two neurons i and j. If w i j > 0 {\displaystyle w_{ij}>0} , the updating rule implies that: Thus,

2415-651: The Hopfield networks, it is implemented in the following manner when learning n {\displaystyle n} binary patterns: w i j = 1 n ∑ μ = 1 n ϵ i μ ϵ j μ {\displaystyle w_{ij}={\frac {1}{n}}\sum _{\mu =1}^{n}\epsilon _{i}^{\mu }\epsilon _{j}^{\mu }} where ϵ i μ {\displaystyle \epsilon _{i}^{\mu }} represents bit i from pattern μ {\displaystyle \mu } . If

2484-456: The Ising model evolving in time, as a process towards thermal equilibrium ( Glauber dynamics ), adding in the component of time. The second component to be added was adaptation to stimulus. Described independently by Kaoru Nakano in 1971 and Shun'ichi Amari in 1972, they proposed to modify the weights of an Ising model by Hebbian learning rule as a model of associative memory. The same idea

2553-601: The actions of others, by showing that, when a person perceives the actions of others, the person activates the motor programs which they would use to perform similar actions. The activation of these motor programs then adds information to the perception and helps predict what the person will do next based on the perceiver's own motor program. A challenge has been to explain how individuals come to have neurons that respond both while performing an action and while hearing or seeing another perform similar actions. Christian Keysers and David Perrett suggested that as an individual performs

2622-536: The associated probability measure , the Gibbs measure , has the Markov property . Hopfield and Tank presented the Hopfield network application in solving the classical traveling-salesman problem in 1985. Since then, the Hopfield network has been widely used for optimization. The idea of using the Hopfield network in optimization problems is straightforward: If a constrained/unconstrained cost function can be written in

2691-404: The attractors of this nonlinear dynamical system are stable, not periodic or chaotic as in some other systems . Therefore, in the context of Hopfield networks, an attractor pattern is a final stable state, a pattern that cannot change any value within it under updating . Training a Hopfield net involves lowering the energy of states that the net should "remember". This allows the net to serve as

2760-1311: The behavior of any neuron in both discrete-time and continuous-time Hopfield networks when the corresponding energy function is minimized during an optimization process. Bruck showed that neuron j changes its state if and only if it further decreases the following biased pseudo-cut. The discrete Hopfield network minimizes the following biased pseudo-cut for the synaptic weight matrix of the Hopfield net. J p s e u d o − c u t ( k ) = ∑ i ∈ C 1 ( k ) ∑ j ∈ C 2 ( k ) w i j + ∑ j ∈ C 1 ( k ) θ j {\displaystyle J_{pseudo-cut}(k)=\sum _{i\in C_{1}(k)}\sum _{j\in C_{2}(k)}w_{ij}+\sum _{j\in C_{1}(k)}{\theta _{j}}} where C 1 ( k ) {\displaystyle C_{1}(k)} and C 2 ( k ) {\displaystyle C_{2}(k)} represents

2829-453: The bits corresponding to neurons i and j are equal in pattern μ {\displaystyle \mu } , then the product ϵ i μ ϵ j μ {\displaystyle \epsilon _{i}^{\mu }\epsilon _{j}^{\mu }} will be positive. This would, in turn, have a positive effect on the weight w i j {\displaystyle w_{ij}} and

Hebb - Misplaced Pages Continue

2898-707: The cells firing B , is increased. The theory is often summarized as " Neurons that fire together, wire together ." However, Hebb emphasized that cell A needs to "take part in firing" cell B , and such causality can occur only if cell A fires just before, not at the same time as, cell B . This aspect of causation in Hebb's work foreshadowed what is now known about spike-timing-dependent plasticity , which requires temporal precedence. The theory attempts to explain associative or Hebbian learning , in which simultaneous activation of cells leads to pronounced increases in synaptic strength between those cells. It also provides

2967-623: The connectivity weight. Updating one unit (node in the graph simulating the artificial neuron) in the Hopfield network is performed using the following rule: s i ← { + 1 if  ∑ j w i j s j ≥ θ i , − 1 otherwise. {\displaystyle s_{i}\leftarrow \left\{{\begin{array}{ll}+1&{\text{if }}\sum _{j}{w_{ij}s_{j}}\geq \theta _{i},\\-1&{\text{otherwise.}}\end{array}}\right.} where: Updates in

3036-437: The electric output of each neuron is not binary but some value between 0 and 1. He found that this type of network was also able to store and reproduce memorized states. Notice that every pair of units i and j in a Hopfield network has a connection that is described by the connectivity weight w i j {\displaystyle w_{ij}} . In this sense, the Hopfield network can be formally described as

3105-433: The energy function decreases monotonically while following the activation rules. A network with asymmetric weights may exhibit some periodic or chaotic behaviour; however, Hopfield found that this behavior is confined to relatively small parts of the phase space and does not impair the network's ability to act as a content-addressable associative memory system. Hopfield also modeled neural nets for continuous values, in which

3174-1167: The evolution in time of the synaptic weight w {\displaystyle w} : Assuming, for simplicity, an identity response function f ( a ) = a {\displaystyle f(a)=a} , we can write or in matrix form: As in the previous chapter, if training by epoch is done an average ⟨ … ⟩ {\displaystyle \langle \dots \rangle } over discrete or continuous (time) training set of x {\displaystyle \mathbf {x} } can be done: d w d t = ⟨ η x x T w ⟩ = η ⟨ x x T ⟩ w = η C w . {\displaystyle {\frac {d\mathbf {w} }{dt}}=\langle \eta \mathbf {x} \mathbf {x} ^{T}\mathbf {w} \rangle =\eta \langle \mathbf {x} \mathbf {x} ^{T}\rangle \mathbf {w} =\eta C\mathbf {w} .} where C = ⟨ x x T ⟩ {\displaystyle C=\langle \,\mathbf {x} \mathbf {x} ^{T}\rangle }

3243-409: The first principal component of the input. This mechanism can be extended to performing a full PCA (principal component analysis) of the input by adding further postsynaptic neurons, provided the postsynaptic neurons are prevented from all picking up the same principal component, for example by adding lateral inhibition in the postsynaptic layer. We have thus connected Hebbian learning to PCA, which

3312-428: The form of the Hopfield energy function E, then there exists a Hopfield network whose equilibrium points represent solutions to the constrained/unconstrained optimization problem.  Minimizing the Hopfield energy function both minimizes the objective function and satisfies the constraints also as the constraints are "embedded" into the synaptic weights of the network. Although including the optimization constraints into

3381-433: The lines of the concept of auto-association, described as follows: If the inputs to a system cause the same pattern of activity to occur repeatedly, the set of active elements constituting that pattern will become increasingly strongly inter-associated. That is, each element will tend to turn on every other element and (with negative weights) to turn off the elements that do not form part of the pattern. To put it another way,

3450-426: The memory of the Hopfield network. It is desirable for a learning rule to have both of the following two properties: These properties are desirable, since a learning rule satisfying them is more biologically plausible. For example, since the human brain is always learning new concepts, one can reason that human learning is incremental. A learning system that was not incremental would generally be trained only once, with

3519-422: The network will eventually converge to a state which is a local minimum in the energy function (which is considered to be a Lyapunov function ). Thus, if a state is a local minimum in the energy function it is a stable state for the network. Note that this energy function belongs to a general class of models in physics under the name of Ising models ; these in turn are a special case of Markov networks , since

SECTION 50

#1732791473474

3588-404: The neuron y ( t ) {\displaystyle y(t)} is usually described as a linear combination of its input, ∑ i w i x i {\displaystyle \sum _{i}w_{i}x_{i}} , followed by a response function f {\displaystyle f} : As defined in the previous sections, Hebbian plasticity describes

3657-467: The pattern as a whole will become 'auto-associated'. We may call a learned (auto-associated) pattern an engram. Work in the laboratory of Eric Kandel has provided evidence for the involvement of Hebbian learning mechanisms at synapses in the marine gastropod Aplysia californica . Experiments on Hebbian synapse modification mechanisms at the central nervous system synapses of vertebrates are much more difficult to control than are experiments with

3726-408: The piano when listening to piano music. Five hours of piano lessons, in which the participant is exposed to the sound of the piano each time they press a key is proven sufficient to trigger activity in motor regions of the brain upon listening to piano music when heard at a later time. Consistent with the fact that spike-timing-dependent plasticity occurs only if the presynaptic neuron's firing predicts

3795-500: The point of view of artificial neurons and artificial neural networks , Hebb's principle can be described as a method of determining how to alter the weights between model neurons. The weight between two neurons increases if the two neurons activate simultaneously, and reduces if they activate separately. Nodes that tend to be either both positive or both negative at the same time have strong positive weights, while those that tend to be opposite have strong negative weights. The following

3864-425: The post-synaptic neuron's firing, the link between sensory stimuli and motor programs also only seem to be potentiated if the stimulus is contingent on the motor program. Hopfield network A Hopfield network (or associative memory ) is a form of recurrent neural network , or a spin glass system, that can serve as a content-addressable memory . The Hopfield network, named for John Hopfield , consists of

3933-574: The postsynaptic neuron). Synaptic modification may not simply occur only between activated neurons A and B, but at neighboring synapses as well. All forms of hetero synaptic and homeostatic plasticity are therefore considered non-Hebbian. An example is retrograde signaling to presynaptic terminals. The compound most commonly identified as fulfilling this retrograde transmitter role is nitric oxide , which, due to its high solubility and diffusivity, often exerts effects on nearby neurons. This type of diffuse synaptic modification, known as volume learning,

4002-436: The presynaptic neuron excites the postsynaptic neuron, the weight between them is reinforced, causing an even stronger excitation in the future, and so forth, in a self-reinforcing way. One may think a solution is to limit the firing rate of the postsynaptic neuron by adding a non-linear, saturating response function f {\displaystyle f} , but in fact, it can be shown that for any neuron model, Hebb's rule

4071-685: The relatively simple peripheral nervous system synapses studied in marine invertebrates. Much of the work on long-lasting synaptic changes between vertebrate neurons (such as long-term potentiation ) involves the use of non-physiological experimental stimulation of brain cells. However, some of the physiologically relevant synapse modification mechanisms that have been studied in vertebrate brains do seem to be examples of Hebbian processes. One such study reviews results from experiments that indicate that long-lasting changes in synaptic strengths can be induced by physiologically relevant synaptic activity working through both Hebbian and non-Hebbian mechanisms. From

4140-421: The set of neurons which are −1 and +1, respectively, at time k {\displaystyle k} . For further details, see the recent paper. The discrete-time Hopfield Network always minimizes exactly the following pseudo-cut The continuous-time Hopfield network always minimizes an upper bound to the following weighted cut where f ( ⋅ ) {\displaystyle f(\cdot )}

4209-400: The sight, sound, and feel of an action and those of the neurons triggering the action should be potentiated. The same is true while people look at themselves in the mirror, hear themselves babble, or are imitated by others. After repeated experience of this re-afference, the synapses connecting the sensory and motor representations of an action are so strong that the motor neurons start firing to

SECTION 60

#1732791473474

4278-454: The sound or the vision of the action, and a mirror neuron is created. Evidence for that perspective comes from many experiments that show that motor programs can be triggered by novel auditory or visual stimuli after repeated pairing of the stimulus with the execution of the motor program (for a review of the evidence, see Giudice et al., 2009 ). For instance, people who have never played the piano do not activate brain regions involved in playing

4347-443: The state (1, −1, 1, −1, 1) is an energy minimum, and we give the network the state (1, −1, −1, −1, 1) it will converge to (1, −1, 1, −1, 1). Thus, the network is properly trained when the energy of states which the network should remember are local minima. Note that, in contrast to Perceptron training, the thresholds of the neurons are never updated. There are various different learning rules that can be used to store information in

4416-487: The statistical properties of the input in a way that can be categorized as unsupervised learning. This can be mathematically shown in a simplified example. Let us work under the simplifying assumption of a single rate-based neuron of rate y ( t ) {\displaystyle y(t)} , whose inputs have rates x 1 ( t ) . . . x N ( t ) {\displaystyle x_{1}(t)...x_{N}(t)} . The response of

4485-524: The synaptic weights in the best possible way is a challenging task, many difficult optimization problems with constraints in different disciplines have been converted to the Hopfield energy function: Associative memory systems, Analog-to-Digital conversion, job-shop scheduling problem, quadratic assignment and other related NP-complete problems, channel allocation problem in wireless networks, mobile ad-hoc network routing problem, image restoration, system identification, combinatorial optimization, etc, just to name

4554-419: The values of i and j will tend to become equal. The opposite happens if the bits corresponding to neurons i and j are different. This rule was introduced by Amos Storkey in 1997 and is both local and incremental. Storkey also showed that a Hopfield network trained using this rule has a greater capacity than a corresponding network trained using the Hebbian rule. The weight matrix of an attractor neural network

4623-559: The values of neurons i and j will converge if the weight between them is positive. Similarly, they will diverge if the weight is negative. Bruck in his paper in 1990   studied discrete Hopfield networks and proved a generalized convergence theorem that is based on the connection between the network's dynamics and cuts in the associated graph. This generalization covered both asynchronous as well as synchronous dynamics and presented elementary proofs based on greedy algorithms for max-cut in graphs. A subsequent paper further investigated

4692-494: Was developed by Dimitry Krotov and Hopfield in 2016 through a change in network dynamics and energy function. This idea was further extended by Demircigil and collaborators in 2017. The continuous dynamics of large memory capacity models was developed in a series of papers between 2016 and 2020.  Large memory storage capacity Hopfield Networks are now called Dense Associative Memories or modern Hopfield networks . In 2024, John J. Hopfield and Geoffrey E. Hinton were awarded

4761-409: Was published by William A. Little  [ de ] in 1974, who was acknowledged by Hopfield in his 1982 paper. See Carpenter (1989) and Cowan (1990) for a technical description of some of these early works in associative memory. The Sherrington–Kirkpatrick model of spin glass, published in 1975, is the Hopfield network with random initialization. Sherrington and Kirkpatrick found that it

#473526