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127-413: 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 the cells firing B ,

254-447: A content-addressable memory . The Hopfield network, named for John Hopfield , consists of 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

381-469: A myelinated axon , which are found periodically interspersed between segments of the myelin sheath. Therefore, at the point of the node of Ranvier, the axon is reduced in diameter. These nodes are areas where action potentials can be generated. In saltatory conduction , electrical currents produced at each node of Ranvier are conducted with little attenuation to the next node in line, where they remain strong enough to generate another action potential. Thus in

508-421: 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

635-406: 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

762-592: 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

889-412: 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,

1016-407: 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

1143-404: 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

1270-495: 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

1397-421: 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

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1524-482: A myelinated axon, action potentials effectively "jump" from node to node, bypassing the myelinated stretches in between, resulting in a propagation speed much faster than even the fastest unmyelinated axon can sustain. An axon can divide into many branches called telodendria (Greek for 'end of tree'). At the end of each telodendron is an axon terminal (also called a terminal bouton or synaptic bouton, or end-foot ). Axon terminals contain synaptic vesicles that store

1651-425: A nerve in the peripheral nervous system can be described as neurapraxia , axonotmesis , or neurotmesis . Concussion is considered a mild form of diffuse axonal injury . Axonal injury can also cause central chromatolysis . The dysfunction of axons in the nervous system is one of the major causes of many inherited and acquired neurological disorders that affect both peripheral and central neurons. When an axon

1778-514: A neurite, causing it to elongate, will make it become an axon. Nonetheless, axonal development is achieved through a complex interplay between extracellular signaling, intracellular signaling and cytoskeletal dynamics. The extracellular signals that propagate through the extracellular matrix surrounding neurons play a prominent role in axonal development. These signaling molecules include proteins, neurotrophic factors , and extracellular matrix and adhesion molecules. Netrin (also known as UNC-6)

1905-505: A neurite, converting it into an axon. As such, the overexpression of phosphatases that dephosphorylate PtdIns leads into the failure of polarization. The neurite with the lowest actin filament content will become the axon. PGMS concentration and f-actin content are inversely correlated; when PGMS becomes enriched at the tip of a neurite, its f-actin content is substantially decreased. In addition, exposure to actin-depolimerizing drugs and toxin B (which inactivates Rho-signaling ) causes

2032-476: 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

2159-424: 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

2286-427: 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

2413-483: 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}}

2540-403: A presynaptic terminal, it activates the synaptic transmission process. The first step is rapid opening of calcium ion channels in the membrane of the axon, allowing calcium ions to flow inward across the membrane. The resulting increase in intracellular calcium concentration causes synaptic vesicles (tiny containers enclosed by a lipid membrane) filled with a neurotransmitter chemical to fuse with

2667-554: A secreted protein, functions in axon formation. When the UNC-5 netrin receptor is mutated, several neurites are irregularly projected out of neurons and finally a single axon is extended anteriorly. The neurotrophic factors – nerve growth factor (NGF), brain-derived neurotrophic factor (BDNF) and neurotrophin-3 (NTF3) are also involved in axon development and bind to Trk receptors . The ganglioside -converting enzyme plasma membrane ganglioside sialidase (PMGS), which

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2794-459: 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

2921-470: 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

3048-478: 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

3175-482: 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

3302-511: 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

3429-469: Is blocked and neutralized, it is possible to induce long-distance axonal regeneration which leads to enhancement of functional recovery in rats and mouse spinal cord. This has yet to be done on humans. A recent study has also found that macrophages activated through a specific inflammatory pathway activated by the Dectin-1 receptor are capable of promoting axon recovery, also however causing neurotoxicity in

3556-488: Is close to 1 millimeter in diameter, the size of a small pencil lead. The numbers of axonal telodendria (the branching structures at the end of the axon) can also differ from one nerve fiber to the next. Axons in the central nervous system (CNS) typically show multiple telodendria, with many synaptic end points. In comparison, the cerebellar granule cell axon is characterized by a single T-shaped branch node from which two parallel fibers extend. Elaborate branching allows for

3683-430: Is crushed, an active process of axonal degeneration takes place at the part of the axon furthest from the cell body. This degeneration takes place quickly following the injury, with the part of the axon being sealed off at the membranes and broken down by macrophages. This is known as Wallerian degeneration . Dying back of an axon can also take place in many neurodegenerative diseases , particularly when axonal transport

3810-403: Is distinct from somatic action potentials in three ways: 1. The signal has a shorter peak-trough duration (~150μs) than of pyramidal cells (~500μs) or interneurons (~250μs). 2. The voltage change is triphasic. 3. Activity recorded on a tetrode is seen on only one of the four recording wires. In recordings from freely moving rats, axonal signals have been isolated in white matter tracts including

3937-468: 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

Hebbian theory - Misplaced Pages Continue

4064-551: 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

4191-604: Is impaired, this is known as Wallerian-like degeneration. Studies suggest that the degeneration happens as a result of the axonal protein NMNAT2 , being prevented from reaching all of the axon. Demyelination of axons causes the multitude of neurological symptoms found in the disease multiple sclerosis . Dysmyelination is the abnormal formation of the myelin sheath. This is implicated in several leukodystrophies , and also in schizophrenia . A severe traumatic brain injury can result in widespread lesions to nerve tracts damaging

4318-677: 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

4445-507: Is involved in the activation of TrkA at the tip of neutrites, is required for the elongation of axons. PMGS asymmetrically distributes to the tip of the neurite that is destined to become the future axon. During axonal development, the activity of PI3K is increased at the tip of destined axon. Disrupting the activity of PI3K inhibits axonal development. Activation of PI3K results in the production of phosphatidylinositol (3,4,5)-trisphosphate (PtdIns) which can cause significant elongation of

4572-431: 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

4699-431: Is one of the many treatments used for different kinds of nerve injury . Some general dictionaries define "nerve fiber" as any neuronal process , including both axons and dendrites . However, medical sources generally use "nerve fiber" to refer to the axon only. Hopfield network A Hopfield network (or associative memory ) is a form of recurrent neural network , or a spin glass system, that can serve as

4826-419: 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

4953-1145: 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 }}

5080-402: Is the corpus callosum that connects the two cerebral hemispheres , and this has around 20 million axons. The structure of a neuron is seen to consist of two separate functional regions, or compartments – the cell body together with the dendrites as one region, and the axonal region as the other. The axonal region or compartment, includes the axon hillock, the initial segment,

5207-410: 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

Hebbian theory - Misplaced Pages Continue

5334-453: 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

5461-411: Is the area formed from the cell body of the neuron as it extends to become the axon. It precedes the initial segment. The received action potentials that are summed in the neuron are transmitted to the axon hillock for the generation of an action potential from the initial segment. The axonal initial segment (AIS) is a structurally and functionally separate microdomain of the axon. One function of

5588-408: 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

5715-426: Is thought to carry a different cargo. The studies on transport in the axon led to the naming of kinesin. In the nervous system, axons may be myelinated , or unmyelinated. This is the provision of an insulating layer, called a myelin sheath. The myelin membrane is unique in its relatively high lipid to protein ratio. In the peripheral nervous system axons are myelinated by glial cells known as Schwann cells . In

5842-449: Is unmyelinated and contains a specialized complex of proteins. It is between approximately 20 and 60 μm in length and functions as the site of action potential initiation. Both the position on the axon and the length of the AIS can change showing a degree of plasticity that can fine-tune the neuronal output. A longer AIS is associated with a greater excitability. Plasticity is also seen in

5969-400: 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 ^{*}}

6096-623: 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

6223-519: 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

6350-736: 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

6477-418: The guidance of neuronal axon growth. These cells that help axon guidance , are typically other neurons that are sometimes immature. When the axon has completed its growth at its connection to the target, the diameter of the axon can increase by up to five times, depending on the speed of conduction required. It has also been discovered through research that if the axons of a neuron were damaged, as long as

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6604-462: The immunoglobulin superfamily. Another set of molecules called extracellular matrix - adhesion molecules also provide a sticky substrate for axons to grow along. Examples of these molecules include laminin , fibronectin , tenascin , and perlecan . Some of these are surface bound to cells and thus act as short range attractants or repellents. Others are difusible ligands and thus can have long range effects. Cells called guidepost cells assist in

6731-406: The nerve cell body . The function of the axon is to transmit information to different neurons, muscles, and glands. In certain sensory neurons ( pseudounipolar neurons ), such as those for touch and warmth, the axons are called afferent nerve fibers and the electrical impulse travels along these from the periphery to the cell body and from the cell body to the spinal cord along another branch of

6858-404: The neurotransmitter for release at the synapse . This makes multiple synaptic connections with other neurons possible. Sometimes the axon of a neuron may synapse onto dendrites of the same neuron, when it is known as an autapse . Some synaptic junctions appear along the length of an axon as it extends; these are called en passant boutons ("in passing boutons") and can be in the hundreds or even

6985-922: The Aα, Aβ, and Aγ nerve fibers, respectively. Later findings by other researchers identified two groups of Aa fibers that were sensory fibers. These were then introduced into a system (Lloyd classification) that only included sensory fibers (though some of these were mixed nerves and were also motor fibers). This system refers to the sensory groups as Types and uses Roman numerals: Type Ia, Type Ib, Type II, Type III, and Type IV. Lower motor neurons have two kind of fibers: Different sensory receptors are innervated by different types of nerve fibers. Proprioceptors are innervated by type Ia, Ib and II sensory fibers, mechanoreceptors by type II and III sensory fibers and nociceptors and thermoreceptors by type III and IV sensory fibers. The autonomic nervous system has two kinds of peripheral fibers: In order of degree of severity, injury to

7112-541: 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

7239-405: 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,

7366-651: 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

7493-452: 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

7620-461: The ability of the AIS to change its distribution and to maintain the activity of neural circuitry at a constant level. The AIS is highly specialized for the fast conduction of nerve impulses . This is achieved by a high concentration of voltage-gated sodium channels in the initial segment where the action potential is initiated. The ion channels are accompanied by a high number of cell adhesion molecules and scaffold proteins that anchor them to

7747-601: 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

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7874-399: The alveus and the corpus callosum as well hippocampal gray matter. In fact, the generation of action potentials in vivo is sequential in nature, and these sequential spikes constitute the digital codes in the neurons. Although previous studies indicate an axonal origin of a single spike evoked by short-term pulses, physiological signals in vivo trigger the initiation of sequential spikes at

8001-528: 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

8128-404: 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

8255-503: The axon length on the molecular level. These studies suggest that motor proteins carry signaling molecules from the soma to the growth cone and vice versa whose concentration oscillates in time with a length-dependent frequency. The axons of neurons in the human peripheral nervous system can be classified based on their physical features and signal conduction properties. Axons were known to have different thicknesses (from 0.1 to 20 μm) and these differences were thought to relate to

8382-400: The axon sometimes consists of several regions that function more or less independently of each other. Axons are covered by a membrane known as an axolemma ; the cytoplasm of an axon is called axoplasm . Most axons branch, in some cases very profusely. The end branches of an axon are called telodendria . The swollen end of a telodendron is known as the axon terminal or end-foot which joins

8509-415: The axon to its target, is one of the six major stages in the overall development of the nervous system . Studies done on cultured hippocampal neurons suggest that neurons initially produce multiple neurites that are equivalent, yet only one of these neurites is destined to become the axon. It is unclear whether axon specification precedes axon elongation or vice versa, although recent evidence points to

8636-399: The axon's membrane and empty their contents into the extracellular space. The neurotransmitter is released from the presynaptic nerve through exocytosis . The neurotransmitter chemical then diffuses across to receptors located on the membrane of the target cell. The neurotransmitter binds to these receptors and activates them. Depending on the type of receptors that are activated, the effect on

8763-416: The axons in a condition known as diffuse axonal injury . This can lead to a persistent vegetative state . It has been shown in studies on the rat that axonal damage from a single mild traumatic brain injury, can leave a susceptibility to further damage, after repeated mild traumatic brain injuries. A nerve guidance conduit is an artificial means of guiding axon growth to enable neuroregeneration , and

8890-1287: 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

9017-453: 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

9144-551: The brain and generate thousands of synaptic terminals. A bundle of axons make a nerve tract in the central nervous system , and a fascicle in the peripheral nervous system . In placental mammals the largest white matter tract in the brain is the corpus callosum , formed of some 200 million axons in the human brain . Axons are the primary transmission lines of the nervous system , and as bundles they form nerves . Some axons can extend up to one meter or more while others extend as little as one millimeter. The longest axons in

9271-455: The brain. The myelin gives the white appearance to the tissue in contrast to the grey matter of the cerebral cortex which contains the neuronal cell bodies. A similar arrangement is seen in the cerebellum . Bundles of myelinated axons make up the nerve tracts in the CNS. Where these tracts cross the midline of the brain to connect opposite regions they are called commissures . The largest of these

9398-443: The cell bodies of the neurons. In addition to propagating action potentials to axonal terminals, the axon is able to amplify the action potentials, which makes sure a secure propagation of sequential action potentials toward the axonal terminal. In terms of molecular mechanisms, voltage-gated sodium channels in the axons possess lower threshold and shorter refractory period in response to short-term pulses. The development of

9525-483: The cell body along the axon, carries mitochondria and membrane proteins needed for growth to the axon terminal. Ingoing retrograde transport carries cell waste materials from the axon terminal to the cell body. Outgoing and ingoing tracks use different sets of motor proteins . Outgoing transport is provided by kinesin , and ingoing return traffic is provided by dynein . Dynein is minus-end directed. There are many forms of kinesin and dynein motor proteins, and each

9652-594: The cell body and terminating at points where the axon makes synaptic contact with target cells. The defining characteristic of an action potential is that it is "all-or-nothing" – every action potential that an axon generates has essentially the same size and shape. This all-or-nothing characteristic allows action potentials to be transmitted from one end of a long axon to the other without any reduction in size. There are, however, some types of neurons with short axons that carry graded electrochemical signals, of variable amplitude. When an action potential reaches

9779-504: The central nervous system the myelin sheath is provided by another type of glial cell, the oligodendrocyte . Schwann cells myelinate a single axon. An oligodendrocyte can myelinate up to 50 axons. The composition of myelin is different in the two types. In the CNS the major myelin protein is proteolipid protein , and in the PNS it is myelin basic protein . Nodes of Ranvier (also known as myelin sheath gaps ) are short unmyelinated segments of

9906-623: 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

10033-418: The cytoskeleton. Interactions with ankyrin-G are important as it is the major organizer in the AIS. The axoplasm is the equivalent of cytoplasm in the cell. Microtubules form in the axoplasm at the axon hillock. They are arranged along the length of the axon, in overlapping sections, and all point in the same direction – towards the axon terminals. This is noted by the positive endings of

10160-412: The dendrite or cell body of another neuron forming a synaptic connection. Axons usually make contact with other neurons at junctions called synapses but can also make contact with muscle or gland cells. In some circumstances, the axon of one neuron may form a synapse with the dendrites of the same neuron, resulting in an autapse . At a synapse, the membrane of the axon closely adjoins the membrane of

10287-437: 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

10414-433: 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

10541-481: The entire process adheres to surfaces and explores the surrounding environment. Actin plays a major role in the mobility of this system. Environments with high levels of cell adhesion molecules (CAMs) create an ideal environment for axonal growth. This seems to provide a "sticky" surface for axons to grow along. Examples of CAMs specific to neural systems include N-CAM , TAG-1  – an axonal glycoprotein  – and MAG , all of which are part of

10668-1167: 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 }

10795-483: The fibers into three main groups using the letters A, B, and C. These groups, group A , group B , and group C include both the sensory fibers ( afferents ) and the motor fibers ( efferents ). The first group A, was subdivided into alpha, beta, gamma, and delta fibers – Aα, Aβ, Aγ, and Aδ. The motor neurons of the different motor fibers, were the lower motor neurons  – alpha motor neuron , beta motor neuron , and gamma motor neuron having

10922-409: 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

11049-428: 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

11176-411: The formation of multiple axons. Consequently, the interruption of the actin network in a growth cone will promote its neurite to become the axon. Growing axons move through their environment via the growth cone , which is at the tip of the axon. The growth cone has a broad sheet-like extension called a lamellipodium which contain protrusions called filopodia . The filopodia are the mechanism by which

11303-433: The hallmark of traumatic brain injuries . Axonal damage is usually to the axon cytoskeleton disrupting transport. As a consequence protein accumulations such as amyloid-beta precursor protein can build up in a swelling resulting in a number of varicosities along the axon. Most axons carry signals in the form of action potentials, which are discrete electrochemical impulses that travel rapidly along an axon, starting at

11430-408: The human body are those of the sciatic nerve , which run from the base of the spinal cord to the big toe of each foot. The diameter of axons is also variable. Most individual axons are microscopic in diameter (typically about one micrometer (μm) across). The largest mammalian axons can reach a diameter of up to 20 μm. The squid giant axon , which is specialized to conduct signals very rapidly,

11557-403: The initial segment is to separate the main part of an axon from the rest of the neuron; another function is to help initiate action potentials. Both of these functions support neuron cell polarity , in which dendrites (and, in some cases the soma ) of a neuron receive input signals at the basal region, and at the apical region the neuron's axon provides output signals. The axon initial segment

11684-451: The integration of synaptic messages at the scale of the neuron. Extracellular recordings of action potential propagation in axons has been demonstrated in freely moving animals. While extracellular somatic action potentials have been used to study cellular activity in freely moving animals such as place cells , axonal activity in both white and gray matter can also be recorded. Extracellular recordings of axon action potential propagation

11811-433: The latter. If an axon that is not fully developed is cut, the polarity can change and other neurites can potentially become the axon. This alteration of polarity only occurs when the axon is cut at least 10 μm shorter than the other neurites. After the incision is made, the longest neurite will become the future axon and all the other neurites, including the original axon, will turn into dendrites. Imposing an external force on

11938-433: 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,

12065-426: 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

12192-419: The microtubules. This overlapping arrangement provides the routes for the transport of different materials from the cell body. Studies on the axoplasm has shown the movement of numerous vesicles of all sizes to be seen along cytoskeletal filaments – the microtubules, and neurofilaments , in both directions between the axon and its terminals and the cell body. Outgoing anterograde transport from

12319-439: The myelin sheath of a myelinated axon. Oligodendrocytes form the insulating myelin in the CNS. Along myelinated nerve fibers, gaps in the myelin sheath known as nodes of Ranvier occur at evenly spaced intervals. The myelination enables an especially rapid mode of electrical impulse propagation called saltatory conduction . The myelinated axons from the cortical neurons form the bulk of the neural tissue called white matter in

12446-656: The network to minimize an energy function, towards local energy minimum states that correspond to stored patterns. Patterns are associatively learned (or "stored") by 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

12573-422: 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

12700-404: 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

12827-499: The neuron. Axons vary largely in length from a few micrometers up to meters in some animals. This emphasizes that there must be a cellular length regulation mechanism allowing the neurons both to sense the length of their axons and to control their growth accordingly. It was discovered that motor proteins play an important role in regulating the length of axons. Based on this observation, researchers developed an explicit model for axonal growth describing how motor proteins could affect

12954-464: 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

13081-407: 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

13208-500: 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

13335-459: 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. Axon An axon (from Greek ἄξων áxōn , axis) or nerve fiber (or nerve fibre : see spelling differences ) is a long, slender projection of a nerve cell, or neuron , in vertebrates , that typically conducts electrical impulses known as action potentials away from

13462-571: 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,

13589-436: 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

13716-684: 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

13843-477: The rest of the axon, and the axon telodendria, and axon terminals. It also includes the myelin sheath. The Nissl bodies that produce the neuronal proteins are absent in the axonal region. Proteins needed for the growth of the axon, and the removal of waste materials, need a framework for transport. This axonal transport is provided for in the axoplasm by arrangements of microtubules and type IV intermediate filaments known as neurofilaments . The axon hillock

13970-455: The same axon. Axon dysfunction can be the cause of many inherited and acquired neurological disorders that affect both the peripheral and central neurons . Nerve fibers are classed into three types – group A nerve fibers , group B nerve fibers , and group C nerve fibers . Groups A and B are myelinated , and group C are unmyelinated. These groups include both sensory fibers and motor fibers. Another classification groups only

14097-761: The sensory fibers as Type I, Type II, Type III, and Type IV. An axon is one of two types of cytoplasmic protrusions from the cell body of a neuron; the other type is a dendrite . Axons are distinguished from dendrites by several features, including shape (dendrites often taper while axons usually maintain a constant radius), length (dendrites are restricted to a small region around the cell body while axons can be much longer), and function (dendrites receive signals whereas axons transmit them). Some types of neurons have no axon and transmit signals from their dendrites. In some species, axons can emanate from dendrites known as axon-carrying dendrites. No neuron ever has more than one axon; however in invertebrates such as insects or leeches

14224-421: 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 )}

14351-400: 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

14478-404: The simultaneous transmission of messages to a large number of target neurons within a single region of the brain. There are two types of axons in the nervous system: myelinated and unmyelinated axons. Myelin is a layer of a fatty insulating substance, which is formed by two types of glial cells : Schwann cells and oligodendrocytes . In the peripheral nervous system Schwann cells form

14605-489: The soma (the cell body of a neuron) is not damaged, the axons would regenerate and remake the synaptic connections with neurons with the help of guidepost cells . This is also referred to as neuroregeneration . Nogo-A is a type of neurite outgrowth inhibitory component that is present in the central nervous system myelin membranes (found in an axon). It has a crucial role in restricting axonal regeneration in adult mammalian central nervous system. In recent studies, if Nogo-A

14732-453: 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

14859-475: The speed at which an action potential could travel along the axon – its conductance velocity . Erlanger and Gasser proved this hypothesis, and identified several types of nerve fiber, establishing a relationship between the diameter of an axon and its nerve conduction velocity. They published their findings in 1941 giving the first classification of axons. Axons are classified in two systems. The first one introduced by Erlanger and Gasser, grouped

14986-443: 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

15113-487: 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

15240-524: 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

15367-411: The target cell can be to excite the target cell, inhibit it, or alter its metabolism in some way. This entire sequence of events often takes place in less than a thousandth of a second. Afterward, inside the presynaptic terminal, a new set of vesicles is moved into position next to the membrane, ready to be released when the next action potential arrives. The action potential is the final electrical step in

15494-473: The target cell, and special molecular structures serve to transmit electrical or electrochemical signals across the gap. Some synaptic junctions appear along the length of an axon as it extends; these are called en passant boutons ("in passing boutons") and can be in the hundreds or even the thousands along one axon. Other synapses appear as terminals at the ends of axonal branches. A single axon, with all its branches taken together, can target multiple parts of

15621-444: The thousands along one axon. In the normally developed brain, along the shaft of some axons are located pre-synaptic boutons also known as axonal varicosities and these have been found in regions of the hippocampus that function in the release of neurotransmitters. However, axonal varicosities are also present in neurodegenerative diseases where they interfere with the conduction of an action potential. Axonal varicosities are also

15748-419: 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

15875-559: 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

16002-494: 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

16129-409: 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

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