Hebbian theory is a neuropsychological theory claiming that an increase in synaptic efficacy arises from a presynaptic cell's repeated and persistent stimulation of a postsynaptic cell. It is an attempt to explain synaptic plasticity, the adaptation of neurons during the learning process. Hebbian theory was introduced by Donald Hebb in his 1949 book The Organization of Behavior. The theory is also called Hebb's rule, Hebb's law, Hebb's postulate, and cell assembly theory. Hebb states it as follows:
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, is increased.
The theory is often summarized as "Neurons that fire together, wire together."Siegrid Löwel, Göttingen University; The exact sentence is: "neurons wire together if they fire together" (Löwel, S. and Singer, W. (1992) Science 255 (published January 10, 1992) 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.
Hebbian 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 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.
Engrams, cell assembly theory, and learning Hebbian theory provides an explanation for how neurons might connect to become engrams, which may be stored in overlapping cell assemblies, or groups of neurons that encode specific information. Initially created as a way to explain recurrent activity in specific groups of cortical neurons, Hebb's theories on the form and function of cell assemblies can be understood from the following:
The general idea 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 using the concept of auto-association, or the brain's ability to retrieve information based on a partial cue, 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, the pattern as a whole will become 'auto-associated'. We may call a learned (auto-associated) pattern an engram.
Research conducted in the laboratory of Nobel laureate Eric Kandel has provided evidence supporting the role of Hebbian learning mechanisms at synapses in the marine gastropod Aplysia californica. Because synapses in the peripheral nervous system of marine invertebrates are much easier to control in experiments, Kandel's research found that Hebbian long-term potentiation along with activity-dependent presynaptic facilitation are both necessary for synaptic plasticity and classical conditioning in Aplysia californica.
While research on invertebrates has established fundamental mechanisms of learning and memory, much of the work on long-lasting synaptic changes between vertebrate neurons 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 review indicates that long-lasting changes in synaptic strengths can be induced by physiologically relevant synaptic activity using both Hebbian and non-Hebbian mechanisms.
Principles In 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 is a formulaic description of Hebbian learning (many other descriptions are possible):
where is the weight of the connection from neuron to neuron , and is the input for neuron . This is an example of pattern learning, where weights are updated after every training example. In a Hopfield network, connections are set to zero if (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 a pattern.
When several training patterns are used, the expression becomes an average of the individuals:
where is the weight of the connection from neuron to neuron , is the number of training patterns and the -th input for neuron . This is learning by epoch, with weights updated after all the training examples are presented and is last term applicable to both discrete and continuous training sets. Again, in a Hopfield network, connections are set to zero if (no reflexive connections).
A variation of Hebbian learning that takes into account phenomena such as blocking and other neural learning phenomena is the mathematical model of Harry Klopf. Klopf's model assumes that parts of a system with simple adaptive mechanisms can underlie more complex systems with more advanced adaptive behavior, such as neural networks.Klopf, A. H. (1972). Brain function and adaptive systems—A heterostatic theory. Technical Report AFCRL-72-0164, Air Force Cambridge Research Laboratories, Bedford, MA.
Relationship to unsupervised learning, stability, and generalization 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 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 , whose inputs have rates . The response of the neuron is usually described as a linear combination of its input, , followed by a response function :
As defined in the previous sections, Hebbian plasticity describes the evolution in time of the synaptic weight :
Assuming, for simplicity, an identity response function , we can write
or in matrix form:
As in the previous chapter, if training by epoch is done an average over discrete or continuous (time) training set of can be done:where is the correlation matrix of the input under the additional assumption that (i.e. the average of the inputs is zero). This is a system of coupled linear differential equations. Since is symmetric, it is also diagonalizable, and the solution can be found, by working in its eigenvectors basis, to be of the form
where are arbitrary constants, are the eigenvectors of and their corresponding eigen values. Since 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 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 , but in fact, it can be shown that for any neuron model, Hebb's rule 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 is the largest eigenvalue of . At this time, the postsynaptic neuron performs the following operation:
Because, again, is the eigenvector corresponding to the largest eigenvalue of the correlation matrix between the s, this corresponds exactly to computing 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 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.
Hebbian learning and mirror neurons Hebbian learning and spike-timing-dependent plasticity have been used in an influential theory of how mirror neurons emerge.Keysers, C. (2011). The Empathic Brain. 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 the actions of others, since when a person perceives the actions of others, motor programs in the person's brain which they would use to perform similar actions are activated, which add information to the perception and help to predict what the person will do next based on the perceiver's own motor program. One limitation of this idea of mirror neuron functions is explaining how individuals develop neurons that respond both while performing an action and while hearing or seeing another perform similar actions.