Therefore, network models of neurons usually employ other learning theories such as BCM theory, Oja's rule, or the generalized Hebbian algorithm. Regardless, 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 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… 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 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 whenev 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: 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}} the k {\displaystyle k} -th input for neuron i {\displaystyle i} . 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 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 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. 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 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 f {\displaystyle f} , 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 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 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.