Hebb rule

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Hebbian learning

A learning rule dating back to D.O. Hebb's classic [a1], which appeared in 1949. The idea behind it is simple. Neurons of vertebrates consist of three parts: a dendritic tree, which collects the input, a soma, which can be considered as a central processing unit, and an axon, which transmits the output. Neurons communicate via action potentials or spikes, pulses of a duration of about one millisecond. If neuron $ j $ emits a spike, it travels along the axon to a so-called synapse on the dendritic tree of neuron $ i $, say. This takes $ \tau _ {ij } $ milliseconds. The synapse has a synaptic strength, to be denoted by $ J _ {ij } $. Its value, which encodes the information to be stored, is to be governed by the Hebb rule.

In [a1], p. 62, one can find the "neurophysiological postulate" that is the Hebb rule in its original form: 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 the efficiency of $ A $, as one of the cells firing $ B $, is increased.

Hebb's postulate has been formulated in plain English (but not more than that) and the main question is how to implement it mathematically. The key ideas are that:

i) only the pre- and post-synaptic neuron determine the change of a synapse;

ii) learning means evaluating correlations. If both $ A $ and $ B $ are active, then the synaptic efficacy should be strengthened. Efficient learning also requires, however, that the synaptic strength be decreased every now and then [a2].

In the present context, one usually wants to store a number of activity patterns in a network with a fairly high connectivity ( $ 10 ^ {4} $ in biological nets). Most of the information presented to a network varies in space and time. So what is needed is a common representation of both the spatial and the temporal aspects. As a pattern changes, the system should be able to measure and store this change. How can it do that?

For unbiased random patterns in a network with synchronous updating this can be done as follows. The neuronal dynamics in its simplest form is supposed to be given by $ S _ {i} ( t + \Delta t ) = { \mathop{\rm sign} } ( h _ {i} ( t ) ) $, where $ h _ {i} ( t ) = \sum _ {j} J _ {ij } S _ {j} ( t ) $. Let $ J _ {ij } $ be the synaptic strength before the learning session, whose duration is denoted by $ T $. After the learning session, $ J _ {ij } $ is to be changed into $ J _ {ij } + \Delta J _ {ij } $ with

$$ \Delta J _ {ij } = \epsilon _ {ij } { \frac{1}{T} } \sum _ { 0 } ^ { T } S _ {i} ( t + \Delta t ) S _ {j} ( t - \tau _ {ij } ) $$

(cf. [a3], [a4]). The above equation provides a local encoding of the data at the synapse $ j \rightarrow i $. The $ \epsilon _ {ij } $ is a constant known factor. The learning session having a duration $ T $, the multiplier $ T ^ {- 1 } $ in front of the sum takes saturation into account. The neuronal activity $ S _ {i} ( t ) $ equals $ 1 $ if neuron $ i $ is active at time $ t $ and $ - 1 $ if it is not. At time $ t + \Delta t $ it is combined with the signal that arrives at $ i $ at time $ t $, i.e., $ S _ {j} ( t - \tau _ {ij } ) $, where $ \tau _ {ij } $ is the axonal delay. Here, $ \{ {S _ {i} ( t ) } : {1 \leq i \leq N } \} $, denotes the pattern as it is taught to the network of size $ N $ during the learning session of duration $ 0 \leq t \leq T $. The time unit is $ \Delta t = 1 $ milliseconds. In the case of asynchronous dynamics, where each time a single neuron is updated randomly, one has to rescale $ \Delta t \pto {1 / N } $ and the above sum is reduced to an integral as $ N \rightarrow \infty $. In passing one notes that for constant, spatial, patterns one recovers the Hopfield model [a5].

Suppose now that the activity $ a $ in the network is low, as is usually the case in biological nets, i.e., $ a \approx - 1 $. Then the appropriate modification of the above learning rule reads

$$ \Delta J _ {ij } = \epsilon _ {ij } { \frac{1}{T} } \sum _ { 0 } ^ { T } S _ {i} ( t + \Delta t ) [ S _ {j} ( t - \tau _ {ij } ) - \mathbf a ] $$

(cf. [a4]). Since $ S _ {j} - a \approx 0 $ when the presynaptic neuron is not active, one sees that the pre-synaptic neuron is gating. One gets a depression (LTD) if the post-synaptic neuron is inactive and a potentiation (LTP) if it is active. So it is advantageous to have a time window [a6]: The pre-synaptic neuron should fire slightly before the post-synaptic one. The above Hebbian learning rule can also be adapted so as to be fully integrated in biological contexts [a6]. The biology of Hebbian learning has meanwhile been confirmed. See the review [a7].

G. Palm [a8] has advocated an extremely low activity for efficient storage of stationary data. Out of $ N $ neurons, only $ { \mathop{\rm ln} } N $ should be active. This seems to be advantageous for hardware realizations.

In summary, Hebbian learning is efficient since it is local, and it is a powerful algorithm to store spatial or spatio-temporal patterns. If so, why is it that good? As to the why, the succinct answer [a3] is that synaptic representations are selected according to their resonance with the input data; the stronger the resonance, the larger $ \Delta J _ {ij } $. In other words, the algorithm "picks" and strengthens only those synapses that match the input pattern.


[a1] D.O. Hebb, "The organization of behavior--A neurophysiological theory" , Wiley (1949)
[a2] T.J. Sejnowski, "Statistical constraints on synaptic plasticity" J. Theor. Biol , 69 (1977) pp. 385–389
[a3] A.V.M. Herz, B. Sulzer, R. Kühn, J.L. van Hemmen, "The Hebb rule: Storing static and dynamic objects in an associative neural network" Europhys. Lett. , 7 (1988) pp. 663–669 (Hebbian learning reconsidered: Representation of static and dynamic objects in associative neural nets, Biol. Cybern. 60 (1989), 457–467)
[a4] J.L. van Hemmen, W. Gerstner, A.V.M. Herz, R. Kühn, M. Vaas, "Encoding and decoding of patterns which are correlated in space and time" G. Dorffner (ed.) , Konnektionismus in artificial Intelligence und Kognitionsforschung , Springer (1990) pp. 153–162
[a5] J.J. Hopfield, "Neural networks and physical systems with emergent collective computational abilities" Proc. Nat. Acad. Sci. USA , 79 (1982) pp. 2554–2558
[a6] W. Gerstner, R. Ritz, J.L. van Hemmen, "Why spikes? Hebbian learning and retrieval of time-resolved excitation patterns" Biol. Cybern. , 69 (1993) pp. 503–515 (See also: W. Gerstner and R. Kempter and J.L. van Hemmen and H. Wagner: A neuronal learning rule for sub-millisecond temporal coding, Nature 383 (1996), 76–78)
[a7] T.H. Brown, S. Chattarji, "Hebbian synaptic plasticity: Evolution of the contemporary concept" E. Domany (ed.) J.L. van Hemmen (ed.) K. Schulten (ed.) , Models of neural networks , II , Springer (1994) pp. 287–314
[a8] G. Palm, "Neural assemblies: An alternative approach to artificial intelligence" , Springer (1982)
How to Cite This Entry:
Hebb rule. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by J.L. van Hemmen (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article