Before we turn to spike-based learning rules, we first review the basic concepts of correlation-based learning in a firing rate formalism. Firing rate models (cf. Ch. 15 ) have been used extensively in the field of artificial neural networks; cf. Hertz et al. ( 215 ) ; Haykin ( 209 ) for reviews.
In order to find a mathematically formulated learning rule based on Hebb’s postulate we focus on a single synapse with efficacy that transmits signals from a presynaptic neuron to a postsynaptic neuron . For the time being we content ourselves with a description in terms of mean firing rates. In the following, the activity of the presynaptic neuron is denoted by and that of the postsynaptic neuron by .
There are two aspects in Hebb’s postulate that are particularly important; these are locality and joint activity . Locality means that the change of the synaptic efficacy can only depend on local variables, i.e., on information that is available at the site of the synapse, such as pre- and postsynaptic firing rate, and the actual value of the synaptic efficacy, but not on the activity of other neurons. Based on the locality of Hebbian plasticity we can write down a rather general formula for the change of the synaptic efficacy,
Here, is the rate of change of the synaptic coupling strength and is a so far undetermined function ( 465 ) . We may wonder whether there are other local variables (e.g., the input potential , cf. Ch. 15 ) that should be included as additional arguments of the function . It turns out that in standard rate models this is not necessary, since the input potential is uniquely determined by the postsynaptic firing rate, , with a monotone gain function .
The second important aspect of Hebb’s postulate is the notion of ‘joint activity’ which implies that pre- and postsynaptic neurons have to be active simultaneously for a synaptic weight change to occur. We can use this property to learn something about the function . If is sufficiently well-behaved, we can expand in a Taylor series about ,
The term containing on the right-hand side of ( 19.2.1 ) is bilinear in pre- and postsynaptic activity. This term implements the AND condition for joint activity. If the Taylor expansion had been stopped before the bilinear term, the learning rule would be called ‘non-Hebbian’, because pre- or postsynaptic activity alone induces a change of the synaptic efficacy and joint activity is irrelevant. Thus a Hebbian learning rule needs either the bilinear term with or a higher-order term (such as ) that involves the activity of both pre- and postsynaptic neurons.
We now switch from rate-based models of synaptic plasticity to a description with spikes. Suppose a presynaptic spike occurs at time and a postsynaptic one at time . Most models of STDP interpret the biological evidence in terms of a pair-based update rule, i.e. the change in weight of a synapse depends on the temporal difference ; cf. Fig. 19.4 F. In the simplest model, the updates are
where describes the dependence of the update on the current weight of the synapse. Usually is positive and is negative. The update of synaptic weights happens immediately after each presynaptic spike (at time ) and each postsynaptic spike (at time ). A pair-based model is fully specified by defining: (i) the weight-dependence of the amplitude parameter ; (ii) which pairs are taken into consideration to perform an update. A simple choice is to take all pairs into account. An alternative is to consider for each postsynaptic spike only the nearest presynaptic spike or vice versa. Note that spikes that are far apart hardly contribute because of the exponentially fast decay of the update amplitude with the interval . Instead of an exponential decay ( 489 ) , some other arbitrary time-dependence, described by a learning window for LTP and for LTD is also possible ( 176; 256 ) .
If we introduce and for the spike trains of pre- and postsynaptic neurons, respectively, then we can write the update rule in the form ( 262 )
where denotes the time course of the learning window while and are non-Hebbian contributions, analogous to the parameters and in the rate-based model of Eq. 19.2.1 . In the standard pair-based STDP rule, we have and ; cf. Eq. ( 19.10 ).
There is considerable evidence that the pair-based STDP rule discussed above cannot give a full account of experimental results with STDP protocols. Specifically, they reproduce neither the dependence of plasticity on the repetition frequency of pairs of spikes in an experimental protocol, nor the results of triplet and quadruplet experiments.
STDP experiments are usually carried out with about pairs of spikes. The temporal distance of the spikes in the pair is of the order of a few to tens of milliseconds, whereas the temporal distance between the pairs is of the order of hundreds of milliseconds to seconds. In the case of a potentiation protocol (i.e. pre-before-post), standard pair-based STDP models predict that if the repetition frequency is increased, the strength of the depressing interaction (i.e. post-before-pre) becomes greater, leading to less net potentiation. However, experiments show that increasing the repetition frequency leads to an increase in potentiation ( 483; 468 ) . Other experimentalists have employed multiple-spike protocols, such as repeated presentations of symmetric triplets of the form pre-post-pre and post-pre-post ( 53; 160; 542; 159 ) . Standard pair-based models predict that the two sequences should give the same results, as they each contain one pre-post pair and one post-pre pair. Experimentally, this is not the case.
One simple approach to modeling STDP which addresses the issues of frequency dependence is the triplet rule developed by Pfister and Gerstner ( 394 ) . In this model, LTP is based on sets of three spikes (one presynaptic and two postsynaptic). The triplet rule can be implemented with local variables as follows. Similarly to pair-based rules, each spike from presynaptic neuron contributes to a trace at the synapse:
where denotes the firing times of the presynaptic neuron. Unlike pair-based rules, each spike from postsynaptic neuron contributes to a fast trace and a slow trace at the synapse:
where , see Fig. 19.7 . The new feature of the rule is that LTP is induced by a triplet effect: the weight change is proportional to the value of the presynaptic trace evaluated at the moment of a postsynaptic spike and also to the slow postsynaptic trace remaining from previous postsynaptic spikes:
where indicates that the function is to be evaluated before it is incremented due to the postsynaptic spike at . LTD is analogous to the pair-based rule, given in 19.14 , i.e. the weight change is proportional to the value of the fast postsynaptic trace evaluated at the moment of a presynaptic spike.
The triplet rule reproduces experimental data from visual cortical slices ( 483 ) that increasing the repetition frequency in the STDP pairing protocol increases net potentiation ( 19.8 ). It also gives a good fit to experiments based on triplet protocols in hippocampal culture ( 542 ) .
The main functional advantage of such a triplet learning rule is that it can be mapped to the BCM rule of Eqs. ( 19.8 ) and ( 19.9 ): if we assume that the pre- and postsynaptic spike trains are governed by Poisson statistics, the triplet rule exhibits depression for low postsynaptic firing rates and potentiation for high postsynaptic firing rates ( 394 ) ; see Exercises. If we further assume that the triplet term in the learning rule depends on the mean postsynaptic frequency, a sliding threshold between potentiation and depression can be defined. In this way, the learning rule matches the requirements of the BCM theory and inherits the properties of the BCM learning rule such as the input selectivity (see exercises). From the BCM properties, we can immediately conclude that the triplet model should be useful for receptive field development ( 58 ) .
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