Donnald Hebb postulated way back in 1949:
‘When an axon of cell A is near enough to excite cell B or repeatedly or persistently takes part in firing it, some growth process ormetabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.’
In other words ‘neurons that fire together wire together’ by increasing the strength of the synapse through which signals travel from neuron A to neuron B. B is thus more likely to fire when stimulated by A.
Spike timing dependant plasticity (STDP) is a refinement of the Hebbian learning principal for spiking neural networks based on empirical measurement from natural systems. Long-term plasticity depends on the exact timing relation, on the time scale of milliseconds, of the spikes from the presynaptic neuron A and the spikes from the postsynaptic neuron B. Different types of synapse do have different plasticity properties. Weight dependence and spike-pairing scheme can have significant consequences for the behaviour of the model. These properties are discussed here.
In order to consider STDP as a local rule based system it is useful to postulate a trace being left at the synapse by spikes passing through. Such a trace may be imagined as an update of an internal hidden variable when a spike travels across the synapse connecting neuron A to neuron B. When neuron B fires, it back propagates a signal to the synapse connecting neuron A to neuron B causing further changes. The hidden variable evolves with some time constant (τ) so that it implements a kind of low-pass filter. The trace left by a spike may not scale linearly with the rate, for example, instead of updating by the same amount each time, we may introduce saturation so that the trace is bound between two limits, 0 and MAX.
The traces left by spikes are then used to update the synapses long term weight. There are different schemes of combining pre and post synaptic spikes. Pair based rules use a combination of either presynaptic spike followed by a postsynaptic spike or a postsynaptic spike followed by a presynaptic spike to manifest changes to the long term weight. The general rule is that a pre-post combination will potentate the synapse and a post-pre combination will depress the synapse, although evidence has been found of the opposite occurring in some cell assemblies. Triplet rules have also been investigated for combinations such as pre-post-pre and post-pre-post, and again differing empirical results have been found. Here we will focus on pair based update rules.
When the postsynaptic neuron fires it initiates the synaptic weight update rule. The update rule considers presynaptic spikes within a given window with an exponential weighting as can be seen in the figure below where F=exp(-|Δt|/τ).
The rule may use an ‘all to all’ scheme in which all presynaptic spikes are considered within the STDP window or a ‘nearest neighbour’ scheme in which only the spike temporally nearest the time of the postsynaptic spike is considered. It is often assumed that the scheme used makes no difference as the inter spike interval of cortical network models is typically an order of magnitude larger than the time constant of the STDP window, however some have argued that this is not true.
For pre-post potentiating the change in weight (Δw) is affected by the exponential of the time difference (Δt) and a learning rate constant 
:
for post-pre depression a similar formula is used with a different learning rate 
:
The term 
in the depression rule represents the dependence on the existing weight value, and the term 
in the potentiation rule also represents the weight dependence but more specifically dependence on the difference between the existing weight and the maximum possible weight value. By setting µ to 0 the terms disappear so there is no dependence and we have what is called an additive rule. By setting µ to 1 there is a dependence and we have what is called a multiplicative rule. An power law fit is achieved by setting µ to different values.
Interesting effects can be seen in the choice of weight dependence. An additive rule makes the weight tend to saturate near to either 0 or MAX so a bimodal distribution develops, whereas in the case of multiplicative STDP the equilibrium distribution is unimodal as weights vary around a singular value. Intermediate values of µ result in rules which have an intermediate dependence on the synaptic strength. The distribution of synaptic weights as µ varies can be seen in the figure below.
Morrison et al note that ‘the critical value for µ at which bimodal distributions appear decreases as the effective population size Nrτ increases, where N is the number of synapses converging onto the postsynaptic neuron, r is the rate of the input spike trains in Hz and τ is the time constant of the STDP window’. The available evidence suggests that both potentiation and depression are dependent on the weight, therefore in the absence of fresh experimental evidence supporting an additive rule, weight dependent rules may be considered as more standard.
The post synaptic firing rate is not sensitive to timing of presynaptic firing but instead total input. The STDP sampling of presynaptic spikes is based on the postsynaptic triggers and therefore averaged over the entire potentiating and depressing spiking influence of the presynaptic neuron around that time. The asymmetry in areas under the positive and negative portions of the STDP modification curve will prefer depression. As synapses are weakened the postsynaptic firing is near potential and tightly correlated with the presynaptic spikes. Therefore postsynaptic action potentials are triggered through statistical fluctuation, at which time there tend to be more excitatory presynaptic spikes before than after a postsynaptic response.
Peak synaptic conductances is achieved when the excess of presynaptic action potentials before postsynaptic firing compensates for the asymmetry under the STDP curve. Song et al explain that ‘STDP thus modifies excitatory synaptic strengths until there is a sufficiently, but not excessively, high probability of a presynaptic action potential occurring before a postsynaptic spike. This causes the neuronal response to be sensitive to the timing of input fluctuations’. If presynaptic neurons fire with varying latencies that extend over a period longer than that required to evoke postsynaptic spike STDP will strengthen the shorter-latency excitatory inputs and weakening those with longer latencies. The effect is to make the postsynaptic neuron respond more quickly.
STDP forms input clusters based on correlations between different inputs provided that these inputs decay rapidly enough as a function of time not to incur depression. These clusters suppress synapses that are uncorrelated or have temporal correlations that last over longer time periods. STDP displays strengthening of correlated groups of synapses, the basic feature of Hebbian learning. In addition the desirable features of firing-rate independence and stability and a novel dependence on correlation decay time are also present.
Sorry about the maths 🙁