Spike-timing dependent plasticity

From Scholarpedia

Author: Dr. Jesper Sjöström, University College London, UK
Author: Dr. Wulfram Gerstner, EPFL, Lausanne, Switzerland

Spike Timing Dependent Plasticity (STDP) is a temporally asymmetric form of Hebbian learning induced by tight temporal correlations between the spikes of pre- and postsynaptic neurons. As with other forms of synaptic plasticity, it is widely believed that it underlies learning and information storage in the brain, as well as the development and refinement of neuronal circuits during brain development (e.g. Bi and Poo, 2001; Sjostrom et al, 2008). With STDP, repeated presynaptic spike arrival a few milliseconds before postsynaptic action potentials leads in many synapse types to long-term potentiation (LTP) of the synapses, whereas repeated spike arrival after postsynaptic spikes leads to long-term depression (LTD) of the same synapse. The change of the synapse plotted as a function of the relative timing of pre- and postsynaptic action potentials is called the STDP function or learning window and varies between synapse types. The rapid change of the STDP function with the relative timing of spikes suggests the possibility of temporal coding schemes on a millisecond time scale.

Experimental STDP Protocol

In a typical STDP protocol (Markram and Sakmann, 1997; Bi and Poo 1998; Sjostrom et al 2001), a synapse is activated by stimulating a presynaptic neuron (or presynaptic pathway) shortly before or shortly after making the postsynaptic neuron fire by injection of a short current pulse. The pairing is repeated for 50-100 times at a fixed frequency (for example 10 Hz). The weight of the synapse is measured as the amplitude (or initial slope) of the postsynaptic potential. The change of the synaptic weight is plotted as a function of the relative timing between presynaptic spike arrival and postsynaptic firing, see Fig.1. The resulting plot is the STDP function or learning window. It is worth noting that different synapse types can have quite different forms of STDP function (Abbott and Nelson, 2000; Bi and Poo, 2001). Compared to many other synaptic plasticity induction protocols, STDP is especially attractive since it is believed to be biologically plausible. In the intact brain, action potentials are often quite precisely timed to stimuli in the outside world, although this is not true for all brain regions and cell types. Nevertheless, STDP is very likely to be induced under such circumstances and many studies provide strong evidence that this is indeed the case (Zhang et al, 1998; Allen et al 2003; Jacob et al, 2007; Meliza and Dan, 2006).

Standard STDP Model

The weight change of a synapse from a presynaptic neuron depends on the relative timing between presynaptic spike arrivals and postsynaptic spikes. Let us name the presynaptic spike arrival times at synapse by where counts the presynaptic spikes. Similarly, with labels the firing times of the postsynaptic neuron. The total weight change induced by a stimulation protocol with pairs of pre- and postsynaptic spikes is then (Gerstner and al. 1996, Kempter et al. 1999)

(1)

where denotes one of the STDP functions (also called learning window) illustrated in Fig.1.

A popular choice for the STDP function

(2)

which has been used in fits to experimental data (Zhang et al. 1998) and models (e.g, Song et al. 2000). The parameters and may depend on the current value of the synaptic weight . The time constants are in the range of and

Variants of STDP Models

Mechanistic implementation of STDP models

Spike-Timing Dependent Plasticity with an STDP function as in Eq. (2) can be implemented in an on-line update rule using the following assumptions. Each presynaptic spike arrival leaves a trace which is updated by an amount at the moment of spike arrival and decays exponentially in the absence of spikes:

(3)

The biophysical nature of the variable need not to be specified, but potential candidates are the amount of glutamate bound to postsynaptic receptors; or the fraction of NMDA receptors in the open state. Similarly, each postsynaptic spike leaves a trace

(4)

which increases by an amount at the moment of postsynaptic spikes. This trace could possibly be interpreted as the voltage at the synapse caused by a backpropagating action potential; or by calcium entry due to a backpropagating action potential. The weight change is then

(5)

Thus, the weight is increased at the moment of postsynaptic firing by an amount that depends on the value of the trace left by the presynaptic spike. Similarly, the weight is depressed at the moment of presynaptic spikes by an amount proportional to the trace left by previous postsynaptic spikes. Integration of Eq. (5) yields Eq. (2).

Weight dependence: hard bounds and soft bounds

For biological reasons, it is desirable to keep the synaptic weights in a range . This can be achieved by an appropriate choice of the functions and . A choice

with positive constants and is called soft bounds or multiplicative weight dependence. The choice

is called hard bounds. Here denotes the Heaviside step function. In practice, hard bounds mean that an update rule with fixed parameters and is used until the bounds are reached (Gerstner et al. 1996, Kempter et al. 1999, Roberts et al. 1999, Song et al. 2000). Soft bounds mean that, for large weights, synaptic depression dominates over potentiation (Kistler and van Hemmen 2000; van Rossum et al. 2000, Rubin et al. 2001). It is possible to interpolate between the two cases (Guetig et al. 2003).

Temporal all-to-all versus nearest-neighbor spike-interaction

If the sum in Eq. (2) goes over all presynaptic spike arrivals and all postsynaptic spikes, then all spike pairs contribute equally. This case has been called all-to-all spike interaction.

It is also possible to restrict the interactions so that only nearest spikes interact. In the mechanistic update rule of Eq. (5), nearest-neighbor interaction can be implemented as follows. The potentiation at the moment of the postsynaptic spike should depend only on the time since the most recent presynaptic spike. To achieve this, suppose that the trace variable is increased at the moment of presynaptic spikes by an amount where denotes the value of the variable just before the update. In other words, the update of is not cumulative but goes always to a fixed value of one, so that the influence of previous spikes is cancelled. see Morrison et al. (2008) for a review.

Triplet rule of STDP

Pair-wise interaction between spikes as in Eq. () would predict that 60 repetitions of pre-post pairings (say, presynaptic spikes 10 ms before postsynaptic ones) give the same result independent of whether the pairing is repeated at 1 Hz or 5Hz. At frequencies above 25 Hz, a pair-wise interaction model would predict a reduced potentiation, since in addition to the pre-post pair at 10ms virtual post-pre pairs at 30ms are created - that should lead to depression. However, in experiments the opposite is observed (Senn et al. 2001; Sjöström et al. 2001). The frequency-dependence of STDP experiments can be accounted for if one assumes that the basic building block of potentiation during STDP experiments is not a pair-wise interaction as assumed in Eq. (), but a triplet interaction between two postsynaptic spikes and one presynaptic spike. Such a triplet interaction can be implemented in the mechanistic model if one works with two postsynaptic traces and with two different time constants, rather than a single trace (Pfister and Gerstner, 2006). Such a model is also compatible with explicit triplet experiments (Wang et al. 2005) while a pair-based model is not.

Homeostatic terms

In addition to the pair-based and triplet-based STDP effects mentioned above, one can also consider STDP models where a isolated postsynaptic or presynaptic spikes give induce a small change of the synaptic weight, even if not paired with another spike. These terms can be used in models to yield a homeostatic control of the total input into the postsynaptic neuron (Kempter et al. 1999, van Rossum 2001).

Another possibility to implement homeostasis into STDP models is by making the parameter in Eq. (2) depend on the mean firing rate calculated as a running average over a time scale of seconds (Pfister and Gerstner 2006).

Voltage dependence

Experiments and Models of Spike-Timing Dependent Plasticity suggest that synaptic weight changes are caused by the tight temporal correlations between pre- and postsynaptic spikes. However, other experimental protocols where presynaptic spikes are paired with a fixed depolarization of the postsynaptic neuron (e.g. under voltage clamp) show that postsynaptic spikes are not necessary to induce long-term potentiation and depression of the synapse (Artola et al., 1990; Ngezahayo et al., 2000; Sjöström et al., 2004). Moreover, the voltage of the postsynaptic neuron just before generation of action potentials influences the direction of change of the synapse, even if the spike timing is held fixed (Sjöström et al., 2001), suggesting that postsynaptic voltage is more fundamental than spike timing. Indeed, a model of synaptic plasticity that postulates pairing between presynaptic spike arrival and postsynaptic voltage contains STDP models as a special case (Brader et al., 2007, Clopath et al., 2008).

Biophysical models

Since signaling chains involved in the induction of Long-Term Potentiation and Depression are partially unknown, most models of STDP are phenomenological models. However, some models attempt to indentify variables such as the traces and in the above mechanistic model with specific biophysical quantities. A few examples:

• Senn-Markram-Tsodyks model. The model shares features with the mechanistic triplet model above and identifies some of the variables with internal states of the NMDA receptor and unspecified second messengers (Senn et al. 1997, 2001).

• Karmarkar-Buonomano model. The model emphasizes the fact that the pathways for upregulation and downregulation are independent and give interpretations of internal variables in terms of NMDA receptor, calcium, and backpropagating action potentials (Karmarkar and Buonomano, 2002).

• Shouval model. The model of Shouval starts from the hypothesis that the intracellular calcium concentration in the vicinity of the synapse controls the up- and downregulation of synaptic weights (Shouval et al., 2002).

• Rubin et al. model. The model gives a detailed account of some of the signaling steps translating the calcium time course into synaptic weight changes (Rubin et al. 2005).

• Lisman model. The model focuses on the autophosphorilation of CaMKII as a critical step for memory formation (Lisman and Zhabotinsky, 2001; Lisman 2003). The calcium based model can be simplified and shows STDP (Graupner and Brunel, 2007).

Relation of STDP to other learning rules

STDP and Hebbian learning rules

STDP can be seen as a spike-based formulation of a Hebbian learning rule. Hebb formulated that a synapse should be strengthened if a presynaptic neuron 'repeatedly or persistently takes part in firing' the postsynaptic one (Hebb 1949). This formulation suggests a potential causal relation between the firing of the two neurons. Causality requires that the presynaptic neuron fires slightly before the postsynaptic one. Indeed, in standard STDP experiments on synapses onto pyramidal neurons, potentiation of the synapse occurs for pre-before-post timing, in agreement with Hebbs postulate. Hebb did not, however, postulate the existence of synaptic weakening (Hebb 1949). The existence of a temporal window for weakening of connective strength in the typical STDP learning rule must thus be viewed as an extension to the Hebbian postulate.

STDP versus Rate based learning rules

Under the assumption of stationary Poisson statistics for the firing pre- and postsynaptic neurons, the most relevant aspect of the STPD function is its integral and an STDP model can mapped to an equivalent rate-based learning rule (Kempter et al., 1999). Under the assumption of independence between pre- and postsynaptic firing, the total weight change is where and denote the firing rate of pre- and postsynaptic neurons averaged over some time and is the integral over the learning window. If the integral is positive, STDP is identical to standard rate-based Hebbian learning. For negative integral, as often used in modeling, STDP corresponds to a anti-Hebbian rate rule.

However, the assumption of independence of pre- and postsynpatic firing is obviously wrong since it neglects the causal correlations generated be the interaction of the two neurons. A more precise mapping to rate models can be achieved if the postsynaptic neuron is described as an inhomogeneous Poisson Process with a rate where denotes the spike times of a presynaptic neuron generated by a Poisson process of rate and for describes the time course of a postsynpatic potential. The total weight change in a period is then where is the integral over the 'causal' part of the learning window, i.e., over all times with 'pre-before-post' relation (Kempter et al. 1999). For standard STDP models , i.e., presynaptic spike arrival leads on average to a positive change of the synapse, because it is likely to cause postsynaptic firing. This is then often combined with a negative integral over the STDP function so that random pairings of pre- and postsynaptic firings leads to a decrease of the synapse (Gerstner et al. 1996, Song et al. 2000). The functional consequences of such a choice are discussed below (see Rate normalization).

STDP and Bienenstock-Cooper-Munro (BCM) rule

STDP can also be related to a nonlinear rate model where the weight change depends linearly on the presynaptic rate, but nonlinearly on the postsynaptic rate (Bienenstock et al. 1982). This can be achieved in two different ways. The first possibility is to implement standard STDP with nearest-neighbor instead of all-to all coupling (see above). This leads to a nonlinearity consistent with the BCM rule (Izikhevich and Desai, 2003). The second possibility is to use triplet STDP rule (see above) instead of the standard pair-based rule. If potentiation requires a triplet of two postsynaptic spike and one presynaptic spike (with post-pre-post or pre-post-post firings in temporal proximity) while depression is modeled by the interaction of a post-pre-pair, then the equivalent rate model under a Poisson firing assumption as above is where describes the minimal postsynaptic frequency for potentiation and is a quadratic function (Pfister and Gerstner, 2006). If the amount of depression increases with the mean postsynaptic frequency, then the threshold shifts with the mean postsynaptic rate. In this case the triplet rule of STDP becomes identical to BCM rule (Bienenstock et al. 1982).

Functional Consequences

As described above, STDP models can be related to rate models under the assumption of Poisson firing of both pre- and postsynaptic neurons. Hence STDP rules inherit functional consequences known for rate models. In particular, the potential of synaptic learning to principal component analysis; to receptive field development; to clustering and map formation does not change fundamentally if one switches from rate-based to spike-based models (Kempter et al. 1999; Song and Abbott 2001). In the rest of this section we focus on aspects that are specific to STPD and to beyond known features of rate-based learning.

Spike-Spike correlations

The postsynaptic depolarization caused by spike arrival at an excitatory synapse makes the postsynaptic neuron more likely to fire. In all spiking neuron models (including Poisson models driven by presynaptic input) this leads to a correlation of the spikes of pre- and postsynaptic neurons on the timescale of milliseconds. These spike-spike correlations contribute to learning in STDP models (Kempter et al. 1999), but are completely neglected in standard rate models of learning. See the section 'STDP versus rate based learning rules'.

Firing early

Suppose a postsynaptic neuron is connected to N presynaptic neurons that fire one after another in a sequence 1-2-3-...-N over several milliseconds. Suppose that the synaptic input makes the postsynaptic neuron fire between the firings of presynaptic neurons N-1 and N. As a result of STDP the connection from neuron N to the postsynaptic neuron is weakened (because of the post-before-pre timing) whereas the connections from neurons N-1, N-2, N-3 ... to the postsynaptic neuron are reinforced (because of appropriate pre-before-post firing). After several repetitions of the same stimulus, the postsynaptic neuron fires earlier, because of the stronger input. Hence the timing of the postsynaptic spike shifts forward in time (Song et al. 2000, Mehta et al. 2000).

Temporal coding

Since STDP is sensitive to spike timing on the millisecond rate, it can be used in temporal coding paradigms. Examples include tuning of synaptic connections in a model of sound source localisation in the auditory pathway (Gerstner et al. 1996); learning of spatio-temporal spike patterns in a model of associative memory (Gerstner et al. 1993); suppression of predictable signals in a model of the electrosensory system of electric fish (Roberts and Bell, 2000); learning time-order codes (Guyonneau et al. 2005); amongst others.

Rate normalization

Rate-based Hebbian learning is intrinsically unstable: synaptic inputs that drive the neuron to a high firing rate will be strengthend further. On one hand, such an instability is necessary to make the neuron detect, and become sensitive, to weak correlations in the input. On the other hand, this leads no only to a growth of individual synapses, but also to an explosion of the firing rate of the postsynaptic neuron. In practice, in rate based learning the growth of synapses and firing rates is controlled by (i) introducing upper and lower bounds for individual weights and (ii) renormaliation of the weights of each time step or each episode. Renormalization can alternatively be implemented online by a rate-dependent decay term of the weights (Oja 1982). Surprisingly, STDP models with an appropriate set of parameters do not need such an explicit normalisation step (Kempter et al. 1999, Song et al. 2000, Kempter et al. 2001).

As discussed above in section 'STDP versus Rate based learning rules', the equivalent rate model of a standard STDP rule is . For a choice of parameter where the integral over the STDP function is negative () and pre-before-post firings lead to potentiation (), the firing rate of the postsynaptic neuron has a stable fixed point, while the learning rule is sensitive to the temporal correlations between pre- and postsynaptic neurons (Kempter et al. 2001).

Experimental results and open questions

Diversity of STDP

STDP varies tremendously across synapse types and brain regions (Abbott and Nelson, 2000). Even so, it is worth recollecting that the temporal assymmetry of classical STDP is also remarkably well preserved and is found in species as different as rat, frog, locust, zebra finch, electric fish, cat, and probably also humans (reviewed in Sjöström et al., 2008; Caporale and Dan, 2008). In mammals, STDP has also been uncovered in multiple brain regions, such as prefrontal, entorhinal, somatosensory, and visual cortices, hippocampus, striatum, the cochlear nucelus, and the amygdala (cf. Sjöström et al., 2008; Caporale and Dan, 2008). The activity requirements that govern STDP at many of these different synapses, however, is variable. For example, the width of the temporal windows for LTD and LTP appear to be roughly equal at hippocampal excitatory synapses (Bi and Poo, 1998; Nishiyama et al., 2000; Zhang et al., 1998), whereas the LTD timing window is considerably wider than that of LTP at several neocortical synapses (Feldman et al., 2000; Sjöström et al., 2001).

For some synapses, the STDP timing windows is inverted as compared to the classical form of STDP, so that pre-before-post timings result in LTD whereas the opposite temporal order results in LTP. This is the case at e.g. inhibitory connections onto neocortical L2/3 pyramidal neurons (Holmgren and Zilberter, 2001), at corticostriatal synapses (Fino et al., 2005) as well as in the electrosensory lobe of the mormyrid electric fish (Bell et al., 1997). The timing requirements for STDP at connections between spiny stellate cells in rat somatosensory cortex are yet again different: Here, synapses undergo LTD seemingly regardless of temporal order (Egger et al., 1999). In neocortical layer-5 pyramidal neurons, the timing requirements also depend critically on synapse location in the dendritic tree: Whereas proximal inputs undergo classical STDP, distal synapses are subject to a "temporally inverted" STDP rule (Letzkus et al., 2006). These same inputs also undergo non-Hebbian LTD or Hebbian LTP depending on the state of depolarization of the apical dendrite (Sjöström and Häusser, 2006).

The activity requirements of STDP may thus vary considerably not only across brain regions and synapse types, but also within a cell, in different dendritic compartments. One open question is what this variability is good for. Since it is well established that synaptic plasticity underpins neural circuit development (Katz and Shatz, 1996), this implies that the STDP rules engaged during development determine circuit functionality in the mature brain. In other words, this variability of STDP is most likely not coincidental.

Voltage dependence

Role of backpropagating action potentials

Biophysical and biochemical mechanism

Influence of neuromodulators

Induction versus maintenance of Long-Term Potentiation

Discrete or continuous synapses

History of STDP

The first experiments with precisely timed pre- and postsynaptic spikes at a millisecond temporal resolution were performed by Markram et al. (1995,1997) soon followed by others (Bell et al. 1997, Bi and Poo 1998, Debanne et al. 1998, Zhang et al. 1998). While the first true STDP experiments started hence in the mid-nineties temporal requirements for the coincidence of pre- and postsynaptic actvity had already been investigated in 1983 in experiments by Levy and Stuart albeit with a lower temporal resolution using bursts of spikes rather than individual action potentials (Levy and Stuart, 1983). These early experiments can be understood as temporally asymmetric Hebbian learning under a rate coding hypothesis, but also as precursors of modern STDP experiments.

The first model using an STDP function with potentiation and depression at a millisecond resolution was published in 1996 (Gerstner et al. 1996). Modelers and theoreticians interested in Hebbian learning have been interested in temporally asymmetric forms of Hebbian learning in 1996 in the context of sequence learning (Abbott and Blum, 1996) and already already in the mid 1980ies in the context of associative memories (Kleinfeld 1986, Sompolinsky and Kanter 1986, Herz et al. 1988). In standard attractor networks of memory, the learning rule includes terms of the Hebbian form where and denote the firing rate of pre- and postsynaptic neurons in the rate pattern present at time step . In order to learn sequences of patterns (or non-stationary attractors) the learning rule should contain terms of the form , i.e., a form of temporally asymmetric Hebbian learning that correlates the firing rate of the postsynaptic neuron at time with that of a presynaptic neuron during the time step (Kleinfeld 1986, Sompolinsky and Kanter 1986, Herz et al. 1988). In 1993 Gerstner and van Hemmen started to translate ideas from sequence learning in discrete-time rate models to the case of spiking neurons in continuous time and formulated a learning rule where presynaptic spikes arrival a few milliseconds before postsynaptic firing leads to a potentiation of synapses (Gerstner et al., 1993). Depression of synapses was unspecific and not part of the spike-timing dependent learning rule. For purely theoretical reasons Gerstner and colleagues postulated in a paper submitted to Nature in 1995 that presynaptic spike arrival before postsynaptic firing should lead to potentiation whereas the reverse timing should lead to depression. Referees of that paper asked whether there was any experimental support for this speculation. In the mean time Markram et al. published an abstract in the Society of Neuroscience meeting of 1995, which was then cited by Gerstner et al. so as to convince the referees and the paper was published in 1996 (Gerstner et al. 1996). To our knowledge, this is the first paper that plotted synaptic plasticity as a function of the relative timing of individual pre- and postsynaptic action potentials.

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See Also

Learning, Models of synaptic plasticity, Long-term depression, Long-term potentiation, Memory, Neuron, Synapse, Synaptic plasticity