# Models of synaptic plasticity

Synaptic plasticity was first proposed as a mechanism for learning and memory on
the basis of theoretical analysis (Hebb, 1949). The plasticity rule proposed by Hebb postulates that when one neuron drives the activity of another neuron, the connection between these neurons is potentiated. Theoretical analysis indicates that not only Hebbian like synaptic potentiation is necessary but also depression between two neurons that
are not sufficiently coactive (Stent, 1973, Sejnowski 1977). Depression is necessary for several reasons, among them to prevent all synapses from saturating to their maximal values and thereby loosing their selectivity, and to prevent a positive feedback loop between network activity and synaptic weights. The experimental correlates of these theoretically proposed forms of synaptic plasticity are called long-term potentiation (LTP) and long-term depression (LTD). Two broad classes of **models of synaptic plasticity** can be described: 1) Phenomenological models: These are very simple models that are typically based on an input-output relationship between neuronal activity and synaptic plasticity. Phenomenological models are typically used in simulations to account for higher level phenomena such as the formation of memory, or the development of neuronal selectivity. 2) Biophysical models: These more detailed models
incorporate more of the cellular and synaptic biophysics of neurons, and are typically used to account for controlled synaptic plasticity experiments.

## Phenomenological models

**Phenomenological models** are characterized by treating the process governing synaptic plasticity as a black box. The black box takes in as input a set of variables, and produces as
output a change in synaptic efficacy. No explicit modeling of the biochemistry and physiology leading to synaptic plasticity is implemented. Two different classes of phenomenological models, rate based and spike based, differ in the type of their input variables.

### Rate Based Models

Many of the phenomenological models of synaptic plasticity that have been proposed over the years are rate based models (Dayan and Abbott, 2001). In these models it is assumed that the rate of pre and postsynaptic firing measured over some time period, determines the sign and magnitude of synaptic plasticity. This can be formulated as: \[\tag{1} {dW_i\over dt}=f(x_i,y,W_i,other), \]

where \(W_i\)is the synaptic efficacy of synapse \(i, x_i\) is the rate of the neuron presynaptic to synapse \(i\ ,\) and \(y\ ,\) is the rate of the postsynaptic neuron. Other variables, for example global variables such as reward, or long time averages of the rate variables can also have an effect. A simple example of a rate based model (Linsker, 1986) has the following form:

\[\tag{2} {dW_i\over dt}=\eta(x_i-x_0)(y-y_0), \]

where \(\eta\) is a small learning rate and \(x_0; y_0 \) are constants. Such simple models might result in uncontrolled weight growth and might produce receptive fields that are not sufficiently selective. Therefore, additional normalizing and competitive factors are often added to obtain more stability or selectivity. One example is the plasticity model proposed by Oja (Oja, 1982) in which Hebbian plasticity is augmented with a decay term. Synaptic weights in this model converge to the first principal component of the input data (PCA). Another example is the BCM model, which has both LTP and LTD regions, and a sliding threshold separating them in order to overcome problems of positive feedback (Bienenstock et al., 1982).

### Spike timing based models

The discovery of spike timing dependent plasticity (STDP) has prompted the development of models that depend on the timing difference between pre and postsynaptic spikes. Most such models depend only on the relative timing between spike pairs, however recently a model that depends on spike triplets has been published (Pfister and Gerstner, 2006). Under certain assumptions about pre and postsynaptic spike statistics, and about the horizon of spike interactions that contributes to plasticity (all to all or nearest neighbor), these STDP models can be averaged and reduced to rate based models (Kempter et al., 1999, Izhikevich and Desai, 2003). It has been shown that assuming an all to all interaction STDP models can be reduced to the correlational form described in equation 2, (Kempter et al., 1999) whereas the nearest neighbor horizon results in models that are similar to BCM (Izhikevich and Desai, 2003). Phenomenological STDP models are typically used to account for higher level phenomena such as the development of receptive fields (Song et al., 2000, Kempter et al., 1999).

Although spike timing dependent plasticity is often thought of as described by a simple curve such as shown in figure 1i, this is not really the case. STDP depends on many other factors such a the frequency at which pairs are delivered (Markram et al., 1997, Sjostrom et al. 2001), the level of local postsynaptic depolarization (Sjostrom et. al. 2001) or the initial synaptic state (Bi and Poo, 1998). In addition there are many other induction protocols such as rate dependent protocols, and pairing induced protocols exist. It is difficult to account for all of these protocols simply on the basis of a phenomenological model that depends only on the relative timing between pairs of spikes. For example, pairing induced plasticity (Fig. 1d-f) is produced with no postsynaptic spikes, how could such a protocol be accounted for by the relative timing between pairs of pre and postsynaptic spikes?

## Biophysical models of synaptic plasticity

**Biophysical models**, in contrast to phenomenological models, concentrate on modeling the biochemical and physiological processes that lead to the induction and expression of synaptic plasticity. However, since it is not possible to implement precisely every portion of the physiological and biochemical networks leading to synaptic plasticity, even the biophysical models rely on many simplifications and abstractions.

Different cortical regions, such as Hippocampus and Visual cortex have somewhat different forms of synaptic plasticity. Even in the same cortical region, different types of cells, or even cells within different layers, or even synapses on the same cell that connect to different types of cells, might have varying forms of synaptic plasticity. In principle a biophysical model for a given system should account for the results of induction protocols in this single system, and might not be able to account for results in a different system. Hopefully some of the fundamental mechanisms will be preserved in the different systems. In this article results obtained from various systems are discussed.

### Calcium dependent models of bidirectional synaptic plasticity

Calcium influx into the postsynaptic spine is crucial for the induction of many forms of bidirectional synaptic plasticity. Much of the calcium entering the postsynaptic spine comes through NMDA receptors. Blocking NMDA receptors pharmacologically can eliminate both LTP and LTD, and a partial block of NMDA receptors can convert an LTP protocol to LTD. Moreover, experimental results show that a strong postsynaptic calcium transient, in the absence of a presynaptic stimulus can produce LTP while a prolonged moderate calcium transient results in LTD (Yang et al., 1999).

The dependence of LTP on calcium influx through NMDA receptors was the basis for some early models of synaptic plasticity (Gamble and Koch, 1987, Zador et al., 1990, Holmes and Levy, 1990). In these models, that simulated calcium transients during an LTP induction protocol, a large calcium transient is taken to be the correlate of LTP. These models embedded a spine with NMDA receptors into a compartmental model of a dendrite and simulated how stimulation of multiple synapses on the same dendritic branch at various frequencies affects the resulting calcium transients in a single spine. Interestingly these different studies came to a common conclusion that to generate large calcium transients during LTP the calcium buffers must be saturated, thus amplifying the small signal observed in the calcium currents. However, these studies did not take into account back propagating action potentials (BPAP) or active dendritic conductances.

An influential hypothesis, the calcium control hypothesis, postulates that a large calcium transient produces LTP whereas a moderate increase in calcium results in LTD (Lisman, 1989). This hypothesis, which was first proposed on the basis of theoretical considerations, has subsequently received significant experimental support. Several different models for the induction of synaptic plasticity are based, either explicitly or implicitly, on this hypothesis (Lisman, 1989, Shouval et al., 2002, Karmarkar and Buonomano,2002).

In order to simulate synaptic plasticity given different calcium transients, it is necessary to define rules that translate calcium dynamics in postsynaptic spines to changes in synaptic strength. A simple choice (Karmarkar and Buonomano, 2002) is to assume that the peak of the calcium transients determines the sign and magnitude of the synaptic weight change. This type of rule however is not a dynamical system, and has some unnatural consequences, for example that the width of calcium transients have no effect on the magnitude of plasticity. A simple dynamical system that implements the calcium control hypothesis has the form (Shouval et al., 2002): \[\tag{3} {dW_i\over dt}=\eta(Ca)\left(\Omega(Ca)-\lambda W_i\right). \]

Here \(W_i\) is the synaptic efficacy of synapse \(i\ ,\) \(\Omega\) (Fig 2a) determines the sign magnitude of synaptic plasticity as a function of \(Ca\) levels, \(\eta\) is a calcium dependent learning rate, which is typically a monotonically increasing function of calcium, and \(\lambda\) is decay constant which can be generically set to \(\lambda=0\ ,\) no decay, or \(\lambda=1\ .\) If \(\lambda=1\ ,\) for sustained calcium elevation, the synaptic efficacy converges to the value of \(\Omega\) and the rate of convergence depends on \(\eta\ .\)

On the basis of the calcium control hypothesis, and mathematical models of NMDA receptors, which are both glutamate dependent and voltage dependent it is very easy to simulate pairing induced plasticity (Fig. 1d-f). Calcium transients induced by paring presynaptic stimulation with either a moderate or a large postsynaptic depolarization (Fig. 3a) are significantly different due to the voltage dependence of NMDA receptors. Consequently (Fig. 3b) the change in the simulated synaptic weights, after 100 presynaptic stimuli, is a function of the level of the postsynaptic voltage during induction, in qualitative agreement with experimental results. In these simple simulations all calcium influx is assumed to originate from NMDA receptors. Some models that take into account calcium influx through voltage gated calcium channels (VGCC) as well produce results that are qualitatively similar for this induction protocol. Note that with this induction protocol there are no postsynaptic spikes.

Rate dependent induction protocols, although experimentally easy to induce, are actually difficult to model, since the details and parameters of the modeled postsynaptic neuron will significantly effect when postsynaptic spikes occur, and therefore significantly effect the resulting plasticity curves. However, parameters can be chosen for the postsynaptic model that the model produces plasticity curves consistent with experimental results (Shouval et al., 2002).

Calcium dependent models that simulate STDP (Fig 1g-i) must take into account how the voltage in the postsynaptic spine depends on the action potential generated in the postsynaptic neuron. Information about the postsynaptic spike is typically assumed to be conveyed via the BPAP. However, a narrow BPAP is unable to account for why LTD is induced when \(\Delta t < 0\ .\) To account for such LTD a BPAP with a wide tail potential is either explicitly assumed, or implicitly included by the parameter choice of the postsynaptic neuron. Calcium based models that account for STDP typically result in another form of LTD at \(\Delta t > 0\ ,\) which is a consequence of the continuity of the magnitude of Ca transients as a function of \(\Delta t\) (Fig. 4). Continuity implies that if Ca influx is large with small \( \Delta t \) and zero with very large \( \Delta t \ ,\) then for moderately large \( \Delta t Ca \) influx must pass through the range of moderate increase that would induce LTD. One model, proposed by Rubin and co. workers (Rubin et. al., 2005) suggests that at these intermediate values of \(\Delta t\) that naturally produce LTD, another biochemical process is initiated which essentially prevents this LTD. Another possibility is that stochastic properties of calcium influx might significantly reduce the magnitude of LTD at \( \Delta t > 0\) (Shouval and Kalantzis, 2005). Additionally, two-coincidence models have been proposed to eliminate LTD at \( \Delta t > 0\) as described below. Experimental evidence is divided regarding the existence of LTD at \( \Delta t > 0\) (Bi and Poo, 1998, Wittenberg and Wang, 2006).

An alternative to the calcium control hypothesis (Froemke et al., 2005), which has not yet been rigorously modeled, assumes that baseline stimulation produce calcium transients with an intermediate amplitude, which results in no plasticity (Fig. 2c), LTP is caused by a large elevation of Calcium, however LTD is induced by calcium transients smaller than baseline. This alternative can account for time dependent LTD on the basis of the assumption that during a post-pre induction protocol, the postsynaptic action potential causes an inactivation of the NMDAR that results in a smaller calcium influx during post-pre protocols than during presynaptic stimulation alone. Although such a model can account for STDP, it is unclear how it can account for paradigms in which a presynaptic stimulus is paired with postsynaptic depolarization (Fig. 1d-f).

### Modeling the signal transduction pathways associated with synaptic plasticity.

The first influential model of the molecular network leading to LTP and LTD was constructed by Lisman (1989) to address the question of how the same molecule, calcium, can trigger both LTP and LTD. The correlate of synaptic strength in Lisman's model is the activation level of CaMKII. Lisman's model postulates that moderate calcium preferentially activates phosphatases, which dephosphorylate CaMKII, whereas high calcium levels cause a net phosphorylation of CaMKII, thus accounting for bidirectional synaptic plasticity. Recent experimental evidence has shown that a correlate of synaptic plasticity is the phosphorylation state of the AMPA receptors: LTP is correlated with phosphorylation at s831 a CaMKII site and LTD is correlated with dephosphorylation at s845, a PKA site. Models by Castellani et. al. (2005), simulated some of the kinases and phosphatases involved in the signal transduction pathways leading from calcium transients to synaptic plasticity. These models show that under various assumptions, bidirectional synaptic plasticity could indeed be accounted for by these pathways. Under certain conditions, such enzymatic models can be approximated by the simpler calcium dependent models described by equation 3.

Both the models of Lisman (1989) and subsequent models by Castellani represent only a limited portion of the extensive signal transduction pathway or pathways related to synaptic plasticity. Several papers by Bhalla and co workers (Ajay and Bhalla, 2004) have modeled more components of this signal transduction pathway. Other models of the signal transduction pathways assume a simpler more phenomenological approach. For example Abarbanel et. al. (2003) postulate a dynamical variable that induces depression (D), which can be taken as an analog for phosphatases and a potentiation inducing variable (P), analogous to a kinase. Both these variables are assumed to be directly calcium dependent similarly to Lisman 1989. These kinases and phosphatases are the basis of a dynamical equation determining the change in synaptic weight, via the following equation:

\[\tag{4} {dW\over dt}=g_0\{P(t)D(t)^\gamma-D(t)P(t)^\gamma\}, \]

where \( g_0\) is a scaling constant and the parameter \( \gamma \ ,\) which is set in the
range of \( 2-4 \ ,\) enforces competition between the P and D variables.
In this plasticity equation phosphatases and kinases
are in competition with each other, in order to determine if LTP or LTD will be induced.
This equation is set in a purely phenomenological manner, and no attempt is made to justify its
form on the basis of the underlying biophysics, or to use experimentally obtained kinetic coefficients. This model, with appropriate parameters, can account for many induction protocols, including induction by postsynaptic calcium transients alone (Yang et al., 1999) and STDP, and like the calcium based models of the previous section it produces a pre-post form of LTD.

The methodology used by all these models is based on deterministic ordinary differential equations. Such an approach implicitly assumes big and well mixed compartments, assumptions that might not hold for single spines. In addition, detailed molecular of the signal transduction pathways leading to synaptic plasticity (Ajay and Bhalla, 2004, Castellani et al., 2005) assume we know the key molecules and kinetic coefficients of their interactions, assumptions that might not currently hold (Castellani et al., 2005). Moreover, synapses are small and structured, implying that the mass action approach that underlies single compartment ODE models might not be appropriate.

### Two coincidence models and other pathways for induction

Most calcium based biophysical models described above predict an STDP curve in which there is a pre-post form of LTD in addition to the standard post-pre form of LTD (but see, Kalantzisn and Shouval, 2005, Rubin et al., 2005). There are indications that pre-post LTD exists in hippocampal slices (Wittenberg and Wang, 2006), but not in neocortical slices (Sjostrom et al., 2001). An alternative to the single coincidence models describe above are two-coincidence models , which postulate that LTP is triggered by calcium influx through NMDA receptors, whereas LTD by a second coincidence detector that is sensitive to post before pre activation. One alternative is that the second coincidence detector is implemented by calcium influx through VGCC concurrent with the activity of metabotropic glutamate receptors (mGluR) (Karmarkar and Buonomano, 2002). Experimental work has indeed shown that postsynaptic NMDA receptors might not be necessary for spike timing dependent LTD (Sjostrom et al., 2003, Bender et al., 2006), and that mGluR (Bender et al., 2006) and VGCC (Bi and Poo, 1998, Bender et al., 2006) are necessary for spike timing dependent LTD. Some experiments also found that the activation of cannabinoid receptors is necessary for this LTD (Sjostrom et al., 2003, Bender et al., 2006). Although elements of the two coincidence model have received support, the biophysical details of this second coincidence detector have yet to be elucidated.

An early computational model that can be interpreted as a two coincidence model was proposed by Senn et. al. (2001). In that model NMDAR's can be in three states, inactive, a state resulting in LTP triggered by pre-post stimuli, and a state leading to LTD triggered by post-pre stimuli. The equation governing the dynamics of this NMDA receptor are quite different from what we know of the biophysics of NMDA receptors, therefore this NMDA receptor is better interpreted as a phenomenological construct.

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**Internal references**

- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Wolfram Schultz (2007) Reward. Scholarpedia, 2(3):1652.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

## External links

## See Also

Learning, Long-Term Depression, Long-Term Potentiation, Memory, Neuron, Synapse, STDP, Synaptic Plasticity