# Short Term Plasticity, Biophysical Models

**DOI:**https://doi.org/10.1007/978-1-4614-7320-6_358-1

## Keywords

Deterministic Model Release Site Synaptic Efficacy Docked Vesicle Presynaptic Spike## Definition

**Short-term plasticity** refers to changes in synaptic efficacy in response to presynaptic spiking that persist for a few seconds at most but more often decay over a timescale of a few hundred milliseconds. There are two primary types of short-term plasticity: short-term depression and short-term facilitation. Short-term depression is a reduction in synaptic efficacy that is often, but not always, caused by a transient depletion of neurotransmitter vesicles. Short-term facilitation is an increase in synaptic efficacy often caused by a transient increase in the number of vesicles released by presynaptic action potentials. The detailed biophysical mechanisms underlying synaptic facilitation are discussed in Facilitation, Biophysical Models, and we therefore focus on more phenomenological models of facilitation here.

## Detailed Description

### Stochastic Models of Short-Term Depression Arising from Neurotransmitter Depletion

Short-term depression is often believed to arise primarily from the depletion of neurotransmitter vesicles, and therefore a large proportion of short-term depression models rely solely on the mechanism of vesicle depletion. It should be noted, though, that some synapses exhibit short-term depression that is inconsistent with vesicle depletion alone (Branco and Staras 2009; Zucker and Regehr 2002; Wong et al. 2003; Xu and Wu 2005).

We begin by describing a model of short-term depression that takes into account the stochastic nature of vesicle release and recovery. The model was introduced in Vere-Jones (1966), and various formulations of it have since been used extensively (Senn et al. 2001; Zador 1998; Wang 1999; Matveev and Wang 2000; Fuhrmann et al. 2002; Goldman et al. 2002; Goldman 2004; de la Rocha and Parga 2005; Pfister et al. 2010; Rosenbaum et al. 2012, 2013, 2014; Reich and Rosenbaum 2013). A presynaptic neuron makes *M* functional contacts onto a postsynaptic neuron, and each functional contact contains *N* _{0} release sites that can each dock at most one vesicle, so that the maximum number of docked vesicles is *M* × *N* _{0}. Of course, the model can easily be extended to allow heterogeneity in the number of release sites at each contact. Here, the term “functional contact” (hereafter, simply “contact”) refers to a group of vesicle release sites where the grouping is chosen so that release sites at different contacts release vesicles independently from one another, but release sites on the same contact are not necessarily independent. Functional contacts need not necessarily correspond to anatomical contacts in a one-to-one fashion. The most widely assumed dependence between release sites arises from the “univesicular hypothesis” which states that each contact can release at most one vesicle in response to a single presynaptic spike. If this constraint is to be met, then release sites at the same contact cannot be independent.

Since release sites at different contacts are independent, we need only specify the probability distribution of the number of vesicles released by a presynaptic spike at each contact. A widely used rule (Vere-Jones 1966; Wang 1999; Matveev and Wang 2000; Goldman et al. 2002; de la Rocha and Parga 2005; Rosenbaum et al. 2014) that is consistent with the univesicular hypothesis states one vesicle is released at a particular contact with probability 1 − (1 − *p*)^{ n }; otherwise, no vesicles are released. Here p is a number between 0 and 1, and *n*∈[0,*N* _{0}] is a dynamical variable representing the number of docked vesicles at the contact in question at the moment when the presynaptic spike occurs. When *n* = 0, the release probability is zero, as expected. When *n* = 1, the release probability is p. In general, the probability of release increases with the number of docked vesicles.

While the rule above is straightforward to simulate numerically, it is difficult to treat analytically due to the non-independence of release sites. Some analytical tractability can be gained by assuming that the release of a vesicle at each release site is independent. This is equivalent to assuming that each functional contact has exactly one release site (i.e., making the substitutions, *M* = *M* × *N* _{0} and *N* _{0} = 1). In this widely used version of the model (Zador 1998; Matveev and Wang 2000; Fuhrmann et al. 2002; Goldman 2004; Pfister et al. 2010; Rosenbaum et al. 2012, 2013; McDonnell et al. 2013; Reich and Rosenbaum 2013), each presynaptic spike releases each docked vesicle independently with probability p. Thus, the number of vesicles released by a presynaptic spike is a binomial random variable with parameters n and p, where n is the number of docked vesicles when the spike arrives.

When a vesicle is released at a contact, the number of docked vesicles at that contact is decremented. The waiting time until a released vesicle is recovered is an exponentially distributed random variable with mean *τ* _{ u }. Equivalently, it is the waiting time for the first event in a Poisson process with rate 1/*τ* _{ u }. Due to the memoryless property of Poisson processes, this rule allows the modeler to keep track of only the *number* of docked vesicles at each contact and ignore the time at which empty release sites became empty. In particular, the probability that one empty release site is filled during the time interval (*t*,*t* + *dt*), i.e., that the number of releasible vesicles is incremented during that time interval, is given by (*N* _{0} − *n*)*dt* + *o*(*dt*), where n is the number of docked vesicles at that release site and *o*(*dt*)/*dt* → 0 as *dt* → 0.

Each released vesicle induces a change in conductance of the postsynaptic neuron’s membrane. This is often modeled by setting the postsynaptic conductance to \( g(t)={\displaystyle \sum_j{w}_j\alpha \Big(t-}{t}_j\Big) \), where *t* _{ j } is the time of the *j*th presynaptic spike, *w* _{ j } is the number of vesicles released by the *j*th spike, and *α*(*t*) is a stereotyped synaptic conductance waveform representing the change in postsynaptic conductance elicited by the release of a single vesicle. In general, *w* _{ j }∈[0,*M* × *N* _{0}] but *w* _{ j }∈[0,*M*] whenever the univesicular hypothesis is satisfied.

Pseudo-code for implementing this and similar models of stochastic vesicle dynamics is provided in McDonnell et al. (2013) along with a review and discussion of some effects of stochastic vesicle dynamics on neural coding and the statistics of the synaptic response.

### Deterministic Models of Short-Term Facilitation and Depression

The stochastic model of short-term depression discussed above can be difficult to analyze mathematically, and it can also be difficult to fit the model’s parameters to data. Moreover, the model assumes that short-term depression is due solely to the depletion of neurotransmitter vesicles even though other factors sometimes play a role. Finally, the model does not account for the effects of short-term facilitation on synaptic efficacy.

These difficulties are partly overcome by a deterministic model in which synaptic efficacy is treated as an abstract, continuous variable that is modified by each presynaptic spike. This model is mostly agnostic to the precise mechanisms responsible for changes in synaptic efficacy, but it can accurately reproduce experimental data after the model parameters are fit to the data using statistical fitting algorithms. Due to its amenability to mathematical analysis and its ability to reproduce experimental data, this deterministic model and its variations are widely used in both theoretical and experimental studies (Tsodyks and Markram 1997; Varela et al. 1997; Tsodyks et al. 1998; Markran et al. 1998; Chance et al. 1998; Maas and Zador 1999; Fuhrmann et al. 2002; Hanson and Jaeger 2002; Cook et al. 2003; Grande and Spain 2005; Rothman et al. 2009; Lindner et al. 2009; Merkel and Lindner 2010; Pfister et al. 2010; Rosenbaum et al. 2012; Scott et al. 2012; Mohan et al. 2013).

We describe the model as presented in Varela et al. (1997) and then discuss some common variations of the model. The efficacy of a synapse is defined by *A*(*t*) = *A* _{0} *F*(*t*)*D* _{1}(*t*)*D* _{2}(*t*)*D* _{3}(*t*), where *A* _{0} is a constant representing a baseline efficacy, *F*(*t*) is a dynamic variable representing facilitation, and each *D* _{ k }(*t*) is a dynamic variable representing depression.

Immediately after each presynaptic spike, each depression variable is updated according to the rule *D* _{ k } ← *D* _{ k } *d* _{ k } for *k* = 1, 2, 3, where each *d* _{ k } is a number between 0 and 1 representing the amount of depression evoked by each spike. Between presynaptic spikes, the depression variables decay exponentially back to their maximal values, which can be normalized to one without loss of generality, so that \( {\tau}_{Dk}\frac{d{D}_k}{ dt}=1-{D}_k \), where *τ* _{ Dk } is the timescale of recovery for the *k*th depression variable.

Similarly, the facilitation variable is updated after each presynaptic spike according to *F* ← *F* + *f*, and the evolution between spikes is defined by \( {\tau}_F\frac{ DF}{d\mathsf{t}}=1-F. \)

The postsynaptic conductance (or current for current-based modeling) can then be written in terms of the synaptic efficacy as \( g(t)={\displaystyle \sum_jA\left({t}_j\right)\alpha \left(t-{t}_j\right).} \), where *α*(*t*) is a stereotyped waveform and *t* _{ j } is the time of the *j*th presynaptic spike.

This model is commonly modified by changing the number of facilitation or depression variables. A common variation uses only one facilitation and one depression variable so that *A*(*t*) = *A* _{0} *F*(*t*)*D*(*t*). Some synapses exhibit short-term depression but not facilitation, in which case the facilitation variable can be removed completely and we can simply write *A*(*t*) = *A* _{0} *D*(*t*). These two variations of the model are amenable to mathematical analysis (Lindner et al. 2009; Merkel and Lindner 2010; Rosenbaum et al. 2012).

In the form of the model described above, the depression variables are updated multiplicatively, while the facilitation variable is updated additively. Updating facilitation multiplicatively can cause synaptic efficacy to grow unrealistically large at high presynaptic rates (Varela et al. 1997), but a saturating multiplicative update rule was applied in Hanson and Jaeger (2002) and accurately captured recorded data. In other variations of the model, updates to the depression variables depend on the value of the facilitation variables, for example, in Lindner et al. (2009) and Merkel and Lindner (2010), the update rule for the depression is given by, *D* _{ k } ← (1 − *F*)*D* _{ k }.

It should be noted that the deterministic model with one depression variable and no facilitation variables is equivalent to the mean field of the stochastic model of short-term depression with *N* _{0} = 1. Specifically, if the stochastic model is simulated over several trials with the same presynaptic spike train, then the trial average of the postsynaptic conductance it produces is proportional to the postsynaptic conductance produced by one trial of the deterministic model driven by the same presynaptic spike train whenever *τ* _{ u } = *τ* _{ D } and *p* _{ r } = *d*.

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## Further Reading

- Branco T, Staras K (2009) The probability of neurotransmitter release: variability and feedback control at single synapses. Nat Rev Neurosci 10:373–383PubMedCrossRefGoogle Scholar
- Senn W, Markram H, Tsodyks M (2001) An algorithm for modifying neurotransmitter release probability based on pre- and postsynaptic spike timing. Neural Comput 13:35–67PubMedCrossRefGoogle Scholar