File:Fig1 comp.jpg

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 pos- tulates that when one neuron drives the activity of another neuron, the connection be- tween 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 suffciently coactive [Stent, 1973, Rochester et al., 1956, von der Malsburg, 1973, Sejnowski, 1977, Bienenstock et al., 1982]. The experimental correlates of these theoretically proposed forms of synaptic plasticity are called long term potentiation (LTP) and long- term depression (LTD) [Bliss and Lomo, 1973, Dudek and Bear, 1992]. Signi¯cant experi- mental evidence has accumulated to support the hypothesis that synaptic plasticity is indeed a basis for learning memory and various forms of development [Martin and Morris, 2002, Heynen et al., 2003, Whitlock et al., 2006]. Two broad classes of synaptic plasticity models can be described: 1) Phenomenological models: These are very simple model 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. This article focuses on biophysical models.

Phenomenological models

Phenomenological models are characterized by treating the process governing synaptic plas- ticity 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 physiol- ogy 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. 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:

<math eq:General_rate>

{dW_i\over dt}=f(x_i,y,W_i,other),      
</math>

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 [Bienenstock et al., 1982] can also have an effect. A simple example of a rate based model [Linsker, 1986] has the following form:

<math eq:Linsker>

{dW_i\over dt}=\eta(x_i-x_0)(y-y_0),    </math>    

where ´ is a small learning rate and \(x_0; y_0 \) are constants. Such simple models might not be stable, or sufficiently selective [MacKay and Miller, 1994] and additional normalizing factors are often added to ob- tain more stability or selectivity [Rochester et al., 1956, von der Malsburg, 1973, Oja, 1982]. An alternative approach to attain selectivity and stability was proposed by the BCM [Bienenstock et al., 1982, Intrator and Cooper, 1992] model, in which plasticity is a non-linear function of the postsynaptic variable and competition between different input patterns is accomplished via the sliding threshold mechanism.

Spike timing based models

The discovery of spike timing dependent plasticity (STDP) has prompted the develop- ment 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 [Song et al., 2000, Kempter et al., 1999], however recently a model that depends on spike triplets has been pub- lished [Pfister and Gerstner, 2006]. Under different 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 [Izhevitch and Desai, 2003]. Phenomenological STDP models are typically used to account for higher level phenomena such as the development of of receptive fields [Song et al., 2000, Kempter et al., 1999].

Figure 1: Different plasticity induction protocols. a. In rate based induction, extracellular stimulating and recording electrodes are used. b. First a baseline fEPSP is established using a very low frequency stimulus. Next a low frequency stimulation (hundreds of pulses at ~1-10Hz) is used to elicit LTD or a high frequency stimulus (~30-200 Hz) is used to elicit LTP. c. A schematic diagram of plasticity as a function of frequency. d. In paring based plasticity an extracellular stimulation is paired with an intracellular voltage clamp controlled depolarization. e. If the cell is slightly depolarized to -50 mV LTD is induces, whereas strong depolarization to 0 mV produces LTP. f. Plasticity curve as a function of postsynaptic depolarization. g. In STDP induction protocols one presyanptic and one postsyanptic cell are patch clamped. h. If the presynaptic cell is induced to repetitively fire 10 ms before the postsyanptic cell, LTP is induced. If the firing order is reversed LTD is induced. i. A schematic plasticity curve as a function of $\Delta t$. (All diagrams are schematic illustrations)

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 physi- ological and biochemical networks leading to synaptic plasticity, even the biophysical models rely on many simplifications and abstractions.

Calcium dependent models of bidirectional synaptic plasticity

Calcium influx into the postsynaptic spine is crucial for the induction of many forms of bidirec- tional synaptic plasticity. Much of the calcium entering the postsynaptic spine comes through NMDA receptors [Sabatini et al., 2002]. Blocking NMDA receptors pharmacologically can eliminate both LTP [Wigstrom and Gustafsson, 1986] and LTD [Dudek and Bear, 1992], and a par- tial block of NMDA receptors can convert an LTP protocol to LTD [Cummings et al., 1996]. 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, ]. In these models, that simulated calcium transients during an LTP in- duction 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 fre- quencies affect 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 cal- cium 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 LTD [Lisman, 1989, Bear et al., 1987]. This hypothesis, which was first proposed on the basis of theoretical considerations, has subsequently received signifcant experimental support [Yang et al., 1999, Cormier et al., 2001, Cummings et al., 1996]. Several different models for the induction of synaptic plasticity are based, either explicitly or implicitly, on this hypothesis [Lisman, 1989, Kitajima and Hara, , Shouval et al., 2002, Karmarkar and Buonomano, , Abarbanel et al., 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 simples 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]:

<math eq:CADP>

   {dW_i\over dt}=\eta(Ca)\left(\Omega(Ca)-\lambda W_i\right).   
   

</math> 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 \(\eta\).

Figure 2: Calcium dependent function controlling synaptic plasticity. a. The $\Omega$ function controls the sign and magnitude of synaptic plasticity. At low calcium concentrations corresponding to baseline activation, no chance in synaptic weight occurs. At intermediate calcium concentration LTD is induced, and at high calcium LTP is induced. b. The $\eta$ function, which controls the rate of convergence to the steady state, is a monotonically increasing function of calcium. c. An alternative theory \cite{FroemkeEtAl05} assumes that baseline produces intermediate calcium transients that cause no change, that LTD protocols produce weaker transients and LTP protocol, larger transients.

On the basis of the calcium control hypothesis, and mathematical models of NMDA re- ceptors, which are both glutamate dependent and voltage dependent [Jahr and Stevens, 1990] it is very easy to simulate pairing induce 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. Conse- quently (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 [Abarbanel et al., 2003, Kitajima and Hara, 2000] that take into account calcium influx through voltage gated calcium channels (VGCC) as well produce results that are qualitatively similar for this induction protocol.

Figure 3: Pairing induced plasticity a. Calcium transients induced by a single presynaptic stimulation at at resting membrane potential (black-dashed) at slightly depolarize potential (-50 mV, red) and for highly depolarized potential (0 mV, blue). b. The plasticity curve shows the magnitude of plasticity s a function of postsynaptic voltage. (Schematic diagrams)

Rate dependent induction protocols, although experimentally easy to induce, are actually diffcult to model, since the details and parameters of the modeled postsynaptic neuron will significantly effect when postsynaptic spikes occur, and therefore signifficantly effect the result- ing plasticity curves [Abarbanel et al., 2003, Kitajima and Hara, 2000, Shouval et al., 2002]. However, parameters can be chosen for the postsynaptic model that produce 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 postsynap- tic 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 s induced when \(\Delta t < 0\). To account for such LTD a BPAP with a wide tail potential is either explicitly assumed [Shouval et al., 2002, Karmarkar and Buonomano, 2002], or implicitly included by the pa- rameter choice of the postsynaptic neuron [Kitajima and Hara, 2000, Abarbanel et al., 2003]. Calcium based models that account for STDP typically result in another form of LTD at \(\Delta t > 0\), [Shouval et al., 2002, Karmarkar and Buonomano, 2002, Abarbanel et al., 2003,Kitajima and Hara, 2000] which in a consequence of the continuity of the magnitude of Ca transients as a function of \(\Delta t\) (Fig. 4) . However, 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, Feldman, 2000, Nishiyama et al., 2000, Wittenberg and Wang, 2006].

An alternative to the calcium control hypothesis [Froemke et al., 2005], which has not yet been rigouresly modeld, 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 pairing induced plasticity.

Figure 4: Spike timing dependent plasticity curves. a. Some experimental results indicate STDP curves with a single LTD region at $\Delta t<0$, and a single LTP region at $\Delta t>0$. b. Other experiments find an additional LTD region at larger values of $\Delta t>0$. Note that in all cases results of single experiments (+) diverge significantly from the simple fit curves. (Schematic diagrams)}

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 Lismans model is the activation level of CaMKII. Lismans model postulates that moderate calcium preferentially activates phosphatases, which dephosphorylate CaMKII, whereas high calcium levels cause a net phos- phorylation of CaMKII, thus accounting for bidirectional synaptic plasticity. Recent experimental evidence has shown that a correlate of synaptic plasticity is the phophorylation state of the AMPA receptors: LTP is correlated with phosphorylation at s831 [Barria et al., 1997, Lee et al., 2003] a CaMKII site and LTD is correlated with dephos- phorylation at s845, a PKA site [Lee et al., 1998, Lee et al., 2003]. Models by Castellani et. al. [Castellani et al., 2001, Castellani et al., 2005], simulate some of the Kinases and phos- phatases involved in the signal transduction pathways leading from calcium transients to synaptic plasticity. These models show that under various assumptions, bidirectional synap- tic plasticity could indeed be accounted for by these pathways. Under certain conditions, such enzymatic models enzymatic models can be approximated by the simpler calcium dependent models described by equation 3 [Shouval et al., 2002]. Both the models of Lisman (1989) and subsequent models by Castellani represent only a limited portion of the extensive signal transduction pathway or pathways re- lated to synaptic plasticity. Several papers by Bhalla and co workers [Bhalla et al., 2002, Ajay and Bhalla, 2004] have modeled more components of this signal transduction pathway. Other models of the signal transduction pathways assume a simpler more phenomeno- logical approach [Abarbanel et al., 2003, Rubin et al., 2005, D'Alcantara et al., 2003]. 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:

<math eq:Abarbanel>

{dW\over dt}=g_0\{P(t)D(t)^\gamma-D(t)P(t)^\gamma\}, </math>

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, assump- tions that might not hold for single spines. In addition, such models assume we know the key molecules and kinetic coe±cients of their interactions, assumptions that might not currently hold [Castellani et al., 2005].

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 [Shouval and Kalantzis, 2005, Rubin et al., 2005]). There are indications that pre-post LTD exists in Hippocampal slices [Nishiyama et al., 2000, Wittenberg and Wang, 2006], but not in neocortical slices [Sjostrom et al., 2001, Froemke and Dan, 2002, Feldman, 2000]. 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 detec- tor 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, Nevian and Sakmann, 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.

Alternatives to the idea that the second coincidence detector leads to LTD have been proposed. One idea is that calcium influx through VGCC suppress LTD [Zhou et al., 2005], or that specific calcium temporal profile invoke a veto mechanism to suppress LTD [Rubin et al., 2005], eliminating the pre-post form of LTD. 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 trigged 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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