Subsections

• 10.4.1 A Simple Mechanistic Model
  • 10.4.1.1 Example: LTP and LTD
  
  
• 10.4.2 A Kinetic Model based on NMDA Receptors
  • 10.4.2.1 Example: Low Rates
  • 10.4.2.2 Example: High Rates
  
  
• 10.4.3 A Calcium-Based Model
  • 10.4.3.1 NMDA receptor as a coincidence detector
  • 10.4.3.2 The calcium control hypothesis
  • 10.4.3.3 Dynamics of the postsynaptic neuron
  • 10.4.3.4 Results

10.4 Detailed Models of Synaptic Plasticity

In the previous section we have introduced a purely phenomenological model for spike-time dependent synaptic plasticity which is at least qualitatively in agreement with experimental results. In this section we take a slightly different approach and discuss how the core idea of this model, the learning window, arises from elementary kinetic processes. We start in Section 10.4.1 with a simple mechanistic model and turn then, in Section 10.4.2 to a more detailed model with saturation. A calcium-based model is the topic of Section 10.4.3. All three models give a qualitative explanation for the learning dynamics on the level of individual spikes.

10.4.1 A Simple Mechanistic Model

The AND condition in Hebb's postulate suggests that two biochemical components are involved in the induction of LTP. We do not wish to speculate on the nature of these components, but simply call them a and b. We assume that the first component is generated by a chemical reaction chain triggered by presynaptic spike arrival. In the absence of further input, the concentration [a] decays with a time constant $\tau_{a}^{}$ back to its resting level [a] = 0. A simple way to describe this process is

$${{\text{d}}\over {\text{d}}t} {[a]\over \tau_a} \sum_{{f}}^{} \delta$$ (10.28)

where the sum runs over all presynaptic firing times t_j^(f). Equation (10.29) states that [a] is increased at each arrival of a presynaptic spike by an amount d_a. A high level of [a] sets the synapse in a state where it is susceptible to changes in its weight. The variable [a] by itself, however, does not yet trigger a weight change.

To generate the synaptic change, another substance b is needed. The production of b is controlled by a second process triggered by postsynaptic spikes,

$${{\text{d}}\over {\text{d}}t} {[b]\over \tau_b} \sum_{{f}}^{} \delta$$ (10.29)

where $\tau_{b}^{}$ is another time constant. The sum runs over all postsynaptic spikes t_i^(f). Note that the second variable [b] does not need to be a biochemical quantity; it could, for example, be the electrical potential caused by the postsynaptic spike itself.

Hebbian learning needs both `substances' to be present at the same time, thus

$${{\text{d}}\over {\text{d}}t} \gamma$$ (10.30)

with some rate constant $\gamma$. The upper index corr is intended to remind us that we are dealing only with the correlation term on the right-hand side of Eq. (10.14) or Eq. (10.24).

Let us now consider the synaptic change caused by a single presynaptic spike at t_j^(f)$\ge$ 0 and a postsynaptic spike a t_i^(f) = t_j^(f) - s. Integration of Eqs. (10.29) and (10.30) yields

$$\tau_{a}^{} \Theta$$ [a(t)]

$$\tau_{b}^{} \Theta$$ [b(t)] (10.31)

where $\Theta$(.) denotes the Heaviside step function as usual. The change caused by the pair of pulses (t_i^(f), t_j^(f)), measured after a time T, is

$$\int_{0}^{T} \left(\vphantom{{{\text{d}}\over {\text{d}}t} w_{ij}^{\rm corr}}\right. {{\text{d}}\over {\text{d}}t} \left.\vphantom{{{\text{d}}\over {\text{d}}t} w_{ij}^{\rm corr}}\right) \gamma \int_{{\max\{t_j^{(f)}, t_i^{(f)}\}}}^{T} \left[\vphantom{-{t-t_j^{(f)}\over \tau_a} - {t-t_i^{(f)}\over \tau_b}}\right. {t-t_j^{(f)}\over \tau_a} {t-t_i^{(f)}\over \tau_b} \left.\vphantom{-{t-t_j^{(f)}\over \tau_a} - {t-t_i^{(f)}\over \tau_b}}\right]$$

The integral over t can be calculated explicitely. The total weight change that is obtained for T $\gg$ $\tau_{a}^{}$,$\tau_{b}^{}$ can be identified with the learning window. Thus we find

$$\left\{\vphantom{ \begin{array}{*{2}{c@{\quad}}c} A\,\exp[s/\tau_... ...{\rm for} & s<0 \\ A\,\exp[-s/\tau_b]& {\rm for} & s>0 \, \end{array} }\right. \begin{array}{*{2}{c@{\quad}}c} A\,\exp[s/\tau_a] & {\rm for} & s<0 \\ A\,\exp[-s/\tau_b]& {\rm for} & s>0 \, \end{array}$$ (10.32)

with s = t_j^(f) - t_i^(f) and A = $\gamma$ d_a d_b $\tau_{a}^{}$$\tau_{b}^{}$/($\tau_{a}^{}$ + $\tau_{b}^{}$). As expected, the change of the synaptic efficacy depends only on the time difference between pre- and postsynaptic spike (Gerstner et al., 1998); cf. Fig. 10.8.

Figure 10.8: Exponential learning window W as a function of the time difference s = t_j^(f) - t_i^(f) between presynaptic spike arrival and postsynaptic firing. The time constants for exponential decay are $\tau_{1}^{}$ = 20ms for s < 0 and $\tau_{2}^{}$ = 10ms for s > 0.

Equation (10.34) describes the change caused by a single pair of spikes. Given a train of presynaptic input spikes and a set of postsynaptic output spikes, many combinations of firing times (t_i^(f), t_j^(f)) exist. Due to the linearity of the learning equation (10.31), the total change is additive, which is consistent with Eq. (10.14).

The combination of two kinetic processes a and b thus yields an exponential learning window as in Eq. (10.16) but with A₊ = A_-. The learning window either describes LTP ($\gamma$ > 0) or LTD ($\gamma$ < 0), but not both. If we want to have an anti-symmetric learning window with LTP and LTD we need additional processes as detailed below.

10.4.1.1 Example: LTP and LTD

For a learning window incorporating both LTP and LTD, we need more microscopic variables. Let us suppose that, as before, we have variables [a] and [b] that contribute to LTP according to (10.31), viz.,

$${{\text{d}}\over {\text{d}}t} \gamma^{{\rm LTP}}_{}$$ (10.33)

Similarly, we assume that there is a second set of variables [c] and [d], that initiate LTD according to

$${{\text{d}}\over {\text{d}}t} \gamma^{{\rm LTD}}_{}$$ (10.34)

The variables [c] and [d] have a dynamics analogous to Eq. (10.29) and Eq. (10.30) with amplitudes d_c and d_d, and time constants $\tau_{c}^{}$ and $\tau_{d}^{}$. The total weight change is the sum of both contributions,

w_ij^corr = w_ij^LTP + w_ij^LTD , (10.35)

and so is the learning window, i.e.,

$$\left\{\vphantom{ \begin{array}{*{2}{c@{\quad}}c} A_+\, \exp[s/\t... ...p[-s/\tau_b]- A_- \, \exp[-s/\tau_d] & {\rm for} & s>0 \\ \end{array} }\right. \begin{array}{*{2}{c@{\quad}}c} A_+\, \exp[s/\tau_a] - A_-\, \exp... ... A_+\,\exp[-s/\tau_b]- A_- \, \exp[-s/\tau_d] & {\rm for} & s>0 \\ \end{array}$$ (10.36)

with A₊ = $\gamma^{{\rm LTP}}_{}$ d_a d_b $\tau_{a}^{}$$\tau_{b}^{}$/($\tau_{a}^{}$ + $\tau_{b}^{}$) and A_- = $\gamma^{{\rm LTD}}_{}$ d_c d_d $\tau_{c}^{}$$\tau_{d}^{}$/($\tau_{c}^{}$ + $\tau_{d}^{}$) (Gerstner et al., 1996a,1998).

We now set d_b = 1/$\tau_{b}^{}$ and d_c = 1/$\tau_{c}^{}$. In the limit of $\tau_{b}^{}$$\to$ 0 and $\tau_{c}^{}$$\to$ 0, we find the asymmetric two-phase learning window introduced in Eq. (10.16). Weight changes are now instantaneous. A postsynaptic spike that is triggered after a presynaptic spike arrival reads out the current value of [a] and induces LTP by an amount

$$\gamma^{{\rm LTP}}_{} \left(\vphantom{ - {t_i^{(f)}- t_j^{(f)}\over \tau_a}}\right. {t_i^{(f)}- t_j^{(f)}\over \tau_a} \left.\vphantom{ - {t_i^{(f)}- t_j^{(f)}\over \tau_a}}\right)$$ (10.37)

A presynaptic spike t_j^(f) that arrives after a postsynaptic spike reads out the current value of [d] and induces LTD by an amount

$$\gamma \left(\vphantom{ - {t_j^{(f)}- t_i^{(f)}\over \tau_d}}\right. {t_j^{(f)}- t_i^{(f)}\over \tau_d} \left.\vphantom{ - {t_j^{(f)}- t_i^{(f)}\over \tau_d}}\right)$$ (10.38)

10.4.2 A Kinetic Model based on NMDA Receptors

A model for LTP and LTD that is slightly more elaborate than the simplistic model discussed in the previous section has been developed by Senn et al. (2001b,1997). This model is based on the assumption that NMDA receptors can be in one of three different states, a resting state, an `up' and a `down' state. Transitions between these states are triggered by presynaptic spike arrival (`rest'$\to$ `up') and postsynaptic firing (`rest'$\to$ `down'). The actual induction of LTP or LTD, however, requires another step, namely the activation of so-called second-messengers. The model assumes two types of second-messenger, one for LTP and one for LTD. If a presynaptic spike arrives before a postsynaptic spike, the upregulation of the NMDA receptors in combination with the activitation S₁$\to$S^up of the first second-messenger triggers synaptic changes that lead to LTP. On the other hand, if the presynaptic spike arrives after postsynaptic firing, the NMDA receptors are downregulated and the activation S₂$\to$S^dn of the other second-messenger triggers LTD; cf. Fig. 10.9

Figure 10.9: Upper part of panel: A presynaptic spike shifts NMDA receptors from the rest state N^rest to the up-regulated state N^up. If a postsynaptic spike arrives shortly afterwards, a second messenger S¹ will be activated ( S^up). Depending on the amount of activated second messengers S^up, postsynaptic spikes lead to LTP. Lower part: Postsynaptic spikes down-regulate NMDA receptors ( N^dn). In the presence of N^dn, presynaptic spikes activate another second messenger ( S^dn) leading to LTD.

The variables N^up, N^dn, and N^rest describe the portion of NMDA receptors that are in one of the three possible states ( N^up + N^dn + N^rest = 1). In the absence of pre- and postsynaptic spikes, all receptors return to the rest state,

$${{\text{d}}\over {\text{d}}t} {N^{\text{up}}\over {\tau_{N^{\rm up}}}} {N^{\text{dn}} \over {\tau_{N^{\rm dn}}}}$$ (10.39)

N^up and N^dn decay with time constants $\tau_{{N^{\rm up}}}^{}$ and $\tau_{{N^{\rm dn}}}^{}$, respectively. Whenever a presynaptic spike arrives, NMDA receptors are up-regulated from rest to the `up'-state according to

$${{\text{d}}\over {\text{d}}t} \sum_{f}^{} \delta {N^{\text{up}}(t)\over{\tau_{N^{\rm up}}}}$$ (10.40)

where t_j^(f) is the arrival time of a presynaptic spike and r^up is the proportion of receptors in the `rest' state that are up-regulated. Since presynaptic spike arrival triggers release of the neurotransmitter glutamate, which is then bound to the NMDA receptors, the `up'-state can be identified with a state where the receptor is saturated with glutamate.

Firing of a postsynaptic spike at time t_i^(f) leads to a down-regulation of NMDA receptors via

$${{\text{d}}\over {\text{d}}t} \sum_{f}^{} \delta {N^{\text{dn}}(t) \over{\tau_{N^{\rm dn}}}}$$ (10.41)

Senn et al. (2001b) suggest that down-regulation of the NMDA receptor is mediated by the intracellular calcium concentration that changes with each postsynaptic spike. Note that, since N^rest = 1 - N^up - N^dn, Eqs. (10.42) and (10.43) account for saturation effects due to a limited number of NMDA-receptors.

The secondary messenger S^up, which finally leads to LTP, is activated by postsynaptic spikes, but only if up-regulated NMDA channels are available. In the absence of postsynaptic spikes the concentration of second messengers decays with time constant $\tau_{{S^{\rm up}}}^{}$. Thus

$${{\text{d}}\over {\text{d}}t} {S^{\text{up}}(t) \over {\tau_{S^{\rm up}}}} \sum_{{f}}^{} \delta$$ (10.42)

where r_S is a rate constant. Since N^up(t) > 0 requires that a presynaptic spike has occurred before t, the activation of S^up effectively depends on the specific timing of pre- and postsynaptic spikes (`first pre, then post').

Similarly, the other second messenger S^dn is activated by a presynaptic spike provided that there are receptors in their down regulated state, i.e.,

$${{\text{d}}\over {\text{d}}t} {S^{\text{dn}} (t) \over {\tau_{S^{\rm dn}}}} \sum_{{f}}^{} \delta$$ (10.43)

where $\tau_{{S^{\rm dn}}}^{}$ is another decay time constant. The second messenger S^dn is therefore triggered by the sequence `first post, then pre'. The factors [1 - S^up] in Eq. (10.44) and [1 - S^dn] in Eq. (10.45) account for the limited amount of second messengers available at the synapse.

Long-term potentiation (weight increase) depends on the presence of S^up, long-term depression (weight decrease) on S^dn. This is described by

$${{\text{d}}\over {\text{d}}t} \gamma_{{\rm LTP}}^{} \theta^{{\rm up}}_{} \sum_{f}^{} \delta \Delta$$ =

$$\gamma_{{\rm LTD}}^{} \theta^{{\rm d}}_{} \sum_{f}^{} \delta \Delta$$ (10.44)

with certain parameters $\gamma_{{\text{LTP/D}}}^{}$ and $\theta^{{\rm up/dn}}_{}$. Here, [x]₊ = x $\Theta$(x) denotes a piecewise linear function with [x]₊ = x for x > 0 and zero otherwise. The delay 0 < $\Delta$ $\ll$ 1 ensures that the actual weight change occurs after the update of S^up, S^dn. Note that this is a third-order model. The variable S^up > 0, for example, is already second-order, because it depends on presynaptic spikes followed by postsynaptic action potentials. In Eq. (10.46) the postsynaptic spike is then used again in order to trigger the weight change.

10.4.2.1 Example: Low Rates

For low pre- and postsynaptic firing rates, saturation effects can be neglected and Eq. (10.46) is equivalent to the elementary model discussed in Section 10.4.1. Let us assume that a single spike induces a small change ( r^dn, r^up $\ll$ 1) so that we can use N^rest $\approx$ 1 in Eqs. (10.42) and (10.43). The equations for N^up and N^dn are then identical to those for the `substances' [a] and [d] in Eqs. (10.29), (10.35), and (10.36).

Let us furthermore assume that interspike intervals are long compared to the decay time constants $\tau_{{S^{\rm up}}}^{}$,$\tau_{{S^{\rm dn}}}^{}$ in Eqs. (10.44) and (10.45). Then S^up is negligible except during and shortly after a postsynaptic action potential. At the moment of postsynaptic firing, S^up `reads out' the current value of N<sup>up</sup>; cf. Eq. (10.44). If this value is large than $\Theta_{{\rm up}}^{}$, it triggers a positive weight change; cf. Eq. (10.46). Similarly, at the moment of presynaptic spike arrival S<sup>dn</sup> `reads out' the value of N^dn and triggers a weight decrease. Thus, in this limit, the model of Senn et al. corresponds to an exponential time window

$$\left\{\vphantom{ \begin{array}{*{2}{c@{\qquad}}c} ~A_+(w_{ij})\,... ...ij}) \, \exp[-s/{\tau_{N^{\rm dn}}}] & {\rm for} & s>0 \\ \end{array} }\right. \begin{array}{*{2}{c@{\qquad}}c} ~A_+(w_{ij})\, \exp[+s/{\tau_{N^... ... A_-(w_{ij}) \, \exp[-s/{\tau_{N^{\rm dn}}}] & {\rm for} & s>0 \\ \end{array}$$ (10.45)

with A₊(w_ij) = r^up r_S (1 - w_ij) and A_-(w_ij) = - r^dn r_S w_ij.

10.4.2.2 Example: High Rates

If we assume that all decay time constants are much longer than typical interspike intervals then the variables N^up/dn and S^up/dn will finally reach a steady state. If we neglect correlations between pre- and postsynaptic neuron by replacing spike trains by rates, we can solve for these stationary states,

$$\infty {{\tau_{N^{\rm up}}}\,{r^{\rm up}}\,\nu_j \over 1 + {\tau_{N^{\rm up}}}\,{r^{\rm up}}\,\nu_j+ {\tau_{N^{\rm dn}}}{r^{\rm dn}}\nu_i}$$ = (10.46)

$$\infty {{\tau_{S^{\rm up}}}\, r_S\, N^{\text{up}}\, \nu_i \over 1+ {\tau_{S^{\rm up}}}r_S\, N^{\text{up}} \, \nu_i}$$ = (10.47)

and similar equations for N^dn and S^dn. Note that S^up_$\infty$ is a function of $\nu_{i}^{}$ and $\nu_{j}^{}$. If we put the equations for S^up_$\infty$ and S^dn_$\infty$ in Eq. (10.46) we get an expression of the form

$${{\text{d}}\over {\text{d}}t} \gamma_{{\rm LTP}}^{} \nu_{i}^{} \nu_{j}^{} \nu_{i}^{} \gamma_{{\rm LTD}}^{} \nu_{i}^{} \nu_{j}^{} \nu_{j}^{}$$ (10.48)

with functions f_LTP and f_LTD. We linearize f_LTP with respect to $\nu_{j}^{}$ about a reference value $\overline{{\nu}}$ > 0 and evaluate f_LTD at $\nu_{j}^{}$ = $\overline{{\nu}}$ in order to make the right-hand side of Eq. (10.50) linear in the input $\nu_{j}^{}$. The result is

$${{\text{d}}\over {\text{d}}t} \phi \nu_{i}^{} \nu_{j}^{}$$ (10.49)

with $\phi$(w_ij;0) = $\phi$(w_ij;$\nu_{\theta}^{}$) = 0 for some value $\nu_{\theta}^{}$ and d$\phi$/d$\nu_{i}^{}$ < 0 at $\nu_{i}^{}$ = 0. Equation (10.51) is a generalized Bienenstock-Cooper-Monroe rule where $\phi$ does not only depend on the postsynaptic rate $\nu_{i}^{}$ but also on the individual synaptic weight; cf. Eq. (10.12). For details see Senn et al. (2001b); Bienenstock et al. (1982).

10.4.3 A Calcium-Based Model

It has been recognized for a long time that calcium ions are an important second messenger for the induction of LTP and LTD in Hippocampus (Malinow et al., 1989; Malenka et al., 1988; Lisman, 1989) and cerebellum (Konnerth and Eilers, 1994; Lev-Ram et al., 1997). Particularly well investigated are `NMDA synapses' in the hippocampus (Dudek and Bear, 1992; Bliss and Collingridge, 1993; Bindman et al., 1991; Collingridge et al., 1983) where calcium ions can enter the cell through channels that are controlled by a glutamate receptor subtype called NMDA (N-methyl-D-aspartic acid) receptor; cf. Section 2.4.2. These channels are involved in transmission of action potentials in glutamatergic (excitatory) synapses. If an action potential arrives at the presynaptic terminal, glutamate, the most common excitatory neurotransmitter, is released into the synaptic cleft and diffuses to the postsynaptic membrane where it binds to NMDA and AMPA receptors. The binding to AMPA receptors results in the opening of the associated ion channels and hence to a depolarization of the postsynaptic membrane. Channels controlled by NMDA receptors, however, are blocked by magnesium ions and do not open unless the membrane is sufficiently depolarized so as to remove the block. Therefore, calcium ions can enter the cell only if glutamate has been released by presynaptic activity and if the postsynaptic membrane is sufficiently depolarized. The calcium influx is the first step in a complex bio-chemical pathway that leads ultimately to a modification of the glutamate-sensitivity of the postsynaptic membrane.

Biophysical models of Hebbian plasticity (Schiegg et al., 1995; Zador et al., 1990; Holmes and Levy, 1990; Shouval et al., 2001; Gold and Bear, 1994; Lisman, 1989) contain two essential components, viz. a description of intracellular calcium dynamics, in particular a model of calcium entry through NMDA synapses; and a hypothesis of how the concentration of intracellular calcium influences the change of synaptic efficacy. In this section we give a simplified account of both components. We start with a model of NMDA synapses and turn then to the so-called calcium control hypothesis of Shouval et al. (2001).

10.4.3.1 NMDA receptor as a coincidence detector

We have emphasized in Sections 10.1-10.3 that all Hebbian learning rules contain a term that depends on the correlation between the firings of pre- and postsynaptic neurons. The signaling chain that leads to a weight change therefore has to contain a nonlinear processing step that requires that pre- and postsynaptic neurons are active within some short time window. Synaptic channels controlled by NMDA receptors are an excellent candidate for a biophysical implementation of this condition of `coincidence' because the opening of the channel requires both the presence of glutamate which reflects presynaptic activity and, in order to remove the magnesium block, a depolarization of the postsynaptic membrane (Mayer et al., 1984; Nowak et al., 1984). A strong depolarization of the postsynaptic membrane does occur, for example, during the back propagation of an action potential into the dendritic tree (Stuart and Sakmann, 1994; Linden, 1999), which is a signature for postsynaptic activity.

Figure 10.10: NMDA-synapse. A. Vesicles in the presynaptic terminal contain glutamate as a neurotransmitter (filled triangles). At resting potential, the NMDA receptor mediated channel (hatched) is blocked by magnesium (filled circle). B. If an action potential (AP) arrives at the presynaptic terminal the vesicle merges with the cell membrane, glutamate diffuses into the synaptic cleft, and binds to NMDA and non-NMDA receptors on the postsynaptic membrane. At resting potential, the NMDA receptor mediated channel remains blocked by magnesium whereas the non-NMDA channel opens (bottom). C. If the membrane of the postsynaptic neuron is depolarized, the magnesium block is removed and calcium ions can enter into the cell. D. The depolarization of the postsynaptic membrane can be caused by a back propagating action potential (BPAP).

In a simple model of NMDA-receptor controlled channels, the calcium current through the channel is described by

$$\alpha$$ (10.50)

cf. Chapter 2.4. Here g_Ca is the maximal conductance of the channel and E_Ca is the reversal potential of calcium. The time course of NMDA binding at the receptors is described by $\alpha$(t - t_j^(f)) where t_j^(f) is the time of spike arrival at the presynaptic terminal. The function

$${1\over 1 + 0.28 \, e^{-0.062\,u}}$$ (10.51)

describes the unblocking of the channel at depolarized levels of membrane potential.

The time course of $\alpha$ is taken as a sum of two exponentials with the time constant of the slow component in the range of 100ms. If there are several presynaptic spikes within 100ms, calcium accumulates inside the cell. The change of the intracellular calcium concentration [Ca²⁺] can be described by

$${{\text{d}}\over {\text{d}}t} {[{\rm Ca}^{2+}](t)\over \tau_{\rm Ca}}$$ (10.52)

where $\tau_{{\rm Ca}}^{}$ = 125 ms is a phenomenological time constant of decay. Without any further presynaptic stimulus, the calcium concentration returns to a resting value of zero. More sophisticated models can take calcium buffers, calcium stores, and ion pumps into account (Schiegg et al., 1995; Zador et al., 1990; Gamble and Koch, 1987).

10.4.3.2 The calcium control hypothesis

While the dynamics of NMDA synapses is fairly well understood in terms of the biophysical processes that control receptor binding and channel opening, much less is known about the complex signaling chain that is triggered by calcium and finally leads to a regulation of the synaptic efficacy (Lisman, 1989). Instead of a developing a detailed model, we adopt a phenomenological approach and assume that the change of the synaptic efficacy w_ij is fully determined by the intracellular calcium concentration [Ca²⁺]; an assumption that has been called `calcium control hypothesis' (Shouval et al., 2001). More specifically, we write the weight change as

$${{\text{d}}\over {\text{d}}t} {\Omega([{\rm Ca}^{2+}]) - w_{ij} \over \tau([{\rm Ca}^{2+}])}$$ (10.53)

For constant calcium concentration, the weight w_ij reaches an asymptotic value $\Omega$([Ca²⁺]) with time constant $\tau$([Ca²⁺]).

Figure 10.11: Calcium control hypothesis. The asymptotic weight value w_ij = $\Omega$([Ca²⁺]) (A) and the time constant $\tau$([Ca²⁺]) of weight changes (B) as a function of the calcium concentration; cf. Eq. (10.55) [adapted from Shouval et al. (2001)].

Figure 10.11 shows the graph of the function $\Omega$([Ca²⁺]) as it is used in the model of Shouval et al. (2001). For a calcium concentration below $\theta_{0}^{}$, the weight assumes a resting value of w₀ = 0.5. For calcium concentrations in the range $\theta_{0}^{}$ < [Ca²⁺] < $\theta_{m}^{}$, the weight tends to decrease, for [Ca²⁺] > $\theta_{m}^{}$ it increases. Qualitatively, the curve $\Omega$([Ca²⁺]) reproduces experimental results that suggest that a high level of calcium leads to an increase whereas an intermediate level of calcium leads to a decrease of synaptic weights. We will see below that the BCM rule of Eq. (10.12) is closely related to the function $\Omega$[Ca²⁺].

The time constant $\tau$([Ca²⁺]) in the model equation (10.55) decreases rapidly with increasing calcium concentration; cf. Fig. 10.11B. The specific dependence has been taken as

$$\tau {{\tau_0} \over [{\rm Ca}^{2+}]^3 + 10^{-4}}$$ (10.54)

where $\tau_{0}^{}$ = 500ms and [Ca²⁺] is the calcium concentration in $\mu$mol/l. At a low level of intracellular calcium ( [Ca²⁺]$\to$ 0), the response time of the weight w_ij is in the range of hours while for [Ca²⁺]$\to$1 the weight changes rapidly with a time constant of 500 ms. In particular, the effective time constant for the induction of LTP is shorter than that for LTD.

10.4.3.3 Dynamics of the postsynaptic neuron

In order to complete the definition of the model, we need to introduce a description of the membrane potential u_i of the postsynaptic neuron. As in the simple spiking neuron model SRM₀ (cf. Chapter 4), the total membrane potential is described as

$$\eta \hat{{t}}_{i}^{} \sum_{f}^{} \epsilon$$ (10.55)

Here $\epsilon$(t - t_j^(f)) is the time course of the postsynaptic potential generated by a presynaptic action potential at time t_j^(f). It is modeled as a double exponential with a rise time of about 5ms and a duration of about 50ms. The action potential of the postsynaptic neuron is described as

$$\eta \left(\vphantom{ 0.75\, e^{-{s / \tau_{\rm fast}}} +0.25\, e^{-{s/ \tau_{\rm slow}}} }\right. \tau_{{\rm fast}} \tau_{{\rm slow}} \left.\vphantom{ 0.75\, e^{-{s / \tau_{\rm fast}}} +0.25\, e^{-{s/ \tau_{\rm slow}}} }\right)$$ (10.56)

Here u_AP = 100 mV is the amplitude of the action potential and $\hat{{t}}_{i}^{}$ is the firing time of the last spike of the postsynaptic neuron. In contrast to the model SRM₀, $\eta$ does not describe the reset of the membrane potential at the soma, but the form of the back propagating action potential (BPAP) at the site of the synapse. It is assumed that the BPAP has a slow component with a time constant $\tau_{{\rm slow}}^{}$ = 35ms. The fast component has the same rapid time constant (about 1 ms) as the somatic action potential. The somatic action potential is not described explicitly.

10.4.3.4 Results

Given the above components of the model, we can understand intuitively how calcium influx at NMDA synapses leads to spike-time dependent plasticity. Let us analyze the behavior by comparing the calcium-based model with the elementary model of Section 10.4.1; cf. Eqs. (10.29)-(10.31). Binding of glutamate at NMDA receptors plays the role of the component a that is triggered by presynaptic firing; the back propagating action potential plays the role of the component b that is triggered by postsynaptic firing. As a result of the depolarization caused by the BPAP, the magnesium block is removed and calcium ions enter the cell. The calcium influx is proportional to the product of the NMDA-binding, i.e., the factor $\alpha$ in Eq. (10.52), and the unblocking, i.e., the factor B(u). Finally, the increase in the calcium concentration leads to a weight change according to Eq. (10.55).

Figure 10.12: Spike-time dependent plasticity in a calcium-based model. A. Calcium transient generated by a presynaptic spike in the absence of postsynaptic firing (bottom) and in the presence of a postsynaptic spike 10ms before (middle) or 10ms after (top) presynaptic spike arrival. Only for the sequence `pre-before-post' the threshold $\theta_{m}^{}$ for LTP can be reached. B. The final weights obtained after several thousands pre- and postsynaptic spikes that are generated at a rate of 1 Hz (solid line) or 3 Hz (dashed line). The weights are given as a function of the time difference between presynaptic spikes t_j^(f) and postsynaptic spikes t_i^(f). C. Pairing of presynaptic spikes with postsynaptic depolarization. The weights w_ij that are obtained after several hundreds of presynaptic spikes (at a rate of $\nu_{j}^{}$ = 0.5 Hz) as a function of the depolarization of the postsynaptic membrane. [Adapted from Shouval et al. (2001)].

A single presynaptic spike (without a simultaneous postsynaptic action potential) leads to a calcium transient that stays below the induction threshold $\theta_{0}^{}$; cf. Fig. 10.12A. If a postsynaptic spike occurs 10ms before the presynaptic spike arrival, the calcium transient has a somewhat larger amplitude that attains a level above $\theta_{0}^{}$. As a consequence, the weight w_ij is reduced. If, however, the postsynaptic spike occurs one or a few milliseconds after the presynaptic one, the calcium transient is much larger. The reason is that the blocking of the NMDA synapse is removed during the time when the NMDA receptors are almost completely saturated by glutamate. In this case, the calcium concentration is well above $\theta_{m}^{}$ so that weights increase. Since the time constant $\tau$([Ca²⁺]) is shorter in the regime of LTP induction than in the regime of LTD induction, the positive weight change is dominant even though the calcium concentration must necessarily pass through the regime of LTD in order to reach the threshold $\theta_{m}^{}$. The resulting time window of learning is shown in Fig. 10.12B. It exhibits LTP if the presynaptic spike precedes the postsynaptic one by less than 40ms. If the order of spiking is inverted LTD occurs. LTD can also be induced by a sequence of `pre-before-post' if the spike time difference is larger than about 40ms. The reason is that in this case the removal of the magnesium block (induced by the BPAP) occurs at a moment when the probability of glutamate binding is reduced; cf. the factor $\alpha$ in Eq. (10.52). As a consequence less calcium enters the cell - enough to surpass the threshold $\theta_{0}^{}$, but not sufficient to reach the threshold $\theta_{1}^{}$ of LTP. We emphasize that the form of the learning window is not fixed but depends on the frequency of pre- and postsynaptic spike firing; cf. Fig. 10.12B.

LTP and LTD can also be induced in the absence of postsynaptic spikes if the membrane potential of the postsynaptic neuron is clamped to a constant value. A pure spike-time dependent learning rule defined by a learning window W(t_j^(f) - t_i^(f)) is obviously not a suitable description of such an experiment. The calcium-based model of Shouval et al. (2001), however, can reproduce voltage-clamp experiments; cf. Fig. 10.12C. Presynaptic spike arrival at low frequency ( $\nu_{j}^{}$ = 0.5 Hz) is `paired' with a depolarization of the membrane potential of the postsynaptic neuron to a fixed value u₀. If u₀ is below -70mV, no significant weight change occurs. For -70 mV < u₀ < -50 mV LTD is induced, while for u₀ > - 50mV LTP is triggered. These results are a direct consequence of the removal of the magnesium block at the NMDA synapses with increasing voltage. The mean calcium concentration - and hence the asymptotic weight value - is therefore a monotonously increasing function of u₀.

Finally, we would like to emphasized the close relation between Fig. 10.12C and the function $\phi$ of the BCM learning rule as illustrated in Fig. 10.5. In a simple rate model, the postsynaptic firing rate $\nu_{i}^{}$ is a sigmoidal function of the potential, i.e., $\nu_{i}^{}$ = g(u_i). Thus, the mapping between the two figures is given by a non-linear transformation of the horizontal axis.