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.
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 back to its resting level [a] = 0. A simple way to describe this process is
To generate the synaptic change, another substance b is needed. The production of b is controlled by a second process triggered by postsynaptic spikes,
Hebbian learning needs both `substances' to be present at the same time, thus
Let us now consider the synaptic change caused by a single presynaptic spike at t_{j}^{(f)} 0 and a postsynaptic spike a t_{i}^{(f)} = t_{j}^{(f)} - s. Integration of Eqs. (10.29) and (10.30) yields
[a(t)] | = d_{a} exp[- (t - t_{j}^{(f)})/] (t - t_{j}^{(f)}) | |
[b(t)] | = d_{b} exp[- (t - t_{i}^{(f)})/] (t - t_{i}^{(f)}) , | (10.31) |
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 ( > 0) or LTD ( < 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.
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.,
We now set d_{b} = 1/ and d_{c} = 1/. In the limit of 0 and 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
W(t_{j}^{(f)} - t_{i}^{(f)}) = d_{a} exp - for t_{j}^{(f)} < t_{i}^{(f)} . | (10.37) |
W(t_{j}^{(f)} - t_{i}^{(f)}) = - ^{LTD}d_{d} exp - for t_{j}^{(f)} > t_{i}^{(f)} . | (10.38) |
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' `up') and postsynaptic firing (`rest' `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 S_{1}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 S_{2}S^{dn} of the other second-messenger triggers LTD; cf. Fig. 10.9
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,
Firing of a postsynaptic spike at time t_{i}^{(f)} leads to a down-regulation of NMDA receptors via
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 . Thus
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.,
Long-term potentiation (weight increase) depends on the presence of
S^{up},
long-term depression (weight decrease) on
S^{dn}. This is described by
w_{ij} | = | (1 - w_{ij}) [S^{up} - ]_{+} (t - t_{i}^{(f)} - ) | |
- w_{ij} [S^{dn} - ]_{+} (t - t_{j}^{(f)} - ) | (10.44) |
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} 1) so that we can use N^{rest} 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 , 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^{up}; cf. Eq. (10.44). If this value is large than , it triggers a positive weight change; cf. Eq. (10.46). Similarly, at the moment of presynaptic spike arrival S^{dn} `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
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,
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).
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.
In a simple model of NMDA-receptor controlled channels, the calcium current through the channel is described by
The time course of 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 [Ca^{2+}] can be described by
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 [Ca^{2+}]; an assumption that has been called `calcium control hypothesis' (Shouval et al., 2001). More specifically, we write the weight change as
Figure 10.11 shows the graph of the function ([Ca^{2+}]) as it is used in the model of Shouval et al. (2001). For a calcium concentration below , the weight assumes a resting value of w_{0} = 0.5. For calcium concentrations in the range < [Ca^{2+}] < , the weight tends to decrease, for [Ca^{2+}] > it increases. Qualitatively, the curve ([Ca^{2+}]) 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 [Ca^{2+}].
The time constant ([Ca^{2+}]) in the model equation (10.55) decreases rapidly with increasing calcium concentration; cf. Fig. 10.11B. The specific dependence has been taken as
([Ca^{2+}]) = | (10.54) |
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 SRM_{0} (cf. Chapter 4), the total membrane potential is described as
u_{i}(t) = (t - ) + (t - t_{j}^{(f)}) . | (10.55) |
(s) = u_{AP}0.75 e^{-s/} +0.25 e^{-s/} . | (10.56) |
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 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).
A single presynaptic spike (without a simultaneous postsynaptic action potential) leads to a calcium transient that stays below the induction threshold ; 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 . 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 so that weights increase. Since the time constant ([Ca^{2+}]) 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 . 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 in Eq. (10.52). As a consequence less calcium enters the cell - enough to surpass the threshold , but not sufficient to reach the threshold 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 ( = 0.5 Hz) is `paired' with a depolarization of the membrane potential of the postsynaptic neuron to a fixed value u_{0}. If u_{0} is below -70mV, no significant weight change occurs. For -70 mV < u_{0} < -50 mV LTD is induced, while for u_{0} > - 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 u_{0}.
Finally, we would like to emphasized the close relation between Fig. 10.12C and the function of the BCM learning rule as illustrated in Fig. 10.5. In a simple rate model, the postsynaptic firing rate is a sigmoidal function of the potential, i.e., = g(u_{i}). Thus, the mapping between the two figures is given by a non-linear transformation of the horizontal axis.
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