2.1 Linear SBC rule

In general, we would like to find a rule that describes the change in a synaptic weight over time as a function of the calcium concentration, i.e. some differential equation of the form \(\frac=f\left(Ca\left(t\right)\right)\), where \(w\) is the synaptic weight, \(Ca\left(t\right)\) is the calcium at that synapse at time \(t\), and \(f\) is some function. It is also possible to use a discrete-time formulation appropriate for numerical computer simulation using a first-order Taylor approximation, where the value of the weight at each time step, \(w\left(t\right)\), is equal to the weight at the previous time-step plus a modification that depends both on the present weight and the weight at the current time step, i.e. \(w\left(t\right)=w\left(t-1\right)+\Delta w\), where \(\Delta w= f\left(Ca\left(t\right), w\left(t\right)\right)\). (In the equations below, for parsimony, we will generally write \(w\) instead of \(w\left(t\right)\), but the synaptic weight is always a function of time).

We first consider the calcium-based plasticity rules of (Shouval et al., 2002), starting with the simplest formulation. In this rule, synaptic strength is modified in a straightforward manner according to the depression and potentiation thresholds: at any time step \(t\), if the calcium concentration [Ca2+] is in the pre-depressive range (\(\left[^\right]< _\)) the synaptic weight \(w\) remains unchanged. If the calcium concentration is in the depressive range (\(_ \le \left[^\right]\le _\)), \(w\) is decreased, and if the [Ca2+] is in the potentiating range (\(\left[^\right]> _\)), \(w\) is increased. Formally, the change in the synaptic weight \(\Delta w\) is given by:

$$\Delta w=\eta \Omega \left(Ca\left(t\right)\right)$$

(2.1.1)

where \(\eta\) is the learning rate and \(\Omega\) is the two-threshold calcium based plasticity function described above, which can be expressed most simply as a step function:

$$\Omega\left(Ca\left(t\right)\right)=\left\\begin0,&\;\;Ca(t)<\theta_D\end\\\begink_D,&\theta_D\leq Ca(t)\leq\theta_P\end\\\begink_P,&Ca(t)>\theta_P\end\end\right.$$

(2.1.2)

where \(_\) and \(_\) are the signed rates of depression and potentiation, respectively (\(_<0, _>0)\), and \(_\) and \(_\) represent the thresholds for depression and potentiation. If smooth transitions between regions are desired, this can also be expressed with a soft threshold using the sum of sigmoids (slightly modified from Shouval et al., 2002):

$$\Omega \left(Ca\left(t\right)\right)=\frac_}^_\left(Ca\left(t\right)-_\right)}}+\frac__}^_\left(Ca\left(t\right)-_\right)}}$$

(2.1.3)

where \(_}\) and \(_}\) control the sharpness of the transitions between regions in the \(\Omega\) function.

In the linear SBC rule, synaptic weights increase linearly at a rate of \(_\) in the potentiative region of \(\left[^\right]\), decrease linearly at rate of \(_\) in the depressive region of \(\left[^\right]\) and remain stable in the pre-depressive region of \(\left[^\right]\). The change in synaptic weight depends only on the calcium concentration, not on the present value of the synaptic weight (Fig. 1B1-1B2). When applying our canonical calcium step stimuli or a potentiative or depressive level of calcium, we observe that the synaptic weights linearly increase or decrease while the calcium stimulus is active, and then the weights cease to change when the \(\left[^\right]\) is dropped to 0 (Fig. 1B3).

(We note that it may be more biologically plausible to implement the plasticity as a delayed function of the calcium signal, in which case one may substitute \(Ca\left(t-D\right)\) in place of \(Ca\left(t\right)\), where \(D\) indicates the duration of the temporal delay between the calcium signal and the plastic effect. For simplicity, however, we will assume that there is no such delay, i.e. \(D=0\).)

2.2 SBC rule with weight decay

To prevent synapses becoming arbitrarily large or small, (Shouval et al., 2002) added a weight decay term to the original plasticity rule.

$$\Delta w=\eta \left(\Omega \left(Ca\left(t\right)\right)- \lambda w\right)$$

(2.2.1)

where \(\lambda\) is the rate of decay. Importantly, the change in the weight \(\Delta w\) now depends on both the [Ca2+] and the present value of the weight \(w\). It can be instructive to visualize \(\Delta w\) as a function of both \(w\) and [Ca2+]. We show how the SBC rule with weight decay differs from the rule without it. When weight decay is added, the magnitude of the weight change depends both on the calcium level and the current weight (Fig. 1C1-1C2).

The difference in dynamical behavior between the SBC rule with weight decay (Eq. 2.2.1) and the linear version (Eq. 2.1.1.) become clear when applying the calcium step experiment described above. In the linear SBC rule without weight decay, the weight linearly increases (or decreases) for as long as the calcium pulse is present, then immediately stops changing (i.e., remains stable) when the calcium is turned off. By contrast, when weight decay is used, in the pre-depressive range of [Ca2+], \(w\) decreases (or increases, if it is negative) asymptotically to the fixed point of 0 at a rate of \(\eta\uplambda\). In the depressive [Ca2+] range, \(w\) decreases asymptotically to the fixed point of \(\frac_}\) (or increases if the weight is below that point). In the potentiating range of [Ca2+], \(w\) increases asymptotically to the fixed point of \(\frac_}\) (or decreases if the weight is above the fixed point) (Fig. 1C3). Note that in this framework, there is no way to independently change the asymptotic behavior without changing either the rates of potentiation and depression (\(_\) and \(_\)) or the decay rate \(\uplambda\).

2.3 SBC rule with weight decay and Ca 2+-dependent learning rate

Weight decay that drifts quickly toward 0 in the absence of plasticity-inducing calcium may be an undesirable feature of a plasticity rule if we wish to model synapses that are potentiated or depressed for a long duration. (Shouval et al., 2002) therefore introduced a sigmoidal [Ca2+]-dependent learning rate, \(\eta \left(Ca\left(t\right)\right)\), to mitigate the effect of the weight decay in the absence of calcium. The SBC rule with the calcium-dependent learning rate is thus defined as:

$$\Delta w=\eta \left(Ca\left(t\right)\right)*(\Omega \left(Ca\left(t\right)\right)- \lambda w)$$

(2.3.1)

The basic idea is that instead of having a constant learning rate, the learning rate (including the rate of the weight decay) increases in a sigmoidal fashion with the amount of calcium (Fig. 1D). Thus, at pre-depressive levels of [Ca2+] the weight will decay slowly, allowing for greater stability over long time horizons. (Fig. 1E1-1E3).

2.4 Fixed point – learning rate version of the SBC rule

In the FPLR framework, we propose a modified version of the SBC rule which allows the modeler to specify the fixed points and learning rates explicitly in all three regions of [Ca2+]:

$$\Delta w=\eta \left(Ca\left(t\right)\right)*\left(F\left(Ca\left(t\right)\right)-w\right)$$

(2.4.1)

Here, instead of \(\Omega \left(Ca\left(t\right)\right)\), we use \(F\left(Ca\left(t\right)\right)\), a step function which describes the fixed points of the weights as a function of the [Ca2+]. (We note that if \(\lambda\) is fixed to 1 in the original SBC rule [Eq. 2.3.1)], \(\Omega \left(Ca\left(t\right)\right)\) also specifies the fixed points of the weights, however \(F\left(Ca\left(t\right)\right)\) is defined this way explicitly.) Similar to the SBC rule, \(\eta \left(Ca\left(t\right)\right)\) here is also a 3-valued step function which determines the rate at which the weight asymptotically approaches the fixed point for each calcium level. (We require \(\eta \left(Ca\right)\le 1\) for all values of \(Ca\) to prevent oscillations; \(\eta \left(Ca\left(t\right)\right)=1\) is a special case in which the synapse immediately jumps to the fixed point specified by \(F\left(Ca\left(t\right)\right)\), which can be useful for modeling discrete-state synapses.)

In this framework, the learning rate \(\eta \left(Ca\left(t\right)\right)\) defines the fraction of the difference between the current weight and the fixed point \(F\left(Ca\left(t\right)\right)\) which is traversed at each time step. We call this the “one-dimensional” version of the FPLR rule, because both the fixed points and the learning rates depend only the calcium concentration at that time step.

For example, we might have:

$$F\left(Ca(t)\right)=\left\0.5,&Ca(t)<\theta_D\\0,&\theta_D\leq Ca(t)<\theta_P\\1,&Ca(t)\geq\theta_P\end\right.$$

(2.4.2)

And

$$\eta\left(Ca(t)\right)=\left\0.015,&Ca(t)<\theta_D\\0.15,&\theta_D\leq Ca(t)<\theta_P\\0.25,&Ca(t)\geq\theta_P\end\right.$$

(2.4.3)

This means that synapses with pre-depressive calcium concentrations (\(Ca\left(t\right)< _\)) eventually drift toward a “neutral” state of 0.5 at a rate of 0.015, synapses with a depressive calcium concentration (\(_\le Ca\left(t\right)<_\)) will depress towards 0 at a rate of 0.15, and synapses with a potentiative [Ca2+] (\(Ca\left(t\right)\ge _\)) will be potentiated towards 1 at a rate of 0.25 (Fig. 1F, 1G). (See (Enoki et al., 2009) for experimental evidence that synapses at baseline can be either potentiated or depressed. Note that the fixed points and rates used here and in subsequent sections are specified in arbitrary units and meant to illustrate qualitative dynamics of the synapse only; see Fig. 5 and Methods for biologically plausible parameters).

The FPLR rule is structurally similar to the SBC rule with weight decay and calcium-dependent learning rate (Eq. 2.3.1) (Fig. 1D-E). The advantage of the FPLR formulation, however, is the explicit interpretation of each piece of the equation. In the FPLR formulation there is no ‘weight decay’, there are rather only fixed points \(F\left(Ca\left(t\right)\right)\), and learning rates, \(\eta \left(Ca\left(t\right)\right)\), which are independently specified.

We can also turn off the drift in the pre-depressive region entirely by setting \(\eta \left(Ca\left(t\right)<_\right)=0\), thus allowing for synapses that are stable at every weight unless modified by depressive or potentiative calcium concentrations (Fig. 1H,1I). This version of the rule can be useful for contexts in which the modeler is primarily interested in understanding the early phase of plasticity, or for theoretical work exploring the consequences of synaptic weights that potentiate and depress asymptotically. (Turning off the pre-depressive drift can also potentially affect the behavior of the synaptic weights in the early phase of plasticity in plasticity protocols where synaptic stimulation is relatively slow, e.g. on the order of 1 Hz. However, because the time scale of the drift is 2–3 orders of magnitude slower than the time scale of the early phase plasticity, these effects are likely to be negligible when modeling plasticity during the early phase of plasticity.)

We note that if we use a differential equation interpretation of Eq. (2.4.1), i.e. we consider weight changes over time to be continuous such that \(\frac=\Delta w\), it is possible to find a closed-form solution to the synaptic weight for a constant pulse of calcium of magnitude \(C\) that starts at time \(_\) and ends at time \(_\):

$$w\left(_\right)=F(C)+\left(w\left(_\right)-F\left(C\right)\right)^_-_\right)}$$

(2.4.4)

Equation (2.4.4) describes an exponential growth or decay process (depending on whether the target fixed point \(F\left(C\right)\) is above or below the initial weight \(w\left(_\right)\)) that asymptotically drives the synaptic weight toward the fixed point \(F\left(C\right)\) with rate \(\eta\). This behavior is apparent when we apply our calcium step stimuli. This formulation is also computationally useful; if we are interested in merely finding the final value of a synaptic weight following some extensive plasticity protocol and not calculating the weight change at each time point, it is only necessary to apply Eq. (2.4.4) once for each time the [Ca2+] crosses one of the plasticity thresholds to calculate the final synaptic weight value.

2.5 Versatility of the FPLR framework

Until now, all of the above rules have assumed that there are only three regions of [Ca2+] relevant for plasticity and that the depression threshold is lower than the potentiation threshold, i.e. \(_<_\). There are some experimental results that complicate this picture. For example, there is evidence that in Purkinje neurons, the depression threshold is higher than the potentation threshold, i.e. \(_>_\) (Coesmans et al., 2004). This can also easily be implemented in the FPLR rule by changing the fixed points in each [Ca2+] region. (Fig. 2A, 2B).

Fig. 2

Incorporating multiple thresholds with the FPLR rule. (A) Possible fixed points and learning rates for Purkinje cells, where the potentiation threshold is lower than the depression threshold (\(_\)< \(_)\).(B1) \(\Delta w\) in modified SBC rule for Purkinje cells using the fixed points and learning rates from (A). (B2) Phase plane heatmap for Purkinje rule (B3) Stimulation protocol for Purkinje cells. Here, the \(\left[^\right]\) used for potentiation (red line) is less than the \(\left[^\right]\) used to induce depression. (B4) Weights over time in Purkinje cells for the stimulation protocols in (B3). Dotted lines represent fixed points. (C) Fixed points and learning rates, defined using a soft-threshold step function, with two additional regions of \(\left[^\right]\) where \(\eta =0\) and therefore no plasticity is induced. \(_\): no man’s land, \(_\): post-potentiative neutral zone. (D1) \(\Delta w\) in FPLR rule using the fixed points and learning rates from (C). Note the U-shaped dependence of depression magnitude on the \(\left[^\right]\) resulting in a no man’s zone between the depressive and potentiative regions of \(\left[^\right]\). Irregularities near the boundaries of \(\left[^\right]\) regions are due to the soft transitions in the fixed point and plasticity rate functions. (D2) Phase plane heatmap for rule shown in (C). (D3) Stimulation protocol for each region of \(\left[^\right]\) shown in (F). (D4) Weights over time for the stimulation protocols in (D3). Note that when the \(\left[^\right]\) is in the no man’s land (green line) depression occurs at a much slower rate, and in the post-potentiative neutral zone (black line) nearly no plasticity occurs

Moreover, even within hippocampal and cortical cells, there may be additional regions of [Ca2+] where the plasticity dynamics change. Cho et al. (Cho et al., 2001) found that within the depressive region of [Ca2+], the magnitude of depression exhibits a U-shaped relationship with the calcium concentration, such that a “No man’s land” appears at the boundary between the depressive and potentiative region of [Ca2+], where the [Ca2+] is too large to induce depression but too small to induce potentiation (Cho et al., 2001; Lisman, 2001). It may make sense for a modeler to include this no-man's land more explicitly in the plasticity rule. There is also some evidence (see Fig. 3 in Tigaret et al., 2016) that there is some maximum level of [Ca2+] beyond which potentiation mechanisms are inactivated. It would be worthwhile to be able to incorporate this “post-potentiative neutral zone” as well. These “neutral zones” can easily be incorporated into the FPLR by adding additional thresholds into the step function from Eq. (2.4.1) and setting the learning rate to 0 in those regions (the fixed points in the no-plasticity regions can be chosen arbitrarily, as they are irrelevant if the learning rate in those regions is 0).

Fig. 3

Graupner-Brunel rule and simplification. (A1-A3) \(\Delta w\) as a function of \(w\) for the original GB rule (green line) and for the simplified GB rule (pink line) in the pre-depressive, depressive, and potentiative regions of \(\left[^\right]\), respectively. Pink circle in A1 indicates unstable fixed point. (B1-B3) Weights over time in the original GB rule in each region of calcium (as in A1-A3) for synapses with different initial weights. (C1-C3) Weight changes, phase plane heatmap, and response to step stimuli (See Figure 1A) in the GB model. Note that weights drift in different directions after the end of the stimulus due to their final weights at time E. (D1-D3) As in (B1-B3) for the simplified version of the GB model. (E1-E3) As in (C1-C3) for the simplified version of the GB model

If desired, we can also explicitly model the U-shaped dependency between [Ca2+] and synaptic depression. Although we have used a hard threshold step function to implement the previous examples of the FPLR rule, in principle both \(F\left(Ca\left(t\right)\right)\) and \(\eta \left(Ca\left(t\right)\right)\) can be arbitrary functions as long as \(\eta \left(Ca\left(t\right)\right)<1\) for all values of \(Ca\left(t\right)\). To implement a U-shaped region, we can extend the soft threshold step function from Eq. (2.1.3) into a general form to include arbitrary regions of [Ca2+]:

$$\eta \left(Ca\left(t\right)\right)= \sum_^\frac_-_}^_\left(Ca\left(t\right)-_\right)}}+_$$

(2.5.1)

where \(_\dots _\dots _\) are the thresholds that differentiate between regions of [Ca2 +] ordered such that \(_< _\), \(_\dots b}_\dots _\) determine the steepness of transition between each [Ca2 +] region, \(_\dots \eta }_\dots _\) are the learning rates in each [Ca2 +] region, and \(_\) is the learning rate in the region between 0 and \(_\). This representation approaches the equivalent step function as the values for \(_\) become sufficiently large. (A similar equation can be used for \(F\left(Ca\left(t\right)\right)\)). Using Eq. (2.5.1), we can easily create the U-shaped relationship between [Ca2+] and synaptic depression (Fig. 2C, 2D).

2.6 Graupner and Brunel model

One drawback of the SBC plasticity rule (Eq. 2.3.1) is that even in the final version with a calcium-dependent learning rate, synaptic weights eventually trend toward 0 in the presence of pre-depressive levels of calcium. There is some experimental evidence, however, that synapses are bistable, existing in a potentiated (UP) state with weight \(_\) or a depressed (DOWN) state with weight \(_\), and that synaptic strengths slowly trend toward one of those two states depending on the early synaptic strength after inducing a plasticity protocol (Bagal et al., 2005; O’Connor et al., 2005a; Petersen et al., 1998). (Graupner & Brunel, 2012) proposed a model that captured these dynamics. In their model, the synaptic efficacy, \(\rho\) (\(\rho\) is linearly mapped to \(w\) according to the equation \(w= _+ \rho \left(_- _\right)\)) asymptotically decreases to the DOWN state (at \(\rho =0\)) in the presence of depressive [Ca2+] or increases asymptotically to the UP state (near \(\rho =1\)) in the presence of potentiating [Ca2+]. When the [Ca2+] is pre-depressive, \(\rho\) either increases toward 1 or decreases toward 0 on a very slow time scale depending only on the present value of \(\rho\). Specifically, if \(\rho\) is larger than the value of an unstable fixed point \(_\) (set to 0.5), \(\rho\) trends toward the UP state whereas if \(\rho\) is smaller than this value, \(\rho\) trends toward the DOWN state. Formally, we have:

$$\begin\tau \frac = -\rho \left(1-\rho \right)\left(_-\rho \right)+ _\left(1-\rho \right)\Theta \left[Ca\left(t\right)-_\right]\\ - _\rho\Theta [Ca\left(t\right)- _]+ Noise(t)\end$$

(2.6.1)

where \(\tau\) is the overall time constant of synaptic change and \(_\) and \(_\) are parameters that denote the rate of potentiation and depression, respectively.

For consistency with the SBC rule, without loss of generality, we can replace \(\rho\) in Eq. (2.6.1) with \(w\) (\(_\) is the unstable drift fixed point of \(w\)), replace \(_\) and \(_\) with \(_\) and \(_\), respectively (\(\tau\) = \(\frac\), representing a global time constant for the rule), and we can restate Eq. (2.6.1) as a discretized formulation expressed as a step function (ignoring the noise term):

$$\tau \Delta w = \left\-w(1-w)(w_* - w), & Ca(t) < \theta_D \\-w(1-w)(w_* - w) - \eta_D w, & \theta_D \leq Ca(t) < \theta_P \\ -w(1-w)(w_* - w) - \eta_D w + \eta_P(1-w), & Ca(t) \geq \theta_P\end\right.$$

(2.6.2)

The first line in Eq. (2.6.2) expresses the calcium-independent dynamics of the slow drift to the UP or DOWN state in a hyperbolic fashion depending on the value of \(w\) (Fig. 3A1, 3B1, green lines). The second line in Eq. (2.6.2) contains two terms; the first term is the calcium-independent drift term from before, and the second term is only active while the [Ca2+] is above the depression threshold and describes the asymptotic depressive dynamics. Because the hyperbolic drift term is always active, it contributes somewhat to the dynamics of the weight change in the depressive region, although if \(_>1\) the depressive dynamics will dominate the drift dynamics (Fig. 3A2, 3B2, green lines). The third line in Eq. (2.6.2) contains three terms: the first two terms are the drift and depressive terms, as before, and the final term describes the asymptotic potentiative dynamics. As such, both the drift term and the depressive term contribute to the dynamics of plasticity in the potentiative region of [Ca2+], but if \(_>_\), the potentiative effect can dominate, but only for relatively low weights. (Fig. 3A3, 3B3, green lines).

The overall profile of the GB rule, as well as its response to calcium step stimuli, appear in (Fig. 3C1-3C3). When a depressive calcium step stimulus is applied to the synapse, the synapse starts depressing toward the DOWN state (0), and when the stimulus turns off, the weight continues to drift downward toward 0, but at a slower pace. When a potentiative stimulus is applied, however, the synaptic weight asymptotes at a point substantially below the UP value (1), because the fixed point in the potentiative region of [Ca2+] is affected by the depressive process which is still active during potentiation, as described above. Nevertheless, once the calcium step is turned off, the potentiated synapse drifts toward the UP value. (Fig. 3C3). This behavior differs from that of the SBC model and one-dimensional FPLR rule, where the potentiated synapse would also eventually drift down toward 0 after the potentiative stimulus is turned off.

2.7 Simplified Graupner and Brunel model

While the GB rule (Eq. 2.6.2) can describe a variety of experimental results, its dynamics can be complicated by the fact that multiple processes are active simultaneously – that is, the slow calcium-independent drift of the first term is always active, and the depressive process is always active when the potentiation process is active. From a modeling standpoint, this aspect of the GB rule may not be desirable, as the dynamics in each region of [Ca2+] do not exhibit simple asymptotic behavior, the fixed point for potentiation is not trivial to specify (note that \(w=1\) is not a fixed point of \(w\) if \(a\left(t\right)>_\), see Fig. 3B3), and specifying \(_\) is insufficient to know the actual rate of potentiation because the depressive term \(_\) also affects the potentiation rate.

From a biological standpoint, it is also questionable whether depressive and potentiating processes are active simultaneously. While some studies have shown that depressive and potentiative mechanisms are operative at the same time and compete with each other (Burrell & Li, 2008; O’Connor et al., 2005b), another study (Cho et al., 2001) argues that once the [Ca2+] reaches the potentiation threshold, the depressive mechanisms are turned off. The slow bistable drift mechanisms in the first term of Eq. (2.6.2) may also not be perpetually active; the long-term stabilization mechanisms required for late LTP/LTD (L-LTP and L-LTD) have been shown to be protein synthesis dependent, and this protein synthesis may only occur after the induction of early-LTP/LTD (Barco et al., 2008; Frey & Morris, 1997; Redondo et al., 2010).

We therefore propose a simplified version of the Graupner-Brunel rule which only has a single term active for any given concentration of calcium and synaptic efficacy value, resulting in a rule which achieves a qualitatively similar result but with more straightforward dynamics. Here, the rate of change within each region of the \(\left[^\right]\) is independently specified by its own learning rate (\(_, _,_\)).

$$\Delta w=\left\-\eta_w,&Ca(t)<\theta_D\;\text\;w<w^\ast\\\eta_(1-w),&Ca(t)<\theta_D\;\text\;w>w^\ast\\0,&Ca(t)<\theta_D\;\text\;w=w^\ast\\-\eta_Dw,&\theta_D\leq Ca(t)<\theta_P\\\eta_P(1-w),&Ca(t)\geq\theta_P\end\right.$$

(2.7.1)

In this rule, in the pre-depressive region of \(\left[^\right]\), \(w\) drifts asymptotically (exponentially, not sigmoidally, as in the original GB rule) toward 0 at a rate of \(_\) if \(w\) is below \(^\) (first line), or asymptotically toward 1 at a rate of \(_\) if \(w\) is above \(^\) (second line). If w is exactly equal to \(^\), no weight change occurs (unstable fixed point) (Fig. 3A1, 3D1, pink lines). In the depressive region of calcium (fourth line), \(w\) trends asymptotically toward the fixed point of 0 at a rate of \(_\) (Fig. 3A2, 3D2, pink lines), and in the potentiative region of calcium (fifth line), \(w\) trends asymptotically toward the fixed point of 1 (unlike the original GB rule) at a rate of \(_\) Fig. 3A3, 3D3, pink lines). The profile of this plasticity rule and its response to calcium step stimuli are shown in Fig. E1-E3. Note that when the potentiative calcium step is applied, in contrast to the original GB rule, the synaptic weight is potentiated asymptotically toward 1, and it continues to drift toward 1 after the stimulus is turned off (Fig. 1E3).

Astute readers may notice that the simplified GB rule is similar in structure to the FPLR rule we described earlier. In fact, the last two lines of Eq. (2.7.1) are identical to the modified SBC rule with fixed points \(\_\le Ca(t)\le _) = 0, F(Ca(t)>_) =1)\}\) and learning rates \(\_\le Ca(t)\le _) = _, \eta (Ca(t)>_) =_)\}\). The first two lines of Eq. (2.7.1), however, add a new feature, namely the dependence of the fixed points in the pre-depressive region on the present weight \(w\), in addition to the [Ca2+].

2.8 Two-dimensional FPLR rule

It is possible to generalize the simplified GB rule into a fully generic two-dimensional FPLR plasticity rule that specifies the fixed points and learning rates as a function of both the synaptic [Ca2+] and the current weight. Similar to the SBC rule, we have:

$$\Delta w=\eta \left(Ca\left(t\right),w\right)*\left(F\left(Ca\left(t\right),w\right)-w\right)$$

(2.8.1)

Here, both the learning rates \(\eta\) and the fixed points \(F\) are two-dimensional step functions of both the [Ca2+] and the current weight \(w\), as opposed to a one-dimensional step function of only the [Ca2+] in the SBC rule. Usually, we will keep the standard fixed points and learning rates for depression and potentiation as only dependent on the [Ca2+], so we still have \(F\left(_<Ca\left(t\right)< _ \right)=0\) and \(F\left(Ca\left(t\right)> _ \right)=1\). However, for the pre-depressive region, we can now specify an arbitrary number of weight-dependent fixed points and their associated learning rates, which allows the synapse to drift to one of several final stable states.

Each fixed point for the weights in the pre-depressive region of calcium has a basin of attraction, i.e. a range of weights surrounding that fixed point that eventually converge to the stable weight. When specifying fixed points of the weights as a function of the present weights, care must be taken to avoid overlapping basins of attraction. For example, if the fixed point for a synapse with a weight of \(w= 0.8\) is \(w= 1\), the fixed point for weight \(w = 0.9\) must also be \(w= 1\), because \(w= 0.8\) must pass \(w= 0.9\) on its way to \(w= 1\).

To enforce this constraint, one must assign \(N\) fixed points and \(N+1\) boundaries of the basins of attraction within each region of [Ca2+] such that the fixed points are always inside the closest basin boundaries on either side (Fig. 4C). For example, we can consider a rule that incorporates a tri-stable pre-depressive drift, where strong synapses drift to an UP state of \(w= 0.8\), weak synapses drift to a DOWN state of \(w= 0.2\), and synapses which aren’t particularly strong or weak drift toward a MIDDLE state of \(w= 0.5\) (instead of an unstable fixed point of \(w= 0.5\)). We intentionally choose fixed points in the pre-depressive region of [Ca2+] that are different from those in the potentiative (\(w= 1)\) and depressive (\(w= 0)\) regions of [Ca2+] to reflect experimental results that synaptic weights during the early phase of LTP and LTD can overshoot/undershoot the eventual weights to which they are stabilized (Manahan-Vaughan et al., 2000; Redondo et al., 2010).

Fig. 4

The two-dimensional FPLR rule. (A) Defining fixed points and basins of attraction for pre-depressive drift in the two-dimensional FPLR rule. We can define an arbitrary number of stable fixed points (filled circles) as a function of the current weight \(w\), as well as the boundaries of their respective non-overlapping basins of attraction (open circles). Blue lines indicate change in weight \((\Delta w)\) as a function of the present weight \(w\). (B1) Representation of fixed points as a function of both \(\left[^\right]\) and current value of the synaptic weight \(w\). Values in the pre-depressive region as in (A) (B2) Learning rates in two-dimensional FPLR rule as a