In Equation (1) we set g(υ) = 0 and the input current is the sum of current-based synaptic inputs. A spike of neuron j in the input population I at time t_j arrives at the postsynaptic neuron without delay and is modeled by a Dirac delta scaled by the synaptic weight w_j, i.e., Isyn=∑j∈I∑tjwj·δ(t-tj) (see Figure 1Aa).

We consider both deterministic and probabilistic synapses (Branco and Staras, 2009) for which a transmission probability p_r < 1 is tantamount to multiplying presynaptic spikes by the outcome of a Bernoulli trial with probability p_r.

In Equation (1) the leak is captured by the term g(υ) = −g_L(υ − E_L) and I_syn is the sum of conductance-based synaptic inputs. The inputs decay exponentially with time constant τ_syn and have a reversal potential E_e, that is I_syn = g_e(t) · (V(t) − E_e) where:

The sum ∑tjδ(t-tj-tdelay) corresponds to presynaptic spikes of neuron j ∈ I, each occurring at time t_j and contributing to postsynaptic depolarization in an amount of w_j after a delay t_delay (see Figure 1Ab).

We assume that presynaptic neurons fire together in short pulses, hereafter termed as “volleys” (Figure 1Ca). In particular, the postsynaptic neuron potential is at rest (i.e., 0) at the start of a volley, and for a volley of duration T_U, at time t, each presynaptic neuron independently selects a spike uniformly at random in the interval [t, t + T_U]. The time between two consecutive volleys is T_D ms. Most of our analysis is carried out under the random order assumption that volley spikes are uniformly distributed, although we also consider deviations from this assumption in Section 3.1.2.

Each presynaptic neuron fires according to an inhomogeneous Poisson process (unless differently specified). The rate of the process alternates between DOWN phases of low rate values (λ_D) and UP phases of high rate values (λ_U). The duration of each DOWN and UP phase corresponds to the parameters T_D and T_U, respectively (Figure 1Cb). The volleys in the model without leak may be regarded as the limiting case of short UP phases and they lead to analytical tractability of the model. In one setting when we study robustness against input parameter variations we shift the input rate function of each input neuron by some random delay, independently drawn from a uniform distribution in [0, σ].

Each synapse tracks pre- and postsynaptic spikes by means of a scalar memory trace m(t) which is updated similarly to classical STDP (Morrison et al., 2008). Accordingly, presynaptic spikes that happen in a short interval T_early before a postsynaptic spike, all increase m(t) promoting synaptic potentiation. Conversely, presynaptic spikes following a postsynaptic spike in a time window T_late, decrease m(t) which can lead to synaptic depression (see green and red shaded regions in Figure 1C). We refer to these spike pair events as potentiation and depression signals and we denote them by the binary indicator variables 𝟙_↑(t) and, respectively, 𝟙_↓(t).

For volley input, these signals simply correspond to the spike order of a synapse in each volley, that is 𝟙_↑(t) is 1 at the time of a postsynaptic spike if it was preceded by a presynaptic spike in that volley and similarly 𝟙_↓(t) is 1 at the time of a presynaptic spike if it was preceded by a postsynaptic spike in that volley.

The memory trace in this setting is given by the moving average of learning signals which are potentiation signals (see Figure 1Da), that is, let S^(signals)(M) be a set containing the time of the last M learning signals, then

The main difference in this setting is that the neurons can spike more than once in an UP phase instead of at most once in a volley. The memory trace is updated for spikes pairs corresponding to potentiation and depression signals as in the model without leak. However, we restrict the pairs which trigger such signals to those which involve the first postsynaptic spike in an UP phase (purple stars in Figure 1). We will hereafter dub such postsynaptic spikes as distinguished spikes (DSPs). This restriction accounts for the fact that STDP may involve more complex interactions between pre- and postsynaptic firing rather than those considered in classic doublet STDP models (Pfister and Gerstner, 2006). To define the memory trace and its update rules formally we therefore first define all relevant variables (for an overview, see Figure 1Db).

For a synapse, we denote by 𝟙_pre(t) and 𝟙_post(t) the indicator variables which are 1 at time t if the corresponding pre- or postsynaptic neuron spiked and 0 otherwise. We use these variables to derive indicator variables for potentiation and depression signals, however, for the derivation, we also require variables which monitor whether a DSP recently occurred and whether presynaptic spikes occur in a time window around it. In this regard, we introduce the binary variable o_burst(t) which is 1 if a DSP occurred in [t − T_burst, t), and 0 otherwise. We define it recursively as follows:

where Spost(t1,t2) denotes the set of all postsynaptic spikes in the interval [t₁, t₂). Similarly, we monitor the depression signal window by the binary variable o_late(t) which is 1 in a time window T_late after a DSP and 0 otherwise, i.e.,

Accordingly, the depression and potentiation signal indicator variables are given by

where Spre(t1,t2) denotes the set of all presynaptic spikes in the interval [t₁, t₂).

Whenever 𝟙_↓(t) is 1 we update the memory trace m(t) as follows

where γ defines the attenuation of the pre-existing memory trace. Similarly, whenever 𝟙_↑(t) is 1 we apply

for each presynaptic spike in the interval [t − T_early, t), i.e., we apply the update |Spre(tpost-Tearly,tpost)| times.

The plasticity of the synapse depends on the value of m(t). In both the model with and without leak, the update of the synaptic weight is applied similarly. The main difference is that in the model without leak the weight update rule is applied only after every L-th update to the memory trace (L ≥ M) whereas for the model with leak the weight update follows every memory trace update. In a weight update, the weight can change if and only if m(t) > θ_P or m(t) < θ_D, where θ_P and θ_D are the depression and potentiation thresholds (the interval [θ_D, θ_P] is shown as a shaded green region in Figure 1Db). Formally, the change in synaptic weight is probabilistic, which has been considered before (Standage and Trappenberg, 2006), and, in particular, is subjected to the outcome of a Bernoulli trial with probability p_s→w for depression and p_w→s for potentiation. This approach is necessary to prevent many synapses from changing their weight together with the risk of destabilizing postsynaptic firing (see Section 3.2.1). The weight update rule is given as follows

With binary synapses, which have weights w_s and w_w, the weight changes according to:

We start our analysis by deriving some formal results in the model without leak. We restrict our analysis to the scenario of a single neuron which receives d excitatory synaptic inputs from presynaptic neurons in a population I, of which d_s are strong. We show that there is an equilibrium input of ds* strong synapses such that when the total input weight is ≫ ds*, then all synapses have a high chance to decrease their weights. On the other hand, if the total input weight is ≪ ds*, then all synapses are more likely to increase their weights. Thus, the equilibrium ds* is stable. The analysis also shows an interval around ds* in which changes of the input weights happen only with very low frequency per volley.

For a single synapse, the total input weight of the postsynaptic neuron determines the expected ratio of potentiation to depression signals. We capture this property by deriving the probability that a synapse receives a potentiation signal instead of a depression signal as a function of the input weight. First, recall that each presynaptic neuron emits exactly one spike per input volley which, in the generic scenario, may be transmitted to the postsynaptic neurons by some probability p_r ≤ 1 (Section 2.1) and, therefore, the postsynaptic neuron spikes at most once per volley. Hence, a potentiation (depression) signal corresponds to a pre/post (post/pre) spike pair within a volley. It should be noted that learning signals are only triggered if both the presynaptic neuron j and the postsynaptic neuron spike and, therefore, one needs to condition on a postsynaptic spike. Furthermore, recall from Equation (3) that the memory of the synapse is the moving average of the number of potentiation signals amongst the last M learning signals. These signals can be viewed as M Bernoulli random variables, where the probability of them being 1 (0) corresponds to the probability of a potentiation (depression) signal. Denote by X the number of strong synapses that transmit a spike in a volley. The postsynaptic neuron spikes if X ≥ θ_v. For a synapse from the j-th input neuron, when d_s out of d synapses are strong, X is binomially distributed such that X ~ Bi(d_s, p_r); accordingly, the probability of the synapse receiving a potentiation signal instead of a depression signal is given by the following expression:

It thus follows that if the pre- or postsynaptic neuron do not spike in a volley, the memory of the synapse does not change. Hence, the probability of a depression signal is 1−Pr[pot. signal]. The expressions in Equation (14) are shown in Figure 2A for deterministic synapses (p_r = 1, orange) and probabilistic synapses (p_r = 0.5, purple). The small difference between a weak and a strong synapse is caused by the factor θvi for strong synapses and θvi+1 for weak ones. For strong synapses this factor corresponds to being amongst the first θ_v input signals out of i strong input signals whereas for weak synapses this factor corresponds to the signal arriving before any of the first θ_v strong input signals where the number of possible places in the order is i + 1.

The memory trace m(t) from Equation (3) is an unbiased estimator of Pr[pot. signal] from Equation (14). Furthermore, the probability depends in a monotone fashion on d_s (Figure 2A). This monotone dependence is the principle behind the intrinsic homeostasis effect of the learning rule which allows a synapse to estimate if the total input weight toward the target neuron is too large or too small by just observing and keeping track of recent spike pair orders within a volley. This mechanism results in a stabilizing effect on d_s that we quantify below.

In the model without leak, recall that we apply the weight update rule after every L-th volley. For deterministic synapses, we set L = M. This assumption makes the weight updates independent and, thus, greatly simplifies analysis (see Appendix in Supplementary Material). In the setting with probabilistic synapses, this property holds with high probability by choosing L a bit larger. There, we set L=2Mpr·Pr[X≥θv] where, as above, X is the number of synapses that transmit a spike. L depends on d_s for probabilistic synapses because the target neuron cannot be reliably activated if d_s is small.

Denote by Δ_s→w(d_s) (Δ_w→s(d_s)) the number of strong (weak) synapses that turn weak (strong) after applying the weight update rule. Recall that the weights are binary 0 and 1 in the model without leak, so the number of strong synapses is the same as the total input weight. The expected number of weight changes of each type is given by

The expected number of weight changes of each type is shown in Figure 2B for deterministic (orange) and probabilistic synapses (p_r = 0.5, purple). The total input weight change (or “weight drift”) after a learning step is given by

For ε > 0, the expected number of weight changes Δ_w→s(d_s) + Δ_s→w(d_s) for a fixed number of strong input synapses can be bounded by choosing

and for some δ > 0 by choosing

A detailed derivation of Equation (19) may be found in the Appendix (Section A2 in Supplementary Material). For this choice of d_s, the expected fraction of synapses that change their weight is less than δ; i.e., 𝔼[Δ_w→s(d_s) + Δ_s→w(d_s)] ≤ δd. It may be noted that this choice of M is independent of the parameters p_s→w and p_w→s. Their roles in the learning rule are simply to reduce the weight drift. Details on how to choose them are given in Section 3.1.2.

In the following, we argue that in the model with leak, the memory trace is a biased estimate of the expected relative spike time r of a DSP within an UP phase. Because of this feature we will refer to the learning rule as the spike latency normalization (SLN) rule. Assume we are in a static setting where the weights are fixed such that the expected relative spike time is r. Let Y_i denote the result of the i-th learning signal: 1 if it was a potentiation signal and 0 otherwise. By ignoring the effect of a specific input neuron on the spike time of the post neuron (which is valid if θ_v is large with respect to w_s) and by assuming that the input neuron spikes only once in the interval [0, T_U] (which is valid if T_U is small compared with 1/λ_U), a synapse expects a potentiation signal with probability r. That is, the variables Y_i can be assumed to be Bernoulli random variables with parameter r. The expected value of the memory trace after ℓ learning signals depends linearly on r and is given by

Since the trials are independent, the variance of the distribution can also be easily computed and equals

An example of the memory trace distribution is presented in Section 3.2. Furthermore, in Section 3.2.5 we show the expected value of the memory trace 𝔼[m(t)] as a function of d_s for a fixed UP phase length T_U. 𝔼[m(t)] decreases monotonously as a function of d_s. This monotone dependence is the principle behind the intrinsic homeostasis mechanism just as p_early is in the model without leak.

In the model without leak, our rule converges to a value ds* of the total input weight that represents the stable state (or attractor) of the learning process. This state minimizes the expected learning error, which may be quantified by the absolute value of the mean weight drift, that is

The stable state is shown with an arrow in Figure 2B. Equations (15) and (16) show that the stable state depends on the parameters θ_P, θ_D, p_s→w, and p_w→s.

In the previous section, M, θ_P, and θ_D were chosen to enforce low expected weight change in some state d_s. We will apply these choices to the steady state ds*. Observe that since 𝔼[Δ_w→s(d_s)] is a decreasing function and 𝔼[Δ_s→w(d_s)] is an increasing function (Figure 2B) we can scale both expectations independently by changing p_w→s and p_s→w, respectively. In this way, ds* can be adapted.

Starting with ds>(1+2ε)ds*, it may be shown that (Appendix A3 in Supplementary Material) the expected number of weight updates T applied, until ds<(1+2ε)ds* is upper bounded as follows

That is, the average number of weight updates is logarithmic in the distance from the target value and inversely proportional to p_s→w, which is the probability of a strong synapse turning weak when m(t) < θ_D. To avoid dangerous oscillations on synaptic weights however, this latter probability should be small so as to prevent the total input weights from dropping to 0. A similar consideration, could also be made for p_w→s insofar as a large value of this probability could result in an overshooting (i.e., strong global potentiation) of synaptic weights, with the important difference however, that, in our formulation, strong potentiation still allows the postsynaptic neuron to fire, whereas strong depression would ultimately result in the shutdown of synaptic inputs. In this catastrophic scenario, the postsynaptic neuron would not be able to fire further, and would become disconnected.

To avoid this possibility and unnecessary weight fluctuations, we present a heuristic criterion to choose p_s→w and p_w→s which allow to upper bound the expected weight change by ⌈ε1ds*⌉ if ds∈[(1−2ε1)ds*,(1+2ε1)ds*] for some ε₁ > 0. For ds=(1+2ε1)ds* the expected number of synapses that depress when a weight update is applied is given by 𝔼[Δs→w((1−2ε1)ds*)] as in Equation (16). Similarly, the number of synapses that potentiate in when a weight update is applied is given by 𝔼[Δs→w((1−2ε1)ds*)] as in Equation (15). To prevent undershooting when ds>(1+2ε1)ds*, it suffices to choose ps→w<c′(ds*−θv)d for a sufficiently small c′ (e.g., c′ = 2 in this study). The above conditions translate to setting the model parameters such that:

Convergence to a stable state where p_s→w and p_w→s are set as in Equations (24) and (25) is shown in Figure 2C, for both deterministic (orange) and probabilistic (purple) synapses in the model without leak. The distribution of the total input weight at equilibrium is shown for the deterministic synapses in Figure 2D, where it is compared with a setting where potentiation is turned off, i.e., where the rule only provides negative feedback. The result in Figure 2D shows that if negative feedback is the only requirement, then it suffices to use the memory trace exclusively for weight depression. If the input to the neuron increases too much, e.g., through some other form of short-term Hebbian learning, then the negative feedback provided by the memory mechanism of the strong synapses can quickly reduce the input weight back to a stable state without undershooting it. In both settings described above, the initial total input weight was d_s = 30 and the distribution in the figures is shown after 200 weight updates have been applied.

Figure 3A shows how the stable state ds* is affected by probabilistic synapses. In particular, in this case ds* increases by a factor of pr-1 if other parameters remain fixed. The intuition behind this relation comes from two observations. First, for that state, the expected total input weight is the stable state for deterministic synapses. Second, if the number of strong synapses d_s is large enough, then the total input X = Bi(d_s, p_r) is concentrated around its mean μ_X = d_sp_r with standard deviation σX=dspr(1-pr). Asymptotically for growing θ_v, d_s, and d, the main contribution in the sum in Equation (14) comes from terms close to the mean of the form

In Equation (26) above, the notation f = Θ(g) means that f is bounded by g both above and below asymptotically.

Our hitherto analysis has considered the input order to be uniformly distributed (i.e., “random order” assumption), yet one may also ask how the SLN rule performs if this assumption is relaxed. Indeed, spikes are known to have a precise repeating temporal structure in some cortical areas related to sensory processing (Bair and Koch, 1996; Heil, 1997; Fellous et al., 2004) and standard STDP is known to tune to the first spike for such repeating spike patterns (Guyonneau et al., 2005). To model the transition from an orderly spike distribution to a uniform one the expected relative spike time of every input neuron in a volley is fixed but the deviation of the spike time is varied. Formally, for the i-th neuron a normal random variable Zi~N(i/d,σZ) is sampled. These variables impose an order on the input spikes within a volley in a natural way. For the resulting order, the input neurons spike in an equi-spaced manner. This approach is valid since we ignore membrane leakage and, therefore, the spike order is the only temporal structure that has any influence on the distribution of potentiation and depression signals. For σ_Z = 0, the ordering is fixed as 1, 2, …, d and for a growing σ_Z, the ordering approaches a uniform distribution. Figure 3B shows how this choice of input spike distribution affects the stable state ds*. Not surprisingly, if the order is fixed (σ_Z = 0), then only the first θ_v synapses get strengthened as was the case in Guyonneau et al. (2005) where classic STDP by a repeating spike pattern turned the neuron into a detector for the start of the pattern. However, already a small value of σ_Z quickly shifts the stable state toward the same one as for a random input order. For σ_Z = 2.0 the first 20 input neurons are close to a random permutation since the probability of the 20th neuron being earlier than the first neuron is approximately 0.47. Figure 3C shows the mean index of strong synapses. Even though the stable state remains close to the same for small values of σ_Z, the early synapses are the ones that are preferably strengthened in agreement with Guyonneau et al. (2005). For other choices of spike order distributions, the negative feedback of the intrinsic homeostasis mechanism still applies (Appendix A4 in Supplementary Material).

As in the model without leak, the number of weight changes in the model with leak is mainly determined by two factors: (i) the number of synapses whose memory trace lies outside the target interval [θ_D, θ_P] and (ii) the weight update parameters p_s→w and p_w→s. How the width of the interval [θ_D, θ_P] affects plasticity is shown in Figures 4A–C, whereas an example of how p_s→w and p_w→s can affect plasticity is shown in Figure 4D. The width of the target interval [θ_D, θ_P] determines how many synapses can change their weight in the stable state. Such intervals are laid over the memory trace distribution in Figure 4A, which was sampled by fixing d_s = 20. The tails of the distribution outside the interval correspond to the synapses that can change their weight. Since these tails fall off exponentially if the learning signals are independent Bernoulli trials, the fraction of synapses that can update their weights in the stable state can be made arbitrarily small by choosing a large interval. It may be noted that the parameter γ also needs to be chosen sufficiently close to 1, which corresponds to choosing M larger in the model without leak. Therefore, one can choose parameters such that a region around the stable state induces almost no weight changes, thus avoiding unnecessary fluctuations of total input weight. However, choosing a large interval also leads to diminished plasticity in regions further away from the stable state (Figure 4B). This suggests that a trade-off exists between synaptic weight plasticity and the time to reach close to the stable state.

Figure 4C shows such trade-off where the orange curve represents the expected number of weight updates per UP phase when d_s = 20 (over all synapses) whereas the purple curve shows the mean time to remain with 25 strong synapses starting from 80. For Figures 4A–C, the parameters p_s→w and p_w→s were set to 0.25. In combination with a small target interval, a bad choice of p_s→w and p_w→s can lead to plasticity over- or undershooting as in the model without leak (see Section 3.1.2). Undershooting the input weight in the stable state is a concern if it results in the weights being too small to activate the target neuron. Overshooting is less of a concern for activation, but it is wasteful since fewer weight updates can be used to reach the stable state. An example of undershooting is shown in Figure 4D, where θ_P = −1.1 and θ_D = −5.1. Over- and undershooting happen in this setting because the memory trace depends on learning signals from the past that are irrelevant when the weights are close to converging. To handle this effect, weight updates can be limited by choosing p_s→w and p_w→s small enough. Furthermore, since input neurons in the model with leak do not always spike in an UP phase only a fraction of the input synapses update their memory trace and trigger the weight update rule. This results in a natural tie-breaking mechanism that prevents all synapses from updating their weights simultaneously.

The SLN rule achieves normalization quickly. Though speed is a desirable property (see the work of Zenke et al., 2013 for a discussion on the necessity of a fast-homeostatic mechanism), it is also important to be energy efficient; that is, the total number of synaptic weight changes should be small. Different synaptic weight distributions can result in the same value of expected relative postsynaptic spike time. A feature of stability in the SLN rule is that this does not result in further weight changes. We highlight these features in Figures 4E,F. Figure 4E shows two different synaptic weight distributions for the strong weight synapses. One is normally distributed with mean 40 nS and standard deviation 5 nS (dark orange). The other is obtained by sampling from a normal distribution as well, with the same standard deviation, by first flipping a fair coin to decide if the mean should be 35 or 45 nS (light orange). The resulting distribution of the memory trace is shown in Figure 4F when p_s→w = p_w→s = 0 and d_s = 24. By setting θ_D = −16 and θ_P = 5, the synaptic weights remain unchanged over 1 min of input activity (data not shown). That is, both weight distributions are stable.

For short UP phases the SLN rule normalizes the expected relative spike time r of the postsynaptic neuron within an UP phase whereas for long UP phases the rule normalizes the absolute expected latency of the postsynaptic neuron, which is equivalent to normalizing the total input weight in this setting. Figures 5A,B demonstrate how the number of strong input synapses d_s and the relative spike time r change when varying the length of the UP phase if we start with d_s = 25 and run the process until the weights converge. We use a log-scale for the x-axis to show data for very long UP phases. Figure 5A shows that for long UP phases the input weight is normalized whereas Figure 5B shows that for short UP phases the relative spike time is normalized. We show UP phases of length 20.0, 25.0, 30.0, 35.0, 40.0, 45.0, 50.0, 60.0, 80.0, 100.0, 500.0, and 1000.0 ms. The length of the down phase is set to 1000.0 ms and we set T_burst = 1000.0 such that there can be only one DSP per UP phase. We use the plasticity parameters in Table 3 for r = 1/3 (dark purple) and r = 1/2 (dark orange) where the lighter colored curves correspond to a setting with p_w→s = 0.0 as in Section 3.1.2. The schematic in Figure 5C explains the reason for this difference. The figure shows two different rate functions with short (30 ms) and long (500 ms) UP phases. Postsynaptic spikes are drawn on top of the rate function for reference where darker spikes represent DSPs. The windows for potentiation and depression signals are drawn in black over the DSPs. For short UP phases the windows for potentiation and depression signals cover the whole UP phase. In this setting, the fraction of potentiation signals represents the relative spike time within an UP phase. In contrast, for long UP phases, the window of depression lies completely within the UP phase. To compensate for more depression signals the input weight needs to decrease such that the ratio of potentiation to depression signals is within the target regime. Figure 5D shows the weight over time for short UP phases of 30 ms with d_s = 17 at t = 0 (left panel) and long UP phases of 0.5 s with d_s = 27 at t = 0 (right panel). In both panels the DOWN phases are of length 1 s. We use the plasticity parameters for r = 1/3 in Table 3 which corresponds to the dark purple curve in panels A-B. Each timescale captures 400 UP phases and we show 20 example runs with one highlighted in orange for clarity.

For long UP phases we chose T_burst sufficiently large to permit only one DSP per UP phase. However, for short DOWN phases this leads to a DSP blocking first spikes of subsequent UP phases becoming DSPs. As an alternative, one can assume that a DSP is only triggered if the membrane potential was recently in a low state of activity and a DSP blocks further DSPs until a low activity state is reached again. One can further assume that spike pairs are only considered to trigger learning signals if the subthreshold activity remains above some threshold between both spikes. This voltage based approach fixes the issue above with short DOWN phases and it implies that for constant or high rate input (as we study in Section 3.2.6), where the subthreshold potentiation remains large, neither the memory nor, hence, the synaptic weight, would change.

The SLN rule also works for heterogeneous and multimodal synapses although the exposition so far was restricted to binary synapses. This more general scenario is discussed in the model without leak of mere SIF neurons, with results from simulations presented in Figure 6.

In the setting with heterogeneous weights w_w = 0 for all synapses, whereas w_s is drawn independently for each synapse from a distribution as in Barbour et al. (2007). The distribution is the one reported in Loewenstein et al. (2011), where ws=eN(μ,σ) with μ = 1.74 and σ² = 0.1002. The mean of the distribution is 10^μ+σ2/2 ≈ 5.726. To compare this with the binary weight setting where θ_v = 10, the threshold is set to θv=10·eμ+σ2/2. Convergence to a stable state is shown in Figure 6A. Initially 20, 40, or 80 synapses are strong. After convergence, the weight distribution of synapses selected to be strong is compared with the lognormal distribution from above (Figure 6B). The empirical distribution is composed of 5.000 trials and the neuron and synapse parameters (except θ_v and the weight) are in Table 1.

Figures 6C,D compare convergence and weight distribution for the two different setups from Section 2.5. The plasticity parameters are the same as in Figure 2. The additive weight update rule is slower to converge since the weight updates are smaller for large input weights compared with the multiplicative rule. Figure 6D shows that in this specific setting, the synaptic weights do not need to be explicitly upper bounded. This holds because the input arrives in a random order and the causative effect of a single synapse is small. Figure 6D also shows that the additive (multiplicative) weight update rule converges to a bimodal (unimodal) distribution. Both uni- and bimodal distributions have been observed for earlier plasticity rules, whereas bimodal distributions are considered a sign of competition since some synapses being strong force others to be weak.

We show that the stability of the rule does not depend on the exact input parameters and that perturbations do not qualitatively affect it. Figure 7 illustrates this robustness in the model with leak. The different panels show two different plasticity parameter settings that correspond to r = 1/3 (purple) and r = 1/2 (orange) in Table 3. In Figures 7A–C, the lengths of UP and DOWN phases (T_U and T_D), and the number of inputs d are varied. The process is first simulated for 200 UP phases starting with d_s = 25 such that the input weight reaches the stable state. The data in the figures is based on UP phases 201–400 in 40 trials. Figure 7A shows that the SLN rule fixes the relative spike time r of the postsynaptic neuron. The stable value of r mildly depends on T_U, which arises from the choice of synaptic delay t_delay = 1 ms. To explain this, denote by t₁ the spike time of the pre spike which triggers the postsynaptic spike at time t₂. Any presynaptic neuron that spikes in the interval (t₁, t₂) receives a potentiation signal, which leads to a mild bias toward potentiation that is stronger for short UP phases. The variable r determines the number of spikes per UP phase (Figure 7B) since the number of spikes is inversely proportional to r. The number of spikes per UP phase has a mild dependence on T_U since the refractory period has a stronger effect for short UP phases and the reset potential makes repeated spikes easier. Figure 7C shows that the intrinsic homeostasis mechanism does not fix the rate of the output neuron since it is determined by T_U and T_D.

Figure 7D shows the effects of single parameter variations. The darker colored curves correspond to the same plasticity parameters as in Figures 7A–C and the lighter colored curves correspond to a setting without positive feedback, i.e., when p_w→s = 0 (see Section 3.1.2 for a similar setting in the model without leak). Observe that most single parameter variations in this setting do not cause large plasticity undershooting. The main effects caused by T_U and T_D are covered above so we do not cover the first two rows here (the lower order effects of T_U on r are covered in the next paragraph). The third row corresponds to varying the phase shift of the inputs randomly (see Section 2.2). For that setting we replace T_U by T_U + σ in the definition of r. We see r increase due to a reduced input weight like we observed for more orderly input distributions in the model without leak (see Section 3.1.2). Rows four and five correspond to the number of input neurons and noise rate, respectively. Variations on both of these parameters have similar effects, as increasing these parameters helps activate the target neuron. Consequently, d_s must decrease to compensate (D 4,2 and D 5,2). The last row corresponds to varying the membrane capacitance C_m. Increasing the capacitance decreases the leak conductance, but it also reduces the excitatory synaptic conductance, which makes it harder to activate the neuron. Despite large variations, r remains fixed.

To conclude this section, let us consider the lower order effects from T_U, d, and noise strength on r. For simplicity assume that θ_P and θ_D are chosen such that a δ fraction of the strong (weak) synapses satisfies m(t) ∉ [θ_D, θ_P] in the stable state. For the corresponding input weight ds*, the parameters p_s→w and p_w→s satisfy

However, if we increase T_U, the noise, or d, then ds* will change to compensate such increase. If ds* needs to decrease by x to obtain the correct value of r, then the left-hand side in Equation (27) increases by xδp_w→s whereas the right-hand side decreases by xδp_s→w. Therefore, the number of strong synapses will not converge to exactly ds* − x. However, if the interval [θ_D, θ_P] is large enough such that δ is small, then this effect is reduced.

The principle of sampling the expected spike time of the postsynaptic neuron in an UP phase also applies if the synapse considers all pre/post and post/pre spike pairs within an UP phase to constitute potentiation and depression signals. In this section, we show the advantage of using a single DSP: it improves the signal-to-noise ratio. Furthermore, later postsynaptic spikes in an UP phase have more preceding presynaptic spikes in the UP phase and, thus, carry a weaker signal. Figure 8A compares the memory value for these two rules in a static setting where synapses have fixed weights. The figure shows the advantage of DSPs. They reduce the variance of the memory trace and increase its range, which makes the two different values of d_s more distinguishable. It may be noted that both versions of the rule perform similarly for small values of d_s because then only one postsynaptic spike is expected (i.e., both act as with DSPs). However, as d_s grows, and the postsynaptic rate increases the memory traces for the two rules start to differ. The difference arises from the fact that if the postsynaptic neuron spikes for example five times, then the update to the memory trace for DSPs is only −1 or 1 (for a single presynaptic spike), whereas it can range from −5 to 5 in the all-to-all setting. Furthermore, depression signals are more common in the DSP setting for large d_s, whereas potentiation dominates in the all-to-all setting. This effect comes from the spike time distribution of the postsynaptic neuron within an UP phase.

So far we assumed that the input signal oscillates between high and low states of activity. One may also ask, however, what happens if the input rate is constant, which is a common assumption in modeling studies (Burkitt, 2006a). In this regard, the behavior of the SLN rule is determined by the ratio of potentiation and depression signals as before. However, for constant input rate the rule does not have a frame of reference to measure the input strength anymore. The ratio of potentiation and depression signals is, therefore, determined by the parameters T_early and T_late. If we choose the parameters θ_P, θ_D such that the memory trace lies entirely within the interval [θ_D, θ_P] for constant input rate, then the weights do not change. However, if a larger fraction of the memory trace distribution lies below θ_D than above θ_P, then the total input weight decreases up to a point where activation of the neuron becomes unreliable. For the opposite case, all the synapses get strengthened, which is not desirable. Figure 8B illustrates this by showing how the system responds to constant input for three different sets of plasticity parameters. For parameters that result in r > 1/2, the weights decrease to a point where the target neuron shows almost no reaction to the input. Indeed, this seems a reasonable response as such an input does not carry any information. On the other hand, for parameters that result in r < 1/2 the weights and the firing rate of the target neuron grow to the maximum value, which is undesirable if the activity of the neuron should be limited. These observations indicate that the parameters should fix r to be at most 1/2, which corresponds to several spikes of a neuron in an UP phase (≈ 1/r) and is in good agreement with experimental findings (Connors and Gutnick, 1990). Furthermore, the synapses should not expect disproportionally more potentiation signals than depression signals.

The SLN rule bears some resemblance to the metaplasticity rule proposed by Brader et al. (2007). Common to both rules is a synapse specific memory trace represented by a real value which is updated for certain spike events. Other metaplasticity rules depend on discrete states within the synapses such as the synaptic integration rule by Elliott and Lagogiannis (2012) which can be considered a multimodal version of the cascade rule by Fusi et al. (2005). However, these metaplasticity rules have not been studied in the context of SLN for UP/DOWN state input activity where, in particular, the fast and robust weight convergence for such activity has not been demonstrated before.

The effect of standard STDP on spike timing was studied and analyzed in detail in the context of volley-like input spike distributions in Gerstner and Kistler (2002, Chapter 12.2). Furthermore, if the relative effect of potentiation is reduced then STDP leads to convergence of the input weights to a stable distribution for random input activity (Van Rossum et al., 2000). For both these cases, the synaptic weights perform random walks in the stable setting whereas the SLN rule provides a stronger stability guarantee since it implicitly fixes the weights at equilibrium and the weight distribution at convergence is not unique. Hence, the SLN rule is economical in the sense that synaptic weights are only changed when it is necessary which is in agreement with other studies since not all spike pairs necessarily lead to a weight change (Yger and Gilson, 2015).

Other theoretical studies have considered repeating spike inputs as the ones presented in Section 3.1.2. Guyonneau et al. (2005) studied standard STDP for repeated stimulation of a neuron with the exact same spike pattern. Through such repeated exposure the postsynaptic neuron becomes a coincidence detector for the given pattern. Further studies showed that this feature is robust to noise (Masquelier et al., 2008, 2009).

Perhaps the most prominent example of homeostasis in plasticity models is the BCM rule (Bienenstock et al., 1982). The weight update in the BCM rule normalizes postsynaptic rate by using a threshold which varies slowly with the postsynaptic rate. Nearest-neighbor STDP has been shown to perform a simplified version of this operation, i.e., without the sliding threshold, when pre- and postsynaptic activity is weakly correlated (Izhikevich and Desai, 2003).

Other studies have focused more directly on preventing single weights from being driven to extreme values. Gütig et al. (2003) presented a plasticity model where the plasticity rule is weight dependent and, therefore, leads to an implicit upper bound on synaptic weights through a mixture of additive and multiplicative weight changes. Babadi and Abbott (2010) had an even more implicit approach by shifting the effective STDP temporal window by roughly 2 ms such that pre/post pairs that are close in time lead to depression instead of potentiation. This effect works against synapses with a large causative spiking effect. It may be noted that the same principle can also be applied for the SLN rule.

Another variable which could be controlled through homeostasis is the type of activity in a neural network. In this regard, STDP is known to sustain a stable background state in balanced networks (Kempter et al., 1999, 2001; Song et al., 2000; Van Rossum et al., 2000; Rubin et al., 2001) and to regulate activity (Kempter et al., 1999, 2001; Song and Abbott, 2001; Pfister et al., 2006; Watt and Desai, 2010) but these models have been criticized for parameter fine-tuning (for a review see Chistiakova et al., 2015). Theoretical investigations showed that fast detection of postsynaptic rate changes can solve the problem (Zenke et al., 2013; Yger and Gilson, 2015) but these studies do not consider the scenario of UP and DOWN phases.

It is plausible that homeostatic plasticity is controlled by more than a single mechanism (Tononi et al., 1999; Watt and Desai, 2010) as many alternatives, like we discussed, which have a homeostatic effect on various parameters of neurons or network activity have been proposed (Bienenstock et al., 1982; Rabinowitch and Segev, 2006; Vogels et al., 2011; Remme and Wadman, 2012). It is known that homeostatic plasticity is affected by a complex web of signaling processes many of which are likely undiscovered (Pozo and Goda, 2010) and which might even be shared between, or belong to, different homeostasis mechanisms. As an example, weight normalization is considered biologically plausible because a neuron might have finite resources dedicated to maintaining its synapses, but there is a lack of experimental evidence to support this.