The differential Hebbian learning rule is a rather simple algorithm for weight changes (Kosko, 1986). The synapse changes proportional to the product of the presynaptic activity and the temporal derivative of the postsynaptic activity. For spiking neurons, however, this makes little sense, and one has to introduce some kind of low pass filtering of neuronal activities to gain a signal suitable to calculate synaptic change. As usual, we use delta pulses to model neuronal spike trains:

where i ∈ {pre, post} denotes the location of the spiking event. Each spikes leaves an exponential trace y_i on its synaptic side, which can be described by the differential equation

We use the dot notation to denote temporal derivatives. The weight change is given by

This simple system of equations is equal to (balanced) spSTDP, as we show now. Consider the solution of Equation (2) to a single spike at time t_i:

Here, Θ(t) is the Heaviside function, i.e., Θ(t) = 0 for t < 0 and Θ(t) = 1 everywhere else. The weight change is calculated via

where we introduce the constant of proportionality c_w. The weight change resulting from a pair of one pre- and one postsynaptic spike is given by

This is the standard STDP window for balanced spSTDP, where the areas under the LTP and LTD part of the curve are of exactly equal size, and the decay time constants are given by τ_pre and τ_post for LTP and LTD, respectively. Due to the linearity of the equations, the learning rule is also completely linear, and every spike pair in a given spike pattern is treated the same by the learning rule.

The STDP window in differential Hebbian learning is determined by three parameters. To scale the LTD and LTP parts of the window relative to each other a fourth parameter is needed. We split the weight change into a depression and a potentiation part by inserting Equation (2) into (3) and introduce a scale parameter q:

This manipulation changes the STDP window to:

Adjusting q scales the LTP part of the STDP window as required. For example, setting q = 1/(1 + τ_post/τ_pre) cancels LTP for every possible spike pattern. There are experiments which show that for low frequencies of spike pair induction, pre-post pairs do not change the synapse, while post-pre pairs still depress the synapse (Sjöström et al., 2001; Nevian and Sakmann, 2006; Wittenberg and Wang, 2006). However, other spike patterns in these studies potentiate the synapse, which suggests that in order to generalize this description of STDP, one has to turn q into a function of time.

For the CD model we use the differential Hebbian learning rule described above as basis, and extend it by several new equations. First, we introduce an adaptation variable u_i for each synaptic side. This dynamics resemble those of the presynaptic resources in models of synaptic short term depression (Tsodyks and Markram, 1997; Tsodyks et al., 1998). Its effect is the attenuation of the impact of rapid spiking on the synapse, and we model it by

The update of the trace Equation (2) is now changed to

where we write u_i (t − 0) to emphasize that in order to update each variable in case of a spike, one has to use the value of u_i shortly before the spike.

If the learning rule Equation (3) was left unchanged, the effect of the adaptation variables u_i would be a rescaling (shrinking) of the STDP window with consecutive spikes, and relaxation during silence. This property removes the linearity of the original STDP learning rule, as the influence of each spike on the synapse depends on the history of spiking of the respective neuron. We introduce an additional non-linearity, by allowing q (Equation 7) to vary over time:

where Θ is again the Heaviside function. q is a trace of the postsynaptic activity conditional on the presynaptic trace: Only if y_pre > ϑ_q at the time of a postsynaptic spike, q increases. For all other times, it relaxes back to q_min. The actual weight change is finally given by

The specific formulation of q is motivated by its simplicity—linear ordinary differential equation of first order for the decay term—and several observations in the data of Nevian and Sakmann (2006). In these experiments, a pre-post pair does not change the synapse, but the pre-post-post triplet does. The translation to an STDP framework is that the LTP part of the STDP window needs to vanish when the synapse is relaxed (any previous activity took place relatively long ago), but reappear in reaction to certain activity patterns. In this example (pre-post vs. pre-post-post), the desired outcome can be achieved by Equation (11) without the Heaviside function: q˙ = (q_min − q)/τ_q + c_qx_post. However, in the case of a post-post-pre-post pattern (Figure 2C third data point from left) this would lead to a huge upscaling of the LTP part, which was not observed. This prompted us to install the threshold such that recent presynaptic activity gates the increase of q. We chose the all-or-none threshold to exclude any non-linear effects of q on the STDP window. Because of the upregulation of potentiation, we call q the activation variable.

Figure 1 gives an overview over the components of the CD model.

To evaluate the ability of the CD model to reproduce experimental findings, we matched its parameters to the following four in vitro data sets:

Visual cortex of young rats, thick tufted cells in layer 5 [VC5, Sjöström et al. (2001)].

In these experimental studies, the change of synaptic efficacy is given as the ratio of the isolated EPSP (which we assume to be proportional to the synaptic weight) after and before the induction protocol:

We identify Δw with the weight change in the CD model. Additionally, we assume that the synaptic weight before the induction has a fixed value. The goal of the fitting process is to compare the experimental synaptic change with the model prediction Δw^CD and find the set of parameters π = {τ^rec_pre, c_pre, τ^rec_post, …} which minimizes the error

where N is the number of experiments in the data set, i the index of the experiment and SEM_i the published standard error of the mean for experiment i. The minimization of the error was done by a brute force search in the space of parameters. The bounds we defined for the search are given in Table 1 (see Appendix for additional remarks on the bounds). For each data set, some parameters were set by hand to fixed values as follows: We set q_min according to the outcome of the pre-post spike pair experiment found in every data set. If the pre-post spike pair resulted in no change of synaptic efficacy, q_min = 1/(1 + τ_post/τ_pre), otherwise q_min = 1. The time constants τ_pre and τ_post turn up as the decay constants of the STDP window (see Equation 8). For the experiments in VC23 and HC these values were explicitly given, so we did not change them. For the other two cortical data sets, we used the values of the experiment in VC23. As each spike pattern contained only one presynaptic spike, no information could be obtained for presynaptic adaptation for the data from SC23. This left for fitting here τ^rec_post, c_post, τ_q, c_q, ϑ_q and c_w. In all other data sets, τ^rec_pre and c_pre were additionally fitted.

The CD model is structurally similar to the Triplet model conceived by Pfister and Gerstner (2006). The latter is a set of linear differential equations of first order that describe traces of activity at the synapse. Each spike leaves two traces at the synapse: r₁, r₂ for the presynapse and o₁, o₂ for the postsynapse, which interact to determine the weight change. For the update of the traces, there are two possible choices. The first one is that each spike increases its respective traces by one; this is equivalent to the y_i dynamics of Equation (2). Second, at the time of a spike the respective traces always jump to unity. The equation of the traces changes to ẏ_i = −y_i/τ_i + (1 − y_i (t − 0))x_i. The first update rule is called “all to all interactions,” the second “nearest neighbor interactions.” The weight change in the Triplet model then consists of a standard spike pair STDP rule plus the spike triplet interaction, which is proportional the product r₁ · o₂ (o₁ · r₂) at the time of a postsynaptic (presynaptic) spike for LTP (LTD). The main differences between the two models are that in the CD model y_pre and y_post are subject to spike amplitude adaptation and that the triplet interactions are replaced by the (conditional postsynaptic) activation q. For comparison, we fitted the triplet model to the data sets VC23, SC23, and VC5. The Triplet model has been fitted to the HC data set, the parameters can be found in Pfister and Gerstner (2006). It has also already been fitted to the VC5 data set (same article). However, the spike induction protocol used for fitting was uniformly 60 spike pairs with Δt = ± 10 ms delivered at different frequencies (1–40 Hz). In the study of Sjöström et al. (2001), spike pairs for frequencies greater than 1 Hz were delivered in 15 bursts of 5 pairs with varying intra burst frequency, with bursts being 10 s apart. We re-fitted the Triplet model to VC5 to better compare the two models. In contrast to the original article, we furthermore allowed the triplet interaction parameters A⁺₃ and A⁻₃ to become negative to account for adaptation in the data. The fitting procedure was similar to the fit of the CD model; in particular, the STDP window time constants τ₊ and τ₋ were not fitted, but set to predetermined values. The bounds defined for the parameters are given in Table 2.

The formulation of spSTDP as differential Hebbian learning allows for a simple analytical treatment of continuous firing rates rather than spike events. Under the assumption of poissonian spiking and vanishing correlations between pre- and postsynaptic spikes, one can easily compute the mean of the traces y_i:

where r_i (t) = 〈x_i〉 is the continuous and time-dependent firing rate of neuron i. Because of the vanishing spike-spike correlations, both traces combine to give the weight change as

where 〈y_i〉 is the solution of differential Equation (15) for a given time course of the firing rates r_i (t).

In the non-linear models of STDP [CD model, Triplet model, the three Calcium models (Shouval et al., 2002; Cai et al., 2007; Graupner and Brunel, 2012)], the numerous non-linearities in each model did not allow to compute and solve the mean field equations. We therefore computed the average weight change for a given stimulation protocol and model by generating many realizations of the same continuous and time dependent firing rates from inhomogeneous poisson processes (Dayan and Abbott, 2005), which we fed into each model. As in the analytical calculations for spSTDP, we assumed poissonian firing with vanishing spike-spike correlations from synaptic transmission. In this case the probability of finding a spike in a time bin of width Δt is given by

where r_i(t) is the firing rate of neuron i as a function of time.

We hypothesize that the link between theta oscillations and learning lies in certain filter properties of the synapse, which likely depend on the model of synaptic plasticity used. We investigate synaptic filter properties in a variety of different models: spSTDP, the CD model, the Triplet model, and three different calcium models, with the aim to carve out prerequisites for a synaptic filter. We used a sinusoidal oscillation to model the firing rate. For the case of spSTDP, the firing rate is given by:

Here, ε ∈ [0, 1] is a parameter that controls the amplitude of the oscillation, ω_mod = 2πf_mod is the modulation frequency, and ϕ_i is the phase of the oscillation. Because only relative phase is important for the weight change, we set ϕ_pre = 0 and ϕ_post = Δφ. We do not specify an absolute baseline firing rate for spSTDP, because it is just a scale factor and does not qualitatively change the results. The value we report is the weight change per time averaged over one period of oscillation. This is constant after transients from the onset of neuronal activity died out:

T = 1/f_mod is the period of the modulatory oscillation, and t' » τ_pre, τ_post is chosen such that any transient behavior in the traces y_i due to switching on the activity are gone. We derive the analytical solution for Equation (19) in the appendix, and use it to generate the plots in Figure 4.

In the case of the non-linear models of STDP a baseline firing rate (the firing rate averaged over one period of oscillation) has to be specified. The respective firing rates of the pre- and postsynaptic neurons change to

To simplify the analysis, both neurons had the same baseline firing rate and the same modulation frequency. Similar to spSTDP, in the CD model and the Triplet model we calculated the weight change per time by averaging over an integer multiple of the oscillation period starting after enough time has passed to settle the transient:

where model ∈ {CD, Triplet} refers to the model used. In all simulations, t' = 2 s and NT = 98 s; we simulated only integer values of the modulation frequencies, so NT is always a multiple of the period.

For the three calcium-based models the procedure was different. All models have inherent weight limits. As a consequence, the rate of weight change itself is a function of time which does not settle into an equilibrium other than saturation. Therefore it is not feasible to calculate an average weight change rate as with the spike pair models. We rather let the two neurons fire with periodic firing rates for some time [2 s and 5 s in the model of Shouval et al. (2002), 10 s in the model of Cai et al. (2007), 5 s in the model of Graupner and Brunel (2012)], after which we silenced the neurons, but continued to simulate until the synapse settled into an equilibrium. With all models, we report the final weight, with w = 1 being the initial weight.

The fitting procedure and numerical simulations (Monte-Carlo-Simulations) were done with custom-made programs in Matlab (Mathworks Inc., Natick, MA, USA). Numerical integration of non-linear differential equations was done with Eulers method and a step size of 0.1 ms. Linear differential equations were solved analytically, and time evolution of the variables was calculated based on the spike times.

The CD model describes the non-linear interactions between spikes on either synaptic side acting on the contributions to synaptic changes. To evaluate its ability to capture synaptic changes, we chose four studies from the literature which measure synaptic changes in response to complex spike patterns, and matched the model parameters to the experimental results. Because the experiments were conducted under different conditions (brain region, presynaptic stimulation method), the parameters had to be fitted separately for each data set. In the following we describe the experiments and how the CD model recreates them in relative detail, to illustrate the action of the different contributions.

In all experimental studies, spikes were artificially induced by application of current pulses to patched neurons, or sometimes in the case of presynaptic spikes by stimulating the tissue close to the dendritic tree of the postsynaptic neuron.

Several studies found that the presence of oscillations with high theta power or large scale theta synchronization in EEG or LFP enhance learning. (Mölle et al., 2002; Seager et al., 2002; Guderian et al., 2009). A possible explanation for these findings could be an underlying filter property of the single synapse, i.e., synaptic change depends on oscillating activity of both neurons in a way specific to the oscillation frequency. To investigate this, we assume a very simple model system: Two connected neurons fire stochastically with vanishing spike-spike correlations, i.e., correlations induced when a presynaptic neuronal activity changes the probability of postsynaptic spikes. Theta oscillations found in EEG or LFP are modeled by periodic sinosoidal modulation of the baseline neuronal activity. Neurons fire spikes with an average rate that is independent of the modulatory rate. Such a modulation could be e.g., induced by periodic inhibition delivered by external sources. As a first step we investigate the filter properties of spSTDP. In Figure 4 we display the rate of weight change as a function of modulation frequency f_mod and phase difference Δφ for five different values of q, corresponding to differently biased STDP. For a given f_mod in balanced spSTDP the plot shows that synaptic change shows the greatest difference between minimum and maximum (malleability or susceptibility) in the theta range (4–10 Hz). For high or low frequencies the change decays back to zero. An analytical calculation (see Appendix) shows that the maximally effective modulation frequency lies at

The frequency of maximum efficiency is a function of the time constants of the STDP window. For the parameters from VC23 used in Figure 4, τ_pre = 14 ms and τ_post = 42 ms, f_max = 6.56 Hz. For the time constants from HC, τ_pre = 17 ms, τ_post = 34 ms, f_max = 6.62 Hz. In general, for physiological parameters the most effective frequency lies in the theta band.

This is a band-pass filter property, which discards too slow or too fast oscillations, and uses intermediate oscillation frequencies as signals for synaptic change. This is contrasted by strongly biased spSTDP (Figures 4C,E). Here, the region of maximal phase dependency of synaptic plasticity on relative phase (i.e., the region of high susceptibility) is not cut off anymore for low modulation frequencies. The synapse acts as a low pass filter.

We chose five different models of synaptic plasticity to compare the filter properties between them. First we examine the effects of extending spSTDP with realistic non-linearities, with the CD model and the Triplet model. Furthermore, we examine three calcium-based models. The first one is the model of Shouval et al. (2002) (“Shouval model” in the following), which introduced a formalization of the calcium control hypothesis. This hypothesis states that moderately elevated levels of calcium in the postsynaptic spine lead to synaptic depression, while high levels lead to potentiation. The goal is then to model the calcium dynamics at the synapse. For reference, we repeat the equations of the Shouval model in the appendix. The second model is an extension of the Shouval model with pre- and postsynaptic adaptation and presynaptic facilitation (Cai et al., 2007, Cai model). The adaptation is shared with the CD model. The third calcium model was developed by Graupner and Brunel (2012) (“Graupner model”). This model makes similar use of the calcium control hypothesis, however, it combines it with a bistable synapse model (Graupner and Brunel, 2007). All models start from biologically plausible first principles and derive the STDP window as a consequence. Also, each model has inherent weight limits, which force us to change the induction protocol for them (see Methods). For all models, we do the analysis with all available parameter sets.

We constrain ourselves to models which use only spikes (spike times) as relevant signals, and derive all relevant variables from them. There exist models which explicitly take subthreshold neuronal dynamics into account (see e.g., Clopath et al., 2010). Although this type of model is potentially more accurate at describing experimental results, it has to make specific assumptions about neuronal dynamics, which we want to avoid here.

For the CD model, the rate of weight change as a function of modulation frequency and phase difference is shown in Figure 5. Because the data in SC23 can not give information about presynaptic adaptation, we use τ^rec_pre and c_pre from area VC23 instead. Three of the plots show a very similar behavior. The weight change is positive almost everywhere, and the zone of maximal LTP depending on modulation frequency is tilted such that for a modulation frequency of 1 Hz highest potentiation occurs at Δφ ≈ 0. A comparison with Figure 4 shows that in the case of a slight bias toward LTP (q = 1.4) the picture looks similar. In the parameters for VC23 and HC, q_min = 1, which means that q(t) ≥ 1 and the requirement for potentiation-biased STDP is fulfilled. In SC23, the STDP window is biased toward LTD (q_min = 0.25), however, the parameters for the activation q show that contributions to it are strong (c_q = 8.5) and last long (τ_q = 0.5 s). Therefore the bias toward LTP results from the baseline activity. The rate of weight change in VC5 deviates strongly from this, it shows an asymmetry between maximal and minimal weight change, and is strongly biased toward LTD.

The characteristics of the susceptibility in the Triplet model are different from the CD model (Figure 6). In three parameter sets (HC, SC23, VC23), the weight change is depression only, and the tilt of maximal weight change (closest to zero) is inverted compared to the CD model. None of these three parameter sets shows a pronounced susceptibility specific to a certain frequency range. Increasing the baseline rate does not change the weight change qualitatively, but rather scales it up (Not shown for SC23, VC23). Comparison with spSTDP (Figure 4; q = 0.7) suggests that the respective parameters are biased toward LTD. Like in the CD model, area VC5 stands out. At 5 Hz the weight change is the same as in the CD model. For an increased baseline rate of 20 Hz, it changes from depression to strong potentiation, as comparison with Figure 4C shows.

In the Shouval model, the susceptibility depends little on the parameters of the induction protocol (stimulation time 2 or 5 s, baseline firing rates different from 5 Hz (not shown), Figure 7). The synapse potentiates for all parameters, and shows a phase dependence for slow oscillations (<3 Hz). However, the difference between maximum and minimum weight is small. The situation is similar in the Graupner model: In VC5 the phase dependence of weight change reaches into the theta band, in HC up to 20 Hz (Figure 7). The synapse, however, shows no band-pass properties. In contrast to the other two models, in the Cai model the synapse is susceptible to oscillations in the theta band. However, as the baseline firing rate is increased, the synapse shifts to a low pass filter.

The CD model is the only model of synaptic plasticity tested here which retains a susceptibility specific for a certain frequency range beyond the linear regime (low firing rates). To illustrate the behavior in the non-linear regime, we computed the rate of weight change for different baseline firing rates for the parameter set of SC23 (Figures 8A–C). At low firing rates, the susceptibility is very similar to that of spSTDP with a moderate bias toward depression (see Figure 4B). One could expect that with increasing firing rate, the bias simply gets stronger until the weight change is similar to Figure 4C. However, the specific susceptibility gets slightly more pronounced, and the maximum of potentiation moves toward (and for very high firing rates beyond) zero phase shift. This effect is stronger for low modulation frequencies (≈3 Hz, Figure 8C). Interestingly, this property resembles the experimental observation that presynaptic stimulation repeatedly delivered at the peak of a theta oscillation potentiates the synapse, while stimulation at the trough depresses it (Hyman et al., 2003).

Next, we investigate the effect of increasing oscillation amplitude on the synapse. We keep the modulation frequency fixed at 5 Hz with Δφ = 0, and vary the oscillation scaling parameter ε. In the case of the SC23 parameters (Figure 9A), the synapse potentiates for constant (unmodulated) firing for baseline rates greater than 5 Hz. Theta oscillations in the firing rates simply lead to an upscaling of this potentiation. We do the same analysis for an altered set of parameters (Figure 9B). We change τ_q to 50 ms instead of 500 ms. This tones down the activation and therefore potentiation, however, it affects the fitting error only slightly (increase from 0.81 to 1.1). In this case, the weight change for constant firing is negative everywhere. Introducing a periodic modulation then increases weight change, turning depression into potentiation, but only for baseline firing rates greater than 5 Hz.

To elucidate what parts of the model are responsible for this susceptibility, we did the analysis for confined versions of the CD model. The resulting weight change as a function of modulation frequency and phase difference is shown in Figures 8D–F, where we removed presynaptic adaptation, postsynaptic adaptation and activation q, respectively. The weight change shows that presynaptic adaptation as well as activation are both important for theta susceptibility. Removing either one results in a low pass filter synapse. The link between these two variables is the threshold ϑ_q, as Figure 9D illustrates. Here, we show the weight change for the full model and parameters of SC23, except that ϑ_q < 0. As a consequence, the synaptic susceptibility has no cutoff for low frequencies anymore. Interestingly, removing the threshold removes large part of the sensitivity to increasing oscillation amplitude (Figure 9D), which further underlines the importance of the interplay of presynaptic adaptation and activation for theta susceptibility.

Although most of the components of the CD model are only loosely guided by biophysical considerations, it is possible to relate them to specific perisynaptic processes, and to envision experiments for a more direct parameter estimation.

The spike traces y are very similar to the dynamics of bound glutamate at postsynaptic receptors and calcium dynamics in the synaptic bouton, which are essentially low-pass filtered action potentials. Due to the differential Hebbian learning rule at the core of the CD model, the decay constants of the spike traces determine the shape of the classical exponential STDP window; therefore, they can be directly estimated from varying the timing of a single pre- and postsynaptic spike pair.

The adaptive suppression u, which leads to a sublinear summation of synaptic change, has a dynamics reminiscent of presynaptic short-term depression (Tsodyks and Markram, 1997; Tsodyks et al., 1998). Its parameters can be determined by measuring the change of the synapse in response to adding leading presynaptic (postsynaptic) spikes to a pre-post (post-pre) spike pair and comparing the experimental result to the prediction of a spike pair model. Presynaptic short-term depression is a well-understood phenomenon, which has been shown to be present in many different cell types (Zucker and Regehr, 2002). Interestingly, the results of the fits of the reduced model indicate that short term depression considerably influences synaptic change and should be taken into account in quantitative models of synaptic plasticity. Some synapses show facilitation as well, and the CD model can easily be accommodated to include this also by adding one equation in a manner similar to Tsodyks et al. (1998). On the postsynaptic side, however, the mechanism behind the adaptive process has been studied much less (Froemke et al., 2006; Gasparini, 2011); the details of the formalization will possibly have to be adapted as soon as more quantitative data becomes available.

The conditional activating variable q is a coincidence detector, which primes the synapse for supralinear potentiation. An interpretation is that q, if dependent on the presynaptic trace, reflects the calcium trace from NMDA receptors, which rely on the coincidence of glutamate binding and postsynaptic depolarization to lift the Mg²⁺ block (Clarke and Johnson, 2006). A negative threshold could mean that less calcium is needed for induction of LTP, or influx of calcium through voltage dependent calcium channels is sufficient for an elevated trace. The relaxed state value q_min on the other hand tunes the balance of LTP and LTD at the synapse. Under certain conditions, like in SC23, a pre-post pair leads to calcium influx which does not exceed the threshold for induction of LTP. A second postsynaptic spike added after the pair then “rides” on an elevated level of calcium, and the summed calcium contributions exceed this threshold. To estimate the parameters of activation with known postsynaptic adaptation, trailing postsynaptic spikes can be added to a pre-post-pair, with an additional post-pre-post triplet to find the sign of the threshold.

In the last decade, a number of models for synaptic plasticity in response to spiking activity have been developed. Some of them are extensions of spSTDP, like the Triplet model, others are grounded on more biophysically plausible considerations, like the calcium models in general. The only model we know of which was fitted to the same four data sets as the CD model here is the recent calcium model by Uramoto and Torikai (2013). They used a different way to calculate the error of their fit, as they did not normalize the experimental results with the standard error of the mean (SEM). We repeated the calculation of the error with normalization by SEM, with the parameters given in the original article. The resulting errors are 0.85 for VC5, 2.4 for HC, 0.27 for SC23, and 27.39 for VC23. See Tables 3, 5 for comparison. The last value mostly results from the omission of the “1−5 × pre-post” experiments by the authors [Figure 4D in the original article of Froemke et al., (2006)], although the error without these experiments is still 9.2. These experiments most prominently highlight the role of presynaptic adaptation: Adding more presynaptic spikes in front of the pre-post pair reliably decreases the magnitude of synaptic potentiation. The Uramoto model has no mechanism which results in less postsynaptic calcium (less LTP) with increasing number of presynaptic spikes, a property shared with the Triplet model, Graupner model and Shouval model. Our quantitative analysis in the CD model showed that in the data sets where applicable (SC23 used only one presynaptic spike in each induction pattern), adding presynaptic adaptation considerably improves the error. However, presynaptic (and postsynaptic) adaptation can easily be implemented in all these models, for example by introducing Equation (9). Different formalizations are realized by Cai et al. (2007) and Kumar and Mehta (2011). We propose that adaptation is a mechanism that should in general be considered for the quantitative modeling of synaptic plasticity.

Among the existing phenomenological models of STDP, the Triplet model is the one most similar to our CD model. Both show similar fitting performance on visual cortex layer 5 and hippocampal data sets. Although it lacks true adaptation, two properties of the Triplet model partially mimic it: (1) with nearest neighbor interactions a spike causes the trace to attain a certain, constant value (from where it relaxes back). If the trace still is greater than zero, the impact of the subsequent spike is reduced; (2) a negative value for the triplet interaction A⁺₃ works in the opposite direction of the normal spike pair interaction, leading to a sublinear summation of potentiation. In total, the CD model reaches a lower error than the Triplet model on all data sets, in SC23 and VC23 by a considerable margin. This suggests that the CD model generalizes better than the Triplet model. In addition, the components of the CD model have a more straightforward interpretation.

We investigated the filter properties of synaptic plasticity in a range of different models. For balanced spSTDP, we can give an explanation of the origin of the susceptibility to intermediate neuronal activity oscillation. spSTDP is equivalent to the formulation as differential Hebbian learning, Equations (2), (3). The traces y_i are driven by the spike trains x_i, and they “smear” out the spike over time, which is basically the action of a low-pass filter. However, the weight change is proportional to the product of the presynaptic trace and the temporal derivative of the postsynaptic trace, ẏ_post. A temporal derivative accentuates (fast) changes, and its effect is similar to a high-pass filter. The result is a band-pass filter with an oscillation frequency of maximal efficiency given by Equation (22). If a moderate bias is introduced (see Figures 4B,C), the basic finding is distorted only slightly.

For non-linear models of synaptic plasticity that are based on spSTDP, the picture in general is similar as long as the average firing rate stays low, which keeps the dynamic equations in the linear regime. If the firing rate gets high enough that spikes in a neuron start to interact with each other, non-linear interactions will start to distort the susceptibility. For a mean firing rate of 5 Hz, the CD model stays in a near-linear regime, while in the Triplet model non-linear effects abandon susceptibility. Interestingly enough, with the right parameter choice the non-linearities in the CD model retain a band-pass behavior similar to the linear regime (Figures 8A–C). The susceptibility in the non-linear regime depends on the interplay of u_pre and q with a threshold for activation ϑ greater than zero. The action of presynaptic adaptation is to suppress y_pre for sustained constant firing of the presynaptic neuron. In fact, a mean field calculation shows that for parameters in VC23 in equilibrium and in the limit of high firing rates 〈y^∞_pre 〉 = 0.033 < ϑ_q. However, with oscillating neuronal firing, y_pre reaches a maximum early during the rise of the rate. The condition on the activation q leads to maximal increase if the postsynaptic firing rate is maximal at the same time. Therefore, q is maximal for slightly negative phases, which leads to the observed phase shift in the transition to the non-linear regime (increasing baseline firing rates).

We found that in contrast to spSTDP-based models, the synapse in the two calcium models without adaptation is simply a low pass filter, and prefers oscillations of both neurons that are in phase. This is due to the extensive low-pass filtering in the dynamical equations in these models. In both models, the contributions to the calcium concentration are low-pass filtered spike signals, which get low-pass filtered again in the calcium dynamics. As the calcium concentration depends on the sum of pre- and postsynaptic contributions, it is not surprising that maximal LTP occurs close to zero phase and slow oscillations. The Cai model is an example of a calcium model with synaptic short term dynamics. Here, the synapse shows a susceptibility to oscillatory modulation in the theta band, if the average firing rate is sufficiently low. Interestingly, our result seemingly conflicts with previous results. In the study of Kumar and Mehta (2011), it was shown that with the Shouval model the STDP window shows maximal malleability, that is the difference between maximal and minimal synaptic change, if the spike pairs are delivered with a repetition frequency of 5–15 Hz. This is very similar to our definition of susceptibility, where there exists a region of f_mod with pronounced and maximal difference between maximal and minimal weight change. However, in our study neuronal firing rate and oscillatory modulation (f_mod) are decoupled, while in the aforementioned study they are equal. Furthermore, we investigated the malleability under the condition of stochastic spiking. Here, spSTDP based models in a near linear regime show a preferred window of modulation frequency, while calcium based models prefer slow oscillations.

Theta band (4–8 Hz) oscillations of both cortical (Landfield et al., 1972) and hippocampal (Berry and Thompson, 1978) local field potentials have been associated with memory processes early on. Later studies extended these findings across species and spatial scales, i.e., from intracellular membrane potential fluctuations in the rodent hippocampus (Harvey et al., 2009) to intracranial recordings in monkey cortex (Liebe et al., 2012) and extracranial EEG in humans (Kahana et al., 2001). Despite being observed throughout the brain, theta band oscillations appears to be generated by a network of hippocampal oscillators (Colgin, 2013), which is then transferred into cortical areas.

Although many studies have established a correlation between activity in the theta frequency band, so far no direct explanation for how theta rhythms influence memory processes has been found (Colgin, 2013). Some studies (Berry and Thompson, 1978; Seager et al., 2002; Nokia et al., 2008) report that the indicator for learning success is the increased oscillation amplitude before the onset of a trial. In other words, theta can be present without being linked to a certain task and still be beneficial. Others find that bursts delivered at theta frequency are optimal for induction of LTP (Larson et al., 1986), and LTD and LTP are inducted by bursting at different phases of a background theta oscillation: Presynaptic bursts at the peak of the oscillation potentiate the synapse, while bursts at the trough lead to synaptic depression (Pavlides et al., 1988; Hyman et al., 2003). In humans though, the situation is not as clear. Some studies find that increased theta power predicts learning success (Sederberg et al., 2003; Guderian et al., 2009; Lega et al., 2012), others emphasize theta synchronization and sometimes find decreased theta power (Mölle et al., 2002; Burke et al., 2013). One experimental study found that in successful learning single neurons show enhanced phaselocking to a background theta oscillation in the LFP, with a wide distribution of specific phase relations to this theta oscillation (Rutishauser et al., 2010). All these observations make it very likely that theta oscillations play a constructive role in the formation of memory.

The synaptic filter properties of several plasticity models reported here provide an explanation, as they endow the synapse with a susceptibility that is specific to oscillations in the theta range. This susceptibility does not rely on precise spike timing, i.e., a fixed phase relation of repeated spikes to an ongoing background theta oscillation. The distinction between the linear (similar to spSTDP) and non-linear regimes we found in the models makes two different scenarios likely of how theta susceptibility plays a role in learning. With low baseline firing rates [<10 Hz, reported in Rutishauser et al. (2010)] and a wide distribution of pairwise relative phases the synaptic changes are also expected to show a wide distribution of values. In a neuronal population firing at higher baseline rates the interplay of presynaptic adaptation and conditional activation shifts the phase requirement for strongest LTP to synchronous (phase zero) oscillation. How can a synapse capitalize on that? The scenario in Figure 9 provides a possible answer. The synapse depresses uniformly for neuronal constant firing. Introduction of theta band oscillations (5 Hz) shift up the weight change, but only for elevated baseline firing rates. The result is Hebbian learning (“those who fire together wire together”), as synapses between neurons which get no external excitation slightly depress. In this scenario, theta oscillations can be present before external stimulation, preparing the synapses for learning correlations of neuronal firing.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.