Basic modeling setup

In order to investigate how the early-phase of homosynaptic plasticity (ELTP and ELTD) is transformed into the late-phase (LLTP and LLTD), we assume a system consisting of one spiking neuron with one incoming synapse stimulated by a Poisson spike train with given frequency (Fig 1B). Next, to reproduce cross-tagging experiments, comparable to the experimental design [19, 22], we consider one neuron with two incoming synapses (red and blue; Fig 1C) each stimulated by an independent Poisson spike train with different frequency and stimulation pattern. By setting the stimulation of one synapse to zero, we analyze the pure heterosynaptic effect of a stimulated synapse (blue) on an unstimulated synapse (red). This is generalized to n stimulated and one unstimulated synapse (Fig 1D). Furthermore, to analyze the spatial restriction of heterosynaptic plasticity, we also considered a setup where synapses are spatially distributed along a dendritic branch with certain inter-synapse distances d (Fig 1E).

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Fig 1. Schematics of setups used throughout this work.

(A) Schematic illustration of the interactions between axonal, dendritic and soma-specific quantities used in the model. (B) One neuron receives stimuli from one synapse (small colored circle). We stimulate this synapse by four different protocols (WTET, WLFS, STET, SLFS; for details see main text) to induce different dynamics of synaptic plasticity (viz. ELTP, ELTD, LLTP, LLTD). (C) For cross-tagging and heterosynaptic plasticity experiments, two synapses (S1 in red and S2 in blue) are connected to one neuron and stimulated independently. (D) Similar to setup (C) but with n stimulated synapses (S1, S2, …, Sn) and one non-stimulated synapse (Sn + 1). (E) For distance-dependent heterosynaptic plasticity protocols we considered that n synapses are stimulated (blue). This leads to the induction of a global postsynaptic calcium signal C_post. Synapses (red) near the stimulated ones receive in addition a diffusive calcium signal C_diff which depends on the distance d between stimulated (blue) and non-stimulated synapses (red).

https://doi.org/10.1371/journal.pone.0161679.g001

Neuron model

For simulating the dynamics of the postsynaptic neuron (Fig 1) we use the biological plausible multi-timescale adaptive threshold (MAT) model [34, 35]. In particular, the MAT model reproduces and predicts precisely neuronal spike timings. Amongst others, such spike timings are a key factor influencing plasticity [31, 36]. The MAT model consists of two parts: i) the membrane potential V(t) and ii) the spike-history-dependent threshold θ_V(t). The dynamics of the membrane potential is given by a non-resetting leaky integrator: (1) The part is the sum over all inputs dependent on the neuronal resistance R. Synapse i contributes to the input current when there is a spike at time point . The synaptic input is weighted by two parts; one describes the influences of the early-phase synaptic weight ρ_i and the other the late-phase synaptic weight Z_i scaled with a factor ρ₀. The dynamics of ρ_i and Z_i will be described later.

The dynamics of the adaptive threshold θ_V(t) is determined by the following equation: (2) The threshold θ_V is composed of a constant ω and of multiple time scales dependent on the spiking history of the neuron: τ₁ = 10 ms and τ₂ = 200 ms (τ₁ represents fast transient Na⁺ currents and delayed rectifier K⁺ currents; τ₂ represents non-inactivating K⁺ currents, hyper-polarization-activated cation currents and Ca²⁺-dependent K⁺ currents [37]). If, at time point t_k, the voltage V(t) is larger than the threshold θ_V(t), a postsynaptic spike will be elicited and the threshold increases to impede the generation of further spikes. We use the same values for the parameters as in [35], see also Table 1.

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Table 1. Used parameters if not stated differently elsewhere.

https://doi.org/10.1371/journal.pone.0161679.t001

Calcium-based early-phase synaptic plasticity

The arrival of presynaptic action potentials and/or the depolarization of the postsynaptic membrane at a synapse induces the entry of postsynaptic calcium through NMDA receptors and voltage-dependent calcium channels [38–42] triggering signaling cascades including protein kinase (LTP) and phosphatase (LTD). These activity-dependent calcium transients are the basic substrate of LTP and LTD induction. The dynamics of the postsynaptic calcium c_i(t) at synapse i is described by the following equation [31]: (3) (4) The parameters c_pre, c_post are the spike-evoked increases of calcium currents by pre- and postsynaptic spikes at time points t_k and t_j, respectively. The time constant D = 18.8 ms accounts for the time delay of presynaptic effects on the postsynaptic calcium concentration. We keep the time scale for the calcium dynamics at τ_c = 48.8 ms.

Experimental evidences show that, if the calcium concentration is high, LTP occurs while moderate calcium concentrations induce LTD [38–42]. In other words, the calcium concentration determines the (early-phase) change of the synaptic efficacy ρ_i(t) as given in the following [31]: (5)

The early-phase synaptic strength ρ_i(t) and postsynaptic calcium concentration c_i(t) are dimension free. If the postsynaptic calcium concentration is θ_d < c_i(t) < θ_p, then ρ_i(t) decreases by γ_d ⋅ ρ_i(t) (ELTD); if the calcium concentration is c_i(t) > θ_p, then ρ_i(t) will increase by γ_p − (γ_p + γ_d) ⋅ ρ_i(t) (ELTP); if the calcium concentration is lower than the threshold θ_d, ρ_i(t) will slowly converge to its initial value ρ_i(0) = ρ₀ with the time scale τ_ρ/0.1 = 6880 seconds or approximately 1.9 hours [19]. The last term is activity-dependent noise with η(t) being Gaussian white noise with unit variance. The Θ- or Heaviside-function is defined as following: Θ[x] = 1, if x ≥ 0 and Θ[x] = 0, if x < 0.

Note, the first term is modified compared to the original model [31] to imply only one attractor (the initial value ρ₀). This modification assures that, similar to experiments, potentiated and depressed synapses will return to the initial weight value within the time scale of one to three hours [19, 22].

We use with being the saturation value by maximally potentiating the synapse (thus, the maximal possible synaptic weight). We define as the change of synaptic strength, so that Δρ_i > 1 means potentiation and Δρ_i < 1 indicates depression.

Protein synthesis, tagging, and late-phase plasticity

Experimental evidences [19] suggest that synaptic consolidation depends on two thresholds. In the following, we will briefly recapitulate these experiments. Consider two independent synaptic inputs connecting to the same neuronal population (similar to Fig 1C). A weak tetanus stimulation was applied to synapse S1 and the resulting synaptic changes decay after one to two hours. Now, a repeated strong tetanus was applied to the second synapse S2 one hour before the weak tetanus to S1. Then, the synaptic changes at S2 get consolidated without effecting S1. Remarkably, with the same stimulation protocol except a slightly stronger stimulation at synapse S1 (using a single strong tetanus instead of a weak tetanus), the changes at S1 are converted from ELTP to LLTP. These experiments indicate two thresholds [19]: one is the high threshold θ_pro for triggering protein synthesis (e.g., repeated strong tetanus) and the second one is the lower threshold θ_tag describing the process of synaptic tagging (slightly stronger stimulation).

Thus, the dynamics for protein synthesis depends on the threshold θ_pro: (6) P(t) represents the number of synthesized proteins, i.e., if the absolute value of the synaptic weight change is larger than θ_pro, P(t) will increase. Thereby, κ dictates how fast the proteins are synthesized which could also depend on the dopamine level [32, 43]. If the sum of synaptic changes is smaller than the threshold θ_pro, P(t) decays with the time scale τ_P to zero.

The protein dynamics P(t) directly influences the dynamics of the late-phase synaptic weight Z_i(t): (7) Z_i(t) represents a complementary synaptic weight for each synapse, i.e., that the total synaptic weight is determined by W_i(t) = ρ_i(t) + Z_i(t)ρ₀. We have included a separate dynamics for Z_i(t) for the synaptic consolidation of LTP and LTD: if the synapse with ELTP grows strong enough to get tagged (ρ_i(t) − ρ₀ > θ_tag), the first term of Eq (7) is non-zero and leads to long-lasting potentiation; vice versa, if the synapse with ELTD is depressed enough to get tagged (ρ₀ − ρ_i(t) > θ_tag), the second term of Eq (7) becomes non-zero and induces late-phase LTD. Note, both processes, the consolidation of LTP and LTD, require proteins P(t) to participate (thus, P(t) ≠ 0) otherwise Z_i(t) will stop changing. As can be seen from the equation, the termination for Z_i(t) has three reasons: i) there are no proteins synthesized P(t) = 0; ii) Z_i(t) reaches a saturating value: Z_i(t) = 1 for LTP and Z_i(t) = −0.5 for LTD; iii) the synapse did not get tagged, i.e., |ρ_i(t) − ρ₀| < θ_tag. Note, for the reasons discussed later, different to other models [32, 44], here we assume that the induction of protein synthesis and synaptic tagging depends directly on the relation of the early-phase synaptic weight to the thresholds θ_tag and θ_pro, thus, the combined model has weight-dependent thresholds. Even though the molecular machinery requires different proteins for tagging LLTP and LLTD [22], for simplicity, we assume that the thresholds θ_tag and θ_pro for LTP and LTD are the same.

For a better overview, all relationships between the variables of the model are summarized in Fig 1A.

Distance-dependency of heterosynaptic plasticity

In the postsynaptic neuron several processes (e.g., backpropagating action potentials (bAPs), calcium spikes) lead to the influx of calcium into the dendrite or the dendritic spines (c_post; see Eq (4)). If this calcium signal alone would be sufficient to trigger the induction of heterosynaptic plasticity, many, if not all, synapses on the dendrite would be affected. However, experimental evidence indicates that this heterosynpatic influence is restricted to a local area (several micrometers) surrounding the presynaptically stimulated synapse [30, 45–47]. Thus, we expect that an additional signal is required to trigger heterosynaptic changes. This additional signal can be, besides secondary messengers, calcium diffusing from the externally activated synapse to other sites (Fig 1E). This calcium diffusion C_diff(d_ij) is locally restricted (several micrometers; [48, 49]) and depends on the distance d_ij between source (j; blue) and target synapse (i; red). For simplicity, we assume that synapse i is not stimulated (C_pre = c_pre∑_tk δ(t − t_k − D) = 0) and only synapse j receives external stimulation and is close enough such that calcium can diffuse to synapse i (C_diff(d_ij) > 0). Interestingly, experimental evidence [50, 51] show that such local calcium currents nonlinearly amplify simultaneous global calcium currents (C_post = c_post∑_tν δ(t − t_ν) with postsynaptic events at time points t_ν). Thus, we assume that the existence of local C_diff could induce such a nonlinear amplification of coinciding C_post or rather that C_diff could serve as a gating signal determining when C_post triggers heterosynaptic changes. Mathematically such gating operations can be formulated as a multiplication with a Heaviside function Θ[x]: (8) with threshold θ_diff defining when the diffusive calcium current is strong enough to nonlinearly amplify C_post. As the diffusion is noisy and locally restricted [48, 49, 51], we consider that the probability , which describes whether enough calcium (C_diff > θ_diff) diffuses from j to i, decreases with larger distance d_ij between the synapses. Thus, Θ[C_diff(d_ij) − θ_diff] can be replaced by with uniform random variable r (from the interval [0, 1]) and Gaussian distributed probability with constant σ_diff: (9)

Note, if synapse i is presynaptically stimulated, this stimulation leads to C_pre > 0 and, thus, Eq (9) can be written as follows: (10) Apparently d_ii = 0 implies that and, thus, and Eq (9) becomes similar to Eq (4). In other words, the basic underlying dynamics of homo- and heterosynaptic plasticity are assumed to be similar.

If a group of synapses n is stimulated simultaneously, the stimulation (n = 1, …, 20) is sparse compared to the total number of synapses on a dendritic tree (about 10⁴). Thus, we consider large distances between activated synapses such that, on average, no synapse receives diffusing calcium currents from two distinct sites (schematically shown in Fig 1E). However, the number of active synapses influences the postsynaptic activity, thus, the postsynaptic calcium influx C_post and, thereby, each synapse has a weak influence on the calcium dynamics and heterosynaptic plasticity at distant synapses.

Stimulation protocols

It is experimentally well known that synaptic consolidation depends strongly on the used stimulation protocols [19, 22]. In this study, four typical stimulation protocols are used similar to experimental studies [19, 22]:

1. Strong TETanus (STET): three trains of Poisson spikes for 1 sec at 100 Hz with 10 min intertrain interval, typical for the induction of LLTP.
2. Weak TETanus (WTET): a 100 Hz Poisson spike train for 0.1 – 0.3 sec (specified in figure legends), typical for the induction of ELTP.
3. Strong Low Frequency Stimulus (SLFS): 900 bursts each consisting of three Poisson spikes at 20 Hz and a interburst interval of 1 sec, typical for the induction of LLTD.
4. Weak Low Frequency Stimulus (WLFS); 900 Poisson spikes at frequency of 1 Hz, typical for the induction of ELTD.

Further protocols are described in the main text or figure captions. All protocols are repeated, comparable to experimental setups, 10 times and their averages with standard deviation are shown. Simulations are performed on a standard desktop PC using Matlab.

Consolidation of homosynaptic changes by synaptic tagging, protein synthesis, and cross-tagging

It is well known that different stimulation protocols at the same synapse (Fig 1B) induce different types of synaptic changes as, for instance, early-phase LTP/LTD and late-phase LTP/LTD. For instance, a weak tetanus stimulation (WTET; Fig 2A) induces a change of ρ (blue) resulting in a change of the synaptic efficacy W (red). Although the early-phase synaptic change Δρ is above the tagging threshold θ_tag and, therefore, the synapse is tagged (details can be seen in next section), the change does not become late-phase or consolidated as protein synthesis is not initiated (Δρ < θ_pro) and W decays back to its initial value. The induced plasticity effect decays after a few hours (ELTP). If the stimulation is stronger or longer (STET; Fig 2B), the synaptic change reaches the threshold (Δρ > θ_pro) and, in turn, triggers protein synthesis P (purple) inducing long-term synaptic changes Z (green; LLTP). These changes lead to long-lasting alterations of the total synaptic strength W. Similar effects arise for low frequency stimulations (WLFS and SLFS) which are used for the induction of LTD (Fig 2C and 2D).

Experimental results indicate that the initiation of postsynaptic protein synthesis is not the only requirement to be fulfilled to consolidate a synaptic change [19]. In addition, the synapse itself has to receive a strong enough presynaptic stimulus so that the synapse will be tagged [19, 23, 32]. Comparable to the initiation of protein synthesis, in the combined model, the linketween the strength of a stimulus and the tagging of the corresponding synapse is given by the early-phase change Δρ. Consider a neuron with two incoming synapses (Fig 1C), one stimulated by a weak tetanus. Independent of the stimulus duration (WTET1: 0.1 sec; WTET2: 0.2 sec), the WTET induces an early-phase change Δρ (Fig 3A and 3C) which decays after two or three hours as protein synthesis is not initiated (similar to Fig 2A). Given a strong stimulus (STET) at the second synapse S2 (red in Fig 3B and 3D), protein synthesis is initiated (similar to Fig 2B) and synaptic changes at each incoming synapse of this neuron can be consolidated (which happens for S2). However, if the weak tetanus at S1 (blue) is too short (WTET1), the early-phase change does not reach the tagging threshold (Δρ_S1 < θ_tag) and, therefore, the synaptic change cannot become LLTP (Fig 3B). In contrast, if the WTET is longer (WTET2), early-phase changes are large enough to pass the tagging threshold (Δρ_S1 > θ_tag) and the synapse is (cross-)tagged and consolidated by the proteins initiated by the second stimulus (Fig 3D). Thus, by linking the synaptic tagging and protein synthesis with the stimulus strength via the early-phase change Δρ, the combined model shows similar dynamics as measured in in vitro experiments [19].

The assumption of having two thresholds θ_tag and θ_pro and the ability of cross-tagging (Fig 3D) enables us to reproduce also other experimental cross-tagging protocols ([22]; Fig 3E–3H). In each protocol a strong stimulus is given to synapse S2 (SLFS in Fig 3E and 3G and STET in Fig 3F and 3H) initiating the synthesis of proteins (similar to Fig 2B and 2D). As the protein synthesis is initiated for about three hours (Fig 2B and 2D), tagged synaptic changes induced by a weak stimulation at another synapse (here S1), also one hour after the strong stimulus, are consolidated (WTET in Fig 3D and 3G and WLFS in Fig 3E and 3F). Thus, early-phase potentiation (depression) can not only be consolidated or cross-tagged by late-phase potentiation (depression) at another synapse (connecting the same neuron) but also by late-phase depression (potentiation).

Note, in Fig 3H the stimulation sequence is reversed: the WTET is provided one hour before the strong stimulus. However, due to the fact, that the tag at the synapse stays as long as the early-phase change is above the tagging threshold, the synaptic change can be consolidated. This predicts that the timing could also be reversed for other cross-tagging protocols (Fig 4). For instance, a WTET induced ELTP can be consolidated by a SLFS one hour later (Fig 4A). We expect similar dynamics for early-phase depression (Fig 4 and 4C). Of course, if the timing between weak and strong stimulus is given in a way that either the tag or protein synthesis are not present anymore, the transfer of early-phase changes to late-phase is not possible (data not shown).

In summary, the combined model of calcium-based plasticity and synaptic consolidation with two weight-dependent thresholds for protein synthesis and synaptic tagging shows similar synaptic dynamics as experimental measurements [19, 21, 22]. A synaptic change is transferred from early- to late-phase if a tag is created and the postsynaptic neuron initiates protein synthesis (Figs 2 and 3A–3D; [19]). Thereby, the initiation of protein synthesis can be triggered by the tagged synapse itself (Fig 2) or by another synapse leading to cross-tagging (Fig 3). Whether the synaptic efficacy is depressed or potentiated is not relevant (Fig 3D–3G; [22]) as well as the sequence of tagging and protein synthesis events (Fig 4). However, so far the induced changes are all homosynaptic. In the following we will show the dynamics of heterosynaptic plasticity.

The induction and consolidation of heterosynaptic plasticity

As shown before, the combined model reproduces experimental results of homosynaptic consolidation and cross-tagging. However, given two synapses connecting to the same postsynaptic neuron (Fig 1C and 1D), they can interfere with each other by the dynamics of this shared neuron. For instance, if synapse S1 receives a strong presynaptic stimulus (Fig 1D), then this stimulus can activate the postsynaptic neuron triggering changes in the postsynaptic calcium concentration [31] effecting both synapses inducing heterosynaptic plasticity at the second synapse S2. Thereby, it is not essential whether synapse S2 also receives presynaptic inputs or is silent.

Thus, the firing rate of the postsynaptic neuron is one important factor determining the induction of heterosynaptic plasticity. Obviously, the postsynaptic activation is influenced by the strength of the incoming stimulus. Amongst others, this input strength can be regulated by the number of stimulated synapses (see S1 and S2 Figs for regulating the resistance instead), as a neuron can receive the stimulus from a population of synchronized presynaptic neurons (e.g., [52]). Thus, a correlated input from a group of neurons can reach a postsynaptic neuron by many synapses enhancing the postsynaptic activity inducing heterosynaptic plasticity. Furthermore, this number of synchronized synapses is variable according to activity-dependent structural changes [53, 54]. Thus, in the following, we test whether the number of stimulated synapses influences the induction and consolidation of heterosynaptic changes.

Consider a postsynaptic neuron with n + 1 incoming synapses (Fig 1E) whereby n of these synapses receive independent Poisson spike trains of the same frequency and one no stimulation. Thus, the system receives a rate-coded stimulus via n inputs. In the following, we will consider again the four protocols WTET, STET, WLFS, and SLFS. Varying the number of synapses enables for these stimulation protocols a rich repertoire of homo- and heterosynaptic changes (Fig 5). A WTET stimulation induces for 3 to 19 input synapses heterosynaptic depression (Fig 5A). However, this change is not tagged and, therefore, does not become consolidated. In contrast, a STET stimulation induces late-phase heterosynaptic depression (Fig 5B, 5E and 5F). For one or two incoming synapses there is no heterosynaptic effect at all (Fig 5B). If 3 ≤ n ≤ 5, the stimulus induces an early-phase heterosynaptic depression (ELTD; Fig 5E) comparable to experimental observations [55]. For 6 ≤ n ≤ 9 heterosynaptic depression is induced, tagged and, thereby, consolidated by cross-tagging becoming LLTD (Fig 5F). Note, in contrast to a stimulus-dependent synaptic tag [32, 44], only the dependency of the synaptic tag on the early-phase synaptic weight, used in this model, enables the consolidation of heterosynaptic changes. Thus, a group of active presynaptic neurons enables the induction and consolidation of homosynaptic potentiation and heterosynaptic depression which could enable stabilization of dynamics on the network level. However, if the stimulus is transmitted by about n = 11 synapses, remarkably, the unstimulated synapse is even potentiated for a short duration (ELTP; Fig 5G). If the number of input synapses exceeds 12, the stimulus induces heterosynaptic plasticity potentiating the unstimulated synapse and, furthermore, inducing a synaptic tag. By this, the unstimulated synapse underwent an heterosynaptic late-phase potentiation (LLTP; Fig 5H). Similar heterosynaptic effects also arise for low frequency stimulation inducing (for low synapse numbers) homosynaptic LTD (Fig 5C, 5D and 5I–5L). Thus, by varying the number of synapses transmitting the stimulus by, for instance, adapting the number of synapses between two neurons (e.g., [54]) or changing the number of neurons encoding the same stimulus (e.g., [56]), the effect of the stimulus on the synapses transmitting the stimulus and on other synapses can be changed accordingly.

Clearly, the heterosynaptic effect depends on the postsynaptic activity enabling the influx of calcium at the unstimulated synapse. The postsynaptic activity, in turn, is influenced by the stimulus frequency of the activated synapses. Besides the frequency (rate-code), however, the correlation [57] between the spike trains affects heterosynaptic plasticity (Fig 6, S3 and S4 Figs). For a high frequency stimulation (STET and WTET in Fig 6A and 6B), correlations in input spiking do not influence significantly the induction of heterosynaptic potentiation. This is due to the fact that the high frequency input is so intensive such that correlations cannot effect the post neuronal firing level. However, for low frequency stimuli (SLFS and WLFS; Fig 6C and 6D), heterosynaptic plasticity is strongly influenced by the correlations in inputs. Especially for the SLFS stimulation protocol at a specific correlation coefficient a transition from heterosynaptic depression to potentiation occurs. With increasing number of correlated input synapses this transition point shifts to weaker correlations. For the WLFS stimulation protocol, although no transition occurs, still the correlation influences the magnitude of heterosynaptic plasticity (depression). Thus, these results imply that synaptic changes are only sensitive to correlations if the average firing rate is relatively low. This can be particularly important considering dynamics in neural networks [57, 58].

However, for all protocols above, the heterosynaptic effect influences all synapses at the postsynaptic dendritic tree. In contrast, experiments show that such effects are limited to only the neighboring synapses of the stimulated one [30, 45–47]. Therefore, we assume that from the stimulated synapse messengers can diffuse to the unstimulated. Amongst others, the increased amount of calcium in the stimulated spine could be such a messenger [48, 49, 51]. If these calcium ions diffuse and the resulting local calcium current coincides with global calcium currents initialized by, for instance, backpropagating action potentials, nonlinear dendritic dynamics lead to additional calcium influx [50, 51] into the unstimulated synapse enabling hetersynaptic plasticity. As diffusion is spatially restricted (by the diffusion parameter σ_diff based on [48, 49]) and the calcium concentration falls with increasing distance d from the stimulated synapse, the nonlinear response and, therefore, heterosynaptic plasticity is also distance-dependent and restricted to a subset of synapses which are close enough to the stimulated ones (Fig 7). This can lead to the fact that, given a homosynaptic potentiation (Fig 7E–7H), nearest-neighbor synapses receive enough diffusive calcium such that they are also potentiated while unstimulated synapses with larger distances receive less calcium and are depressed. As the diffusion is spatially restricted, far away synapses do not receive enough diffusive calcium and the nonlinear increase in calcium influx by coincidence with global signals fail to appear. Thus, these synapses are unaffected and their synaptic strength stays constant. Such a ‘Mexican-hat’-like plasticity profile (Fig 7E–7H) is also found experimentally [33]. Furthermore, given that some of the resulting local synaptic changes (at the nearby synapses) could be tagged while others not, one can get a complex pattern of homo- and heterosynaptic changes over a broad variety of time scales.

In summary, the combined model of calcium-based plasticity and synaptic consolidation with weight-dependent thresholds enables the induction of a rich repertoire of homo- and heterosynaptic plasticity. Thereby, dependent on the stimulation protocol and the number of input synapses, heterosynaptic depression and potentiation can be consolidated in a similar way as homosynaptic changes.