Effect of the Initial Synaptic State on the Probability to Induce Long-Term Potentiation and Depression

Abstract

Long-term potentiation (LTP) and long-term depression (LTD) are the two major forms of long-lasting synaptic plasticity in the mammalian neurons, and are directly related to higher brain functions such as learning and memory. Experimentally, they are characterized by a change in the strength of a synaptic connection induced by repetitive and properly patterned stimulation protocols. Although many important details of the molecular events leading to LTP and LTD are known, experimenters often report problems in using standard induction protocols to obtain consistent results, especially for LTD in vivo. We hypothesize that a possible source of confusion in interpreting the results, from any given experiment on synaptic plasticity, can be the intrinsic limitation of the experimental techniques, which cannot take into account the actual state and peak conductance of the synapses before the conditioning protocol. In this article, we investigate the possibility that the same experimental protocol may result in different consequences (e.g., LTD instead of LTP), according to the initial conditions of the stimulated synapses, and can generate confusing results. Using biophysical models of synaptic plasticity and hippocampal CA1 pyramidal neurons, we study how, why, and to what extent the phenomena observed at the soma after induction of LTP/LTD reflects the actual (local) synaptic state. The model and the results suggest a physiologically plausible explanation for why LTD induction is experimentally difficult to obtain. They also suggest experimentally testable predictions on the stimulation protocols that may be more effective.

Introduction

The two major forms of long-lasting synaptic plasticity in the mammalian neurons, long-term potentiation (LTP) and long-term depression (LTD), are characterized by changes in the strength of a synaptic connection induced by a repetitive and properly patterned electrical activity. Discovered ∼40 years ago (1,2), for both phenomena there is now compelling evidence that they represent a basic step to understand neuronal development, circuit reorganization, and learning and memory mechanisms (3–6). The underlying molecular mechanisms are starting to be unraveled (7) and are under intense experimental and theoretical scrutiny, especially in the CA1 region of the hippocampus. Experimental studies on LTP and LTD are usually performed using in vitro or in vivo preparations (e.g., Buschler et al. (8) and Goh and Manahan-Vaughan (9)), exploiting specific stimulation patterns of synaptic inputs to induce long-lasting changes in the synaptic strength.

In almost all cases, the overall amount of LTP or LTD is measured at the soma, using specific and precisely defined experimental conditioning protocols that have been found to be particularly effective in inducing synaptic plasticity, such as constant frequency or θ-burst stimulation (10–14). A possible major limitation with this approach is that, even when it is carried out using more-sophisticated electrophysiological recording techniques (15,16), it cannot take into account the actual state of a given synapse before the conditioning protocol. From a general point of view, it cannot be excluded that the same experimental protocol delivered to different synapses may result in opposite consequences (e.g., LTD instead of LTP), according to the state of the synapse at the time of conditioning. The interpretation of the overall results measured at the soma may thus be misleading, especially for LTD, which is well known to be difficult to induce (9,17) for unclear reasons. A number of questions on the interplay among stimulation patterns, preconditioning synaptic strength, and dendritic location, thus remain open.

The main aim of this work is to gain insight into the relation between the stimulation patterns inducing local (dendritic) synaptic plasticity and the excitatory postsynaptic potential (EPSP) heights observed at the soma. Using biophysical models of synaptic plasticity and hippocampal CA1 pyramidal neurons, we studied how, why, and to what extent EPSPs observed at the soma after induction of LTP/LTD reflects the actual (local) synaptic state. The model and the results suggest a physiologically plausible explanation of why LTD induction is experimentally difficult, and they offer experimentally testable predictions on the stimulation protocols that may be more effective.

Materials and Methods

Simulations were carried out using the NEURON simulation environment (Ver. 7.3 (18)). In most cases, an IBM Blue Gene/Q (Armonk, NY) supercomputer (the FERMI system at Cineca, Bologna, Italy) was used to run simulations in parallel. A typical set of 18,564, 90-s long, simulations required ∼8 h using 2048 processors. Model and simulation files will be available for public download on the MODELDB section of the SENSELAB suite (Accession No. 157339; The SenseLab Project, http://senselab.med.yale.edu/ModelDB/).

A morphologically realistic three-dimensional model of a hippocampal CA1 pyramidal neuron (Fig. 1 A, cell 5,038,804, originally downloaded from the public archive, http://www.neuromorpho.org) was used in all cases. The model neuron included uniform passive properties (τ_m = 28 ms, R_m = 28 kΩ × cm², and R_a = 150 Ω × cm), and a set of active properties (voltage-dependent ionic channels, kinetic and distribution) identical to those described in a previous article (Migliore et al. (19), MODELDB accession No. 87535). Briefly, the model included a sodium (g_NA) and a delayed rectifier potassium (g_KDR) conductance, uniformly distributed throughout the dendrites, whereas g_KA and I_h linearly increased with the distance from soma. The neuron model has already been validated against several experimental findings on electrophysiological and synaptic integration properties of CA1 neurons (e.g., Migliore (20) and Gasparini et al. (21)). One example is shown in the inset of Fig. 1 A, where we show the model’s reproduction of the classic experimental finding on distance-independent synaptic integration in CA1 pyramidal neurons (22,23). The traces show somatic membrane potential in response to a short train of five synaptic stimuli delivered at 20 Hz to either a distal (304 μm from the soma, distal) or a proximal (23 μm from the soma, proximal) dendritic compartment. As in the experiments (22), the amount of temporal summation occurring at the soma was independent from the synaptic input location.

Figure 1.

(A) The three-dimensional reconstruction of the CA1 hippocampal pyramidal neuron used for all simulations (top). The traces are somatic membrane potential during a train of stimulations of a distal (red) or proximal (green) synaptic input. (B) Schematic representation of the model synapse; in all cases we used: U_SE = 0.36, τ_in = 3 ms, τ_rec = 50 ms, A_SE = 250 pA, R_in = 10 MΩ, f = 0.05 × 10⁻³ mV⁻¹, δ_i = 400 ms⁻¹, λ_P = 10⁻³ ms⁻¹, λ_D = 2 × 10⁻³ ms⁻¹, ν_P = 0.0987 ms⁻¹, ν_D = 0.07 ms⁻¹, η = 2 × 10⁻³ ms⁻¹, γ = 0.2 ms⁻¹, β = 0.5 × 10⁻⁶ ms⁻¹, τ_m = 40 ms, g₂ = 43 μS, M_i = 3 × 10⁻³ mV/ms, A_P = 2 mV², and A_D = 0.5 mV². (C) Dendritic (black line) and somatic (green line) membrane potential in response to test or conditioning stimuli; stimulation patterns correspond to the experimental TBS (left) or constant frequency protocols (right) for LTP and LTD. To see this figure in color, go online.

To be more closely related to what can happen in vivo, and in contrast to the widely used experimental practice (in vitro) to pharmacologically block the generation of action potentials, our model cell was in all cases a fully active neuron. Synaptic weights eliciting dendritic or somatic action potentials during test pulses were excluded from the analysis.

The model synapse

To investigate synaptic plasticity, we started from a model previously proposed for LTP and LTD induction (24,25), adapting its scheme as shown in Fig. 1 B to also take into account experimental findings on heterosynaptic LTP (26) and depotentiation (7). The presynaptic part was modeled using the phenomenological model of neocortical synapses discussed by Tsodyks and Markram (27) and Abbott et al. (28), described by the equations

$$\frac{dx}{dt}=\frac{z}{{\tau }_{\text{rec}}}-I{U}_{SE}x,$$

$$\frac{dy}{dt}=-\frac{z}{{\tau }_{in}}+I{U}_{SE}x,\text{and}$$

$$z=1-x-y,$$

which reproduce the stereotypical synaptic response dynamics between pyramidal neurons under physiological conditions. The variables x, y, and z are the fraction of resources in the recovered, active, and inactive states, respectively. The parameters used for the presynaptic mechanisms (see legend of Fig. 1) reproduced experimental findings on pyramidal CA1 neurons under control conditions (22).

When an input stimulus I is delivered, the active fraction y of the presynaptic resources generates a synaptic current, I_syn = A_SEw_Sy, whose effects are described by the postsynaptic equations

$$\frac{dg}{dt}=-\frac{g}{{\tau }_{in}},$$

$$\frac{d{v}_S}{dt}=-\frac{v_S}{{\tau }_m}+R_{in}{A}_{SE}g\left[\frac{1}{{\tau }_m}+f\left({\delta }_P{N}_P-{\delta }_D{N}_D\right)\right],$$

$$\frac{dC}{dt}=\gamma v_S-\eta C+\beta \left(V_m+65\right),$$

$$\frac{d{N}_i}{dt}={\nu }_{i}C-\left({\lambda }_i+g{\delta }_i\right)N_i+\frac{M_i{N}_i^2}{A_i+N_i^2},$$

where i ≡ P, D, and g = w_Sy. These equations derive from the synaptic transmission scheme illustrated in Fig. 1 B. A synaptic current, I_syn, generates an effective postsynaptic membrane potential v_s (and an effective post synaptic current, I_s = g₂v_s), which is further modulated by autocatalytic processes, N_P and N_D, representing all those postsynaptic mechanisms that could be involved with LTP and LTD induction and maintenance, such as protein autophosphorylation (29–31). The details and rationale for this kind of implementation are discussed elsewhere (24,25,32). Briefly, the signal generated by a presynaptic event produces a postsynaptic depolarization, with an eventual contribution from additional local sources (e.g., depolarization spread from nearby dendrites). Under the appropriate presynaptic stimulation pattern, this process activates N_P and N_D, which modulate the overall postsynaptic response, increasing (LTP) or decreasing (LTD) the amount of signal generated at each stimulus. The operation of the model can be better understood by considering that this set of coupled nonlinear differential equations implements a bistable switch for N_P and N_D, independently controlled by the postsynaptic depolarization and each one characterized by two steady-state conditions (ground and high). The high state for N_P and N_D is responsible for the induction and maintenance of LTP and/or LTD.

In agreement with experimental suggestions (33,34), following the appropriate conditioning protocol a given synapse will change its state in an all-or-none manner. The model has already been validated against experimental findings (32). This model has suggested experimentally testable predictions (24,25). In this work we have also taken into account experimental findings on heterosynaptic plasticity, by adding a term (β(V_m + 65), in the equations above) that explicitly depends on the local membrane depolarization (V_m) from rest (see below). It should be noted that this model does not take into account the relatively slow subcellular processes (such as protein turnover) underlying LTP/LTD expression and modulation after induction, because they are out of the scope of this work. In our case, once a synapse reaches a stable LTP/LTD state, it does not spontaneously decay to ground.

Results

Stimulation protocols

Experimentally, a widely used protocol is the θ-burst stimulation (TBS), mimicking the endogenous θ-frequency EEG activity seen in the rat hippocampus during exploratory learning tasks (35,36). It consists of trains of short bursts separated by interburst intervals. With TBS protocols, LTP is typically induced by four pulses at 100 Hz repeated at 200-ms intervals, and LTD by two pulses at 100 Hz repeated at 1-s intervals. Typical examples of LTP and LTD induction in our model are shown in Fig. 1 C (left), where we plot the somatic and dendritic membrane potential in response to two test stimuli delivered to a single synapse before and after conditioning stimulation patterns corresponding to the experimental TBS protocol for LTP and LTD. Corresponding traces for the two variables N_P and N_D are shown in Fig. S1 in the Supporting Material. Another typical experimental protocol that we used in this article is tetanic stimulation at constant frequency. In this case, LTP is typically induced with a high-frequency stimulation (e.g., 50 or 100 Hz (10,13,37)), whereas LTD can be induced by a train of stimuli at low frequency (typically in the range of 1–5 Hz (9,10,38)). We will refer to these protocols as HFS or LFS, respectively (10,11,37,39). Typical simulations findings for both cases are shown in Fig. 1 C (right). In this work, we considered only experimental protocols involving presynaptic stimulation alone. Other protocols used to study synaptic plasticity induced by paired pre- and postsynaptic activity (such as spike-time-dependent plasticity) were not considered.

Model validation against experimental findings

We start by validating the model against experimental findings on LTP and LTD. For this purpose we first activated individual synapses, randomly located in the proximal apical trunk (<110 μm from soma), with a test pulse delivered every 50 ms. In this way we can assess a control baseline for the local (dendritic) and somatic EPSP (Fig. 2 A, top left, symbols for t < 750 ms). A 5-s conditioning stimulation was then independently delivered to each synapse, according to the relative experimental protocols for TBS LTP or LTD induction. Finally, another set of test stimuli were delivered to measure the amount of synaptic plasticity observed locally and at the soma (Fig. 2 A, top middle, symbols for 6000 < t < 7000 ms). The results show, on average, 165% potentiation and 35% depression, in qualitative agreement with experimental findings (13,40). We also tested the ability of the model to erase LTP after its induction, an effect called depotentiation and experimentally found in CA1 neurons (41,42). To this purpose, all the synapses that were previously potentiated with the TBS LTP protocol received an additional 40-s conditioning period at a constant frequency of 1 Hz (42). As shown in Fig. 2 A (top-right plot), LTP was erased and all synapses returned to their control state.

Figure 2.

(A) Recordings from a representative simulation in which (top) homosynaptic long-term depression (LTP) and (bottom) homosynaptic long-term potentiation (LTD) were induced by a θ-burst stimulation (TBS) of synapses placed on the apical trunk within 110 μm from the soma. Each point represents the percent change of the somatic (solid squares) and local (open circles) peak EPSP amplitude evoked by test pulses delivered to one random synapse before (t < 750 ms) and after (6000 < t < 7000 ms) a LTP or LTD conditioning period. In both cases, a depotentiation protocol was applied at t = 7000 ms for 40 s. (B) Heterosynaptic LTP, occurring in synapses that were not directly stimulated, as a function of the relative distance from a stimulated synapse. To see this figure in color, go online.

We found that a depotentiation protocol was also able to erase LTD, switching the synapses to their control state (Fig. 2 A, bottom plot). Finally, we tested our model for heterosynaptic LTP, which occurs in unconditioned synapses spatially close to synapses conditioned with a TBS LTP protocol. Experimentally, it has been observed in CA1 neuron excitatory synapses up to ≈70 μm from the stimulation location (43). We selected couples of synapses (n = 3 ± 7) located on oblique dendrites at different relative distance, and conditioned one of them with an LTP protocol. As in the experiments, heterosynaptic LTP was observed up to ≈70 μm from the site of stimulation (Fig. 2 B). It should be stressed that, to the best of our knowledge, our model is the only one able to reproduce such a wide set of experimental protocols and findings.

These results show that our model is able to take into account the most relevant experimental findings on LTP and LTD, validating its use to gain more information on the relation between what is observed at the soma and what is really occurring at the synaptic locations.

The effect of induction protocols and role of the initial synaptic state

To test the efficacy of TBS protocols in inducing LTP and LTD in synapses with a relatively uniform distribution of peak conductance, we run a set of simulations stimulating random groups of 15 synapses uniformly distributed over the dendritic tree. Their weights (peak conductances) were drawn from a normal distribution (with average values in the range 0.1–1.3 nS and a variance of 0.01 nS²), and assumed to be in their ground control state (i.e., neither potentiated nor depressed, N_P = N_D = 0). For each average value, we ran 10 simulations, randomly redistributing location and peak conductance of all synapses. The average (mean ± SE) EPSP change measured at the soma is shown in Fig. 3 (solid symbols). It clearly show the robustness of the LTP protocol (solid red squares) for synaptic weights larger than ≈0.25 nS, whereas the TBS LTD protocol (solid blue circles) was effective only for a relatively small range of peak conductance (∼0.6–1.1 nS). Note that both protocols lead to LTP for a peak synaptic conductance larger than 1.2 nS. The qualitative results did not change using a more localized (e.g., proximal/distal) dendritic distribution of the synapses (not shown).

Figure 3.

Peak EPSP amplitude, measured at the soma during a test pulse after TBS LTP (red squares) or LTD (blue circles) protocol on 15 synapses uniformly distributed over the entire neuron, as a function of the peak synaptic conductance. Synapses started from their ground state (solid symbols) or in a potentiated state (open symbols). To see this figure in color, go online.

We also hypothesize that the actual initial synaptic state can affect the amount of observed LTP or LTD. To test this hypothesis, we repeated the simulations assuming that the initial synaptic weights were the result of a previously induced LTP on all synapses. The results (Fig. 3, open symbols) show how big the discrepancy can be between the expected and actual result measured at the soma. Whereas a LTD protocol would be observed as LTD for a much wider range of synaptic strength (Fig. 3, open blue symbols), there can combinations that would result in an outcome that appears to be the opposite of what is expected—for example, an LTP protocol would be observed as LTD or no effects (Fig. 3, open red symbols). These findings suggest that the induction of LTD can be difficult because it strongly depends on the actual peak conductance and initial state of the synapse at the time of the experiment: weak synapses can be unaffected by the stimulation protocol, whereas those with higher strength can be potentiated. Although some information is available on the strength and distribution of synaptic currents in CA1 neurons (e.g., Ito and Schuman (44)), the actual state of individual synapses before any given experiment is carried out cannot be identified. Induction protocols may thus be applied to synapses that are already in the appropriate state, which will not be affected by the conditioning pattern. This can generate misleading results.

An overall picture of what can be observed at the soma, after a conditioning period, can be obtained by considering groups of synapses under control conditions (i.e., in their ground state) uniformly distributed over the entire neuron but with a wider distribution of peak conductance, as it would occur in a real experiment. For this purpose, we ran a series of simulations using groups of 15 randomly distributed synapses conditioned with a TBS LTD (left part of Fig. 4 A) or a TBS LTP protocol (right part of Fig. 4 A). In each case, the synapses were initialized to their ground state, with weights randomly drawn from a uniform distribution within increasing ranges, in 0.4 nS bins.

Figure 4.

(A) Probability distribution function to obtain LTP (right) or LTD (left) for different synaptic weight ranges. (B) Two typical cases illustrating the state of all synapses for two synaptic weight ranges after a TBS LTD conditioning period. (C) Schematic representation of the overall result that can be obtained by TBS LTP or LTD protocols for different values of the initial synaptic peak conductance. (Colored lines) Different ranges tested in the simulations. To see this figure in color, go online.

The probability distribution of somatic EPSP change was calculated from 10 trials carried out for each range and stimulation protocol (randomly redistributing both the synaptic location and the peak conductance). The normalized distributions of the final peak conductance change (relative to control) are reported in Fig. 4 A. Each distribution (colored lines in Fig. 4 A) indicates the probability to observe LTD (left side) or LTP (right side) for the different range of peak conductance that was considered. The model confirmed that LTP is an extremely robust process, resulting in 160 ± 170% increase in synaptic strength for the entire range tested (0.5–1.3 nS). In contrast, the overall effect elicited by a TBS LTD protocol will drastically differ according to the initial synaptic weight. The maximum effectiveness of a TBS LTD protocol would be obtained only for initial synaptic weights within a rather narrow range (0.7–1.1 nS, in our case).

This effect depends on the final state of the individual synapses that, according to their initial weight, may end up being in different states, as schematically represented in Fig. 4 B, where we show two typical results for the LTD protocol and different synaptic weight range. The overall picture, for synapses starting from a ground state, is depicted in Fig. 4 C: for a LTD protocol, initial synaptic weights that are too low or too high would result in a significant proportion of synapses being unaffected or potentiated, respectively. Taken together, these results suggest that, when a group of synapses is subjected to a LTP protocol, the overall effect observed at the soma could effectively reflect the proportion of potentiated synapses. Instead, LTD protocols will result in an overall effect that cannot be directly interpreted in the same way. The reason for this apparently inconsistent result can be understood by considering the dynamics of the variable C in the equations for the synaptic transmission; it is one of the main determinants for the activation of the autocatalytic processes and could correspond to the intracellular calcium concentration, which is experimentally known to have this role in synaptic plasticity (reviewed in Lisman and McIntyre (45)).

In Fig. 5 A we summarize the possible results that can be obtained by conditioning a single synapse in its ground state with a TBS LTD protocol: no change, LTD, or LTP if the initial peak conductance is low or intermediate, high, or very high, respectively. The local dendritic membrane potential for each case is shown in Fig. 5 B, whereas the corresponding time course of the variables C, N_p, and N_d is shown in Fig. 5 C. For a low or intermediate initial peak conductance (Fig. 5, B and C, green traces), the current generated by the synaptic input during the conditioning stimulation is not able to raise the level of C enough to activate N_p and/or N_d. At the end of the conditioning period they all return to the ground state, and this is observed as a “no-change” in the amplitude of the test pulse. For a higher initial peak conductance (Fig. 5, blue traces), C, N_p, and N_d reach a higher level. However, although this is sufficient to switch N_d to its high state, it is not enough for N_p (compare blue traces in Fig. 5 C, middle and bottom plots). The end result will be observed as a depression of the test pulse. For even higher (but still subthreshold) initial synaptic peak conductance (Fig. 5, red traces), C raises to a level sufficient to switch both N_p, and N_d to their high state; this will be observed as a potentiation. These results explain why the interpretation of experimental findings, on the amount of LTD elicited by standard protocols, can be misled by the initial conditions of the stimulated synapses.

Figure 5.

(A) Schematic representation of the results obtained by conditioning a single synapse in its ground state with a TBS LTD protocol as a function of the peak conductance. (B) Corresponding local dendritic potential for a low or intermediate (green line), high (blue line), and very high (red line) peak conductance. (C) Time course of the variables C, N_p, and N_d for each case. To see this figure in color, go online.

Finally, to test these results with another of the widely used induction protocols, we explored LTP and LTD induced by a tetanic stimulation at a constant HFS or LFS (see Fig. 1 C). For this purpose, we carried out a series of simulations on a synapse at 150 μm from the soma, using a wide range of initial peak conductance (0.1–1.3 nS) and a range of stimulation frequency at ∼4 Hz for LTD, the most common constant frequency protocol experimentally used to induce LTD (Goh and Manahan-Vaughan (9) and references therein), and >50–100 Hz for LTP. The results for LTP were rather robust and consistent with those found for the TBS protocol (Fig. 3): LTP was consistently induced over practically the entire range of frequency tested (not shown).

Also consistently with what we found for a TBS LTD protocol, our model predicts that a synapse starting from its ground state (Fig. 6 A) would exhibit LTD after a LFS conditioning only for a rather limited region of the parameter space (blue area in Fig. 6 A). The initial synaptic state also was particularly important for a correct interpretation of the results after a conditioning period: the model suggests that synapses already in their potentiated state will exhibit LTD for selected ranges of peak conductance and conditioning frequency (Fig. 6 B), whereas initially depressed synapses would most likely appear as strongly potentiated after an LFS period (Fig. 6 C). Taken together, these results suggest that LTD induction using LFS protocols may be experimentally difficult to achieve because synaptic parameters (in our case, initial state and peak conductance) need to start within a small subset of the physiological range.

Figure 6.

Results for LFS conditioning of a synapse located 150 μm from the soma; (A) synapse initially in the ground state; (B) synapse initially in the potentiated state; (C) synapse initially in the depressed state. To see this figure in color, go online.

Discussion

The main aim of this article was to investigate the relation between the actual state of synapses subjected to typical experimental protocols for LTP and LTD induction, and what is observed at the soma. Our model pointed out that the outcome of an experiment, testing the amount of synaptic LTP/LTD plasticity that can be induced, strongly depends (at least) on the initial synaptic state and peak conductance. We think that this is an important issue because, no matter how sophisticated these experimental techniques might be, the actual state and peak conductance of individual synapses before any given conditioning period cannot be easily measured. Following the results obtained in this work we suggest that, to obtain results as consistent as possible among different preparations and experimental conditions, it would be a good experimental practice to use a stimulation paradigm able to preset the synapses in a preconditioning state that would give, in principle, the best results.

Our model suggests that this preconditioning state should be LTP before a LTD protocol, and depotentiated before a LTP protocol. The rationale for these choices can be understood by considering the overall model findings, discussed below.

Before an LTD conditioning protocol, one would like to avoid synapses that are either depressed or in the ground state, because a number of them will change to a potentiated state, confusing the overall result. This could be obtained by preconditioning the synapses with an LTP protocol. In this case, our model predicts that starting from a potentiated state an LTD protocol will result in LTD for the widest range of initial peak conductance. Even if at the end of the conditioning period individual synapses can be in the ground or depressed state, according to their initial strength, the overall result in most cases will be a depression, giving the experimenter a more precise and consistent idea of how much that particular synaptic pathway can be depressed.

For synapses undergoing a LTP protocol, in principle one can expect that the best preconditioning state would be a depressed state. The problem with this approach is that LTD protocols are not able to consistently switch synapses in their depressed state. Because of the role also played by the initial peak conductance, an attempt to depress a random group of synapses would result in a mix of synaptic states that would confuse the interpretation of the experimental findings. The model predicts that the best preconditioning option in this case would be a depotentiation, because of its ability to bring a synapse to its ground state, no matter the initial state (Fig. 2 A); a LTP protocol will thus result in LTP in most cases.

Finally, the model explains why LTD induction may be more critical to be obtained, with respect to LTP, at least in vivo. The reason is directly related to the dynamics of critical postsynaptic processes. A LTP protocol is rather efficient in generating enough signal to activate the autocatalytic processes responsible for LTP, and it does not have a ceiling: a stronger signal would simply saturate the overall amount of observed LTP. A LTD protocol instead is more critical, because it would generate LTD only when the combination of the involved pre- and postsynaptic quantities reach values that are just enough to activate the autocatalytic process for LTD (but not that for LTP, which has a higher threshold). Although in vitro these conditions can be more easily obtained by controlling many electrophysiological parameters (background activity, bathing solutions, stimulation locations, etc.), in vivo it may be much more difficult.