The suppression model.

<p><b>A.</b> Schematic illustration of spike interactions in the suppression model, in which the effect of the presynaptic spike in a pair is suppressed by a previous presynaptic spike (top), and the effect of the postsynaptic spike is suppressed by a previous postsynaptic spike (bottom). <b>B.</b> Plasticity in the suppression model induced by triplets of spikes: pre-post-pre triplets induce potentiation (top left), and post-pre-post triplets induce depression (bottom right).</p

The triplet model.

<p><b>A.</b> Schematic illustration of spike interactions in the triplet model in which previous presynaptic spikes induce extra depression (top) and previous postsynaptic spikes induce extra potentiation (bottom). <b>B.</b> Plasticity due to triplets of spikes: pre-post-pre triplets induce depression or weak potentiation (top left), and post-pre-post ordering induces mostly potentiation (bottom right). This figure is based on parameters fit to hippocampal data (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004750#pcbi.1004750.t002" target="_blank">Table 2</a>).</p

Dynamics of reciprocal synapses with rightward shifted STDP.

<p><b>A.</b> When the baseline firing rates of the two neurons are 1.8 Hz, a saddle node exists out of the allowed range, schematically illustrated at the top right. Arrows show the movement of trajectories. Initial conditions starting within the red area end up at the attractor at the top right corner, which corresponds to strong recurrent connections. This increases the baseline firing rate of the embedding network and pushes the network into the regime shown in B. <b>B.</b> When the baseline firing rates of the two neurons are 37 Hz, a single stable fixed point exists within the allowed range of synaptic weights. All initial conditions end up at this fixed point, resulting in a recurrent reciprocal connection. <b>C.</b> When the baseline firing rates of the two neurons are 50 Hz, a stable fixed point exists out of the allowed range, schematically illustrated at the bottom left. Movement of trajectories toward the stable fixed point results in connectivity loss, regardless of the initial condition. This effect reduces the rate of the embedding network and pushes the system into the regime shown in B. It is not necessary to impose upper bounds in this case, so they are depicted as dotted lines.</p

Stability and competition in the suppression model.

<p><b>A.</b> Fixed points of ⟨<i>w</i>⟩ as functions of the ratio between the potentiation and depression time constants. The stable fixed point disappears beyond the critical value <i>τ</i>₊/<i>τ</i>₋ < 1.2. When the ratio approaches the critical value, the fixed point grows rapidly (gray area), leading to a stable distribution. <b>B.</b> The average drift when <i>τ</i>₊/<i>τ</i>₋ = 1. The solid curve shows the analytical result (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004750#pcbi.1004750.e018" target="_blank">Eq 6</a>) and the boundaries of gray shading is obtained by simulations. The filled circle is the stable fixed point. <b>C.</b> The average drift when <i>τ</i>₊/<i>τ</i>₋ = 1.1. The stable fixed point moves to larger values than in B. <b>D.</b> The average drift when <i>τ</i>₊/<i>τ</i>₋ = 1.5. No nontrivial fixed point exists. <b>E.</b> The partially stable bimodal steady-state distribution of weights corresponding to the parameters of B. <b>F.</b> The stable steady-state distribution of weights corresponding to the parameters of C. <b>G.</b> The unstable steady-state distribution of weights clustered around the upper bound corresponding to the parameters of D, when no stable fixed point exists. <b>H-J.</b> Competition between correlated and uncorrelated synapses with parameter corresponding to E-G. The competition is anti-Hebbian in all cases.</p

The pair-based STDP model.

<p><b>A.</b> Top-left: The STDP window with <i>A</i>₊ < <i>A</i>₋. Top-right: a triplet of spikes composed of two pre-post pairs with intervals Δ<i>t</i>₁ and Δ<i>t</i>₂. Bottom: the amount of synaptic modification in response to triplets, which is symmetric in the pair-based model. <b>B.</b> The average drift induced by the pair-based model on a population of excitatory synapses converging onto a single postsynaptic neuron, when <i>A</i>₊ < <i>A</i>₋. The black curve is a numerical evaluation of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004750#pcbi.1004750.e003" target="_blank">Eq 2</a> and the gray area is the simulation results. The half-width of the gray area is the standard error. The filled circle is the stable fixed point. The inset shows the <i>w</i>-dependent drift (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004750#pcbi.1004750.e005" target="_blank">Eq 3</a>) <b>C.</b> The steady state distribution of synaptic weights obtained by simulation when <i>A</i>₊ < <i>A</i>₋. <b>D.</b> The steady state distribution of weights when half of the synapses receive correlated input (magenta) and the other half receive uncorrelated input (cyan). When <i>A</i>₊ < <i>A</i>₋ correlated synapses are strengthened. <b>E-H.</b> The same as A-D, but for <i>A</i>₊ > <i>A</i>₋. Note that there is no stable fixed point in <b>F</b>, and that all the synapses are pushed to the upper bound in <b>G</b> and <b>H</b>. For these simulations, the constants of the STDP model were <i>τ</i>₊ = <i>τ</i>₋ = 20 <i>ms</i>, <i>A</i>₊ = 0.005 <i>mV</i> and <i>A</i>₋ = 1.0.1 <i>A</i>₊ in A-D and <i>A</i>₋ = 0.005 <i>mV</i> and <i>A</i>₊ = 1.0.1 <i>A</i>₋ in E-H.</p

Summary of stability/plasticity in STDP models.

<p>Summary of stability/plasticity in STDP models.</p

Original parameters of the multi-spike STDP models used to generate Figs 2B, 4B, 7B and 9.

<p>Original parameters of the multi-spike STDP models used to generate Figs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004750#pcbi.1004750.g002" target="_blank">2B</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004750#pcbi.1004750.g004" target="_blank">4B</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004750#pcbi.1004750.g007" target="_blank">7B</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004750#pcbi.1004750.g009" target="_blank">9</a>.</p

Neuronal, synaptic, and plasticity parameters.

<p>Neuronal, synaptic, and plasticity parameters.</p

The shifted triplet model.

<p><b>A</b>. The final distribution of weights for different values of maximum triplet potentiation () and depression (). Except for very high depression values, the distribution is unimodal and stable. We used the representative value of for both and (red dotted box) for the remaining results in this figure. <b>B</b>. The shift stabilizes the distribution of synaptic weights. The horizontal axis is the value of the shift, the vertical axis is the synaptic strength, and the gray level is the probability density of the strengths (as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000961#pcbi-1000961-g002" target="_blank">figure 2</a>), obtained by simulation. <b>C</b>. The steady-state firing rate of the postsynaptic neuron as a function of the excitatory and inhibitory input rates. <b>D</b>. The shift in the triplet model can implement both Hebbian and anti-Hebbian competition. As in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000961#pcbi-1000961-g004" target="_blank">figure 4</a>, the top panel shows the distribution of the uncorrelated synapses (cyan) and the bottom panel shows the distribution of the correlated ones (magenta), as a function of the inhibitory input rate. The transition from anti-Hebbian to Hebbian competition occurs at an inhibitory input rate of 16 Hz.</p

Dynamics of reciprocal synapses when STDP is potentiation dominated.

<p><b>A.</b> When the baseline firing rates of the two neurons are both 20 Hz, an unstable fixed point exists out of the allowed range, schematically illustrated at the bottom left. Arrows show that the trajectories drift away from this outlying fixed point. Initial conditions starting within the red area end up at the attractor at the top-right corner, which corresponds to recurrent connections. Trajectories that hit the boundaries perpendicularly delineate the borders of the basins of attraction (solid curves). Initial conditions in the yellow area end up at the attractor at the bottom right, corresponding to a unidirectional connection from neuron 1 to neuron 2. Initial conditions within the green area go to the attractor at top left, corresponding to a unidirectional connection from neuron 2 to neuron 1. <b>B.</b> The same as <b>A</b> when the baseline firing rates are 50 Hz. The basin of attraction for recurrent connections (red area) becomes larger when the baseline firing rate increases.</p