Kernel function .

<p>(A) Anticonvolution of a fictive correlogram (red curve) and a typical kernel function (blue curve). The amount of LTP/LTD corresponds to the area under the curve of the product of the two functions. (B) Plot of a typical kernel as a function of (blue curve). It corresponds to (4) for log-STDP with the baseline parameters in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002584#pcbi-1002584-t001" target="_blank">Table 1</a>, namely rise and decay time constants and in (29), respectively, and a purely axonal delay . The related STDP learning window is plotted in black dashed line and the mirrored PSP response in pink solid line. The effect of the axonal delay shifts both the and the PSP in the same direction, which cancels out. (C) Variants of for longer PSP time constants, and (purple curve); and for a dendritic delay (green dashed-dotted curve). In contrast to that does not play a role in (4), shifts to the right. The arrows indicate .</p

From PCA to ICA.

<p>The plots show the mutual information between each correlation source and the neuronal output firing after learning as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002584#pcbi-1002584-g006" target="_blank">Fig. 6D–F</a>. The neuron is stimulated by pools that mix three correlation sources as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002584#pcbi-1002584-g005" target="_blank">Fig. 5</a>. The two columns compare log-STDP with different degrees of weight dependence: (1) and (2) that induces stronger competition via weaker LTD. Each row corresponds to a different combination of single-spike contributions: (A) plain log-STDP meaning and log-STDP+SCC with (B) and ; (C) and ; (D) and ; (E) and . The scale on the y-axis is identical for all plots. The ratio of between and is indicated by .</p

Transmission of the correlated activity after learning by STDP.

<p>The results are averaged over 10 neurons and 100 s with the same configuration as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002584#pcbi-1002584-g005" target="_blank">Fig. 5</a>. Comparison of the PSTHs of the response to each correlated event of (A) before and (B) after learning for (red), (green) and (blue). Note the change of scale for the y-axis; the curves in A are reproduced in B in dashed line. (C) Ratio of the learning-related increase of mean firing rate (black) and PSTHs in B with respect to A (same colors). For each PSTH, only the area above its baseline is taken into account. (D) Mutual information between a correlated event and the firing of two spikes, as defined in (14). For each reference, the left (right) bar indicates before (after) learning. The crosses correspond to the theoretical prediction using (16) as explained in the text. (E) Example of neuron selective to with weight means for each pool set by hand to ; and . The bars correspond to the simulated similar to D. (F) Same as E with a neuron selective to and ; and .</p

Spectral Analysis of Input Spike Trains by Spike-Timing-Dependent Plasticity

<div><p>Spike-timing-dependent plasticity (STDP) has been observed in many brain areas such as sensory cortices, where it is hypothesized to structure synaptic connections between neurons. Previous studies have demonstrated how STDP can capture spiking information at short timescales using specific input configurations, such as coincident spiking, spike patterns and oscillatory spike trains. However, the corresponding computation in the case of arbitrary input signals is still unclear. This paper provides an overarching picture of the algorithm inherent to STDP, tying together many previous results for commonly used models of pairwise STDP. For a single neuron with plastic excitatory synapses, we show how STDP performs a spectral analysis on the temporal cross-correlograms between its afferent spike trains. The postsynaptic responses and STDP learning window determine kernel functions that specify how the neuron “sees” the input correlations. We thus denote this unsupervised learning scheme as ‘kernel spectral component analysis’ (kSCA). In particular, the whole input correlation structure must be considered since all plastic synapses compete with each other. We find that kSCA is enhanced when weight-dependent STDP induces gradual synaptic competition. For a spiking neuron with a “linear” response and pairwise STDP <em>alone</em>, we find that kSCA resembles principal component analysis (PCA). However, plain STDP does not isolate correlation sources in general, e.g., when they are mixed among the input spike trains. In other words, it does not perform independent component analysis (ICA). Tuning the neuron to a single correlation source can be achieved when STDP is paired with a homeostatic mechanism that reinforces the competition between synaptic inputs. Our results suggest that neuronal networks equipped with STDP can process signals encoded in the transient spiking activity at the timescales of tens of milliseconds for usual STDP.</p> </div

Principal component analysis for mixed correlation sources.

<p>(A) The postsynaptic neuron is excited by pools of 50 inputs each with the global input correlation matrix in (13). The thickness of the colored arrows represent the correlation strengths from each reference to each input pool. The input synapses are modified by log-STDP with . The simulation parameters are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002584#pcbi-1002584-t001" target="_blank">Table 1</a>. (B) Spectrum and (C) eigenvectors of . The eigenvalues sorted from the largest to the smallest one correspond to the solid, dashed, dashed-dotted and dotted curves, respectively. (D) Evolution of the weights (gray traces) and the means over each pool (thick black curves) over 500 s. (E) Evolution of the weights in the basis of spectral components (eigenvectors in C). (F) Weight structure at the end of the learning epoch. Each weight is averaged over the last 100 s. The purple curve represents the dominant spectral component (solid line in C).</p

Influence of the STDP parameters.

<p>(A) The neuron is stimulated by four pools. From left to right, pool is stimulated by the correlation source with correlation strength in (11). Pools and are related to the correlation source with ; pool tends to fire after pool . (B) Input cross-correlograms for first three pools described in A. The simulation time is 1000 s and the spike counts have been rescaled by the time bin equal to 1 ms. The peak is predicted by (11). Note the shift of the peak for the cross correlograms between inputs from pools and . (C) Spectrum of the correlation matrix corresponding to B. (D–L) Comparison of the final weight distribution for different STDP models. The two strongest spectral components of the correlation structure in red and green thick lines; they are rescaled between the minimum and maximum weights obtained in the simulation. For STDP+SCC in F, H and K, the single-spike contributions are and . (D) log-STDP with , and ; (E) log-STDP as in D with ; (F) log-STDP+SCC with the same parameters as D; (G) nlta-STDP with , and ; (H) nlta-STDP+SCC with the same parameters as G; (I) mlt-STDP with , (J) add-STDP with , ; (K) add-STDP+SCC with the same parameters as J; (L) add-STDP+SCC2 with and .</p

Existence of a fixed point for the weight dynamics.

<p>(A) Curves of the zeros of (39) for weights in the case of positively correlated inputs. The two curves have an intersection point, as the equilibrium curve for in red spans all , while that for in blue spans all . The arrows indicate the signs of the derivatives and in each quadrant (red and blue, resp.). (B) Similar to A with negative input correlations, for which the curves do not intersect.</p

Variables and parameters that describe the neuronal learning system.

<p>The variable denotes the time, whereas indicates the spike-time difference (or time lag) used in correlations and covariances.</p

Single neuron with STDP-plastic excitatory synapses.

<p>(A) Schematic representation of the neuron (top gray-filled circle) and the synapses (pairs of black-filled semicircles) that are stimulated by the input spike trains (bottom arrows). (B) Detail of synapse , whose weight is , postsynaptic response kernel , axonal and dendritic delays and , respectively. The arrows indicate that describes the propagation along the axon to the synapse, while relates to both conduction of postsynaptic potential (PSP) toward soma and back-propagation of action potential toward the synaptic site. (C) Example of temporally Hebbian weight-dependent learning window that determines the STDP contribution of pairs of pre- and postsynaptic spikes. The curve corresponds to (22). Darker blue indicates a stronger value for , which leads to less potentiation and more depression. (D) Schematic evolution of the weight for given pre- and postsynaptic spike trains and . The size of each jump is indicated by the nearby expression. Comparison between plain STDP for which only pairs contribute and STDP+SCC where single spikes also modify the weight via the terms . Here only the pair of latest spikes falls into the temporal range of STDP and thus significantly contributes to STDP. (E) Scaling functions of that determine the weight dependence for LTP and LTD. In the left panel, the blue solid curve corresponds to log-STDP <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002584#pcbi.1002584-Gilson3" target="_blank">[16]</a> with , and in (23). The parameter controls the saturation of the LTD curve: the dashed curve corresponds to and the dashed-dotted curve to . In the right panel, the red solid curves represent for nlta-STDP <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002584#pcbi.1002584-Gtig1" target="_blank">[13]</a> with and in (24); the black dashed-dotted horizontal lines indicate the add-STDP that is weight independent; the green dashed line corresponds to a linearly dependent LTD for mlt-STDP <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002584#pcbi.1002584-vanRossum2" target="_blank">[38]</a>.</p

Neuronal and learning parameters.

<p>Unless specified, the above parameters are used in numerical simulation.</p