Figure 1.
Example of PCA performed by Oja's rate-based plasticity rule.
(A) Theoretical cross-correlation matrix of 100 inputs induced by two Gaussian distributions of rates. Darker pixels indicate higher correlations. The circles indicate where the correlation for each source drops below 10% of its maximum (see the Gaussian profiles in D). (B) Traces of the weights modified by Oja's rule [26] in (1). At random times, the input rates follow either Gaussian rate profile corresponding to the green and blue curves in D; white noise is added at each timestep. (C) The asymptotic distribution of the weights (red) is close to the principal component of the matrix in A (black solid curve), but distinct from the second component (black dashed curve). (D) The final weight distribution (red) actually overlaps both Gaussian rate profiles in green dashed and blue dashed-dotted lines that induce correlation in the inputs. The green and blue curves correspond to the small and large circles in A, respectively.
Figure 2.
Single neuron with STDP-plastic excitatory synapses.
(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 [16] 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 [13] 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 [38].
Table 1.
Neuronal and learning parameters.
Figure 3.
(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 Table 1, 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
.
Table 2.
Variables and parameters that describe the neuronal learning system.
Figure 4.
Existence of a fixed point for the weight dynamics.
(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.
Figure 5.
Principal component analysis for mixed correlation sources.
(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 Table 1. (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).
Figure 6.
Transmission of the correlated activity after learning by STDP.
The results are averaged over 10 neurons and 100 s with the same configuration as in Fig. 5. 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
.
Figure 7.
The plots show the mutual information between each correlation source and the neuronal output firing after learning as in Fig. 6D–F. The neuron is stimulated by
pools that mix three correlation sources as in Fig. 5. 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
.
Figure 8.
Influence of the STDP parameters.
(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
.
Figure 9.
Influence of the PSP response on the kernel and the resulting weight structure for log-STDP.
(A) The neuron is stimulated by input pools. The first two pools have the same reference for correlations with strength
and pool
tends to fire
after pool
. Pool
has no correlation. For all inputs, the firing rate is
. (B) Short PSP response with
and
, as well as purely axonal delays
. (C) Long PSP response with
and
, as well as purely axonal delays
. (D) Short PSP response with
and
, as well as purely dendritic delays
. For each configuration, we present (1) a schematic diagram of the synaptic parameters, (2) the eigenvalues of
and (3) the resulting weight specialization. As in Fig. 5, the purple curves represent the expression in (10). The dotted horizontal line indicates
, the equilibrium weight for log-STDP. Two eigenvalues are roughly equal to zero in D2.