Figure 1.
The network model and its probabilistic interpretation.
A Circuit architecture. External input variables are encoded by populations of spiking neurons, which feed into a Winner-take-all (WTA) circuit. Neurons within the WTA circuit compete via lateral inhibition and have their input weights updated through STDP. Spikes from the WTA circuit constitute the output of the system. B Generative probabilistic model for a multinomial mixture: A vector of external input variables is dependent on a hidden cause, which is represented by the discrete random variable
. In this model it is assumed that the
's are conditionally independent of each other, given
. The inference task is to infer the value of
, given the observations for
. Our neuronal network model encodes the conditional probabilities of the graphical model into the weight vector
, such that the activity of the network can be understood as execution of this inference task.
Figure 2.
Under the simple STDP model (red curve), potentiation occurs only if the postsynaptic spike falls within a time window of length (typically
ms) after the presynaptic spike. The convergence properties of this simpler version in conjunction with rectangular non-additive EPSPs are easier to analyze. In our simulations we use the more complex version (blue dashed curve) in combination with EPSPs that are modeled as biologically realistic
-kernels (with plausible time-constants for rise and decay of
respectively
ms).
Figure 3.
Example for the emergence of Bayesian computation through STDP and adaptation of neural excitability.
A, B: Visualization of hidden structure in the spike inputs shown in D, E: Each row in panels A and B shows two results of drawing pixels from the same Gauss distribution over a 28×28 pixel array. Four different Gauss distributions were used in the four rows, and the location of their center represents the latent variable behind the structure of the input spike train. C: Transformation of the four 2D images in B into four linear arrays, resulting from random projections from 2D locations to 1D indices. Black lines indicate active pixels, and pixels that were active in less than 4
of all images were removed before the transformation (these pixels are white in panel H). By the random projection, both the 2D structure of the underlying pixel array and the value of the latent variable are hidden when the binary 1D vector is encoded through population coding into the spike trains
that the neural circuit receives. D: Top row: Spike trains from 832 input neurons that result from the four linear patterns shown in panel C (color of spikes indicates which of the four hidden processes had generated the underlying 2D pattern, after 50 ms another 2D pattern is encoded). The middle and bottom row show the spike output of the four output neurons at the beginning and after 500 s of unsupervised learning with continuous spike inputs (every 50 ms another 2D pattern was randomly drawn from one of the 4 different Gauss distributions, with different prior probabilities of 0.1, 0.2, 0.3, and 0.4.). Color of spikes indicates the emergent specialization of the four output neurons on the four hidden processes for input generation. Black spikes indicate incorrect guesses of hidden cause. E: Same as D, but with a superimposed 20 Hz oscillation on the firing rates of input neurons and membrane potentials of the output neurons. Fewer error spikes occur in the output, and output spikes are more precisely timed. F: Internal models (weight vectors
) of output neurons
after learning (pixel array). G: Autonomous learning of priors
, that takes place simultaneously with the learning of internal models. H: Average “winner” among the four output neurons for a test example (generated with equal probability by any of the 4 Gaussians) when a particular pixel was drawn in this test example, indicating the impact of the learned priors on the output response. I: Emergent discrimination capability of the output neurons during learning (red curve). The dashed blue curve shows that a background oscillation as in E speeds up discrimination learning. Curves in G and I represent averages over 20 repetitions of the learning experiment.
Figure 4.
Relationship between the continuous-time SEM model and experimental data on synaptic plasticity.
A–C: The effect of the continuous-time plasticity rule in Eq. (18) at a single synapse for different stimulation frequencies and different time-differences between pre- and post-synaptic spike pairs. Only time-intervals without overlapping pairs are shown. A: For very low stimulation frequencies (1 Hz) the standard shape of the complex learning rule from Fig. 2 is recovered. B: At a stimulation frequency of 20 Hz the plasticity curve shifts more towards LTP, and depression is no longer time independent, due to overlapping EPSPs. C: At high stimulation frequencies of 40 Hz or above, the STDP curve shifts towards only LTP, and thus becomes similar to a rate-based Hebbian learning rule. D: Cumulative effect of pre- and post-synaptic burst stimulation (50 Hz bursts of 5 pre-synaptic and 4 post-synaptic spikes) with different onset delays of -120, -60, 10, 20, 30, 80 and 140 ms (time difference between the onsets of the post- and pre-synaptic bursts). As in [55], the amount of overlap between bursts determines the magnitude of LTP, rather than the exact temporal order of spikes.
Figure 5.
Emergence of orientation selective cells for visual input consisting of oriented bars with random orientations.
A Examples of -pixel input images with oriented bars and additional background noise. B Internal models (weight vectors of output neurons
) that are learned through STDP after the presentation of
input images (each encoded by spike trains for 50 ms, as in Fig. 3). C, D Plot of the most active neuron for
images of bars with orientations from
to
in
steps. Colors correspond to the colors of
neurons in B. Before training (C), the
output neurons fire without any apparent pattern. After training (D) they specialize on different orientations and cover the range of possible angles approximately uniformly. E: Spike train encoding of the 10 samples in A. F,G: Spike trains produced by the
output neurons in response to these samples before and after learning with STDP for 200 s. Colors of the spikes indicate the identity of the output neuron, according to the color code in B.
Figure 6.
Emergent discrimination of handwritten digits through STDP.
A: Examples of digits from the MNIST dataset. The third and fourth row contain test examples that had not been shown during learning via STDP. B: Spike train encoding of the first 5 samples in the third row of A. Colors illustrate the different classes of digits. C, D: Spike trains produced by the output neurons before and after learning with STDP for 500 s. Colored spikes indicate that the class of the input and the class for which the neuron is mostly selective (based on human classification of its generative model shown in F) agree, otherwise spikes are black. E: Temporal evolution of the self-organization process of the 100 output neurons (for the complex version of STDP-curve shown in Fig. 1B), measured by the conditional entropy of digit labels under the learned models at different time points. F: Internal models generated by STDP for the 100 output neurons after 500 s. The network had not received any information about the number of different digits that exist and the colors for different ways of writing the first 5 digits were assigned by the human supervisor. On the basis of this assignment the test samples in row 3 of panel A had been recognized correctly.
Figure 7.
Output neurons self-organize via STDP to detect and represent spatio-temporal spike patterns.
A: Sample of the Poisson input spike trains at 20 Hz (only 100 of the 500 input channels are shown). Dashed vertical lines mark time segments of 50 ms length where spatio-temporal spike patterns are embedded into noise. B: Same spike input as in A, but spikes belonging to five repeating spatio-temporal patterns (frozen Poisson spike patterns at 15 Hz) are marked in five different colors. These spike patterns are superimposed by noise (Poisson spike trains at 5 Hz), and interrupted by segments of pure noise of the same statistics (Poisson spike trains at 20 Hz) for intervals of randomly varying time lengths. C, D: Firing probabilities and spike outputs of 6 output neurons (z-neurons in Fig. 1A) for the spike input shown in A, after applying STDP for 200 s to continuous spike trains of the same structure (without any supervision or reward). These 6 output neurons have self-organized so that 5 of them specialize on one of the 5 spatio-temporal patterns. One of the 6 output neurons (firing probability and spikes marked in black) only responds to the noise between these patterns. The spike trains in A represent test inputs, that had never been shown during learning.
Figure 8.
Ideal dependence of weight potentiation under STDP on the initial value of the weight (solid lines).
Open circles represent results of samples from this ideal curve with 100% noise, that can be used in the previously discussed computer experiments with almost no loss in performance. A: Dependence of weight potentiation on initial weight according to the STDP rule in Eq. (5). B: Same with an additional factor .
Figure 9.
STDP learning curves with time-dependent LTD.
Under the simple STDP model (red curve), weight-dependent LTP occurs only if the postsynaptic spike falls within a time window of length after the presynaptic spike, and LTD occurs in a time window of the same length, but for the opposite order of spikes. This can be extended to a more complex STDP rule (blue dashed curve), in which both LTP and LTD follow
-kernels with different time constants, typically with longer time-constants for LTD.