Bayesian Computation Emerges in Generic Cortical Microcircuits through Spike-Timing-Dependent Plasticity

The principles by which networks of neurons compute, and how spike-timing dependent plasticity (STDP) of synaptic weights generates and maintains their computational function, are unknown. Preceding work has shown that soft winner-take-all (WTA) circuits, where pyramidal neurons inhibit each other via interneurons, are a common motif of cortical microcircuits. We show through theoretical analysis and computer simulations that Bayesian computation is induced in these network motifs through STDP in combination with activity-dependent changes in the excitability of neurons. The fundamental components of this emergent Bayesian computation are priors that result from adaptation of neuronal excitability and implicit generative models for hidden causes that are created in the synaptic weights through STDP. In fact, a surprising result is that STDP is able to approximate a powerful principle for fitting such implicit generative models to high-dimensional spike inputs: Expectation Maximization. Our results suggest that the experimentally observed spontaneous activity and trial-to-trial variability of cortical neurons are essential features of their information processing capability, since their functional role is to represent probability distributions rather than static neural codes. Furthermore it suggests networks of Bayesian computation modules as a new model for distributed information processing in the cortex.

with parameters a and b Let w = log q, than w is distributed as follows: In order to calculate E[w] and E[w 2 ] we use the moment-generating function of p(w) The first and the second derivative of M w read which can be simplified using the approximations ψ(x) ≈ log(x) and ψ 1 (x) ≈ 1 x to

Adaptation to changing input distributions
In this computer experiment, 10 output neurons learned implicit generative models for images of handwritten digits from the MNIST database. The same procedure for encoding the images by spike trains as in Fig. 6 was used. Initially, only images representing the digits 0 and 3 were presented, and the WTA circuit learned accurate probabilistic models for these images. After 100 seconds of learning, the input distribution was changed, and a third class of inputs, images of handwritten digits 4, was introduced. Through the adaptive learning rate from Eq. (71), the z k neurons spontaneously reorganized, and two output neurons changed their internal models to represent the new digit 4. In the end, an accurate generative model for all three types of input images was learned.

Simulation Parameters
All simulations were carried out in MATLAB, with a simulation time step of 1 ms. The time constant of the OU process that modeled background synaptic inputs was set to 5 ms, its variance to 2500. Fig. 3:

Input generation:
For each input image pixels were drawn over a 28 x 28 array from one of 4 symmetrical Gaussians with σ 2 = 10 and centers at (14,8), (16,22), (9,15), (20,14), with maximal probability 0.3 for any pixel to be drawn (causing high variability of samples from the same Gaussian). In addition any pixel was drawn with probability 0.03 (added noise).
When an output neuron z k fired, on average only 8.6% of the input neurons y i had fired during the preceding 10 ms (the time window for potentiation according to the STDP rule in Eq. (5). Hence for over 90% of the pixels no spike was received within that time window from either one of the two neurons y i that encoded the value of this pixel by population coding. The corresponding average activity level of all input synapses was at 0.182.  Fig. 7. They demonstrate that the emergent discrimination ability of these 6 output neurons automatically generalizes to time-warped input patterns (embedded into noise).
In the variation with superimposed background oscillations at 20 Hz the firing rates of input neurons y i did not rise, but the average synaptic activity level at the time of an output spike rose to 0.215, an increase of around 18%. This leads to an increased learning rate.
The mean (offset) µ ou of the OU-noise was set to 200, the initial value A inh of lateral inhibition (caused by a firing of a z-neuron) was set to 3000, its resting value O inh to 550. For the version with background oscillations (at 20 Hz) the amplitude of the oscillation was set to 500 (mean = 0), and the phase was shifted by 5 ms for the z-neurons, A inh = 3000, O inh = 650.

Simulation for
In Figs. 4 A-C pre-and post-synaptic neurons were forced to fire at frequencies of 1, 20, and 40 Hz with different time delays. The weight was kept fixed at w = 3.5 for c = e −5 , and the learning rate was kept fixed at η = 0.5. For Fig. 4D we simulated a pre-synaptic burst consisting of 5 spikes with 20 ms time difference, and a post-synaptic burst of 4 spikes, also with 20 ms time difference. The starting points of these bursts were shifted relative to each other. We kept the weight fixed at w = 3.5 for c = e −5 , and the learning rate fixed at η = 0.1, and added up the resulting weight changes for all 4 postsynaptic spikes.