Coin-Flipping In The Brain: Statistical Learning with Neuronal Assemblies

TL;DR

This work studies the emergence of statistical learning in NEMO, a biologically plausible computational model of the brain based on stylized neurons and synapses, plasticity, and inhibition, and shows that connections between assemblies record statistics, and ambient noise can be harnessed to make probabilistic choices between assemblies.

Abstract

How intelligence arises from the brain is a central problem in science. A crucial aspect of intelligence is dealing with uncertainty -- developing good predictions about one's environment, and converting these predictions into decisions. The brain itself seems to be noisy at many levels, from chemical processes which drive development and neuronal activity to trial variability of responses to stimuli. One hypothesis is that the noise inherent to the brain's mechanisms is used to sample from a model of the world and generate predictions. To test this hypothesis, we study the emergence of statistical learning in NEMO, a biologically plausible computational model of the brain based on stylized neurons and synapses, plasticity, and inhibition, and giving rise to assemblies -- a group of neurons whose coordinated firing is tantamount to recalling a location, concept, memory, or other primitive item of cognition. We show in theory and simulation that connections between assemblies record statistics, and ambient noise can be harnessed to make probabilistic choices between assemblies. This allows NEMO to create internal models such as Markov chains entirely from the presentation of sequences of stimuli. Our results provide a foundation for biologically plausible probabilistic computation, and add theoretical support to the hypothesis that noise is a useful component of the brain's mechanism for cognition.

Figures (8)

• Figure 1: On the left we exhibit the plasticity increment, $\min\{\alpha , e^{\lambda (1 + \beta - w(t))} \}$, as a function of the weight; on the right is the total weight versus the number of rounds of training. Here, $\alpha = \beta + \log \lambda / \lambda$ as we assume in all of our proofs and experiments.
• Figure 2: Choosing one of two assemblies to fire, based on context. The probability distribution depends on the weights of each assembly from the context (Theorem \ref{['thm:coinflip']}).
• Figure 3: Variation in the probability that assembly $A$ wins as its weight from the context assembly is varied, while the weights to $B$ and $C$ are held constant (and equal). On the left the weight is directly adjusted and compared against a best-fitting softmax function (dashed line); on the right all weights are updated incrementally according to a plasticity rule which approximately recovers the observed frequency (dashed line). Dark center line is the mean, while shaded area is the $[0.05, 0.95]$ quantile range, across 100 trials.
• Figure 4: Variation in the probability of an assembly winning, as the support of the training distribution grows to include more assemblies, under a plasticity rule that approximately recovers the training distribution. For two different sequences of distributions, we plot the empirical probability of assembly $A_1$ firing (blue dotted line) along with the target probability from the training distribution (black dashed). In (a) and (b) $A_2, \ldots, A_m$ fire $5$ times during training, for each $m=2,\ldots,10$, while in (a) $A_1$ fires $5(m-1)$ times and in (b) $A_1$ fires $10$ times for each $m$.
• Figure 5: Variation in the error of the learned distribution over two assemblies as the cap size increases. For each of $20$ trials, a new connectivity graph was sampled, the assemblies $A$ and $B$ were made to fire exactly the same number of times, and the resulting distribution was estimated from $500$ samples. The absolute deviation of the frequency with which $A$ won from $1/2$ was recorded for each trial. The dark center line is the mean of this error over trials, the shaded area is the range, and the dashed line is the expected maximum if the samples were truly unbiased (i.e. the probability that $A$ comes to fire is exactly $1/2$).

Theorems & Definitions (22)

• Theorem 1: Conditional Coin-Flipping
• Corollary 2
• Lemma 3
• Corollary 4: Markov Chain
• Remark
• proof : Proof of Theorem \ref{['thm:coinflip']}
• proof : Proof Sketch of Corollary \ref{['cor:target']}
• proof : Proof Sketch of Lemma \ref{['lemma:plasticity']}
• proof : Proof Sketch of Corollary \ref{['cor:markovchain']}
• Lemma 5