Scale-Free Memory to Swiftly Generate Fuzzy Future Predictions

Abstract

A flexible access to how the current state of memory would evolve into the future is extremely valuable in predicting events to occur in distant future. We propose a neural mechanism to non-destructively translate the current state of temporal memory into the future, so as to construct a timeline of future predictions. In a two-layer memory network that encodes the Laplace transform of the past, time-translation can be accomplished by modulating the weights between the layers. Computationally, such a network appears to be extremely resource-conserving, and could prove useful in AI systems. We hypothesize that such a mechanism is neurally realistic in the sense that the brain performs it. We propose that within each cycle of hippocampal theta oscillations, the memory state is swept through a range of time-translations to yield a future timeline of predictions. A physical constraint requiring coherence in time-translation across memory nodes results in a Weber-Fechner spacing for the representation of both past (memory) and future (prediction) timelines.

Appendix

This appendix provides an explicit recipe for computing $\mathbf {L}^{{{-1}}}_{\text {k}}$ for the distribution of $s_n$ derived in the text and $k=2$. A more general derivation can be found in [14]. When $s_n = s_o (1+c)^{-n}$ the connection strengths in the $\mathbf {L}^{{{-1}}}_{\text {k}}$ operator takes a special form–for every n, the local connectivity from the $(2k+1)$ $\mathbf {t} _{ }$-nodes to the n-th $\mathbf {T} _{ }$-node has an identical form multiplied by $s_n$. For example, with $k=2$, the connection strengths to the n-th $\mathbf {T} _{ }$ node from the $\mathbf {t} _{ }$ nodes in the local neighborhood are given by

$$\begin{aligned} \mathbf {t} _{n+2 }\rightarrow & {} s_n \frac{(c+1)^5}{c^2(c+2)^2} \\ \mathbf {t} _{n+1 }\rightarrow & {} s_n \frac{-(c+1)^2}{c} \\ \mathbf {t} _{n }\rightarrow & {} s_n \frac{c^4+3c^3 +c^2 -4c-2}{c^2(c+2)^2} \\ \mathbf {t} _{n-1 }\rightarrow & {} s_n \frac{1}{c^2+c} \\ \mathbf {t} _{n-2 }\rightarrow & {} s_n \frac{1}{c^2(c+1)(c+2)^2} \\ \end{aligned}$$

The factor $s_n$ can be treated as a post-synaptic weight and the rest (which only depend on c) can be treated as pre-synaptic weight which are constant along the dorsoventral axis.