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Neural Coding as a Statistical Testing Problem

Guilherme Ost, Patricia Reynaud-Bouret

TL;DR

It is shown, through the notion of adaptation, that a fixed place cell system can have a minimum discrimination time that decreases when the stimuli are further away, which could be a considerable advantage for the placecell system that could complement the grid cell system, which is able to discriminate stimuli that are much closer than place cells.

Abstract

We take the testing perspective to understand what the minimal discrimination time between two stimuli is for different types of rate coding neurons. Our main goal is to describe the testing abilities of two different encoding systems: place cells and grid cells. In particular, we show, through the notion of adaptation, that a fixed place cell system can have a minimum discrimination time that decreases when the stimuli are further away. This could be a considerable advantage for the place cell system that could complement the grid cell system, which is able to discriminate stimuli that are much closer than place cells.

Neural Coding as a Statistical Testing Problem

TL;DR

It is shown, through the notion of adaptation, that a fixed place cell system can have a minimum discrimination time that decreases when the stimuli are further away, which could be a considerable advantage for the placecell system that could complement the grid cell system, which is able to discriminate stimuli that are much closer than place cells.

Abstract

We take the testing perspective to understand what the minimal discrimination time between two stimuli is for different types of rate coding neurons. Our main goal is to describe the testing abilities of two different encoding systems: place cells and grid cells. In particular, we show, through the notion of adaptation, that a fixed place cell system can have a minimum discrimination time that decreases when the stimuli are further away. This could be a considerable advantage for the place cell system that could complement the grid cell system, which is able to discriminate stimuli that are much closer than place cells.
Paper Structure (29 sections, 18 theorems, 145 equations, 3 figures)

This paper contains 29 sections, 18 theorems, 145 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Discrimination time
  3. Model and notation
  4. Place cells code $\mathcal{P}$.
  5. Grid cells code $\mathcal{G}((n_i,\lambda_i)_{i=1,...,m})$.
  6. Unspecified grid cells code $\bar{{\mathcal{G}}}$.
  7. General binary code
  8. The statistical testing problem
  9. Order of magnitude of $p_{e}$
  10. Minimal discrimination time
  11. Results for place cells code
  12. Lower bound on the minimax discrimination time
  13. Upper bounds and minimax codes
  14. A particular adaptive code
  15. Random codes are adaptive as well
  16. ...and 14 more sections

Key Result

proposition 1

For all $T>0$, $\mu>1$ and $s_1,s_2 \in \mathbb{S}^1$ with $C_{\mu}=\frac{(\mu-1)^2}{4}\min\left\{\frac{1}{2\mu}, \frac{3}{5+\mu}\right\}$ and $\tilde{C}_{\mu}=(\mu-1)\log(\mu).$

Figures (3)

  • Figure 1: Visual representation of the main notions. In A, $\mathbb{S}^u$, the circle of radius $u$ with a point $a$ and its corresponding argument $\theta_a\in [0,u)$ In B, the visual representation of the intervals. In red and green, the visual representation of the function $s \mapsto g(s)=\mu_{\llbracket a,b\llbracket }(s)+1_{\llbracket a,b\llbracket ^c}(s)$, with the value $\mu$ in red and $1$ in green. In C and D, visual representation of the action of $mod$. In these pictures, $\alpha=a~\mathrm{mod}~1/4=a_1~\mathrm{mod}~ 1/4=a_2~\mathrm{mod}~1/4 = a_3~\mathrm{mod}~1/4$. Also in D, the representation of the function $s\mapsto g_{1/4}(s)=\mu 1_{\llbracket \alpha,\beta\llbracket }(s)+1_{\llbracket \alpha,\beta\llbracket^c}(s)$ with the same color code as in B. In C, the representation of the periodic function $s \mapsto g(s)=g_{1/4}(s \ \text{mod} \ u) = \mu 1_{\llbracket\alpha,\beta\llbracket}(s~\mathrm{mod}~1/4)+1_{\llbracket \alpha,\beta\llbracket ^c}(s~\mathrm{mod}~1/4)$
  • Figure 2: Visual representation of Examples 3. For an easier visualisation, the different half red circles have a different radius. Nevertheless, each of them correspond a certain cell $i$ of the code, and more precisely to the locations $s$ such that $f_i(s)=\mu$. Two couples are considered $(s_1,s_2)$ (standing for two positions that are very close) and $(s_1,s'_2)$ (standing for positions that are very far). We see that $\Delta_{s_1,s'_2}=n-1>>\Delta_{s_1,s_2}=1$.
  • Figure 3: Discrimination time as a function of $\Delta^f_{s_1,s_2}$ and $\rho$. On the left, $T_{min}(f,s_1,s_2,\alpha)$ as a function of $1/\Delta^f_{s,s'}$ for the 5 different codes. On the right, $T_{min}(f,\rho,\alpha)$ as a function of $\rho$ for the 5 different codes.

Theorems & Definitions (37)

  • definition 1
  • definition 2
  • definition 3
  • proposition 1
  • proof
  • proposition 2
  • proof
  • proposition 3
  • proof
  • corollary 1
  • ...and 27 more