Resolution of Nested Neuronal Representations Can Be Exponential in the Number of Neurons

Collective computation is typically polynomial in the number of computational elements, such as transistors or neurons, whether one considers the storage capacity of a memory device or the number of floating-point operations per second of a CPU. However, we show here that the capacity of a computational network to resolve real-valued signals of arbitrary dimensions can be exponential in $N$, even if the individual elements are noisy and unreliable. Nested, modular codes that achieve such high resolutions mirror the properties of grid cells in vertebrates, which underlie spatial navigation.

Figure 1

Example of nested modules. (a) All modules, except for the coarsest one, have periodic tuning curves ${\Omega }_i(x-c_i)$. A module consists of a set of tuning curves with the same period but different phases $c_i$. The spatial period for modules 2 and 3 are ${\lambda }_2=0.45$ and ${\lambda }_3=0.3$, respectively. In each module, we highlight a single tuning curve by a solid line to show the period. Shifted but otherwise identical tuning curves are dashed. Nested modules successively refine the representation of the stimulus. Periodicity implies that the map from stimulus to population response is not one-to-one within a single module. Only the ensemble response provides a unique representation of $x$. (b) A unimodal tuning curve $\Omega$ in two dimensions, shown at the top, can be rescaled and periodically extended using Eqs. (5, 6). The periodic tuning curves ${\Omega }_{\Gamma }$ in the lower panel is based on a rectangular lattice $\Gamma$ spanned by $v_1=\lambda (1,0{)}^{\prime }$ and $v_2=\lambda (0,1{)}^{\prime }$, with $\lambda =1/2$. For this lattice, a fundamental domain $U$ is depicted.

Figure 2

(a) Grid codes (GC) outperform place codes (PC), regardless of the number of stimulus dimensions. A population of $N=3\times {10}^5$ neurons is divided into one, two, or three modules, according to Eq. (8). The neurons’ tuning width is fixed as $\sigma =2$. (b) The Fisher information for a neuronal population with $M={10}^5$ neurons per module and $D=3$. For a nested modular code with $L$ modules, the Fisher information grows exponentially in $N=L·M$, whereas it is linear in $N$ for a place code. (c) The error of the estimator that minimizes $(x-\widehat{x}{)}^2$ for place and grid codes in $D=3$ dimensions, based on sampling the stochastic response 1200 times. Each module comprises $M=8^3$ equidistantly spaced cells. The lattice lengths for different modules are scaled according to Eq. (8), with a safety factor $C=20$. The Cramér-Rao bound of Eq. (10) is tight for both the grid and place codes. (d) When the lattice lengths contract more strongly than allowed by Eq. (8), such that $C=1$, the error fails to improve in a nested modular code. Although the Fisher information predicts an error even lower than in (c), the uncertainty derived from the first module’s response is larger than the lattice length scale of the next finer-grained module, so that adding modules with finer spacing does not improve the resolution.