Figure 4.
Neurobiologically useful properties of the modulo code. *A*, Similarly sized registers. Six registers can represent numbers in the range 0 to (10^{6} − 1) in the decimal fixed-base positional numeral system. However, the leftmost register represents quantities that are one million times larger than the rightmost register, and increments one million times more slowly as the represented quantity is varied. Registers in the modulo code may also span a large range (middle row) but, importantly, may be chosen to be similar in size (bottom row). *B*, Similar update rates. With similarly sized moduli (shown in gray), all registers are equally important for representing position at all scales (compare the representations of 45 and 800,000), and all registers increment at similar rates as position is varied (compare the representations of 800,000 and 800,001). *C*, Parallel, carry-free position updating. Left, Summation of 97 with entails a carry operation when the 1's register wraps around in the decimal fixed-base numeral system. Right, In the modular phase representation with moduli (7, 6, 5), the same numbers [97 ≡ (6, 1, 2) and 4 ≡ (4, 4, 4)] sum to 101 ≡ (3, 5, 1). The register corresponding to the modulus 5 wraps around because 2 + 4 mod 5 = 6 mod 5 = 1, as does the register corresponding to 7, because 6 + 4 mod 7 = 10 mod 7 = 3. However, no information is carried to other registers to produce the correct result. (The examples above use integer moduli, but the principle holds for reals.)