Directional interference patterns (see ), showing the positive part of a single directional interference pattern (A, rightward preferred direction), and the product of two (B), three (C) or (6) such patterns oriented at multiples of 60° to each other. i) Pattern generated by straight runs at 30cm/s from the bottom left hand corner to each point in a 78×78cm square. ii) Pattern generated by averaging the values generated at each locations during 10 minutes of a rat's actual trajectory while foraging for randomly scattered food in a 78cm cylinder (white spaces indicate unvisited locations). iii) As ii) but shown with 5cm boxcar smoothing for better comparison with experimental data. All oscillations are set to be in phase (*φ*_{i} = 0) at the initial position (i: bottom left corner; ii: start of actual trajectory – indicated by an arrow in ). The plots show *f*(*x*(*t*)) = Θ(Π_{i=1}^{n} cos(*w*_{i} t+*φ*_{i}) + cos(*w*_{s} t)), for *n*=1 (A), 2 (B), 3 (C) and 4 (D), with *w*_{i} = *w*_{s} + *βs*cos(*ø - ø*_{i}), where *s* is running speed, spatial scaling factor *β* = 0.05×2π rad/cm (i.e., 0.05 cycles/cm), preferred directions: *ø*_{1} = 0° (i.e. rightwards), *ø*_{2} = 60°, *ø*_{3} = 120°, *ø*_{4} = 180°, *ø*_{5} = 240°, *ø*_{6} = 310°. Θ is the Heaviside function. All plots are auto-scaled so that red is the maximum value and blue is zero.

## PubMed Commons