Figure 2.

Directional interference patterns (see equation 5), 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 Figure 5). The plots show f(x(t)) = Θ(Π_i=1ⁿ 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 + βscos(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: ø₁ = 0° (i.e. rightwards), ø₂ = 60°, ø₃ = 120°, ø₄ = 180°, ø₅ = 240°, ø₆ = 310°. Θ is the Heaviside function. All plots are auto-scaled so that red is the maximum value and blue is zero.