Deriving the distance encoding component from a leaky-integrator model

The leaky integrator model we considered is given as follows:

where l′(t) is the encoded distance which is reduced proportionally to its current value (leaky) and incremented by the walking speed (integration). β is the rate of memory decay (if β>0; otherwise, it represents the rate of memory gain, but for convenience of description, we term the above equation as a leaky integrator). k is a speed gain, with k=1 representing that the speed is converted perfectly into the instantaneous walking distance. The equation represents that participants continuously update their internal estimation of traversed distance by using an estimation of their walking speed.

By assuming that a participant's walking speed is constant on a given outbound path (the speed may vary across trials), then, a general solution to Equation 9 is:

where v is a constant, and c is obtained by the limits of integration, for which l′(0)=0. Therefore:

which gives c=ln(−kvβ). Thus, the full solution to Equation 9 is given by:

Since v=l(t)t, Equation 12 can be re-written as:

Therefore, when participants finish walking the second outbound leg, the length of the second leg is encoded as:

with l2 the actual walking distance and T2 the total time spent on the second outbound leg. We further assume that the encoded distance of the first leg keeps updating when participants walk on the second leg, and thus, the distance of the first leg is encoded as:

with l1 the actual walking distance and T1 the total time spent on the first outbound leg. e−βT2 represents the exponential decay of the memorized walking distance on the first leg after participants spend T2 on the second leg. Equations 14 and 15 show the encoded distance used in our generative model (see Equation 2).