Figure 2.
Constructing a function of past time. The top row shows the history leading up to the present moment (τ = 0, right). There are no receptors that can directly detect events that happened in the past. To solve this problem, we assume that at each moment the present value of the stimulus (black dot) provides input to all of the cells in an intermediate representation F(s). In the figure, the input is zero at τ = 0 but was non-zero at two instants in the past (vertical lines). Those past values of f(τ) provided input to all of the cells in F(s). The middle row shows F(s) as a function of the rate constant s at τ = 0. The time constant of each cell in F(s) is just 1/s. The value of ∞ for s corresponds to a time constant of zero and is simply meant to align the axis of this schematic figure correctly. The cells in F(s) with short time constants (right) have decayed almost back to zero; the cells with longer time constants (left) have not decayed nearly as much. In principle, the pattern of activation across cells in F(s) contains complete information about the history of the inputs from prior moments. The bottom row shows the estimate of the reconstruction f̃(τ*). At each moment f̃ is constructed from F(s) via a matrix of feedforward connections Lk−1. These connections from F to f̃ can be understood as several projections with lateral inhibition in series, analogous to sensory processing (see text for details). Each cells' value of τ* is aligned with the time constant of the corresponding cell in F(s). We can see that at time τ = 0, f̃(τ*) estimates the past values of f(τ < 0); across cells f̃(τ*) contains a smeared estimate of f(τ).