  Many cognitive models, including those for predicting the time of future
events, can be mapped onto a particular form of neural representation in which
activity across a population of neurons is restricted to manifolds that specify
the Laplace transform of functions of continuous variables. These populations
coding Laplace transform are associated with another population that inverts
the transform, approximating the original function. This paper presents a
neural circuit that uses continuous attractor dynamics to represent the Laplace
transform of a delta function evolving in time. One population places an edge
at any location along a 1-D array of neurons; another population places a bump
at a location corresponding to the edge. Together these two populations can
estimate a Laplace transform of delta functions in time along with an
approximate inverse transform. Building the circuit so the edge moves at an
appropriate speed enables the network to represent events as a function of log
time. Choosing the connections appropriately within the edge network make the
network states map onto Laplace transform with exponential change as a function
of time. In this paper we model a learned temporal association in which one
stimulus predicts another at some fixed delay $T$. Shortly after $t=0$ the
first stimulus recedes into the past. The Laplace Neural Manifold representing
the past maintains the Laplace transform $\exp(-st)$. Another Laplace Neural
Manifold represents the predicted future. At $t=0$, the second stimulus is
represented a time $T$ in the future. At each moment between 0 and $T$, firing
over the Laplace transform predicting the future changes as $\exp[-s(T-t)]$.
Despite exponential growth in firing, the circuit is robust to noise, making it
a practical means to implement Laplace Neural Manifolds in populations of
neurons for a variety of cognitive models.
