Efficient Neural Computation in the Laplace Domain
Neural Information Processing Systems
Cognitive computation ought to be fast, efficient and flexible, reusing the same neural mechanisms to operate on many different forms of information. In order to develop neural models for cognitive computation we need to develop neurallyplausible implementations of fundamental operations. If the operations can be applied across sensory modalities, this requires a common form of neural coding. Weber-Fechner scaling is a general representational motif that is exploited by the brain not only in
... ion and audition, but also for efficient representations of time, space and numerosity. That is, for these variables, the brain appears to support functions f (x) by placing receptors at locations x i such that x i − x i−1 ∝ x i . The existence of a common form of neural representation suggests the possibility of a common form of cognitive computation across information domains. Efficient Weber-Fechner representations of time, space and number can be constructed using the Laplace transform, which can be inverted using a neurally-plausible matrix operation. Access to the Laplace domain allows for a range of efficient computations that can be performed on Weber-Fechner scaled representations. For instance, translation of a function f (x) by an amount δ to give f (x + δ) can be readily accomplished in the Laplace domain. We have worked out a neurally-plausible mapping hypothesis between translation and theta oscillations. Other operations, such as convolution and cross-correlation are extremely efficient in the Laplace domain, enabling the computation of addition and subtraction of neural representations. Implementation of neural circuits for these elemental computations would allow hybrid neural-symbolic architectures that exhibit properties such as compositionality and productivity. This paper argues that 1. The brain represents functions of many quantities, including time, using a common form of coding that we refer to as Weber-Fechner scaling. 2. Some of these quantities can be efficiently computed using the Laplace domain and a neurally-plausible mechanism for approximating the inverse Laplace transform.