Power law or multiscale learning and adaptation

It turns out that processes of learning and adaptation in various neural and molecular systems show multiscale or power-law properties, where the memory is infinite and power-law weighted into past. This is in contrast to the usual exponential scaling that one gets from literally any physical process not at criticality. While the power-law functional behavior is well-characterized, biophysical mechanisms responsible for it are hardly known.

Relevant papers

1. P Drew and L Abbott. Models and Properties of Power-Law Adaptation in Neural Systems. J Neurophysiol 96: 826–833, 2006. PDF.

   Abstract

   Many biological systems exhibit complex temporal behavior that cannot be adequately characterized by a single time constant. This dynamics, observed from single channels up to the level of human psychophysics, is often better described by power-law rather than exponential dependences on time. We develop and study the properties of neural models with scale-invariant, power-law adaptation and contrast them with the more commonly studied exponential case. Responses of an adapting firing-rate model to constant, pulsed, and oscillating inputs in both the power-law and exponential cases are considered. We construct a spiking model with power-law adaptation based on a nested cascade of processes and show that it can be “programmed” to produce a wide range of time delays. Finally, within a network model, we use power-law adaptation to reproduce longterm features of the tilt aftereffect.

   Comments

   The paper discusses power law (in their case 1 / (t + t₀)) adapatation. A decent review of relevant papers, arguing that much adaptation is power law in single ion channels, in synapses, in single neuron activity, in retinal ciruitry and auditory processing, in local field potentials, and in behavioral responses, such as time estimation, tilt aftereffect, forgetting. The paper shows that power law adaptation may fit data better than exponential, and it suggests a modification of the reagular integrate and fire neuron to explain it. However, the modification is not solving the problem, as the power law multiscale dependence is put in by hand (although they do it in a minimal way, getting a few extra orders of magnitude of adaptation out). They study examples that the neuron would be able to explain, like a programmable timer, or tilt aftereffect. However, in my view, the importance of the paper is in elucidating the problem, and as a review (not as a solution -- they give none).