  To guide behavior, the brain extracts relevant features from high-dimensional
data streamed by sensory organs. Neuroscience experiments demonstrate that the
processing of sensory inputs by cortical neurons is modulated by instructive
signals which provide context and task-relevant information. Here, adopting a
normative approach, we model these instructive signals as supervisory inputs
guiding the projection of the feedforward data. Mathematically, we start with a
family of Reduced-Rank Regression (RRR) objective functions which include
Reduced Rank (minimum) Mean Square Error (RRMSE) and Canonical Correlation
Analysis (CCA), and derive novel offline and online optimization algorithms,
which we call Bio-RRR. The online algorithms can be implemented by neural
networks whose synaptic learning rules resemble calcium plateau potential
dependent plasticity observed in the cortex. We detail how, in our model, the
calcium plateau potential can be interpreted as a backpropagating error signal.
We demonstrate that, despite relying exclusively on biologically plausible
local learning rules, our algorithms perform competitively with existing
implementations of RRMSE and CCA.
