Maneesh Sahani
Gatsby Computational
Neuroscience Unit
University College, London
Techniques
from the field of systems identification have been very productive in
experimental neuroscience as a means of characterizing either the encoding
transfer functional of a neuron F[I(t)] =s(t) (Marmarelis and Marmarelis,
1978), or the associated decoding functional G[s(t)] =I(t) (Rieke et al.,
1997); here I(t) is the sensory input, while s(t) is the spike train. The simplest form of systems identification
involves estimation of the coefficients in the Volterra series expansion of the
desired functional. However, due to
data limitations and the high dimensionality of higher-order coefficients,
estimation of more than the first two terms has proven intractable. With discretized signals, it is easily seen
that the estimation of the Volterra coefficients is equivalent to linear
regression of the system output on non-linear functions of lag-vectors of the
input (note “system output” and “input” may be reversed with respect to the
neuron, depending on whether we are expanding the encoding or decoding
functional). In the first part of this
talk, I will discuss the use of techniques form the linear regression
literature, specifically ridge-regression and automatic relevance determination
(ARD), to stabilize estimates of higher-order Volterra coefficients in the face
of limited data. In the second part, I
will argue that regression using the conventional Volterra series expansion is
equivalent to kernel regression (in the sense of support-vector or
relevance-vector regression) using a series of polynomial kernels. This viewpoint allows for a relatively compact
representation of the coefficients in terms of data vectors, and, in
combination with ARD, allows estimation of still higher-order Volterra
coefficients. Furthermore, regression
using kernels other than polynomial, leads to non-linear expansions of the
transfer functional different from the Volterra series. These alternative expansions may provide
additional insight into the neural code.