Back to homepage of Richard Hahnloser
Hahnloser, R.H.R. and Seung, H.S. and Slotine, J.-J. (2002).
Permitted and forbidden sets in symmetric threshold-linear networks.
Neural Computation, 15 (3). (ps or pdf).
Abstract
The richness and complexity of recurrent cortical circuits is an inexhaustible
source of inspiration for thinking about high-level biological computation.
In past theoretical studies, constraints on the synaptic connection patterns
of threshold-linear networks were found that guaranteed bounded network dynamics,
convergence to attractive fixed points, and multistability, all fundamental
aspects of cortical information processing. However, these conditions were
only sufficient and it remained unclear which were the
minimal (necessary) conditions for convergence and multistability.
We show that symmetric threshold-linear networks converge to a set of attractive
fixed point if and only if the
network matrix is copositive. Furthermore, the set of attractive fixed points
is nonconnected (the network is multiattractive) if and only if the network
matrix is not positive semidefinite. There are permitted sets of neurons
that can be coactive at a stable steady state and forbidden sets that cannot.
Permitted sets are clustered in the sense that subsets of permitted sets
are permitted and supersets of forbidden sets are forbidden. By viewing permitted
sets as memories stored in the synaptic connections, we provide a formulation
of long-term memory that is more general than the traditional perspective
of fixed-point attractor networks. There is a close correspondence
between threshold-linear networks and networks defined by the generalized
Lotka-Volterra equations.