Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions.

We establish two conditions that ensure the nondivergence of additive recurrent networks with unsaturating piecewise linear transfer functions, also called linear threshold or semilinear transfer functions. As Hahnloser, Sarpeshkar, Mahowald, Douglas, and Seung (2000) showed, networks of this type can be efficiently built in silicon and exhibit the coexistence of digital selection and analog amplification in a single circuit. To obtain this behavior, the network must be multistable and nondivergent, and our conditions allow determining the regimes where this can be achieved with maximal recurrent amplification. The first condition can be applied to nonsymmetric networks and has a simple interpretation of requiring that the strength of local inhibition match the sum over excitatory weights converging onto a neuron. The second condition is restricted to symmetric networks, but can also take into account the stabilizing effect of nonlocal inhibitory interactions. We demonstrate the application of the conditions on a simple example and the orientation-selectivity model of Ben-Yishai, Lev Bar-Or, and Sompolinsky (1995). We show that the conditions can be used to identify in their model regions of maximal orientation-selective amplification and symmetry breaking.

Computer simulation of the hydrostatic skeleton. The physical equivalent, mathematics and application to worm-like forms.

The functional principles of a hydrostatic skeleton were combined to obtain a physical model which includes geometry, number and length-tension relationships of the elastic elements in the body wall, internal volume and internal pressure. The model skeleton with pre-set internal volume assumes a certain shape and develops a specific internal pressure in order to minimize the potential energy stored in the elastic elements. This shape is calculated as equilibrium state by using finite element methods and optimization techniques. This model is flexible enough to accommodate different geometries and length-tension-relationships of the elastic elements. Presently, the model is implemented with linear length-tension relationships and certain geometrical restrictions, such as uniform width over the entire animal, and rectangular cross sections; the general case is outlined. First simulations with the "unit-worm" yield stable solutions, i.e. stable shapes for all combinations of parameters tested so far. They define the conditions for bringing all muscles to an optimal operating point. We detected a pressure maximum with increasing volume, assessed the contribution of circular muscles to bending, and determined the shapes of animals with different muscle activations in each body half (Chapman-matrix). We summarize our results by the volume rule and stabilization rule, two simple concepts which predict changes in shape as the result of muscle activation.