Scaling law for the transient behavior of type-II neuron models.

We study the transient regime of type-II biophysical neuron models and determine the scaling behavior of relaxation times tau near but below the repetitive firing critical current, tau approximately or equal to C(I(c)-I)(-Delta). For both the Hodgkin-Huxley and Morris-Lecar models we find that the critical exponent is independent of the numerical integration time step and that both systems belong to the same universality class, with Delta=1/2. For appropriately chosen parameters, the FitzHugh-Nagumo model presents the same generic transient behavior, but the critical region is significantly smaller. We propose an experiment that may reveal nontrivial critical exponents in the squid axon.

Phase-induced stability in a parametric dimer.

We report results on a model of two coupled oscillators that undergo periodic parametric modulations with a phase difference straight theta. Being to a large extent analytically solvable, the model reveals a rich straight theta dependence of the regions of parametric resonance. In particular, the intuitive notion that antiphase modulations are less prone to parametric resonance is confirmed for sufficiently large coupling and damping. Some general results concerning synchronization properties in this system are presented. We also compare our results to a recently reported mean-field model of collective parametric instability, showing that the two-oscillator model captures much of the qualitative behavior of the infinite system.

Unsupervised learning of binary vectors: a Gaussian scenario.

We study a model of unsupervised learning where the real-valued data vectors are isotropically distributed, except for a single symmetry-breaking binary direction Bin¿-1,+1¿(N), onto which the projections have a Gaussian distribution. We show that a candidate vector J undergoing Gibbs learning in this discrete space, approaches the perfect match J=B exponentially. In addition to the second-order "retarded learning" phase transition for unbiased distributions, we show that first-order transitions can also occur. Extending the known result that the center of mass of the Gibbs ensemble has Bayes-optimal performance, we show that taking the sign of the components of this vector (clipping) leads to the vector with optimal performance in the binary space. These upper bounds are shown generally not to be saturated with the technique of transforming the components of a special continuous vector, except in asymptotic limits and in a special linear case. Simulations are presented which are in excellent agreement with the theoretical results.