11 research outputs found
Beyond the Fokker-Planck equation: Pathwise control of noisy bistable systems
We introduce a new method, allowing to describe slowly time-dependent
Langevin equations through the behaviour of individual paths. This approach
yields considerably more information than the computation of the probability
density. The main idea is to show that for sufficiently small noise intensity
and slow time dependence, the vast majority of paths remain in small space-time
sets, typically in the neighbourhood of potential wells. The size of these sets
often has a power-law dependence on the small parameters, with universal
exponents. The overall probability of exceptional paths is exponentially small,
with an exponent also showing power-law behaviour. The results cover time spans
up to the maximal Kramers time of the system. We apply our method to three
phenomena characteristic for bistable systems: stochastic resonance, dynamical
hysteresis and bifurcation delay, where it yields precise bounds on transition
probabilities, and the distribution of hysteresis areas and first-exit times.
We also discuss the effect of coloured noise.Comment: 37 pages, 11 figure
The Temporal Winner-Take-All Readout
How can the central nervous system make accurate decisions about external stimuli
at short times on the basis of the noisy responses of nerve cell populations? It
has been suggested that spike time latency is the source of fast decisions.
Here, we propose a simple and fast readout mechanism, the temporal
Winner-Take-All (tWTA), and undertake a study of its accuracy. The tWTA is
studied in the framework of a statistical model for the dynamic response of a
nerve cell population to an external stimulus. Each cell is characterized by a
preferred stimulus, a unique value of the external stimulus for which it
responds fastest. The tWTA estimate for the stimulus is the preferred stimulus
of the cell that fired the first spike in the entire population. We then pose
the questions: How accurate is the tWTA readout? What are the parameters that
govern this accuracy? What are the effects of noise correlations and baseline
firing? We find that tWTA sensitivity to the stimulus grows algebraically fast
with the number of cells in the population, N, in contrast to
the logarithmic slow scaling of the conventional rate-WTA sensitivity with
N. Noise correlations in first-spike times of different
cells can limit the accuracy of the tWTA readout, even in the limit of large
N, similar to the effect that has been observed in
population coding theory. We show that baseline firing also has a detrimental
effect on tWTA accuracy. We suggest a generalization of the tWTA, the
n-tWTA, which estimates the stimulus by the identity of the
group of cells firing the first n spikes and show how this
simple generalization can overcome the detrimental effect of baseline firing.
Thus, the tWTA can provide fast and accurate responses discriminating between a
small number of alternatives. High accuracy in estimation of a continuous
stimulus can be obtained using the n-tWTA