Path planning in control systems: The Combinatorics of high order iterated integrals
DateFriday, February 8, 2019 - 3:00pm
AbstractWe consider path planning problems for highly nonlinear control systems that are governed by finite dimensional differential equations. Think of parallel parking as a prototypical, simple example that every driver understands. Common strategies involve nilpotent approximating systems and steering with sinusoids. For low order terms, both iterated Lie brackets and iterated integral functionals, for most practical purposes, are characterized by their finely homogeneous degrees. But starting with length five, the internal combinatorial structures present new challenges to deciding controllability, and in steering algorithms, e.g. the labeled map to independent Lie brackets. We review some combinatorial and algebraic structures, including Zinbiel and combinatorial Hopf algebras. For systems with two control inputs we present novel choices for steering with sinusoids, that are orders of magnitude more efficient than the most general known algorithm for generating motions in most directions.
Multiscale Stochastic Transcription Models
DateFriday, February 15, 2019 - 3:00pm
AbstractThe formalism of stochastic reaction networks (SRNs) provides building blocks for number of models in mathematical biology both at molecular and population levels (e.g., gene transcription or epidemic outbreak). In particular the SRNs allow to naturally incorporate both delay and multi-scale phenomena. In the first case the resulting models may be often expressed in the language of queuing theory, in the second case they lead to stochastic diffusions and ODE/PDE approximations. In this talk I will provide a brief overview of the applications of SRNs to modeling molecular biological systems emphasizing the recent work on multi-scaling for simple stochastic gene transcription model.
Resonance-based mechanisms of generation of oscillations in networks of non-oscillatory neurons
DateFriday, February 22, 2019 - 3:00pm
AbstractSeveral neuron types have been shown to exhibit (subthreshold) membrane potential resonance (MPR), defined as the occurrence of a peak in their voltage amplitude response to oscillatory input currents at a preferred (resonant) frequency. MPR has been investigated both experimentally and theoretically. However, whether MPR is simply an epiphenomenon or it plays a functional role for the generation of neuronal network oscillations, and how the latent time scales present in individual, non-oscillatory cells affect the properties of the oscillatory networks in which they are embedded are open questions. We address these issues by investigating a minimal network model consisting of (i) a non-oscillatory linear resonator (band-pass filter) with 2D dynamics, (ii) a passive cell (low-pass filter) with 1D linear dynamics, and (iii) nonlinear graded synaptic connections (excitatory or inhibitory) with instantaneous dynamics. We demonstrate that (i) the network oscillations crucially depend on the presence of MPR in the resonator, (ii) they are amplified by the network connectivity, (iii) they develop relaxation oscillations for high enough levels of mutual inhibition/excitation, and (iv) the network frequency monotonically depends on the resonator’s resonant frequency. We explain these phenomena using a reduced adapted version of the classical phase-plane analysis that helps uncovering the type of effective network nonlinearities that contribute to the generation of network oscillations. Our results have direct implications for network models of firing rate type and other biological oscillatory networks (e.g, biochemical, genetic).