Topological Properties of the Immediate Basins of Attraction for the Secant Method

We study the discrete dynamical system defined on a subset of $$R^2$$ R 2 given by the iterates of the secant method applied to a real polynomial p. Each simple real root $$\alpha $$ α of p has associated its basin of attraction $${\mathcal {A}}(\alpha )$$ A ( α ) formed by the set of points converging towards the fixed point $$(\alpha ,\alpha )$$ ( α , α ) of S. We denote by $${\mathcal {A}}^*(\alpha )$$ A ∗ ( α ) its immediate basin of attraction, that is, the connected component of $${\mathcal {A}}(\alpha )$$ A ( α ) which contains $$(\alpha ,\alpha )$$ ( α , α ) . We focus on some topological properties of $${\mathcal {A}}^*(\alpha )$$ A ∗ ( α ) , when $$\alpha $$ α is an internal real root of p. More precisely, we show the existence of a 4-cycle in $$\partial {\mathcal {A}}^*(\alpha )$$ ∂ A ∗ ( α ) and we give conditions on p to guarantee the simple connectivity of $${\mathcal {A}}^*(\alpha )$$ A ∗ ( α ) .

Module-based kinetostatic analysis and optimization of a modular reconfigurable robot

n this thesis, a newly developed kinetostatic model for modular reconfigurable robots (MRRs) is presented. First, a kinetmatic computational method was created to allow for simple connectivity between modules which included the possibilities of angular offsets. Then, a flexibility analysis was performed to determine the static and dynamic flexibility of link and join modules and the regions of flexibility were plotted to determine exactly which of the components can be considered flexible or rigid, depending on their sizes. Afterwards, the kinetostatic model was developed and compared to a finite element model and results give essentially the same tip deflections between the two models. This kinetostatic model was then used to determine the maximum allowable payload and maximum deflection position for a given MRR. Additionally, a direct method was created to determine the cross section properties of all modules in a given MRR for a given payload and maximum desirable tip deflection.

Module-based kinetostatic analysis and optimization of a modular reconfigurable robot

n this thesis, a newly developed kinetostatic model for modular reconfigurable robots (MRRs) is presented. First, a kinetmatic computational method was created to allow for simple connectivity between modules which included the possibilities of angular offsets. Then, a flexibility analysis was performed to determine the static and dynamic flexibility of link and join modules and the regions of flexibility were plotted to determine exactly which of the components can be considered flexible or rigid, depending on their sizes. Afterwards, the kinetostatic model was developed and compared to a finite element model and results give essentially the same tip deflections between the two models. This kinetostatic model was then used to determine the maximum allowable payload and maximum deflection position for a given MRR. Additionally, a direct method was created to determine the cross section properties of all modules in a given MRR for a given payload and maximum desirable tip deflection.

Topology of random -dimensional cubical complexes

We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

The spectrum of covariance matrices of randomly connected recurrent neuronal networks

A key question in theoretical neuroscience is the relation between connectivity structure and collective dynamics of a network of neurons. Here we study the connectivity-dynamics relation as reflected in the distribution of eigenvalues of the covariance matrix of the dynamic fluctuations of the neuronal activities, which is closely related to the network’s Principal Component Analysis (PCA) and the associated effective dimensionality. We consider the spontaneous fluctuations around a steady state in randomly connected recurrent network of spiking neurons. We derive an exact analytical expression for the covariance eigenvalue distribution in the large network limit. We show analytically that the distribution has a finitely supported smooth bulk spectrum, and exhibits an approximate power law tail for coupling matrices near the critical edge. Effects of adding connectivity motifs and extensions to EI networks are also discussed. Our results suggest that the covariance spectrum is a robust feature of population dynamics in recurrent neural circuits and provide a theoretical predictions for this spectrum in simple connectivity models that can be compared with experimental data.

Simple connectivity in polar spaces

We settle the simple connectivity of the geometry opposite a chamber in a polar space of rank 3 by completing the job for the non-embeddable polar spaces and some polar spaces with small parameters.

Counting on dis-inhibition: a circuit motif for interval counting and selectivity in the anuran auditory system

Information can be encoded in the temporal patterning of spikes. How the brain reads these patterns is of general importance and represents one of the greatest challenges in neuroscience. We addressed this issue in relation to temporal pattern recognition in the anuran auditory system. Many species of anurans perform mating decisions based on the temporal structure of advertisement calls. One important temporal feature is the number of sound pulses that occur with a species-specific interpulse interval. Neurons representing this pulse count have been recorded in the anuran inferior colliculus, but the mechanisms underlying their temporal selectivity are incompletely understood. Here, we construct a parsimonious model that can explain the key dynamical features of these cells with biologically plausible elements. We demonstrate that interval counting arises naturally when combining interval-selective inhibition with pulse-per-pulse excitation having both fast- and slow-conductance synapses. Interval-dependent inhibition is modeled here by a simple architecture based on known physiology of afferent nuclei. Finally, we consider simple implementations of previously proposed mechanistic explanations for these counting neurons and show that they do not account for all experimental observations. Our results demonstrate that tens of millisecond-range temporal selectivities can arise from simple connectivity motifs of inhibitory neurons, without recourse to internal clocks, spike-frequency adaptation, or appreciable short-term plasticity.