Özkan Karabacak

Stability of Coupled Map Networks with Delays

We consider networks of coupled scalar maps, with weighted connections which may include a time delay, and study the stability of equilibria with respect to the delays and connection structure. We prove that the largest eigenvalue of the graph Laplacian determines the effect of the connection topology on stability. The stability region in the parameter plane shrinks with increasing values of the largest eigenvalue, or of the time delay of the same parity. In particular, all bipartite graphs have an identical stability region, regardless of the delay or graph size, which is also the smallest stability region among those of all graphs. Furthermore, for certain parameter ranges, unstable (and possibly chaotic) maps can be stabilized via diffusive coupling with an odd time delay, provided that the network does not have a nontrivial and connected bipartite component. On the other hand, stabilization is not possible for even values of the delay or for bipartite networks.

Dwell time and average dwell time methods based on the cycle ratio of the switching graph

Abstract A new method to find an upper bound on dwell time and average dwell time for switched linear systems is proposed. The method is based on computing the maximum cycle ratio and the maximum cycle mean of the directed graph that governs switchings. For planar switched systems, an upper bound for dwell time and average dwell time can be estimated by considering only the cycles of length two.

Criteria for robustness of heteroclinic cycles in neural microcircuits

We introduce a test for robustness of heteroclinic cycles that appear in neural microcircuits modeled as coupled dynamical cells. Robust heteroclinic cycles (RHCs) can appear as robust attractors in Lotka-Volterra-type winnerless competition (WLC) models as well as in more general coupled and/or symmetric systems. It has been previously suggested that RHCs may be relevant to a range of neural activities, from encoding and binding to spatio-temporal sequence generation.The robustness or otherwise of such cycles depends both on the coupling structure and the internal structure of the neurons. We verify that robust heteroclinic cycles can appear in systems of three identical cells, but only if we require perturbations to preserve some invariant subspaces for the individual cells. On the other hand, heteroclinic attractors can appear robustly in systems of four or more identical cells for some symmetric coupling patterns, without restriction on the internal dynamics of the cells.

A dwell time approach to the stability of switched linear systems based on the distance between eigenvector sets

In this article, a sufficient condition on the minimum dwell time that guarantees the stability of switched linear systems is given. The proposed method interprets the stability of switched linear systems through the distance between the eigenvector sets of subsystem matrices. Thus, an explicit relation in view of stability is obtained between the family of the involved subsytems and the set of admissible switching signals.

A Computational Model of Cortico-Striato-Thalamic Circuits in Goal-Directed Behaviour

A connectionist model of cortico-striato-thalamic loops unifying learning and action selection is proposed. The aim in proposing the connectionist model is to develop a simple model revealing the mechanisms behind the cognitive process of goal directed behaviour rather than merely obtaining a model of neural structures. In the proposed connectionist model, the action selection is realized by a non-linear dynamical system, while learning that modifies the action selection is realized similar to actor-critic model of reinforcement learning. The task of sequence learning is solved with the proposed model to make clear how the model can be implemented.

A dynamical model of a cognitive function: Action selection

Abstract A model of cortex-basal ganglia-thalamus-cortex loop is presented. Even though the mentioned loop has been proposed to take part in different cognitive processes, the model exploits only the “action selection” function. The analysis of the model is based on the stability analysis of a non-linear dynamical system. Thus a biologically valid model of a cognitive function is given and its analysis is accomplished using system theory tools.

Robust Heteroclinic Behaviour, Synchronization, and Ratcheting of Coupled Oscillators

This review examines some recent work on robust heteroclinic networks that can appear as attractors for coupled dynamical systems. We focus on coupled phase oscillators and discuss a number of nonlinear dynamical phenomena that are atypical in systems without some coupling structure. The phenomena we discuss include heteroclinic cycles and networks between partially synchronized states. These networks can be attracting and robust to perturbations in parameters and system structure as long as the coupling structure is preserved. We discuss two related effects; extreme sensitivity to detuning (strongly coupled oscillators may lose their frequency synchrony for very small detunings) and heteroclinic ratchet where the sensitivity may only appear for detunings of one sign.

A computational model for the effect of dopamine on action selection during stroop test

Based on a connectionist model of cortex-basal ganglia-thalamus loop recently proposed by authors, a simple connectionist model realizing the Stroop effect is established. The connectionist model of cortex-basal ganglia-thalamus loop is a nonlinear dynamical system and the model is not only capable of revealing the action selection property of basal ganglia but also is capable of modelling the effect of dopamine on action selection. While the interpretation of action selection function is based on solutions of nonlinear dynamical system, the effect of dopamine is modelled by a parameter. The effect of dopamine in inhibiting the habitual behaviour corresponding to word reading in Stroop test and letting the novel one occur corresponding to colour naming is investigated using the model established in this work.

On statistical attractors and the convergence of time averages

There are various notions of attractor in the literature, including measure (Milnor) attractors and statistical (Ilyashenko) attractors. In this paper we relate the notion of statistical attractor to that of the essential ω-limit set and prove some elementary results about these. In addition, we consider the convergence of time averages along trajectories. Ergodicity implies the convergence of time averages along almost all trajectories for all continuous observables. For non-ergodic systems, time averages may not exist even for almost all trajectories. However, averages of some observables may converge; we characterize conditions on observables that ensure convergence of time averages even in non-ergodic systems.

The role of synchronization of neural structures on neuron dynamics

Due to the measurement techniques developed, it is now possible to observe signals at different levels in neural structures. These observations give rise to increasing works dealing with setting up the relation between such signals and cognitive processes, neurodegenerative diseases and neurobehavioral disorders. Computational neuroscience is becoming more influential in investigating the mechanisms underlining these signals. Especially oscillations in different neural structures and the role of these oscillations on the dynamics of a neuron that is in connection with these structures have drawn the attention of computational neuroscience recently. In this work, the direct pathway in basal ganglia circuits, which takes place in action selection has been considered. The role of synchrony in striatum - the input structure of basal ganglia circuits, on the dynamics of the neurons in thalamus - the output structure of basal ganglia has been investigated. This work reveals that this effect depends on the dynamics of thalamus neurons and explains the role of the rebound activity in thalamus.