| dc.description.abstracteng | In this thesis we aimed at understanding the fundamental properties of features commonly used in models of theoretical neuroscience from a dynamical systems' perspective. Dynamical system theory was developed while studying classical physical systems, such as planetary movements. Such systems can often be described by smooth ordinary differential equations (ODEs) and the dynamics arising from such equations are relatively well understood. However, in recent years the quantitative study of biological systems, including neuronal systems, has become a major research field. Models of these systems often include features that are not common to classical physics and still lack a good understanding from a dynamical systems' perspective. We contribute towards closing this gap.
In theoretical neuroscience models often encompass networks with many interacting neuronal units. These networks exhibit interesting dynamical phenomena. At the same time the neuronal units themselves may include dynamical features, such as pulse-coupling or adaptation, which may be able to produce rich dynamics already in very simple network architectures. If so it may be possible to consider extremely simple systems including these neural features and observe complex dynamics often associated with complex network models.
In this thesis we considered three very simple systems including features such as adaptation and pulse-coupling and analyzed their behavior. In all cases we found that these simple systems can produce surprisingly rich dynamics.
In a minimal adaptive system described by only two ordinary differential equations, we found a phase space portrait of presumably infinitely many nested limit cycles. In a simplified system with the same qualitative phase space portrait of nested limit cycles, we identified a structure, called funnel structure, which separates the flow in finite time into a set of discrete trajectories. Further, we showed that these trajectories are cycles, hence proving the nested limit cycle behavior in a certain parameter regime and hence providing analytic insights into the global dynamics of a nonlinear system.
Building on work by Kielblock, Kirst and Timme [1], we analyzed symmetrical all-to-all pulse-coupled phase oscillator networks. While corresponding ODE systems necessarily show order conservation of the oscillators, this is not true if pulse-coupling is introduced. We showed that gradually introducing self-loops may restore order conservation. In a system with delayed delta-pulse-coupling we were able to analytically understand the transition to order conservation and to uncover the mechanism behind the reordering process. Depending on system parameters simple or quasi-chaotic reordering patterns were observed. We discuss how it is possible that pulse-coupled oscillators can circumvent the dynamical restrictions present in ODE systems and break order conservation. Further, we discuss why order conservation may be restored by introducing self-loops.
Finally we described a system of only two delayed delta-pulse-coupled phase oscillators, that shows chaotic behavior characterized by a beautiful orbit diagram. We found that the observed chaos can be related to the period-doubling route to chaos arising in unimodal maps, such as the logistic map.
We have seen that the neural features we focused on indeed are able to produce very rich dynamics, even if embedded into very simple systems.
Studying these systems has lead to a better understanding of the dynamic properties of these features. At the same time it turns out that the features themselves comprise a collection of different subfeatures with different dynamical properties, i.e. Pulse-coupling can have quite different properties depending on the shape of the pulse used. We also encountered conceptual difficulties, e.g. what do we mean by adaptation in a dynamical context? Or how can we compare trajectories in a pulse-coupled system, if the dimensionality of the space containing them changes dynamically?
It is clear that our systems studied here are only a starting point towards gaining a full understanding of the dynamical implications of adaptation and pulse-coupling.
If more systems are studied encompassing these dynamical features hopefully patterns and principles can be discovered to develop a better theory of which topologies are possible and under which conditions. And maybe new dynamical structures are discovered altogether.
[1] H. Kielblock, C. Kirst, and M. Timme, Chaos 21, 025113 (2011). | de |