In the past decade, our laboratory has been successful in establishing that dynamics from single neurons, small ensembles of neurons, and from intact human cortical recordings show significant degrees of nonlinear determinism. This result was derived using concepts from nonlinear dynamics theory; specifically that a system with relatively few degrees of freedom can exhibit highly complex behavior,and that such dynamics can be decomposed into a hierarchy of unstable periodic orbits (UPOs). These orbits are the equilibrium states of the system, but, since they are unstable, the actual dynamics tend to be a series of close approaches and subsequent divergences from them. We have developed a highly effective method for extracting hierarchies of UPOs from experimental data, along with approximations of the dynamics near them. These hierarchies form predictive models of the full dynamics.
|An abstract ''dynamical landscape'' of the unstable periodic orbits (UPOs) extracted from measurements of a single neuron within an ensemble of neurons. This picture includes UPOs with period 1 and 2 only. Period 2 orbits appear in pairs and they are indicated here by connected green dots. The single green dot in the middle is the period 1 orbit. The temporal evolution of the neuron’s dynamics can be abstractly visualized as a trajectory moving within this “landscape”. When the trajectory is near one of these UPOs, the neuron will brust with a periodicity approximately given by the UPOs and the duration of this transient periodicity is determined by the local stability of these UPOs.|
Our recent work suggests that UPOs form a natural symbolic code for characterizing complex systems including neuronal networks. Previous theoretical work has been done establishing the connection between ergodic dynamical measures such as the Kolmogorov entropy and the fractal dimension of a complex dynamical system with the underlining UPO structure. Preliminary work from our group has indicated the synchronization process between two uni-directionally coupled dynamical systems might be analyzed in terms of a sequence of topological changes in the UPO structure of the driven system. Furthermore, these structural changes in UPOs affect changes in the entropy measure of the system. An in-depth theoretical understanding of the mechanisms and processes of information flow between coupled systems, including neurons within an ensemble, is important. A quantitative description of the dynamical interplay between two simple coupled models is only the first step in pursuing this goal. It is our near term goal to extend this theoretical description of coupled systems to more complex and realistic ensembles. It is our vision that a new statistical analysis of neuronal systems can be developed based on the assumption that UPOs are the microstates within a "thermodynamical" formalism for their dynamics.
EXTRA: A write up on UPOs in Scholarpedia by P. SO