Relating structure and dynamics in neural networks with applications to brain imaging
Final Report Abstract
In summary, in our project we have set out to study the relationship between structure and dynamics in neural networks by investigating the concept of a catalyst and its representation in the brain. We have developed a framework that allows to study systematically and in great generality the relation between structure and dynamics in neural networks, which was a central motivation for our project. Moreover, because of its generality, this framework is also suitable to include the notion of function. Brain function is dynamic and hence best viewed as a process. From an abstract level brain function can be described in terms of dynamical systems. To provide an example, a bistable dynamical system with two fixed points can describe the famous “face - vase” illusion where the same picture can be seen either as two faces facing each other or as a vase, depending on which part of the picture is considered to be the background. Since the framework developed in this project is suitable for creating network representations for arbitrary dynamical systems, it ties together function, dynamics and structure of a neural network. Moreover, our framework allows to create network models of brain function at different levels of description. For instance, the firing rate network chosen for the representation of the catalyst represents a model at the neural population level. Known facts about the network connectivity in the brain can be easily implemented within our framework by fixing at zero the appropriate entries of the connectivity matrix and estimating only the remaining entries. Up to now analyses of the relation between structure and dynamics in networks have been performed mostly on the dynamics of networks of known topology, be it regular such as feedforward or lattices, or irregular such as random, scale-free, or small-world networks. These analyses have been mostly statistical in nature, giving only a coarse picture of how dynamics is related to network structure. The urgency of having more detailed knowledge is readily apparent if we think, for instance, of a surgeon who needs to know the effect of severing connections in the brain of a particular patient. Our framework provides a first step towards such detailed knowledge. Moreover our framework could serve to perform lesion studies which pinpoint the most vulnerable connections, i. e. the connections which, if lesioned, lead to the largest deviation from the original dynamics. Then our framework could identify alternative network structures which restore the original function, for instance by finding connectivity matrices which are similar to the lesioned matrix in the sense of least rewiring cost, but exhibit dynamics which are equivalent to the original dynamics. Further possible applications of our framework could be envisioned in robotic control, for instance in the implementation of a desired motor behavior in terms of an electrical circuit.
