Frequency-doubling bifurcations in neuronal networks – a means of cross-frequency interactions
Early Stage Researcher: Bastian Pietras
Principle Investigators: Andreas Daffertshofer, Bob van Dijk (Amsterdam, VUA – major institution), Aneta Stefanovska, Peter McClintock (Lancaster, ULANC – partner institution)
Neuronal populations often display oscillatory activity. Prominent examples for this are ~10 Hz alpha oscillations in visual cortex and ~ 20 Hz beta oscillations in motor cortex. This project addresses whether switches between frequency bands on a macroscopic scale can be interpreted as frequency-doubling bifurcations or should be viewed as discrete switches between two different oscillations. Such sequential links will shed light on dynamic correlations between oscillatory activity in the human brain: by classifying bifurcation routes we can constrain symmetries and types of interactions between neurons or neural ensembles and thereby confine models. Different neurons can exhibit qualitatively different oscillatory behavior1. This difference has been ascribed to mechanisms of excitability, treating the amplitude of the injected current in Hodgkin’s experiment as a bifurcation parameter2,3 Similarly, a variety of neural networks have been proposed that are capable of generating oscillations in a range of different frequencies observed in empirical data. This leads to the hypotheses:
- Large-scale oscillatory activities at distinct frequencies emerge through a generic type of bifurcation on a more local level.
- Switches between frequencies on the level of a large-scale network can be cast as a combination of sub- and supercritical Hopf-bifurcations rather than frequency- or period-doubling bifurcations.
The project will take small networks that are known for their bifurcation schemes and combine them to build larger ensembles79. The constraints for two Hopf-bifurcations to appear in parallel will be analytically determined using simple, tractable neural models. Large systems will be analyzed via mean-field approaches and comparable statistical approximations focusing on the bifurcation characteristics. How do local bifurcation parameters transform to bifurcation parameters of the ensemble dynamics and do the co-dimensions match across levels? The analytic approach will be complemented with numerical estimates of the large-scale dynamics allowing for more physiologically realistic conductance-based neurons.
- Hodgkin, A. L. The local electric changes associated with repetitive action in a non-medullated J Physiol 107, 165-181 (1948).
- Rinzel, J. & Ermentrout, G. B. in Methods of neural modeling: from synapses to networks (eds C. Koch & I. Segev) (MIT Press, 1989).
- Hoppensteadt, F. C. An introduction to the mathmatics of neurons. (Cambridge University Press, 1997).