Mathematical neuron models: isolated and coupled neurons
Prof. Roberto Barrio
Departamento de Matematica Aplicada (Computational Dynamics Group), University of Zaragoza
DEIB - PT1 Room
September 6th, 2016
11.00 am
Contact:
Fabio Dercole
Research Line:
Dynamics of complex systems
Departamento de Matematica Aplicada (Computational Dynamics Group), University of Zaragoza
DEIB - PT1 Room
September 6th, 2016
11.00 am
Contact:
Fabio Dercole
Research Line:
Dynamics of complex systems
Abstract
In this talk we present some approaches to mathematical neuroscience, begining with detailed mathematical analysis of isolated neurons (Hodgkin-Huxley, Hindmarsh-Rose and leech neuron models) to small neuron networks. The studies are based on detailed biparametric "roadmaps'' that provide an exhaustive information about the dynamics of a single neuron that one must have to build small neuron networks and to study rhythmogenesis in central pattern generators (CPG). In the first application we characterize the systematic changes in the topological structure of chaotic attractors that occur at spike-adding and homoclinic bifurcations in the slow-fast dynamics of neuron models.
This phenomenon is detailed in the phenomenological Hindmarsh-Rose neuron model and a reduced model of the leech heart interneuron. In the second application, we reveal the existence of heteroclinic cycles between saddle fixed points (FP) and invariant circles (IC) in a 3-cell CPG network (leech heart neurons). Such a cycle underlies a robust "jiggling'' behavior in bursting synchronization. To study biologically plausible CPG models we employ novel techniques based on the Poincare return maps for phase lags between coupled bursters. Using the combination of these techniques we are able to aggregate big data to parametrically continue FPs and ICs of the maps and to fully disclose their bifurcation unfoldings as the network configuration is varied and how to control the different synchronization patterns. Note that some of these synchronization patterns of individual neurons are related with some undesirable neurologic diseases, and they are believed to play a crucial role in the emergence of pathological rhythmic brain activity in different diseases, like Parkinson's disease.
This phenomenon is detailed in the phenomenological Hindmarsh-Rose neuron model and a reduced model of the leech heart interneuron. In the second application, we reveal the existence of heteroclinic cycles between saddle fixed points (FP) and invariant circles (IC) in a 3-cell CPG network (leech heart neurons). Such a cycle underlies a robust "jiggling'' behavior in bursting synchronization. To study biologically plausible CPG models we employ novel techniques based on the Poincare return maps for phase lags between coupled bursters. Using the combination of these techniques we are able to aggregate big data to parametrically continue FPs and ICs of the maps and to fully disclose their bifurcation unfoldings as the network configuration is varied and how to control the different synchronization patterns. Note that some of these synchronization patterns of individual neurons are related with some undesirable neurologic diseases, and they are believed to play a crucial role in the emergence of pathological rhythmic brain activity in different diseases, like Parkinson's disease.