Mathematical Modeling in Neuroscience

Research proposal from Dr. Andres Fraguela Collar

Objective: This project is to contribute to linking developments in mathematics and physics with studies in physiological process, collaborating with fundamental and numerical research in physical-mathematical problems derived from research made in Neuroscience. We have mainly focused attention on the physiological information of Cortex and Basal Ganglia.

Starting point: We studied physical-mathematical problems that arise from certain models at cellular level (neurons from Basal Ganglia) and at global level (cerebral mass and cortex).

  1. Problems studied at cellular level:
    Basal Ganglion specialists are interested in learning intrinsic properties of specific neurons (Ionic conductance, membrane capacitance) starting from measurements that can be carried out from their discharge patterns.
    In this project necessary mathematical techniques are developed to offer solutions for inverse problems of identification of coefficients which correspond to the parameters of interest.
    In this project necessary mathematical techniques are developed to offer solutions for inverse problems of identification of coefficients which correspond to the parameters of interest.
  2. Models at Global level:
    Here we mainly analyze models that describe the origin and spatial temporary evolution of the electroencephalogram (EEG) and that can be correlated to the brain’s activity.

Scope of the research:

  1. Developing methods that allow the brain’s stimulated zones to be located
  2. Interactive diagrams between the cortex’s different zones, which describe their physiological activity in a given EEG measurement.
  3. Circuits that appear in Basal Ganglia which are related to motor activity.
  4. Qualitative study of new models which describe the development of new methods and algorithms in order to study the inverse problems that arise here.
  5. Brain's functional activities, such as, knowledge, thought and learning which develop in the cerebral cortex, or more precisely in its most evolved part- the neo-cortex.
  6. How the brain works globally and the mechanisms that give rise to the EEG, explaining the origins of its spatial and temporal variation.
  7. To reproduce logic operations similar to those carried out by the brain.
  8. Patterns of activity of isolated real neurons or of finite sets of real neurons interacting with each other.
  9. To carry out correlation between EEG measurements and the active zones in the cerebral cortex.
  10. To resolve inverse problems of determination of the diagram of functional connections and the function of neuronal firing corresponding to a given physiological state.

Expected results:

  1. Construction of basic firing mechanism models of the medium spiny neurons from the striate (fundamental component of Basal Ganglia), following the ideas developed by Hodgkin and Huxley. These models should include the modulator effect of the main neuro-transmitters (dopamine and acetilcoline) whose balance defines these neurons’ state of activity.
  2. Building a lattices of the activity of certain neuronal circuits and for understanding mechanisms of the generation of many pathologies in the nervous system.

Methods used in the research:

  1. Methods from the theory of bifurcation are used, in which the parameters of bifurcation correspond to concentrations of neuromodulators.
  2. Techniques of average and small parameter that lead to reduction of the model, which is important due to the great number of ionic currents that flow through these neurons’ membranes.
  3. Algorithms of identification of coefficients are constructed for these models allowing identification of significative parameters (ionic conductivities, times of activation, steady states of activation, etc.) through measurements of membrane potential with no need to apply the usual techniques of blockage of channel or clamp.

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