Knowledge of the anatomical organisation of the brain is now fairly advanced. Thanks to technological advances in recent decades (such as functional MRI, massively parallel recording by multi-electrode arrays or optical imaging) there has been also a sharp increase of our knowledge of the functional organisation in many areas of the brain. Mathematical modeling has become a vital tool in linking the anatomical and physiological data with that on neural function and providing insight into the mechanisms of information processing in the brain, as well as brain development.
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In the past half a century a multitude of models have been developed and applied with great success in generating dynamics that is observed in biological experiments, at all levels of neural and brain functioning. Some examples include the Hodgkin-Huxley model for the action-potential propagation along the axon, the application of cable theory to the dendritic tree by Rall, the model of Núñez for EEG rhythms, the population mean field model of Wilson and Cowan and the later unification of the two by Jirsa and Haken.
It has been recognised that the higher-level mechanisms of information processing and storage in the brain are implemented through the temporal dynamics and spatial distribution of neuronal activity. Since spatiotemporal evolution of activity patterns is a function primarily of the architecture of interneuronal connectivity, this has prompted extensive theoretical research on neural networks. In the perspective of theoretical neuroscience, a network is a collection of coupled dynamical systems, each describing an individual processing unit, a neuron. The latter typically are lower-dimensional approximations of some biologically realistic model of the single neuron that capture only the features thought to be most significant within the network context. However, in contrast to artificial neural networks studied in computer science, neuroscientists try to preserve as much biological relevance as possible. A limiting factor is the mathematical or computational tractability as well as the gaps in our knowledge on the finer details of the real system.
Common approaches in neuroscience are to use the methods of statistical mechanics and information theory to study properties of general networks or the theory of dynamic systems for networks whose connectivity is specified to a varying degree of biological detail. I am involved with the dynamic system approach, investigating networks that have been averaged to produce mathematically elegant integral models. Averaging similar to that in statistical mechanics is thought to be valid since in cortex there are huge number of cells per unit volume and the dense interconnections of nearby cells result in coherent distributed population responses to electrical stimulation. The macroscopic description derived by this method turns out to be a continuous nonlinear spatially extended system, which has as state variable the mean firing rate of the neuronal populations at a given location.

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