The human brain is a highly complex system that is composed of about hundred billion neurons each of which may interact with up to ten thousand other neurons. The brain serves many purposes as recognition of our surrounding, steering of movement, emotions, etc. Roughly speaking, the brain consists of individual areas that serve specific purposes. But these areas are again strongly interconnected. The neurons can fulfill the tasks only by a high degree of cooperation. But who or what steers the neurons? Since two decades I have been propagating the idea that the brain acts by self-organization; a point of view that is accepted more and more in the scientific community. In particular, concepts of synergetics can be applied to brain function. Some typical aspects will be illustrated by examples from visual perception, such as recognition of faces and facial expressions, perception of ambiguous figures, hysteresis, etc.
I first present some basic features about the structure and dynamics of individual neurons including cell body, axons, dendrites, and synapses. The signal processing via spikes will be discussed. Then I will study the interplay between dendritic currents and axonal spikes (pulses) via the light-house model. The spikes are described by a phase angle, increasing in the course of time in analogy to a rotating light beam from a light- house through which the intervals between spikes are determined. The rotation speed depends on the inputs from other neurons. After elimination of the dendritic currents, we find a special form of an integrate and fire model, whereupon I will discuss a number of different forms that take into account arbitrary strengths of synapses and time-lags.
I analytically study the solutions of a network of integrate and fire neurons of a general form. In particular, I study the circumstances under which phase locking becomes possible and discuss cases of coexistence between phase-locked neurons and other neurons.
The general equations of the integrate and fire neurons under different sensory inputs are time-averaged over short intervals. It is shown how by suitable choices of the synaptic connections these models give rise to associative memory. Associative memory means that a set of incomplete data is completed by the system to a well-defined set depending on the partly given data. A connection with Kaniza figures is established, as well as with other models of associative memory.
Again using suitable time-averages instead of the equations for spikes, rate equations for spike sequences as well as for dendritic currents are established. Such equations are related to the Nunez equations in a simplified form and have been derived along different lines by Jirsa and Haken. When the dendritic currents are eliminated, we reobtain the Wilson- Cowan equations that in turn allow the derivation of spatio-temporal patterns of brain activity in terms of firing rates. If, on the other hand, the axonal pulses or pulse-rates are eliminated, we obtain a new type of equation first derived by Jirsa and Haken for the dendritic currents, which, in turn, are responsible for the electric and magnetic fields measured in EEGs and MEGs.
References
http://www.theo1.physik.uni-stuttgart.de/en/mitarbeiter/haken/