By Hermann Haken

This publication addresses a wide number of versions in mathematical and computational neuroscience. it really is written for the specialists in addition to for graduate scholars wishing to go into this interesting box of analysis. the writer reviews the behaviour of enormous neural networks composed of many neurons coupled by way of spike trains. An research of section locking through sinusoidal couplings resulting in different types of flow coordination is integrated.

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**Extra info for Brain Dynamics: Synchronization and Activity Patterns in Pulse-Coupled Neural Nets with Delays and Noise**

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On the other hand, we obviously obtain < (±1)j (±1)k > = < 1 > = 1 for j = k . 102). s. 105) is evaluated by assuming a so-called Poisson process for the times of the kicks. For our purposes it 54 4. Spikes, Phases, Noise: How to Describe Them Mathematically? s. by taking an average over the time T of a game, multiplied by the number N of kicks during T . Then we obtain T 1 < ... 106) 0 which can be evaluated to yield < ... >= N δ(t − t ) . 107) can be easily veriﬁed. 108) our ﬁnal result reads < F (t)F (t ) >= Qδ(t − t0 ) .

To model this phenomenon, we invoke the mechanical example of a soccer ball that is kicked by a soccer player and rolls over grass, whereby its motion will be slowed down. In this case, it is rather obvious how to describe the whole process. Our starting point is Newton’s law according to which the velocity v of a particle with mass m changes according to the equation m dv = force . 72) In order to get rid of superﬂuous constants, at least for the time being, we put m = 1 . s. consists of the damping force of the grass that we assume to be proportional to the velocity v and the individual kick of the soccer player.

Scheme of representation of EEG measurement. Each box is a plot of voltage V versus time, corresponding to the electrodes of Fig. 5 (after Lehmann, private communication) vald is optical imaging of the visual cortex that requires, however, the opening of the skull of the animal. In view of the models we will discuss in our book, we ﬁrst mention other methods that allow us to study rhythmic or oscillatory phenomena of large groups of neurons. These are the electroencephalogram (EEG) and the magnetoencephalogram (MEG).