Discrete Dynamics in Nature and Society
Volume 4 (2000), Issue 3, Pages 187-200

Quasi-discreted dynamics of a neural net: The lighthouse model

Hermann Haken

Institute for Theoretical Physics 1, Center of Synergetics, Pfaffenwaldring 57/4, Stuttgart D-70550, Germany

Received 30 March 1999

Copyright © 2000 Hermann Haken. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper studies the features of a net of pulse-coupled model neurons, taking into account the dynamics of dendrites and axons. The axonal pulses are modelled by δ-functions. In the case of small damping of dendritic currents, the model can be treated exactly and explicitly. Because of the δ-functions, the phase-equations can be converted into algebraic equations at discrete times. We first exemplify our procedure by two neurons, and then present the results for N neurons. We admit a general dependence of input and coupling strengths on the neuronal indices. In detail, the results are

(1) exact solution of the phase-locked state;

(2) stability of phase-locked state with respect to perturbations, such as phase jumps and random fluctuations, the correlation functions of the phases are calculated;

(3) phase shifts due to spontaneous opening of vesicles or due to failure of opening;

(4) effect of different sensory inputs on axonal pulse frequencies of coupled neurons.