Session 7. Difference equations and their application in the mathematical modeling

Behaviour of solutions of one neuron models

Inese Bula, University of Latvia and Institute of Mathematics and Computer Science of University of Latvia, Latvia
The talk is based on the joint work with Aija Anisimova and Maruta Avotina.
Neural networks are complex and large-scale nonlinear dynamical systems. As the proverbial forest can not be seen through because of the trees, a detailed study of single neuron is an interesting subject itself, but it is not necessary to understand the macroscopic dynamics and role of neural networks. In the literature [2] a delay differential equation $$ x' (t)=-g(x(t-\tau)) \label{eq:1} $$ is used as a model for a single neuron with no internal decay where \(g: {\bf R}\to {\bf R}\) is either a sigmoid or a piecewise linear signal function and \(\tau \le 0\) is a synaptic transmission delay. From equation \eqref{eq:1} we obtain a difference equation $$ x_{n+1}=\beta x_n-g(x_{n}). \label{eq:2} $$ By [2] \(x\) denotes the activation level of a neuron, \(\beta\) is interpreted as an internal decay rate and \(g\) is a signal function. Accordingly to the parameter \(\beta\) we obtain different behaviour of solutions of difference equation (\ref{eq:1}). Idea of finding periodic orbits of the model first was demonstrated in [3]. Signal function play an important role in the investigation. In our work we used step functions with two and three thresholds (see [1]) therefore in fact we investigated one dimensional discontinuous piecewise linear map. We will present some results about the solutions of model \eqref{eq:2} with different signal functions.
References
  1. A. Anisimova, M. Avotina, I. Bula, Periodic Orbits of Single Neuron Models with Internal Decay Rate \(0< \beta \le 1\) , Mathematical Modelling and Analysis 18, 2013, 325--345.
  2. J.Wu, Introduction to Neural Dynamics and Signal Transmission Delay , De Gruyter, Berlin, 2001.
  3. Z.Zhou, Periodic Orbits on Discrete Dynamical Systems , Computers and Mathematics with Applications 45, 2003, 1155-1161.
Print version