Mathematical Sciences: "Analog Neural Networks and Delay Equations".

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Gedeon 9622722 Analog neural networks admit stationary, periodic and chaotic solutions. So far the stationary solutions have received most attention of researchers. However, in recent years the importance of non-stationary patterns in brain became clear. The investigator studies networks which admit periodic orbits and addresses the apparent discrepancy between discrete and continuous time networks of this type. While in continuous networks one expects generally a convergence to a periodic orbit, in discrete time networks there are stable stationary solutions. The project investigates chaotic behaviour of a class of networks with step-function nonlinearities. These can be viewed as a limiting case of a steep sigmoid nonlinearities and are easier to handle analytically. The delay in synaptic response of a neuron is a recognized fact; if one wants to account for its affect the appropriate model must be expressed in terms of delay differential equation. The investigator studies the discretization of this equation and shows that this finite-dimensional approximation captures the essential features of the dynamics of the delay equation. Artificial neural networks is an area on a crossroads between artificial inteligence and neurobiology. Mathematical models play an important role in this area. Neural networks try to model some basic functions of brain (learning, recognizing trained patterns) without trying to understand the intricate details of how the neurons work on a chemical or physiological level. Neural networks are used today in many areas of apllied science and engineering. The network can be trained to recognize a certain pattern, given slightly perturbed data. This pattern must be stationary, i.e. fixed in time. On the other hand, there is hardly any process in the brain which is stationary, and most processes are periodic. The investigator studies networks that admit periodic patterns, to understand the relationship between differen t models and see which one is the most suitable for applications. If one were able to train the networks to recognize complicated patterns (in space and time), that would have an immediate impact on robotics and artificial inteligence.