NEUROSTOCHSIM · Neural Field Equations: Stochastic Approach and Numerical Simulations

Neural Field Equations: Stochastic Approach and Numerical Simulations

REPORT OF THE PROJECT NEUROSTOCHSIM PERIOD: 15/09/2014-14-09-2015 PROJECT WEBSITE: http://www.math.tecnico.ulisboa.pt/~plima/neuro/ PUBLISHABLE SUMMARY In the frame of this project, the researcher Pedro Lima has moved for one year from the University of Lisbon, his home institution, to the Institute of Stochastics of the Johannes Kepler University (JKU), in Linz. The whole period of the project was 1 year, so the summary of the final report coincides with the summary of the first year report. According to the application the research objectives of the project were: 1. Development of numerical methods for Neural Field Equations (NFE) in the deterministic case 2. Extension to the stochastic case 3. Numerical experiments and analysis of results Concerning the first objective, we have created and analysed a new numerical algorithm for computing approximate solutions of the two-dimensional neural field equations with delay. The stability, convergence and complexity of this algorithm have been analysed and a set of numerical examples has been considered. The numerical experiments are in agreement with the theoretical results and confirm that this algorithm can be successfully used for the solution of problems in Neuroscience and Robotics. The main advantages of the described algorithm are its stability and accuracy, which was illustrated by many examples. Moreover, due to the use of a rank reduction technique, it is efficient when dealing with the two-dimensional case. A research paper containing the description and analysis of the algorithm was accepted for publication in the SIAM Journal of Scientific Computing [1]. The results were also disseminated in poster presentations at the First International Conference on Mathematical Neuroscience [2] and Conference on Computational Neuroscience [3]. Finally, this was the subject of a plenary talk at the International Conference on Modelling of Nonlinear Processes and Systems [4] (This corresponds to Milestone 1 of the project propoposal). The work on the second objective has started in the fifth month of the project, with the analysis of existing numerical algorithms for stochastic partial differential equations and stochastic models for Neural Field Equations. This analysis enabled us to choose a stochastic model for the neural field equation and a numerical method for its approximation. The work on the numerical algorithm has continued till the end of the project period and the first results were presented at the International Workshop on Analysis and Numerical Approximation of Singular Problems [5] (this corresponds to Milestone 2 of the project proposal). Concerning the third objective, intensive work has been carried out in applying the algorithm to the numerical simulation of real world problems. In particular, we have obtained numerical results for the propagation of a localized stimulus in the visual cortex and spatially localized oscillations (breathers); the analysis of such objects is of crucial importance for the understanding of a wide range of neurobiological phenomena, such as the encoding of visual stimuli and neurological disorders like epilepsy. We have also carried out numerical simulations of multibump solutions of neural fields, which are very important in Cognitive Robotics. The results of all these numerical experiments have been reported at the above mentioned conferences [2][3][4] and at the international conference on Mathematics and Computers in Science and Industry [6], currently a summary is uploaded to arXiv [15]. The researcher has reported his research results to Dr. Patricia Figueiredo, leader of a research team in Neuroscience at the Institute of Systems and Robotics at the University of Lisbon; they have discussed plans of future collaboration (joint publications, joint participation in research grants) after the return of Pedro Lima to the University of Lisbon. The researcher has also discussed with Dr. Wolfram Erlhagen, from the University of Minho (Portugal), about the possible application of his algorithm to numerical simulations in Cognitive Robotics and plans of future collaboration were also set in this field. The project had as main training objectives to develop expertise of the researcher in the following topics: 1. Stochastic numerical analysis; 2. Modelling with NFEs; 3. Mathematical Neuroscience. Concerning the first topic the researcher has attended the course of Stochastic Differential Equations, read by Prof Evelyn Buckwar at the JKU during the spring term of 2014-2015. He has also carried out independent study of numerical methods for this kind of equations, using literature resources available a the JKU, in particular lecture notes for this course by Prof. Evelyn Buckwar. Moreover, he has attended the seminar of Theory of Probability and Mathematical Statistics, held weekly at the JKU, where researchers of the JKU and other Universities report about their recent results on actual topics of stochastic analysis and modelling. Finally, he has participated in the 10th IMACS Seminar on Monte Carlo Methods, an important event of international level which took place at the JKU [7]. In order to carry out the research work referred above, the researcher had to study intensively the second topic. The knowledge on this topic was obtained by reading, attending conferences on Mathematical Neuroscience (as mentioned above) and discussions with the Host and Colleagues of JKU working in related topics. Finally, the third topic was developed mainly based on the participation in the conferences [2] and [3], as well as in discussions with Colleagues that the researcher has met during these conferences. This topic also required intensive reading. Another important objective of the project was to give the researcher the possibility of interacting with scientists from different countries working in the same fields (mathematical neuroscience and stochastic analysis) and creating a network of sicentific contacts. This became possible thanks to the contacts of the Researcher with the team of the Institute of Stochastics, with the visitors of the Institute and with many other reserchers he has known during the participation in conferences and workshops. Concluding, we can say that the main objectives of the project were completely attained and no doubt it will have a very positive impact in the future career of the researcher.

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