The Random Neural Network: the model and some of its applications

1. The Random Neural Network: the model and some of its applications Erol Gelenbe1,2 and Varol Kaptan2 1 www.ee.imperial.ac.uk/gelenbe Denis Gabor Professor Head of Intelligent Systems and Networks 2 Dept of Electrical and Electronic Engineering Imperial College London SW7 2BT

3. Thank you to the agencies and companies who have generously supported RNN work over the last 15 yrs France (1989-97): ONERA, CNRS C3, Esprit Projects QMIPS, EPOCH and LYDIA USA (1993-): ONR, ARO, IBM, Sandoz, US Army Stricom, NAWCTSD, NSF, Schwartz Electro-Optics, Lucent UK (2003-): EPSRC, MoD, General Dynamics UK Ltd, EU FP6 for grant awards for the next three years, hopefully more ..

4. Outline Biological Inspiration for the RNN The RNN Model Some Applications modeling biological neuronal systems texture recognition and segmentation image and video compression multicast routing

5. Random Spiking Behaviour of Neurons

6. Random Spiking Behaviour of Neurons

7. Random Spiking Behaviour of Neurons

8. Random Spiking Behaviour of Neurons

9. Queuing Networks: Exploiting the Analogy

10. Queuing Network <-> Random Neural Network

12. Mathematical Model: A �neural� network with n neurons Internal State of Neuron i at time t, is an Integer xi(t) > 0 Network State at time t is a Vector x(t) = (x1(t), � , xi(t), � , xk(t), � , xn(t)) If xi(t)> 0, we say that Neuron i is excited and it may fire at t+ (in which case it will send out a spike) Also, if xi(t)> 0, the Neuron i will fire with probability riDt in the interval [t,t+Dt] If xi(t)=0, the Neuron cannot fire at t+ When Neuron i fires: - It sends a spike to some Neuron j, w.p. pij - Its internal state changes xi(t+) = xi(t) - 1

13. Mathematical Model: A �neural� network with n neurons The arriving spike at Neuron j is an: - Excitatory Spike w.p. pij+ - Inhibitory Spike w.p. pij - - pij = pij+ + pij- with Snj=1 pij < 1 for all i=1,..,n From Neuron i to Neuron j: - Excitatory Weight or Rate is wij+ = ri pij+ - Inhibitory Weight or Rate is wij- = ri pij- - Total Firing Rate is ri = Snj=1 (wij+ + wij�) To Neuron i, from Outside the Network - External Excitatory Spikes arrive at rate Li - External Inhibitory Spikes arrive at rate li

14. State Equations

17. Random Neural Network Neurons exchange Excitatory and Inhibitory Spikes (Signals) Inter-neuronal Weights are Replaced by Firing Rates Neuron Excitation Probabilities obtained from Non-Linear State Equations Steady-State Probability is Product of Marginal Probabilities Separability of the Stationary Solution based on Neuron Excitation Probabilities Existence and Uniqueness of Solutions for Recurrent Network Learning Algorithms for Recurrent Network are O(n3) Multiple Classes (1998) and Multiple Class Learning (2002)

18. Some Applications Modeling Cortico-Thalamic Response � Supervised learning in RNN (Gradient Descent) Texture based Image Segmentation Image and Video Compression Multicast Routing

20. Rat Brain Modeling with the Random Neural Network

23. Comparing Measurements and Theory:Calibrated RNN Model and Cortico-Thalamic Oscillations

24. Oscillations Disappear when Signaling Delay in Cortex is Decreased

25. Thalamic Oscillations Disappear when Positive Feedback from Cortex is removed

26. When Feedback in Cortex is Dominantly Negative, Cortico-Thalamic Oscillations Disappear

27. Summary of findings

28. Gradient Computation for the Recurrent RNN is O(n3)

29. Texture Based Object Identification Using the RNN, US Patent �99 (E. Gelenbe, Y. Feng)

30. 1) MRI Image Segmentation

31. MRI Image Segmentation

32. Brain Image Segmentation with RNN

33. Extracting Abnormal Objects from MRI Images of the Brain

34. 2) RNN based Adaptive Video Compression:Combining Motion Detection and RNN Still Image Compression

35. Neural Still Image CompressionFind RNN R that Minimizes|| R(I) - I ||Over a Training Set of Images {I }

36. RNN based Adaptive Video Compression

40. 3) Analytical Annealing with the RNN: Multicast Routing(Similar Results with the Traveling Salesman Problem) Finding an optimal many-to-many communications path in a network is equivalent to finding a Minimal Steiner Tree which is an NP-Complete problem The best heuristics are the Average Distance Heuristic (ADH) and the Minimal Spanning Tree (MSTH) for the network graph RNN Analytical Annealing improves the number of optimal solutions found by ADH and MST by approximately 10%

41. Selected References E. Gelenbe, ``Random neural networks with negative and positive signals and product form solution,'' Neural Computation, vol. 1, no. 4, pp. 502-511, 1989. E. Gelenbe, ``Stability of the random neural network model,'�Neural Computation, vol. 2, no. 2, pp. 239-247, 1990. E. Gelenbe, A. Stafylopatis, and A. Likas, ``Associative memory operation of the random network model,'' in {\it Proc. Int. Conf. Artificial Neural Networks}, Helsinki, pp. 307-312, 1991. E. Gelenbe, F. Batty, ``Minimum cost graph covering with the random neural network,'' Computer Science and Operations Research, O. Balci (ed.), New York, Pergamon, pp. 139-147, 1992. E. Gelenbe, ``Learning in the recurrent random neural network,'� Neural Computation, vol. 5, no. 1, pp. 154-164, 1993. E. Gelenbe, V. Koubi, F. Pekergin, ``Dynamical random neural network approach to the traveling salesman problem,'' Proc. IEEE Symp. Syst., Man, Cybern., pp. 630-635, 1993. A. Ghanwani, ``A qualitative comparison of neural network models applied to the vertex covering problem,'' Elektrik, vol. 2, no. 1, pp. 11-18, 1994. E. Gelenbe, C. Cramer, M. Sungur, P. Gelenbe ``Traffic and video quality in adaptive neural compression'', Multimedia Systems, Vol. 4, pp. 357--369, 1996. �

42. Selected References � �C. Cramer, E. Gelenbe, H. Bakircioglu ``Low bit rate video compression with neural networks and temporal subsampling,'' Proceedings of the IEEE, Vol. 84, No. 10, pp. 1529--1543, October 1996. �E. Gelenbe, T. Feng, K.R.R. Krishnan ``Neural network methods for volumetric magnetic resonance imaging of the human brain,'' Proceedings of the IEEE, Vol. 84, No. 10, pp. 1488--1496, October 1996. �E. Gelenbe, A. Ghanwani, V. Srinivasan, ``Improved neural heuristics for multicast routing,'' IEEE J. Selected Areas in Communications, vol. 15, no. 2, pp. 147-155, 1997. �E. Gelenbe, Z. H. Mao, and Y. D. Li, ``Function approximation with the random neural network,'' IEEE Trans. Neural Networks, vol. 10, no. 1, January 1999. �E. Gelenbe, J.M. Fourneau ``Random neural networks with multiple classes of signals,'' Neural Computation, vol. 11, pp. 721--731, 1999. E. Gelenbe, E. Seref, and Z. Xu,�``Simulation with learning agents�� Proceedings of the IEEE, 89 (2) pp.148-157 (2001) E. Gelenbe, K. Hussain ``Learning in the multiple class random neural network��, IEEE Trans. Neural Networks, vol. 13, no. 6, pp. 1257�1267, 2002. E. Gelenbe, R. Lent and A. Nunez ``Self-aware networks and QoS ��, Proceedings of the IEEE, 92 ( 9) pp. 1479-1490 (2004)