1. Grid-based Simulations of Mammalian Visual System Grzegorz M. Wójcik and Wiesław A. Kamiński Maria Curie-Sklodowska University, Lublin, Poland. Abstract Large biological neural networks are examined. Ensembles of simulated microcircuits model behaviour of mammalian visual system in some detail. All neural cells are simulated according to Hodgkin-Huxley theory. In that model each neuron is treated as a set of several non-linear differential equations. Good simulation of large groups of Hodgkin-Huxley neurons usually requires high computational powers. The modular structure of visual system seems to be appropriate task for grid computations. In this paper we report first results of CLUSTERIX grid application to modelling of vision processes. MPGENESIS simulator is used for all simulations. We investigate networks consisting 16 thousands of Hodgkin-Huxley neurons. First simulations were run on the local cluster with 24 nodes. Consequently, in the next stage of experiments, we are going to check the time of simulation for larger number of processors, using CLUTERIX grid resources. Such number of simulated neurons allowed us to observe liquid computing phenomena. In theory cortical microcircuits are treated as Liquid State Machines (LSM). The work of each machine resembles behaviour of particles in a liquid. Though, we also present some results confirming the thesis that neural liquids tend to be in different states for different, changing in time stimulations and that such biological structures can have computational power. Introduction Human brain built of about 1011 neural cells is a hard object of simulation even for contemporary super-computers. Some idea of whole brain modelling was suggested by Maass and since then it has been called Liquid State Machine (LSM) [1-3]. In general, the brain (or a fragment of it) is treated as a liquid. Mammalian brains’ cortex is built of neurons organised in microcircuits [4]. Microcircuits form columns, and the function of each column depends on its location in the brain. Cortical microcircuits turn out to be very good “liquids” for computing on perturbations because of the large diversity of their elements, neurons and synapses, and the large variety of mechanisms and time constants characterising their interactions, involving recurrent connections on multiple spatial scales. Like Turing machine, the model of LSM is based on strict mathematical framework that guarantees under ideal conditions universal computational power [3]. RETINA 256 NEURONS 16 (4×4) PATCHES READOUT 256 NEURONS 16 (4×4) PATCHES LIQUID 16 (4×4) COLUMNS 16×1024 NEURONS Fig. 1. Scheme of Simulated Visual System We investigate the model consisting 16896 neurons (as the Liquid is simulated by ensemble of 16 HHLSM columns). 30% of randomly chosen retinal cells are stimulated and the signal is transformed by the Liquid. As a result we obtain some activity of the Readout device. We simulate 20 ms of biological work for such system. The main objective of the referred research is to check the dependence of simulation time from the number of processors used for simulation and from the percentage number of connections established among neurons of the liquid. As we mentioned the architecture of the network implies that the problem can be most effectively parallelised into sixteen main nodes with one controlling node. Results confirm our expectations (see. Fig. 2). Simulation’s time reaches its minimum when the model parallelised for 17 nodes runs on 17 processors. Note that increasing the number of processors is useless for the algorithm with 17-nodes parallelisation implemented. Model of mammalian visual system Discussed model of mammalian visual system consists three main modules (Fig. 1). The Retina is built on 16×16 square-shaped grid and it is divided into 16 patches (4×4). Each patch is connected with HHLSM column which simulates the Lateral Geniculate Nuclei (LGN) and ensemble of cortical microcircuits. HHLSM consists 1024 cells put on 8×8×16 grid. There are layers arranged in each column. Set of columns simulates the Liquid which is connected with so-called Readout device. The Readout’s architecture is similar to the Retina, in analogy it is divided into 16 patches with 16 cells in each patch. Connections among layers and neurons of each layer are established with some probability i.e. p=10%. All simulations discussed in this paper are conducted in parallel version of GENESIS for MPI environment [6]. Such a model can be easily scaled to multiprocessor simulation. In referred research each column and its corresponding retinal or readout patches should be simulated on one node. Note that in that case we require 16 processors for the best realisation of the model and additional one for control of simulation. However, both the Retina and the Readout may be easily divided into 4 (2×2), 64 (8×8) or 256 (16×16) patches, depending on the number of processors available. Thus, if each patch is connected with a corresponding HHLSM column we will have possibility to conduct a simulation of about 256 thousands Hodgkin-Huxley neural cells. Fig. 2 Time of Simulation as a function of processors’ number Next, 500 ms of the real system’s work was simulated. 30% of retinal cells were stimulated with random, varying in time spike trains. The signal was then transformed by a liquid. As a result some activity appeared on the readout. We investigated whether there is a significant distance of Liquid states evolving in high-dimensional space. Fig. 3 shows the typical Euclidean distance of two liquid states calculated for two different stimulating patterns. Fig. 3. Distance of the Liquid obtained for two different stimulating patterns. One can note some peak of characteristic liquid activity appearing in about first 50 ms of time. Such peaks are also observed in Maass works [1]. That proves that liquid computing appears also in Hodgkin-Huxley neurons and its properties are quite similar to these known from networks of integrate and fire neurons. After the abovementioned time of 50 ms some oscillations still can be observed, however, they are on the same level until the end of simulation. That implies that the network is in two completely different states for two different stimulations. Fig. 3. Distance of the liquid states for two different stimulating patterns. References [1] Maass W., Natschlager T., Markram H.: Real-time Computing Without Stable States: A New Framework for Neural Computation Based on perturbations. Neural Computations, 14(11):2531- 2560. (2002). [2] Kamiński W.A., Wójcik G.M.: Liquid State Machine Built of Hodgkin ‑ Huxley Neurons – Pattern Recognition and Informational Entropy. Annales Informatica UMCS, vol.1, Lublin, (2003). [3] Wójcik G.M., Kamiński W.A.: Liquid State Machine Built of Hodgkin ‑ Huxley Neurons and Pattern Recognition, Neurocomputing vol. 239, (2004). [4] Gupta A., Wang Y., Markram H.: Organizing principles for a diversity of GABAergic interneurons and synapses in the neocortex, Science 287, 273-278. (2000). [5] Hodgkin A., Huxley A.: Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligio. J. Physiol., London 1952. [6] Bower J. M., Beeman D.: The Book of GENESIS – Exploring Realistic Neural Models with the GEneral NEural SImulation System. Telos, New York 1995. [7] CLUSTERIX: http://www.clusterix.pcz.pl