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Grid-based Simulations
of Mammalian Visual System
Grzegorz M. Wójcik and Wiesław A. Kamiński
Maria Curie-Sklodowska University, Lublin, Poland.
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
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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
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.
Time of Simulation [s]
Abstract
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Number of Processors
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.
READOUT
256 NEURONS
16 (4×4) PATCHES
LIQUID
16 (4×4) COLUMNS
16×1024 NEURONS
Fig. 1. Scheme of Simulated Visual System
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. 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):25312560. (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