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Scientific Report 2007-2009
Theoretical physics
T15. From Artificial Neural Networks to Neurobiology
Neural networks have at least a double meaning.
In one sense they are algorithms which solve certain
tasks in the other they are models of realistic biological
neurons. Our group works on the two approaches in
this field since many years as well as in the application
of mathematical methods to biology. The great
variety of topics connected with the research on Neural
Networks is the cause of a great spread of different
mathematical tools used in the investigation. Neurons
are very complex biological objects and there are many
different ways to schematize them according to the
aims that one wants to achieve. We describe only few
aims of the many considered by us. One great topic
is sinchronization. In the papers [1] and [3] this
theme was analyzed from very different point of view.
In [1] the neurons are described by a set of first order
linear differential equations with an n × n interaction
matrix with independent and gaussian distributed
N(0, 1/ √ n) random elements. The problem of stability
of the motion for large values of n has been solved in
this paper using the cavity method of the theory of
disordered systems. In this work it is assumed that
many properties of a large system of neurons depend on
the connections more than the biological structure of the
neurons. This strategy involves a lot of mathematical
tools in the theory, mainly nice and intriguing probability
estimates. But this point is rather controversial and
so we have developed also a more biological approach
in the paper [3], where we have modeled the behavior
of the oxytocin neurons of the hypothalamus when they
emit the oxytocin hormone. In this model there are the
ion currents characteristic of these neurons and all the
interactions are through Poisson processes describing
the synaptic inputs. This model cannot be solved
analytically because of the large set of equations with
many Poisson inputs, so it has been solved numerically
with results in good agreement with the measures of
electrical activity of these kind of neurons. These
encouraging results convinced us that the best approach
for finding general properties of large system of neurons
are semi-phenomenological models where inputs are
described by Poisson processes with activity found in
the experiments and currents given by experimental
measures of the patch-clamp type. Another good
example of this approach is given in the paper [4] where
the problem of the control of the movement of the eye
( saccadic movements) is considered. The unexpected
fact of the nature is that the smooth movements of
any part of the body is controlled with the stochastic
firing activity of the neurons! So the search for the
minimimum of the variance of the motion becomes
a problem of the theory of the stochastic control. In
[4] it is shown that the control function can be found
analytically by solving a simple differential equation,
while usually the theory of stochastic optimization ends
with the hard Hamilton-Jacobi equations which usually
cannot be solved analytically. Another interesting
fact is that the control function found in this paper
gives the usual motion and velocity of the saccadic eye
movement! Another interesting theme of our research
has been connected with a pure biological question.
Since we have developed the tools of extreme value
theory of statistics we were able to apply it to biological
questions, reinforcing the conviction that probability
and statistics are the most useful instruments for dealing
with the large complex systems of the biology. Thus in
the paper [2] we applied this nice theory for finding the
the motifs or the place in the precursor of the DNA, the
set of blocks where the transcription factors of proteins
that need to be reproduced bind starting the process
of reproduction. These binding sites are the sites
with maximal probability of binding. In the current
literature the probability distribution of these sites was
considered to be gaussian and so the distribution of the
maximum was assumed to be a Gumbel distribution
while we discovered using the statistic tools that the
distribution of the maximum was a Weibull. This result
brought to the identification of new binding sites and
also to the distribution of couple of binding sites.
References
1. J.F. Feng et al., Comm. Pure Appl. Anal., 7, 249 (2008).
2. D. Bianchi et al., Europhys. Lett., 84, (2008).
3. E. Rossoni et al., PLOS Comp. Biol., 4, e1000123 (2008).
4. J.F. Feng et al., Math. Comp. Modelling, 46, 680 (2007).
Authors
B. Tirozzi, D. Bianchi
http://pamina.phys.uniroma1.it/
Sapienza Università di Roma 38 Dipartimento di Fisica