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Nano-Modelling and Computation in Bio and Brain Dynamics
Paolo Di Sia1,2,3
, Ignazio Licata3,4
1 University of Verona, Lungadige Porta Vittoria 17, 37129 Verona, Italy
2 [email protected]
3 ISEM, Institute for Scientific Methodology, Palermo, Italy
4 [email protected]
Abstract: The study of brain dynamics utilizes today the new features of nano-bio-
technology. Neurons are connected by synapses in an electronic way, therefore we have a
charge transport in the brain. Electronic systems have voltages and currents sources and
complex interconnected impedances; the particular behaviour of current generates wanted
voltages transformation with the same impedance. Considering the neural computation as the
study of the classes of trajectories on a manifold geometry defined under appropriate
constraints, different types of brain geometries change the form of straight line and can be
modeled by the electrical power. Locally a new theoretical analytical transport model is able
to provide interesting predictions relatively to an appropriate use of nanomaterials for the
brain. This clearly indicates that nano-modelling in bio and brain contexts requires a recovery
of the concept of natural computation.
Keywords: Neuroscience, Cognitive Science, Nanoscience, Brain, Carrier Transport,
Theoretical Modelling, Neural Geometry, Memristor, Electrical Circuits.
1. Introduction
The great potential of nano-bio-technology is based also on the ability to deal with complex
hierarchically structured systems from the macroscale to the nanoscale; this requires novel
theoretical approaches and the competence to create models able to explain the dynamics at
this scale. Geometric and analytical approaches seem to be very promising in all scientific
areas, like for the study of brain processes.
The comprehension and adaptiveness bring to a change of the internal brain parameters
(conductance of synapses), in order to mimic the external transformation by the appropriate
use of sensors and effectors. The essential mathematical aspects can be illustrated with the use
of a toy model related to Network Resistors with Adaptive Memory (Memristors). Designed
by Chua in 1971 (Chua, 1971, 2013), only in recent years it has been possible to develop
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effective realisations (Tetzlaff, 2013; Thomas, 2013). As known, the memristor is an
electrical circuit with “analogic“ properties, able to vary the resistance after a variation of the
current and therefore to preserve the last state at the interruption of the energy flow. In a toy
model of the brain, this introduces an element of memory which is sufficient to keep into
account of the enormous non-linear complexity of the omeocognitive equilibrium states. This
fact suggests the utility of going back to models of natural computation and therefore of
looking at the Turing computation as a “coarse grain“ of processes which are best suited to be
described with geometric manifolds (Nugent & Molter, 2014; Resconi & Licata, 2015; Licata,
2007, 2012; MacLennan, 2004, 2010).
Technology advancement provides a much finer modelling, new solutions and capabilities for
the active interaction between environment, machines and humans and does not necessarily
scale as per Moore’s law (Alivisatos et al., 2013; Wang, 2010; Hao & Foster, 2008; Kim et
al., 2009).
The dynamics of substances in the brain is based on transport models; the improvement of
them is a mandatoty step for the deep comprehension of the brain functioning. An advance in
analytical modelling, working from sub-pico to macro-level, is able to adequately study the
nano-dynamics in the brain and is giving interesting ideas for future developments.
The learning experiences produce a “chain action” of signaling among a group of neurons in
some areas of the brain, with the modification of connections among neurons in determined
areas of the brain and a consequent reorganization of these areas. Research on brain plasticity
and circuitry also indicates that the brain is always learning, in both formal and informal
contexts.
The following paper aims to highlight some areas of interest for research at the intersection
among natural computing, nanotechnology and brain modelization, and is structured as
follows: after an overview about the nanoscience in the brain (Par. 2), we consider technical
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details of a recently appeared analytical transort model (Par. 3). In Par. 4 examples of
appliaction concerning geometrical image in neural spaces and nano-diffusion are considered,
ending therefore with conclusions (Par. 5).
2. About Nanoscience in the Brain
Chemical communication and key bio-molecular interactions in the brain occur at the
nanoscale, therefore the idea of taking advantages of nanoscience for advances in the study of
the brain structure and function is increasing in interest. In the human brain there are 85
billion neurons and estimated about 100 trillion synapses (Mishra, 2013); experimental brain
nano-techniques three complementary approaches are considered:
1) snapshots of connections in the brain by making thin physical slices and stacking
electron microscopy images of these slices. This technique does not provide dynamic and
chemical information;
2) dynamic voltage map of the brain, dealing with the brain as close relative of a
computer (Azevedo et al., 2009; Alivisatos et al., 2012), with the aim to make accessible the
emergent properties underlying the use and storage of information by mapping network
activity versus single or small numbers of multiple unit recordings currently available;
3) the attempt to obtain functional chemical maps of the neurotransmitter systems in
the brain, for investigating the genetic and chemical heterogeneity of the brain and the
interactions between various neurotransmitter systems.
In all cases, the length scale ranges from the centimeter scale (cm) (in mapping brain regions
and networks), to the micrometer scale (µm) (cells and local connectivity), to the nanometer
scale (nm) (synapses), to the single-molecule scale. The current ability to perform
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neurochemical and electrophysiological measurements needs to be miniaturized, sped up, and
multiplexed. Electrical measurements at time scales of milliseconds are uncomplicated, but
getting to the nanometer scale and making tens of thousands measurements simultaneously in
vivo remains arduous. Obtaining dynamic chemical maps at these scales is an even bigger
challenge, so as problems in analysis, interpretation and visualization of data.
3. Transport Processes at Nano-level: Technical Details
Research at theoretical level always helped science in all sectors. Recently it has been
performed a new analytical model generalizing the Drude-Lorentz model, one of the most
utilized models for transport processes in solid state physics and soft condensed matter (Jones,
2002). The model provides analytical time-dependent expressions of the three most important
quantities related to the transport processes:
a) the velocities correlation function of a system Tvtv >⋅< )0()(rr
at the temperature T,
from which it is possible to obtain the velocity of a carrier at generic time t;
b) the mean squared deviation of position R2(t), defined as
22 )]0()([)( RtRtRrr
−= , from
which the position of a carrier in time is obtainable;
c) the diffusion coefficient D, defined as dt
tdRtD
2
)()(
2
= , which gives important
information about the propagation in time of carriers inside a nanostructure (Di Sia, 2011a,
2011b, 2012a).
With this model it is possible both to fit experimental data, confirming known results, and to
discover “a priori” new characteristics and details with mathematical modelling. The presence
of a gauge factor in the model permits its use from sub-pico-level to macro-level (Di Sia,
2012b, 2013, 2014a, 2014b).
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Starting by the time-dependent perturbation theory, analytical calculation leads to relations for
the velocities correlation function, the mean square deviation of position and the diffusion
coefficient of carriers moving in a nanostructure. The general calculation is performed via
contour integration in the complex plane. With the analytical expression of the velocities
correlation function , the mean square deviation of position:
( ) )0()'(''2)(0
2 vtvttdttR
trr⋅−= ∫ (1)
and the diffusion coefficient:
)0()'(')(
0
vtvdttD
trr⋅= ∫ (2)
allows a complete dynamical study of carriers.
The classical analytical expressions of the diffusion coefficient D are as follows:
⋅
=
I
B
m
TktD
ατ
*)(
+−−
−−
τα
τα tt II
2
)1(exp
2
)1(exp (3)
−
=
ττα
ατ
2exp
2sin2
*
tt
m
TKD R
R
B (4)
The parameters of the model are two real numbers defined in this way:
20
241 ωτα −=I ; (5)
142
02 −= ωταR (6)
with τ and 0ω relaxation time and center frequency respectively, and m* effective mass of the
carrier. The model keeps into account also of quantum (Di Sia, 2012) and relativistic effects
(Di Sia, 2014).
4. Geometrical Image in Neural Spaces and Nano-diffusion: Examples of Application
a) An inertial system is used to describe the ODE types based on standard physical
terminology, which is well defined. An example of non trivial inertial system is the geodesic
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(kinetic energy) for a mechanical rotatory system with the inertial moment Ii,j. In this case the
geodesic is a non inertial geodesic with invariant given by the respective computable
geodesic.
b) For taking into account a minimum of biological plausibility, it is necessary to
introduce a simplified membrane model. In a toy membrane just having, for example, three
potassium channels (twelve gates), nine open gates can be configured into a variety of
topological states with the possible results that none of the channels is open, one is open, or
two are open. Given a vector q in the channels state, we can compute the probability for the
given configuration q of states in the channels. Associating a probability at any configuration,
there exist configurations with very low probability and configurations with high probability.
Given the join probability P, we compute the variation of the probability with respect to the
state qj. Given the current ji , we can compute the flux of states jΦ for the current as a
random variable. Assuming the invariant form 0=+Φ PD jj λ , the flux is controlled by the
probability in an inverse way and it is zero when the probability is a constant value. In
conclusion we have three different powers: W1, the ordinary power for the ionic current
without noise, W1,2, the flux of power from current to the noise current, and W2 as power in
the noise currents.
c) About the Fisher information in neurodynamics, computing the average of the
power as the cost function which value must be minimum, we can consider a parameterized
family of probability distributions S = {P (x, t ,q1, q2 ,..., qn)}, with x and t random variables
and qi real vector parameters specifying the distribution, the family is regarded as a n-
dimensional manifold having q as a coordinate system is a Riemannian manifold and G(q)
play the role of a Fisher information matrix. The underlying idea of information geometry is
to think the family of probability distributions as a space, each distribution being a “point”,
while the parameters q play the role of coordinates. There is a natural unique way for
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measuring the extent to which neighbouring “points” can be distinguished from each other,
which has all wished properties for imposing upon a measure of distance, making therefore
“distinguishability“ the distance. Considering the well known Kullback-Leibler divergence,
with noise equal to zero the Fisher information assumes the maximum value and the geodesic
is equal to the classical geodesic. In the case of noise, the information approaches zero and the
cost function is reduced. Resistor networks provide the natural generalization of the lattice
models for which percolation thresholds and percolation probabilities can be considered. The
geodesic results composed by two parts: the synchronic and crisp geodesic and the noise
change of the crisp and synchronic geodesic (figure 1).
Therefore the electrical neural activity can be represented in manifold state space, where the
minimum path (geodesic) between two points in the multi-space of currents is function of the
neural parameters as resistors with or without memory. Simple cases given by electrical
activity of one little part of the membrane of axons, dendrites or soma, ignoring the presence
of the voltage-gated channels in the membrane, can be done. So the power is comparable to
the Lagrangian in mechanics (Hamilton principle) or the Fermat principle in optics (minimum
time). In the context of Freeman’s neurodynamics, we hypothesize that the minimum
condition in any neural network gives us the meaning of “Intentionality”. A neural network
changes the reference and the neurodynamics in a trajectory with minimum dissipation of
power or geodesic. Therefore any neural network or the equivalent electrical circuit generates
a deformation of the currents space and geodesic trajectories. For every part of the neural
network it is possible to give the similar electrical circuit in this way.
d) In bio-molecules it is easy to understand that the natural computation is the most
suitable than the Turing computation. The proposed nano-bio-approach takes back the
attention toward geometric patterns and attractors. As examples of application, we consider a
nanomaterial of great interest in nanomedicine, the fullerene in tubular form, i.e. carbon
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nanotubes (CNTs), and we study the behaviour of their diffusion using the new proposed
analytical model for transport processes performed in these last years. About the utilized
values, it has been fixed the temperature T = 310 K, three values of the parameter Iα (0.1,
0.5, 0.9), an average relaxation time sav1310−=τ (the relaxation time in soft condensed matter
takes values of order of s1412 1010 −− − ) and two values of m* in relation to the variation of the
chiral vector (n,m) (Marulanda & Srivastava, 2008):
(a) (n,m) = (3,1) → meff = 0.507 me
(b) (n,m) = (7,3) → meff = 0.116 me
(me = mass of the electron = 9.109 × 10–31
kg).
Figures 2, 3 present the variation of the diffusion in time for cases (a) and (b) respectively. It
is significant to underline that the variation of the parameter Iα (or Rα ) implies also a
variation of the shape of diffusion, because of the appearance of Iα (or Rα ) in the arguments
of exponentials of Eq. (3), (4).
About the peak values of diffusion (in cm2/s) evaluable by Figures 2, 3, we have:
CTNs (a): 6.74 - 7.05 - 8.15
CTNs (b): 28.89 – 30.29 - 35.15
5. Conclusions
In this work we have considered two interesting theoretical approaches related to brain
dynamics:
a) the memory in the neural network, as not passive element for storing information,
Memory is integratedd in the neural parameters as synaptic conductances which give the
geodesic trajectories in the non orthogonal space of the free states. The optimal non-linear
dynamics is a geodesic inside the deformed space that directs the neural computation.
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This approach provides the ability to mathematically set up the Freeman hypothesis on the
intentionality as optimal emergency in the “system / environment” relations (Freeman, 1997).
b) A new transport model, which presents analytical expressions of the three most
important parameters related to transport processes, can help at nano-level in relation to nano-
substances injectable in the human body. It holds both for the motion of carriers inside a
nanostructure, as studied in this work, and for the motion of nanoparticles inside the human
body, because it contains a gauge factor, allowing its use from sub-pico-level to macro-level.
The model can be used to understand and manage existing data, but also to make predictions
concerning, as example, the best nanomaterial for use in a particular situation with peculiar
characteristics.
Other possibilities in the direction of a required variation of diffusion regard a variation in
temperature, the variation of the effective mass through the doping and in connection to the
chiral vector, the variation of the parameters , which are functions of the frequency and the
relaxation time. The diffusion rapidity of a nano-substance travelling in the human body is an
important parameter for a fast diagnosis of possible diseases, bringing to a rapid treatment.
We underline that the proposed nano-bio approach to brain directs the attention to geometric
patterns and attractors, a general return to analogic-geometric models, which are allowed by
the fineness of advances in nano-modelling.
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Figures (Paper Di Sia_Licata)
Figure 1: In evidence the geodesic as solution of the ODE. With noise the geodesic is
transformed in a more complex structure which is related to the Fisher information. The
total effect results the percolation random geodesic.
Figure 2: D vs t for CNTs with (n,m) = (3,1). Iα = 0.1 = blue solid line; Iα = 0.5 = red dashed
line; Iα = 0.9 = green dot line.
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Figure 3: D vs t for CNTs with (n,m) = (7,3). Iα = 0.1 = blue solid line; Iα = 0.5 = red dashed
line; Iα = 0.9 = green dot line.
Paolo Di Sia is a Theoretical Physicist, currently Adjunct Professor by the University of Verona (Italy) and
member of ISEM, Institute for Scientific Methodology, Palermo (Italy). Research areas: Classical Quantum
Relativistic Nanophysics, Planck Scale Physics, Supergravity, Quantum Relativistic Information, Nano-Neuro-
Science, Mind Philosophy, Quantum Relativistic Econophysics, Philosophy of Science. He wrote 180 works at
today, is reviewer of 2 mathematics books, reviewer of many international journals, 5 Awards obtained, included
in “Who's Who” in the World 2015 and 2016, member of 5 scientific societies and of 22 International
Advisory/Editorial Boards.
Ignazio Licata is a Theoretical Physicist, Scientific Director of ISEM, Institute for Scientific Methodology,
Palermo (Italy), School of Advanced International Studies on Theoretical and non Linear Methodologies of
Physics, Bari (Italy) and International Institute for Applicable Mathematics & Information Sciences (IIAMIS),
B.M. Birla Science Centre, Adarsh Nagar, Hyderabad 500 (India). Editor of EJTP, Entropy Biosystems and “Il
Nucleare” he is member of CiE, Computability in Europe. Research areas: Foundation of Quantum Mechanics,
Quantum Cosmology , Dissipative Quantum Field Theories, Quantum Information, Physics of Emergence and
Organization. His recent books are ”Quantum Potential. Physics, Geometry, Algebra” (with D. Fiscaletti),
Springer 2013, and “Beyond Peaceful Coexistence.The Emergence of Space, Time and Quantum” (editor), ICP
2015.
