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The study of brain dynamics currently utilizes the new features of nanobiotechnology and bioengineering. New geometric and analytical approaches appear very promising in all scientific areas, particularly in the study of brain processes. Efforts to engage in deep comprehension lead to a change in the inner brain parameters, in order to mimic the external transformation by the proper use of sensors and effectors. This paper highlights some crossing research areas of natural computing, nanotechnology, and brain modeling and considers two interesting theoretical approaches related to brain dynamics: (a) the memory in neural network, not as a passive element for storing information, but integrated in the neural parameters as synaptic conductances; and (b) a new transport model based on analytical expressions of the most important transport parameters, which works from sub-pico-level to macro-level, able both to understand existing data and to give new predictions. Complex biological systems are highly dependent on the context, which suggests a “more nature-oriented” computational philosophy.
The great potential of nanobiotechnology is based on the ability to deal with complex hierarchically structured systems from the macroscale to the nanoscale ; it requires novel theoretical approaches and the competence of creating models, able to explain the dynamics at such a scale [2,3,4,5]. Geometric and analytical approaches seem to be very promising in all scientific areas, including the study of brain processes.
Deep comprehension and adaptiveness cause a change in the inner brain parameters (conductance of synapses), in order to mimic the outer transformation by the appropriate use of sensors and effectors. The basic 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 , only in recent years it has been possible to develop effective realizations [7,8]. The memristor is an electrical circuit with “analogic” properties, able to vary the resistance after a variation in the current and 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 that takes into account the enormous non-linear complexity of the homeo-cognitive equilibrium states. This promotes 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 described by geometric manifolds [9,10,11,12,13].
Technology advancement provides a finer modeling, new solutions, and a capability of active interaction between the environment, machines, and humans and the possibility of not necessarily scaling, as per Moore’s law [14,15,16,17].
Dynamics in the brain are based on transport models; their improvement is a mandatory step in deep comprehension of brain functioning. Advances in analytical modeling can adequately study the nano-dynamics in the brain and lead to interesting ideas for future developments.
Learning experiences produce a “chain action” of signaling among neurons in some areas of the brain, with the modification of neuron connections in particular brain areas, resulting in reorganization. Research on brain plasticity and circuitry also indicates that the brain is always learning, in both formal and informal contexts.
Natural computing refers to observed computational processes and “human-designed/inspired-by-nature” computing. Analyzing complex natural phenomena in terms of computational processes, we reinforce the understanding of both nature and essence of computation. Peculiar to this kind of approach is the use of concepts, principles, and mechanisms underlying natural systems.
This paper aims to highlight some areas of interest for research, combining natural computing, nanotechnology, and brain modelization. It is structured as follows: after an overview of the nanoscience in the brain (Section 2), we consider the technical details of a recently appeared analytical transport model (Section 3). In Section 4, examples of application concerning geometrical images in neural spaces and nano-diffusion together with results are considered, which is followed by conclusions (Section 5).
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 brain structure and function is becoming increasingly popular. In the human brain, there are 85 billion neurons and an estimated 100 trillion synapses ; as experimental “nano-brain” techniques, we remember:
In all these 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 in performing neurochemical and electrophysiological measurements needs to be miniaturized, sped up, and multiplexed. Electrical measurements at time scales of milliseconds are not complicated, but getting to the nanometer scale and simultaneously making tens of thousands in vivo measurements is very difficult. Obtaining dynamic chemical maps at this scale is a bigger challenge, as there are problems in analysis, interpretation, and visualization of data.
Research at the theoretical level help science in all sectors. Recently, a new analytical model that generalizes the Drude-Lorentz and Smith models for transport processes in solid-state physics and soft condensed matter has been used [21,22,23]. It provides analytical time-dependent expressions of the three most important parameters related to transport processes:
With this model, it is possible both to fit experimental data, confirming known results, and to discover new features and details. The presence of a gauge factor inside the model allows its use from sub-pico-level to macro-level [25,26].
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. Analytical expressions of the velocities correlation function , the mean square deviation of position:
and the diffusion coefficient:
allow a complete dynamical study of carriers.
The classical analytical expressions of the diffusion coefficient D are as follows:
Therefore, the electric neural activity can be represented in the manifold state space, where the minimum path (geodesic) between two points in the multi-space of currents is a function of the neural parameters as resistors, with or without memory. Simple cases given by electric activity of a little part of the membrane of axons, dendrites, or soma, thus ignoring the presence of the voltage-gated channels in the membrane, can be done. 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 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 electric circuit, generates a deformation of the current space and geodesic trajectories. For every part of the neural network, it is possible to give a similar electric circuit in this way.
Figure 2 and Figure 3 show the variation of the diffusion vs. time for cases (a) and (b) respectively. It is important to emphasize that the variation of the parameter (or ) also implies a variation of the shape of diffusion because of the appearance of (or ) in the arguments of exponentials of Equations (3) and (4).
CTNs (a): 6.74 – 7.05 – 8.15;
CTNs (b): 28.89 – 30.29 – 35.15
The considered examples clearly show:
In this work we have considered two interesting theoretical approaches related to brain dynamics:
Other possibilities in the direction of a required variation of diffusion concern variations in temperature, variations of the effective mass through doping and chiral vector, the variation of the parameters and which are functions of the frequency and the relaxation time. The diffusivity of a nano-substance traveling in the human body is an important parameter for a fast diagnosis of possible diseases, potentially leading to a rapid treatment.
We emphasize that the proposed nanobio approach to the brain directs attention to geometric patterns and attractors, which is a general return to analogic-geometric models, made possible by the fine advances in nano-modeling. Despite the simulations using Turing computation, it is clear that the complex biological systems are highly dependent on the context, which suggests a computationally philosophy, more oriented to natural computation [29,30].
These authors contributed equally to this work.
The authors declare no conflict of interest.