Mathematica Eterna
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Optical Intensity Differentiator Differential Equations

Constantinos Siettos^{* *}Correspondence: Constantinos Siettos, School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Athens, Greece, Email:

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We propose associated through AN experiment demonstrate AN all-optical differentiator-based computation system used for determination constant-coefficient first-order linear traditional differential equations. It consists of associate all-optical intensity person and a wavelength device, every supported a semiconductor optical equipment (SOA) associated AN optical filter (OF). The equation is resolved for varied values of the constant-coefficient and a pair of thought-about input waveforms, namely, super- Gaussian and scientist signals. An outstanding agreement between the numerical simulation and conjointly the experimental results is obtained [1].

The extreme complexity of the brain naturally desires mathematical modeling approaches on AN outsized style of scales; the spectrum ranges from single somatic cell dynamics over the behavior of groups of neurons to neural network activity. Thus, the affiliation between the microscopic scale (single somatic cell activity) to gross behavior (emergent behavior of the collective dynamics) and also the different means around could also be a key to grasp the brain in its complexity. throughout this work, we've an inclination to do a review of an oversized vary of approaches, ranging from the modeling of single somatic cell dynamics to machine learning. The models embrace biophysical moreover as data-driven phenomenological models. The mentioned models embrace Hodgkin-Huxley, FitzHugh-Nagumo, coupled oscillators (Kuramoto oscillators, Rössler oscillators, and conjointly the Hindmarsh-Rose neuron), Integrate and fireside, networks of neurons, and neural field equations. In addition to the mathematical models, necessary mathematical ways in multiscale modeling and reconstruction of the borne in upon property ar sketched [2].

Several mathematical approaches to learning analytically the dynamics of neural networks settle for mean-field approximations, that ar strictly applicable exclusively to networks of infinite size. However, all existing real biological networks have finite size, and much of of them, like microscopic circuits in invertebrates, ar composed exclusively of some tens of neurons. Thus, it is necessary to be able to bit small-size networks our ability to see analytically neural dynamics. Analytical solutions of the dynamics of smallsize neural networks have remained elusive for many decades, as a results of the powerful ways of maths analysis, just like the central limit theorem and conjointly the law of giant numbers, do not apply to very little networks. Throughout this text, we've an inclination to critically review recent progress on the study of the dynamics of very little networks composed of binary neurons. specifically, we've an inclination to review the mathematical techniques we've an inclination to developed for learning the bifurcations of the network dynamics, the philosophical system between neural activity and membrane potentials, cross-neuron correlations, and pattern storage in random networks. Then, we've an inclination to check our results with existing mathematical techniques for learning networks composed of a finite kind of neurons. Finally, we've an inclination to focus on key challenges that keep open, future directions for added progress, and potential implications of our results for biological science [3].

We construct embedded purposeful property networks (FCN) from benchmark resting-state purposeful resonance imaging (rsfMRI) data learned from patients with psychopathy and healthy controls supported linear and nonlinear manifold learning algorithms, namely, multidimensional Scaling, Isometric Feature Mapping, Diffusion Maps, regionally Linear Embedding and kernel PCA. Moreover, supported key international graph-theoretic properties of the embedded FCN, we've an inclination to check their classification potential victimization machine learning. We’ve an inclination to together assess the performance of two metrics that ar wide used for the event of FCN from resonance imaging, notably the mathematician distance and conjointly the cross correlation metric. We’ve an inclination to indicate that diffusion maps with the cross correlation metric crush the other combos [4].

References

1. Constantinos S, Jens S. Multiscale modeling of brain dynamics: from single neurons and networks to mathematical tools. Wiley Interdiscip Rev Syst Biol Med. 2016;8(5):438-58.
2. Diego F, Stefano P. Mathematical studies of the dynamics of finite-size binary neural networks: A review of recent progress. Math Biosci Eng. 2019;16(6):8025-8059.
3. Ioannis KG, Evangelos G, Constantinos IS. Construction of embedded fMRI resting-state functional connectivity networks using manifold learning. Cogn Neurodyn. 2021;15(4):585-608.
4. Sisi T, Zhao W, Lei L, Shoujin H, Jianji D, Xinliang Z. All-optical computation system for solving differential equations based on optical intensity differentiator. Opt Express. 2013;21(6):7008-13.