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Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C2: Dynamical Systems".

Deadline for manuscript submissions: closed (31 May 2023) | Viewed by 25553

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Special Issue Editor

Prof. Dr. Ilya V. Sysoev

E-Mail Website1 Website2
Guest Editor

1. Department of Nano- and Biomedical Technologies, Saratov State University, 410012 Saratov, Russia
2. Saratov Branch of Kotel'nikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, 410019 Saratov, Russia
Interests: mathematical modeling; time series analysis; coupling analysis; model construction; mathematical biology

Special Issue Information

Dear Colleagues,

Understanding brain dynamics and function is one of the most relevant topics nowadays. A large number of experimental results based on the measurement of electromagnetic activity of the brain have been obtained and published in recent years due to advances in measurement techniques and devices. Coupling between brain subsystems and the impact of different cell types and synapses was reported by many groups all over the world and these phenomena need to be summarized as mathematical models since adequate mathematical modeling has been always considered as a significant step in understanding phenomena of nature. Though many models have been already constructed, they are still far from covering most observed phenomena. This Special Issue is devoted to answering the following specific questions:

Models of neurons and synapses, including new models, comparison of existing ones, computational issues, fitting models to data, etc.
Models of brain subsystems, i.e. limbic system, thalamocortical system, etc.
Models of sleep and wakefulness and the transitions between them.
Memory models.
Models of disorders and disease, including epilepsy, Parkinsonism, thalamocortical dysrhythmia, Alzheimer’s disease, etc.
Approaches to model (re)construction from experimental data.

Submissions on any other topics related to the general subject of this Special Issue are also welcome.

Prof. Dr. Ilya V. Sysoev
Guest Editor

Manuscript Submission Information

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Keywords

Biological neuron models
Models of neuron synapses
Brain networks
Models of brain dysfunction
Memory models
Systems identification
Neuro-experimental data
Neuroimaging analysis
Computational neuroscience

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Published Papers (11 papers)

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Editorial

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1 pages, 149 KiB

Open AccessEditorial

Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications

by Ilya V. Sysoev

Mathematics 2023, 11(20), 4304; https://doi.org/10.3390/math11204304 - 16 Oct 2023

Viewed by 1465

Abstract

Mathematical modeling is a necessary step in understanding real world phenomena after experimental data are obtained and analyzed [...] Full article

(This article belongs to the Special Issue Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications)

Research

Jump to: Editorial

13 pages, 3219 KiB

Open AccessArticle

Convolutional Neural Network Outperforms Graph Neural Network on the Spatially Variant Graph Data

by Anna Boronina, Vladimir Maksimenko and Alexander E. Hramov

Mathematics 2023, 11(11), 2515; https://doi.org/10.3390/math11112515 - 30 May 2023

Cited by 3 | Viewed by 2127

Abstract

Applying machine learning algorithms to graph-structured data has garnered significant attention in recent years due to the prevalence of inherent graph structures in real-life datasets. However, the direct application of traditional deep learning algorithms, such as Convolutional Neural Networks (CNNs), is limited as they are designed for regular Euclidean data like 2D grids and 1D sequences. In contrast, graph-structured data are in a non-Euclidean form. Graph Neural Networks (GNNs) are specifically designed to handle non-Euclidean data and make predictions based on connectivity rather than spatial structure. Real-life graph data can be broadly categorized into two types: spatially-invariant graphs, where the link structure between nodes is independent of their spatial positions, and spatially-variant graphs, where node positions provide additional information about the graph’s properties. However, there is limited understanding of the effect of spatial variance on the performance of Graph Neural Networks. In this study, we aim to address this issue by comparing the performance of GNNs and CNNs on spatially-variant and spatially-invariant graph data. In the case of spatially-variant graphs, when represented as adjacency matrices, they can exhibit Euclidean-like spatial structure. Based on this distinction, we hypothesize that CNNs may outperform GNNs when working with spatially-variant graphs, while GNNs may excel on spatially-invariant graphs. To test this hypothesis, we compared the performance of CNNs and GNNs under two scenarios: (i) graphs in the training and test sets had the same connectivity pattern and spatial structure, and (ii) graphs in the training and test sets had the same connectivity pattern but different spatial structures. Our results confirmed that the presence of spatial structure in a graph allows for the effective use of CNNs, which may even outperform GNNs. Thus, our study contributes to the understanding of the effect of spatial graph structure on the performance of machine learning methods and allows for the selection of an appropriate algorithm based on the spatial properties of the real-life graph dataset. Full article

(This article belongs to the Special Issue Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications)

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20 pages, 4363 KiB

Open AccessArticle

Mathematical Modelling of Physiological Effects Caused by a Glycine Receptors Post-Synaptic Density Spatial Polymorphism

by Yaroslav R. Nartsissov and Leonid A. Ivontsin

Mathematics 2023, 11(11), 2499; https://doi.org/10.3390/math11112499 - 29 May 2023

Cited by 2 | Viewed by 1472

Abstract

Synaptic transmission is the main process providing cross-connecting activity among neurons in the central nervous system (CNS). In the present study, the 3D mathematical model of a neuronal bouton with a cluster localization of glycine receptors (GlyRs) on the post-synaptic membrane was developed. The number and eventual position of the receptors are defined by the structural data of the GlyR-gephyrin complex. Furthermore, the forming of inhibitory post-synaptic potential (IPSP) and an electro-diffusion of chloride ions were evaluated by applying the boundary problems for a Poisson’s equation and a non-steady-state diffusion equation, respectively. It was shown that local changes in the chloride ion concentration near the post-synaptic membrane, mediated by GlyRs activation, can raise up to 80–110% from the initial level. The average value of the concentration increase was as high as 10% in a pike of activity under the full activation of GlyRs. The central spatial localization of GlyRs in the cluster had a considerable difference both in the chloride ion concentration changes (6%) and IPSP (17%) compared to the divided or rear localization. Thus, a spatial polymorphism of the post-synaptic density of GlyRs is important to form a physiological response to a neuromediator release. Full article

(This article belongs to the Special Issue Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications)

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11 pages, 2740 KiB

Open AccessArticle

Dynamics in the Reduced Mean-Field Model of Neuron–Glial Interaction

by Sergey M. Olenin, Tatiana A. Levanova and Sergey V. Stasenko

Mathematics 2023, 11(9), 2143; https://doi.org/10.3390/math11092143 - 3 May 2023

Cited by 18 | Viewed by 2244

Abstract

The goal of this study is to propose a new reduced phenomenological model that describes the mean-field dynamics arising from neuron–glial interaction, taking into account short-term synaptic plasticity and recurrent connections in the presence of astrocytic modulation of the synaptic connection. Using computer simulation and numerical methods of nonlinear dynamics, it is shown that the proposed model reproduces a rich set of patterns of population activity, including spiking, bursting and chaotic temporal patterns. These patterns can coexist for specific regions in the parameter space of the model. The main focus of this study was on bifurcation mechanisms that lead to the occurrence of the described types of mean-field dynamics. The proposed phenomenological model can be used to reproduce various patterns of population activity of neurons in a wide range of studies of dynamic memory and information processing. One of the possible applications of such research is the development of new effective methods for the treatment of neurological diseases associated with neuron–glial interactions. Full article

(This article belongs to the Special Issue Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications)

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17 pages, 2636 KiB

Open AccessArticle

Bursting Dynamics of Spiking Neural Network Induced by Active Extracellular Medium

by Sergey V. Stasenko and Victor B. Kazantsev

Mathematics 2023, 11(9), 2109; https://doi.org/10.3390/math11092109 - 28 Apr 2023

Cited by 8 | Viewed by 2598

Abstract

We propose a mathematical model of a spiking neural network (SNN) that interacts with an active extracellular field formed by the brain extracellular matrix (ECM). The SNN exhibits irregular spiking dynamics induced by a constant noise drive. Following neurobiological facts, neuronal firing leads to the production of the ECM that occupies the extracellular space. In turn, active components of the ECM can modulate neuronal signaling and synaptic transmission, for example, through the effect of so-called synaptic scaling. By simulating the model, we discovered that the ECM-mediated regulation of neuronal activity promotes spike grouping into quasi-synchronous population discharges called population bursts. We investigated how model parameters, particularly the strengths of ECM influence on synaptic transmission, may facilitate SNN bursting and increase the degree of neuronal population synchrony. Full article

(This article belongs to the Special Issue Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications)

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9 pages, 370 KiB

Open AccessArticle

Mathematical Model of a Main Rhythm in Limbic Seizures

by Maksim V. Kornilov and Ilya V. Sysoev

Mathematics 2023, 11(5), 1233; https://doi.org/10.3390/math11051233 - 3 Mar 2023

Cited by 2 | Viewed by 1581

Abstract

While synchronization in the brain neural networks has been studied, the emergency of the main oscillation frequency and its evolution at different normal and pathological states remains poorly investigated. We propose a new concept of the formation of a main frequency in limbic epilepsy. The idea is that the main frequency is not a result of the activity of a single cell, but is formed due to collective dynamics in a ring of model neurons connected with delay. The individual cells are in an excitable mode providing no self-oscillations without coupling. We considered the ring of a different number of Hodgkin–Huxley neurons connected with synapses with time delay. We have shown that the proposed circuit can generate oscillatory activity with frequencies close to those experimentally observed. The frequency can be varied by changing the number of model neurons, time delay in synapses and coupling strength. The linear dependence of the oscillation period on both coupling delay and the number of neurons in the ring was hypothesized and checked by means of fitting the values obtained from the numerical experiments to the empirical formula for a constant value of coupling coefficient. Full article

(This article belongs to the Special Issue Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications)

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18 pages, 3238 KiB

Open AccessArticle

A Moving Target Detection Model Inspired by Spatio-Temporal Information Accumulation of Avian Tectal Neurons

by Shuman Huang, Xiaoke Niu, Zhizhong Wang, Gang Liu and Li Shi

Mathematics 2023, 11(5), 1169; https://doi.org/10.3390/math11051169 - 27 Feb 2023

Cited by 3 | Viewed by 2016

Abstract

Moving target detection in cluttered backgrounds is always considered a challenging problem for artificial visual systems, but it is an innate instinct of many animal species, especially the avian. It has been reported that spatio-temporal information accumulation computation may contribute to the high efficiency and sensitivity of avian tectal neurons in detecting moving targets. However, its functional roles for moving target detection are not clear. Here we established a novel computational model for detecting moving targets. The proposed model mainly consists of three layers: retina layer, superficial layers of optic tectum, and intermediate-deep layers of optic tectum; in the last of which motion information would be enhanced by the accumulation process. The validity and reliability of this model were tested on synthetic videos and natural scenes. Compared to EMD, without the process of information accumulation, this model satisfactorily reproduces the characteristics of tectal response. Furthermore, experimental results showed the proposed model has significant improvements over existing models (EMD, DSTMD, and STMD plus) on STNS and RIST datasets. These findings do not only contribute to the understanding of the complicated processing of visual motion in avians, but also further provide a potential solution for detecting moving targets against cluttered environments. Full article

(This article belongs to the Special Issue Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications)

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20 pages, 3827 KiB

Open AccessArticle

Computational Model of Noradrenaline Modulation of Astrocyte Responses to Synaptic Activity

by Andrey Verisokin, Darya Verveyko, Artem Kirsanov, Alexey Brazhe and Dmitry Postnov

Mathematics 2023, 11(3), 628; https://doi.org/10.3390/math11030628 - 26 Jan 2023

Cited by 7 | Viewed by 3474

Abstract

The mathematical modeling of synaptically connected neuronal networks is an established instrument for gaining insights into dynamics of neuronal ensembles and information processing in the nervous system. Recently, calcium signaling in astrocytes—glial cells controlling local tissue metabolism and synapse homeostasis—and its corresponding downstream effect on synaptic plasticity and neuromodulation appeared in the limelight of modeling studies. Here, we used mechanism-based mathematical modeling to disentangle signaling pathways and feedback loops in the astrocytic calcium response to noradrenaline, an important neuromodulator marking periods of heightened alertness and arousal. The proposed model is based on an experiment-based 2D representation of astrocyte morphology, discrete random glutamate synapses with placement probability defined by the morphology pattern, and spatially heterogeneous noradrenaline sources, reflecting axonal varicosities of the adrenergic axons. Both glutamate and noradrenaline drive Ca

^{2 +}

dynamics in the astrocyte in an additive or synergistic manner. Our simulations replicate the global activation of astrocytes by noradrenaline and predict the generation of high-frequency Ca

^{2 +}

waves in a dose-dependent manner and the preferred Ca

^{2 +}

wave origination near noradrenaline release sites if they colocalise with high-density clusters of glutamate synapses. We tested positive feedback loops between noradrenaline release and glutamate spillover directly or mediated by gliotransmitter release from the activated astrocyte. The simulations suggest that the coupled stochastic drive by glutamate and noradrenaline release converges on the graded modulation of the IP

_{3}

level, which is translated into whole-cell Ca

^{2 +}

waves of different frequencies. Thus, the proposed approach is supported by experimental data and can be used to address situations inaccessible directly by experiment, and is a starting point for a more detailed model that includes other signaling mechanisms providing negative feedback. Full article

(This article belongs to the Special Issue Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications)

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12 pages, 17266 KiB

Open AccessArticle

Coexisting Attractors and Multistate Noise-Induced Intermittency in a Cycle Ring of Rulkov Neurons

by Irina A. Bashkirtseva, Alexander N. Pisarchik and Lev B. Ryashko

Mathematics 2023, 11(3), 597; https://doi.org/10.3390/math11030597 - 23 Jan 2023

Cited by 1 | Viewed by 1541

Abstract

We study dynamics of a unidirectional ring of three Rulkov neurons coupled by chemical synapses. We consider both deterministic and stochastic models. In the deterministic case, the neural dynamics transforms from a stable equilibrium into complex oscillatory regimes (periodic or chaotic) when the coupling strength is increased. The coexistence of complete synchronization, phase synchronization, and partial synchronization is observed. In the partial synchronization state either two neurons are synchronized and the third is in antiphase, or more complex combinations of synchronous and asynchronous interaction occur. In the stochastic model, we observe noise-induced destruction of complete synchronization leading to multistate intermittency between synchronous and asynchronous modes. We show that even small noise can transform the system from the regime of regular complete synchronization into the regime of asynchronous chaotic oscillations. Full article

(This article belongs to the Special Issue Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications)

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20 pages, 4153 KiB

Open AccessArticle

Central Nervous System: Overall Considerations Based on Hardware Realization of Digital Spiking Silicon Neurons (DSSNs) and Synaptic Coupling

by Mohammed Balubaid, Osman Taylan, Mustafa Tahsin Yilmaz, Ehsan Eftekhari-Zadeh, Ehsan Nazemi and Mohammed Alamoudi

Mathematics 2022, 10(6), 882; https://doi.org/10.3390/math10060882 - 10 Mar 2022

Cited by 4 | Viewed by 2263

Abstract

The Central Nervous System (CNS) is the part of the nervous system including the brain and spinal cord. The CNS is so named because the brain integrates the received information and influences the activity of different sections of the bodies. The basic elements of this important organ are: neurons, synapses, and glias. Neuronal modeling approach and hardware realization design for the nervous system of the brain is an important issue in the case of reproducing the same biological neuronal behaviors. This work applies a quadratic-based modeling called Digital Spiking Silicon Neuron (DSSN) to propose a modified version of the neuronal model which is capable of imitating the basic behaviors of the original model. The proposed neuron is modeled based on the primary hyperbolic functions, which can be realized in high correlation state with the main model (original one). Really, if the high-cost terms of the original model, and its functions were removed, a low-error and high-performance (in case of frequency and speed-up) new model will be extracted compared to the original model. For testing and validating the new model in hardware state, Xilinx Spartan-3 FPGA board has been considered and used. Hardware results show the high-degree of similarity between the original and proposed models (in terms of neuronal behaviors) and also higher frequency and low-cost condition have been achieved. The implementation results show that the overall saving is more than other papers and also the original model. Moreover, frequency of the proposed neuronal model is about 168 MHz, which is significantly higher than the original model frequency, 63 MHz. Full article

(This article belongs to the Special Issue Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications)

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15 pages, 1736 KiB

Open AccessArticle

Controlling Effects of Astrocyte on Neuron Behavior in Tripartite Synapse Using VHDL–AMS

by Osman Taylan, Mona Abusurrah, Ehsan Eftekhari-Zadeh, Ehsan Nazemi, Farheen Bano and Ali Roshani

Mathematics 2021, 9(21), 2700; https://doi.org/10.3390/math9212700 - 25 Oct 2021

Cited by 3 | Viewed by 2783

Abstract

Astrocyte cells form the largest cell population in the brain and can influence neuron behavior. These cells provide appropriate feedback control in regulating neuronal activities in the Central Nervous System (CNS). This paper presents a set of equations as a model to describe the interactions between neurons and astrocyte. A VHDL–AMS-based tripartite synapse model that includes a pre-synaptic neuron, the synaptic terminal, a post-synaptic neuron, and an astrocyte cell is presented. In this model, the astrocyte acts as a controller module for neurons and can regulates the spiking activity of them. Simulation results show that by regulating the coupling coefficients of astrocytes, spiking frequency of neurons can be reduced and the activity of neuronal cells is modulated. Full article

(This article belongs to the Special Issue Mathematical Modeling of Neurons and Brain Networks: Fundamental Principles and Special Applications)

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