Mathematical Modeling for a Better Understanding of Neural Complexity

Background

Understanding the complex dynamics of the nervous system represents one of the fundamental challenges to resolving several research questions in neuroscience.

Compared to other research fields, the latter received great attention recently due to the unprecedented increase in the incidence and prevalence of neurological disorders. Given the intricate nature of these disorders, it is crucial to adopt multidisciplinary techniques capable of addressing specific research questions to create meaningful answers.

During the last 15 years, the integration of mathematical modeling in neuroscience has had impactful outcomes as it provides quantitative frameworks to most of the complex networks in the brain (Figure 1).

This field has expanded significantly and shaped the future direction for identifying more targeted diagnostic and therapeutic potentials for neurological disorders. Its diversified nature allows the designing of comprehensive research projects in neural modeling including single neuron and network models.

In addition, analyzing functional imaging data (e.g. fMRI, EEG, and MEG) using mathematical models has leveraged the outcomes and enhanced the interpretation of complex mechanisms within the nervous system.

Designs and protocols

A wide range of advanced techniques can be used in mathematical modeling within the neuroscience field. This includes numerical stimulation which is required to solve differential equations and to monitor the behavior of the system over time using finite difference, finite element, and Monte Carlo methods (Figure 2).

In addition, optimization techniques like gradient descent and genetic algorithms help in identifying parameter values that best match predictive models. Moreover, linear algebra methods such as matrices, and eigenvectors are considered fundamental techniques for the analysis of connectivity and exploring the stability in neural networks.

Furthermore, the study of the spatial organization of neurons within the brain requires a set of specialized and advanced techniques, and computational geometry represents an appropriate technique to model the spatial aspects of the neuronal structural networks (Figure 2).

Selected free full text articles

• Modirshanechi A, Kondrakiewicz K, Gerstner W, Haesler S. Curiosity-driven exploration: foundations in neuroscience and computational modeling. Trends Neurosci. 2023 Dec;46(12):1054-1066. doi: 10.1016/j.tins.2023.10.002. Epub 2023 Nov 2. PMID: 37925342. https://pubmed.ncbi.nlm.nih.gov/37925342/
  
  
• Finotelli P, Eustache F. Mathematical modeling of human memory. Front Psychol. 2023 Dec 22;14:1298235. doi: 10.3389/fpsyg.2023.1298235. PMID: 38187417; PMCID: PMC10771340. https://pubmed.ncbi.nlm.nih.gov/38187417/
  
  
• Sebastian A, Forstmann BU, Matzke D. Towards a model-based cognitive neuroscience of stopping – a neuroimaging perspective. Neurosci Biobehav Rev. 2018 Jul;90:130-136. doi: 10.1016/j.neubiorev.2018.04.011. Epub 2018 Apr 13. PMID: 29660415. https://pubmed.ncbi.nlm.nih.gov/29660415/
  
  
• Bassett DS, Sporns O. Network neuroscience. Nat Neurosci. 2017 Feb 23;20(3):353-364. doi: 10.1038/nn.4502. PMID: 28230844; PMCID: PMC5485642. https://pubmed.ncbi.nlm.nih.gov/28230844/
  
  
• Tewari SG, Gottipati MK, Parpura V. Mathematical Modeling in Neuroscience: Neuronal Activity and Its Modulation by Astrocytes. Front Integr Neurosci. 2016 Feb 4;10:3. doi: 10.3389/fnint.2016.00003. PMID: 26869893; PMCID: PMC4740383. https://pubmed.ncbi.nlm.nih.gov/26869893/