The heterogeneity of neuronal networks plays a crucial role in shaping emergent dynamics. In this work, we introduced a kinetic modeling approach to describe the activity of heterogeneous neuronal networks through transition probabilities and adjacency matrices. The model explicitly accounts for both structural and functional heterogeneity by considering two interacting neuronal populations—excitatory pyramidal neurons and inhibitory interneurons—distributed across network slices. The transition probabilities encode the binary stochastic interactions between neurons, capturing both the neuronal types involved (excitatory or inhibitory) and the connectivity structure within and between slices. Complementarily, adjacency matrices define the weighted connections among neurons, specifying the structural organization of each slice and the interactions across slices. Together, these two components characterize the functional and the structural heterogeneity of the system. From this framework, we derived a system of nonlinear ordinary differential equations describing the mesoscopic dynamics of the network. First, for the one-slice model, we provided analytical results on the existence and stability of equilibrium states. Then, we presented numerical simulations for two- and four-slice networks to investigate the role of functional and structural heterogeneity. In particular, after defining the excitatory-, inhibitory-, and balanced count regimes and introducing an a priori criterion for their identification, we demonstrated how heterogeneity influences both the short- and long-term dynamics of the network. Our findings revealed that increasing heterogeneity not only alters the proportion of active neurons but also induces more complex dynamical behaviors, potentially driving shifts between excitatory-count- and inhibitory-count-dominated regimes.
Citation: M. Menale, C. Tribuzi, R. Shah, C. A. Lupascu, A. Marasco. Kinetic modeling approach for a heterogeneous neuronal network activity using adjacency matrices[J]. Networks and Heterogeneous Media, 2025, 20(4): 1292-1332. doi: 10.3934/nhm.2025056
The heterogeneity of neuronal networks plays a crucial role in shaping emergent dynamics. In this work, we introduced a kinetic modeling approach to describe the activity of heterogeneous neuronal networks through transition probabilities and adjacency matrices. The model explicitly accounts for both structural and functional heterogeneity by considering two interacting neuronal populations—excitatory pyramidal neurons and inhibitory interneurons—distributed across network slices. The transition probabilities encode the binary stochastic interactions between neurons, capturing both the neuronal types involved (excitatory or inhibitory) and the connectivity structure within and between slices. Complementarily, adjacency matrices define the weighted connections among neurons, specifying the structural organization of each slice and the interactions across slices. Together, these two components characterize the functional and the structural heterogeneity of the system. From this framework, we derived a system of nonlinear ordinary differential equations describing the mesoscopic dynamics of the network. First, for the one-slice model, we provided analytical results on the existence and stability of equilibrium states. Then, we presented numerical simulations for two- and four-slice networks to investigate the role of functional and structural heterogeneity. In particular, after defining the excitatory-, inhibitory-, and balanced count regimes and introducing an a priori criterion for their identification, we demonstrated how heterogeneity influences both the short- and long-term dynamics of the network. Our findings revealed that increasing heterogeneity not only alters the proportion of active neurons but also induces more complex dynamical behaviors, potentially driving shifts between excitatory-count- and inhibitory-count-dominated regimes.
| [1] |
S. Dura-Bernal, B. Herrera, C. Lupascu, B. M. Marsh, D. Gandolfi, A. Marasco, et al., Large-scale mechanistic models of brain circuits with biophysically and morphologically detailed neurons, J. Neurosci., 44 (2024), e1236242024. https://doi.org/10.1523/JNEUROSCI.1236-24.2024 doi: 10.1523/JNEUROSCI.1236-24.2024
|
| [2] |
D. Golomb, C. Yue, Y. Yaari, Contribution of persistent Na+ current and M-type K+ current to somatic bursting in CA1 pyramidal cells: combined experimental and modeling study, J. Neurophysiol., 96 (2006), 1912–1926. https://doi.org/10.1152/jn.00205.2006 doi: 10.1152/jn.00205.2006
|
| [3] |
D. Bianchi, A. Marasco, A. Limongiello, C. Marchetti, H. Marie, B. Tirozzi, et al., On the mechanisms underlying the depolarization block in the spiking dynamics of CA1 pyramidal neurons, J. Comput. Neurosci., 33 (2012), 207–225. https://doi.org/10.1007/s10827-012-0383-y doi: 10.1007/s10827-012-0383-y
|
| [4] |
R. Migliore, C. A. Lupascu, L. L. Bologna, A. Romani, J. Courcol, S. Antonel, et al., The physiological variability of channel density in hippocampal CA1 pyramidal cells and interneurons explored using a unified data-driven modeling workflow, PLoS Comput. Biol., 14 (2018), e1006423. https://doi.org/10.1371/journal.pcbi.1006423 doi: 10.1371/journal.pcbi.1006423
|
| [5] | D. Cai, L. Tao, A. V. Rangan, D. W. McLaughlin, Kinetic theory for neuronal network dynamics, Commun. Math. Sci., 4 (2006), 97–127. |
| [6] |
A. V. Rangan, G. Kovačič, D. Cai, Kinetic theory for neuronal networks with fast and slow excitatory conductances driven by the same spike train, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 77 (2008), 041915. https://doi.org/10.1103/PhysRevE.77.041915 doi: 10.1103/PhysRevE.77.041915
|
| [7] |
A. Ferrara, D. Angulo-Garcia, A. Torcini, S. Olmi, Population spiking and bursting in next-generation neural masses with spike-frequency adaptation, Phys. Rev. E, 107 (2023), 024311. https://doi.org/10.1103/PhysRevE.107.024311 doi: 10.1103/PhysRevE.107.024311
|
| [8] |
E. Montbrió, D. Pazó, A.Roxin, Macroscopic description for networks of spiking neurons, Phys. Rev. X, 5 (2015), 021028. https://doi.org/10.1103/PhysRevX.5.021028 doi: 10.1103/PhysRevX.5.021028
|
| [9] |
G. Kovačič, L. Tao, A. V Rangan, D. Cai, Fokker-Planck description of conductance-based integrate-and-fire neuronal networks, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys., 80 (2009), 021904. https://doi.org/10.1103/PhysRevE.80.021904 doi: 10.1103/PhysRevE.80.021904
|
| [10] |
L. Chen, S. A. Campbell, Exact mean-field models for spiking neural networks with adaptation, J. Comput. Neurosci., 50 (2022), 445–469. https://doi.org/10.1007/s10827-022-00825-9 doi: 10.1007/s10827-022-00825-9
|
| [11] |
J. D. Kopsick, C. Tecuatl, K. Moradi, S. M. Attili, H. J. Kashyap, J. Xing, et al., Robust resting-state dynamics in a large-scale spiking neural network model of area CA3 in the mouse hippocampus, Cognit. Comput., 15 (2023), 1190–1210. https://doi.org/10.1007/s12559-021-09954-2 doi: 10.1007/s12559-021-09954-2
|
| [12] |
A. Marasco, C. Tribuzi, A. Iuorio, M. Migliore, Mathematical generation of data-driven hippocampal CA1 pyramidal neurons and interneurons copies via A-GLIF models for large-scale networks covering the experimental variability range, Math. Biosci., 371 (2024a), 109179. https://doi.org/10.1016/j.mbs.2024.109179 doi: 10.1016/j.mbs.2024.109179
|
| [13] |
A. Marasco, C. Tribuzi, C. A. Lupascu, M. Migliore, Modeling realistic synaptic inputs of CA1 hippocampal pyramidal neurons and interneurons via adaptive generalized leaky integrate-and-fire models, Math. Biosci., 372 (2024b), 109192. https://doi.org/10.1016/j.mbs.2024.109192 doi: 10.1016/j.mbs.2024.109192
|
| [14] | E. Spera, Intrinsic network dynamics in a full-scale point-neuron model of the CA1 hippocampal microcircuit incorporating excitability heterogeneity through multiple realistic A-GLIF instantiations, in preparation. |
| [15] |
N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons, J. Comput. Neurosci., 8 (2000), 183–208. https://doi.org/10.1023/a:1008925309027 doi: 10.1023/a:1008925309027
|
| [16] |
Q. L. Gu, S. Li, W. P. Dai, D. Zhou, D. Cai, Balanced active core in heterogeneous neuronal networks, Front. Comput. Neurosci., 12 (2019), 109. https://doi.org/10.3389/fncom.2018.00109. doi: 10.3389/fncom.2018.00109
|
| [17] |
C. van Vreeswijk, H. Sompolinsky, Chaotic balanced state in a model of cortical circuits, Neural Comput., 10 (1998), 1321–1371. https://doi.org/10.1162/089976698300017214 doi: 10.1162/089976698300017214
|
| [18] |
Z. C. Xiao, K. K Lin, L. S. Young, A data-informed mean-field approach to mapping of cortical parameter landscapes, PLOS Comput. Biol., 17 (2021), e1009718. https://doi.org/10.1371/journal.pcbi.1009718 doi: 10.1371/journal.pcbi.1009718
|
| [19] |
Y. Zerlaut, S. Chemla, F. Chavane, A. Destexhe, Modeling mesoscopic cortical dynamics using a mean-field model of conductance-based networks of adaptive exponential integrate-and-fire neurons, J. Comput. Neurosci., 44 (2018), 45–61. https://doi.org/10.1007/s10827-017-0668-2 doi: 10.1007/s10827-017-0668-2
|
| [20] |
R. Gast, S. A. Solla, A. Kennedy, Neural heterogeneity controls computations in spiking neural networks, Proc. Natl. Acad. Sci., 121 (2024), e2311885121. https://doi.org/10.1073/pnas.2311885121 doi: 10.1073/pnas.2311885121
|
| [21] |
R. Gast, H. Schmidt, T. R Knösche, A mean-field description of bursting dynamics in spiking neural networks with short-term adaptation, Neural Comput., 32 (2020), 1615–1634. https://doi.org/10.1162/neco_a_01300 doi: 10.1162/neco_a_01300
|
| [22] |
G. Gigante, M. Mattia, P. D. Giudice, Diverse population-bursting modes of adapting spiking neurons, Phys. Rev. Lett., 98 (2007), 148101. https://doi.org/10.1103/PhysRevLett.98.148101 doi: 10.1103/PhysRevLett.98.148101
|
| [23] |
W. Nicola, S. A. Campbell, Mean-field models for heterogeneous networks of two-dimensional integrate and fire neurons, Front. Comput. Neurosci., 7 (2013), 184. https://doi.org/10.3389/fncom.2013.00184 doi: 10.3389/fncom.2013.00184
|
| [24] |
M. Khanjanianpak, N. Azimi-Tafreshi, A. Valizadeh, Emergence of complex oscillatory dynamics in the neuronal networks with long activity time of inhibitory synapses, iScience, 27 (2024a), 109401. https://doi.org/10.1016/j.isci.2024.109401 doi: 10.1016/j.isci.2024.109401
|
| [25] |
T. Sasaki, N. Matsuki, Y. Ikegaya, Metastability of active CA3 networks, J. Neurosci., 27 (2007), 517–528. https://doi.org/10.1523/JNEUROSCI.4514-06.2007 doi: 10.1523/JNEUROSCI.4514-06.2007
|
| [26] |
J. Pretel, V. Buendía, J. J. Torres, M. A. Muñoz, From asynchronous states to Griffiths phases and back: Structural heterogeneity and homeostasis in excitatory-inhibitory networks, Phys. Rev. Res., 6 (2024), 023018. https://doi.org/10.1103/PhysRevResearch.6.023018 doi: 10.1103/PhysRevResearch.6.023018
|
| [27] |
R. C. López, V. Buendía, M. A. Muñoz, Excitatory-inhibitory branching process: A parsimonious view of cortical asynchronous states, excitability, and criticality, Phys. Rev. Res., 4 (2022), L042027. https://doi.org/10.1103/PhysRevResearch.4.L042027 doi: 10.1103/PhysRevResearch.4.L042027
|
| [28] |
T. Murakami, Spatial dynamics of spontaneous activity in the developing and adult cortices, Neurosci. Res., 212 (2025), 1–10. https://doi.org/10.1016/j.neures.2024.12.002 doi: 10.1016/j.neures.2024.12.002
|
| [29] |
D. B. Larremore, W. L. Shew, E. Ott, F. Sorrentino, J. G. Restrepo, Inhibition causes ceaseless dynamics in networks of excitable nodes, Phys. Rev. Lett., 112 (2014a) 138103. https://doi.org/10.1103/PhysRevLett.112.138103 doi: 10.1103/PhysRevLett.112.138103
|
| [30] | M. Di Volo, A. Destexhe, Optimal responsiveness and information flow in networks of heterogeneous neurons, Sci. Rep., 11 (2022), 1–11. |
| [31] |
C. G. Fink, Simulating synchronization in neuronal networks, Am. J.Phys., 84 (2016), 467–473. https://doi.org/10.1119/1.4945009 doi: 10.1119/1.4945009
|
| [32] | D. J. Watts, S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440–442. |
| [33] | E. Bullmore, O. Sporns, Complex brain networks: Graph theoretical analysis of structural and functional systems, Nat. Rev. Neurosci., 10 (2009), 186–198. |
| [34] |
F. V. Farahani, W. Karwowski, N. R. Lighthall, Application of graph theory for identifying connectivity patterns in human brain networks: A systematic review, Front. Neurosci., 13 (2019), 439505. https://doi.org/10.3389/fnins.2019.00585 doi: 10.3389/fnins.2019.00585
|
| [35] |
O. Sporns, Graph theory methods: Applications in brain networks, Dialogues Clin. Neurosci., 20 (2018), 111–121. https://doi.org/10.31887/DCNS.2018.20.2/osporns doi: 10.31887/DCNS.2018.20.2/osporns
|
| [36] |
A. L. Barabási, R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509–512. https://doi.org/10.1126/science.286.5439.509 doi: 10.1126/science.286.5439.509
|
| [37] |
M. Khanjanianpak, A. Valizadeh, Optimizing information capacity in modular neural networks through excitatory and inhibitory connectivity, Neurocomputing, 640 (2025), 130313. https://doi.org/10.1016/j.neucom.2025.130313 doi: 10.1016/j.neucom.2025.130313
|
| [38] | J. Banasiak, A. Błoch, Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability, Networks Heterogen. Media, 17 (2022), 73–99. |
| [39] | A. Barrat, M. Barthelemy, A. Vespignani, Dynamical Processes on Complex Networks, Cambridge University Press, 2008. |
| [40] |
J. Gao, N. Yang, Some bounds on the largest eigenvalue of degree-based weighted adjacency matrix of a graph, Discrete App. Math., 356 (2024), 21–31. https://doi.org/10.1016/j.dam.2024.05.011 doi: 10.1016/j.dam.2024.05.011
|
| [41] |
N. Bellomo, D. Burini, N. Outada, Multiscale models of Covid-19 with mutations and variants, Networks & Heterogen. Media, 17 (2022), 293–310. https://doi.org/10.3934/nhm.2022008 doi: 10.3934/nhm.2022008
|
| [42] |
N. Bellomo, D. Burini, G. Dosi, L. Gibelli, D. Knopoff, N. Outada, et al., What is life? A perspective of the mathematical kinetic theory of active particles, Math. Models Methods Appl. Sci., 31 (2021), 1821–1866. https://doi.org/10.1142/S0218202521500408 doi: 10.1142/S0218202521500408
|
| [43] |
N. Bellomo, A. Bellouquid, J. Nieto, J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems, Math. Models Methods Appl. Sci., 20 (2010), 1179–1207. https://doi.org/10.1142/S0218202510004568 doi: 10.1142/S0218202510004568
|
| [44] |
A. Bellouquid, M. Delitala, Mathematical methods and tools of kinetic theory towards modelling complex biological systems, Math. Models Methods Appl. Sci., 15 (2005), 1639–1666. https://doi.org/10.1142/S0218202505000923 doi: 10.1142/S0218202505000923
|
| [45] |
M. J. Cáceres, J. A. Carrillo, L. Tao, A numerical solver for a nonlinear Fokker–Planck equation representation of neuronal network dynamics, J.Comput. Phys., 230 (2011), 1084–1099. https://doi.org/10.1016/j.jcp.2010.10.027 doi: 10.1016/j.jcp.2010.10.027
|
| [46] |
B. Perthame, D. Salort, Derivation of a voltage density equation from a voltage-conductance kinetic model for networks of integrate-and-fire neurons, Commun. Math. Sci., 17 (2019), 1193–1211. https://doi.org/10.4310/CMS.2019.v17.n5.a2 doi: 10.4310/CMS.2019.v17.n5.a2
|
| [47] | K. D. Harris, G. M. G. Shepherd, The neocortical circuit: Themes and variations, Nature Neurosci., 18 (2015), 170–181. |
| [48] |
A. M. Bastos, W. M. Usrey, R. A. Adams, G. R. Mangun, P. Fries, K. J. Friston, Canonical microcircuits for predictive coding, Neuron, 76 (2012), 695–711. https://doi.org/10.1016/j.neuron.2012.10.038 doi: 10.1016/j.neuron.2012.10.038
|
| [49] |
A. H. Gittis, A. C Kreitzer, Striatal microcircuitry and movement disorders, Trends Neurosci., 35 (2012), 557–564. https://doi.org/10.1016/j.tins.2012.06.008 doi: 10.1016/j.tins.2012.06.008
|
| [50] |
B. Hangya, H. J. Pi, D. Kvitsiani, S. P. Ranade, A. Kepecs, From circuit motifs to computations: Mapping the behavioral repertoire of cortical interneurons, Curr. Opin. Neurobiol., 26 (2014), 117–124. https://doi.org/10.1016/j.conb.2014.01.007 doi: 10.1016/j.conb.2014.01.007
|
| [51] |
M. Khona, I. R. Fiete, Attractor and integrator networks in the brain, Nature Rev. Neurosci., 23 (2022), 744–766. https://doi.org/10.1038/s41583-022-00642-0 doi: 10.1038/s41583-022-00642-0
|
| [52] |
D. Gandolfi, J. Mapelli, S. Solinas, R. De Schepper, A. Geminiani, C. Casellato, et al., A realistic morpho-anatomical connection strategy for modelling full-scale point-neuron microcircuits, Sci. Rep., 12 (2022), 246–276. https://doi.org/10.1038/s41598-022-18024-y doi: 10.1038/s41598-022-18024-y
|
| [53] |
M. L. Schiavo, Discrete kinetic cellular models of tumors immune system interactions, Math. Models Methods Appl. Sci., 6 (1996), 1187–1209. https://doi.org/10.1142/S021820259600050X doi: 10.1142/S021820259600050X
|
| [54] |
M. L. Bertotti, M. Delitala, On a discrete generalized kinetic approach for modelling persuader's influence in opinion formation processes, Math. Comput. Modell., 48 (2008), 1107–1121. https://doi.org/10.1016/j.mcm.2007.12.021 doi: 10.1016/j.mcm.2007.12.021
|
| [55] |
A. Bellouquid, E. De Angelis, From kinetic models of multicellular growing systems to macroscopic biological tissue models, Nonlinear Anal. Real World Appl., 12 (2011), 1111–1122. https://doi.org/10.1016/j.nonrwa.2010.09.005 doi: 10.1016/j.nonrwa.2010.09.005
|
| [56] |
D. Burini, N. Chouhad, Hilbert method toward a multiscale analysis from kinetic to macroscopic models for active particles, Math. Models Methods Appl. Sci., 27 (2017), 1327–1353. https://doi.org/10.1142/S0218202517400176 doi: 10.1142/S0218202517400176
|
| [57] |
R. C. López, V. Buendía, M. A. Muñoz, Excitatory-inhibitory branching process: A parsimonious view of cortical asynchronous states, excitability, and criticality, Phys. Rev. Res., 4 (2022), L042027. https://doi.org/10.1103/PhysRevResearch.4.L042027 doi: 10.1103/PhysRevResearch.4.L042027
|
| [58] |
L. M. van Kessenich, D. Berger, L. de Arcangelis, H. J. Herrmann, Pattern recognition with neuronal avalanche dynamics, Phys. Rev. E, 99 (2019), 010302. https://doi.org/10.1103/PhysRevE.99.010302 doi: 10.1103/PhysRevE.99.010302
|
| [59] |
D. B. Larremore, W. L. Shew, E. Ott, F. Sorrentino, J. G. Restrepo, Inhibition causes ceaseless dynamics in networks of excitable nodes, Phys. Rev. Lett., 112 (2014b), 138103. https://doi.org/10.1103/PhysRevLett.112.138103 doi: 10.1103/PhysRevLett.112.138103
|
| [60] |
A. V. Goltsev, F. V. De Abreu, S. N. Dorogovtsev, J. F. F. Mendes, Stochastic cellular automata model of neural networks, Phys. Rev. E, 81 (2010), 061921. https://doi.org/10.1103/PhysRevE.81.061921 doi: 10.1103/PhysRevE.81.061921
|
| [61] |
J. D. Cowan, J. Neuman, W. van Drongelen, Wilson–Cowan equations for neocortical dynamics, J. Math. Neurosci., 6 (2016), 1. https://doi.org/10.1186/s13408-015-0034-5 doi: 10.1186/s13408-015-0034-5
|
| [62] |
J. Pérez-Ortega, T. Alejandre-García, R. Yuste, Long-term stability of cortical ensembles, Elife, 10 (2021), e64449. https://doi.org/10.7554/eLife.64449 doi: 10.7554/eLife.64449
|
| [63] |
P. Theodoni, G. Kovács, M. W. Greenlee, G. Deco, Neuronal adaptation effects in decision making, J. Neurosci., 31 (2011), 234–246. https://doi.org/10.1523/JNEUROSCI.2757-10.2011 doi: 10.1523/JNEUROSCI.2757-10.2011
|
| [64] |
C. D. Verdugo, S. Myren-Svelstad, E. Aydin, E. Van Hoeymissen, C. Deneubourg, S. Vanderhaeghe, et al., Glia-neuron interactions underlie state transitions to generalized seizures, Nature Commun., 10 (2019), 3830. https://doi.org/10.1038/s41467-019-11739-z doi: 10.1038/s41467-019-11739-z
|
| [65] |
S. Fernández-García, J. G. Orlandi, G. A. García-Díaz Barriga, M. J. Rodríguez, M Masana, J. Soriano, et al., Deficits in coordinated neuronal activity and network topology are striatal hallmarks in Huntington's disease, BMC Biol., 18 (2020), 58. https://doi.org/10.1186/s12915-020-00794-4 doi: 10.1186/s12915-020-00794-4
|
| [66] |
N. Brunel, Storage capacity of neural networks: Effect of the fluctuations of the number of active neurons per memory, J. Phys. A: Math. Gen., 27 (1994), 4783. https://doi.org/10.1088/0305-4470/27/14/009 doi: 10.1088/0305-4470/27/14/009
|
| [67] |
E. Estévez-Priego, M. Moreno-Fina, E. Monni, Z. Kokaia, J. Soriano, D. Tornero, Long-term calcium imaging reveals functional development in hiPSC-derived cultures comparable to human but not rat primary cultures, Stem Cell Rep., 18 (2005), 205–219. https://doi.org/10.1016/j.stemcr.2022.11.014 doi: 10.1016/j.stemcr.2022.11.014
|
| [68] |
K. Blinowska, G. Müller-Putz, V. Kaiser, L. Astolfi, K. Vanderperren, S. Van Huffel, et al., Multimodal imaging of human brain activity: Rational, biophysical aspects and modes of integration, Comput. Intell. Neurosci., 1 (2009), 813607. https://doi.org/10.1155/2009/813607 doi: 10.1155/2009/813607
|
| [69] |
M. L. Bertotti, M. Delitala, From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences, Math. Models Methods Appl. Sci., 14 (2004), 1061–1084. https://doi.org/10.1142/S0218202504003544 doi: 10.1142/S0218202504003544
|
| [70] |
A. Marasco, E. Spera, V. De Falco, A. Iuorio, C. A. Lupascu, S. Solinas, et al., An adaptive generalized leaky integrate-and-fire model for hippocampal CA1 pyramidal neurons and interneurons, Bull. Math. Biol., 85 (2023), 109. doi: https://doi.org/10.1007/s11538-023-01206-8 doi: 10.1007/s11538-023-01206-8
|
| [71] |
M. Khanjanianpak, N. Azimi-Tafreshi, A. Valizadeh, Emergence of complex oscillatory dynamics in the neuronal networks with long activity time of inhibitory synapses, Iscience, 27 (2024b), 109401. https://doi.org/10.1016/j.isci.2024.109401 doi: 10.1016/j.isci.2024.109401
|
| [72] |
D. Gandolfi, J. Mapelli, S. M. G. Solinas, P. Triebkorn, E. D'Angelo, V. Jirsa, et al., Full-scale scaffold model of the human hippocampus CA1 area, Nature Comput. Sci., 3 (2023), 264–276. https://doi.org/10.1038/s43588-023-00417-2 doi: 10.1038/s43588-023-00417-2
|
| [73] | C. Godsil, G. F. Royle, Algebraic Graph Theory, Springer Science & Business Media, 2013. |
| [74] | L. Pareschi, G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, OUP Oxford, 2013. |
| [75] |
S. I. Amari, Characteristics of random nets of analog neuron-like elements, IEEE Trans. Syst. Man Cybern., SMC-2 (1972a), 643–657. https://doi.org/10.1109/TSMC.1972.4309193 doi: 10.1109/TSMC.1972.4309193
|
| [76] |
S. I. Amari, Learning patterns and pattern sequences by self-organizing nets of threshold elements, IEEE Trans, Comput, C-21 (1972b), 1197–1206. https://doi.org/10.1109/T-C.1972.223477 doi: 10.1109/T-C.1972.223477
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M. Menale, C. Tribuzi, R. Shah, C. A. Lupascu, A. Marasco. Kinetic modeling approach for a heterogeneous neuronal network activity using adjacency matrices[J]. Networks and Heterogeneous Media, 2025, 20(4): 1292-1332. doi: 10.3934/nhm.2025056
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