CNS*2025 Florence

Kinetic modeling approach for a heterogeneous neuronal network activity using adjacency matrices

M. Menale^*a, C. Tribuzi^b, R. Shah^a, C. A. Lupascu^c, and A. Marasco<sup>a,c

a</sup> Department of Mathematics and Applications, University of Naples Federico II, Naples, Italy
^b Nova Analysis, Brescia, Italy
^c Institute of Biophysics, National Research Council, Palermo, Italy

^*Email: marco.menale@unina.it

Introduction


The brain can be modeled
as a complex network composed of billions of interacting neurons¹. The result
of such nonlinear interactions is the emergence of intricate spatio-temporal
patterns^2-6. The heterogeneity of neuronal networks plays a crucial role in these
dynamics, as these networks display strong neuron-to-neuron
heterogeneity in physiology, dynamics, and connectivity, supporting functional
diversity and adaptability^7,8. Therefore, we introduce a kinetic modeling framework to investigate how
heterogeneity in neuronal networks shapes emergent activity patterns locally
and globally in the network^5,9. The approach enables a multiscale
description, that account for both structural and functional heterogeneity of
the network.
Methods

The network is divided
into slices that introduce a double heterogeneity: they can both group neurons
with similar firing properties and represent discrete spatial regions within
the network. Each neuron can play an inhibitory (interneuron) or excitatory
role (pyramidal neuron). The somatic membrane potential is the microscopic
variable of the system, and it identifies whether a neuron is active or not. Structural
heterogeneity is encoded via weighted adjacency matrices, while functional
heterogeneity is captured through transition probabilities, that model binary
interactions among neurons. These interactions may
modify the state of a neuron or leave it unchanged. Thus, the nonlinear ODEs system models the evolution of active neurons⁹

Results
After the
well-posedness of the system, we then focus on the setting where interactions
depend only on slice membership, in one-, two-, and four-slice cases. For
one-slice case, we perform an analytical study of coexistence equilibria. In the
other two cases, we examine and compare how homogeneous and heterogeneous
connectivity influence the dynamics of active neurons over the time, both overall
and within slices. Numerical simulations highlight the difference between
global and local dynamics induced by heterogeneity, along with its influence on
short- and long-term behavior. Finally, in the four-slice case, we show that heterogeneous
connectivity may lead to transitions between excitatory- and
inhibitory-dominated regimes (Fig.1)


Discussion
The proposed kinetic
framework provides a tractable yet expressive model of heterogeneous neuronal
networks^5,9. Our findings reveal that increasing heterogeneity not only
alters the proportion of active neurons but also leads to more intricate
dynamical patterns, potentially driving shifts between excitatory- and
inhibitory-dominated regimes. As a future direction, we aim to introduce
time-dependent adjacency matrixes and transition probabilities. This generalization
will introduce further analytical and numerical challenges, as the system
becomes non-autonomous. Nevertheless, such developments are essential to
capture, among others, plasticity and homeostasis, that are relevant features
in neuroscience






This research has been funded by the Italian National Recovery and Resilience Plan (NRRP), M4C2, funded by the European Union - NextGenerationEU (Project IR0000011, CUP B51E22000150006, EBRAINS-Italy), and by the EU Horizon Europe Program under the specific Grant Agreement 101147319, EBRAINS 2.0 project.

[1] DOI: https://doi.org/10.1523/JNEUROSCI.1236-24.2024

[2] DOI: https://doi.org/10.1016/j.isci.2024.109401

[3] DOI: https://doi.org/10.1523/JNEUROSCI.4514-06.2007

[4] DOI: https://doi.org/10.1073/pnas.2311885121

[5] DOI: https://doi.org/10.1103/PhysRevResearch.6.023018
[6] DOI: https://doi.org/10.1103/PhysRevResearch.4.L042027
[7] DOI: https://doi.org/110.1038/s41598-022-18024-y

[8] DOI: https://doi.org/10.1038/s43588-023-00417-2

[9] DOI: https://doi.org/10.1007/978-0-8176-4600-4