Abstract
Realistic single-cell neuronal dynamics are typically obtained by solving models that involve solving a set of differential equations similar to the Hodgkin-Huxley (HH) system. However, realistic simulations of neuronal tissue dynamics —especially at the organ level, the brain— can become intractable due to an explosion in the number of equations to be solved simultaneously. Consequently, such efforts of modeling tissue- or organ-level systems require a lot of computational time and the need for large computational resources. Here, we propose to utilize a cellular automata (CA) model as an efficient way of modeling a large number of neurons reducing both the computational time and memory requirement. First, a first-order approximation of the response function of each HH neuron is obtained and used as the response-curve automaton rule. We then considered a system where an external input is in a few cells. We utilize a Moore neighborhood (both totalistic and outer-totalistic rules) for the CA system used. The resulting steady-state dynamics of a two-dimensional (2D) neuronal patch of size 1, 024 × 1, 024 cells can be classified into three classes: (1) Class 0–inactive, (2) Class 1–spiking, and (3) Class 2–oscillatory. We also present results for different quasi-3D configurations starting from the 2D lattice and show that this classification is robust. The numerical modeling approach can find applications in the analysis of neuronal dynamics in mesoscopic scales in the brain (patch or regional). The method is applied to compare the dynamical properties of the young and aged population of neurons. The resulting dynamics of the aged population shows higher average steady-state activity 〈a(t → ∞)〉 than the younger population. The average steady-state activity 〈a(t → ∞)〉 is significantly simplified when the aged population is subjected to external input. The result conforms to the empirical data with aged neurons exhibiting higher firing rates as well as the presence of firing activity for aged neurons stimulated with lower external current.
Highlights
Since the development of the first neuronal model by Louis Lapicque in 1907, most neuronal models we have today use a set of ordinary differential equations (ODEs) to model the dynamics of neurons (Lapicque, 1907; Brunel and Van Rossum, 2007)
We found that a value of 100 timesteps is enough to achieve steady-state for any initial state, and that randomizing the initial location of active cells does not affect the dynamical results of our model (Ramos and Bantang, 2018; Ramos, 2019)
We found a significant difference in the dynamics between young and aged neuronal systems
Summary
Since the development of the first neuronal model by Louis Lapicque in 1907, most neuronal models we have today use a set of ordinary differential equations (ODEs) to model the dynamics of neurons (Lapicque, 1907; Brunel and Van Rossum, 2007). We propose simple cellular automata models to simulate many interconnected neurons that will help investigate integrated dynamics of up to millions (106) of neurons using lower CPU and GPU requirements. CA models can employ a look-up-table-based algorithm that is usually faster than solving ODEs. Cellular automaton modeling paradigm was first developed in the late 1940’s by Stanislaw Ulam and John von Neumann (von Neumann, 1966). We perform different analyses (spatiotemporal, cobweb, bifurcation) on the CA system to classify the observed dynamics This lays the groundwork of our proposed model that can be extended to future directions. The empirical data from the study is used as an application of our CA model
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