Simplified cellular automata model of neuronal patch dynamics with generalized non-linear cell response

Abstract

Hodgkin-Huxley (HH) and other computational neuronal models are adequate methods in describing the dynamics of a single neuron. However, the HH model consist of four ordinary differential equations (ODEs) to solve for the neuronal response. Hence, simulating a large network of HH neurons would require very powerful computing devices. A model based on cellular automata (CA) has been recently made possible resulting to a simplistic way of simulating many neurons without requiring large computational resource. In this study, we introduce a non-linear CA that generalizes the response of the neuron compatible with empirical data via a nonlinearity parameter b. The resulting simple model is used to study a neuronal patch of 10⁴ (N = 10000) neurons in a 2D lattice with periodic boundary conditions. The phase space diagrams show that in the limit that the activation function becomes linear (b → 1⁺), the system transitions from an active steady-state (Class 1) to a quiescent steady-state (Class 0) at a₀ ≈ 0.5. Actual spatio-temporal neuronal responses simulations are obtained by translating the activation probability into dynamical transitions between the three standard neuronal cell (discrete) states: 1) Q = quiescent or inactive; 2) F = firing or spiking; 3) R = refractory period, such that Q → F → R → Q.