# Biological Basis of the Computational Model The provided code is a computational implementation of a model that simulates the dynamics of a single population of excitatory neurons featuring synapses with dynamic properties such as depression and facilitation. The goal is to find the fixed points of the system, which are crucial in understanding the steady-state behavior of the neuronal network. ## Key Biological Concepts ### Neuronal Populations - **Excitatory Neurons**: The model simulates a population of neurons that predominantly use glutamate as a neurotransmitter to enhance synaptic activity, hence they are termed excitatory neurons. ### Synaptic Dynamics - **Depression**: Refers to a temporary reduction in synaptic strength following repeated stimulation. The viability of synaptic vesicles decreases due to depletion, which is represented by the variables `xD` (depletion of resources) and `uD` (utilization factor). - **Facilitation**: A temporary increase in synaptic strength with repetitive stimulation, as more calcium ions enter the presynaptic terminal, increasing neurotransmitter release probability. It is modeled with the variables `xF` (availability of resources) and `uF` (utilization factor for facilitation). ### Model Components - **Threshold-Linear Response Function**: The output firing rate of neurons (`E(h)`) depends on the membrane potential `h`, determined by a threshold (`theta`). The firing rate only increases linearly when the potential exceeds this threshold, mimicking physiological neuronal response. - **Parameters**: - `J`: Synaptic weight or strength determining how effectively presynaptic activity influences postsynaptic neurons. - `I`: External input current affecting the overall excitability of the neuron. - `U`, `D`, `F`: Parameters governing the utilization and recovery rates in depression (`Ud`, `Dd`, `Fd`) and facilitation (`Uf`, `Df`, `Ff`). ### Synaptic Plasticity - This model integrates short-term synaptic plasticity, with differential equations describing how synaptic efficacy changes with time and stimulus, capturing complex real-world dynamics of synaptic behavior. ### Equations - **Differential Equations**: These describe the temporal evolution of synaptic states: - The rate of change of `h` is governed by synaptic input modulated by factors `u`, `x`, `E(h)`. - Equations (2) to (5) describe the dynamics of synaptic depression and facilitation resources and usage, reflecting how these processes recover or are depleted over time due to neural activity. ### Fixed Points - **Fixed Points of the System**: The model searches for stable solutions (fixed points) where neuron states do not change over time, providing insights into potential steady states of network activity. The code is a powerful tool for investigating how short-term plasticity mechanisms influence overall network dynamics, potentially affecting phenomena like memory and learning in neural circuits.