The code provided models the dynamics of a neural network consisting of two populations: excitatory and inhibitory neurons. This kind of modeling is rooted in computational neuroscience, which seeks to understand the functioning of neural circuits and the emergent properties of networks of neurons, often by using mathematical and computational models. ### Biological Basis #### Neuron Types: - **Excitatory Neurons** (e.g., RS-cells or Regular Spiking cells): These neurons are modeled to primarily increase the likelihood of firing action potentials in connected neurons. They typically release neurotransmitters such as glutamate which bind to receptors leading to excitatory post-synaptic potentials (EPSPs). - **Inhibitory Neurons** (e.g., FS-cells or Fast Spiking cells): These neurons decrease the likelihood of firing action potentials in connected neurons, often using neurotransmitters such as GABA (gamma-aminobutyric acid), leading to inhibitory post-synaptic potentials (IPSPs). #### Populations and Parameters: - **Population Size:** The code considers large neuronal populations with parameters `Ne` and `Ni`, representing the number of excitatory (8,000) and inhibitory (2,000) neurons respectively. This reflects the biological distribution commonly seen in cortical areas where excitatory neurons vastly outnumber inhibitory ones. - **Transfer Functions (TF):** These mathematical functions characterize the input-output relationship of the neurons. They are used to describe how synaptic inputs are converted into firing rates, a crucial aspect of neural signaling. #### Dynamics and Equations: - The code models network dynamics through differential equations, capturing how the population firing rates change over time given synaptic inputs. This is inspired by biological processes where individual neuronal activities sum up to population-level phenomena. - **First and Second Order Systems:** The model incorporates both first-order and second-order differential systems to account for the basic dynamics and finite-size effects in the network, respectively. This corresponds to capturing simple rate dynamics (first-order) and the impact of fluctuations or variability in neuronal firing (second-order), which are significant in real brain networks. #### Fixed Points: - The use of fixed points in the code is an approach to find a steady state of the network where the system's dynamics no longer change. Biologically, this corresponds to the idea of network homeostasis or stable states where a neural circuit can maintain consistent activity levels. #### Numerical Derivatives: - The code uses numerical derivatives to find the slope of the transfer functions at given points, reflecting dynamic changes in neuronal output in response to changing synaptic inputs. This is tied to the concept of neuronal responsiveness and plasticity in biological systems. By abstracting neuronal populations and their interactions into a set of equations and functional relationships, the model aims to replicate essential aspects of neural behavior observed in biological networks. The differential operators and transfer functions integrated into this computational model allow for a nuanced examination of how populations of neurons could behave under various conditions, contributing to our understanding of complex brain functions.