The provided code models a mean-field approximation of two interconnected neural populations. This approach simplifies the dynamics of large numbers of neurons into a few collective variables, capturing the average behavior of neuronal ensembles rather than individual neural activity. Here are the key biological aspects of the model: ## Biological Basis ### Neural Populations - **Two Populations**: The model represents two distinct populations of neurons, which can be thought of as simplifying a more complex network of cortical neurons into manageable components. The populations interact via synaptic connections, each with their own set of parameters. ### Membrane Potential Dynamics - **Membrane Potential (\(v_{\text{mean}}\))**: Represents the average voltage across the membrane of the neurons in the population. The equation for \(v_{\text{mean}}\) includes contributions from synaptic inputs, intrinsic neuronal properties, and external inputs. ### Firing Rate (r) - **Average Firing Rate (\(r\))**: This variable reflects the average rate at which neurons within a population are firing action potentials. The firing rate equation includes a balance between synaptic inputs and intrinsic neuron activity. ### Synaptic Activity (s) - **Synaptic Dynamics**: The variables \(s\) and their equations describe the dynamics of synaptic inputs to a neuron population, including neurotransmitter binding and release. This is modeled with decay terms and input-driven increases based on firing rates. ### Adaptation and Plasticity - **Adaptation Variables (\(w_{\text{mean}}\))**: These variables capture the effect of activity-dependent processes that adjust neuron excitability and synaptic strength gradually over time. These processes often manifest biologically as potassium currents or other ionic currents that regulate neuron excitability in response to sustained activity. ### Synaptic Coupling - **Synaptic Strength Parameters (\(gsyn\))**: These parameters (e.g., `gsyn11`, `gsyn12`, etc.) define the strength of connectivity between and within the two populations, influencing how activity propagates through the network. ### Input Heterogeneity - **Intrinsic Heterogeneity**: The model incorporates heterogeneity in neuronal properties with parameters like `mu` (average current) and `hw` (which likely refers to the half-width of a distribution, e.g., Lorentzian, describing variation in those currents across neurons). This captures biological variability in neural populations. ### External Inputs - **External Current Inputs (\(I_{\text{ext}}\))**: External inputs represent sensory inputs or other external influences on the neural populations, akin to stimulus-driven neural activity in cerebral cortex experiments. ## Summary This mean-field model abstracts complex neural interactions to depict how average properties like firing rate, synaptic activity, and membrane potential evolve over time in interconnected neuronal populations. It provides a mechanistic framework for understanding the emergent dynamics of brain regions at the population level, focusing on the interplay of synaptic inputs, neural excitability, and adaptive feedback processes.