The provided code is designed to simulate a spatially extended model of macroscopic Spike-Wave Discharges (SWDs), which are characteristic patterns observed in electroencephalography (EEG) recordings, typically associated with absence epilepsy. In this model, three primary components are represented: excitatory activity (E), and two types of inhibitory activity (I1 and I2). These components interact to produce the emergent SWD pattern seen in the simulations. ### Biological Basis: 1. **Spike-Wave Discharges (SWDs):** - SWDs are oscillatory patterns characterized by synchronous, rhythmic, spike-and-wave complexes. They typically manifest at 2.5-4 Hz in EEG readings and are associated with generalized absence seizures. - The model seeks to replicate the propagation and dynamics of these discharges at a macroscopic level, potentially providing insights into their underlying mechanisms. 2. **Neuronal Populations:** - **Excitatory Neurons (E):** These neurons depolarize postsynaptic neurons, increasing the likelihood of firing. In the model, the term `e1` represents the average excitatory input in the network. - **Inhibitory Neurons (I1 and I2):** These neurons hyperpolarize postsynaptic neurons, decreasing the likelihood of firing. Their activities are modeled as `if1` and `is1`, each responding differently to the input. 3. **Feedback and Coupling:** - The code uses a combination of weighted interconnections (excitatory and inhibitory) to reflect synaptic interactions in the brain. - Parameters `h1`, `h2`, and `h3` provide baseline levels of excitation/inhibition which can be adjusted to explore different pathological and non-pathological states. 4. **Neuronal Dynamics:** - The model implements a first-order differential equation system, with constants such as `tau1` and `tau2` representing time constants for neuronal processing, capturing the temporal dynamics of excitatory and inhibitory responses. 5. **Nonlinear Transformation:** - A `sigfunc` or sigmoid-like function normalizes the input dynamics, simulating saturation effects, where increasingly high activity levels lead to diminishing returns in neuronal firing rate increase, resembling neural activity limits. 6. **Temporal Evolution:** - The model solves these differential equations numerically to observe how excitatory and inhibitory interactions evolve over time to generate SWDs. This model provides an abstract representation of neural circuitry involved in SWDs, allowing researchers to explore the complex interplay of excitation and inhibition in generating these pathophysiological states.