The provided code is a computational neuroscience model that simulates the behavior of excitatory neural networks, which could represent simplified versions of cortical microcircuits. The model consists of two classes, `network1D` and `network2D`, representing networks with one or two excitatory populations, respectively. Here is a description of its biological basis: ### Biological Basis #### Neural Populations - **Excitatory Neurons:** The focus of the model is on excitatory neuronal populations. In the 1D model, there is a single excitatory population, while the 2D model includes two interacting excitatory populations. This represents basic cortical networks where excitatory neurons form the principal component of neural circuits, responsible for propagating excitation. #### Network Structure - **Recurrent Connectivity:** The model includes recurrent connections within the excitatory neuron populations. This captures the biological feature where excitatory neurons within a cortical area are interconnected, facilitating the propagation and modulation of signals. - **External Stimulation:** Both models can incorporate external inputs, simulating sensory inputs or other forms of stimulation that cortical areas receive from different brain regions or sensory organs. #### Dynamic Parameters - **Rate-Based Neuron Models:** The function `Phi` is used to calculate firing rates based on membrane potential fluctuations characterized by mean (`mu`) and standard deviation (`sigma`) parameters. These parameters arise from the theory linked to neural network activities and help define how neurons respond to input. - **Synaptic Dynamics:** Parameters such as synaptic time constants (e.g., `tau_m`, `tau_syn_ex`) describe the temporal dynamics of synaptic potentials, important for understanding how neurons integrate incoming signals over time. - **Threshold and Reset Potentials:** Neuronal firing properties are modulated by threshold (`V_th`) and reset potentials (`V_reset`) relative to resting potential (`E_L`). These are critical properties governing action potential initiation and recovery in biological neurons. #### Computation of Rates - **Mean-Field Approximations:** The code employs functions such as `nu0_fb` for firing rate computation under mean-field theory. This approximation assumes that the collective interactions of a large number of neurons can be described by averaged properties, a common method in theoretical neuroscience to simplify the description of large networks. - **Firing Rate Equilibria:** The code uses numerical solvers (e.g., `fsolve`) to find equilibrium firing rates, which represent stable firing patterns in response to connectivity and external input. Equilibrium states in networks can relate to sustained activities, such as those seen during cognitive tasks or resting states. #### Two-Dimensional Dynamics - **Population Interaction:** The `network2D` class models the interaction between two excitatory populations, introducing the ability to simulate phenomena such as competitive dynamics, synchronization, or pattern formation. These interactions reflect the complex interplay found in larger cortical circuits where multiple neuronal ensembles may influence each other's activities. - **Vector Fields and Nullclines:** These constructs help visualize the dynamic behavior and stability of the network. Nullclines indicate points where the rate of change of activity is zero, corresponding to stable states or attractors that reflect natural operating points of biological networks. This model highlights core properties of cortical networks: excitatory neuronal dynamics, recurrent synaptic integration, and external inputs, forming a framework to study emergent phenomena such as sustained activity, oscillations, or bistability, relevant to understanding brain function and information processing.