The provided code is intended to plot a two-dimensional bifurcation diagram of a thalamocortical neural mass model. This type of model simulates neural dynamics in the thalamocortical network, which plays a crucial role in generating and regulating brain rhythms, particularly during different sleep stages such as Non-Rapid Eye Movement (NREM) sleep. ### Biological Basis of the Model #### Thalamocortical Neural Dynamics - **Thalamic Connectivity**: The thalamus acts as a relay station for sensory information and is critical in modulating neural signals sent to the cortex. This model likely involves key thalamic mechanisms that influence cortical excitability and rhythm generation. - **Neural Mass Modeling**: Neural mass models are used to mimic the collective behavior of large populations of neurons. Instead of focusing on single neuron dynamics, these models capture average properties of neuron clusters, making them useful for investigating macroscopic brain activities observed in electrophysiological recordings such as EEG. #### Key Variables in the Model - **Conductances (\(\bar{g}_{LK}\) and \(\bar{g}_{h}\))**: - \(\bar{g}_{LK}\) denotes the leak potassium conductance. Potassium ion channels play an essential role in setting the membrane potential and influence neuronal excitability. - \(\bar{g}_{h}\) represents the hyperpolarization-activated cyclic nucleotide-gated (HCN) channels, also known as "h-channels". These channels activate during hyperpolarization and influence rhythmic activity and have a significant role in sleep oscillations. #### Bifurcation Analysis in Neuronal Models - **Period-doubling, Saddle-node, Torus, and Hopf Bifurcations**: - These bifurcations are mathematical frameworks to describe changes in system dynamics due to parameter variations. In neural systems, bifurcations help to identify transitions between different activity patterns, such as stable steady states, oscillations, or complex rhythms. #### EEG Dynamics and Sleep Spindles - **Spindle and Delta Regimes**: - The plot lines represent zones where certain rhythms dominate. Sleep spindles, characterized by 11-16 Hz oscillations, occur during NREM sleep and are critical for certain cognitive functions like memory consolidation. - Delta waves, slower oscillations, underpin deep sleep and restorative processes. The boundaries in the plot likely delineate parameter spaces where these rhythms predominate. This model, through the adjustment of conductance parameters, simulates the thalamocortical network's response under different conditions, potentially offering insights into how auditory stimuli affect sleep dynamics and how various neural rhythms transition during sleep stages.